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A strong convergence theorem for solutions of equilibrium problems and asymptotically quasi-ϕ-nonexpansive mappings in the intermediate sense

Fixed Point Theory and Applications20132013:305

https://doi.org/10.1186/1687-1812-2013-305

Received: 9 September 2013

Accepted: 25 October 2013

Published: 21 November 2013

Abstract

In this paper, common solutions to an equilibrium problem and a nonlinear operator equation involving a finite family of asymptotically quasi-ϕ-nonexpansive mappings in the intermediate sense are discussed. Strong convergence theorems of common solutions are established in a uniformly smooth and strictly convex Banach space which also enjoys the Kadec-Klee property.

MSC:47H09, 47J25.

Keywords

asymptotically quasi-ϕ-nonexpansive mappingasymptotically quasi-ϕ-nonexpansive mapping in the intermediate sensegeneralized projectionequilibrium problemfixed point

1 Introduction

Equilibrium problems have been revealed as a very powerful and important tool in the study of nonlinear phenomena. They have turned out to be very useful in studying optimization problems, differential equations, and minimax theorems and in certain applications to mechanics and economic theory; see [127] and the references therein. Important practical situations motivate the study of a system of equilibrium problems. For instance, the flow of fluid through a fissured porous medium and certain models of plasticity led to such problems; see, for instance, [28]. Because of their importance, they have been extensively analyzed. The aim of this paper is to present an iterative method for solving solutions of an equilibrium problem, which is known as the Ky Fan inequality, and a nonlinear operator equation involving a finite family of asymptotically quasi-ϕ-nonexpansive mappings in the intermediate sense.

The organization of this paper is as follows. In Section 2, we provide some necessary preliminaries. In Section 3, an iterative algorithm is presented. A strong convergence theorem is established in a reflexive Banach space. Some results in Hilbert spaces are also discussed.

2 Preliminaries

Let E be a real Banach space. 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. Recall that E is said to be strictly convex if x + y 2 < 1 for all x , y E with x = y = 1 and x y . It is said to be uniformly convex if 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 . Let U E = { x E : x = 1 } be the unit sphere of E. Then the Banach space E is said to be smooth if lim t 0 x + t y x t exists for each x , y U E . It is said to be uniformly smooth if 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 if E is uniformly smooth if and only if E is uniformly convex.

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 . 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 [29] 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 follows from the properties of the functional ϕ ( x , y ) and strict monotonicity of the mapping J. In Hilbert spaces, Π C = P C . It is obvious from the definition of function ϕ that
( x y ) 2 ϕ ( x , y ) ( y + x ) 2 , x , y E
(2.1)
and
ϕ ( x , y ) = ϕ ( x , z ) + ϕ ( z , y ) + 2 x z , J z J y , x , y , z 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 [29] and the reference therein.

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 equilibrium problem. Find p C such that
f ( p , q ) + A p , q p 0 , q C .
(2.3)
We use EP ( f , A ) to denote the solution set of inequality (2.3). That is,
EP ( 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. Find p C such that
f ( p , q ) 0 , q C .
(2.4)
We use EP ( f ) to denote the solution set of inequality (2.4). That is,
EP ( 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 VI ( C , A ) to denote the solution set of inequality (2.5). That is,
VI ( 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 semi-continuous.

     
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 the 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 such that lim n x n T x n = 0 . The set of asymptotic fixed points of T will be denoted by F ˜ ( T ) .

A mapping T is said to be relatively nonexpansive iff
F ˜ ( T ) = F ( T ) , ϕ ( p , T x ) ϕ ( p , x ) , x C , p F ( T ) .
A mapping 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 .

Remark 2.2 The class of relatively asymptotically nonexpansive mappings was first considered in Agarwal et al. [30].

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 2.3 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. Quasi-ϕ-nonexpansive mappings and asymptotically quasi-ϕ-nonexpansive mappings do not require the restriction F ( T ) = F ˜ ( T ) ; for more details, see [3133] the reference therein.

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 asymptotically quasi-ϕ-nonexpansive in the intermediate sense iff F ( T ) and the following inequality holds:
lim sup n sup p F ( T ) , x C ( ϕ ( p , T n x ) ϕ ( p , x ) ) 0 .
(2.6)
Put
ξ n = max { 0 , sup p F ( T ) , x C ( ϕ ( p , T n x ) ϕ ( p , x ) ) } .
It follows that ξ n 0 as n . Then (2.6) is reduced to the following:
ϕ ( p , T n x ) ϕ ( p , x ) + ξ n , p F ( T ) , x C .
(2.7)

Remark 2.5 The class of asymptotically quasi-ϕ-nonexpansive mappings in the intermediate sense was first considered by Qin and Wang [34]; see also [35].

Remark 2.6 The class of asymptotically quasi-ϕ-nonexpansive mappings in the intermediate sense is a generalization of the class of asymptotically quasi-nonexpansive mappings in the intermediate sense, which was considered by Kirk [36] in the framework of Banach spaces.

In order to state our main results, we also need the following lemmas.

Lemma 2.7 [29]

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 2.8 [29]

Let E be a reflexive, strictly convex, and smooth Banach space, let 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 2.9 [37]

Let E be a smooth, strictly convex, and reflexive Banach space, and let 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 let 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 ) = EP ( 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)

    EP ( f , A ) is closed and convex.

     

Lemma 2.10 [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 let C be a nonempty closed and convex subset of E. Let f be a bifunction from C × C to satisfying (A1)-(A4), and let N be some positive integer. Let A : C E be a κ i -inverse-strongly monotone mapping. Let T i : C C be an asymptotically quasi-ϕ-nonexpansive mapping in the intermediate sense for every 1 i N . Assume that T i is closed asymptotically regular on C and i = 1 N F ( T i ) EP ( f , A ) is nonempty and bounded. Let { x n } be a sequence generated in the following manner:
{ x 0 C chosen arbitrarily , y n = J 1 ( α n , 0 J x n + i = 1 N α n , i J T i n x n ) , u n C  such that  f ( u n , y ) + A u n + y u n + 1 r n y u n , J u n J y n 0 , y C , H n = { z C : ϕ ( z , u n ) ϕ ( z , x n ) + N ξ n } , W n = { z C : x n z , J x 0 J x n 0 } , x n + 1 = Π H n W n x 0 ,

where ξ n = max { 0 , sup p i = 1 N F ( T i ) , x C ( ϕ ( p , T i n x ) ϕ ( p , x ) ) } , { α n , i } is a real number sequence in ( 0 , 1 ) for every 1 i N , { r n } is a real number sequence in [ k , ) , where k is some positive real number. Assume that i = 0 N α n , i = 1 and lim inf n α n , 0 α n , i > 0 for every 1 i N . Then the sequence { x n } converges strongly to Π i = 1 N F ( T i ) EP ( f , A ) x 0 , where Π i = 1 N F ( T i ) EP ( f , A ) is the generalized projection from E onto i = 1 N F ( T i ) EP ( f , A ) .

Proof Since F i is closed and convex for every 1 i N , we obtain from Lemma 2.9 that the common element set i = 1 N F ( T i ) EP ( f , A ) is closed and convex. Next, we show that both H n and W n are closed and convex. From the definition of H n and W n , it is obvious that H n is closed and W n is closed and convex. We show that H n is convex since ϕ ( z , u n ) ϕ ( z , x n ) + N ξ n is equivalent to
2 z , J x n J u n x n 2 u n 2 + N ξ n .
It follows that H n is convex. This in turn shows that Π H n W n x 0 is well defined. Putting u n = k r n y n , from Lemma 2.9 we see that K r n is quasi-ϕ-nonexpansive. Now, we are in a position to prove that i = 1 N F ( T i ) EP ( f , A ) H n W n . Let w i = 1 N F ( T i ) EP ( f , A ) ,
ϕ ( w , u n ) = ϕ ( w , K r n y n ) ϕ ( w , y n ) = ϕ ( w , J 1 ( α n , 0 J x n + i = 1 N α n , i J T i n x n ) ) = w 2 2 w , α h , 0 J x n + i = 1 N α n , i J T i n x n + α n , 0 J x n + i = 1 N α n , i J T i n x n 2 w 2 2 α n , 0 w , J x n 2 i = 1 N α n , i w , J T i n x n + α n , 0 x n 2 + i = 1 N α n , i T i n x n 2 = α n , 0 ϕ ( w , x n ) + i = 1 N α n , i ϕ ( w , T i n x n ) α n , 0 ϕ ( w , x n ) + i = 1 N α n , i ϕ ( w , x n ) + i = 1 N α n , i ξ n = ϕ ( w , x n ) + i = 1 N α n , i ξ n ϕ ( w , x n ) + N ξ n .
(3.1)
We have w H n . This implies that i = 1 N F ( T i ) EP ( f , A ) H n . On the other hand, we see that i = 1 N F ( T i ) EP ( f , A ) H 0 W 0 . Suppose that i = 1 N F ( T i ) EP ( f , A ) H m W m for some m. There exists an element x m + 1 H m W m such that x m + 1 = Π H m W m x 0 . In view of Lemma 2.7, we find that
x m + 1 w , J x 0 J x m + 1 0 , w H m W m .
Since i = 1 N F ( T i ) EP ( f , A ) H m W m , we arrive at
x m + 1 w , J x 0 J x m + 1 0
(3.2)

for every w i = 1 N F ( T i ) EP ( f , A ) . We therefore find that i = 1 N F ( T i ) EP ( f , A ) W m + 1 . It follows that i = 1 N F ( T i ) EP ( f , A ) H m + 1 W m + 1 . This shows that the sequence { x n } is well defined.

Next, we prove that the sequence { x n } is bounded. It follows from the definition of W n and Lemma 2.7 that x n = Π W n x 0 . In view of Lemma 2.8, we find that
ϕ ( x n , x 0 ) = ϕ ( Π W n x 0 , x 0 ) ϕ ( w , x 0 ) ϕ ( w , x n ) ϕ ( w , x 0 )
for each w i = 1 N F ( T i ) EP ( f , A ) W n . This implies that { ϕ ( x n , x 0 ) } is bounded. It follows from (2.1) that { x n } is also bounded. Since x n + 1 = Π H n W n x 0 W n , we find from Lemma 2.7 that ϕ ( x n , x 0 ) ϕ ( x n + 1 , x 0 ) . Therefore, we obtain that { ϕ ( x n , x 0 ) } is nondecreasing. So there exists the limit of ϕ ( x n , x 0 ) . It follows from Lemma 2.8 that
ϕ ( x n + 1 , x n ) = ϕ ( x n + 1 , Π W n x 0 ) ϕ ( x n + 1 , x 0 ) ϕ ( Π W n x 0 , x 0 ) = ϕ ( x n + 1 , x 0 ) ϕ ( x n , x 0 ) .
This shows that lim n ϕ ( x n + 1 , x n ) = 0 . Since x n + 1 = Π H n W n x 0 H n , we find that lim n ϕ ( x n + 1 , u n ) = 0 . Since the space is reflexive, we may assume, without loss of generality, that x n x ¯ . Since W n is closed and convex, we find that x ¯ W n . This implies from x n = Π W n x 0 that ϕ ( x n , x 0 ) ϕ ( x ¯ , x 0 ) . On the other hand, we see from the weakly lower semicontinuity of that
ϕ ( x ¯ , x 0 ) = x ¯ 2 2 x ¯ , J x 0 + x 0 2 lim inf n ( x n 2 2 x n , J x 0 + x 0 2 ) = lim inf n ϕ ( x n , x 0 ) lim sup n ϕ ( x n , x 0 ) ϕ ( x ¯ , x 0 ) ,
which implies that lim n ϕ ( x n , x 0 ) = ϕ ( x ¯ , x 0 ) . Hence, we have lim n x n = x ¯ . In view of the Kadec-Klee property of E, we find that x n x ¯ as n . In view of (2.1), we see that lim n ( x n + 1 u n ) = 0 . This implies that lim n u n = x ¯ . That is,
lim n J u n = lim n u n = J x ¯ .
(3.3)
This implies that { J u n } is bounded. Note 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 .
Taking lim inf n on the both sides of the equality above yields that
0 x ¯ 2 2 x ¯ , u + u 2 = x ¯ 2 2 x ¯ , J u + J u 2 = x ¯ 2 2 x ¯ , J u + u 2 = ϕ ( x ¯ , u ) .
That is, x ¯ = u , which in turn implies that u = J x ¯ . It follows that J u n J x ¯ E . Since E enjoys the Kadec-Klee property, we obtain from (3.3) that lim n J u n = J x ¯ . Since J 1 : E E is demi-continuous and E enjoys the Kadec-Klee property, we obtain that u n x ¯ , as n . Note that x n u n x n x ¯ + x ¯ u n . It follows that
lim n x n u n = 0 .
(3.4)
Since J is uniformly norm-to-norm continuous on any bounded sets, we have
lim n J x n J u n = 0 .
(3.5)
On the other hand, we have
ϕ ( w , x n ) ϕ ( w , u n ) = x n 2 u n 2 2 w , J x n J u n x n u n ( x n + u n ) + 2 w J x n J u n .
We, therefore, find that
lim n ( ϕ ( w , x n ) ϕ ( w , u n ) ) = 0 .
(3.6)
Since E is uniformly smooth, we know that E is uniformly convex. In view of Lemma 2.10, we find that
ϕ ( w , u n ) = ϕ ( w , K r n y n ) ϕ ( w , y n ) = ϕ ( w , J 1 ( α n , 0 J x n + i = 1 N α n , i J T i n x n ) ) = w 2 2 w , α n , 0 J x n + i = 1 N α n , i J T i n x n + α n , 0 J x n + i = 1 N α n , i J T i n x n 2 w 2 2 α n , 0 w , J x n 2 i = 1 N α n , i w , J T i n x n + α n , 0 x n 2 + i = 1 N α n , i T i n x n 2 α n , 0 α n , 1 g ( J x n J T 1 n x n ) = α n , 0 ϕ ( w , x n ) + i = 1 N α n , i ϕ ( w , T i n x n ) α n , 0 α n , 1 g ( J x n J T 1 n x n ) α n , 0 ϕ ( w , x n ) + i = 1 N α n , i ϕ ( w , x n ) + i = 1 N α n , i ξ h α n , 0 α n , 1 g ( J x n J T 1 n x n ) = ϕ ( w , x n ) + i = 1 N α n , i ξ n α n , 0 α n , 1 g ( J x n J T 1 n x n ) ϕ ( w , x n ) + N ξ n α n , 0 α n , 1 g ( J x n J T 1 n x n ) .
It follows that α n , 0 α n , 1 g ( J x n J T 1 n x n ) ϕ ( w , x n ) ϕ ( w , u n ) + ξ n . In view of the restriction on the sequences, we find from (3.6) that lim n g ( J x n J T 1 n x n ) = 0 . It follows that lim n J x n J T 1 n x n = 0 . In the same way, we obtain that lim n J x n J T i n x n = 0 , 2 i N . Notice that J T i n x n J x ¯ J T i n x n J x n + J x n J x ¯ . It follows that lim n J T i n x n J x ¯ = 0 . The demicontinuity of J 1 : E E implies that T i n x n x ¯ . Note that
| T i n x n x ¯ | = | J T i n x n J x ¯ | J T i n x n J x ¯ .
This implies that lim n T i n x n = x ¯ . Since E has the Kadec-Klee property, we obtain that lim n T i n x n x ¯ = 0 . On the other hand, we have
T i n + 1 x n x ¯ T i n + 1 x n T i n x n + T i n x n x ¯ .
It follows from the asymptotic regularity of T i that lim n T i n + 1 x n x ¯ = 0 . That is, T i T i n x n x ¯ . From the closedness of T i , we find x ¯ = T i x ¯ for every 1 i N . This proves x ¯ i = 1 N F ( T i ) . Now, we state that x ¯ EP ( f , A ) . In view of Lemma 2.9, we find that
ϕ ( u n , y n ) ϕ ( w , y n ) ϕ ( w , u n ) ϕ ( w , x n ) + N ξ n ϕ ( w , u n ) .
It follows from (3.6) that lim n ϕ ( u n , y n ) = 0 . This implies that lim n ( u n y n ) = 0 . It follows from (3.4) that lim n y n = x ¯ . It follows that
lim n J y n = lim n y n = x ¯ = J x ¯ .
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 + J y n 2 . Taking lim inf n on the both sides of the equality above yields that
0 x ¯ 2 2 x ¯ , y + y 2 = x ¯ 2 2 x ¯ , J y + J y 2 = x ¯ 2 2 x ¯ , J y + y 2 = ϕ ( x ¯ , y ) .
That is, x ¯ = y , which in turn implies that y = J x ¯ . It follows that J y n J x ¯ E . Since E enjoys the Kadec-Klee property, we obtain that J y n J x ¯ 0 as n . Note that J 1 : E E is demi-continuous. It follows that y n x ¯ . Since E enjoys the Kadec-Klee property, we obtain that y n x ¯ as n . Note that u n y n u n x ¯ + x ¯ 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 assumption r n k , we see that
lim n J u n J y n r n = 0 .
(3.7)
Since u n = K r n y n , we find that
F ( u n , y ) + 1 r n y u n , J u n J y n 0 , y C ,
where F ( u n , y ) = f ( u n , y ) + A u n + y u n for every y C . It follows from (A2) that
y u n J u n J y n r n 1 r n y u n , J u n J y n F ( y , u n ) , y C .
It follows from (3.7) that
F ( y , x ¯ ) 0 , y C .
For 0 < t < 1 and y C , define y t = t y + ( 1 t ) x ¯ . It follows that y t C , which yields that F ( y t , x ¯ ) 0 . It follows from (A1) and (A4) that
0 = F ( y t , y t ) t F ( y t , y ) + ( 1 t ) F ( y t , x ¯ ) t F ( y t , y ) .
That is,
F ( y t , y ) 0 .
Letting t 0 , we obtain from (A3) that F ( x ¯ , y ) 0 , y C . That is, f ( u n , y ) + A u n + y u n 0 for every y C . This implies that x ¯ EP ( f , A ) . This completes the proof that x ¯ i = 1 N F ( T i ) EP ( f , A ) . Since x n + 1 = Π H n W n x 0 . In view of Lemma 2.7, we find that
x n + 1 w , J x 0 J x n + 1 0 , w H n W n .
Since i = 1 N F ( T i ) EP ( f , A ) H n W n , we arrive at
x n + 1 w , J x 0 J x n + 1 0 .
Letting n in the above inequality, we obtain that
x ¯ w , J x 0 J x ¯ 0 , w i = 1 N F ( T i ) EP ( f , A ) .

It follows from Lemma 2.7 that x ¯ = Π i = 1 N F ( T i ) EP ( f , A ) x 0 . This completes the proof. □

Remark 3.2 Theorem 3.1 includes the corresponding results in the literature as special cases. Since every uniformly convex Banach space enjoys the Kadec-Klee property, the framework of the space can be applicable to L p , p 1 .

Next, we state a result on Ky Fan inequality (2.4) and a single asymptotically quasi-ϕ-nonexpansive mapping in the intermediate sense.

Corollary 3.3 Let E be a uniformly smooth and strictly convex Banach space which also enjoys the Kadec-Klee property, and let C be a nonempty closed and convex subset of E. Let f be a bifunction from C × C to satisfying (A1)-(A4). Let T : C C be an asymptotically quasi-ϕ-nonexpansive mapping in the intermediate sense. Assume that T is closed asymptotically regular on C and F ( T ) EP ( f ) is nonempty and bounded. Let { x n } be a sequence generated in the following manner:
{ x 0 C chosen arbitrarily , y n = J 1 ( α n J x n + ( 1 α n ) J T n x n ) , u n C  such that  f ( u n , y ) + 1 r n y u n , J u n J y n 0 , y C , H n = { z C : ϕ ( z , u n ) ϕ ( z , x n ) + N ξ n } , W n = { z C : x n z , J x 0 J x n 0 } , x n + 1 = Π H n W n x 0 ,

where ξ n = max { 0 , sup p F ( T ) , x C ( ϕ ( p , T n x ) ϕ ( p , x ) ) } , { α n } is a real number sequence in ( 0 , 1 ) , { r n } is a real number sequence in [ k , ) , where k is some positive real number. Assume that lim inf n α n ( 1 α n ) > 0 . Then the sequence { x n } converges strongly to Π F ( T ) EP ( f ) x 0 , where Π F ( T ) EP ( f ) is the generalized projection from E onto F ( T ) EP ( f ) .

If the mapping T is quasi-ϕ-nonexpansive, we find from Corollary 3.3 the following.

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

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

Finally, we give a result in the framework of Hilbert spaces based on Theorem 3.1.

Corollary 3.5 Let E be a Hilbert space, and let C be a nonempty closed and convex subset of E. Let f be a bifunction from C × C to satisfying (A1)-(A4), and let N be some positive integer. Let A : C E be a κ i -inverse-strongly monotone mapping. Let T i : C C be an asymptotically quasi-nonexpansive mapping in the intermediate sense for every 1 i N . Assume that T i is closed asymptotically regular on C and i = 1 N F ( T i ) EP ( f , A ) is nonempty and bounded. Let { x n } be a sequence generated in the following manner:
{ x 0 C chosen arbitrarily , y n = α n , 0 x n + i = 1 N α n , i T i n x n , u n C  such that  f ( u n , y ) + A u n + y u n + 1 r n y u n , u n y n 0 , y C , H n = { z C : z u n 2 z x n 2 + N ξ n } , W n = { z C : x n z , x 0 x n 0 } , x n + 1 = Proj H n W n x 0 ,

where ξ n = max { 0 , sup p i = 1 N F ( T i ) , x C ( p T i n x 2 p x 2 ) } , { α n , i } is a real number sequence in ( 0 , 1 ) for every 1 i N , { r n } is a real number sequence in [ k , ) , where k is some positive real number. Assume that i = 0 N α n , i = 1 and lim inf n α n , 0 α n , i > 0 for every 1 i N . Then the sequence { x n } converges strongly to Proj i = 1 N F ( T i ) EP ( f , A ) x 0 , where Proj i = 1 N F ( T i ) EP ( f , A ) is the metric projection from E onto i = 1 N F ( T i ) EP ( f , A ) .

Proof Since ϕ ( x , y ) = x y 2 and J = I in the framework of Hilbert spaces, we draw the desired conclusion immediately. □

Declarations

Acknowledgements

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

Authors’ Affiliations

(1)
Department of Mathematics, Hangzhou Normal University
(2)
School of Mathematics and Information Science, Shangqiu Normal University

References

  1. Park S:A review of the KKM theory on ϕ A -space or GFC-spaces. Adv. Fixed Point Theory 2013, 3: 355–382.Google Scholar
  2. Cho SY, Li W, Kang SM: Convergence analysis of an iterative algorithm for monotone operators. J. Inequal. Appl. 2013., 2013: Article ID 199Google Scholar
  3. 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
  4. Wang ZM, Lou W: A new iterative algorithm of common solutions to quasi-variational inclusion and fixed point problems. J. Math. Comput. Sci. 2013, 3: 57–72.Google Scholar
  5. Wang G, Sun S: Hybrid projection algorithms for fixed point and equilibrium problems in a Banach space. Adv. Fixed Point Theory 2013, 3: 578–594.Google Scholar
  6. 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
  7. 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
  8. Qin X, Cho SY, Kang SM: Convergence of an iterative algorithm for systems of variational inequalities and nonexpansive mappings with applications. J. Comput. Appl. Math. 2009, 233: 231–240. 10.1016/j.cam.2009.07.018MathSciNetView ArticleGoogle Scholar
  9. Lv S: Strong convergence of a general iterative algorithm in Hilbert spaces. J. Inequal. Appl. 2013., 2013: Article ID 19Google Scholar
  10. Yuan Q: Some results on asymptotically quasi- ϕ -nonexpansive mappings in the intermediate sense. J. Fixed Point Theory 2012., 2012: Article ID 1Google Scholar
  11. Cho SY, Qin X, Kang SM: Iterative processes for common fixed points of two different families of mappings with applications. J. Glob. Optim. 2013. 10.1007/s10898-012-0017-yGoogle Scholar
  12. He RH: Coincidence theorem and existence theorems of solutions for a system of Ky Fan type minimax inequalities in FC -spaces. Adv. Fixed Point Theory 2012, 2: 47–57.Google Scholar
  13. Tanaka Y: A constructive version of Ky Fan’s coincidence theorem. J. Math. Comput. Sci. 2012, 2: 926–936.MathSciNetGoogle 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. Al-Bayati AY, Al-Kawaz RZ: A new hybrid WC-FR conjugate gradient-algorithm with modified secant condition for unconstrained optimization. J. Math. Comput. Sci. 2012, 2: 937–966.MathSciNetGoogle Scholar
  16. Qin X, Cho SY, Kang SM: Iterative algorithms for variational inequality and equilibrium problems with applications. J. Glob. Optim. 2010, 48: 423–445. 10.1007/s10898-009-9498-8MathSciNetView ArticleGoogle Scholar
  17. Chen JH: Iterations for equilibrium and fixed point problems. J. Nonlinear Funct. Anal. 2013., 2013: Article ID 4Google Scholar
  18. Yang S: A proximal point algorithm for zeros of monotone operators. Math. Finance Lett. 2013., 2013: Article ID 7Google Scholar
  19. Song J, Chen M: On generalized asymptotically quasi- ϕ -nonexpansive mappings and a Ky Fan inequality. Fixed Point Theory Appl. 2013., 2013: Article ID 237Google Scholar
  20. Cho SY, Kang SM: Approximation of common solutions of variational inequalities via strict pseudocontractions. Acta Math. Sci. 2012, 32: 1607–1618.View ArticleGoogle Scholar
  21. Takahashi W, Yao JC: Nonlinear operators of monotone type and convergence theorems with equilibrium problems in Banach spaces. Taiwan. J. Math. 2011, 15: 787–818.MathSciNetGoogle Scholar
  22. Takahashi W, Wong NC, Yao JC: Iterative common solutions for monotone inclusion problems, fixed point problems and equilibrium problems. Fixed Point Theory Appl. 2012., 2012: Article ID 181Google Scholar
  23. Noor MA, Noor KI, Waseem M: Decomposition method for solving system of linear equations. Eng. Math. Lett. 2013, 2: 34–41.Google Scholar
  24. Qing Y, Kim JK: Weak convergence of algorithms for asymptotically strict pseudocontractions in the intermediate sense and equilibrium problems. Fixed Point Theory Appl. 2012., 2012: Article ID 132Google Scholar
  25. Kim JK: A new iterative algorithm of pseudomonotone mappings for equilibrium problems in Hilbert spaces. J. Inequal. Appl. 2013., 2013: Article ID 128Google Scholar
  26. 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
  27. Wu C: Mann iteration for zero theorems of accretive operators. J. Fixed Point Theory 2013., 2013: Article ID 3Google Scholar
  28. Showalter RE Math. Surveys Monogr. 49. In Monotone Operators in Banach Space and Nonlinear Partial Differential Equations. Am. Math. Soc., Providence; 1997.Google Scholar
  29. 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
  30. 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
  31. 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
  32. 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
  33. 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
  34. Qin X, Wang L: On asymptotically quasi- ϕ -nonexpansive mappings in the intermediate sense. Abstr. Appl. Anal. 2012., 2012: Article ID 636217Google Scholar
  35. Hecai Y, Aichao L: Projection algorithms for treating asymptotically quasi- ϕ -nonexpansive mappings in the intermediate sense. J. Inequal. Appl. 2013., 2013: Article ID 265Google Scholar
  36. Kirk WA: Fixed point theorems for non-Lipschitzian mappings of asymptotically nonexpansive type. Isr. J. Math. 1974, 17: 339–346. 10.1007/BF02757136MathSciNetView ArticleGoogle Scholar
  37. Chang SS, Chan CK, Lee HWJ: 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
  38. Zǎlinescu C: On uniformly convex functions. J. Math. Anal. Appl. 1983, 95: 344–374. 10.1016/0022-247X(83)90112-9MathSciNetView ArticleGoogle Scholar

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© Qing and Lv; licensee Springer. 2013

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