Open Access

Iterative algorithms for quasi-variational inclusions and fixed point problems of pseudocontractions

Fixed Point Theory and Applications20142014:82

https://doi.org/10.1186/1687-1812-2014-82

Received: 27 February 2014

Accepted: 17 March 2014

Published: 28 March 2014

Abstract

In this paper, quasi-variational inclusions and fixed point problems of pseudocontractions are considered. An iterative algorithm is presented. A strong convergence theorem is demonstrated.

MSC:49J40, 47J20, 47H09, 65J15.

Keywords

quasi-variational inclusionsfixed point problempseudocontractionsmaximal monotonefirmly nonexpansive mappings

1 Introduction

Let C be a nonempty closed convex subset of a real Hilbert space H. Let A : C H be a single-valued nonlinear mapping and B : H 2 H be a multi-valued mapping. The ‘so called’ quasi-variational inclusion problem is to find an u 2 H such that
0 A u + B u .
(1.1)
The set of solutions of (1.1) is denoted by ( A + B ) 1 ( 0 ) . A number of problems arising in structural analysis, mechanics, and economics can be studied in the framework of this kind of variational inclusions; see for instance [14]. For related work, see [510]. The problem (1.1) includes many problems as special cases.
  1. (1)
    If B = ϕ : H 2 H , where ϕ : H R + is a proper convex lower semi-continuous function and ∂ϕ is the subdifferential of ϕ, then the variational inclusion problem (1.1) is equivalent to finding u H such that
    A u , y u + ϕ ( y ) ϕ ( u ) 0 , y H ,

    which is called the mixed quasi-variational inequality (see [11]).

     
  2. (2)
    If B = δ C , where C is a nonempty closed convex subset of H and δ C : H [ 0 , ] is the indicator function of C, i.e.,
    δ C = { 0 , x C , + , x C ,
     
then the variational inclusion problem (1.1) is equivalent to finding u C such that
A u , v u 0 , v C .

This problem is called the Hartman-Stampacchia variational inequality (see [12]).

Let T : C C be a nonlinear mapping. The iterative scheme of Mann’s type for approximating fixed points of T is the following: x 0 C and
x n + 1 = α n x n + ( 1 α n ) T x n ,
for all n 1 , where { α n } is a sequence in [ 0 , 1 ] ; see [13]. For two nonlinear mappings S and T, Takahashi and Tamura [14] considered the following iteration procedure: x 0 C and
x n + 1 = α n x n + ( 1 α n ) S ( β n x n + ( 1 β n ) T x n ) ,

for all n 1 , where { α n } and { β n } are two sequences in [ 0 , 1 ] . Algorithms for finding the fixed points of nonlinear mappings or for finding the zero points of maximal monotone operators have been studied by many authors. The reader can refer to [1519]. Especially, Takahashi et al. [20] recently gave the following convergence result.

Theorem 1.1 Let C be a closed and convex subset of a real Hilbert space H. Let A be an α-inverse strongly monotone mapping of C into H and let B be a maximal monotone operator on H, such that the domain of B is included in C. Let J λ B = ( I + λ B ) 1 be the resolvent of B for λ > 0 and let T be a nonexpansive mapping of C into itself, such that F ( T ) ( A + B ) 1 0 . Let x 1 = x C and let { x n } C be a sequence generated by
x n + 1 = β n x n + ( 1 β n ) T ( α n x + ( 1 α n ) J λ n B ( x n λ n A x n ) ) ,
for all n 0 , where { λ n } ( 0 , 2 α ) , { α n } ( 0 , 1 ) and { β n } ( 0 , 1 ) satisfy
0 < a λ n b < 2 α , 0 < c β n d < 1 , lim n ( λ n + 1 λ n ) = 0 , lim n α n = 0 and n α n = .

Then { x n } converges strongly to a point of F ( T ) ( A + B ) 1 0 .

Recently, Zhang et al. [21] introduced a new iterative scheme for finding a common element of the set of solutions to the inclusion problem and the set of fixed points of nonexpansive mappings in Hilbert spaces. Peng et al. [22] introduced another iterative scheme by the viscosity approximate method for finding a common element of the set of solutions of a variational inclusion with set-valued maximal monotone mapping and inverse strongly monotone mappings, the set of solutions of an equilibrium problem, and the set of fixed points of a nonexpansive mapping.

Motivated and inspired by the works in this field, the purpose of this paper is to consider the quasi-variational inclusions and fixed point problems of pseudocontractions. An iterative algorithm is presented. A strong convergence theorem is demonstrated.

2 Notations and lemmas

Let H be a real Hilbert space with inner product , and norm , respectively. Let C be a nonempty closed convex subset of H. It is well known that in a real Hilbert space H, the following equality holds:
t x + ( 1 t ) y 2 = t x 2 + ( 1 t ) y 2 t ( 1 t ) x y 2
(2.1)

for all x , y H and t [ 0 , 1 ] .

Recall that a mapping T : C C is called

(D1) L-Lipschitzian there exists L > 0 such that T x T y L x y for all x , y C ; in the case of L = 1 , T is said to be nonexpansive;

(D2) Firmly nonexpansive T x T y 2 x y 2 ( I T ) x ( I T ) y 2 T x T y 2 T x T y , x y for all x , y C ;

(D3) Pseudocontractive T x T y , x y x y 2 T x T y 2 x y 2 + ( I T ) x ( I T ) y 2 for all x , y C ;

(D4) Strongly monotone there exists a positive constant γ ˜ such that T x T y , x y γ ˜ x y for all x , y C ;

(D5) Inverse strongly monotone T x T y , x y α T x T y 2 for some α > 0 and for all x , y C .

Let B be a mapping of H into 2 H . The effective domain of B is denoted by dom ( B ) , that is, dom ( B ) = { x H : B x } . A multi-valued mapping B is said to be a monotone operator on H iff
x y , u v 0

for all x , y dom ( B ) , u B x , and v B y . A monotone operator B on H is said to be maximal iff its graph is not strictly contained in the graph of any other monotone operator on H. Let B be a maximal monotone operator on H and let B 1 0 = { x H : 0 B x } .

For a maximal monotone operator B on H and λ > 0 , we may define a single-valued operator J λ B = ( I + λ B ) 1 : H dom ( B ) , which is called the resolvent of B for λ. It is known that the resolvent J λ B is firmly nonexpansive, i.e.,
J λ B x J λ B y 2 J λ B x J λ B y , x y

for all x , y C and B 1 0 = Fix ( J λ B ) for all λ > 0 .

Usually, the convergence of fixed point algorithms requires some additional smoothness properties of the mapping T such as demi-closedness.

Recall that a mapping T is said to be demiclosed if, for any sequence { x n } which weakly converges to x ˜ , and if the sequence { T ( x n ) } strongly converges to z, then T ( x ˜ ) = z . For the pseudocontractions, the following demiclosed principle is well known.

Lemma 2.1 ([23])

Let H be a real Hilbert space, C a closed convex subset of H. Let U : C C be a continuous pseudo-contractive mapping. Then
  1. (i)

    Fix ( U ) is a closed convex subset of C,

     
  2. (ii)

    ( I U ) is demiclosed at zero.

     

Lemma 2.2 ([24])

Let { r n } be a sequence of real numbers. Assume { r n } does not decrease at infinity, that is, there exists at least a subsequence { r n k } of { r n } such that r n k r n k + 1 for all k 0 . For every n N , define an integer sequence { τ ( n ) } as
τ ( n ) = max { i n : r n i < r n i + 1 } .
Then τ ( n ) as n , and for all n N
max { r τ ( n ) , r n } r τ ( n ) + 1 .

Lemma 2.3 ([25])

Assume { a n } is a sequence of nonnegative real numbers such that
a n + 1 ( 1 γ n ) a n + δ n γ n ,
where { γ n } is a sequence in ( 0 , 1 ) and { δ n } is a sequence such that
  1. (1)

    n = 1 γ n = ;

     
  2. (2)

    lim sup n δ n 0 or n = 1 | δ n γ n | < .

     

Then lim n a n = 0 .

In the sequel we shall use the following notations:
  1. 1.

    ω w ( u n ) = { x : u n j x  weakly } denote the weak ω-limit set of { u n } ;

     
  2. 2.

    u n x stands for the weak convergence of { u n } to x;

     
  3. 3.

    u n x stands for the strong convergence of { u n } to x;

     
  4. 4.

    Fix ( T ) stands for the set of fixed points of T.

     

3 Main results

In this section, we consider a strong convergence theorem for quasi-variational inclusions and fixed point problems of pseudocontractive mappings in a Hilbert space.

Algorithm 3.1 Let C be a nonempty closed and convex subset of a real Hilbert space H. Let A be an α-inverse strongly monotone mapping of C into H and let B be a maximal monotone operator on H, such that the domain of B is included in C. Let J λ B = ( I + λ B ) 1 be the resolvent of B for λ. Let F : C H be an L 1 -Lipschitzian and ς strongly monotone mapping and f : C C be a ρ-contraction such that ρ < max { 1 , ς / 2 } . Let T : C C be an L 2 ( > 1 ) -Lipschitzian pseudocontraction. For x 0 C , define a sequence { x n } as follows:
{ z n = J λ B ( I λ A ) x n , y n = ν z n + ( 1 ν ) T ( ( 1 ζ ) z n + ζ T z n ) , x n + 1 = α n x n + ( 1 α n ) ( β n f ( x n ) + ( I β n F ) y n ) ,
(3.1)

for all n N , where λ, ν and ζ are three constants, { α n } and { β n } are two sequences in [ 0 , 1 ] .

Now, we demonstrate the convergence analysis of the algorithm (3.1).

Theorem 3.2 Suppose Γ : = Fix ( T ) ( A + B ) 1 ( 0 ) . Assume the following conditions are satisfied:

(C1) α n [ a , b ] ( 0 , 1 ) ;

(C2) lim n β n = 0 and n = 1 β n = ;

(C3) λ ( 0 , 2 α ) and 0 < 1 ν ζ < 1 1 + L 2 2 + 1 .

Then the sequence { x n } defined by (3.1) converges strongly to u = P Γ ( I F + f ) u .

Proof Let x Fix ( T ) ( A + B ) 1 ( 0 ) . Then, we get x = J λ B ( I λ A ) x = T x . From (3.1), we have
z n x 2 = J λ B ( I λ A ) x n J λ B ( I λ A ) x 2 x n x λ ( A x n A x ) 2 = x n x 2 2 λ A x n A x , x n x + λ 2 A x n A x 2 x n x 2 2 λ α A x n A x 2 + λ 2 A x n A x 2 = x n x 2 λ ( 2 α λ ) A x n A x 2 x n x 2 .
(3.2)
It follows that
z n x x n x .
(3.3)
Since x Fix ( T ) , we have from (D3) that
T x x 2 x x 2 + T x x 2 ,
(3.4)

for all x C .

Thus,
T ( ( 1 ζ ) I + ζ T ) z n x 2 ( 1 ζ ) ( z n x ) + ζ ( T z n x ) 2 + ( ( 1 ζ ) I + ζ T ) z n T ( ( 1 ζ ) I + ζ T ) z n 2 .
(3.5)
By (3.4), (3.5), and (2.1), we obtain
T ( ( 1 ζ ) I + ζ T ) z n x 2 ( 1 ζ ) ( z n x ) + ζ ( T z n x ) 2 + ( ( 1 ζ ) I + ζ T ) z n T ( ( 1 ζ ) I + ζ T ) z n 2 = ( 1 ζ ) ( z n T ( ( 1 ζ ) I + ζ T ) z n ) + ζ ( T z n T ( ( 1 ζ ) I + ζ T ) z n ) 2 + ( 1 ζ ) ( z n x ) + ζ ( T z n x ) 2 = ( 1 ζ ) z n T ( ( 1 ζ ) I + ζ T ) z n 2 + ζ T z n T ( ( 1 ζ ) I + ζ T ) z n 2 ζ ( 1 ζ ) z n T z n 2 + ( 1 ζ ) z n x 2 + ζ T z n x 2 ζ ( 1 ζ ) z n T z n 2 ( 1 ζ ) z n x 2 + ζ ( z n x 2 + z n T z n 2 ) 2 ζ ( 1 ζ ) z n T z n 2 + ( 1 ζ ) z n T ( ( 1 ζ ) I + ζ T ) z n 2 + ζ T z n T ( ( 1 ζ ) I + ζ T ) z n 2 .
Noting that T is L 2 -Lipschitzian and z n ( ( 1 ζ ) I + ζ T ) z n = ζ ( z n T z n ) , we have
T ( ( 1 ζ ) I + ζ T ) z n x 2 ( 1 ζ ) z n x 2 + ζ ( z n x 2 + z n T z n 2 ) 2 ζ ( 1 ζ ) z n T z n 2 + ( 1 ζ ) z n T ( ( 1 ζ ) I + ζ T ) z n 2 + ζ 3 L 2 2 z n T z n 2 = z n x 2 + ( 1 ζ ) z n T ( ( 1 ζ ) I + ζ T ) z n 2 ζ ( 1 2 ζ ζ 2 L 2 2 ) z n T z n 2 .
(3.6)
Since ζ < 1 1 + L 2 2 + 1 , we have 1 2 ζ ζ 2 L 2 2 > 0 . From (3.6), we can deduce
T ( ( 1 ζ ) I + ζ T ) z n x 2 z n x 2 + ( 1 ζ ) z n T ( ( 1 ζ ) I + ζ T ) z n 2 .
(3.7)
Hence,
y n x 2 = ν z n + ( 1 ν ) T ( ( 1 ζ ) I + ζ T ) z n x 2 = ν ( z n x ) + ( 1 ν ) ( T ( ( 1 ζ ) I + ζ T ) z n x ) 2 = ν z n x 2 + ( 1 ν ) T ( ( 1 ζ ) I + ζ T ) z n x 2 ν ( 1 ν ) T ( ( 1 ζ ) I + ζ T ) z n z n 2 ν z n x 2 + ( 1 ν ) [ z n x 2 + ( 1 ζ ) z n T ( ( 1 ζ ) I + ζ T ) z n 2 ] ν ( 1 ν ) T ( ( 1 ζ ) I + ζ T ) z n z n 2 = z n x 2 + ( 1 ν ) ( 1 ζ ν ) T ( ( 1 ζ ) I + ζ T ) z n z n 2 .
(3.8)
By (C3) and (3.8), we obtain
y n x z n x .
(3.9)
Let u n = β n f ( x n ) + ( I β n F ) y n for all n 0 . Then, we have
u n x = β n f ( x n ) + ( I β n F ) y n x β n f ( x n ) F x + ( I β n F ) y n ( I β n F ) x β n f ( x n ) f ( x ) + β n f ( x ) F x + ( I β n F ) y n ( I β n F ) x β n ρ x n x + β n f ( x ) F x + ( I β n F ) y n ( I β n F ) x .
(3.10)
Since F is L 1 -Lipschitzian and ς strongly monotone, we have
( I β n F ) y n ( I β n F ) x 2 = ( y n x ) β n ( F y n F x ) 2 = y n x 2 2 β n F y n F x , y n x + β n 2 F y n F x 2 y n x 2 2 β n ς y n x 2 + β n 2 L 1 2 y n x 2 = ( 1 2 β n ς + β n 2 L 1 2 ) y n x 2 .
(3.11)
Noting that L 1 ς and lim n β n = 0 , without loss of generality, we assume that β n < ς L 1 2 ς / 4 for all n 0 . Thus, 1 2 β n ς + β n 2 L 1 2 ( 1 β n ς 2 ) 2 . So,
( I β n F ) y n ( I β n F ) x ( 1 β n ς 2 ) y n x .
(3.12)
We have from (3.9), (3.10), and (3.12)
u n x β n ρ x n x + β n f ( x ) F x + ( 1 β n ς 2 ) x n x = [ 1 ( ς 2 ρ ) β n ] x n x + β n f ( x ) F x .
(3.13)
From (3.1) and (3.13), we have
x n + 1 x = α n ( x n x ) + ( 1 α n ) ( u n x ) ( 1 α n ) ( [ 1 ( ς 2 ρ ) β n ] x n x + β n f ( x ) F x ) + α n x n x = [ 1 ( ς 2 ρ ) ( 1 α n ) β n ] x n x + ( 1 α n ) β n f ( x ) F x .
(3.14)
By the definition of x n , we have
x n + 1 x n = α n x n + ( 1 α n ) ( β n f ( x n ) + ( I β n F ) y n ) x n = ( 1 α n ) [ β n f ( x n ) β n F y n + y n x n ] .
(3.15)
Hence,
x n + 1 x n , x n x = ( 1 α n ) [ β n f ( x n ) β n F y n + y n x n ] , x n x = ( 1 α n ) β n f ( x n ) , x n x ( 1 α n ) β n F y n , x n x + ( 1 α n ) y n x n , x n x .
(3.16)
Since 2 x n + 1 x n , x n x = x n + 1 x 2 x n x 2 x n + 1 x n 2 and 2 y n x n , x n x = y n x 2 x n x 2 y n x n 2 , it follows from (3.16), (3.3), and (3.9) that
x n + 1 x 2 x n x 2 x n + 1 x n 2 = 2 ( 1 α n ) β n f ( x n ) , x n x 2 ( 1 α n ) β n F y n , x n x + ( 1 α n ) [ y n x 2 x n x 2 y n x n 2 ] 2 ( 1 α n ) β n f ( x n ) , x n x 2 ( 1 α n ) β n F y n , x n x ( 1 α n ) y n x n 2 .
(3.17)
By (3.15), we obtain
x n + 1 x n 2 ( 1 α n ) 2 [ β n f ( x n ) F y n + y n x n ] 2 = ( 1 α n ) 2 [ β n 2 f ( x n ) F y n 2 + y n x n 2 + 2 β n f ( x n ) F y n y n x n ] .
(3.18)
Combining (3.17) and (3.18) to deduce
x n + 1 x 2 x n x 2 2 ( 1 α n ) β n f ( x n ) , x n x 2 ( 1 α n ) β n F y n , x n x ( 1 α n ) y n x n 2 + ( 1 α n ) 2 [ β n 2 f ( x n ) F y n 2 + y n x n 2 + 2 β n f ( x n ) F y n y n x n ] 2 ( 1 α n ) β n f ( x n ) , x n x 2 ( 1 α n ) β n F y n , x n x ( 1 α n ) α n y n x n 2 + ( 1 α n ) 2 [ β n 2 f ( x n ) F y n 2 + 2 β n f ( x n ) F y n y n x n ] .
Hence, we obtain
x n + 1 x 2 x n x 2 + ( 1 α n ) α n y n x n 2 2 ( 1 α n ) β n f ( x n ) , x n x 2 ( 1 α n ) β n F y n , x n x + ( 1 α n ) 2 [ β n 2 f ( x n ) F y n 2 + 2 β n f ( x n ) F y n y n x n ] .
It follows that, hence, we obtain
( 1 α n ) α n y n x n 2 x n x 2 x n + 1 x 2 + 2 ( 1 α n ) β n f ( x n ) , x n x 2 ( 1 α n ) β n F y n , x n x + ( 1 α n ) 2 [ β n 2 f ( x n ) F y n 2 + 2 β n f ( x n ) F y n y n x n ] .
(3.19)

Next we divide our proof into two possible cases.

Case 1. There exists an integer number m such that x n + 1 x x n x for all n m . In this case, we have lim n x n x exists. Since α n [ a , b ] ( 0 , 1 ) and lim n β n = 0 , by (3.19), we derive
lim n y n x n = 0 .
(3.20)
This together with (3.18) implies that
lim n x n + 1 x n = 0 .
(3.21)
Note that
u n y n = β n f ( x n ) + ( I β n F ) y n y n β n f ( x n ) F y n .
So,
lim n u n y n = 0 .
(3.22)
By (3.20) and (3.22), we obtain
lim n u n x n = 0 .
(3.23)
From (3.2) and (3.9), we have
y n x 2 z n x 2 x n x 2 λ ( 2 α λ ) A x n A x 2 .
Hence,
λ ( 2 α λ ) A x n A x 2 x n x 2 y n x 2 x n y n ( x n x + y n x ) .
Therefore,
lim n A x n A x = 0 .
(3.24)
Since J λ B is firmly nonexpansive and A is monotone, we have
z n x 2 = J λ B ( I λ A ) x n J λ B ( I λ A ) x 2 ( I λ A ) x n ( I λ A ) x , z n x = z n x , x n x λ z n x , A x n A x = 1 2 ( z n x 2 + x n x 2 z n x n 2 ) λ x n x , A x n A x λ z n x n , A x n A x 1 2 ( z n x 2 + x n x 2 z n x n 2 ) + λ z n x n A x n A x .
It follows that
z n x 2 x n x 2 z n x n 2 + 2 λ z n x n A x n A x .
(3.25)
By (3.25) and (3.9), we deduce
y n x 2 z n x 2 x n x 2 z n x n 2 + 2 λ z n x n A x n A x .
Therefore,
z n x n 2 x n x 2 y n x 2 + 2 λ z n x n A x n A x x n y n ( x n x + y n x ) + 2 λ z n x n A x n A x .
(3.26)
Equations (3.20), (3.24), and (3.26) imply that
lim n z n x n = 0 .
(3.27)
Notice that F f is ( ς ρ ) strongly monotone. Thus, the variational inequality of finding y Γ such that ( F f ) y , x y 0 for all x Γ has a unique solution, denoted by x , that is, x = P Γ ( I V + F ) ( x ) . Next, we prove that
lim sup n ( f F ) x , u n x 0 .
Since u n is bounded, without loss of generality, we assume that there exists a subsequence { z n i } of { u n } such that u n i x ˜ for some x ˜ H and
lim sup n ( f F ) x , u n x = lim sup i ( f F ) x , u n i x .
Thus, we have that x n i x ˜ and
lim i J λ B ( I λ A ) x n i x n i = 0 .

Therefore, x ˜ Fix ( J λ B ( I λ A ) ) = ( A + B ) 1 ( 0 ) .

Next we show that x ˜ Fix ( T ) . First, we show that Fix ( T ) = Fix ( T ( ( 1 ζ ) I + ζ T ) ) . As a matter of fact, Fix ( T ) Fix ( T ( ( 1 ζ ) I + ζ T ) ) is obvious. Next, we show that Fix ( T ( ( 1 ζ ) I + ζ T ) ) Fix ( T ) .

Take any x Fix ( T ( ( 1 ζ ) I + ζ T ) ) . We have T ( ( 1 ζ ) I + ζ T ) x = x . Set S = ( 1 ζ ) I + ζ T . We have T S x = x . Write S x = y . Then, T y = x . Now we show x = y . In fact,
x y = T y S x = T y ( 1 ζ ) x ζ T x = ζ T y T x ζ L 2 y x .

Since, ζ < 1 1 + L 2 2 + 1 < 1 L 2 , we deduce y = x Fix ( S ) = Fix ( T ) . Thus, x Fix ( T ) . Hence, Fix ( T ( ( 1 ζ ) I + ζ T ) ) Fix ( T ) . Therefore, Fix ( T ( ( 1 ζ ) I + ζ T ) ) = Fix ( T ) .

By (3.1), (3.20), and (3.27), we deduce
lim n T ( ( 1 ζ ) I + ζ T ) x n x n = 0 .
(3.28)

Next we prove that T ( ( 1 ζ ) I + ζ T ) I is demiclosed at 0. Let the sequence { w n } H 2 satisfying w n x and w n T ( ( 1 ζ ) I + ζ T ) w n 0 . Next, we will show that x Fix ( T ( ( 1 ζ ) I + ζ T ) ) = Fix ( T ) .

Since T is L 2 -Lipschizian, we have
w n T w n w n T ( ( 1 ζ ) I + ζ T ) w n + T ( ( 1 ζ ) I + ζ T ) w n T w n w n T ( ( 1 ζ ) I + ζ T ) w n + ζ L w n T w n .
It follows that
w n T w n 1 1 ζ L w n T ( ( 1 ζ ) I + ζ T ) w n .
Hence,
lim n w n T w n = 0 .
Since T I is demiclosed at 0 by Lemma 2.1, we immediately deduce x Fix ( T ) = Fix ( T ( ( 1 ζ ) I + ζ T ) ) . Therefore, T ( ( 1 ζ ) I + ζ T ) I is demiclosed at 0. By (3.28), we deduce x ˜ Fix ( T ) . Hence, x ˜ Γ . So,
lim sup n ( f F ) x , u n x = lim sup i ( f F ) x , u n i x = ( f F ) x , x ˜ x 0 .
(3.29)
Note that
u n x 2 = β n ( f ( x n ) f ( x ) ) + β n ( f ( x ) F x ) + ( I β n F ) ( y n x ) 2 ( I β n F ) ( y n x ) 2 + 2 β n f ( x n ) f ( x ) , u n x + 2 β n f ( x ) F x , u n x ( 1 β n ς 2 ) 2 x n x 2 + 2 β n ρ x n x u n x + 2 β n f ( x ) F x , u n x ( 1 β n ς 2 ) 2 x n x 2 + 2 β n ρ x n x 2 + 1 2 u n x 2 + 2 β n f ( x ) F x , u n x .
It follows that
u n x 2 [ 1 2 ( ς 2 ρ ) β n ] x n x 2 + β n 2 ς 4 2 x n x 2 + 4 β n f ( x ) F x , u n x .
So,
x n + 1 x 2 = α n ( x n x ) + ( 1 α n ) ( u n x ) 2 α n x n x 2 + ( 1 α n ) u n x 2 [ 1 2 ( ς 2 ρ ) ( 1 α n ) β n ] x n x 2 + ( 1 α n ) β n 2 ς 2 4 x n x 2 + 4 ( 1 α n ) β n f ( x ) F x , u n x = [ 1 ( ς 2 ρ ) ( 1 α n ) β n ] x n x 2 + ( ς 2 ρ ) ( 1 α n ) β n { β n ς 2 4 ( ς 2 ρ ) x n x 2 + 4 ς 2 ρ f ( x ) F x , u n x } .
(3.30)

Applying Lemma 2.3 to (3.30) we deduce x n x .

Case 2. Assume there exists an integer n 0 such that x n 0 x x n 0 + 1 x . In this case, we set ω n = { x n x } . Then, we have ω n 0 ω n 0 + 1 . Define an integer sequence { τ n } for all n n 0 as follows:
τ ( n ) = max { l N | n 0 l n , ω l ω l + 1 } .
It is clear that τ ( n ) is a non-decreasing sequence satisfying
lim n τ ( n ) =
and
ω τ ( n ) ω τ ( n ) + 1 ,
for all n n 0 . From (3.19), we get
( 1 α τ ( n ) ) α τ ( n ) y τ ( n ) x τ ( n ) 2 x τ ( n ) x 2 x τ ( n ) + 1 x 2 + 2 ( 1 α τ ( n ) ) β τ ( n ) f ( x τ ( n ) ) , x τ ( n ) x 2 ( 1 α τ ( n ) ) β τ ( n ) F y τ ( n ) , x τ ( n ) x + ( 1 α τ ( n ) ) 2 [ β τ ( n ) 2 f ( x τ ( n ) ) F y τ ( n ) 2 + 2 β τ ( n ) f ( x τ ( n ) ) F y τ ( n ) y τ ( n ) x τ ( n ) ] .
(3.31)
It follows that
lim n y τ ( n ) x τ ( n ) = 0 .
By a similar argument to that of (3.29) and (3.30), we can prove that
lim sup n ( f F ) x , u τ ( n ) x 0 ,
(3.32)
and
ω τ ( n ) + 1 2 [ 1 2 ( ς 2 ρ ) ( 1 α τ ( n ) ) β τ ( n ) ] ω τ ( n ) 2 + ( 1 α τ ( n ) ) β τ ( n ) 2 ς 2 4 ω τ ( n ) 2 + 4 ( 1 α τ ( n ) ) β τ ( n ) f ( x ) F x , u τ ( n ) x .
(3.33)
Since ω τ ( n ) ω τ ( n ) + 1 , we have from (3.33)
ω τ ( n ) 2 16 4 ( ς 2 2 ρ ) ς 2 β τ ( n ) f ( x ) F x , u τ ( n ) x .
(3.34)
Combining (3.33) and (3.34), we have
lim sup n ω τ ( n ) 0 ,
and hence
lim n ω τ ( n ) = 0 .
(3.35)
From (3.33), we also obtain
lim sup n ω τ ( n ) + 1 lim sup n ω τ ( n ) .
This together with (3.35) imply that
lim n ω τ ( n ) + 1 = 0 .
Applying Lemma 2.2 to get
0 ω n max { ω τ ( n ) , ω τ ( n ) + 1 } .

Therefore, ω n 0 . That is, x n x . This completes the proof. □

Declarations

Acknowledgements

Yonghong Yao was supported in part by NSFC 71161001-G0105. Yeong-Cheng Liou was supported in part by NSC 101-2628-E-230-001-MY3 and NSC 101-2622-E-230-005-CC3.

Authors’ Affiliations

(1)
Department of Mathematics, Tianjin Polytechnic University
(2)
Department of Mathematics, Texas A&M University
(3)
Nonlinear Analysis and Applied Mathematics Research Group (NAAM), King Abdulaziz University
(4)
Department of Information Management, Cheng Shiu University
(5)
Center for General Education, Kaohsiung Medical University

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© Yao et al.; licensee Springer. 2014

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