Open Access

The structure of fixed-point sets of Lipschitzian type semigroups

Fixed Point Theory and Applications20122012:163

DOI: 10.1186/1687-1812-2012-163

Received: 15 March 2012

Accepted: 22 August 2012

Published: 25 September 2012

Abstract

The purpose of this paper is to establish some results on the structure of fixed point sets for one-parameter semigroups of nonlinear mappings which are not necessarily Lipschitzian in Banach spaces. Our results improve several known existence and convergence fixed point theorems for semigroups which are not necessarily Lipschitzian.

MSC:47H09, 47H10, 47B20, 54C15.

Keywords

asymptotic center normal structure coefficient pseudo-contractive semigroup sunny nonexpansive retraction uniformly convex Banach space uniformly Lipschitzian semigroup variational inequality

1 Introduction

Let C be a nonempty subset of a Banach space X and T : C C https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq1_HTML.gif a mapping. We use Fix ( T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq2_HTML.gif to denote the set of all fixed points of T. A nonempty closed convex subset D of C is said to satisfy property ( ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq3_HTML.gif with respect to mapping T [1] if

(ω) ω T ( x ) D https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq4_HTML.gif for every x D https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq5_HTML.gif,

where ω T ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq6_HTML.gif denotes the set of all weak subsequential limits of { T n x : n N } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq7_HTML.gif. Moreover, T is said to satisfy the ( ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq3_HTML.gif-fixed point property if T has a fixed point in every nonempty closed convex subset D of C which satisfies property ( ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq3_HTML.gif. For a Lipschitzian mapping S : C C https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq8_HTML.gif, we use the symbol σ ( S ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq9_HTML.gif to denote the exact Lipschitz constant of S, i.e.,
σ ( S ) = inf { k [ 0 , ] : S x S y k x y  for all  x , y C } . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_Equa_HTML.gif
A mapping T : C C https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq10_HTML.gif is said to be
  1. (1)

    nonexpansive if σ ( T ) = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq11_HTML.gif,

     
  2. (2)

    asymptotically nonexpansive [2] if σ ( T n ) 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq12_HTML.gif for all n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq13_HTML.gif and lim n σ ( T n ) = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq14_HTML.gif,

     
  3. (3)

    uniformly L-Lipschitzian if σ ( T n ) = L https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq15_HTML.gif for all n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq13_HTML.gif and for some L ( 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq16_HTML.gif.

     

In general, the fixed-point set of a nonexpansive mapping need not be convex and can be extremely irregular. Suppose that C is a nonempty closed convex bounded subset of a Banach space X and T : C C https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq17_HTML.gif is a nonexpansive mapping with Fix ( T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq18_HTML.gif. Obviously, Fix ( T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq2_HTML.gif is a closed set. Fix ( T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq2_HTML.gif is convex if X is strictly convex (see [3, 4]).

Nonexpansive retracts have been studied in several contexts (for example, convex geometry [5], extension problems [6], fixed point theory [7], optimal sets [8]). It is well known that if C is a nonempty closed convex bounded subset of a Banach space and if a nonexpansive mapping T : C C https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq17_HTML.gif has a fixed point in every nonempty closed convex subset of C which is invariant under T, then Fix ( T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq2_HTML.gif is a nonexpansive retract of C (that is, there exists a nonexpansive mapping R : C Fix ( T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq19_HTML.gif such that R | Fix ( T ) = I https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq20_HTML.gif) (see [[7], Theorem 2]). The Bruck result was extended by Benavides and Ramirez [1] to the case of asymptotically nonexpansive mappings if the space X was sufficiently regular.

The class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk [2] in 1972 and they proved that if C is a nonempty closed convex bounded subset of a uniformly convex Banach space, then every asymptotically nonexpansive self-mapping of C has a fixed point. Several authors have studied the existence of fixed points of asymptotically nonexpansive mappings in Banach spaces having rich geometric structure, see [1, 9, 10].

There is a class of mappings which lies strictly between the class of contraction mappings and the class of nonexpansive mappings. The class of pointwise contractions was introduced in Belluce and Kirk [11], and later it was called ‘generalized contractions’ in [12]. Banach’s celebrated contraction principle was extended to this larger class of mappings as follows:

Theorem 1.1 [11, 12]

Let C be a nonempty weakly compact convex subset of a Banach space and T : C C https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq17_HTML.gif a pointwise contraction. Then T has a unique fixed point x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq21_HTML.gif, and { T n x } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq22_HTML.gif converges strongly to x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq21_HTML.gif for each x C https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq23_HTML.gif.

Kirk [13] combined the ideas of pointwise contraction [11] and asymptotic contraction [14] and introduced the concept of an asymptotic pointwise contraction. He announced that an asymptotic pointwise contraction defined on a closed convex bounded subset of a super-reflexive Banach space has a fixed point. Recently, Kirk and Xu [15] gave a simple and elementary proof of the fact that an asymptotic pointwise contraction defined on a weakly compact convex set always has a unique fixed point (with convergence of Picard iterates). They also introduced the concept of pointwise asymptotically nonexpansive mapping and proved that every pointwise asymptotically nonexpansive mapping defined on a closed convex bounded subset of a uniformly convex Banach space has a fixed point.

Every asymptotically nonexpansive mapping is uniformly L-Lipschitzian, and the ( ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq3_HTML.gif-fixed point property of uniformly L-Lipschitzian mappings is closely related to the class of nonexpansive and asymptotically nonexpansive mappings. In this connection, a deep result of Casini and Maluta [16] was generalized by Lim and Xu [17] as follows:

Theorem LX (Lim and Xu [[17], Theorem 1])

Let X be a Banach space with a uniform normal structure and let N ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq24_HTML.gif be the normal structure coefficient of X. Let C be a nonempty bounded subset of X and T : C C https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq25_HTML.gif a uniformly L-Lipschitzian mapping with L < N ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq26_HTML.gif. Then T satisfies the ( ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq3_HTML.gif-fixed point property.

The ( ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq3_HTML.gif-fixed point property plays a key role in the existence and approximation of solutions of fixed point problems and variational inequality problems, see [1720].

The mapping theory for accretive mappings is closely related to the fixed point theory of pseudo-contractive mappings. Recently, applications of the semigroup result on the existence of solutions to certain partial differential equations have been explored in Hester and Morales [21]. They proved that the semigroup result directly implies the existence of unique global solutions to time evolution equations of the form u = A u https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq27_HTML.gif, where A is a combination of derivatives. In many applications, semigroups are not necessarily Lipschitzian. It is an interesting problem to extend fixed point existence results, namely Theorem LX, for semigroups of nonlinear mappings which are not necessarily Lipschitzian.

Motivated by the results above, in this paper we establish some results on the structure of fixed point sets for one-parameter semigroups of nonlinear mappings which are not necessarily Lipschitzian in Banach spaces. Our theorems significantly extend Theorem LX to more general Banach spaces and to a more general class of operators. We obtain a general convergence theorem for semigroups of non-Lipschitzian pseudo-contractive mappings. Our results improve several known fixed point problems and variational inequality problems for semigroups which are not necessarily Lipschitzian.

2 Preliminaries

Let R + https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq28_HTML.gif denote the set of nonnegative real numbers, and let N 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq29_HTML.gif denote the set of nonnegative integers. Throughout this paper, G denotes an unbounded set of R + : = [ 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq30_HTML.gif such that s + t G https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq31_HTML.gif for all s , t G https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq32_HTML.gif and s t G https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq33_HTML.gif for all s , t G https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq34_HTML.gif with s t https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq35_HTML.gif (often G = N 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq36_HTML.gif or R + https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq37_HTML.gif).

2.1 Lipschitzian type mappings

Let C be a nonempty subset of a Banach space X and T : C C https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq38_HTML.gif a mapping. Then T is called
  1. (i)

    pointwise contractive [11] if there exists a function α : C [ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq39_HTML.gif such that T x T y α ( x ) x y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq40_HTML.gif for all x , y C https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq41_HTML.gif;

     
  2. (ii)

    asymptotic pointwise contractive [13] if for each n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq13_HTML.gif, there exists a function α n : C [ 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq42_HTML.gif such that T n x T n y α n ( x ) x y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq43_HTML.gif for all x , y C https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq41_HTML.gif, where α n α https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq44_HTML.gif pointwise on C;

     
  3. (iii)

    pointwise asymptotically nonexpansive [15] if for each integer n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq45_HTML.gif, T n x T n y α n ( x ) x y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq46_HTML.gif for all x , y C https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq47_HTML.gif, where α n 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq48_HTML.gif pointwise;

     
  4. (iv)
    asymptotically nonexpansive in the intermediate sense [22] provided T is uniformly continuous and
    lim sup n sup x , y C ( T n x T n y x y ) 0 ; https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_Equ1_HTML.gif
    (2.1)
     
  5. (v)
    mapping of asymptotically nonexpansive type [23] if
    lim sup n sup y C ( T n x T n y x y ) 0 for all  x C . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_Equb_HTML.gif
     
Fix a sequence { a n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq49_HTML.gif in [ 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq50_HTML.gif with a n 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq51_HTML.gif. A mapping T : C C https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq1_HTML.gif is said to be nearly Lipschitzian with respect to { a n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq49_HTML.gif [24] if for each n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq52_HTML.gif, there exists a constant k n > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq53_HTML.gif such that
T n x T n y k n ( x y + a n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_Equ2_HTML.gif
(2.2)
for all x , y C https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq54_HTML.gif. The infimum of constants k n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq55_HTML.gif in (2.2) is called nearly Lipschitz constant and is denoted by η ( T n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq56_HTML.gif. A nearly Lipschitzian mapping T with the sequence { ( a n , η ( T n ) ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq57_HTML.gif is called
  1. (i)

    nearly contractive if η ( T n ) < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq58_HTML.gif for all n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq13_HTML.gif,

     
  2. (ii)

    nearly uniformly L-Lipschitzian if η ( T n ) L https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq59_HTML.gif for all n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq13_HTML.gif,

     
  3. (iii)

    nearly uniformly k-contractive if η ( T n ) k < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq60_HTML.gif for all n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq13_HTML.gif,

     
  4. (iv)

    nearly nonexpansive if η ( T n ) = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq61_HTML.gif for all n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq13_HTML.gif,

     
  5. (v)

    nearly asymptotically nonexpansive if η ( T n ) 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq62_HTML.gif for all n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq13_HTML.gif with lim n η ( T n ) = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq63_HTML.gif.

     
The mapping T is said to be demicontinuous if, whenever a sequence { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq64_HTML.gif in C converges strongly to x C https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq65_HTML.gif, then { T x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq66_HTML.gif converges weakly to Tx. The mapping T is said to be weakly contractive if
T x T y x y ψ ( x y ) for all  x , y C , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_Equc_HTML.gif

where ψ : [ 0 , ) [ 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq67_HTML.gif is a continuous and nondecreasing function such that ψ ( 0 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq68_HTML.gif, ψ ( t ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq69_HTML.gif for t > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq70_HTML.gif and lim t ψ ( t ) = https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq71_HTML.gif.

Let C be a convex subset of a Banach space X and D a nonempty subset of C. Then a continuous mapping P from C onto D is called a retraction if P x = x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq72_HTML.gif for all x D https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq5_HTML.gif, i.e., P 2 = P https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq73_HTML.gif. A retraction P is said to be sunny if P ( P x + t ( x P x ) ) = P x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq74_HTML.gif for each x C https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq23_HTML.gif and t 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq75_HTML.gif with P x + t ( x P x ) C https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq76_HTML.gif. If the sunny retraction P is also nonexpansive, then D is said to be a sunny nonexpansive retract of C.

In what follows, we shall make use of the following lemmas:

Lemma 2.1 [3]

Let C be a nonempty closed convex subset of a Banach space X and T : C C https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq38_HTML.gif a continuous strongly pseudo-contractive mapping. Then T has a unique fixed point in C.

Lemma 2.2 (Goebel and Reich [[4], Lemma 13.1])

Let C be a convex subset of a smooth Banach space X, D a nonempty subset of C and P a retraction from C onto D. Then the following are equivalent:
  1. (a)

    P is sunny and nonexpansive.

     
  2. (b)

    x P x , J ( z P x ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq77_HTML.gif for all x C https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq23_HTML.gif, z D https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq78_HTML.gif.

     
  3. (c)

    x y , J ( P x P y ) P x P y 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq79_HTML.gif for all x , y C https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq54_HTML.gif.

     

2.2 Semigroups

Let C be a nonempty subset of a Banach space X. The one-parameter family F : = { T ( t ) : t G } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq80_HTML.gif is said to be a strongly continuous semigroup of mappings from C into itself if
  1. (I)

    T ( 0 ) x = x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq81_HTML.gif for all x C https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq23_HTML.gif;

     
  2. (II)

    T ( s + t ) = T ( s ) T ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq82_HTML.gif for all s , t G https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq83_HTML.gif;

     
  3. (III)

    for each x C https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq23_HTML.gif, the mapping T ( ) x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq84_HTML.gif from G into C is continuous.

     
We denote by Fix ( F ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq85_HTML.gif the set of all common fixed points of , i.e., Fix ( F ) : = t G Fix ( T ( t ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq86_HTML.gif. For a Lipschitzian semigroup , we write
σ ( F ) : = lim inf t σ ( T ( t ) ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_Equd_HTML.gif
If satisfies (I)-(III) and
lim t T ( t ) x T ( s ) T ( t ) x = 0 for all  x C  and  s G , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_Equ3_HTML.gif
(ar)
then is called asymptotically regular on C. If satisfies (I)-(III) and
lim t ( sup x C ˜ T ( t ) x T ( s ) T ( t ) x ) = 0 for all  s > 0  and bounded  C ˜ C , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_Equ4_HTML.gif
(uar)

then is called uniformly asymptotically regular on C.

A Lipschitzian semigroup is called a
  1. (i)

    uniformly L-Lipschitzian semigroup if sup t G σ ( T ( t ) ) = L < https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq87_HTML.gif;

     
  2. (ii)

    nonexpansive semigroup if σ ( T ( t ) ) = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq88_HTML.gif for all t G https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq89_HTML.gif;

     
  3. (iii)

    asymptotically nonexpansive semigroup if σ ( T ( t ) ) 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq90_HTML.gif for all t G https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq89_HTML.gif and lim t σ ( T ( t ) ) = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq91_HTML.gif.

     

2.3 Asymptotic center

Throughout the paper, ( X , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq92_HTML.gif is a Banach space which is assumed not to be Schur. That is, X has weakly convergent sequences that are not norm convergent. Let C be a nonempty closed convex subset of a Banach space X and { x t } t G https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq93_HTML.gif a bounded set in X. Consider the functional r a ( , { x t } t G ) : X R + https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq94_HTML.gif defined by
r a ( x , { x t } t G ) = lim sup G t x t x , x X . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_Eque_HTML.gif
The infimum of r a ( , { x t } t G ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq95_HTML.gif over C is said to be the asymptotic radius of { x t } t G https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq93_HTML.gif with respect to C and is denoted by r a ( C , { x t } t G ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq96_HTML.gif. A point z C https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq97_HTML.gif is said to be an asymptotic center of { x t } t G https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq93_HTML.gif with respect to C if
r a ( z , { x t } t G ) = inf { r a ( x , { x t } t G ) : x C } . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_Equf_HTML.gif
The set of all asymptotic centers of { x t } t G https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq93_HTML.gif with respect to C is denoted by Z a ( C , { x t } t G ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq98_HTML.gif. A number diam a ( { x t } t G ) = lim sup G k ( sup { x s x t : s , t k } ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq99_HTML.gif is called an asymptotic diameter of { x t } t G https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq93_HTML.gif. It is well known that if X is reflexive, then Z a ( C , { x t } t G ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq98_HTML.gif is nonempty closed convex and bounded, and if X is uniformly convex, then Z a ( C , { x t } t G ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq98_HTML.gif consists only of a single point, { z } = Z a ( C , { x t } t G ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq100_HTML.gif, i.e., z C https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq101_HTML.gif is the unique point which minimizes the functional
lim sup G t x t x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_Equg_HTML.gif

over x C https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq23_HTML.gif.

2.4 Normal structure

Normal structure plays a key role in some problems of metric fixed point theory. Let C be a nonempty bounded subset of a Banach space X. We denote by
diam ( C ) = sup x , y C x y , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_Equh_HTML.gif
the diameter of C. Put
r C ( C ) = inf x C { sup y C x y } . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_Equi_HTML.gif
This nonnegative real number is called the Chebyshev radius of C relative to itself. The normal structure coefficient N ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq24_HTML.gif of a Banach space X is defined [25] by
N ( X ) = inf { diam ( C ) r C ( C ) : C  is nonempty bounded convex subset of  X  with  diam ( C ) > 0 } . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_Equj_HTML.gif
The space X is said to have the uniformly normal structure if N ( X ) > 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq102_HTML.gif. It is well known that, for every uniformly convex Banach space X, N ( X ) > 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq102_HTML.gif. A weakly convergent sequence coefficient of X is defined (see [25]) by
WCS ( X ) = sup { k : k lim sup n x n < diam a ( { x n } )  for all  { x n }  in  X  with  x n 0 } . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_Equk_HTML.gif
It is proved in [[26], Theorem 1] that
WCS ( X ) = β ( X ) : = inf { D [ { x n } ] : x n 0 , x n 1 } , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_Equl_HTML.gif
where D [ { x n } ] : = lim sup m ( lim sup n x m x n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq103_HTML.gif. It is readily seen that
1 N ( X ) WCS ( X ) 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_Equm_HTML.gif
The space X is said to have the weak uniformly normal structure if WCS ( X ) > 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq104_HTML.gif. If X is a reflexive Banach space with modulus of convexity δ X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq105_HTML.gif, then
1 1 δ X ( 1 ) N ( X ) WCS ( X ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_Equn_HTML.gif
Thus, if X is a uniformly convex Banach space, then WCS ( X ) > 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq106_HTML.gif and also the equation
α 2 WCS ( X ) δ X 1 ( 1 1 α ) = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_Equo_HTML.gif
has a unique solution α > 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq107_HTML.gif. A general formula for WCS ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq108_HTML.gif in an arbitrary Banach space is not known. In particular, it has been calculated that for a Hilbert space H,
WCS ( H ) = 2 ; https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_Equp_HTML.gif
for p https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq109_HTML.gif ( 1 p < https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq110_HTML.gif),
WCS ( p ) = 2 1 / p https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_Equq_HTML.gif
and for https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq111_HTML.gif,
WCS ( ) = 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_Equr_HTML.gif
Remark 2.3 p https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq109_HTML.gif is an example of a reflexive Banach space such that N ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq112_HTML.gif and WCS ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq113_HTML.gif are different. Indeed,
N ( p ) = 2 ( p 1 ) / p < 2 1 / p = WCS ( p ) , for  1 < p < 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_Equs_HTML.gif
A Banach space X is said to satisfy the Opial condition, if whenever a sequence { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq64_HTML.gif in X converges weakly to x, then
lim inf n x n x < lim inf n x n y for all  y X { x } . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_Equt_HTML.gif
The Opial modulus r X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq114_HTML.gif of X is defined by
r X ( c ) = inf { lim inf n x n + x 1 } , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_Equu_HTML.gif
where c > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq115_HTML.gif and the infimum is taken over all x X https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq116_HTML.gif with x c https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq117_HTML.gif and all sequences { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq64_HTML.gif in X such that w lim n x n = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq118_HTML.gif and lim inf n x n 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq119_HTML.gif. For any Banach space X, we have the following inequality:
1 + r X ( 1 ) WCS ( X ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_Equv_HTML.gif

3 Nonemptiness of common fixed-point sets

First, we introduce some wider classes of semigroups.

Definition 3.1 Let C be a nonempty subset of a normed space X and F : = { T ( t ) : t G } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq80_HTML.gif a strongly continuous semigroup of mappings from C into itself. The semigroup is said to be nearly Lipschitzian if there exist a function a ( ) : G [ 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq120_HTML.gif with lim t a ( t ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq121_HTML.gif and a function η ( ) : G ( 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq122_HTML.gif such that
T ( t ) x T ( t ) y η ( T ( t ) ) ( x y + a ( t ) ) for all  x , y C  and  t G . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_Equw_HTML.gif
For a nearly Lipschitzian semigroup , we write
η ( F ) : = lim inf t η ( T ( t ) ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_Equx_HTML.gif
We say is
  1. (a)
    pointwise nearly Lipschitzian if for each t G https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq123_HTML.gif, there exist a function a ( ) : G [ 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq124_HTML.gif with lim t a ( t ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq121_HTML.gif and a function α t ( ) : C ( 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq125_HTML.gif such that
    T ( t ) x T ( t ) y α t ( x ) ( x y + a ( t ) ) for all  x , y C ; https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_Equy_HTML.gif
     
  2. (b)
    pointwise nearly uniformly α ( ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq126_HTML.gif-Lipschitzian if there exist a function a ( ) : G [ 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq124_HTML.gif with lim t a ( t ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq127_HTML.gif and a function α ( ) : C ( 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq128_HTML.gif such that
    T ( t ) x T ( t ) y α ( x ) ( x y + a ( t ) ) for all  x , y C ; https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_Equz_HTML.gif
     
  3. (c)
    asymptotic pointwise nearly Lipschitzian if for each t G https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq123_HTML.gif, there exist a function a ( ) : G [ 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq124_HTML.gif with lim t a ( t ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq121_HTML.gif and two functions α t ( ) , α ( ) : C ( 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq129_HTML.gif with lim t α t = α https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq130_HTML.gif pointwise such that
    T ( t ) x T ( t ) y α t ( x ) ( x y + a ( t ) ) for all  x , y C . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_Equaa_HTML.gif
     
We say that an asymptotic pointwise nearly Lipschitzian semigroup is pointwise nearly asymptotically nonexpansive (pointwise asymptotically nonexpansive) if α t ( x ) 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq131_HTML.gif for all t G https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq89_HTML.gif and lim t α t = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq132_HTML.gif pointwise ( a ( t ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq133_HTML.gif and α t ( x ) 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq131_HTML.gif for all t G https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq89_HTML.gif and α t ( x ) 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq134_HTML.gif pointwise). Further, we say that an asymptotic pointwise nearly Lipschitzian semigroup is asymptotic pointwise nearly contractive if α t α https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq135_HTML.gif pointwise and α ( x ) k < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq136_HTML.gif for all x C https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq65_HTML.gif. The semigroup is said to be nearly uniformly L-Lipschitzian if there exist a constant L [ 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq137_HTML.gif and a function a ( ) : G [ 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq120_HTML.gif with lim t a ( t ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq121_HTML.gif such that
T ( t ) x T ( t ) y L ( x y + a ( t ) ) for all  x , y C . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_Equab_HTML.gif

The nearly uniformly L-Lipschitzian semigroup will be called nearly nonexpansive semigroup.

Before presenting the main result of this section, we give another definition:

Definition 3.2 Let C be a nonempty weakly compact convex subset of Banach space X, F : = { T ( t ) : t G } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq80_HTML.gif a strongly continuous semigroup of mappings from C into itself. A nonempty closed convex subset D of C is said to satisfy property ( ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq3_HTML.gif with respect to semigroup if

(ω) ω F ( x ) D https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq138_HTML.gif for every x D https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq5_HTML.gif,

where ω F ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq139_HTML.gif denotes the set of all weak limits of { T ( t n ) x : n N } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq140_HTML.gif as t n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq141_HTML.gif.

The semigroup is said to satisfy the ( ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq3_HTML.gif-fixed point property if has a common fixed point in every nonempty closed convex subset D of C which satisfies property ( ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq3_HTML.gif.

We now establish that a semigroup of a certain class of Lipschitzian type mappings satisfies the ( ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq3_HTML.gif-fixed point property.

Theorem 3.3 Let X be a Banach space with weak uniformly normal structure, C a nonempty weakly compact convex subset of X and F = { T ( t ) : t G } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq142_HTML.gif a strongly continuous semigroup of demicontinuous mappings from C into itself. Suppose that for each t G https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq123_HTML.gif, there exist a function a ( ) : G [ 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq124_HTML.gif with lim t a ( t ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq121_HTML.gif and two functions α t ( ) , α ( ) : C ( 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq129_HTML.gif with lim t α t = α https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq130_HTML.gif pointwise and sup x C α ( x ) < WCS ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq143_HTML.gif such that
T ( t ) x T ( t ) y α t ( x ) ( x y + a ( t ) ) for all x , y C . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_Equac_HTML.gif
Also suppose that there exists a nonempty closed convex subset M of C which satisfies property (ω) with respect to . Then:
  1. (a)
    For arbitrary x 0 M https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq144_HTML.gif, there exist a sequence { t n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq145_HTML.gif in G with lim n t n = https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq146_HTML.gif and an iterative sequence { x m } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq147_HTML.gif in M defined by
    x m = w lim n T ( t n ) x m 1 for all m N . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_Equ5_HTML.gif
    (3.1)
     
  2. (b)

    If is asymptotically regular on C, then there exists an element x M Fix ( F ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq148_HTML.gif such that { x m } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq147_HTML.gif converges strongly to x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq21_HTML.gif.

     

Proof (a) Since one can easily construct a nonempty closed convex separable subset C 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq149_HTML.gif of C which is invariant under each T ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq150_HTML.gif (i.e., T ( t ) ( C 0 ) C 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq151_HTML.gif for t G https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq89_HTML.gif), we may assume that C itself is separable.

The separability of C 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq149_HTML.gif makes it possible to select a sequence { T ( t n ) x } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq152_HTML.gif of { T ( t ) x } t G https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq153_HTML.gif such that
{ T ( t n ) x }  is weakly convergent for every  x C https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_Equad_HTML.gif
and
lim n α t n ( x ) = α ( x ) for every  x C . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_Equae_HTML.gif
For any x 0 M C https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq154_HTML.gif, consider a sequence { T ( t n ) x 0 } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq155_HTML.gif in C. Suppose w lim n T ( t n ) x 0 = x 1 C https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq156_HTML.gif. Using property (ω), we obtain that x 1 M https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq157_HTML.gif. Now we can construct a sequence { x m } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq147_HTML.gif in M in the following way:
{ x 0 M arbitrary , x m = w lim n T ( t n ) x m 1 for all  m N . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_Equaf_HTML.gif
  1. (b)
    The weak asymptotic regularity of ensures that x m = w lim n T ( t n + t r ) x m 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq158_HTML.gif, t r G https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq159_HTML.gif. We now show that { x m } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq147_HTML.gif converges strongly to a common fixed point of . Set
    L : = sup x C α ( x ) , D m : = lim sup n x m T ( t n ) x m https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_Equag_HTML.gif
     
and
R m : = lim sup n x m + 1 T ( t n ) x m https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_Equah_HTML.gif
for all m = 0 , 1 , 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq160_HTML.gif . By the property of WCS ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq108_HTML.gif, we have
R m = lim sup n x m + 1 T ( t n ) x m 1 WCS ( X ) D [ { T ( t n ) x m } ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_Equ6_HTML.gif
(3.2)
By the asymptotic regularity of and the w-l.s.c. of the norm https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq161_HTML.gif, we have
D m = lim sup r x m T ( t r ) x m lim sup r ( lim sup s T ( t s ) x m 1 T ( t r ) x m ) lim sup r ( lim sup s ( T ( t s ) x m 1 T ( t r + t s ) x m 1 + T ( t r + t s ) x m 1 T ( t r ) x m ) ) lim sup r ( lim sup s ( α t r ( x m ) ( T ( t s ) x m 1 x m + a ( t r ) ) ) ) = α ( x m ) lim sup s T ( t s ) x m 1 x m L R m 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_Equai_HTML.gif
On the other hand, by the asymptotic regularity of , we have
D [ { T ( t n ) x m } ] = lim sup n ( lim sup r T ( t n ) x m T ( t r ) x m ) lim sup n ( lim sup r ( T ( t n ) x m T ( t n + t r ) x m + T ( t n + t r ) x m T ( t r ) x m ) ) lim sup n ( lim sup r ( α t n ( x m ) ( x m T ( t r ) x m + a ( t n ) ) ) ) = L D m L 2 R m 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_Equaj_HTML.gif
Set λ : = L 2 WCS ( X ) < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq162_HTML.gif. From (3.2), we obtain
R m λ R m 1 λ 2 R m 2 λ m R 0 0 as  m . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_Equak_HTML.gif
For any m N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq163_HTML.gif, one can see that
x m + 1 x m lim sup n ( x m + 1 T ( t n ) x m + T ( t n ) x m x m ) R m + D m = R m + L R m 1 ( λ + L ) R m 1 ( λ + L ) λ m 1 R 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_Equ7_HTML.gif
(3.3)
so it follows that { x m } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq147_HTML.gif is a Cauchy sequence in M. Let lim m x m = v M https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq164_HTML.gif. Observe that
v T ( t n ) v v x m + 1 + x m + 1 T ( t n ) x m + T ( t n ) x m T ( t n ) v v x m + 1 + x m + 1 T ( t n ) x m + α t n ( x m ) ( x m v + a ( t n ) ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_Equal_HTML.gif
Taking the limit superior as n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq165_HTML.gif on both sides, we get
lim sup n v T ( t n ) v v x m + 1 + R m + α ( x m ) x m v v x m + 1 + R m + L x m v 0 as  m . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_Equam_HTML.gif

Hence lim n T ( t n ) v = v https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq166_HTML.gif. Let t G https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq89_HTML.gif. Note T ( t n ) v v https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq167_HTML.gif, so it follows from the demicontinuity of T ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq150_HTML.gif that T ( t ) T ( t n ) v T ( t ) v https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq168_HTML.gif. Observe that T ( t ) T ( t n ) v = T ( t + t n ) v v https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq169_HTML.gif. By the uniqueness of the weak limit of { T ( t ) T ( t n ) v } n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq170_HTML.gif, we have T ( t ) v = v https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq171_HTML.gif. Therefore, v M Fix ( F ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq172_HTML.gif. □

Theorem 3.3 generalizes the result due to Górnicki [27] in the context of the (ω)-fixed point property for a wider class of mappings. Theorem 3.3 also extends corresponding results of Sahu, Agarwal and O’Regan [18], Sahu, Liu and Kang [28] and Sahu, Petruşel and Yao [29] for asymptotic pointwise nearly Lipschitzian semigroups. As N ( X ) WCS ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq173_HTML.gif, and there are Banach spaces for which N ( X ) = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq174_HTML.gif while WCS ( X ) > 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq175_HTML.gif, the following result is an improvement on Casini and Maluta [16] and Lim and Xu [[17], Theorem 1].

Corollary 3.4 Let X be a Banach space with weak uniformly normal structure, C a nonempty weakly compact convex subset of X and F = { T ( t ) : t G } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq142_HTML.gif a strongly continuous semigroup of demicontinuous nearly uniformly L-Lipschitzian of mappings from C into itself. Suppose that L < WCS ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq176_HTML.gif and that there exists a nonempty closed convex subset M of C which satisfies property (ω) with respect to . Then:
  1. (a)
    For arbitrary x 0 M https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq144_HTML.gif, there exist a sequence { t n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq145_HTML.gif in G with lim n t n = https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq146_HTML.gif and an iterative sequence { x m } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq147_HTML.gif in M defined by
    x m = w lim n T ( t n ) x m 1 for all m N . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_Equan_HTML.gif
     
  2. (b)

    If is asymptotically regular on C, then there exists an element x M Fix ( F ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq148_HTML.gif such that { x m } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq147_HTML.gif converges strongly to x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq21_HTML.gif.

     

Corollary 3.5 Let X be a Banach space with weak uniformly normal structure, C a nonempty weakly compact convex subset of X and F = { T ( t ) : t G } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq142_HTML.gif a strongly continuous semigroup of demicontinuous nearly uniformly L-Lipschitzian asymptotically regular mappings from C into itself such that L < WCS ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq177_HTML.gif. Then has a common fixed point in C.

4 Common fixed-point sets as Lipschitzian retracts

Theorem 4.1 Let X be a uniformly Banach space with the Opial condition, C a nonempty closed convex bounded subset of X and F = { T ( t ) : t G } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq142_HTML.gif a strongly continuous semigroup of demicontinuous mappings from C into itself. Suppose that there exists a function a ( ) : G [ 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq124_HTML.gif with lim t a ( t ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq121_HTML.gif such that
T ( t ) x T ( t ) y η ( T ( t ) ) ( x y + a ( t ) ) for all x , y C , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_Equao_HTML.gif

where η ( T ( t ) ) ( 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq178_HTML.gif with η ( F ) : = lim inf t η ( T ( t ) ) < WCS ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq179_HTML.gif. Also suppose that is asymptotically regular on C. Then Fix ( F ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq180_HTML.gif and Fix ( F ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq85_HTML.gif is η ( F ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq181_HTML.gif-Lipschitzian retract of C.

Proof Using similar arguments as in the proof of Theorem 3.3(a), we may select a sequence { T ( t n ) x } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq152_HTML.gif of { T ( t ) x } t G https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq153_HTML.gif such that lim n η ( T ( t n ) ) = η ( F ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq182_HTML.gif and
{ T ( t n ) x }  is weakly convergent for every  x C . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_Equap_HTML.gif
Let A : C C https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq183_HTML.gif denote a mapping which associates with a given x C https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq184_HTML.gif a unique z Z a ( C , { T ( t ) x } t G ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq185_HTML.gif, that is, z = A x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq186_HTML.gif. Since A x = w lim n T ( t n ) x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq187_HTML.gif for all x C https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq65_HTML.gif, it follows from the lower weak semi-continuity of the norm that
A x A y η ( F ) x y for all  x , y C , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_Equaq_HTML.gif

i.e., A is η ( F ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq181_HTML.gif-Lipschitzian mapping. It follows that A is uniformly continuous.

For any x = x 0 C https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq188_HTML.gif, consider a sequence { T ( t n ) x 0 } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq155_HTML.gif in C. Suppose w lim n T ( t n ) x 0 = x 1 C https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq156_HTML.gif. Now we can construct a sequence { x m } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq147_HTML.gif in C in the following way:
{ x = x 0 C arbitrary , x m = w lim n T ( t n ) x m 1 for all  m N . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_Equ8_HTML.gif
(4.1)
From (4.1), we have
x m + 1 = A x m for all  m N 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_Equar_HTML.gif
Set L : = η ( F ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq189_HTML.gif, λ : = L 2 WCS ( X ) < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq162_HTML.gif, D m : = lim sup n x m T ( t n ) x m https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq190_HTML.gif and R m : = lim sup n x m + 1 T ( t n ) x m https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq191_HTML.gif for all m = 0 , 1 , 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq160_HTML.gif . From (3.3), we have
A m + 1 x A m x = x m + 1 x m ( λ + L ) λ m 1 R 0 ( λ + L ) λ m 1 diam ( C ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_Equas_HTML.gif
for x C https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq65_HTML.gif and m N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq163_HTML.gif. It follows that
m = 1 sup x C A m + 1 x A m x < . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_Equat_HTML.gif
Thus, the sequence { A m x } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq192_HTML.gif converges uniformly to a function Q defined by
Q x = lim m A m x for all  x C . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_Equau_HTML.gif
For x = x 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq193_HTML.gif, we have
Q x T ( t n ) Q x Q x x m + x m T ( t n ) x m + T ( t n ) x m T ( t n ) Q x Q x x m + x m T ( t n ) x m + η ( T ( t n ) ) ( x m Q x + a ( t n ) ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_Equav_HTML.gif
Taking the limit superior as n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq165_HTML.gif, we get
lim sup n Q x T ( t n ) Q x ( 1 + η ( F ) ) x m Q x + D m 0 as  m . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_Equaw_HTML.gif

Hence lim n T ( t n ) Q x = Q x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq194_HTML.gif. Let s G https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq195_HTML.gif. From the demicontinuity of T ( s ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq196_HTML.gif, we obtain that T ( s ) Q x = Q x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq197_HTML.gif. One can see that T ( s ) Q x = Q x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq198_HTML.gif for all s G https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq195_HTML.gif. Thus, T ( s ) Q x = Q x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq198_HTML.gif for all x C https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq65_HTML.gif and s G https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq195_HTML.gif. Therefore, Q is a retraction of C onto Fix ( F ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq85_HTML.gif. □

Corollary 4.2 Let X be a uniformly Banach space with the Opial condition, C a nonempty closed convex bounded subset of X and T : C C https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq199_HTML.gif a demicontinuous asymptotically regular nearly Lipschitzian mapping such that η ( T ) : = lim inf n η ( T n ) < WCS ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq200_HTML.gif. Then Fix ( T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq18_HTML.gif and Fix ( T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq2_HTML.gif is a η ( T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq201_HTML.gif-Lipschitzian retract of C.

One sees from Theorem 4.1 that if η ( F ) = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq202_HTML.gif, then Fix ( F ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq85_HTML.gif is a nonexpansive retract of C. In the next section, we show that Fix ( F ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq85_HTML.gif is a sunny nonexpansive retract of C when F = { T ( t ) : t R + } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq203_HTML.gif a strongly continuous semigroup of asymptotically pseudo-contractive mappings (see Theorem 5.6).

5 Common fixed-point sets as sunny nonexpansive retracts

Let C be a nonempty subset of a Banach space X and F = { T ( t ) : t G } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq142_HTML.gif a semigroup of mappings from C into itself. A sequence { x n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq64_HTML.gif in C is said to an approximating fixed point sequence of if lim n ( x n T ( t ) x n ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq204_HTML.gif for all t G https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq89_HTML.gif. The family { I T ( t ) : t G } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq205_HTML.gif is demiclosed at zero if { y n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq206_HTML.gif is a sequence in C weakly converging to z C https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq97_HTML.gif and lim n ( x n T ( t ) y n ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq207_HTML.gif for all t G https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq89_HTML.gif imply z = T ( t ) z https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq208_HTML.gif for all t G https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq89_HTML.gif. Following [18], we say that has property ( A https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq209_HTML.gif) if for every bounded set { x t } t G https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq93_HTML.gif in C, we have
lim t x t T ( t ) x t = 0 implies lim t x t T ( s ) x t = 0 for all  s G . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_Equax_HTML.gif

In [30], Schu introduced the concept of asymptotically pseudo-contractive mapping as follows:

Let H be a real Hilbert space whose inner product and norm are denoted by , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq210_HTML.gif and https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq211_HTML.gif respectively. Let C be a nonempty subset of H and T : C C https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq1_HTML.gif a mapping. Then T is called an asymptotically pseudo-contractive mapping if there exists a sequence { k n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq212_HTML.gif in [ 1 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq213_HTML.gif with lim n k n = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq214_HTML.gif such that
T n x T n y , x y k n x y 2 for all  x , y C  and  n N . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_Equay_HTML.gif

The class of asymptotically pseudo-contractive mappings contain properly the class of asymptotically nonexpansive mappings. The following example shows that a continuous asymptotically pseudo-contractive mapping is not necessarily asymptotically nonexpansive.

Example 5.1 Let X = R https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq215_HTML.gif and C = [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq216_HTML.gif. Define T : C C https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq1_HTML.gif by
T x = ( 1 x 2 / 3 ) 3 / 2 , x C . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_Equaz_HTML.gif

Note that T is a pseudo-contractive mapping which is not Lipschitzian (see [31]). Since T is not Lipschitzian, it is not asymptotically nonexpansive. It is shown in [30] that T is an asymptotically pseudo-contractive mapping with sequence { 1 } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq217_HTML.gif.

Let F = { T ( t ) : t G } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq142_HTML.gif be a semigroup of mappings from C into itself. Then is said to be pseudo-contractive if
T ( t ) x T ( t ) y , x y x y 2 for all  x , y C  and  t G . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_Equba_HTML.gif
Remark 5.2
  1. (i)
    The semigroup is pseudo-contractive if and only if the following holds:
    T ( t ) x T ( t ) y 2 x y 2 + x T ( t ) x ( y T ( t ) y ) 2 for all  x , y C  and  t G . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_Equbb_HTML.gif
     
  2. (ii)

    Every nonexpansive semigroup must be a continuously pseudo-contractive semigroup.

     
We say is asymptotically pseudo-contractive if there exists a function k ( ) : G [ 1 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq218_HTML.gif with lim t k ( t ) = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq219_HTML.gif such that
T ( t ) x T ( t ) y , x y k ( t ) x y 2 for all  x , y C  and  t G . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_Equbc_HTML.gif
Example 5.3 Let X = R https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq215_HTML.gif, b ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq220_HTML.gif, C = [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq216_HTML.gif and G = [ 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq221_HTML.gif. For t > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq70_HTML.gif, define T ( t ) : C C https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq222_HTML.gif by
T ( t ) x = { b t x , if  x [ 0 , 1 / 2 ] ; 0 , if  x ( 1 / 2 , 0 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_Equbd_HTML.gif
and define
T ( 0 ) x = x , x C . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_Eqube_HTML.gif
Set C 1 : = [ 0 , 1 / 2 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq223_HTML.gif and C 2 : = ( 1 / 2 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq224_HTML.gif. Note that
T ( t ) x T ( t ) y , x y = b t | x y | 2 | x y | 2 for all  x , y C 1  and  t > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_Equbf_HTML.gif
and
T ( t ) x T ( t ) y , x y = 0 | x y | 2 for all  x , y C 2  and  t > 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_Equbg_HTML.gif
For x C 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq225_HTML.gif and y C 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq226_HTML.gif, we have x y 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq227_HTML.gif, and hence
T ( t ) x T ( t ) y , x y = ( b t x 0 ) ( x y ) 0 | x y | 2 for all  t > 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_Equbh_HTML.gif
Thus,
T ( t ) x T ( t ) y , x y | x y | 2 for all  x , y C  and  t 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_Equbi_HTML.gif

Therefore, F = { T ( t ) : t 0 } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq228_HTML.gif is an asymptotically pseudo-contractive semigroup with function k = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq229_HTML.gif. Moreover, for each t > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq70_HTML.gif, T ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq150_HTML.gif is discontinuous at x = 1 / 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq230_HTML.gif and hence is not a Lipschitzian semigroup.

We begin with the following:

Theorem 5.4 (Demiclosedness Principle)

Let C be a nonempty closed convex bounded subset of a real Hilbert space H. Let F = { T ( t ) : t G } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq142_HTML.gif be a strongly continuous semigroup of uniformly continuous nearly uniformly L-Lipschitzian asymptotically pseudo-contractive mappings from C into itself. Then the family { I T ( t ) : t G } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq205_HTML.gif is demiclosed at zero.

Proof Assume that { y n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq206_HTML.gif is a sequence in C weakly converging to z and lim n ( y n T ( s ) y n ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq231_HTML.gif for all s G https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq195_HTML.gif. Let t G https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq89_HTML.gif with t 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq232_HTML.gif and let { t m } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq233_HTML.gif be a sequence in G defined by t m = m t https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq234_HTML.gif for all m N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq163_HTML.gif. Notice that z C https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq235_HTML.gif. Fix α ( 0 , 1 / ( 1 + L ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq236_HTML.gif and define
z m = ( 1 α ) z + α T ( t m ) z for all  m N . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_Equbj_HTML.gif

Since T is uniformly continuous, we have y n T ( t m ) z 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq237_HTML.gif as n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq165_HTML.gif for fixed m N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq163_HTML.gif.

Indeed, for fixed m N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq163_HTML.gif, we have
y n T ( t m ) y n y n T ( t ) y n + T ( t ) y n T ( 2 t ) y n + + T ( ( m 1 ) t ) y n T ( m t ) y n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_Equbk_HTML.gif
for all n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq13_HTML.gif. Since is a uniformly continuous semigroup, it follows that lim n y n T ( t m ) y n = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq238_HTML.gif for each fixed m N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq163_HTML.gif. Noticing that is an asymptotically pseudo-contractive semigroup, for fixed m N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq163_HTML.gif, we have
z z m , ( I T ( t m ) ) z m = z y n , ( I T ( t m ) ) z m + y n z m , ( I T ( t m ) ) z m = z y n , ( I T ( t m ) ) z m + y n z m , ( I T ( t m ) ) z m ( I T ( t m ) ) y n + y n z m , ( I T ( t m ) ) y n z y n , ( I T ( t m ) ) z m + ( k ( t n ) 1 ) y n z m 2 + y n z m , ( I T ( t m ) ) y n . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_Equ9_HTML.gif
(5.1)
Since y n z https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq239_HTML.gif and lim n y n T ( t m ) y n = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq238_HTML.gif, it follows from (5.1) that
z z m , ( I T ( t m ) ) z m ( k ( t n ) 1 ) diam ( C ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_Equbl_HTML.gif
Note that
z T ( t m ) z 2 = z T ( t m ) z , z T ( t m ) z = 1 α z z m , z T ( t m ) z = 1 α z z m , ( I T ( t m ) ) z ( I T ( t m ) ) z m + ( I T ( t m ) ) z m = 1 α [ z z m , ( I T ( t m ) ) z ( I T ( t m ) ) z m + z z m , ( I T ( t m ) ) z m ] 1 α [ z z m 2 + z z m T ( t m ) z T ( t m ) z m + ( k ( t m ) 1 ) diam ( C ) ] 1 α [ z z m 2 + L z z m ( z z m + a ( t m ) ) + ( k ( t m ) 1 ) diam ( C ) ] 1 α [ α 2 ( 1 + L ) z T ( t m ) z 2 + ( a ( t m ) L + k ( t m ) 1 ) diam ( C ) ] , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_Equbm_HTML.gif
which implies that
α [ 1 α ( 1 + L ) ] z T ( t m ) z 2 ( a ( t m ) L + k ( t m ) 1 ) diam ( C ) for all  m N . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_Equ10_HTML.gif
(5.2)
Letting m https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq240_HTML.gif in (5.2), we obtain that T ( t m ) z z https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq241_HTML.gif. It follows from the continuity of T ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq150_HTML.gif that
z = lim m T ( t + t m ) z = lim m T ( t ) T ( t m ) z = T ( t ) z . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_Equbn_HTML.gif

Therefore, z = T ( t ) z https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq208_HTML.gif for all t G https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq89_HTML.gif. □

The following result extends the celebrated convergence theorem of Browder [32] and many results concerning Browder’s convergence theorem to a semigroup of uniformly continuous nearly uniformly L-Lipschitzian asymptotically pseudo-contractive mappings.

Theorem 5.5 Let C be a nonempty closed convex bounded subset of a real Hilbert space H and F = { T ( t ) : t R + } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq203_HTML.gif a strongly continuous semigroup of uniformly continuous nearly uniformly L-Lipschitzian asymptotically pseudo-contractive mappings from C into itself. Let { b n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq242_HTML.gif be a sequence in ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq243_HTML.gif and { t n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq145_HTML.gif a sequence in ( 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq244_HTML.gif such that k ( t n ) 1 < b n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq245_HTML.gif for all n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq13_HTML.gif, lim n b n = lim n k ( t n ) 1 b n = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq246_HTML.gif and lim n t n = https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq247_HTML.gif. Then:
  1. (a)
    There exists a sequence { y n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq206_HTML.gif in C defined by
    y n = b n u + ( 1 b n ) T ( t n ) y n , n N . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_Equ11_HTML.gif
    (5.3)
     
  2. (b)
    If has property ( A https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq209_HTML.gif), then Fix ( F ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq180_HTML.gif and { y n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq206_HTML.gif converges strongly to y Fix ( F ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq248_HTML.gif such that
    y u , y v 0 for all v Fix ( F ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_Equ12_HTML.gif
    (5.4)
     
Proof (a) Let F = { T ( t ) : t R + } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq203_HTML.gif be a strongly continuous semigroup of asymptotically pseudo-contractive mappings with a net { k ( t ) : t ( 0 , ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq249_HTML.gif. Set ϱ n : = k ( t n ) 1 b n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq250_HTML.gif. Note k ( t n ) 1 < b n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq245_HTML.gif for all n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq13_HTML.gif, it follows that ϱ n < 1 k ( t n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq251_HTML.gif and hence ( 1 b n ) k ( t n ) < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq252_HTML.gif for all n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq13_HTML.gif. Then, for each n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq253_HTML.gif, the mapping G n : C C https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq254_HTML.gif defined by
G n y : = b n u + ( 1 b n ) T ( t n ) y , y C https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_Equbo_HTML.gif
is continuous and strongly pseudo-contractive. Indeed, for x, y in C, we have
G n x G n y , x y = ( 1 b n ) T ( t n ) x T ( t n ) y , x y ( 1 b n ) k ( t n ) x y 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_Equbp_HTML.gif
Therefore, by Lemma 2.1, there exists a sequence { y n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq206_HTML.gif in C described by (5.3).
  1. (b)

    Assume that has property ( A https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq209_HTML.gif). From (5.3), we have y n T ( t n ) y n 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq255_HTML.gif as n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq165_HTML.gif. The property ( A https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq209_HTML.gif) of gives that y n T ( t ) y n 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq256_HTML.gif as n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq165_HTML.gif for all t > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq257_HTML.gif. Since { y n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq206_HTML.gif is bounded, we can assume that a subsequence { y n i } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq258_HTML.gif of { y n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq206_HTML.gif such that y n i z https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq259_HTML.gif for some z C https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq97_HTML.gif. By Theorem 5.4, we have z Fix ( F ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq260_HTML.gif.

     
For v Fix ( F ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq261_HTML.gif, we have
y n T ( t n ) y n , y n v = y n v + T ( t n ) v T ( t n ) y n , y n v ( k ( t n ) 1 ) y n v 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_Equbq_HTML.gif
From (5.3), we have
y n u , y n v = ( 1 b n ) T ( t n ) y n u , y n v = ( 1 b n ) T ( t n ) y n y n + y n u , y n v , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_Equbr_HTML.gif
so it follows that
y n u , y n v 1 b n b n T ( t n ) y n y n , y n v ( 1 b n ) ϱ n y n v 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_Equ13_HTML.gif
(5.5)
Since lim n ϱ n = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq262_HTML.gif and C is bounded, it follows from (5.5) that
lim sup n y n u , y n v 0 for all  v Fix ( F ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_Equ14_HTML.gif
(5.6)
We claim that the set { y n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq206_HTML.gif is sequentially compact. For v Fix ( F ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq261_HTML.gif, we have
y n v 2 = b n ( u v ) + ( 1 b n ) ( T ( t n ) y n v ) , y n v b n u v , y n v + ( 1 b n ) k ( t n ) y n v 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_Equbs_HTML.gif
which implies that
y n v 2 b n 1 ( 1 b n ) k ( t n ) u v , y n v . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_Equ15_HTML.gif
(5.7)
By the weak compactness of C, there exists a weakly convergent subsequence { y n i } { y n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq263_HTML.gif. Suppose that y n i y C https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq264_HTML.gif as i https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq265_HTML.gif. Since { y n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq206_HTML.gif is an approximating fixed point sequence of , we infer from Theorem 5.4 that y Fix ( F ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq266_HTML.gif. In (5.7), interchange v and y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq267_HTML.gif to obtain that
y n i y 2 b n i 1 ( 1 b n i ) k ( t n i ) u y , y n i y . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_Equbt_HTML.gif

Since y n i y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq268_HTML.gif, we get that y n i y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq269_HTML.gif. Hence the set { y n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq206_HTML.gif is sequentially compact.

Next, we show that y n y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq270_HTML.gif. Suppose, for contradiction, that { y n j } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq271_HTML.gif is another subsequence of { y n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq206_HTML.gif such that y n j z y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq272_HTML.gif. It is easy to see that z Fix ( F ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq273_HTML.gif. Observe that
| y n u , y n z y u , y z | | y n u , y n z y u , y n z | + | y u , y n z y u , y z | y n u ( y u ) y n z + | y u , y n y | for all  n N . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_Equbu_HTML.gif
Since y n i y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq274_HTML.gif, we get
y n i u , y n i z y u , y z . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_Equbv_HTML.gif
From (5.6), we obtain
y u , y z 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_Equ16_HTML.gif
(5.8)
Similarly, we have
z u , z y 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_Equ17_HTML.gif
(5.9)
Adding inequalities (5.8) and (5.9) yields
y z , y z 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_Equbw_HTML.gif

a contradiction. In a similar way it can be shown that each cluster point of the sequence { y n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq206_HTML.gif is equal to y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq267_HTML.gif. Therefore, the entire sequence { y n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq206_HTML.gif converges strongly to y https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq267_HTML.gif. It is easy to see, from (5.6), that the inequality (5.4) holds. □

Theorem 5.6 Let C be a nonempty closed convex bounded subset of a real Hilbert space H and F = { T ( t ) : t R + } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq203_HTML.gif a strongly continuous semigroup of uniformly continuous nearly uniformly L-Lipschitzian asymptotically pseudo-contractive mappings from C into itself. Suppose that has property ( A https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq209_HTML.gif). Then Fix ( F ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq275_HTML.gif and Fix ( F ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq276_HTML.gif is a sunny nonexpansive retract of C.

Proof Assume that F = { T ( t ) : t R + } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq203_HTML.gif is a semigroup of asymptotically pseudo-contractive mappings from C into itself with a function k ( ) : R + [ 1 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq277_HTML.gif with lim t k ( t ) = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq219_HTML.gif. Without loss of generality, we may assume that { b n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq242_HTML.gif in ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq243_HTML.gif and { t n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq145_HTML.gif in ( 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq244_HTML.gif such that k ( t n ) 1 < b n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq245_HTML.gif for all n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq13_HTML.gif, lim n b n = lim n k ( t n ) 1 b n = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq246_HTML.gif and lim n t n = https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq247_HTML.gif. Then, for an arbitrarily fixed element u C https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq278_HTML.gif, there exists a sequence { y n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq206_HTML.gif in C defined by (5.3). By Theorem 5.5(b), Fix ( F ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq275_HTML.gif.

By Theorem 5.5(b), { y n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq206_HTML.gif converges strongly to an element y Fix ( F ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq248_HTML.gif such that the inequality (5.4) holds. Define a mapping Q : C Fix ( F ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq279_HTML.gif by
Q u = lim n y n , u C . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_Equbx_HTML.gif
In view of (5.4), we have
Q u u , Q u v 0 for all  u C  and  v Fix ( F ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_Equby_HTML.gif

Therefore, by Lemma 2.2, we conclude that Q is sunny nonexpansive. □

6 Application

Let C be a nonempty convex subset of a real Hilbert space H and D a nonempty subset of C. For a nonlinear mapping F : C H https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq280_HTML.gif, the variational inequality problem VIP ( F , C ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq281_HTML.gif over D is to find a point x D https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq282_HTML.gif such that
F x , v x 0 for all  v D . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_Equbz_HTML.gif

It is important to note that the theory of variational inequalities has played an important role in the study of many diverse disciplines, for example, partial differential equations, optimal control, optimization, mathematical programming, mechanics, finance, etc.; see, for example, [33, 34] and references therein.

We now turn our attention to dealing with the problem of the existence of solutions of VIP ( C , F ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq283_HTML.gif by sunny nonexpansive retractions.

Following Wong, Sahu and Yao [[35], Proposition 4.6], one can show that the variational inequality problem VIP ( C , F ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq283_HTML.gif with F = I f https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq284_HTML.gif is equivalent to the fixed point problem. Indeed,

Proposition 6.1 Let C be a nonempty convex subset of a smooth Banach space X and F = { T ( t ) : t R + } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq203_HTML.gif a strongly continuous semigroup of mappings from C into itself with Fix ( F ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq180_HTML.gif. Let f : C C https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq285_HTML.gif be a mapping with F = I f https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq284_HTML.gif and let Q be the sunny nonexpansive retraction from C onto Fix ( F ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq286_HTML.gif. Then x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq21_HTML.gif is a solution of variational inequality problem VIP ( C , F ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq287_HTML.gif over Fix ( F ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq286_HTML.gif if and only if x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq21_HTML.gif is a fixed point of Qf.

The following result improves the so-called viscosity approximation method which was first introduced by Moudafi [36] from nonexpansive mappings to a semigroup of pseudo-contractive mappings.

Theorem 6.2 Let C be a nonempty closed convex bounded subset of a real Hilbert space H, f : C C https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq285_HTML.gif a weakly contractive mapping with function ψ and F = { T ( t ) : t R + } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq203_HTML.gif a strongly continuous semigroup of uniformly continuous pseudo-contractive mappings from C into itself. Suppose that has property ( A https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq209_HTML.gif) and is nearly nonexpansive with function a : [ 0 , ) [ 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq288_HTML.gif. Let { b n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq242_HTML.gif be a sequence in ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq243_HTML.gif and { t n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq145_HTML.gif a sequence in ( 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq244_HTML.gif such that lim n b n = lim n a ( t n ) b n = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq289_HTML.gif and lim n t n = https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq247_HTML.gif. Then, we have the following:
  1. (a)

    The variational inequality problem VIP ( C , I f ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq290_HTML.gif over Fix ( F ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq291_HTML.gif has a unique solution in Fix ( F ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq85_HTML.gif.

     
  2. (b)
    There exists a sequence { y n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq206_HTML.gif in C defined by
    y n = b n f ( y n ) + ( 1 b n ) T ( t n ) y n , n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_Equ18_HTML.gif
    (6.1)
     

such that { y n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq206_HTML.gif converges strongly to the unique solution of the variational inequality problem VIP ( C , I f ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq290_HTML.gif.

Proof (a) By Theorem 5.6, there is a sunny nonexpansive retraction Q from C onto Fix ( F ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq85_HTML.gif. Since Qf is a weakly contractive mapping from C into itself, it follows from Rhoades [[39], Theorem 1] that there exists a unique element x C https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq292_HTML.gif such that x = Q f x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq293_HTML.gif. Note x C https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq292_HTML.gif is an element of Fix ( F ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq85_HTML.gif. It follows from Proposition 6.1 that x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq21_HTML.gif is the unique solution of the variational inequality problem VIP ( C , I f ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq290_HTML.gif over Fix ( F ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq286_HTML.gif.
  1. (b)
    For each n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq294_HTML.gif, the mapping F n : C C https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq295_HTML.gif defined by
    F n y : = b n f y + ( 1 b n ) T ( t n ) y , y C https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_Equca_HTML.gif
     
is continuous and strongly pseudo-contractive. In fact, for all x , y C https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq296_HTML.gif and n N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq13_HTML.gif, we have
F n x F n y , x y = b n f x f y , x y + ( 1 b n ) T ( t n ) x T ( t n ) y , x y b n f x f y x y + ( 1 b n ) x y 2 b n [ x y ψ ( x y ) ] x y + ( 1 b n ) x y 2 = x y 2 b n ψ ( x y ) x y . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_Equcb_HTML.gif
Hence each F n https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq297_HTML.gif is continuous b n ψ https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq298_HTML.gif-strongly pseudo-contractive. Therefore, by [37, 38], there exists a sequence { y n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq206_HTML.gif in C described by (6.1). As in Theorem 5.5(a), we may define a sequence { z n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq299_HTML.gif in C by
z n = b n f x + ( 1 b n ) T ( t n ) z n for all  n N . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_Equcc_HTML.gif
By Theorem 5.5, we have that z n x = Q f x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq300_HTML.gif. Observe that
y n z n b n f y n f x + ( 1 b n ) T ( t n ) y n T ( t n ) z n b n ( f y n f z n + f z n f x ) + ( 1 b n ) T ( t n ) y n T ( t n ) z n b n ( y n z n ψ ( y n z n ) + z n x ψ ( z n x ) ) + ( 1 b n ) ( y n z n + a ( t n ) ) y n z n b n ψ ( y n z n ) + b n z n x + a ( t n ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_Equcd_HTML.gif
It follows that
ψ ( y n z n ) z n x + a ( t n ) b n for all  n N . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_Equce_HTML.gif

Thus, y n z n 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq301_HTML.gif. Therefore, x n x = Q f x https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_Article_316_IEq302_HTML.gif. □

Declarations

Authors’ Affiliations

(1)
Department of Mathematics, Banaras Hindu University
(2)
Department of Mathematics, Texas A&M University
(3)
School of Mathematics, Statistics and Applied Mathematics, National University of Ireland Galway

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