Open Access

Strong convergence theorem for quasi-Bregman strictly pseudocontractive mappings and equilibrium problems in Banach spaces

  • Godwin C Ugwunnadi1, 2,
  • Bashir Ali3Email author,
  • Ibrahim Idris3 and
  • Maaruf S Minjibir3
Fixed Point Theory and Applications20142014:231

https://doi.org/10.1186/1687-1812-2014-231

Received: 18 March 2014

Accepted: 16 October 2014

Published: 17 November 2014

Abstract

In this paper, we introduce a new iterative scheme by a hybrid method and prove a strong convergence theorem of a common element in the set of fixed points of a finite family of closed quasi-Bregman strictly pseudocontractive mappings and common solutions to a system of equilibrium problems in reflexive Banach space. Our results extend important recent results announced by many authors.

MSC:47H09, 47J25.

Keywords

Bregman distancequasi-Bregman strictly pseudocontractive mapfixed point

1 Introduction

Let E be a real Banach space and C a nonempty closed convex subset of E. The normalized duality map from E to 2 E ( E is the dual space of E) denoted by J is defined by
J ( x ) = { f E : x , f = x 2 = f 2 } .

Let T : C C be a map, a point x C is called a fixed point of T if T x = x , and the set of all fixed points of T is denoted by F ( T ) . The mapping T is called L-Lipschitzian or simply Lipschitz if there exists L > 0 , such that T x T y L x y , x , y C and if L = 1 , then the map T is called nonexpansive.

Let g : C × C R be a bifunction. The equilibrium problem with respect to g is to find
z C  such that  g ( z , y ) 0 , y C .
The set of solution of equilibrium problem is denoted by EP ( g ) . Thus
EP ( g ) : = { z C : g ( z , y ) 0 , y C } .

Numerous problems in physics, optimization and economics reduce to finding a solution of equilibrium problem. Some methods have been proposed to solve the equilibrium problem in Hilbert spaces; see for example Blum and Oettli [1], Combettes and Hirstoaga [2]. Recently, Tada and Takahashi [3, 4] and Takahashi and Takahashi [5] obtain weak and strong convergence theorems for finding a common element of the set of solutions of an equilibrium problem and set of fixed points of a nonexpansive mapping in Hilbert space. In particular, Takahashi and Zembayashi [4] established a strong convergence theorem for finding a common element of the two sets by using the hybrid method introduced in Nakajo and Takahashi [6]. They also proved such a strong convergence theorem in a uniformly convex and uniformly smooth Banach space.

Reich and Sabach [7] and Kassay et al. [8] proved some convergence theorems for the solution of some equilibrium and variational inequality problems in the setting of reflexive Banach spaces.

Let ϕ : E × E [ 0 , ) denote the Lyapunov functional defined by
ϕ ( x , y ) = x 2 2 x , J y + y 2 , x , y E .
A mapping T : C C is said to be quasi-ϕ strictly pseudocontractive, see [9], if F ( T ) and there exists a constant k ( 0 , 1 ] such that
ϕ ( p , T x ) ϕ ( p , x ) + k ϕ ( x , T x ) , x C  and  p F ( T ) .

Let E be a real reflexive Banach space with norm and E the dual space of E. Throughout this paper, we shall assume f : E ( , + ] is a proper, lower semi-continuous and convex function. We denote by dom f : = { x E : f ( x ) < + } the domain of f.

Let x int dom f ; the subdifferential of f at x is the convex set defined by
f ( x ) = { x E : f ( x ) + x , y x f ( y ) , y E } ,
where the Fenchel conjugate of f is the function f : E ( , + ] defined by
f ( x ) = sup { x , x f ( x ) : x E } .
We know that the Young-Fenchel inequality holds:
x , x f ( x ) + f ( x ) , x E , x E .
A function f on E is coercive [10] if the sublevel set of f is bounded; equivalently,
lim x + f ( x ) = + .
A function f on E is said be strongly coercive [11] if
lim x + f ( x ) x = + .
For any x int dom f and y E , the right-hand derivative of f at x in the direction y is defined by
f ( x , y ) : = lim t 0 + f ( x + t y ) f ( x ) t .

The function f is said to be Gâteaux differentiable at x if lim t 0 + f ( x + t y ) f ( x ) t exists for any y. In this case, f ( x , y ) coincides with f ( x ) , the value of the gradient f of f at x. The function f is said to be Gâteaux differentiable if it is Gâteaux differentiable for any x int dom f . The function f is said to be Fréchet differentiable at x if this limit is attained uniformly in y = 1 . Finally, f is said to be uniformly Fréchet differentiable on a subset C of E if the limit is attained uniformly for x C and y = 1 . It is well known that if f is Gâteaux differentiable (resp. Fréchet differentiable) on int dom f , then f is continuous and its Gâteaux derivative f is norm-to-weak continuous (resp. continuous) on int dom f (see also [12, 13]). We will need the following results.

Lemma 1.1 [14]

If f : E R is uniformly Fréchet differentiable and bounded on bounded subsets of E, then f is uniformly continuous on bounded subsets of E from the strong topology of E to the strong topology of E .

Definition 1.2 [15]

The function f is said to be:
  1. (i)

    essentially smooth, if ∂f is both locally bounded and single-valued on its domain;

     
  2. (ii)

    essentially strictly convex, if ( f ) 1 is locally bounded on its domain and f is strictly convex on every convex subset of dom f ;

     
  3. (iii)

    Legendre, if it is both essentially smooth and essentially strictly convex.

     
Remark 1.3 Let E be a reflexive Banach space. Then we have:
  1. (i)

    f is essentially smooth if and only if f is essentially strictly convex (see [15], Theorem 5.4);

     
  2. (ii)

    ( f ) 1 = f (see [13]);

     
  3. (iii)

    f is Legendre if and only if f is Legendre (see [15], Corollary 5.5);

     
  4. (iv)

    if f is Legendre, then f is a bijection satisfying f = ( f ) 1 , ran f = dom f = int dom f and ran f = dom f = int dom f (see [15], Theorem 5.10).

     

Examples of Legendre functions were given in [15, 16]. One important and interesting Legendre function is 1 p p ( 1 < p < ) when E is a smooth and strictly convex Banach space. In this case the gradient f of f is coincident with the generalized duality mapping of E, i.e., f = J p ( 1 < p < ). In particular, f = I the identity mapping in Hilbert spaces. In the rest of this paper, we always assume that f : E ( , + ] is Legendre.

Let f : E ( , + ] be a convex and Gâteaux differentiable function. The function D f : dom f × int dom f [ 0 , + ) , defined as follows:
D f ( y , x ) : = f ( y ) f ( x ) f ( x ) , y x ,
(1.1)
is called the Bregman distance with respect to f (see [17]). It is obvious from the definition of D f that
D f ( z , x ) = D f ( z , y ) + D f ( y , x ) + f ( y ) f ( x ) , z y .
(1.2)
Recall that the Bregman projection [18] of x int dom f onto the nonempty, closed, and convex set C dom f is the necessarily unique vector P C f ( x ) C satisfying
D f ( P C f ( x ) , x ) = inf { D f ( y , x ) : y C } .

Concerning the Bregman projection, the following are well known.

Lemma 1.4 [19]

Let C be a nonempty, closed, and convex subset of a reflexive Banach space E. Let f : E R be a Gâteaux differentiable and totally convex function and let x E . Then:
  1. (a)

    z = P C f ( x ) if and only if f ( x ) f ( z ) , y z 0 , y C ;

     
  2. (b)

    D f ( y , P C f ( x ) ) + D f ( P C f ( x ) , x ) D f ( y , x ) , x E , y C .

     
Let f : E ( , + ] be a convex and Gâteaux differentiable function. The modulus of the total convexity of f at x int dom f is the function v f ( x , ) : [ 0 , + ) [ 0 , + ] defined by
v f ( x , t ) : = inf { D f ( y , x ) : y dom f , y x = t } .
The function f is called totally convex at x if v f ( x , t ) > 0 whenever t > 0 . The function f is called totally convex if it is totally convex at any point x int dom f and is said to be totally convex on bounded sets if v f ( B , t ) > 0 for any nonempty bounded subset B of E and t > 0 , where the modulus of the total convexity of the function f on the set B is the function v f : int dom f × [ 0 , + ) [ 0 , + ] defined by
v f ( B , t ) : = inf { v f ( x , t ) : x B dom f } .

Lemma 1.5 [20]

If x dom f , then the following statements are equivalent:
  1. (i)

    the function f is totally convex at x;

     
  2. (ii)
    for any sequence { y n } dom f ,
    lim n + D f ( y n , x ) = 0 lim n + y n x = 0 .
     
Recall that the function f called sequentially consistent [19] if for any two sequences { x n } and { y n } in E such that the first one is bounded
lim n + D f ( y n , x n ) = 0 lim n + y n x n = 0 .

Lemma 1.6 [21]

The function f is totally convex on bounded sets if and only if the function f is sequentially consistent.

Lemma 1.7 [22]

Let f : E R be a Gâteaux differentiable and totally convex function. If x 0 E and the sequence { D f ( x n , x 0 ) } is bounded, then the sequence { x n } is bounded too.

Lemma 1.8 [22]

Let f : E R be a Gâteaux differentiable and totally convex function, x 0 E and let C be a nonempty, closed, and convex subset of E. Suppose that the sequence { x n } is bounded and any weak subsequential limit of { x n } belongs to C. If D f ( x n , x 0 ) D f ( P C f ( x 0 ) , x 0 ) for any n R , then { x n } converges strongly to P C f ( x 0 ) .

A mapping T is said to be Bregman firmly nonexpansive [23], if for all x , y C ,
f ( T x ) f ( T y ) , T x T y f ( x ) f ( y ) , T x T y
or, equivalently,
D f ( T x , T y ) + D f ( T y , T x ) + D f ( T x , x ) + D f ( T y , y ) D f ( T x , y ) + D f ( T y , x ) .

A point p C is said to be asymptotic fixed point of a map T, if there exists a sequence { x n } in C which converges weakly to p such that lim n x n T x n = 0 . We denote by F ˆ ( T ) the set of asymptotic fixed points of T. A point p C is said to be strong asymptotic fixed point of a map T, if there exists a sequence { x n } in C which converges strongly to p such that lim n x n T x n = 0 . We denote by F ˜ ( T ) the set of strong asymptotic fixed points of T. Let f : E R , a mapping T : C C is said to be Bregman relatively nonexpansive [24] if F ( T ) , F ˆ ( T ) = F ( T ) and D f ( p , T ( x ) ) D f ( p , x ) for all x C and p F ( T ) . The map T : C C is said to be Bregman weak relatively nonexpansive if F ( T ) , F ˜ ( T ) = F ( T ) and D f ( p , T ( x ) ) D f ( p , x ) for all x C and p F ( T ) . T is said to be quasi-Bregman relatively nonexpansive if F ( T ) , and D f ( p , T ( x ) ) D f ( p , x ) for all x C and p F ( T ) . In [22] quasi-Bregman relatively nonexpansive is called left quasi-Bregman relatively nonexpansive. A map T : C C is called right quasi-Bregman relatively nonexpansive [25] if F ( T ) , and D f ( T ( x ) , p ) D f ( x , p ) for all x C and p F ( T ) . T is said to be quasi-Bregman strictly pseudocontractive if there exist a constant k [ 0 , 1 ) and F ( T ) such that D f ( p , T x ) D f ( p , x ) + k D f ( x , T x ) for all x C and p F ( T ) . In particular, T is said to be quasi-Bregman relatively nonexpansive if k = 0 and T is said to be quasi-Bregman pseudocontractive if k = 1 .

Very recently, Zhou and Gao [9] introduced this definition of a quasi-strict pseudocontraction related to the function ϕ and proved the convergence of a hybrid projection algorithm to a fixed point of a closed and quasi-strict pseudocontraction in a smooth and uniformly convex Banach space. They studied the strong convergence of the following scheme:
{ x 0 E , C 1 = C , x 1 = C 1 ( x 0 ) , C n + 1 = { z C n : ϕ ( x n , T x n ) 2 1 k x n z , J x n J T x n } , x n + 1 = C n + 1 ( x 0 ) ,

where C n + 1 is the generalized projection from E onto C n + 1 . They proved that the sequence { x n } converges strongly to F ( T ) ( x 0 ) .

Recently, Zegeye and Shahzad [26] proved a strong convergence theorem for the common fixed point of a finite family of right Bregman strongly nonexpansive mappings in a reflexive Banach space. Alghamdi et al. [27] proved a strong convergence theorem for the common fixed point of a finite family of quasi-Bregman nonexpansive mappings. Pang et al. [28] proved weak convergence theorems for Bregman relatively nonexpansive mappings. Shahzad and Zegeye [29] proved a strong convergence theorem for multivalued Bregman relatively nonexpansive mappings, while Zegeye and Shahzad [30] proved a strong convergence theorem for a finite family of Bregman weak relatively nonexpansive mappings.

Motivated and inspired by the above works, in this paper, we prove a new strong convergence theorem for a finite family of closed quasi-Bregman strictly pseudocontractive mapping and a system of equilibrium problems in a real reflexive Banach space. These results generalize and improve several recent results. We showed by an example that the class of quasi-Bregman strictly pseudocontractive mappings is a proper generalization of the class of quasi-ϕ-Bregman strictly pseudocontractive mappings.

2 Preliminaries

The next lemma will be useful in the proof of our main results.

Lemma 2.1 Let f : E R be a Legendre function which is uniformly Fréchet differentiable and bounded on subsets of E, let C be a nonempty, closed, and convex subset of E and let T : C C be a quasi-Bregman strictly pseudocontractive mapping with respect to f. Then, for any x C , p F ( T ) and k [ 0 , 1 ) the following hold:
D f ( x , T x ) 1 1 k f ( x ) f ( T x ) , x p .
(2.1)
Proof Let x C , p F ( T ) and k [ 0 , 1 ) , by definition of T, we have
D f ( p , T x ) D f ( p , x ) + k D f ( x , T x )
and, from (1.2), we obtain
D f ( p , x ) + D f ( x , T x ) + f ( x ) f ( T x ) , p x D f ( p , x ) + k D f ( x , T x ) ,
which implies
D f ( x , T x ) 1 1 k f ( x ) f ( T x ) , x p .

This completes the proof. □

Lemma 2.2 [31]

Let E be a real reflexive Banach space, f : E ( , + ] be a proper lower semi-continuous function, then f : E ( , + ] is a proper weak lower semi-continuous and convex function. Thus, for all z E , we have
D f ( z , f ( i = 1 N t i f ( x i ) ) ) i = 1 N t i D f ( z , x i ) .
(2.2)

In order to solve the equilibrium problem, let us assume that a bifunction g : C × C R satisfies the following conditions [1]:

(A1) g ( x , x ) = 0 , x C ;

(A2) g is monotone, i.e., g ( x , y ) + g ( y , x ) 0 , x , y C ;

(A3) lim sup t 0 g ( x + t ( z x ) , y ) g ( x , y ) , x , z , y C ;

(A4) the function y g ( x , y ) is convex and lower semi-continuous.

The resolvent of a bifunction g [2] is the operator Res g f : E 2 C defined by
Res g f ( x ) = { z C : g ( z , y ) + f ( z ) f ( x ) , y z 0 , y C } .
(2.3)

From Lemma 1, in [32], if f : ( , + ] R is a strongly coercive and Gâteaux differentiable function, and g satisfies conditions (A1)-(A4), then dom ( Res g f ) = E . The following lemma gives some characterization of the resolvent Res g f .

Lemma 2.3 [32]

Let E be a real reflexive Banach space and C be a nonempty closed convex subset of E. Let f : E ( , + ] be a Legendre function. If the bifunction g : C × C R satisfies the conditions (A1)-(A4), then the following hold:
  1. (i)

    Res g f is single-valued;

     
  2. (ii)

    Res g f is a Bregman firmly nonexpansive operator;

     
  3. (iii)

    F ( Res g f ) = EP ( g ) ;

     
  4. (iv)

    EP ( g ) is closed and convex subset of C;

     
  5. (v)
    for all x E and for all q F ( Res g f ) , we have
    D f ( q , Res g f ( x ) ) + D f ( Res g f ( x ) , x ) D f ( q , x ) .
    (2.4)
     

3 Main result

Lemma 3.1 Let f : E R be a Legendre function which is uniformly Fréchet differentiable on bounded subsets of E, let C be a nonempty, closed, and convex subset of E and let T : C C be a quasi-Bregman strictly pseudocontractive mapping with respect to f. Then F ( T ) is closed and convex.

Proof Let F ( T ) be nonempty set. First we show that F ( T ) is closed. Let { x n } be a sequence in F ( T ) such that x n z as n , we need to show that z F ( T ) . From Lemma 2.1, we obtain
D f ( z , T z ) 1 1 k f ( z ) f ( T z ) , z x n .
(3.1)

From (3.1), we have D f ( z , T z ) 0 , and from [15], Lemma 7.3, it follows that T z = z . Therefore F ( T ) is closed.

Next, we show that F ( T ) is convex. Let z 1 , z 2 F ( T ) , for any t ( 0 , 1 ) ; putting z = t z 1 + ( 1 t ) z 2 , we need to show that z F ( T ) . From Lemma 2.1, we obtain, respectively,
D f ( z , T z ) 1 1 k f ( z ) f ( T z ) , z z 1
(3.2)
and
D f ( z , T z ) 1 1 k f ( z ) f ( T z ) , z z 2 .
(3.3)
Multiplying (3.2) by t and (3.3) by ( 1 t ) and adding the results, we obtain
D f ( z , T z ) 1 1 k f ( z ) f ( T z ) , z z ,
(3.4)

which implies D f ( z , T z ) 0 , and from [15], Lemma 7.3, it follows that T z = z . Therefore F ( T ) is also convex. This completes the proof. □

We now prove the following theorem.

Theorem 3.2 Let C be a nonempty, closed, and convex subset of a real reflexive Banach space E and f : E R a strongly coercive Legendre function which is bounded, uniformly Fréchet differentiable and totally convex on bounded subset of E. For each k = 1 , 2 , , m , let g k be a bifunction from C × C to satisfying (A1)-(A4) and let { T i = 1 N } be a finite family of L i -Lipschitzian, i = 1 , 2 , 3 , , N , closed and quasi-Bregman strictly pseudocontractive self mappings of C such that F : = ( k = 1 m EP ( g k ) ) ( i = 1 N F ( T i ) ) . Let { x n } n = 1 be a sequence generated by x 1 = x C , C 1 = C and
{ x 1 C , y n = f ( α n f ( x n ) + ( 1 α n ) f ( T n x n ) ) , u j , n = Res g j f y n , j = 1 , 2 , 3 , , m , w n = f ( j = 1 m β j , n f ( u j , n ) ) , C n + 1 = { w C n : D f ( x n , w n ) 1 1 k f ( x n ) C n + 1 = f ( T n x n ) , x n w + f ( T n x n ) f ( w n ) , x n w } , x n + 1 = P C n + 1 f ( x ) , n N ,
(3.5)

where T n = T n ( mod N ) , and k [ 0 , 1 ) , for each i = 1 , 2 , , N , T i is uniformly continuous; suppose { α n } n = 1 and { β j , n } n = 1 , j = 1 , 2 , , m are sequences in ( 0 , 1 ) such that (i) lim inf n ( 1 α n ) > 0 , (ii) j = 1 m β j , n = 1 , n 1 . Then { x n } n = 1 converges strongly to P F f ( x ) , where P F f is the Bregman projection of E onto F.

Proof The proof is divided into six steps.

Step I. Show that F = ( j = 1 m EP ( g j ) ) ( i = 1 N F ( T i ) ) is closed and convex. From Lemma 3.1, i = 1 N F ( T i ) is closed and convex and from (iv) of Lemma 2.3, j = 1 m EP ( g j ) is closed and convex. So, F = ( j = 1 m EP ( g j ) ) ( i = 1 N F ( T i ) ) is closed and convex.

Step II. Show that C n is closed and convex for all n 1 . For n = 1 , C 1 = C is closed and convex. Assume that C h is closed and convex for some h > 1 . For w C h + 1 , one obtains
D f ( x h , w h ) 1 1 k f ( x h ) f ( T h x h ) , x h w + f ( T h x h ) f ( w h ) , x h w ;

using the fact that f ( x h ) f ( T h x h ) , and f ( T h x h ) f ( w h ) , are continuous and linear in E, for h 1 , C h + 1 is closed and convex.

Step III. Show that F C n for every n 1 . Note that F C 1 = C . Suppose F C h , for h 1 , then for all w F C h , since u j , h = Res g j f ( y h ) for each j = 1 , 2 , , m , from (2.2) and Lemma 2.3, we have
D f ( w , w h ) = D f ( w , f ( j = 1 m β j , n f ( u j , n ) ) ) j = 1 m β j h D f ( w , u j h ) j = 1 m β j h D f ( w , y h ) = D f ( w , y h ) ;
(3.6)
also from (2.2) and (2.1), we obtain
D f ( w , y h ) = D f ( w , f ( α h f ( x h ) + ( 1 α h ) f ( T h x h ) ) ) α h D f ( w , x h ) + ( 1 α h ) D f ( w , T h x h ) α h D f ( w , x h ) + ( 1 α h ) [ D f ( w , x h ) + k D f ( x h , T h x h ) ] D f ( w , x h ) + k D f ( x h , T h x h ) D f ( w , x h ) + k 1 k f ( x h ) f ( T h x h ) , x h w .
(3.7)
But, from (1.2),
D f ( w , w h ) = D f ( w , x h ) + D f ( x h , w h ) + f ( x h ) f ( w h ) , w x h .
(3.8)
From (3.6), (3.7), and (3.8), we obtain
D f ( x h , w h ) k 1 k f ( x h ) f ( T h x h ) , x h w + f ( x h ) f ( w h ) , x h w = k 1 k f ( x h ) f ( T h x h ) , x h w + f ( x h ) f ( T h x h ) , x h w + f ( T h x h ) f ( w h ) , x h w = ( k 1 k + 1 ) f ( x h ) f ( T h x h ) , x h w + f ( T h x h ) f ( w h ) , x h w = 1 1 k f ( x h ) f ( T h x h ) , x h w + f ( T h x h ) f ( w h ) , x h w .
(3.9)

This shows that w C h + 1 , which implies F C n for every n 1 .

Step IV. Show that lim n D f ( x n , x ) exists. From (3.5), x n = P C n f x , which from (a) of Lemma 1.4 implies
f ( x ) f ( x n ) , y x n 0 , y C n .
Since F C n , we have
f ( x ) f ( x n ) , w x n 0 , w F .
(3.10)
From (b) of Lemma 1.4 we have
D f ( x n , x ) = D f ( P C n f x , x ) D f ( w , x ) D f ( w , P C n f x ) D f ( w , x ) , n 1 , w F .
(3.11)
This implies that { D f ( x n , x ) } is bounded, from Lemma 1.7, { x n } is bounded. By the construction of C n , we have x m C m C n , and x n = P C n f x , for any positive integer m n . Then we obtain
D f ( x m , x n ) = D f ( x m , P C n f x ) D f ( x m , x ) D f ( P C n f x , x ) = D f ( x m , x ) D f ( x n , x ) .
(3.12)
In particular,
D f ( x n + 1 , x n ) D f ( x n + 1 , x ) D f ( x n , x ) .

Since x n = P C n f x and x n + 1 = P C n + 1 f x C n + 1 C n , we obtain D f ( x n , x ) D f ( x n + 1 , x ) , n 1 . This shows that { D f ( x n , x ) } is nondecreasing and hence the limit lim n D f ( x n , x ) exists. Thus from (3.12), taking the limit as m , n , we obtain lim n D f ( x m , x n ) = 0 . Since f is totally convex on bounded subsets of E, f is sequentially consistent (see [17]). It follows that x m x n 0 as m , n . Hence { x n } is Cauchy sequence in C. As { x n } is Cauchy in a complete space E, there exists p E such that x n p as n . Clearly p C .

Since D f ( x m , x n ) 0 , as m , n , we have in particular
lim n D f ( x n + 1 , x n ) = 0 ,
(3.13)
and this further implies that
lim n x n + 1 x n = 0 .
(3.14)

Step V. Next we show that x n p F .

Since x n + 1 = P C n + 1 f x C n + 1 , we have from (3.5)
D f ( x n , w n ) 1 1 k f ( x n ) f ( T n x n ) , x n x n + 1
(3.15)
+ f ( T n x n ) f ( w n ) , x n x n + 1 ,
(3.16)
which implies that lim n D f ( x n , w n ) = 0 . Since f is totally convex on bounded subsets of E, f is sequentially consistent (see [17]). It follows that
lim n x n w n = 0 .
(3.17)
From (3.14) and (3.17), we have
lim n x n + 1 w n = 0 .
(3.18)
Since f is uniformly Fréchet differentiable, it follows from Lemma 1.1 that f is uniformly continuous and f is uniformly continuous on bounded subsets of E (see [33], Theorem 1.8). Hence
lim n f ( x n + 1 ) f ( w n ) = 0
(3.19)
and
lim n | f ( x n + 1 ) f ( w n ) | = 0 .
(3.20)
Since x n + 1 C n + 1 , it follows from (3.6), (3.7) that
f ( x n + 1 ) f ( w n ) f ( w n ) , x n + 1 w n = D f ( x n + 1 , w n ) D f ( x n + 1 , y n ) D f ( x n + 1 , x n ) + k 1 k f ( x n ) f ( T n x n ) , x n x n + 1 ,
which implies from (3.20), (3.18), (3.13), and (3.14) that
lim n D f ( x n + 1 , y n ) = 0 .
From the sequential consistency of f, we have
lim n x n + 1 y n = 0 ;
(3.21)
from (3.14) and (3.21), we obtain
lim n x n y n = 0 ,
(3.22)
which implies that y n p C , since x n p C . From the uniform continuity of f, we have
lim n f ( x n ) f ( y n ) = 0 .
(3.23)
From (3.5), we have
f ( T n x n ) f ( x n ) = 1 1 α n f ( x n ) f ( y n ) ,
which implies from (3.23) that
lim n f ( T n x n ) f ( x n ) = 0 .
(3.24)
Since f is strongly coercive and uniformly convex on bounded subsets of E, f is uniformly Fréchet differentiable on bounded sets. Moreover, f is bounded on bounded sets, and from (3.24) we obtain
lim n T n x n x n = 0 .
(3.25)
On the other hand, we see that
x n T n + l x n x n x n + l + x n + l T n + l x n + l + T n + l x n + l T n + l x n ( 1 + L ) x n x n + l + x n + l T n + l x n + l
for all l { 1 , 2 , , N } , where L : = sup 1 i N L i . It follows from (3.14) and (3.25) that
lim n x n T n + l x n = 0
for all l { 1 , 2 , , N } . Thus
lim n x n T l x n = 0
(3.26)

for all l { 1 , 2 , , N } . Since x n p as n , by the closedness of T l for each l { 1 , 2 , , N } , we obtain p l = 1 N F ( T l ) .

Also, since y n p as n , we have from Lemma 2.3, for each j = 1 , 2 , , m ,
0 D f ( p , u j n ) = D f ( p , Res g j f y n ) D f ( p , y n ) 0 as  n .
Then we have from Lemma 1.5 that lim n p u j n = 0 , for each j = 1 , 2 , , m . Consequently, we have
u j n y n u j n p + p y n 0 as  n .
(3.27)
From the uniform continuity of f, for each j = 1 , 2 , , m we have
lim n f ( u j n ) f ( y n ) = 0 .
(3.28)
From (2.3), we have, for j = 1 , 2 , , m ,
g j ( u j n , y ) + f ( u j n ) f ( y n ) , y u j n 0 , y C .
Furthermore, using (A2) in the last inequality, we obtain
f ( u j n ) f ( y n ) , y u j n g j ( y , u j n ) , y C .
By (A4), (3.28), and u j n p as n , we have
g j ( y , p ) 0 , y C .
(3.29)
Let z t : = t y + ( 1 t ) p for t ( 0 , 1 ] and y C . This implies that z t C . This yields g j ( z t , p ) 0 . It follows from (A1) and (A4) that
0 = g j ( z t , z t ) t g j ( z t , y ) + ( 1 t ) g j ( z t , p ) t g j ( z t , y ) ,
and hence
0 g j ( z t , y ) .
From condition (A3), we obtain
g j ( p , y ) 0 , y C  and  j { 1 , 2 , 3 , , m } .

This implies that p EP ( g j ) , for each j = 1 , 2 , , m . Thus, p j = 1 m EP ( g j ) . Hence we have p F = ( i = 1 N F ( T i ) ) ( j = 1 m EP ( g j ) ) .

Step VI. Finally, we show that p = P F f x . Setting n in (3.10), we obtain
f ( x ) f ( p ) , w p 0 , w F .

By (a) of Lemma 1.4, we have p = P F f x . □

Here we give an example of a quasi-Bregman strictly peudocontractive mapping which is not quasi-ϕ strictly pseudocontractive mapping; this shows that the former class is a generalization of the latter.

Example 3.3 Let E = R , C = [ 1 , 0 ] and define T , f : [ 1 , 0 ] R by f ( x ) = x and T x = 2 x , for all x [ 1 , 0 ] . We want to show that T is a quasi-Bregman strictly pseudocontractive but not quasi-ϕ strictly pseudocontractive.

Proof From the definition it is clear that f is proper, lower semi-continuous, and convex, and also F ( T ) = { 0 } . By the definition of quasi-Bregman strict pseudocontractivity, we find k [ 0 , 1 ) such that D f ( p , T x ) D f ( p , x ) + k D f ( x , T x ) for all x C and p F ( T ) . Now,
D ( 0 , T x ) = f ( 0 ) f ( T x ) f ( T x ) , 0 T x D ( 0 , T x ) = 0 2 x f ( 2 x ) , 0 2 x D ( 0 , T x ) = 2 x 2 , 2 x D ( 0 , T x ) = 2 x + 4 x = 2 x ,
(3.30)
D ( 0 , x ) = f ( 0 ) f ( x ) f ( x ) , 0 x D ( 0 , x ) = 0 x 1 , x D ( 0 , x ) = x + x = 0
(3.31)
and
D ( x , T x ) = f ( x ) f ( T x ) f ( T x ) , x T x = x 2 x 2 , x 2 x = x + 2 x = x .
(3.32)
From (3.30), (3.31), and (3.32), we obtain
D ( 0 , T x ) = 2 x x 0 + k x , x [ 1 , 0 ] , k [ 0 , 1 ) D ( 0 , x ) + k D ( x , T x ) , x [ 1 , 0 ] , k [ 0 , 1 ) .
Therefore
D ( 0 , T x ) D ( 0 , x ) + k D ( x , T x ) , x [ 1 , 0 ] , k [ 0 , 1 ) .

Hence, T is a quasi-Bregman strictly pseudocontractive map.

Further,
ϕ ( 0 , T x ) = | 0 | 2 2 0 , J ( T x ) + | T x | 2 ϕ ( 0 , T x ) = 0 2 0 , J ( 2 x ) + | 2 x | 2 ϕ ( 0 , T x ) = 4 | x | 2 ,
(3.33)
ϕ ( 0 , x ) = | 0 | 2 2 0 , J ( x ) + | x | 2 ϕ ( 0 , x ) = 0 2 0 , J ( x ) + | x | 2 ϕ ( 0 , x ) = | x | 2
(3.34)
and
ϕ ( x , T x ) = | x | 2 2 x , J ( T x ) + | T x | 2 = | x | 2 2 x , J ( 2 x ) + 4 | x | 2 = | x | 2 4 x , J ( x ) + 4 | x | 2 = | x | 2 4 | x | 2 + 4 | x | 2 = | x | 2 .
(3.35)
Since 4 | x | 2 > | x | 2 + k | x | 2 , for all k [ 0 , 1 ) and for all x [ 1 , 0 ] ,
ϕ ( 0 , T x ) ϕ ( 0 , x ) + k ϕ ( x , T x ) , x [ 1 , 0 ]

cannot hold for any k [ 0 , 1 ) . Hence, T is not a quasi-ϕ strictly pseudocontractive map. □

4 Numerical example

In this section we discuss the direct application of Theorem 3.2 on a typical example on a real line. Consider the following:
E = R , C = [ 1 , 1 ] , g ( z , y ) = y 2 + y z 2 z 2 , f ( x ) = 2 3 x 2 , f ( x ) = 4 3 x , T x = 2 x , f ( x ) = sup { x , x f ( x ) : x E } , f ( z ) = 3 8 z 2 , f ( z ) = 3 4 z , α n = n + 1 4 n , α n f ( x n ) + ( 1 α n ) f ( T x n ) = ( 5 n 3 ) 3 n x n , k = 1 / 2 , x 1 = 1 / 2 C ,
then the scheme can be simplified as
y n = f ( α n f ( x n ) + ( 1 α n ) f ( T x n ) ) , y n = ( 5 n 3 ) 4 n x n , u n = Res g f y n = 4 13 y n , w n = f ( f ( u n ) ) = u n , C n + 1 = { w C n : w x n ( 1 k ) ( x n w n ) 2 2 [ ( 1 + 2 k ) x n ( 1 k ) w n ] } , x n + 1 = P C n + 1 f ( x 1 ) = x n ( 1 k ) ( x n w n ) 2 2 [ ( 1 + 2 k ) x n ( 1 k ) w n ] .

Declarations

Authors’ Affiliations

(1)
Department of Mathematics, Michael Okpara University of Agriculture
(2)
Department of Mathematics, Ahmadu Bello University
(3)
Department of Mathematical Sciences, Bayero University Kano

References

  1. Blum E, Oettli W: From optimization and variational inequalities to equilibrium problems. Math. Stud. 1994, 63: 123–145.MathSciNetGoogle Scholar
  2. Combettes PL, Hirstoaga SA: Equilibrium programming in Hilbert spaces. J. Nonlinear Convex Anal. 2005, 6: 117–136.MathSciNetGoogle Scholar
  3. Tada A, Takahashi W: Strong convergence theorem for an equilibrium problem and a nonexpansive mapping. In Nonlinear Analysis and Convex Analysis. Edited by: Tada A, Takahashi W. Yokohama Publishers, Yokohama; 2007:609–617.Google Scholar
  4. Takahashi W, Zembayashi K: Strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces. Nonlinear Anal., Theory Methods Appl. 2009, 70(1):45–57. 10.1016/j.na.2007.11.031View ArticleMathSciNetGoogle Scholar
  5. Takahashi S, Takahashi W: Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces. J. Math. Anal. Appl. 2003, 1331(1):372–379.Google Scholar
  6. Nakajo K, Takahashi W: Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups. J. Math. Anal. Appl. 2003, 279(2):372–379. 10.1016/S0022-247X(02)00458-4View ArticleMathSciNetGoogle Scholar
  7. Reich S, Sabach S: Three strong convergence theorems regarding iterative methods for solving equilibrium problems in reflexive Banach space. Contemp. Math. 2012, 568: 225–240.View ArticleMathSciNetGoogle Scholar
  8. Kassay G, Reich S, Sabach S: Iterative methods for solving systems of variational inequalities in reflexive Banach space. SIAM J. Optim. 2011, 21: 1319–1344. 10.1137/110820002View ArticleMathSciNetGoogle Scholar
  9. Zhou H, Gao E: An iterative method of fixed points for closed and quasi-strict pseudocontraction in Banach spaces. J. Appl. Math. Comput. 2010, 33: 227–237. 10.1007/s12190-009-0283-0View ArticleMathSciNetGoogle Scholar
  10. Hiriart-Urruty JB, Lemarchal C Grundlehren der Mathematischen Wissenschaften 306. In Convex Analysis and Minimization Algorithms II. Springer, Berlin; 1993.View ArticleGoogle Scholar
  11. Zǎlinescu C: Convex Analysis in General Vector Spaces. World Scientific, River Edge; 2002.View ArticleMATHGoogle Scholar
  12. Asplund E, Rockafellar RT: Gradients of convex functions. Trans. Am. Math. Soc. 1969, 139: 443–467.View ArticleMathSciNetGoogle Scholar
  13. Bonnans JF, Shapiro A: Perturbation Analysis of Optimization Problems. Springer, New York; 2000.MATHView ArticleGoogle Scholar
  14. Reich S, Sabach S: A strong convergence theorem for a proximal-type algorithm in reflexive Banach spaces. J. Nonlinear Convex Anal. 2009, 10: 471–485.MathSciNetGoogle Scholar
  15. Bauschke HH, Borwein JM, Combettes PL: Essential smoothness, essential strict convexity, and Legendre functions in Banach spaces. Commun. Contemp. Math. 2001, 3: 615–647. 10.1142/S0219199701000524View ArticleMathSciNetGoogle Scholar
  16. Bauschke HH, Borwein JM: Legendre functions and the method of random Bregman projections. J. Convex Anal. 1997, 4: 27–67.MathSciNetGoogle Scholar
  17. Censor Y, Lent A: An iterative row-action method for interval convex programming. J. Optim. Theory Appl. 1981, 34: 321–353. 10.1007/BF00934676View ArticleMathSciNetGoogle Scholar
  18. Bregman LM: The relaxation method for finding the common point of convex sets and its application to the solution of problems in convex programming. USSR Comput. Math. Math. Phys. 1967, 7: 200–217.View ArticleGoogle Scholar
  19. Butnariu D, Resmerita E: Bregman distances, totally convex functions and a method for solving operator equations in Banach spaces. Abstr. Appl. Anal. 2006., 2006: Article ID 84919Google Scholar
  20. Resmerita E: On total convexity, Bregman projections and stability in Banach spaces. J. Convex Anal. 2004, 11: 1–16.MathSciNetGoogle Scholar
  21. Butnariu D, Iusem AN: Totally Convex Functions for Fixed Points Computation and Infinite Dimensional Optimization. Kluwer Academic, Dordrecht; 2000.MATHView ArticleGoogle Scholar
  22. Reich S, Sabach S: Two strong convergence theorems for a proximal method in reflexive Banach spaces. Numer. Funct. Anal. Optim. 2010, 31: 22–44. 10.1080/01630560903499852View ArticleMathSciNetGoogle Scholar
  23. Reich S, Sabach S: Existence and approximation of fixed points of Bregman firmly nonexpansive mappings in reflexive Banach spaces. In Fixed-Point Algorithms for Inverse Problems in Science and Engineering. Springer, New York; 2011:301–316.View ArticleGoogle Scholar
  24. Matsushita S, Takahashi W: Weak and strong convergence theorems for relatively nonexpansive mappings in Banach spaces. Fixed Point Theory Appl. 2004, 2004: 37–47.View ArticleMathSciNetGoogle Scholar
  25. Martin-Marquez V, Reich S, Sabach S: Right Bregman nonexpansive operators in Banach spaces. Nonlinear Anal. 2012, 75: 5448–5465. 10.1016/j.na.2012.04.048View ArticleMathSciNetGoogle Scholar
  26. Zegeye H, Shahzad N: Convergence theorems for right Bregman strongly nonexpansive mappings in reflexive Banach spaces. Abstr. Appl. Anal. 2014., 2014: Article ID 584395Google Scholar
  27. Alghamdi MA, Shahzad N, Zegeye H: Strong convergence theorems for quasi-Bregman nonexpansive mappings in reflexive Banach spaces. J. Appl. Math. 2014., 2014: Article ID 580686Google Scholar
  28. Pang CT, Naraghirad E, Wen CF: Weak convergence theorems for Bregman relatively nonexpansive mappings in Banach spaces. J. Appl. Math. 2014., 2014: Article ID 573075Google Scholar
  29. Shahzad N, Zegeye H: Convergence theorem for common fixed points of finite family of multivalued Bregman relatively nonexpansive mappings. Fixed Point Theory Appl. 2014., 2014: Article ID 152Google Scholar
  30. Zegeye H, Shahzad N: Strong convergence theorems for common fixed point of finite family of Bregman weak relatively nonexpansive mappings in reflexive Banach spaces. Sci. World J. 2014., 2014: Article ID 493450Google Scholar
  31. Phelps RP Lecture Notes in Mathematics 1364. In Convex Functions, Monotone Operators, and Differentiability. 2nd edition. Springer, Berlin; 1993.Google Scholar
  32. Reich S, Sabach S: Two strong convergence theorems for Bregman strongly nonexpansive operators in reflexive Banach spaces. Nonlinear Anal. TMA 2010, 73: 122–135. 10.1016/j.na.2010.03.005View ArticleMathSciNetGoogle Scholar
  33. Ambrosetti A, Prodi G: A Primer of Nonlinear Analysis. Cambridge University Press, Cambridge; 1993.MATHGoogle Scholar

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© Ugwunnadi et al.; licensee Springer. 2014

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