The structure of fixed-point sets of Lipschitzian type semigroups

  • DR Sahu1Email author,

    Affiliated with

    • RP Agarwal2 and

      Affiliated with

      • Donal O’Regan3

        Affiliated with

        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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq1_HTML.gif a mapping. We use http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq3_HTML.gif with respect to mapping T[1] if

        (ω) http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq4_HTML.gif  for every http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq5_HTML.gif ,

        where http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq6_HTML.gif denotes the set of all weak subsequential limits of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq7_HTML.gif . Moreover, T is said to satisfy the http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq3_HTML.gif . For a Lipschitzian mapping http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq8_HTML.gif , we use the symbol http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq9_HTML.gif to denote the exact Lipschitz constant of S, i.e.,
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_Equa_HTML.gif

        A mapping http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq10_HTML.gif is said to be

        1. (1)

          nonexpansive if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq11_HTML.gif ,

           
        2. (2)

          asymptotically nonexpansive[2] if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq12_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq13_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq14_HTML.gif ,

           
        3. (3)

          uniformly L-Lipschitzian if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq15_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq13_HTML.gif and for some http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq17_HTML.gif is a nonexpansive mapping with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq18_HTML.gif . Obviously, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq2_HTML.gif is a closed set. http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq17_HTML.gif has a fixed point in every nonempty closed convex subset of C which is invariant under T, then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq2_HTML.gif is a nonexpansive retract of C (that is, there exists a nonexpansive mapping http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq19_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq17_HTML.gif a pointwise contraction. Then T has a unique fixed point http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq21_HTML.gif , and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq22_HTML.gif converges strongly to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq21_HTML.gif for each http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq24_HTML.gif be the normal structure coefficient of X. Let C be a nonempty bounded subset of X and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq25_HTML.gif a uniformly L-Lipschitzian mapping with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq26_HTML.gif . Then T satisfies the http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq3_HTML.gif -fixed point property.

        The http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq28_HTML.gif denote the set of nonnegative real numbers, and let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq29_HTML.gif denote the set of nonnegative integers. Throughout this paper, G denotes an unbounded set of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq30_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq31_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq32_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq33_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq34_HTML.gif with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq35_HTML.gif (often http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq36_HTML.gif or http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq37_HTML.gif ).

        2.1 Lipschitzian type mappings

        Let C be a nonempty subset of a Banach space X and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq38_HTML.gif a mapping. Then T is called

        1. (i)

          pointwise contractive[11] if there exists a function http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq39_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq40_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq41_HTML.gif ;

           
        2. (ii)

          asymptotic pointwise contractive[13] if for each http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq13_HTML.gif , there exists a function http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq42_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq43_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq41_HTML.gif , where http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq44_HTML.gif pointwise on C;

           
        3. (iii)

          pointwise asymptotically nonexpansive[15] if for each integer http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq45_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq46_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq47_HTML.gif , where http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq48_HTML.gif pointwise;

           
        4. (iv)
          asymptotically nonexpansive in the intermediate sense[22] provided T is uniformly continuous and
          http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_Equ1_HTML.gif
          (2.1)
           
        5. (v)
          mapping of asymptotically nonexpansive type[23] if
          http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_Equb_HTML.gif
           
        Fix a sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq49_HTML.gif in http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq50_HTML.gif with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq51_HTML.gif . A mapping http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq1_HTML.gif is said to be nearly Lipschitzian with respect to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq49_HTML.gif [24] if for each http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq52_HTML.gif , there exists a constant http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq53_HTML.gif such that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_Equ2_HTML.gif
        (2.2)

        for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq54_HTML.gif . The infimum of constants http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq55_HTML.gif in (2.2) is called nearly Lipschitz constant and is denoted by http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq56_HTML.gif . A nearly Lipschitzian mapping T with the sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq57_HTML.gif is called

        1. (i)

          nearly contractive if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq58_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq13_HTML.gif ,

           
        2. (ii)

          nearly uniformly L-Lipschitzian if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq59_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq13_HTML.gif ,

           
        3. (iii)

          nearly uniformly k-contractive if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq60_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq13_HTML.gif ,

           
        4. (iv)

          nearly nonexpansive if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq61_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq13_HTML.gif ,

           
        5. (v)

          nearly asymptotically nonexpansive if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq62_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq13_HTML.gif with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq63_HTML.gif .

           
        The mapping T is said to be demicontinuous if, whenever a sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq64_HTML.gif in C converges strongly to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq65_HTML.gif , then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq66_HTML.gif converges weakly to Tx. The mapping T is said to be weakly contractive if
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_Equc_HTML.gif

        where http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq67_HTML.gif is a continuous and nondecreasing function such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq68_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq69_HTML.gif for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq70_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq72_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq5_HTML.gif , i.e., http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq73_HTML.gif . A retraction P is said to be sunny if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq74_HTML.gif for each http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq23_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq75_HTML.gif with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_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)

          http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq77_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq23_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq78_HTML.gif .

           
        3. (c)

          http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq79_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq54_HTML.gif .

           

        2.2 Semigroups

        Let C be a nonempty subset of a Banach space X. The one-parameter family http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq80_HTML.gif is said to be a strongly continuous semigroup of mappings from C into itself if

        1. (I)

          http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq81_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq23_HTML.gif ;

           
        2. (II)

          http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq82_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq83_HTML.gif ;

           
        3. (III)

          for each http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq23_HTML.gif , the mapping http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq84_HTML.gif from G into C is continuous.

           
        We denote by http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq85_HTML.gif the set of all common fixed points of ℱ, i.e., http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq86_HTML.gif . For a Lipschitzian semigroup ℱ, we write
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_Equd_HTML.gif
        If ℱ satisfies (I)-(III) and
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_Equ3_HTML.gif
        (ar)
        then ℱ is called asymptotically regular on C. If ℱ satisfies (I)-(III) and
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_Equ4_HTML.gif
        (uar)

        then ℱ is called uniformly asymptotically regular on C.

        A Lipschitzian semigroup ℱ is called a

        1. (i)

          uniformly L-Lipschitzian semigroup if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq87_HTML.gif ;

           
        2. (ii)

          nonexpansive semigroup if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq88_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq89_HTML.gif ;

           
        3. (iii)

          asymptotically nonexpansive semigroup if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq90_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq89_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq91_HTML.gif .

           

        2.3 Asymptotic center

        Throughout the paper, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq93_HTML.gif a bounded set in X. Consider the functional http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq94_HTML.gif defined by
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_Eque_HTML.gif
        The infimum of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq95_HTML.gif over C is said to be the asymptotic radius of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq93_HTML.gif with respect to C and is denoted by http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq96_HTML.gif . A point http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq97_HTML.gif is said to be an asymptotic center of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq93_HTML.gif with respect to C if
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_Equf_HTML.gif
        The set of all asymptotic centers of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq93_HTML.gif with respect to C is denoted by http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq98_HTML.gif . A number http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq99_HTML.gif is called an asymptotic diameter of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq93_HTML.gif . It is well known that if X is reflexive, then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq98_HTML.gif is nonempty closed convex and bounded, and if X is uniformly convex, then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq98_HTML.gif consists only of a single point, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq100_HTML.gif , i.e., http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq101_HTML.gif is the unique point which minimizes the functional
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_Equg_HTML.gif

        over http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_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
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_Equh_HTML.gif
        the diameter of C. Put
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_Equi_HTML.gif
        This nonnegative real number is called the Chebyshev radius of C relative to itself. The normal structure coefficient http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq24_HTML.gif of a Banach space X is defined [25] by
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_Equj_HTML.gif
        The space X is said to have the uniformly normal structure if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq102_HTML.gif . It is well known that, for every uniformly convex Banach space X, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq102_HTML.gif . A weakly convergent sequence coefficient of X is defined (see [25]) by
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_Equk_HTML.gif
        It is proved in [[26], Theorem 1] that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_Equl_HTML.gif
        where http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq103_HTML.gif . It is readily seen that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_Equm_HTML.gif
        The space X is said to have the weak uniformly normal structure if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq104_HTML.gif . If X is a reflexive Banach space with modulus of convexity http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq105_HTML.gif , then
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_Equn_HTML.gif
        Thus, if X is a uniformly convex Banach space, then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq106_HTML.gif and also the equation
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_Equo_HTML.gif
        has a unique solution http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq107_HTML.gif . A general formula for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq108_HTML.gif in an arbitrary Banach space is not known. In particular, it has been calculated that for a Hilbert space H,
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_Equp_HTML.gif
        for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq109_HTML.gif ( http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq110_HTML.gif ),
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_Equq_HTML.gif
        and for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq111_HTML.gif ,
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_Equr_HTML.gif
        Remark 2.3 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq109_HTML.gif is an example of a reflexive Banach space such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq112_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq113_HTML.gif are different. Indeed,
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_Equs_HTML.gif
        A Banach space X is said to satisfy the Opial condition, if whenever a sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq64_HTML.gif in X converges weakly to x, then
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_Equt_HTML.gif
        The Opial modulus http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq114_HTML.gif of X is defined by
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_Equu_HTML.gif
        where http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq115_HTML.gif and the infimum is taken over all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq116_HTML.gif with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq117_HTML.gif and all sequences http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq64_HTML.gif in X such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq118_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq119_HTML.gif . For any Banach space X, we have the following inequality:
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq80_HTML.gif a strongly continuous semigroup of mappings from C into itself. The semigroup ℱ is said to be nearly Lipschitzian if there exist a function http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq120_HTML.gif with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq121_HTML.gif and a function http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq122_HTML.gif such that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_Equw_HTML.gif
        For a nearly Lipschitzian semigroup ℱ, we write
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_Equx_HTML.gif

        We say ℱ is

        1. (a)
          pointwise nearly Lipschitzian if for each http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq123_HTML.gif , there exist a function http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq124_HTML.gif with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq121_HTML.gif and a function http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq125_HTML.gif such that
          http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_Equy_HTML.gif
           
        2. (b)
          pointwise nearly uniformly http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq126_HTML.gif -Lipschitzian if there exist a function http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq124_HTML.gif with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq127_HTML.gif and a function http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq128_HTML.gif such that
          http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_Equz_HTML.gif
           
        3. (c)
          asymptotic pointwise nearly Lipschitzian if for each http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq123_HTML.gif , there exist a function http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq124_HTML.gif with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq121_HTML.gif and two functions http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq129_HTML.gif with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq130_HTML.gif pointwise such that
          http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_Equaa_HTML.gif
           
        We say that an asymptotic pointwise nearly Lipschitzian semigroup ℱ is pointwise nearly asymptotically nonexpansive (pointwise asymptotically nonexpansive) if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq131_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq89_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq132_HTML.gif pointwise ( http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq133_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq131_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq89_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq134_HTML.gif pointwise). Further, we say that an asymptotic pointwise nearly Lipschitzian semigroup ℱ is asymptotic pointwise nearly contractive if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq135_HTML.gif pointwise and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq136_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq65_HTML.gif . The semigroup ℱ is said to be nearly uniformly L-Lipschitzian if there exist a constant http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq137_HTML.gif and a function http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq120_HTML.gif with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq121_HTML.gif such that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_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, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq3_HTML.gif with respect to semigroup ℱ if

        (ω) http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq138_HTML.gif for every http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq5_HTML.gif ,

        where http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq139_HTML.gif denotes the set of all weak limits of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq140_HTML.gif as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq141_HTML.gif .

        The semigroup ℱ is said to satisfy the http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq3_HTML.gif -fixed point property if ℱ has a common fixed point in every nonempty closed convex subset D of C which satisfies property http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq3_HTML.gif .

        We now establish that a semigroup ℱ of a certain class of Lipschitzian type mappings satisfies the http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq142_HTML.gif a strongly continuous semigroup of demicontinuous mappings from C into itself. Suppose that for each http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq123_HTML.gif , there exist a function http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq124_HTML.gif with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq121_HTML.gif and two functions http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq129_HTML.gif with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq130_HTML.gif pointwise and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq143_HTML.gif such that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_Equac_HTML.gif

        Also suppose that there exists a nonempty closed convex subset M of C which satisfies property (ω) with respect to ℱ. Then:

        1. (a)
          For arbitrary http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq144_HTML.gif , there exist a sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq145_HTML.gif in G with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq146_HTML.gif and an iterative sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq147_HTML.gif in M defined by
          http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_Equ5_HTML.gif
          (3.1)
           
        2. (b)

          Ifis asymptotically regular on C, then there exists an element http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq148_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq147_HTML.gif converges strongly to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq21_HTML.gif .

           

        Proof (a) Since one can easily construct a nonempty closed convex separable subset http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq149_HTML.gif of C which is invariant under each http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq150_HTML.gif (i.e., http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq151_HTML.gif for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq89_HTML.gif ), we may assume that C itself is separable.

        The separability of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq149_HTML.gif makes it possible to select a sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq152_HTML.gif of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq153_HTML.gif such that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_Equad_HTML.gif
        and
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_Equae_HTML.gif
        For any http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq154_HTML.gif , consider a sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq155_HTML.gif in C. Suppose http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq156_HTML.gif . Using property (ω), we obtain that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq157_HTML.gif . Now we can construct a sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq147_HTML.gif in M in the following way:
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_Equaf_HTML.gif
        (b) The weak asymptotic regularity of ℱ ensures that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq158_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq159_HTML.gif . We now show that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq147_HTML.gif converges strongly to a common fixed point of ℱ. Set
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_Equag_HTML.gif
        and
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_Equah_HTML.gif
        for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq160_HTML.gif  . By the property of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq108_HTML.gif , we have
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_Equ6_HTML.gif
        (3.2)
        By the asymptotic regularity of ℱ and the w-l.s.c. of the norm http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq161_HTML.gif , we have
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_Equai_HTML.gif
        On the other hand, by the asymptotic regularity of ℱ, we have
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_Equaj_HTML.gif
        Set http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq162_HTML.gif . From (3.2), we obtain
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_Equak_HTML.gif
        For any http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq163_HTML.gif , one can see that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_Equ7_HTML.gif
        (3.3)
        so it follows that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq147_HTML.gif is a Cauchy sequence in M. Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq164_HTML.gif . Observe that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_Equal_HTML.gif
        Taking the limit superior as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq165_HTML.gif on both sides, we get
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_Equam_HTML.gif

        Hence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq166_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq89_HTML.gif . Note http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq167_HTML.gif , so it follows from the demicontinuity of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq150_HTML.gif that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq168_HTML.gif . Observe that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq169_HTML.gif . By the uniqueness of the weak limit of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq170_HTML.gif , we have http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq171_HTML.gif . Therefore, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq173_HTML.gif , and there are Banach spaces for which http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq174_HTML.gif while http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq142_HTML.gif a strongly continuous semigroup of demicontinuous nearly uniformly L-Lipschitzian of mappings from C into itself. Suppose that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq176_HTML.gif and that there exists a nonempty closed convex subset M of C which satisfies property (ω) with respect to ℱ. Then:
        1. (a)
          For arbitrary http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq144_HTML.gif , there exist a sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq145_HTML.gif in G with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq146_HTML.gif and an iterative sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq147_HTML.gif in M defined by
          http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_Equan_HTML.gif
           
        2. (b)

          Ifis asymptotically regular on C, then there exists an element http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq148_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq147_HTML.gif converges strongly to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq142_HTML.gif a strongly continuous semigroup of demicontinuous nearly uniformly L-Lipschitzian asymptotically regular mappings from C into itself such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq177_HTML.gif . Thenhas 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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq142_HTML.gif a strongly continuous semigroup of demicontinuous mappings from C into itself. Suppose that there exists a function http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq124_HTML.gif with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq121_HTML.gif such that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_Equao_HTML.gif

        where http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq178_HTML.gif with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq179_HTML.gif . Also suppose thatis asymptotically regular on C. Then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq180_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq85_HTML.gif is http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq152_HTML.gif of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq153_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq182_HTML.gif and
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_Equap_HTML.gif
        Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq183_HTML.gif denote a mapping which associates with a given http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq184_HTML.gif a unique http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq185_HTML.gif , that is, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq186_HTML.gif . Since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq187_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq65_HTML.gif , it follows from the lower weak semi-continuity of the norm that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_Equaq_HTML.gif

        i.e., A is http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq181_HTML.gif -Lipschitzian mapping. It follows that A is uniformly continuous.

        For any http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq188_HTML.gif , consider a sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq155_HTML.gif in C. Suppose http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq156_HTML.gif . Now we can construct a sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq147_HTML.gif in C in the following way:
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_Equ8_HTML.gif
        (4.1)
        From (4.1), we have
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_Equar_HTML.gif
        Set http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq189_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq162_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq190_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq191_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq160_HTML.gif  . From (3.3), we have
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_Equas_HTML.gif
        for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq65_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq163_HTML.gif . It follows that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_Equat_HTML.gif
        Thus, the sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq192_HTML.gif converges uniformly to a function Q defined by
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_Equau_HTML.gif
        For http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq193_HTML.gif , we have
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_Equav_HTML.gif
        Taking the limit superior as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq165_HTML.gif , we get
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_Equaw_HTML.gif

        Hence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq194_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq195_HTML.gif . From the demicontinuity of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq196_HTML.gif , we obtain that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq197_HTML.gif . One can see that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq198_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq195_HTML.gif . Thus, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq198_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq65_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq195_HTML.gif . Therefore, Q is a retraction of C onto http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq199_HTML.gif a demicontinuous asymptotically regular nearly Lipschitzian mapping such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq200_HTML.gif . Then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq18_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq2_HTML.gif is a http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq201_HTML.gif -Lipschitzian retract of C.

        One sees from Theorem 4.1 that if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq202_HTML.gif , then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq85_HTML.gif is a nonexpansive retract of C. In the next section, we show that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq85_HTML.gif is a sunny nonexpansive retract of C when http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq142_HTML.gif a semigroup of mappings from C into itself. A sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq64_HTML.gif in C is said to an approximating fixed point sequence of ℱ if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq204_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq89_HTML.gif . The family http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq205_HTML.gif is demiclosed at zero if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq206_HTML.gif is a sequence in C weakly converging to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq97_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq207_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq89_HTML.gif imply http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq208_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq89_HTML.gif . Following [18], we say that ℱ has property ( http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq209_HTML.gif ) if for every bounded set http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq93_HTML.gif in C, we have
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq210_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq211_HTML.gif respectively. Let C be a nonempty subset of H and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq1_HTML.gif a mapping. Then T is called an asymptotically pseudo-contractive mapping if there exists a sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq212_HTML.gif in http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq213_HTML.gif with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq214_HTML.gif such that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq215_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq216_HTML.gif . Define http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq1_HTML.gif by
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq217_HTML.gif .

        Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq142_HTML.gif be a semigroup of mappings from C into itself. Then ℱ is said to be pseudo-contractive if
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_Equba_HTML.gif

        Remark 5.2

        1. (i)
          The semigroup ℱ is pseudo-contractive if and only if the following holds:
          http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_Equbb_HTML.gif
           
        2. (ii)

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

           
        We say ℱ is asymptotically pseudo-contractive if there exists a function http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq218_HTML.gif with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq219_HTML.gif such that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_Equbc_HTML.gif
        Example 5.3 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq215_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq220_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq216_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq221_HTML.gif . For http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq70_HTML.gif , define http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq222_HTML.gif by
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_Equbd_HTML.gif
        and define
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_Eqube_HTML.gif
        Set http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq223_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq224_HTML.gif . Note that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_Equbf_HTML.gif
        and
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_Equbg_HTML.gif
        For http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq225_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq226_HTML.gif , we have http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq227_HTML.gif , and hence
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_Equbh_HTML.gif
        Thus,
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_Equbi_HTML.gif

        Therefore, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq228_HTML.gif is an asymptotically pseudo-contractive semigroup with function http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq229_HTML.gif . Moreover, for each http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq70_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq150_HTML.gif is discontinuous at http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq230_HTML.gif and hence ℱ 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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq205_HTML.gif is demiclosed at zero.

        Proof Assume that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq206_HTML.gif is a sequence in C weakly converging to z and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq231_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq195_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq89_HTML.gif with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq232_HTML.gif and let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq233_HTML.gif be a sequence in G defined by http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq234_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq163_HTML.gif . Notice that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq235_HTML.gif . Fix http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq236_HTML.gif and define
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_Equbj_HTML.gif

        Since T is uniformly continuous, we have http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq237_HTML.gif as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq165_HTML.gif for fixed http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq163_HTML.gif .

        Indeed, for fixed http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq163_HTML.gif , we have
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_Equbk_HTML.gif
        for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq13_HTML.gif . Since ℱ is a uniformly continuous semigroup, it follows that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq238_HTML.gif for each fixed http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq163_HTML.gif . Noticing that ℱ is an asymptotically pseudo-contractive semigroup, for fixed http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq163_HTML.gif , we have
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_Equ9_HTML.gif
        (5.1)
        Since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq239_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq238_HTML.gif , it follows from (5.1) that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_Equbl_HTML.gif
        Note that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_Equbm_HTML.gif
        which implies that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_Equ10_HTML.gif
        (5.2)
        Letting http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq240_HTML.gif in (5.2), we obtain that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq241_HTML.gif . It follows from the continuity of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq150_HTML.gif that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_Equbn_HTML.gif

        Therefore, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq208_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq203_HTML.gif a strongly continuous semigroup of uniformly continuous nearly uniformly L-Lipschitzian asymptotically pseudo-contractive mappings from C into itself. Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq242_HTML.gif be a sequence in http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq243_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq145_HTML.gif a sequence in http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq244_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq245_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq13_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq246_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq247_HTML.gif . Then:
        1. (a)
          There exists a sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq206_HTML.gif in C defined by
          http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_Equ11_HTML.gif
          (5.3)
           
        2. (b)
          Ifhas property ( http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq209_HTML.gif ), then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq180_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq206_HTML.gif converges strongly to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq248_HTML.gif such that
          http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_Equ12_HTML.gif
          (5.4)
           
        Proof (a) Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq203_HTML.gif be a strongly continuous semigroup of asymptotically pseudo-contractive mappings with a net http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq249_HTML.gif . Set http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq250_HTML.gif . Note http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq245_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq13_HTML.gif , it follows that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq251_HTML.gif and hence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq252_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq13_HTML.gif . Then, for each http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq253_HTML.gif , the mapping http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq254_HTML.gif defined by
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_Equbo_HTML.gif
        is continuous and strongly pseudo-contractive. Indeed, for x, y in C, we have
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_Equbp_HTML.gif

        Therefore, by Lemma 2.1, there exists a sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq206_HTML.gif in C described by (5.3).

        (b) Assume that ℱ has property ( http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq209_HTML.gif ). From (5.3), we have http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq255_HTML.gif as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq165_HTML.gif . The property ( http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq209_HTML.gif ) of ℱ gives that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq256_HTML.gif as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq165_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq257_HTML.gif . Since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq206_HTML.gif is bounded, we can assume that a subsequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq258_HTML.gif of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq206_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq259_HTML.gif for some http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq97_HTML.gif . By Theorem 5.4, we have http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq260_HTML.gif .

        For http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq261_HTML.gif , we have
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_Equbq_HTML.gif
        From (5.3), we have
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_Equbr_HTML.gif
        so it follows that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_Equ13_HTML.gif
        (5.5)
        Since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq262_HTML.gif and C is bounded, it follows from (5.5) that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_Equ14_HTML.gif
        (5.6)
        We claim that the set http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq206_HTML.gif is sequentially compact. For http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq261_HTML.gif , we have
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_Equbs_HTML.gif
        which implies that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_Equ15_HTML.gif
        (5.7)
        By the weak compactness of C, there exists a weakly convergent subsequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq263_HTML.gif . Suppose that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq264_HTML.gif as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq265_HTML.gif . Since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq206_HTML.gif is an approximating fixed point sequence of ℱ, we infer from Theorem 5.4 that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq266_HTML.gif . In (5.7), interchange v and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq267_HTML.gif to obtain that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_Equbt_HTML.gif

        Since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq268_HTML.gif , we get that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq269_HTML.gif . Hence the set http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq206_HTML.gif is sequentially compact.

        Next, we show that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq270_HTML.gif . Suppose, for contradiction, that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq271_HTML.gif is another subsequence of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq206_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq272_HTML.gif . It is easy to see that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq273_HTML.gif . Observe that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_Equbu_HTML.gif
        Since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq274_HTML.gif , we get
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_Equbv_HTML.gif
        From (5.6), we obtain
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_Equ16_HTML.gif
        (5.8)
        Similarly, we have
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_Equ17_HTML.gif
        (5.9)
        Adding inequalities (5.8) and (5.9) yields
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_Equbw_HTML.gif

        a contradiction. In a similar way it can be shown that each cluster point of the sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq206_HTML.gif is equal to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq267_HTML.gif . Therefore, the entire sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq206_HTML.gif converges strongly to http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq203_HTML.gif a strongly continuous semigroup of uniformly continuous nearly uniformly L-Lipschitzian asymptotically pseudo-contractive mappings from C into itself. Suppose thathas property ( http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq209_HTML.gif ). Then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq275_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq276_HTML.gif is a sunny nonexpansive retract of C.

        Proof Assume that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq203_HTML.gif is a semigroup of asymptotically pseudo-contractive mappings from C into itself with a function http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq277_HTML.gif with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq219_HTML.gif . Without loss of generality, we may assume that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq242_HTML.gif in http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq243_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq145_HTML.gif in http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq244_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq245_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq13_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq246_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq247_HTML.gif . Then, for an arbitrarily fixed element http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq278_HTML.gif , there exists a sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq206_HTML.gif in C defined by (5.3). By Theorem 5.5(b), http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq275_HTML.gif .

        By Theorem 5.5(b), http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq206_HTML.gif converges strongly to an element http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq248_HTML.gif such that the inequality (5.4) holds. Define a mapping http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq279_HTML.gif by
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_Equbx_HTML.gif
        In view of (5.4), we have
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq280_HTML.gif , the variational inequality problem http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq281_HTML.gif over D is to find a point http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq282_HTML.gif such that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq283_HTML.gif with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq203_HTML.gif a strongly continuous semigroup of mappings from C into itself with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq180_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq285_HTML.gif be a mapping with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq284_HTML.gif and let Q be the sunny nonexpansive retraction from C onto http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq286_HTML.gif . Then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq21_HTML.gif is a solution of variational inequality problem http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq287_HTML.gif over http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq286_HTML.gif if and only if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_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, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq285_HTML.gif a weakly contractive mapping with function ψ and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq203_HTML.gif a strongly continuous semigroup of uniformly continuous pseudo-contractive mappings from C into itself. Suppose thathas property ( http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq209_HTML.gif ) andis nearly nonexpansive with function http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq288_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq242_HTML.gif be a sequence in http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq243_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq145_HTML.gif a sequence in http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq244_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq289_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq247_HTML.gif . Then, we have the following:
        1. (a)

          The variational inequality problem http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq290_HTML.gif over http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq291_HTML.gif has a unique solution in http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq85_HTML.gif .

           
        2. (b)
          There exists a sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq206_HTML.gif in C defined by
          http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_Equ18_HTML.gif
          (6.1)
           

        such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq206_HTML.gif converges strongly to the unique solution of the variational inequality problem http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq290_HTML.gif .

        Proof (a) By Theorem 5.6, there is a sunny nonexpansive retraction Q from C onto http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq292_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq293_HTML.gif . Note http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq292_HTML.gif is an element of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq85_HTML.gif . It follows from Proposition 6.1 that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq21_HTML.gif is the unique solution of the variational inequality problem http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq290_HTML.gif over http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq286_HTML.gif .

        (b) For each http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq294_HTML.gif , the mapping http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq295_HTML.gif defined by
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_Equca_HTML.gif
        is continuous and strongly pseudo-contractive. In fact, for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq296_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq13_HTML.gif , we have
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_Equcb_HTML.gif
        Hence each http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq297_HTML.gif is continuous http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq298_HTML.gif -strongly pseudo-contractive. Therefore, by [37, 38], there exists a sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq206_HTML.gif in C described by (6.1). As in Theorem 5.5(a), we may define a sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq299_HTML.gif in C by
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_Equcc_HTML.gif
        By Theorem 5.5, we have that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq300_HTML.gif . Observe that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_Equcd_HTML.gif
        It follows that
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_Equce_HTML.gif

        Thus, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_316_IEq301_HTML.gif . Therefore, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-163/MediaObjects/13663_2012_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

        References

        1. Benavides TD, Ramirez PL: Structure of the fixed point set and common fixed points of asymptotically nonexpansive mappings. Proc. Am. Math. Soc. 2001, 129:3549–3557.MATHView Article
        2. Goebel K, Kirk WA: A fixed point theorem for asymptotically nonexpansive mappings. Proc. Am. Math. Soc. 1972, 35:171–174.MathSciNetMATHView Article
        3. Agarwal RP, O’Regan D, Sahu DR: Fixed Point Theory for Lipschitzian-type Mappings With Applications. Springer, New York; 2009. [Topological Fixed Point Theory and Its Applications 6]
        4. Goebel K, Reich S: Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings. Dekker, New York; 1984.MATH
        5. Kopecká E, Reich S: Nonexpansive retracts in Banach spaces. Banach Cent. Publ. 2007, 77:161–174.View Article
        6. Reich S: Extension problems for accretive sets in Banach spaces. J. Funct. Anal. 1977, 26:378–395.MATHView Article
        7. Bruck RE Jr.: Properties of fixed-point sets of nonexpansive mappings in Banach spaces. Trans. Am. Math. Soc. 1973, 179:251–262.MathSciNetMATHView Article
        8. Beauzamy B: Projections contractantes dans les espaces de Banach. Bull. Sci. Math. 1978, 102:43–47.MathSciNetMATH
        9. Yao JC, Zeng LC: Fixed point theorems for asymptotically regular semigroups in metric spaces with uniform normal structure. J. Nonlinear Convex Anal. 2007, 8:153–163.MathSciNetMATH
        10. Ceng LC, Xu HK, Yao JC: Uniformly normal structure and uniformly Lipschitzian semigroups. Nonlinear Anal. 2010, 73:3742–3750.MathSciNetMATHView Article
        11. Belluce LP, Kirk WA: Fixed point theorems for certain classes of nonexpansive mappings. Proc. Am. Math. Soc. 1969, 20:141–146.MathSciNetMATHView Article
        12. Kirk WA: Mappings of generalized contractive type. J. Math. Anal. Appl. 1970, 32:567–572.MathSciNetMATHView Article
        13. Kirk WA: Asymptotic pointwise contractions. Plenary Lecture, the 8th International Conference on Fixed Point Theory and Its Applications, Chiang Mai University, Thailand July 16-22 2007.
        14. Kirk WA: Fixed points of asymptotic contractions. J. Math. Anal. Appl. 2003, 277:645–650.MathSciNetMATHView Article
        15. Kirk WA, Xu HK: Asymptotic pointwise contractions. Nonlinear Anal. 2008, 69:4706–4712.MathSciNetMATHView Article
        16. Casini E, Maluta E: Fixed points of uniformly Lipschitzian mappings in spaces with uniformly normal structure. Nonlinear Anal. 1985, 9:103–108.MathSciNetMATHView Article
        17. Lim TC, Xu HK: Fixed point theorems for asymptotically nonexpansive mappings. Nonlinear Anal. 1994,22(11):1345–1355.MathSciNetMATHView Article
        18. Sahu DR, Agarwal RP, O’Regan D: Structure of the fixed point set of asymptotically nonexpansive mappings in Banach spaces with weak uniformly normal structure. J. Anal. Appl. 2011, 17:51–68.MathSciNetView ArticleMATH
        19. Zegeye H, Shahzad N: Strong convergence theorems for continuous semigroups of asymptotically nonexpansive mappings. Numer. Funct. Anal. Optim. 2009,30(7–8):833–848.MathSciNetMATHView Article
        20. Chidume CE, Zegeye H: Strong convergence theorems for common fixed points of uniformly L -Lipschitzian pseudocontractive semi-groups. Appl. Anal. 2007, 86:353–366.MathSciNetMATHView Article
        21. Hester A, Morales CH: Semigroups generated by pseudo-contractive mappings under the Nagumo condition. J. Differ. Equ. 2008, 245:994–1013.MathSciNetMATHView Article
        22. Bruck RE, Kuczumow T, Reich S: Convergence of iterates of asymptotically nonexpansive mappings in Banach spaces with the uniform Opial property. Colloq. Math. 1993, LXV:169–179.MathSciNet
        23. Kirk WA: Fixed point theorems for non-Lipschitzian mappings of asymptotically nonexpansive type. Isr. J. Math. 1974, 17:339–346.MathSciNetMATHView Article
        24. Sahu DR: Fixed points of demicontinuous nearly Lipschitzian mappings in Banach spaces. Comment. Math. Univ. Carol. 2005,46(4):653–666.MathSciNetMATH
        25. Bynum WL: Normal structure coefficient for Banach spaces. Pac. J. Math. 1980, 86:427–436.MathSciNetMATHView Article
        26. Benavides TD, Acedo GL, Xu HK: Weak uniform normal structure and iterative fixed points of nonexpansive mappings. Colloq. Math. 1995, LXVIII:17–23.
        27. Gornicki J: Fixed points of asymptotically regular semigroups in Banach spaces. Rend. Circ. Mat. Palermo 1997, 46:89–118.MathSciNetMATHView Article
        28. Sahu DR, Zeqing L, Kang SM: Existence and approximation of fixed points of nonlinear mappings in spaces with weak uniform normal structure. Comput. Math. Appl. 2012, 64:672–685.MathSciNetMATHView Article
        29. Sahu, DR, Petruşel, A, Yao, JC: On fixed points of pointwise Lipschitzian type mappings. Fixed Point Theory (2012, accepted)Sahu, DR, Petruşel, A, Yao, JC: On fixed points of pointwise Lipschitzian type mappings. Fixed Point Theory (2012, accepted)
        30. Schu J: Iterative construction of fixed points of asymptotically nonexpansive mappings. J. Math. Anal. Appl. 1991, 158:407–413.MathSciNetMATHView Article
        31. Rhoades BE: Comments on two fixed point iteration methods. J. Math. Anal. Appl. 1976, 56:741–750.MathSciNetMATHView Article
        32. Browder FE: Convergence of approximants to fixed points of nonexpansive nonlinear mappings in Banach spaces. Arch. Ration. Mech. Anal. 1967, 24:82–90.MathSciNetMATHView Article
        33. Kinderlehrer D, Stampacchia G: An Introduction To Variational Inequalities and Their Applications. Academic Press, New York; 1980.MATH
        34. Jaillet P, Lamberton D, Lapeyre B: Variational inequalities and the pricing of American options. Acta Appl. Math. 1990, 21:263–289.MathSciNetMATHView Article
        35. Wong NC, Sahu DR, Yao JC: Solving variational inequalities involving nonexpansive type mappings. Nonlinear Anal. 2008, 69:4732–4753.MathSciNetMATHView Article
        36. Moudafi A: Viscosity approximation methods for fixed-points problems. J. Math. Anal. Appl. 2000, 241:46–55.MathSciNetMATHView Article
        37. Xiang CH: Fixed point theorem for generalized F-pseudo-contractive mappings. Nonlinear Anal. 2009, 70:277–279.View Article
        38. Morales CH: Variational inequalities for Φ-pseudo-contractive mappings. Nonlinear Anal. 2012, 75:477–484.MathSciNetMATHView Article
        39. Rhoades BE: Some theorems on weakly contractive maps. Nonlinear Anal. 2001, 47:2683–2693.MathSciNetMATHView Article

        Copyright

        © Sahu et al.; licensee Springer 2012

        This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://​creativecommons.​org/​licenses/​by/​2.​0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.