Open Access

On generalized asymptotically quasi-ϕ-nonexpansive mappings and a Ky Fan inequality

Fixed Point Theory and Applications20132013:237

https://doi.org/10.1186/1687-1812-2013-237

Received: 24 May 2013

Accepted: 12 August 2013

Published: 23 September 2013

Abstract

Generalized asymptotically quasi-ϕ-nonexpansive mappings and a Ky Fan inequality are investigated. A strong convergence theorem for common solutions to a fixed point problem of generalized asymptotically quasi-ϕ-nonexpansive mappings and a Ky Fan inequality is established in a Banach space.

MSC:47H05, 47J25, 90C33.

Keywords

asymptotically quasi-ϕ-nonexpansive mapping generalized asymptotically quasi-ϕ-nonexpansive mapping generalized projection equilibrium problem fixed point

1 Introduction

Iterative algorithms have been studied by many authors. The applications of iterative algorithms are found in a wide range of areas, including economics, image recovery and signal processing. Many well-known problems can be studied by using algorithms which are iterative in their nature; see [114] and the references therein. As an example, in computer tomography with limited data, each piece of information implies the existence of a convex set, in which the required solution lies. The problem of finding a point in the intersection of these convex subsets is then of crucial interest, and it cannot be usually solved directly. Therefore, an iterative algorithm must be used to approximate such a point.

Mann iteration, introduced by Mann [15], is an efficient tool to study fixed point problems of asymptotical nonexpansive mappings. However, Mann iteration is only weak convergence in infinite-dimensional spaces; see [10] and the references therein. The importance of strong convergence is underlined in [16], where a convex function f is minimized via the proximal-point algorithm: it is shown that the rate of convergence of the value sequence { f ( x n ) } is better when { x n } converges strongly than when it converges weakly. Such properties have a direct impact when the process is executed directly in the underlying infinite-dimensional space. To obtain strong convergence of Mann iteration, projection methods, which were first introduced by Haugazeau [17], have been considered for modifying Mann iteration to obtain strong convergence. The advantage of projection methods is that strong convergence of iterative sequences can be guaranteed without any compact assumptions.

The organization of this paper is as follows. In Section 2, we provide some necessary concepts and lemmas. In Section 3, fixed point problems of generalized asymptotically quasi-ϕ-nonexpansive mappings and solutions of a Ky Fan inequality are investigated. A strong convergence theorem is established in a Banach space.

2 Preliminaries

Recall that the normalized duality mapping J from E to 2 E is defined by
J x = { f E : x , f = x 2 = f 2 } ,
where , denotes the generalized duality pairing. Let U E = { x E : x = 1 } be the unit sphere of E. Then the Banach space E is said to be smooth iff
lim t 0 x + t y x t

exists for each x , y U E . It is also said to be uniformly smooth iff the above limit is attained uniformly for x , y U E . It is well known that if E is uniformly smooth, then J is uniformly norm-to-norm continuous on each bounded subset of E. It is also well known that E is uniformly smooth if and only if E is uniformly convex. Recall that E is said to be strictly convex iff x + y 2 < 1 for all x , y E with x = y = 1 and x y . It is said to be uniformly convex iff lim n x n y n = 0 for any two sequences { x n } and { y n } in E such that x n = y n = 1 and lim n x n + y n 2 = 1 . Recall that E enjoys the Kadec-Klee property if for any sequence { x n } E , and x E with x n x , and x n x , then x n x 0 as n . For more details on the Kadec-Klee property, readers can refer to [18] and the references therein. It is well known that if E is a uniformly convex Banach space, then E enjoys the Kadec-Klee property.

Next, we assume that E is a smooth Banach space. Consider the functional defined by
ϕ ( x , y ) = x 2 2 x , J y + y 2 , x , y E .
Observe that in a Hilbert space H, the equality is reduced to ϕ ( x , y ) = x y 2 , x , y H . As we all know, if C is a nonempty closed convex subset of a Hilbert space H and P C : H C is the metric projection of H onto C, then P C is nonexpansive. This fact actually characterizes Hilbert spaces and, consequently, it is not available in more general Banach spaces. In this connection, Alber [19] recently introduced a generalized projection operator Π C in a Banach space E which is an analogue of the metric projection P C in Hilbert spaces. Recall that the generalized projection Π C : E C is a map that assigns to an arbitrary point x E the minimum point of the functional ϕ ( x , y ) , that is, Π C x = x ¯ , where x ¯ is the solution to the minimization problem
ϕ ( x ¯ , x ) = min y C ϕ ( y , x ) .
Existence and uniqueness of the operator Π C follow from the properties of the functional ϕ ( x , y ) and strict monotonicity of the mapping J; see, for example, [18]. In Hilbert spaces, Π C = P C . It is obvious from the definition of a function ϕ that
ϕ ( x , y ) = ϕ ( x , z ) + ϕ ( z , y ) + 2 x z , J z J y ,
(2.1)
and
( x y ) 2 ϕ ( x , y ) ( y + x ) 2 , x , y E .
(2.2)

Remark 2.1 If E is a reflexive, strictly convex and smooth Banach space, then ϕ ( x , y ) = 0 if and only if x = y ; for more details, see [18] and the reference therein.

Let C be a nonempty subset of E and T : C C be a mapping. In this paper, we use F ( T ) to denote the fixed point set of T. T is said to be asymptotically regular on C if for any bounded subset K of C,
lim sup n { T n + 1 x T n x : x K } = 0 .

T is said to be closed if for any sequence { x n } C such that lim n x n = x 0 and lim n T x n = y 0 , then T x 0 = y 0 . In this paper, we use → and to denote strong convergence and weak convergence, respectively. Recall that a point p in C is said to be an asymptotic fixed point of T iff C contains a sequence { x n } which converges weakly to p so that lim n x n T x n = 0 . The set of asymptotic fixed points of T will be denoted by F ˜ ( T ) .

Recall that T is said to be relatively nonexpansive iff
F ˜ ( T ) = F ( T ) , ϕ ( p , T x ) ϕ ( p , x ) , x C , p F ( T ) .
Recall that T is said to be relatively asymptotically nonexpansive iff
F ˜ ( T ) = F ( T ) , ϕ ( p , T n x ) ( 1 + μ n ) ϕ ( p , x ) , x C , p F ( T ) , n 1 ,

where { μ n } [ 0 , ) is a sequence such that μ n 0 as n .

Recall that a mapping T is said to be quasi-ϕ-nonexpansive iff
F ( T ) , ϕ ( p , T x ) ϕ ( p , x ) , x C , p F ( T ) .
Recall that a mapping T is said to be asymptotically quasi-ϕ-nonexpansive iff there exists a sequence { μ n } [ 0 , ) with μ n 0 as n such that
F ( T ) , ϕ ( p , T n x ) ( 1 + μ n ) ϕ ( p , x ) , x C , p F ( T ) , n 1 .

Remark 2.2 The class of asymptotically quasi-ϕ-nonexpansive mappings was considered in Zhou et al. [20] and Qin et al. [21]; see also [22] and [23].

Remark 2.3 The class of quasi-ϕ-nonexpansive mappings and the class of asymptotically quasi-ϕ-nonexpansive mappings are more general than the class of relatively nonexpansive mappings and the class of relatively asymptotically nonexpansive mappings [24]. Quasi-ϕ-nonexpansive mappings and asymptotically quasi-ϕ-nonexpansive mappings do not require the restriction F ( T ) = F ˜ ( T ) .

Remark 2.4 The class of quasi-ϕ-nonexpansive mappings and the class of asymptotically quasi-ϕ-nonexpansive mappings are generalizations of the class of quasi-nonexpansive mappings and the class of asymptotically quasi-nonexpansive mappings in Banach spaces.

Recall that T is said to be generalized asymptotically quasi-ϕ-nonexpansive if F ( T ) , and there exists a sequence { μ n } [ 1 , ) with μ n 1 as n and a sequence { ν n } [ 0 , ) with ν n 0 as n such that ϕ ( p , T x ) μ n ϕ ( p , x ) + ν n for all x C , p F ( T ) and n 1 .

Remark 2.5 The class of generalized asymptotically quasi-ϕ-nonexpansive mappings was considered in Qin et al. [25]; see also [26].

Let f be a bifunction from C × C to , where denotes the set of real numbers, and let A : C E be a mapping. Consider the following Ky Fan inequality which is known as a generalized equilibrium problem. Find p C such that
f ( p , q ) + A p , q p 0 , q C .
(2.3)
We use S ( f , A ) to denote the solution set of inequality (2.3). That is,
S ( f ) = { p C : f ( p , q ) + A p , q p 0 , q C } .
If A = 0 , then problem (2.3) is reduced to the following Ky Fan inequality which is known as an equilibrium problem. Find p C such that
f ( p , q ) 0 , q C .
(2.4)
We use S ( f ) to denote the solution set of inequality (2.4). That is,
S ( f ) = { p C : f ( p , q ) 0 , q C } .
If f = 0 , then problem (2.3) is reduced to the classical variational inequality. Find p C such that
A p , q p 0 , q C .
(2.5)
We use V I ( C , A ) to denote the solution set of inequality (2.5). That is,
V I ( C , A ) = { p C : A p , q p 0 , q C } .
Recall that a mapping A : C E is said to be α-inverse-strongly monotone if there exists α > 0 such that
A x A y , x y α A x A y 2 .
For solving problem (2.3), let us assume that the nonlinear mapping A : C E is α-inverse-strongly monotone and the bifunction f : C × C R satisfies the following conditions:
  1. (A1)

    F ( x , x ) = 0 , x C ;

     
  2. (A2)

    F is monotone, i.e., F ( x , y ) + F ( y , x ) 0 , x , y C ;

     
  3. (A3)
    lim sup t 0 F ( t z + ( 1 t ) x , y ) F ( x , y ) , x , y , z C ;
     
  4. (A4)

    for each x C , y F ( x , y ) is convex and weakly lower semicontinuous.

     

Recently, many authors investigated the solutions of problems (2.3), (2.4) and (2.5) based on iterative methods; see [2737]. In this paper, we investigate generalized asymptotically quasi-ϕ-nonexpansive mappings and problem (2.3). A strong convergence theorem for common solutions to a fixed point problem of generalized asymptotically quasi-ϕ-nonexpansive mappings and problem (2.3) is established in a Banach space.

In order to state our main results, we need the following lemmas, which play an import role in the paper.

Lemma 2.6 [28]

Let E be a smooth, strictly convex and reflexive Banach space and C be a nonempty closed convex subset of E. Let A : C E be an α-inverse-strongly monotone mapping and f be a bifunction satisfying conditions (A1)-(A4). Let r > 0 be any given number and x E be any given point. Then there exists p C such that
f ( p , q ) + A p , q p + 1 r q p , J p J x 0 , q C .

Lemma 2.7 [28]

Let E be a smooth, strictly convex and reflexive Banach space and C be a nonempty closed convex subset of E. Let A : C E be an α-inverse-strongly monotone mapping and f be a bifunction satisfying conditions (A1)-(A4). Let r > 0 be any given number and x E define a mapping K r : C C as follows: for any x C ,
K r x = { p C : f ( p , q ) + A p , q p + 1 r q p , J p J x 0 , q C } .
Then the following conclusions hold:
  1. (1)

    K r is single-valued;

     
  2. (2)
    K r is a firmly nonexpansive-type mapping, i.e., for all x , y E ,
    K r x K r y , J K r x J K r y S r x S r y , J x J y ;
     
  3. (3)

    F ( K r ) = S ( f , A ) ;

     
  4. (4)

    K r is quasi-ϕ-nonexpansive;

     
  5. (5)
    ϕ ( q , K r x ) + ϕ ( K r x , x ) ϕ ( q , x ) , q F ( K r ) ;
     
  6. (6)

    S ( f , A ) is closed and convex.

     

Lemma 2.8 [19]

Let C be a nonempty closed convex subset of a smooth Banach space E and x E . Then x 0 = Π C x if and only if
x 0 y , J x J x 0 0 , y C .

Lemma 2.9 [19]

Let E be a reflexive, strictly convex and smooth Banach space, C be a nonempty closed convex subset of E and x E . Then
ϕ ( y , Π C x ) + ϕ ( Π C x , x ) ϕ ( y , x ) , y C .

Lemma 2.10 [25]

Let E be a uniformly smooth and strictly convex Banach space which also enjoys the Kadec-Klee property and C be a nonempty closed convex subset of E. Let T : C C be a closed generalized asymptotically quasi-ϕ-nonexpansive mapping. Then F ( T ) is closed and convex.

Lemma 2.11 [38]

Let E be a smooth and uniformly convex Banach space and let r > 0 . Then there exists a strictly increasing, continuous and convex function g : [ 0 , 2 r ] R such that g ( 0 ) = 0 and
t x + ( 1 t ) y 2 t x 2 + ( 1 t ) y 2 t ( 1 t ) g ( x y )

for all x , y B r = { x E : x r } and t [ 0 , 1 ] .

3 Main results

Theorem 3.1 Let E be a uniformly smooth and strictly convex Banach space which also enjoys the Kadec-Klee property and C be a nonempty closed convex subset of E. Let T : C C be a generalized asymptotically quasi-ϕ-nonexpansive mapping. Let f be a bifunction from C × C to satisfying (A1)-(A4) and A : C E be an α-inverse-strongly monotone mapping. Assume that T is closed and asymptotically regular on C, and F ( T ) S ( f , A ) is nonempty and bounded. Let { x n } be a sequence generated in the following manner:
{ x 0 E chosen arbitrarily , C 1 = C , x 1 = Π C 1 x 0 , y n = J 1 ( ( 1 α n ) J x n + α n J T n x n ) , u n C such that f ( u n , q ) + A u n + q u n + 1 r n q u n , J u n J y n 0 , q C , C n + 1 = { k C n : ϕ ( k , u n ) ϕ ( k , x n ) + ( μ n 1 ) W n + ν n } , x n + 1 = Π C n + 1 x 0 ,

where W n = sup { ϕ ( p , x n ) : p F ( T ) S ( f , A ) } , { α n } is a real number sequence in ( 0 , 1 ) such that lim inf n α n ( 1 α n ) > 0 and { r n } is a real number sequence such that lim inf n r n > 0 . Then the sequence { x n } converges strongly to Π F ( T ) S ( f , A ) x 1 , where Π F ( T ) S ( f , A ) is the generalized projection from E onto F ( T ) S ( f , A ) .

Proof First, we prove C n is closed and convex so that the projection is well defined. We see that C 1 = C is closed and convex. Assume that C m is closed and convex for some positive integer m. For k C m , we find that
ϕ ( k , u m ) ϕ ( k , x m ) + ( μ m 1 ) W m + ν m ,
which is equivalent to
2 k , J x m J u m x m 2 u m 2 + ( μ m 1 ) W m + ν m .

It is easy to see that C m + 1 is closed and convex. This proves that C n is closed and convex so that Π C n + 1 x 1 is well defined. Set u n = K r n y n . It follows from Lemma 2.7 that K r n is quasi-ϕ-nonexpansive.

Now, we are in a position to prove that F ( T ) S ( f , A ) C n . Indeed, F ( T ) S ( f , A ) C 1 = C is obvious. Assume that F ( T ) S ( f , A ) C m for some positive integer m. Then, for e F ( T ) S ( f , A ) C m , we have
ϕ ( e , u m ) = ϕ ( e , S r m y m ) ϕ ( e , y m ) = ϕ ( e , J 1 ( ( 1 α m ) J x h + α m J T m x m ) ) = e 2 2 e , ( 1 α m ) J x m + α m J T m x m + ( 1 α m ) J x h + α m J T m x m 2 e 2 2 ( 1 α m ) e , J x m 2 α m e , J T m x m + ( 1 α m ) x m 2 + α m T m x m 2 = ( 1 α m ) ϕ ( e , x m ) + α m ϕ ( e , T m x m ) ( 1 α m ) ϕ ( e , x m ) + α m μ m ϕ ( e , x m ) + α m ν m ϕ ( e , x m ) + α m ( μ m 1 ) ϕ ( e , x m ) + α m ν m ϕ ( e , x m ) + ( μ m 1 ) W m + ν m ,
(3.1)
which proves that e C m + 1 . This implies that F ( T ) S ( f , A ) C n . Notice that x n = Π C n x 1 . We find from Lemma 2.8 that x n z , J x 1 J x n 0 for any z C n . Since F ( T ) S ( f , A ) C n , we therefore find that
x n w , J x 1 J x n 0 , w F ( T ) S ( f ) .
(3.2)
It follows from Lemma 2.9 that
ϕ ( x n , x 1 ) ϕ ( Π F ( T ) S ( f , A ) x 1 , x 1 ) ϕ ( Π F ( T ) S ( f , A ) x 1 , x n ) ϕ ( Π F ( T ) S ( f , A ) x 1 , x 1 ) .
This implies that the sequence { ϕ ( x n , x 1 ) } is bounded. This in turn implies that the sequence { x n } is bounded. Since E is a uniform space, we find that E is reflexive. We may assume, without loss of generality, that x n x ˆ . Next, we prove that x ˆ F ( T ) S ( f , A ) . Since C n is closed and convex, we find that x ˆ C n . This implies from x n = Π C n x 1 that ϕ ( x n , x 1 ) ϕ ( x ˆ , x 1 ) . On the other hand, we see from the weakly lower semicontinuity of the norm that
ϕ ( x ˆ , x 1 ) = x ˆ 2 2 x ˆ , J x 1 + x 1 2 lim inf n ( x n 2 2 x n , J x 1 + x 1 2 ) = lim inf n ϕ ( x n , x 1 ) lim sup n ϕ ( x n , x 1 ) ϕ ( x ˆ , x 1 ) ,
which implies that lim n ϕ ( x n , x 1 ) = ϕ ( x ˆ , x 1 ) . Hence, we have lim n x n = x ˆ . Since E enjoys the Kadec-Klee property, we find that x n x ˆ as n . In the light of x n = Π C n x 1 and x n + 1 = Π C n + 1 x 1 C n + 1 C n , we find that ϕ ( x n , x 1 ) ϕ ( x n + 1 , x 1 ) . This shows that { ϕ ( x n , x 1 ) } is nondecreasing. We obtain that lim n ϕ ( x n , x 1 ) exists. It follows that
ϕ ( x n + 1 , x n ) = ϕ ( x n + 1 , Π C n x 1 ) ϕ ( x n + 1 , x 1 ) ϕ ( Π C n x 1 , x 1 ) = ϕ ( x n + 1 , x 1 ) ϕ ( x n , x 1 ) .
This implies that lim n ϕ ( x n + 1 , x n ) = 0 . In view of x n + 1 = Π C n + 1 x 1 C n + 1 , we find that
ϕ ( x n + 1 , u n ) ϕ ( x n + 1 , x n ) + ( μ n 1 ) W n + ν n .
It follows that
lim n ϕ ( x n + 1 , u n ) = 0 .
In view of (2.2), we see that lim n ( x n + 1 u n ) = 0 . This implies that lim n u n = x ˆ . That is,
lim n J u n = lim n u n = J x ¯ .
(3.3)
This implies that { J u n } is bounded. Since both E and E are uniform, we find that both E and E are reflexive. We may assume, without loss of generality, that J u n u E . In view of the reflexivity of E, we see that J ( E ) = E . This shows that there exists an element u E such that J u = u . It follows that
ϕ ( x n + 1 , u n ) = x n + 1 2 2 x n + 1 , J u n + u n 2 = x n + 1 2 2 x n + 1 , J u n + J u n 2 .
It follows that
0 x ˆ 2 2 x ˆ , u + u 2 = x ˆ 2 2 x ˆ , J u + J u 2 = x ˆ 2 2 x ˆ , J u + u 2 = ϕ ( x ˆ , u ) .
That is, x ˆ = u , which in turn implies that u = J x ˆ . It follows that J u n J x ˆ E . Since E is uniformly smooth, we know that E is uniformly convex. Therefore, E enjoys the Kadec-Klee property, we obtain that lim n J u n = J x ˆ . Since J 1 : E E is demicontinuous and E enjoys the Kadec-Klee property, we obtain that u n x ˆ as n . Note that
x n u n x n x ˆ + x ˆ u n .
It follows that
lim n x n u n = lim n J x n J u n = 0 .
(3.4)
Since E is uniformly smooth, we know that E is uniformly convex. In the light of Lemma 2.11, we find that
ϕ ( e , u n ) = ϕ ( e , S r n y n ) ϕ ( e , y n ) = ϕ ( e , J 1 ( ( 1 α n ) J x n + α n J T n x n ) ) = e 2 2 e , ( 1 α n ) J x n + α n J T n x n + ( 1 α n ) J x n + α n J T n x n 2 e 2 2 ( 1 α n ) e , J x n 2 α n e , J T n x n + ( 1 α n ) x n 2 + α n T n x n 2 α n ( 1 α n ) g ( J x n J T n x n ) = ( 1 α n ) ϕ ( e , x n ) + α n ϕ ( e , T n x n ) α n ( 1 α n ) g ( J x n J T n x n ) ( 1 α n ) ϕ ( e , x n ) + α n μ n ϕ ( e , x n ) + α n ν n α n ( 1 α n ) g ( J x n J T n x n ) ϕ ( e , x n ) + α n ( μ n 1 ) ϕ ( e , x n ) + α n ν n α n ( 1 α n ) g ( J x n J T n x n ) ϕ ( e , x n ) + ( μ n 1 ) W n + ν n α n ( 1 α n ) g ( J x n J T n x n ) .
It follows that
α n ( 1 α n ) g ( J x n J T n x n ) ϕ ( e , x n ) ϕ ( e , u n ) + ( μ n 1 ) W n + ν n .
(3.5)
Notice that
ϕ ( e , x n ) ϕ ( e , u n ) = x n 2 u n 2 2 e , J x n J u n x n u n ( x n + u n ) + 2 e J x n J u n .
We find from (3.4) that
lim n ( ϕ ( e , x n ) ϕ ( e , u n ) ) = 0 .
In view of the restriction on the sequences, we find from (3.5) that lim n g ( J x n J T n x n ) = 0 . Notice that
J T n x n J x ˆ J T n x n J x n + J x n J x ˆ .
It follows that
lim n J T n x n J x ˆ = 0 .
The demicontinuity of J 1 : E E implies that T i n x n x ˆ . Note that
| T n x n x ˆ | = | J T n x n J x ˆ | J T n x n J x ˆ .
This implies that lim n T n x n = x ˆ . Since E has the Kadec-Klee property, we obtain that lim n T n x n x ˆ = 0 . Notice that
T n + 1 x n x ˆ T n + 1 x n T n x n + T n x n x ˆ .
It follows from the uniformly asymptotic regularity of T that
lim n T n + 1 x n x ˆ = 0 .
That is, T T n x n x ˆ . From the closedness of T, we find x ˆ = T x ˆ . This proves x ˆ F ( T ) . Next, we show that x ˆ S ( f , A ) . It follows from Lemma 2.9 and (3.1) that
ϕ ( u n , y n ) ϕ ( e , y n ) ϕ ( e , u n ) ϕ ( e , x n ) + ( μ n 1 ) W n + ν n ϕ ( e , u n ) .
This yields that lim n ϕ ( u n , y n ) = 0 . This implies from (2.2) that lim n ( u n y n ) = 0 . It follows that
lim n y n = x ˆ .
We, therefore, find that
lim n J y n = lim n y n = x ˆ = J x ˆ .
This shows that { J y n } is bounded. Since E is reflexive, we may assume that J y n y E . In view of J E = E , we see that there exists y E such that J y = y . It follows that
ϕ ( u n , y n ) = u n 2 2 u n , J y n + y n 2 = u n 2 2 u n , J y n + J y n 2 .
Taking lim inf n on both sides of the equality above yields that
0 x ˆ 2 2 x ˆ , y + y 2 = x ˆ 2 2 x ˆ , J y + J y 2 = x ˆ 2 2 x ˆ , J y + y 2 = ϕ ( x ˆ , y ) .
That is, x ˆ = y , which in turn implies that y = J x ˆ . It follows that J y n J x ˆ E . Since E enjoys the Kadec-Klee property, we obtain that J y n J x ˆ 0 as n . Note that J 1 : E E is demicontinuous. It follows that y n x ˆ . Since E enjoys the Kadec-Klee property, we obtain that y n x ˆ as n . Note that
u n y n u n x ˆ + x ˆ y n .
This implies that lim n u n y n = 0 . Since J is uniformly norm-to-norm continuous on any bounded sets, we have lim n J u n J y n = 0 . In view of the restriction lim inf n r n > 0 , we see that
lim n J u n J y n r n = 0 .
Since u n = K r n y n , we find that
F ( u n , q ) + 1 r n q u n , J u n J y n 0 , q C ,
where
F ( u n , q ) = f ( u n , q ) + A u n , q u n .
It follows from (A2) that
q u n J u n J y n r n 1 r n q u n , J u n J y n F ( q , u n ) , q C .
In view of (A4), we find that
F ( q , x ¯ ) 0 , q C .
For 0 < t < 1 and q C , define q t = t q + ( 1 t ) x ˆ . It follows that q t C , which yields that F ( q t , x ˆ ) 0 . It follows from (A1) and (A4) that
0 = F ( q t , q t ) t F ( q t , q ) + ( 1 t ) F ( q t , x ˆ ) t F ( q t , q ) .
That is,
F ( q t , q ) = f ( q t , q ) + A q t , q u n 0 .

Letting t 0 , we obtain from (A3) that F ( x ˆ , q ) 0 , q C . This implies that x ˆ S ( f , A ) . This completes the proof x ˆ F ( T ) S ( f , A ) .

Finally, what we need to prove is x ˆ = Π F ( T ) S ( f , A ) x 1 .

Letting n in (3.2), we obtain that
x ˆ w , J x 1 J x ˆ 0 , w F ( T ) S ( f , A ) .

From Lemma 2.8, we immediately find that x ˆ = Π F ( T ) S ( f , A ) x 1 . This completes the whole proof. □

Remark 3.2 Since the class of generalized asymptotically quasi-ϕ-nonexpansive mappings is a generalization of the class of asymptotically quasi-ϕ-nonexpansive mappings, Theorem 3.1 includes Kim’s [36] results as a special case.

Remark 3.3 Notice that every uniformly smooth and uniformly convex space is a uniformly smooth and strictly convex Banach space which also enjoys the Kadec-Klee property, and every uniformly convex Banach space enjoys the Kadec-Klee property. We find that Theorem 3.1 is still valid in the framework of every uniformly smooth and uniformly convex space.

Next, we consider the solution of problem (2.4).

If the mapping T is closed quasi-ϕ-nonexpansive, which is more general than relatively nonexpansive mappings, we have the following.

Corollary 3.4 Let E be a uniformly smooth and strictly convex Banach space which also enjoys the Kadec-Klee property and C be a nonempty closed convex subset of E. Let T : C C be a quasi-ϕ-nonexpansive mapping and f be a bifunction from C × C to satisfying (A1)-(A4). Assume that T is closed and F ( T ) S ( f ) is nonempty. Let { x n } be a sequence generated in the following manner:
{ x 0 E chosen arbitrarily , C 1 = C , x 1 = Π C 1 x 0 , y n = J 1 ( ( 1 α n ) J x n + α n J T x n ) , u n C such that f ( u n , q ) + 1 r n q u n , J u n J y n 0 , q C , C n + 1 = { k C n : ϕ ( k , u n ) ϕ ( k , x n ) } , x n + 1 = Π C n + 1 x 0 ,

where { α n } is a real number sequence in ( 0 , 1 ) such that lim inf n α n ( 1 α n ) > 0 and { r n } is a real number sequence such that lim inf n r n > 0 . Then the sequence { x n } converges strongly to Π F ( T ) S ( f ) x 1 , where Π F ( T ) S ( f ) is the generalized projection from E onto F ( T ) S ( f ) .

In the framework of Hilbert spaces, we find from Theorem 3.1 the following.

Theorem 3.5 Let E be a Hilbert space and C be a nonempty closed convex subset of E. Let T : C C be a generalized asymptotically quasi-nonexpansive mapping. Let f be a bifunction from C × C to satisfying (A1)-(A4), and let A : C E be an α-inverse-strongly monotone mapping. Assume that T is closed and asymptotically regular on C, and F ( T ) S ( f , A ) is nonempty and bounded. Let { x n } be a sequence generated in the following manner:
{ x 0 E chosen arbitrarily , C 1 = C , x 1 = P C 1 x 0 , y n = ( 1 α n ) x n + α n T n x n , u n C such that f ( u n , q ) + A u n , q u n + 1 r n q u n , u n y n 0 , q C , C n + 1 = { k C n : k u n 2 k x n 2 + ( μ n 1 ) W n + ν n } , x n + 1 = P C n + 1 x 0 ,

where W n = sup { p x n 2 : p F ( T ) S ( f , A ) } , { α n } is a real number sequence in ( 0 , 1 ) such that lim inf n α n ( 1 α n ) > 0 and { r n } is a real number sequence such that lim inf n r n > 0 . Then the sequence { x n } converges strongly to P F ( T ) S ( f , A ) x 1 , where P F ( T ) S ( f , A ) is the metric projection from E onto F ( T ) S ( f , A ) .

Proof In the framework of Hilbert spaces, we see that ϕ ( x , y ) = x y 2 and the mapping J is reduced to the identity mapping. The desired conclusion can be immediately drawn from Theorem 3.1. □

For problem (2.4), we have the following result.

Corollary 3.6 Let E be Hilbert space and C be a nonempty closed convex subset of E. Let T : C C be a generalized asymptotically quasi-nonexpansive mapping. Let f be a bifunction from C × C to satisfying (A1)-(A4). Assume that T is closed and asymptotically regular on C, and F ( T ) S ( f ) is nonempty and bounded. Let { x n } be a sequence generated in the following manner:
{ x 0 E chosen arbitrarily , C 1 = C , x 1 = P C 1 x 0 , y n = ( 1 α n ) x n + α n T n x n , u n C such that f ( u n , q ) + 1 r n q u n , u n y n 0 , q C , C n + 1 = { k C n : k u n 2 k x n 2 + ( μ n 1 ) W n + ν n } , x n + 1 = P C n + 1 x 0 ,

where W n = sup { p x n 2 : p F ( T ) S ( f ) } , { α n } is a real number sequence in ( 0 , 1 ) such that lim inf n α n ( 1 α n ) > 0 and { r n } is a real number sequence such that lim inf n r n > 0 . Then the sequence { x n } converges strongly to P F ( T ) S ( f ) x 1 , where P F ( T ) S ( f ) is the metric projection from E onto F ( T ) S ( f ) .

Declarations

Acknowledgements

The authors are grateful to the editor and the anonymous reviewers’ suggestions which improved the contents of the article.

Authors’ Affiliations

(1)
School of Mathematics and Sciences, Shijiazhuang University of Economics

References

  1. Cho SY, Kang SM: Zero point theorems for m -accretive operators in a Banach space. Fixed Point Theory 2012, 13: 49–58.MathSciNetGoogle Scholar
  2. Mahato NK, Nahak C: Equilibrium problem under various types of convexities in Banach space. J. Math. Comput. Sci. 2011, 1: 77–88.MathSciNetGoogle Scholar
  3. Iiduka H: Fixed point optimization algorithm and its application to network bandwidth allocation. J. Comput. Appl. Math. 2012, 236: 1733–1742. 10.1016/j.cam.2011.10.004MathSciNetView ArticleGoogle Scholar
  4. Shen J, Pang LP: An approximate bundle method for solving variational inequalities. Commun. Optim. Theory 2012, 1: 1–18.Google Scholar
  5. Qin X, Chang SS, Kang SM: Iterative methods for generalized equilibrium problems and fixed point problems with applications. Nonlinear Anal. 2010, 11: 2963–2972. 10.1016/j.nonrwa.2009.10.017View ArticleGoogle Scholar
  6. Censor Y, Cohen N, Kutscher T, Shamir J: Summed squared distance error reduction by simultaneous multiprojections and applications. Appl. Math. Comput. 2002, 126: 157–179. 10.1016/S0096-3003(00)00144-2MathSciNetView ArticleGoogle Scholar
  7. Abdel-Salam HS, Al-Khaled K: Variational iteration method for solving optimization problems. J. Math. Comput. Sci. 2012, 2: 1475–1497.MathSciNetGoogle Scholar
  8. Noor MA, Noor KI, Waseem M: Decomposition method for solving system of linear equations. Eng. Math. Lett. 2012, 2: 34–41.Google Scholar
  9. Kang SM, Cho SY, Liu Z: Convergence of iterative sequences for generalized equilibrium problems involving inverse-strongly monotone mappings. J. Inequal. Appl. 2010., 2010: Article ID 827082Google Scholar
  10. Bauschke HH, Matouskova E, Reich S: Projection and proximal point methods: convergence results and counterexamples. Nonlinear Anal. 2004, 56: 715–738. 10.1016/j.na.2003.10.010MathSciNetView ArticleGoogle Scholar
  11. Qin X, Cho SY, Zhou H: Common fixed points of a pair of non-expansive mappings with applications to convex feasibility problems. Glasg. Math. J. 2010, 52: 241–252. 10.1017/S0017089509990309MathSciNetView ArticleGoogle Scholar
  12. Zegeye H, Shahzad N: Strong convergence theorem for a common point of solution of variational inequality and fixed point problem. Adv. Fixed Point Theory 2012, 2: 374–397.Google Scholar
  13. Cho SY, Kang SM: Approximation of fixed points of pseudocontraction semigroups based on a viscosity iterative process. Appl. Math. Lett. 2011, 24: 224–228. 10.1016/j.aml.2010.09.008MathSciNetView ArticleGoogle Scholar
  14. Qin X, Cho SY, Kang SM: An extragradient-type method for generalized equilibrium problems involving strictly pseudocontractive mappings. J. Glob. Optim. 2011, 49: 679–693. 10.1007/s10898-010-9556-2MathSciNetView ArticleGoogle Scholar
  15. Mann WR: Mean value methods in iteration. Proc. Am. Math. Soc. 1953, 4: 506–510. 10.1090/S0002-9939-1953-0054846-3View ArticleGoogle Scholar
  16. Güler O: On the convergence of the proximal point algorithm for convex minimization. SIAM J. Control Optim. 1991, 29: 403–409. 10.1137/0329022MathSciNetView ArticleGoogle Scholar
  17. Haugazeau, Y: Sur les inéquations variationnelles et la minimisation de fonctionnelles convexes. PhD Thesis, Université de Paris (1968)Google Scholar
  18. Cioranescu I: Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems. Kluwer, Dordrecht; 1990.View ArticleMATHGoogle Scholar
  19. Alber YI: Metric and generalized projection operators in Banach spaces: properties and applications. In Theory and Applications of Nonlinear Operators of Accretive and Monotone Type. Edited by: Kartsatos AG. Dekker, New York; 1996.Google Scholar
  20. Zhou H, Gao G, Tan B: Convergence theorems of a modified hybrid algorithm for a family of quasi- ϕ -asymptotically nonexpansive mappings. J. Appl. Math. Comput. 2010, 32: 453–464. 10.1007/s12190-009-0263-4MathSciNetView ArticleGoogle Scholar
  21. Qin X, Cho SY, Kang SM: On hybrid projection methods for asymptotically quasi- ϕ -nonexpansive mappings. Appl. Math. Comput. 2010, 215: 3874–3883. 10.1016/j.amc.2009.11.031MathSciNetView ArticleGoogle Scholar
  22. Chang SS, Chan CK, Lee HWJ: Modified block iterative algorithm for quasi- ϕ -asymptotically nonexpansive mappings and equilibrium problem in Banach spaces. Appl. Math. Comput. 2011, 217: 7520–7530. 10.1016/j.amc.2011.02.060MathSciNetView ArticleGoogle Scholar
  23. Qin X, Agarwal RP: Shrinking projection methods for a pair of asymptotically quasi- ϕ -nonexpansive mappings. Numer. Funct. Anal. Optim. 2010, 31: 1072–1089. 10.1080/01630563.2010.501643MathSciNetView ArticleGoogle Scholar
  24. Agarwal RP, Cho YJ, Qin X: Generalized projection algorithms for nonlinear operators. Numer. Funct. Anal. Optim. 2007, 28: 1197–1215. 10.1080/01630560701766627MathSciNetView ArticleGoogle Scholar
  25. Qin X, Agarwal RP, Cho SY, Kang SM: Convergence of algorithms for fixed points of generalized asymptotically quasi- ϕ -nonexpansive mappings with applications. Fixed Point Theory Appl. 2012., 2012: Article ID 58Google Scholar
  26. Agarwal RP, Qin X, Kang SM: An implicit iterative algorithm with errors for two families of generalized asymptotically nonexpansive mappings. Fixed Point Theory Appl. 2011., 2011: Article ID 58Google Scholar
  27. Cho SY, Li W, Kang SM: Convergence analysis of an iterative algorithm for monotone operators. J. Inequal. Appl. 2013., 2013: Article ID 199Google Scholar
  28. Chang SS, Lee HWJ, Chan CK: A new hybrid method for solving a generalized equilibrium problem, solving a variational inequality problem and obtaining common fixed points in Banach spaces, with applications. Nonlinear Anal. 2010, 73: 2260–2270. 10.1016/j.na.2010.06.006MathSciNetView ArticleGoogle Scholar
  29. Reich S, Sabach S: Two strong convergence theorems for Bregman strongly nonexpansive operators in reflexive Banach spaces. Nonlinear Anal. 2010, 73: 122–135. 10.1016/j.na.2010.03.005MathSciNetView ArticleGoogle Scholar
  30. Kassay G, Reich S, Sabach S: Iterative methods for solving systems of variational inequalities in reflexive Banach spaces. SIAM J. Optim. 2011, 21: 1319–1344. 10.1137/110820002MathSciNetView ArticleGoogle Scholar
  31. Martin-Marquez V, Reich S, Sabach S: Bregman strongly nonexpansive operators in reflexive Banach spaces. J. Math. Anal. Appl. 2013, 400: 597–614. 10.1016/j.jmaa.2012.11.059MathSciNetView ArticleGoogle Scholar
  32. Qin X, Cho SY, Kang SM: Strong convergence of shrinking projection methods for quasi- ϕ -nonexpansive mappings and equilibrium problems. J. Comput. Appl. Math. 2010, 234: 750–760. 10.1016/j.cam.2010.01.015MathSciNetView ArticleGoogle Scholar
  33. Hao Y:On generalized quasi- p h i -nonexpansive mappings and their projection algorithms. Fixed Point Theory Appl. 2013., 2013: Article ID 204Google Scholar
  34. Takahashi W, Zembayashi K: Strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces. Nonlinear Anal. 2009, 70: 45–57. 10.1016/j.na.2007.11.031MathSciNetView ArticleGoogle Scholar
  35. Qin X, Cho YJ, Kang SM: Convergence theorems of common elements for equilibrium problems and fixed point problems in Banach spaces. J. Comput. Appl. Math. 2009, 225: 20–30. 10.1016/j.cam.2008.06.011MathSciNetView ArticleGoogle Scholar
  36. Kim JK: Strong convergence theorems by hybrid projection methods for equilibrium problems and fixed point problems of the asymptotically quasi- ϕ -nonexpansive mappings. Fixed Point Theory Appl. 2011., 2011: Article ID 10Google Scholar
  37. Zhang M: Iterative algorithms for common elements in fixed point sets and zero point sets with applications. Fixed Point Theory Appl. 2012., 2012: Article ID 21Google Scholar
  38. Zǎlinescu C: On uniformly convex functions. J. Math. Anal. Appl. 1983, 95: 344–374. 10.1016/0022-247X(83)90112-9MathSciNetView ArticleGoogle Scholar

Copyright

© Song and Chen; licensee Springer. 2013

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.