Open Access

Some results on asymptotically quasi-ϕ-nonexpansive mappings in the intermediate sense and equilibrium problems

Journal of Inequalities and Applications20142014:202

https://doi.org/10.1186/1029-242X-2014-202

Received: 29 January 2014

Accepted: 2 May 2014

Published: 22 May 2014

Abstract

In this paper, we investigate a common fixed point problem of a finite family of asymptotically quasi-ϕ-nonexpansive mappings in the intermediate sense and an equilibrium problem. 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, 90C33.

Keywords

asymptotically quasi-ϕ-nonexpansive mapping asymptotically quasi-ϕ-nonexpansive mapping in the intermediate sense generalized projection equilibrium problem fixed point

1 Introduction-preliminaries

Let E be a real Banach space. 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 .

Recall that E has 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 of the Kadec-Klee property, the readers can refer to [1] and the references therein. It is well known that if E is a uniformly convex Banach space, then E enjoys the Kadec-Klee property.

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. 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.

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 [2] recently introduced a generalized projection operator Π C in a Banach space E which is an analog 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 ) .
The 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, [1, 2]. In Hilbert spaces, Π C = P C . It is obvious from the definition of the function ϕ that
( x y ) 2 ϕ ( x , y ) ( y + x ) 2 , x , y E ,
(1.1)
and
ϕ ( x , y ) = ϕ ( x , z ) + ϕ ( z , y ) + 2 x z , J z J y , x , y , z E .
(1.2)

Remark 1.1 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 [1, 2] and the references therein.

Let C be a nonempty subset of E and let 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 [3] 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 1.2 The class of relatively asymptotically nonexpansive mappings were first considered in [4]; see also, [5] and the references therein.

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 1.3 The class of quasi-ϕ-nonexpansive mappings was considered in [6]. The class of asymptotically quasi-ϕ-nonexpansive mappings which was investigated in [7] and [8] includes the class of quasi-ϕ-nonexpansive mappings as a special case.

Remark 1.4 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 do not require the restriction F ( T ) = F ˜ ( T ) .

Remark 1.5 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 .
(1.3)
Putting
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 (1.3) is reduced to the following:
ϕ ( p , T n x ) ϕ ( p , x ) + ξ n , p F ( T ) , x C .
(1.4)

Remark 1.6 The class of asymptotically quasi-ϕ-nonexpansive mappings in the intermediate sense was first considered by Qin and Wang [9].

Remark 1.7 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 [10], in the framework of Banach spaces.

Let f be a bifunction from C × C to , where denotes the set of real numbers. Recall the following equilibrium problem. Find p C such that
f ( p , y ) 0 , y C .
(1.5)
We use E P ( f ) to denote the solution set of the equilibrium problem (1.5). That is,
E P ( f ) = { p C : f ( p , y ) 0 , y C } .
We remark here that the equilibrium problem was first introduced by Fan [11]. Given a mapping Q : C E , let
f ( x , y ) = Q x , y x , x , y C .
Then p E P ( f ) if and only if p is a solution of the following variational inequality. Find p such that
Q p , y p 0 , y C .
(1.6)

To study the equilibrium problems (1.5), we may assume that F satisfies the following conditions:

(A1) F ( x , x ) = 0 for all x C ;

(A2) F is monotone, i.e., F ( x , y ) + F ( y , x ) 0 for all x , y C ;

(A3) for each x , y , z C ,
lim sup t 0 F ( t z + ( 1 t ) x , y ) F ( x , y ) ;

(A4) for each x C , y F ( x , y ) is convex and weakly lower semi-continuous.

Numerous problems in physics, optimization, and economics reduce to find a solution of (1.5). Recently, many authors have investigated common solutions of fixed point and equilibrium problems in Banach spaces; see, for example, [1233] and the references therein.

In this paper, we consider a projection algorithm for treating the equilibrium problem and fixed point problems of asymptotically quasi-ϕ-nonexpansive mappings in the intermediate sense.

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

Lemma 1.8 [2]

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

Lemma 1.9 [2]

Let C be a nonempty closed convex subset of a smooth Banach space E and let x E . Then x 0 = Π C x if and only if
x 0 y , J x J x 0 0 , y C .
Lemma 1.10 Let C be a closed convex subset of a smooth, strictly convex and reflexive Banach space E. Let f be a bifunction from C × C to satisfying (A1)-(A4). Let r > 0 and x E . Then
  1. (a)
    [34]There exists z C such that
    f ( z , y ) + 1 r y z , J z J x 0 , y C .
     
  2. (b)
    [6, 24]Define a mapping T r : E C by
    S r x = { z C : f ( z , y ) + 1 r y z , J z J x , y C } .
     
Then the following conclusions hold:
  1. (1)

    S r is single-valued;

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

    F ( S r ) = E P ( f ) ;

     
  4. (4)

    S r is quasi-ϕ-nonexpansive;

     
  5. (5)

    ϕ ( q , S r x ) + ϕ ( S r x , x ) ϕ ( q , x ) , q F ( S r ) ;

     
  6. (6)

    E P ( f ) is closed and convex.

     

Lemma 1.11 [35]

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 ] .

2 Main results

Theorem 2.1 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 T i : C C 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 ) E F ( 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 = Π C 1 x 0 , 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 ) + 1 r n y u n , J u n J y n 0 , y C , C n + 1 = { z C n : ϕ ( z , u n ) ϕ ( z , x n ) + ξ n } , x n + 1 = Π C n + 1 x 0 ,

where ξ n = max { 0 , sup p 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 ) E F ( f ) x 1 , where Π i = 1 N F ( T i ) E F ( f ) is the generalized projection from E onto i = 1 N F ( T i ) E F ( f ) .

Proof First, we show that i = 1 N F ( T i ) E F ( f ) is closed and convex. From [9], we find that i = 1 N F ( T i ) is closed and convex, which combines with Lemma 1.10 shows that i = 1 N F ( T i ) E F ( f ) is closed and convex. Next, we show that C n is closed and convex. It is obvious that C 1 = C is closed and convex. Suppose that C h is closed and convex for some positive integer h. For z C h , we see that ϕ ( z , u h ) ϕ ( z , x h ) + ξ h is equivalent to
2 z , J x h J u h x k 2 u k 2 + ξ h .
It is to see that C h + 1 is closed and convex. This proves that C n is closed and convex. This in turn shows that Π C n + 1 x 1 is well defined. Putting u n = S r n y n , we from Lemma 1.10 see that S r n is quasi-ϕ-nonexpansive. Now, we are in a position to prove that i = 1 N F ( T i ) E F ( f ) C n . Indeed, i = 1 N F ( T i ) E F ( f ) C 1 = C is obvious. Assume that i = 1 N F ( T i ) E F ( f ) C h for some positive integer h. Then, for w i = 1 N F ( T i ) E F ( f ) C h , we have
ϕ ( w , u h ) = ϕ ( w , S r h y h ) ϕ ( w , y h ) = ϕ ( w , J 1 ( α h , 0 J x h + i = 1 N α h , i J T i h x h ) ) = w 2 2 w , α h , 0 J x h + i = 1 N α h , i J T i h x h + α h , 0 J x h + i = 1 N α h , i J T i h x h 2 w 2 2 α h , 0 w , J x h 2 i = 1 N α h , i w , J T i h x h + α h , 0 x h 2 + i = 1 N α h , i T i h x h 2 = α h , 0 ϕ ( w , x h ) + i = 1 N α h , i ϕ ( w , T i h x h ) α h , 0 ϕ ( w , x h ) + i = 1 N α h , i ϕ ( w , x h ) + i = 1 N α h , i ξ h = ϕ ( w , x h ) + i = 1 N α h , i ξ h ϕ ( w , x h ) + i = 1 N ξ h ,
(2.1)

which shows that w C h + 1 . This implies that i = 1 N F ( T i ) E F ( f ) C n .

Next, we prove that the sequence { x n } is bounded. Notice that x n = Π C n x 1 . We find from Lemma 1.9 that x n z , J x 1 J x n 0 , for any z C n . Since i = 1 N F ( T i ) E F ( f ) C n , we find that
x n w , J x 1 J x n 0 , w i = 1 N F ( T i ) E F ( f ) .
(2.2)
It follows from Lemma 1.8 that
ϕ ( x n , x 1 ) ϕ ( Π i = 1 N F ( T i ) E F ( f ) x 1 , x 1 ) ϕ ( Π i = 1 N F ( T i ) E F ( f ) x 1 , x n ) ϕ ( Π i = 1 N F ( T i ) E F ( f ) x 1 , x 1 ) .
This implies that the sequence { ϕ ( x n , x 1 ) } is bounded. It follows from (1.1) that the sequence { x n } is also bounded. Since the space is reflexive, we may assume, without loss of generality, that x n x ¯ . Next, we prove that x ¯ i = 1 N F ( T i ) E F ( f ) . Since C n is closed and convex, we find that x ¯ C n . This implies from x n = Π C n x 1 that ϕ ( x n , x 1 ) ϕ ( x ¯ , x 1 ) . On the other hand, we see from the weakly lower semicontinuity of that
ϕ ( x ¯ , x 1 ) = x ¯ 2 2 x ¯ , 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 ¯ , x 1 ) ,
which implies that lim n ϕ ( x n , x 1 ) = ϕ ( x ¯ , x 1 ) . 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 . Since 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 find from its boundedness 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 .
(2.3)
In the light 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 ) + ξ n .
It follows from (2.3) that
lim n ϕ ( x n + 1 , u n ) = 0 .
(2.4)
In view of (1.1), 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 ¯ .
(2.5)
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 both sides of the equality aboven yields
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 (2.5) that lim n J u n = J x ¯ . Since J 1 : E E is demicontinuous and E enjoys the Kadec-Klee property, we obtain 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 .
(2.6)
Since J is uniformly norm-to-norm continuous on any bounded sets, we have
lim n J x n J u n = 0 .
(2.7)
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 .
(2.8)
Since E is uniformly smooth, we know that E is uniformly convex. In view of Lemma 1.11, we find that
ϕ ( w , u n ) = ϕ ( w , S 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 ) + 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 (2.8) 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
lim n J x n J T i n x n = 0 , 1 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 .
(2.9)
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 from (2.9) that lim n T i n x n = x ¯ . Since E has the Kadec-Klee property, we obtain 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 uniformly 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 ) .

Next, we show that x ¯ E F ( f ) . In view of Lemma 1.8, we find that
ϕ ( u n , y n ) ϕ ( w , y n ) ϕ ( w , u n ) ϕ ( w , x n ) + μ n ϕ ( w , u n ) .
It follows from (2.8) that lim n ϕ ( u n , y n ) = 0 . This implies that lim n ( u n y n ) = 0 . It follows from (2.6) 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 v E . In view of J ( E ) = E , we see that there exists v E such that J v = v . 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 the both sides of equality above yields that
0 x ¯ 2 2 x ¯ , v + v 2 = x ¯ 2 2 x ¯ , J v + J v 2 = x ¯ 2 2 x ¯ , J v + v 2 = ϕ ( x ¯ , v ) .
That is, x ¯ = v , which in turn implies that v = J x ¯ . It follows that J y n J x ¯ E . Since E enjoys the Kadec-Klee property, we obtain J y n J x ¯ 0 as n . Note that J 1 : E E is demicontinuous. It follows that y n x ¯ . Since E enjoys the Kadec-Klee property, we obtain 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 . From the assumption r n k , we see that
lim n J u n J y n r n = 0 .
(2.10)
Since u n = S 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 .
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 .
In view of (A4), we find from (2.10) 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 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 . This implies that x ¯ E P ( f ) .

Finally, we turn our attention to proving that x ¯ = Π i = 1 N F ( T i ) E F ( f ) x 1 .

Letting n in (2.2), we obtain
x ¯ w , J x 1 J x ¯ 0 , w i = 1 N F ( T i ) E F ( f ) .

In view of Lemma 1.9, we find that x ¯ = Π i = 1 N F ( T i ) E F ( f ) x 1 . This completes the proof. □

From the definition of quasi-ϕ-nonexpansive mappings, we see that every quasi-ϕ-nonexpansive mapping is asymptotically quasi-ϕ-nonexpansive in the intermediate sense. We also know that every uniformly smooth and uniformly convex space is a uniformly smooth and strictly convex Banach space which also enjoys the Kadec-Klee property (note that every uniformly convex Banach space enjoys the Kadec-Klee property).

Remark 2.2 Theorem 2.1 can be viewed an extension of the corresponding results in Qin et al. [6], Kim [12], Qin et al. [22], Takahashi and Zembayashi [24], respectively. The space  L p , where p > 1 , satisfies the restriction in Theorem 2.1.

3 Applications

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). Let T : C C an asymptotically quasi-ϕ-nonexpansive mapping in the intermediate sense. Assume that T is closed asymptotically regular on C and F ( T ) E F ( 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 = Π C 1 x 0 , y n = J 1 ( α n , 0 J x n + α n , 1 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 , C n + 1 = { z C n : ϕ ( z , u n ) ϕ ( z , x n ) + ξ n } , x n + 1 = Π C n + 1 x 0 ,

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

Proof Putting N = 1 , we draw from Theorem 2.1 the desired conclusion immediately. □

Remark 3.2 If the mapping T in Theorem 3.1 is quasi-ϕ-nonexpansive, then the restrictions that T is closed asymptotically regular on C and F ( T ) E F ( f ) is bounded will not be required anymore.

If T i = I , where I is the identity for every 1 i N , then we find from Theorem 2.1 the following.

Theorem 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). Assume that E F ( 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 , u n C  such that  f ( u n , y ) + 1 r n y u n , J u n J x n 0 , y C , C n + 1 = { z C n : ϕ ( z , u n ) ϕ ( z , x n ) } , x n + 1 = Π C n + 1 x 0 ,

where { r n } is a real number sequence in [ k , ) , where k is some positive real number. Then the sequence { x n } converges strongly to Π E F ( f ) x 1 , where Π E F ( f ) is the generalized projection from E onto E F ( f ) .

Declarations

Acknowledgements

The authors thank the reviewers for useful suggestions which improved the contents of this paper.

Authors’ Affiliations

(1)
School of Mathematics and Information Science, North China University of Water Resources and Electric Power
(2)
Basic Experimental & Teaching Center, Henan University

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© Huang and Ma; licensee Springer. 2014

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