Open Access

Strong convergence theorem for total quasi-ϕ-asymptotically nonexpansive mappings in a Banach space

Fixed Point Theory and Applications20132013:297

https://doi.org/10.1186/1687-1812-2013-297

Received: 11 July 2013

Accepted: 9 October 2013

Published: 11 November 2013

Abstract

In this paper, we prove strong convergence theorems to a point which is a fixed point of multi-valued mappings, a zero of an α-inverse-strongly monotone operator and a solution of the equilibrium problem. Next, we obtain strong convergence theorems to a solution of the variational inequality problem, a fixed point of multi-valued mappings and a solution of the equilibrium problem. The results presented in this paper are improvement and generalization of the previously known results.

Keywords

total quasi-ϕ-asymptotically nonexpansive multi-valued mappingshybrid schemeequilibrium problemvariational inequality problemsinverse-strongly monotone operator

1 Introduction

Let E be a real Banach space with dual E , and let C be a nonempty closed convex subset of E. Let A : C E be an operator. A is called monotone if
A x A y , x y 0 , x , y C ;
α-inverse-strongly monotone if there exists a constant α > 0 such that
A x A y , x y α A x A y 2 , x , y C ;
L-Lipschitz continuous if there exists a constant L > 0 such that
A x A y L x y , x , y C .
If A is α-inverse strongly monotone, then it is 1 α -Lipschitz continuous, i.e.,
A x A y 1 α x y , x , y C .

A monotone operator A is said to be maximal if its graph G ( A ) = { ( x , x ) : x A x } is not properly contained in the graph of any other monotone operator.

Let A be a monotone operator. We consider the problem of finding x E such that
0 A x ,
(1.1)

a point x E is called a zero point of A. Denote by A 1 0 the set of all points x E such that 0 A x . This problem is very important in optimization theory and related fields.

Let A be a monotone operator. The classical variational inequality problem for an operator A is to find z ˆ C such that
A z ˆ , y z ˆ 0 , y C .
(1.2)

The set of solutions of (1.2) is denoted by VI ( A , C ) . This problem is connected with the convex minimization problem, the complementary problem, the problem of finding a point x E satisfying A x = 0 .

The value of x E at x E will be denoted by x , x or x ( x ) . For each p > 1 , the generalized duality mapping J p : E 2 E is defined by
J p ( x ) = { x E : x , x = x p , x = x p 1 }

for all x E . In particular, J = J 2 is called the normalized duality mapping. If E is a Hilbert space, then J = I , where I is the identity mapping.

Consider the functional defined by
ϕ ( y , x ) = y 2 2 y , J x + x 2 for  x , y E ,
(1.3)
where J is the normalized duality mapping. It is obvious from the definition of ϕ that
( y x ) 2 ϕ ( y , x ) ( y + x ) 2 , x , y E .
(1.4)
Alber [1] introduced 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 of the minimization problem
ϕ ( x ¯ , x ) = inf y C ϕ ( y , x ) ,
(1.5)

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.

Iiduka and Takahashi [2] introduced the following iterative scheme for finding a solution of the variational inequality problem for an inverse-strongly monotone operator A in a 2-uniformly convex and uniformly smooth Banach space E: x 1 = x C and
x n + 1 = Π C J 1 ( J x n λ n A x n ) , n 1 ,
(1.6)
where Π C is the generalized projection from E onto C, J is the duality mapping from E into E and { λ n } is a sequence of positive real numbers. They proved that the sequence { x n } generated by (1.6) converges weakly to some element of VI ( A , C ) . In connection, Iiduka and Takahashi [3] studied the following iterative scheme for finding a zero point of a monotone operator A in a 2-uniformly convex and uniformly smooth Banach space E:
{ x 1 = x E chosen arbitrarily , y n = J 1 ( J x n λ n A x n ) , X n = { z E : ϕ ( z , y n ) ϕ ( z , x n ) } , Y n + 1 = { z E : x n z , J x J x n 0 } , x n + 1 = Π X n Y n ( x ) ,
(1.7)

where Π X n Y n is the generalized projection from E onto X n Y n , J is the duality mapping from E into E and { λ n } is a sequence of positive real numbers. They proved that the sequence { x n } converges strongly to an element of A 1 0 . Moreover, under the additional suitable assumption they proved that the sequence { x n } converges strongly to some element of VI ( A , C ) . Some solution methods have been proposed to solve the variational inequality problem; see, for instance, [46].

A mapping T : C C is said to be ϕ-nonexpansive [7, 8] if
ϕ ( T x , T y ) ϕ ( x , y ) , x , y C .
T is said to be quasi-ϕ-nonexpansive [7, 8] if F ( T ) and
ϕ ( p , T x ) ϕ ( p , x ) , x C , p F ( T ) .
T is said to be total quasi-ϕ-asymptotically nonexpansive, if F ( T ) and there exist nonnegative real sequences ν n , μ n with ν n 0 , μ n 0 as n and a strictly increasing continuous function φ : R + R + with φ ( 0 ) = 0 such that
ϕ ( p , T n x ) ϕ ( p , x ) + ν n φ ( ϕ ( p , x ) ) + μ n , n 1 , x C , p F ( T ) .
Let 2 C be the family of all nonempty subsets of C, and let S : C 2 C be a multi-valued mapping. For a point q C , n 1 define an iterative sequence as follows:
S q : = { q 1 : q 1 S q } , S 2 q = S S q : = q 1 S q S q 1 , S 3 q = S S 2 q : = q 2 T 2 q S q 2 , S n q = S S n 1 q : = q n 1 S n 1 q S q n 1 .
A point p C is said to be an asymptotic fixed point of S if there exists a sequence { x n } in C such that { x n } converges weakly to p and
lim n d ( x n , S x n ) : = lim n inf x S x n x n x = 0 .

The asymptotic fixed point set of S is denoted by F ˆ ( S ) .

A multi-valued mapping S is said to be total quasi-ϕ-asymptotically nonexpansive if F ( S ) and there exist nonnegative real sequences ν n , μ n with ν n 0 , μ n 0 as n and a strictly increasing continuous function φ : R + R + with φ ( 0 ) = 0 such that for all x C , p F ( S ) ,
ϕ ( p , w n ) ϕ ( p , x ) + ν n φ ( ϕ ( p , x ) ) + μ n , n 1 , w n S n x .

S is said to be closed if for any sequence { x n } and { w n } in C with w n S x n if x n x and w n w , then w S x .

A multi-valued mapping S is said to be uniformly asymptotically regular on C if
lim n ( sup x C s n + 1 s n ) = 0 , s n S n x .

Every quasi-ϕ-asymptotically nonexpansive multi-valued mapping implies a quasi-ϕ-asymptotically nonexpansive mapping but the converse is not true.

In 2012, Chang et al. [9] introduced the concept of total quasi-ϕ-asymptotically nonexpansive multi-valued mapping and then proved some strong convergence theorem by using the hybrid shrinking projection method.

Let f : C × C R be a bifunction, the equilibrium problem is to find x C such that
f ( x , y ) 0 , y C .
(1.8)

The set of solutions of (1.8) is denoted by EP ( f ) . The equilibrium problem is very general in the sense that it includes, as special cases, optimization problems, variational inequality problems, min-max problems, saddle point problem, fixed point problem, Nash EP. In 2008, Takahashi and Zembayashi [10, 11] introduced iterative sequences for finding a common solution of an equilibrium problem and a fixed point problem. Some solution methods have been proposed to solve the equilibrium problem; see, for instance, [1221].

For a mapping A : C E , let f ( x , y ) = A x , y x for all x , y C . Then x EP ( f ) if and only if T x , y x 0 for all y C ; i.e., x is a solution of the variational inequality.

Motivated and inspired by the work mentioned above, in this paper, we introduce and prove strong convergence of a new hybrid projection algorithm for a fixed point of total quasi-ϕ-asymptotically nonexpansive multi-valued mappings, the solution of the equilibrium problem, a zero point of monotone operators. Moreover, we prove strong convergence to the solution of the variation inequality in a uniformly smooth and 2-uniformly convex Banach space.

2 Preliminaries

A Banach space E with the norm is called strictly convex if x + y 2 < 1 for all x , y E with x = y = 1 and x y . Let U = { x E : x = 1 } be the unit sphere of E. A Banach space E is called smooth if the limit lim t 0 x + t y x t exists for each x , y U . It is also called uniformly smooth if the limit exists uniformly for all x , y U . The modulus of convexity of E is the function δ : [ 0 , 2 ] [ 0 , 1 ] defined by
δ ( ε ) = inf { 1 x + y 2 : x , y E , x = y = 1 , x y ε } .

A Banach space E is uniformly convex if and only if δ ( ε ) > 0 for all ε ( 0 , 2 ] . Let p be a fixed real number with p 2 . A Banach space E is said to be p-uniformly convex if there exists a constant c > 0 such that δ ( ε ) c ε p for all ε [ 0 , 2 ] . Observe that every p-uniform convex is uniformly convex. Every uniformly convex Banach space E has the Kadec-Klee property, that is, for any sequence { x n } E , if x n x E and x n x , then x n x .

Let E be a real Banach space with dual E , E is uniformly smooth if and only if E is a uniformly convex Banach space. If E is a uniformly smooth Banach space, then E is a smooth and reflexive Banach space.

Remark 2.1

  • If E is uniformly smooth, then J is uniformly norm-to-norm continuous on each bounded subset of E.

  • If E is reflexive smooth and strictly convex, then the normalized duality mapping J is single-valued, one-to-one and onto.

  • If E is a reflexive strictly convex and smooth Banach space and J is the duality mapping from E into E , then J 1 is also single-valued, bijective and is also the duality mapping from E into E and thus J J 1 = I E and J 1 J = I E .

See [22] for more details.

Remark 2.2 If E is a reflexive, strictly convex and smooth Banach space, then ϕ ( x , y ) = 0 if and only if x = y . It is sufficient to show that if ϕ ( x , y ) = 0 , then x = y . From (1.3) we have x = y . This implies that x , J y = x 2 = J y 2 . From the definition of J, one has J x = J y . Therefore, we have x = y (see [22, 23] for more details).

Lemma 2.3 (Beauzamy [24] and Xu [25])

If E is a 2-uniformly convex Banach space, then, for all x , y E , we have
x y 2 c 2 J x J y ,

where J is the normalized duality mapping of E and 0 < c 1 .

The best constant 1 c in the lemma is called the p-uniformly convex constant of E.

Lemma 2.4 (Beauzamy [24] and Zalinescu [26])

If E is a p-uniformly convex Banach space, and let p be a given real number with p 2 , then, for all x , y E , J x J p ( x ) and J y J p ( y ) ,
x y , J x J y c p 2 p 2 p x y p ,

where J p is the generalized duality mapping of E and 1 c is the p-uniformly convex constant of E.

Lemma 2.5 (Kamimura and Takahashi [27])

Let E be a uniformly convex and smooth Banach space, and let { x n } , { y n } be two sequences of E. If ϕ ( x n , y n ) 0 and either { x n } or { y n } is bounded, then x n y n 0 .

Lemma 2.6 (Alber [1])

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 2.7 (Alber [1])

Let E be a reflexive strictly convex and smooth Banach space, 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 2.8 (Chang et al. [9])

Let C be a nonempty, closed and convex subset of a uniformly smooth and strictly convex Banach space E with the Kadec-Klee property. Let S : C 2 C be a closed and total quasi-ϕ-asymptotically nonexpansive multi-valued mapping with nonnegative real sequence ν n and μ n with ν n 0 , μ n 0 as n and a strictly increasing continuous function φ : R + R + with φ ( 0 ) = 0 . If μ 1 = 0 , then the fixed point set F ( S ) is a closed convex subset of C.

For solving the equilibrium problem for a bifunction f : C × C R , let us assume that f satisfies the following conditions:
  1. (A1)

    f ( x , x ) = 0 for all x C ;

     
  2. (A2)

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

     
  3. (A3)
    for each x , y , z C ,
    lim t 0 f ( t z + ( 1 t ) x , y ) f ( x , y ) ;
     
  4. (A4)

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

     

Lemma 2.9 (Blum and Oettli [28])

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), and let r > 0 and x E . Then there exists z C such that
f ( z , y ) + 1 r y z , J z J x 0 , y C .

Lemma 2.10 (Takahashi and Zembayashi [11])

Let C be a closed convex subset of a uniformly smooth, strictly convex and reflexive Banach space E, and let f be a bifunction from C × C to satisfying conditions (A1)-(A4). For all r > 0 and x E , define a mapping T r : E C as follows:
T r x = { z C : f ( z , y ) + 1 r y z , J z J x 0 , y C } .
Then the following hold:
  1. (1)

    T r is single-valued;

     
  2. (2)
    T r is a firmly nonexpansive-type mapping [29], that is, for all x , y E ,
    T r x T r y , J T r x J T r y T r x T r y , J x J y ;
     
  3. (3)

    F ( T r ) = EP ( f ) ;

     
  4. (4)

    EP ( f ) is closed and convex.

     

Lemma 2.11 (Takahashi and Zembayashi [11])

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), and let r > 0 . Then, for x E and q F ( T r ) ,
ϕ ( q , T r x ) + ϕ ( T r x , x ) ϕ ( q , x ) .
Let A be an inverse-strongly monotone mapping of C into E which is said to be hemicontinuous if for all x , y C , the mapping h of [ 0 , 1 ] into E , defined by h ( t ) = A ( t x + ( 1 t ) y ) , is continuous with respect to the weak topology of E . We define by N C ( v ) the normal cone for C at a point v C , that is,
N C ( v ) = { x E : v y , x 0 , y C } .
(2.1)

Theorem 2.12 (Rockafellar [30])

Let C be a nonempty, closed convex subset of a Banach space E, and let A be a monotone, hemicontinuous operator of C into E . Let B E × E be an operator defined as follows:
B v = { A v + N C ( v ) , v C ; , otherwise .
(2.2)

Then B is maximal monotone and B 1 0 = VI ( A , C ) .

Theorem 2.13 (Takahashi [31])

Let C be a nonempty subset of a Banach space E, and let A be a monotone, hemicontinuous operator of C into E with C = D ( A ) . Then
VI ( A , C ) = { u C : v u , A v 0 , v C } .
(2.3)

It is obvious that the set VI ( A , C ) is a closed and convex subset of C and the set A 1 0 = VI ( A , E ) is a closed and convex subset of E.

Theorem 2.14 (Takahashi [31])

Let C be a nonempty compact convex subset of a Banach space E, and let A be a monotone, hemicontinuous operator of C into E with C = D ( A ) . Then VI ( A , C ) is nonempty.

We make use of the following mapping V studied in Alber [1]:
V ( x , x ) = x 2 2 x , x + x 2 , x E , x E ,
(2.4)

that is, V ( x , x ) = ϕ ( x , J 1 ( x ) ) .

Lemma 2.15 (Alber [1])

Let E be a reflexive strictly convex smooth Banach space, and let V be as in (2.4). Then we have
V ( x , x ) + 2 J 1 ( x ) x , y V ( x , x + y ) , x E , x , y E .

Lemma 2.16 (Beauzamy [24] and Xu [25])

If E is a 2-uniformly convex Banach space, then, for all x , y E , we have
x y 2 c 2 J x J y ,

where J is the normalized duality mapping of E and 0 < c 1 .

Lemma 2.17 (Cho et al. [32])

Let E be a uniformly convex Banach space, and let B r ( 0 ) = { x E : x r } be a closed ball of E. Then there exists a continuous strictly increasing convex function g : [ 0 , ) [ 0 , ) with g ( 0 ) = 0 such that
λ x + μ y + γ z 2 λ x 2 + μ y 2 + γ z 2 λ μ g ( x y )

for all x , y , z B r ( 0 ) and λ , μ , γ [ 0 , 1 ] with λ + μ + γ = 1 .

Lemma 2.18 (Pascali and Sburlan [33])

Let E be a real smooth Banach space, and let A : E 2 E be a maximal monotone mapping. Then A 1 0 is a closed and convex subset of E and the graph G ( A ) of A is demiclosed in the following sense: if { x n } D ( A ) with x n x E and y n A x n with y n y E , then x D ( A ) and y A x .

3 Main results

Theorem 3.1 Let C be a nonempty closed and convex subset of a uniformly smooth and 2-uniformly convex Banach space E. Let f be a bifunction from C × C to satisfying conditions (A1)-(A4), and let A be an α-inverse-strongly monotone mapping of E into E . Let S : C 2 C be a closed and total quasi-ϕ-asymptotically nonexpansive multi-valued mapping with nonnegative real sequences ν n , μ n with ν n 0 , μ n 0 as n and a strictly increasing continuous function ψ : R + R + with ψ ( 0 ) = 0 . Assume that S is uniformly asymptotically regular on C with μ 1 = 0 and F : = F ( S ) EP ( f ) A 1 0 . For arbitrary x 1 C , C 1 = C , generate a sequence { x n } by
{ z n = J 1 ( J x n λ n A x n ) , u n = T r n z n , y n = J 1 ( α n J x n + β n J w n + γ n J u n ) , w n S n x n , C n + 1 = { v C n : ϕ ( v , y n ) ϕ ( v , z n ) ϕ ( v , x n ) + K n } , x n + 1 = Π C n + 1 x 1 , n N ,
(3.1)
where K n = ν n sup q F ψ ( ϕ ( q , x n ) ) + μ n . Assume that the control sequences { α n } , { β n } , { γ n } , { λ n } and { r n } satisfy the following conditions:
  1. 1.

    { α n } , { β n } and { γ n } are sequences in ( 0 , 1 ) such that α n + β n + γ n = 1 , lim inf n α n β n > 0 ,

     
  2. 2.

    { λ n } [ a , b ] for some a, b with 0 < a < b < c 2 α 2 and 1 c is the 2-uniformly convex constant of E,

     
  3. 3.

    { r n } [ d , ) for some d > 0 ,

     

then { x n } converges strongly to Π F x 1 .

Proof We will show that C n is closed and convex for all n N . Since C 1 = C is closed and convex. Suppose that C n is closed and convex for all n N . For any v C n , we know that ϕ ( v , y n ) ϕ ( v , x n ) + K n is equivalent to
2 v , J x n J y n x n 2 y n 2 + K n .

That is, C n + 1 is closed and convex, hence C n is closed and convex for all n N .

We show by induction that F C n for all n N . It is obvious that F C = C 1 . Suppose that F C n where n N . Let q F , we have
ϕ ( q , z n ) = ϕ ( q , J 1 ( J x n λ n A x n ) ) = V ( q , J x n λ n A x n ) V ( q , ( J x n λ n A x n ) + λ n A x n ) 2 J 1 ( J x n λ n A x n ) q , λ n A x n = V ( q , J x n ) 2 λ n J 1 ( J x n λ n A x n ) q , A x n = ϕ ( q , x n ) 2 λ n x n q , A x n + 2 J 1 ( J x n λ n A x n ) x n , λ n A x n .
(3.2)
Since A is an α-inverse-strongly monotone mapping, we get
2 λ n x n q , A x n = 2 λ n x n q , A x n A q 2 λ n x n q , A q 2 λ n x n q , A x n A q = 2 α λ n A x n A q 2 .
(3.3)
It follows from Lemma 2.17 that
2 J 1 ( J x n λ n A x n ) x n , λ n A x n = 2 J 1 ( J x n λ n A x n ) J 1 ( J x n ) , λ n A x n 2 J 1 ( J x n λ n A x n ) J 1 ( J x n ) λ n A x n 4 c 2 J J 1 ( J x n λ n A x n ) J J 1 ( J x n ) λ n A x n = 4 c 2 J x n λ n A x n J x n λ n A x n = 4 c 2 λ n A x n 2 = 4 c 2 λ n 2 A x n 2 4 c 2 λ n 2 A x n A q 2 .
(3.4)
Replacing (3.2) by (3.3) and (3.4), we get
ϕ ( q , z n ) ϕ ( q , x n ) 2 α λ n A x n A q 2 + 4 c 2 λ n 2 A x n A q 2 = ϕ ( q , x n ) + 2 λ n ( 2 c 2 λ n α ) A x n A q 2 ϕ ( q , x n ) .
(3.5)
From Lemma 2.11, we know that
ϕ ( q , u n ) = ϕ ( q , T r n z n ) ϕ ( q , z n ) ϕ ( q , x n ) .
(3.6)
Since S is a total quasi-ϕ-asymptotically nonexpansive multi-valued mapping and w n S n x n , it follows that
ϕ ( q , y n ) = ϕ ( q , J 1 ( α n J x n + β n J w n + γ n J u n ) ) = q 2 2 q , α n J x n + β n J w n + γ n J u n + α n J x n + β n J w n + γ n J u n 2 α n ϕ ( q , x n ) + β n ϕ ( q , w n ) + γ n ϕ ( q , u n ) α n ϕ ( q , x n ) + β n ϕ ( q , x n ) + β n ν n ψ ( ϕ ( q , x n ) ) + β n μ n + γ n ϕ ( q , u n ) α n ϕ ( q , x n ) + β n ϕ ( q , x n ) + ν n sup q F ψ ( ϕ ( q , x n ) ) + μ n + γ n ϕ ( q , u n ) = α n ϕ ( q , x n ) + β n ϕ ( q , x n ) + γ n ϕ ( q , u n ) + K n α n ϕ ( q , x n ) + β n ϕ ( q , x n ) + γ n ϕ ( q , T r n z n ) + K n α n ϕ ( q , x n ) + β n ϕ ( q , x n ) + γ n ϕ ( q , z n ) + K n α n ϕ ( q , x n ) + β n ϕ ( q , x n ) + γ n ϕ ( q , x n ) + K n ϕ ( q , x n ) + K n ,
(3.7)

where K n = ν n sup q F ψ ( ϕ ( q , x n ) ) + μ n .

This shows that q C n + 1 , which implies that F C n + 1 . Hence F C n for all n N and the sequence { x n } is well defined.

From the definition of C n + 1 with x n = Π C n x 1 and x n + 1 = Π C n + 1 x 1 C n + 1 C n , it follows that
ϕ ( x n , x 1 ) ϕ ( x n + 1 , x 1 ) , n 1 ,
(3.8)
that is, { ϕ ( x n , x 1 ) } is nondecreasing. By Lemma 2.7, we get
ϕ ( x n , x 1 ) = ϕ ( Π C n x 1 , x 1 ) ϕ ( q , x 1 ) ϕ ( q , x n ) ϕ ( q , x 1 ) , q F .
(3.9)

This implies that { ϕ ( x n , x 1 ) } is bounded and so lim n ϕ ( x n , x 1 ) exists. In particular, by (1.4), the sequence { ( x n x 1 ) 2 } is bounded. This implies { x n } is also bounded. So, we have { u n } , { z n } and { y n } are also bounded.

Since x m = Π C m x 1 C m C n for all m , n 1 with m > n , by Lemma 2.7, we have
ϕ ( x m , x n ) = ϕ ( x m , Π C n x 1 ) ϕ ( x m , x 1 ) ϕ ( Π C n x 1 , x 1 ) = ϕ ( x m , x 1 ) ϕ ( x n , x 1 ) ,
taking m , n , we have ϕ ( x m , x n ) 0 . This implies that { x n } is a Cauchy sequence. From Lemma 2.5, it follows that x n x m 0 and { x n } is a Cauchy sequence. By the completeness of E and the closedness of C, we can assume that there exists p C such that
lim n x n = p ,
(3.10)
we also get that
lim n K n = lim n ν n sup q F ψ ( ϕ ( q , x n ) ) + μ n = 0 .
(3.11)

Next, we show that p F : = F ( S ) A 1 0 EP ( f ) .

(a) We show that p F ( S ) . By the definition of Π C n x 1 , we have
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 ) .
Since lim n ϕ ( x n , x 1 ) exists, we get
lim n ϕ ( x n + 1 , x n ) = 0 .
(3.12)
It follows from Lemma 2.5 that
lim n x n + 1 x n = 0 .
(3.13)
From the definition of C n + 1 and x n + 1 = Π C n + 1 x 1 C n + 1 C n , we have ϕ ( x n + 1 , y n ) ϕ ( x n + 1 , x n ) + K n 0 as n . By Lemma 2.5, it follows that
lim n x n + 1 y n = 0 .
(3.14)
From lim n x n = p , we also have
lim n y n = p .
(3.15)
By using the triangle inequality, we get x n y n x n x n + 1 + x n + 1 y n 0 as n . Since J is uniformly norm-to-norm continuous, we obtain J x n J y n 0 as n . On the other hand, we note that
ϕ ( q , x n ) ϕ ( q , y n ) = x n 2 y n 2 2 q , J x n J y n x n y n ( x n + y n ) + 2 q J x n J y n .
In view of x n y n 0 and J x n J y n 0 as n , we obtain that
ϕ ( q , x n ) ϕ ( q , y n ) 0 as  n .
(3.16)
From Lemma 2.17, we have
ϕ ( q , y n ) = ϕ ( q , J 1 [ α n J x n + β n J w n + γ n J u n ] ) q 2 2 q , α n J x n + β n J w n + γ n J u n + α n J x n + β n J w n + γ n J u n 2 α n β n g ( J x n J w n ) = α n ϕ ( q , x n ) + β n ϕ ( q , w n ) + γ n ϕ ( q , u n ) α n β n g ( J x n J w n ) ϕ ( q , x n ) + K n α n β n g ( J x n J w n ) .
(3.17)
It follows from lim inf n α n β n > 0 , (3.16), (3.11) and the property of g that
lim n J x n J w n = 0 .
Since J 1 is uniformly norm-to-norm continuous, we obtain
lim n x n w n = 0 .
(3.18)
From (3.10) it follows that
lim n w n p = 0 .
(3.19)
For w n S n x n , generate a sequence { s n } by
s 2 S w 1 S 2 x 1 , s 3 S w 2 S 3 x 2 , s 4 S w 3 S 4 x 3 , s n + 1 S w n S n + 1 x n .
On the other hand, we have s n + 1 p s n + 1 w n + w n p . Since S is uniformly asymptotically regular, it follows that
lim n s n + 1 p = 0 ,
(3.20)
we have
lim n S n + 1 x n p = 0 ,
(3.21)

that is, S S n x n p as n . From the closedness of S, we have p F ( S ) .

(b) We show that p A 1 0 .

From the definition of C n + 1 and x n + 1 = Π C n + 1 x 1 C n + 1 C n , we have ϕ ( x n + 1 , z n ) ϕ ( x n + 1 , x n ) + K n 0 as n . By Lemma 2.5, it follows that lim n x n + 1 z n = 0 . By the triangle inequality, we get x n z n x n x n + 1 + x n + 1 z n 0 as n . From lim n z n x n = 0 and from (3.10), it follows that
lim n z n = p .
(3.22)
Since J is uniformly norm-to-norm continuous, we also have
lim n J z n J x n = 0 .
(3.23)
Hence, from the definition of the sequence { z n } , it follows that
A x n = J z n J x n λ n .
(3.24)
From (3.23) and the definition of the sequence { λ n } , we have
lim n A x n = 0 ,
(3.25)
that is,
lim n A x n = 0 .
(3.26)
Since A is Lipschitz continuous, it follows from (3.10) that
A p = 0 .
(3.27)

Again, since A is Lipschitz continuous and monotone so it is maximal monotone. It follows from Lemma 2.18 that p A 1 0 .

(c) We show that p EP ( f ) .

From x n , y n 0 and K n 0 as n and applying (3.7) for any q F , we get lim n ϕ ( q , u n ) ϕ ( q , p ) , it follows that
ϕ ( u n , x n ) = ϕ ( T r n , x n ) ϕ ( q , x n ) ϕ ( q , T r n x n ) = ϕ ( q , x n ) ϕ ( q , u n ) .
Taking limit as n on the both sides of the inequality, we have lim n ϕ ( u n , x n ) = 0 . From Lemma 2.5, it follows that
lim n u n x n = 0
(3.28)
and
lim n u n = p .
(3.29)
Since J is uniformly norm-to-norm continuous on bounded subsets of E, we obtain
lim n J u n J z n = 0 .
Since r n > 0 for all n 1 , we have J u n J z n r n 0 as n and
f ( u n , y ) + 1 r n y u n , J u n J z n 0 , y C .
From (A2), the fact that
y u n J u n J z n r n 1 r n y u n , J u n J z n f ( u n , y ) f ( y , u n ) , y C ,
taking the limit as n in the above inequality and from the fact that u n p as n , it follows that f ( y , p ) 0 for all y C . For any 0 < t < 1 , define y t = t y + ( 1 t ) p . Then y t C , which implies that f ( y t , p ) 0 . Thus it follows from (A1) that
0 = f ( y t , y t ) t f ( y t , y ) + ( 1 t ) θ ( y t , p ) t f ( y t , y ) ,

and so f ( y t , y ) 0 . From (A3) we have f ( p , y ) 0 for all y C and so p EP ( f ) . Hence, by (a), (b) and (c), that is, p F ( S ) A 1 0 EP ( f ) .

Finally, we show that p = Π F x 1 . From x n = Π C n x 1 , we have J x 1 J x n , x n z 0 for all z C n . Since F C n , we also have
J x 1 J x n , x n p ˆ 0 , p ˆ F .
Taking limit n , we obtain
J x 1 J p , p p ˆ 0 , p ˆ F .

By Lemma 2.6, we can conclude that p = Π F x 1 and x n p as n . The proof is completed. □

Next, we define z n = Π C J 1 ( J x n λ n A x n ) and assume that A y A y A u for all y C and u VI ( A , C ) . We can prove the strong convergence theorem for finding the set of solutions of the variational inequality problem in a real uniformly smooth and 2-uniformly convex Banach space.

Remark 3.2 (Qin et al. [7])

Let Π C be the generalized projection from a smooth strictly convex and reflexive Banach space E onto a nonempty closed convex subset C of E. Then Π C is a closed quasi-ϕ-nonexpansive mapping from E onto C with F ( Π C ) = C .

Corollary 3.3 Let C be a nonempty closed and convex subset of a uniformly smooth and 2-uniformly convex Banach space E. Let f be a bifunction from C × C to satisfying conditions (A1)-(A4), and let A be an α-inverse-strongly monotone mapping of C into E satisfying A y A y A u for all y C and u VI ( A , C ) . Let S : C 2 C be a closed and total quasi-ϕ-asymptotically nonexpansive multi-valued mapping with nonnegative real sequences ν n , μ n with ν n 0 , μ n 0 as n and a strictly increasing continuous function ψ : R + R + with ψ ( 0 ) = 0 . Assume that S is uniformly asymptotically regular on C with μ 1 = 0 and F : = F ( S ) EP ( f ) VI ( A , C ) . For arbitrary x 1 C , C 1 = C , generate a sequence { x n } by
{ z n = Π C J 1 ( J x n λ n A x n ) , u n = T r n x n , y n = J 1 ( α n J x n + β n J w n + γ n J u n ) , w n S n x n , C n + 1 = { v C n : ϕ ( v , y n ) ϕ ( v , z n ) ϕ ( v , x n ) + K n } , x n + 1 = Π C n + 1 x 1 , n N ,
(3.30)
where K n = ν n sup q F ψ ( ϕ ( q , x n ) ) + μ n . Assume that the control sequences { α n } , { β n } , { γ n } , { λ n } and { r n } satisfy the following conditions:
  1. 1.

    { α n } , { β n } and { γ n } are sequences in ( 0 , 1 ) such that α n + β n + γ n = 1 , lim inf n α n β n > 0 ,

     
  2. 2.

    { λ n } [ a , b ] for some a, b with 0 < a < b < c 2 α 2 and 1 c is the 2-uniformly convex constant of E,

     
  3. 3.

    { r n } [ d , ) for some d > 0 ,

     

then { x n } converges strongly to Π F x 1 .

Proof For q F and Π C is quasi-ϕ-nonexpansive mapping, we have
ϕ ( q , z n ) = ϕ ( q , Π C J 1 ( J x n λ n A x n ) ) ϕ ( q , J 1 ( J x n λ n A x n ) ) .

So, we can show that p VI ( A , C ) .

Define B E × E by Theorem 2.14, B is maximal monotone and B 1 0 = VI ( A , C ) . Let ( z , w ) G ( B ) . Since w B z = A z + N C ( z ) , we get w A z N C ( z ) .

From z n C , we have
z z n , w A z 0 .
(3.31)
On the other hand, since z n = Π C J 1 ( J x n λ n A x n ) . Then, by Lemma 2.6, we have
z z n , J z n ( J x n λ n A x n ) 0 ,
and thus
z z n , J x n J z n λ n A x n 0 .
(3.32)
It follows from (3.31) and (3.32) that
z z n , w z z n , A z z z n , A z + z z n , J x n J z n λ n A x n = z z n , A z A x n + z z n , J x n J z n λ n = z z n , A z A z n + z z n , A z n A x n + z z n , J x n J z n λ n z z n z n x n α z z n J x n J z n a M ( z n x n α + J x n J z n a ) ,

where M = sup n 1 z z n . From x n z n 0 as n and (3.23), taking lim n on the both sides of the equality above, we have z p , w 0 . By the maximality of B, we have p B 1 0 , that is, p VI ( A , C ) . From Theorem 3.1, we have p F ( S ) EP ( f ) VI ( A , C ) . The proof is completed. □

Let A be a strongly monotone mapping with constant k, Lipschitz with constant L > 0 , that is,
A x A y L x y , x , y D ( A ) ,
which implies that
1 L A x A y x y , x , y D ( A ) .
It follows that
A x A y , x y k x y 2 k L A x A y 2

hence A is α-inverse-strongly monotone with α = k L . Therefore, we have the following corollaries.

Corollary 3.4 Let C be a nonempty closed and convex subset of a uniformly smooth and 2-uniformly convex Banach space E. Let f be a bifunction from C × C to satisfying conditions (A1)-(A4), and let A : E E be a strongly monotone mapping with constant k, Lipschitz with constant L > 0 . Let S : C 2 C be a closed and total quasi-ϕ-asymptotically nonexpansive multi-valued mapping with nonnegative real sequences ν n , μ n with ν n 0 , μ n 0 as n and a strictly increasing continuous function ψ : R + R + with ψ ( 0 ) = 0 . Assume that S is uniformly asymptotically regular on C with μ 1 = 0 and F : = F ( S ) EP ( f ) A 1 0 . For arbitrary x 1 C , C 1 = C , a sequence { x n } is generated by
{ z n = J 1 ( J x n λ n A x n ) , u n = T r n z n , y n = J 1 ( α n J x n + β n J w n + γ n J u n ) , w n S n x n , C n + 1 = { v C n : ϕ ( v , y n ) ϕ ( v , z n ) ϕ ( v , x n ) + K n } , x n + 1 = Π C n + 1 x 1 , n N ,
(3.33)
where K n = ν n sup q F ψ ( ϕ ( q , x n ) ) + μ n . Assume that the control sequences { α n } , { β n } , { γ n } , { λ n } and { r n } satisfy the following conditions:
  1. 1.

    { α n } , { β n } and { γ n } are sequences in ( 0 , 1 ) such that α n + β n + γ n = 1 , lim inf n α n β n > 0 ,

     
  2. 2.

    { λ n } [ a , b ] for some a, b with 0 < a < b < c 2 k 2 L and 1 c is the 2-uniformly convex constant of E,

     
  3. 3.

    { r n } [ d , ) for some d > 0 ,

     

then { x n } converges strongly to Π F x 1 .

Corollary 3.5 Let C be a nonempty closed and convex subset of a uniformly smooth and 2-uniformly convex Banach space E. Let f be a bifunction from C × C to satisfying conditions (A1)-(A4), and let A : C E be a strongly monotone mapping with constant k,Lipschitz with constant L > 0 satisfying A y A y A u for all y C and u VI ( A , C ) . Let S : C 2 C be a closed and total quasi-ϕ-asymptotically nonexpansive multi-valued mapping with nonnegative real sequences ν n , μ n with ν n 0 , μ n 0 as n and a strictly increasing continuous function ψ : R + R + with ψ ( 0 ) = 0 . Assume that S is uniformly asymptotically regular on C with μ 1 = 0 and F : = F ( S ) EP ( f ) VI ( A , C ) . For arbitrary x 1 C , C 1 = C , generate a sequence { x n } by
{ z n = Π C J 1 ( J x n λ n A x n ) , u n = T r n x n , y n = J 1 ( α n J x n + β n J w n + γ n J u n ) , w n S n x n , C n + 1 = { v C n : ϕ ( v , y n ) ϕ ( v , z n ) ϕ ( v , x n ) + K n } , x n + 1 = Π C n + 1 x 1 , n N ,
(3.34)
where K n = ν n sup q F ψ ( ϕ ( q , x n ) ) + μ n . Assume that the control sequences { α n } , { β n } , { γ n } , { λ n } and { r n } satisfy the following conditions:
  1. 1.

    { α n } , { β n } and { γ n } are sequences in ( 0 , 1 ) such that α n + β n + γ n = 1 , lim inf n α n β n > 0 ,

     
  2. 2.

    { λ n } [ a , b ] for some a, b with 0 < a < b < c 2 k 2 L and 1 c is the 2-uniformly convex constant of E,

     
  3. 3.

    { r n } [ d , ) for some d > 0 ,

     

then { x n } converges strongly to Π F x 1 .

Let F be a Fréchet differentiable functional in a Banach space E and F be the gradient of F, denote ( F ) 1 0 = { x E : F ( x ) = min y E F ( y ) } . Baillon and Haddad [34] proved the following lemma.

Lemma 3.6 (Baillon and Haddad [34])

Let E be a Banach space. Let F be a continuously Fréchet differentiable convex functional on E and F be the gradient of F. If F is 1 α -Lipschitz continuous, then F is an α-inverse strongly monotone mapping.

We replace A in Theorem 3.1 by F, then we can obtain the following corollary.

Corollary 3.7 Let C be a nonempty closed and convex subset of a uniformly smooth and 2-uniformly convex Banach space E. Let f be a bifunction from C × C to satisfying conditions (A1)-(A4). Let F be a continuously Fréchet differentiable convex functional on E and F be 1 α -Lipschitz continuous. Let S : C 2 C be a closed and total quasi-ϕ-asymptotically nonexpansive multi-valued mapping with nonnegative real sequences ν n , μ n with ν n 0 , μ n 0 as n and a strictly increasing continuous function ψ : R + R + with ψ ( 0 ) = 0 . Assume that S is uniformly asymptotically regular on C with μ 1 = 0 and F : = F ( S ) F ( T ) EP ( f ) A 1 0 . For an initial point x 1 E , C 1 = C , define the sequence { x n } by
{ z n = J 1 ( J x n λ n F x n ) , u n = T r n x n , y n = J 1 ( α n J x n + β n J w n + γ n J u n ) , w n S n x n , C n + 1 = { v C n : ϕ ( v , y n ) ϕ ( v , z n ) ϕ ( v , x n ) + K n } , x n + 1 = Π C n + 1 x 1 , n N ,
(3.35)

where μ n = sup { μ n S , μ n T } , ν n = sup { ν n S , ν n T } , ψ = sup { ψ S , ψ T } , k n = ν n sup q F ψ ( ϕ ( q , x n ) ) + μ n .

Assume that the control sequences { α n } , { β n } , { γ n } , { λ n } and { r n } satisfy the following conditions:
  1. 1.

    { α n } , { β n } and { γ n } are sequences in ( 0 , 1 ) such that α n + β n + γ n = 1 , lim inf n α n β n > 0 and lim inf n α n γ n > 0 ,

     
  2. 2.

    { λ n } [ a , b ] for some a, b with 0 < a < b < c 2 α 2 and the 2-uniformly convex constant 1 c of E,

     
  3. 3.

    { r n } [ d , ) for some d > 0 ,

     

then { x n } converges strongly to Π F x 1 .

Declarations

Acknowledgements

This work was supported by Thaksin University Research Fund. Moreover, the author also would like to thank Faculty of Science, Thaksin University.

Authors’ Affiliations

(1)
Department of Mathematics and Statistics, Faculty of Science, Thaksin University (TSU),

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© Saewan; licensee Springer. 2013

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