Open Access

On a hybrid algorithm for a family of total quasi-ϕ-asymptotically nonexpansive mappings in Banach spaces

Fixed Point Theory and Applications20122012:70

https://doi.org/10.1186/1687-1812-2012-70

Received: 23 December 2011

Accepted: 1 May 2012

Published: 1 May 2012

Abstract

The purpose of this article is to introduce the concept of total quasi- ϕ-asymptotically nonexpansive mapping which contains many kinds of mappings as its special cases and we prove a strong convergence theorem by using a hybrid method for finding a common element of the set of solutions for a generalized mixed equilibrium problems, the set of fixed points of a family of total quasi- ϕ-asymptotically nonexpansive mappings in uniformly smooth and strictly convex Banach space with the Kadec-Klee property. The results presented in the article improve and extend some recent results.

MR (2000) Subject Classification: 47H09; 47H05.

Keywords

total quasi- ϕ-asymptotically nonexpansive mapping quasi- ϕ-asymptotically nonexpansive mapping generalized mixed equilibrium problem hybrid method.

1 Introduction

Let E be a Banach space and C be a closed convex subsets of E. Let F be an equilibrium bifunction from C × C into R, ψ: CR be a real-valued function and A: CE* be a nonlinear mapping. The "so-called" generalized mixed equilibrium problem is to find z C such that
F ( z , y ) + A z , y - z + ψ ( y ) - ψ ( z ) 0 , y C .
(1.1)
The set of solutions of (1.1) is denoted by GMEP, i.e.,
G M E P = { z C : F ( z , y ) + A z , y - z + ψ ( y ) - ψ ( z ) 0 , y C } .
Special examples:
  1. (I)
    If A = 0, then the problem (1.1) is equivalent to find z C such that
    F ( z , y ) + ψ ( y ) - ψ ( z ) 0 , y C .
    (1.2)
     
which is called the mixed equilibrium problem, see [1]. The set of solutions of (1.2) is denoted by MEP.
  1. (II)
    If F = 0, then the problem (1.1) is equivalent to find z C such that
    A z , y - z + ψ ( y ) - ψ ( z ) 0 , y C .
    (1.3)
     
which is called the mixed variational inequality of Browder type. The set of solutions of (1.3) is denoted by V I(C, A, ψ).
  1. (III)
    If ψ = 0, then the problem (1.1) is equivalent to find z C such that
    F ( z , y ) + A z , y - z 0 , y C .
    (1.4)
     
which is called the generalized equilibrium problem, see [2]. The set of solutions of (1.4) is denoted by EP.
  1. (IV)
    If A = 0, ψ = 0, then the problem (1.1) is equivalent to find z C such that
    F ( z , y ) 0 , y C .
    (1.5)
     

which is called the equilibrium problem. The set of solutions of (1.5) is denoted by EP(F).

These show that the problem (1.1) is very general in the sense that numerous problems in physics, optimization, and economics reduce to finding a solution of (1.1). Recently, some methods have been proposed for the generalized mixed equilibrium problem in Banach spaces (see, for example [17]).

Let E be a smooth, strictly convex, and reflexive Banach spaces and C be a nonempty closed convex subsets of E. Throughout this article, we denote by ϕ the function defined by
ϕ ( y , x ) = | | y | | 2 - 2 y , J x + | | x | | 2 , x , y E ,
(1.6)

where J: E → 2 E* is the normalized duality mapping.

Let T: CC be a mapping and F (T) be the set of fixed points of T .

Recall that a point p C is said to be an asymptotic fixed point of T if there exists {x n } in C which converges weakly to p and limn||x n - Tx n || = 0. We denote the set of all asymptotic fixed point of T by F ̃ ( T ) . A point p C is said to be a strong asymptotic fixed point of T if there exists {x n } in C such that x n p and limn||x n - Tx n || = 0 We denote the set of all strongly asymptotic fixed point of T by F ^ ( T ) .

A mapping T: CC is said to be nonexpansive, if
| | T x - T y | | | | x - y | | , x , y C .
A mapping T: CC is said to be relatively nonexpansive if F (T) ≠ Ø, F ( T ) = F ̃ ( T ) , and
ϕ ( u , T x ) ϕ ( u , x ) , u F ( T ) , x C
A mapping T: CC is said to be weak relatively nonexpansive if F(T) ≠ Ø, F ( T ) = F ^ ( T ) , and
ϕ ( u , T x ) ϕ ( u , x ) , u F ( T ) , x C
A mapping T: CC is said to be quasi-ϕ-nonexpansive, if F(T) ≠ Ø and
ϕ ( u , T x ) ϕ ( u , x ) , x C , u F ( T )
A mapping T: CC is said to be quasi-ϕ-asymptotically nonexpansive, if there exists some real sequence {k n } with k n 1 and k n → 1 and F (T) ≠ such that
ϕ ( u , T n x ) k n ϕ ( u , x ) , n 1 , x C , u F ( T )
(1.7)

A mapping T: CC is said to be closed, if for any sequence {x n } C with x n x and Tx n y, then Tx = y.

Definition 1.1 (1) A mapping T: CC is said to be total quasi-ϕ-asymptotically nonexpansive, if F (T) ≠ and there exist nonnegative real sequences {υ n }, {μ n } with υ n → 0, μ n → 0 (as n) and a strictly increasing continuous function ζ: + + with ζ(0) = 0 such that for all x C, p F(T)
ϕ ( p , T n x ) ϕ ( p , x ) + v n ζ ( ϕ ( p , x ) )  +  μ n , n 1 .
(1.8)
  1. (2)
    A family of mappings {T λ }λΛ: CC is said to be uniformly total quasi-ϕ-asymptotically nonexpansive, if ∩λΛ F (T λ ) ≠ and there exist nonnegative real sequences {υ n }, {μ n } with υ n 0, μ n 0(as n) and a strictly increasing continuous function ζ: + + with ζ(0) = 0 such that for all x C, p λΛ F (T λ )
    ϕ ( p , T i n x ) ϕ ( p , x ) + v n ζ ( ϕ ( p , x ) )  +  μ n , n 1 .
    (1.9)
     
Remark 1.1 From the definitions, it is easy to know that
  1. (1)

    Each relatively nonexpansive mapping is closed;

     
  2. (2)
    Taking ζ(t) = t, t ≥ 0, υ n = (k n - 1) and μ n = 0,, then (1.7) can be rewritten as
    ϕ ( p , T n x ) ϕ ( p , x ) + v n ζ ( ϕ ( p , x ) )  +  μ n , n 1 , x C , p F ( T ) .
    (1.10)
     
This implies that each quasi-ϕ-asymptotically nonexpansive mapping must be a total quasi-ϕ-asymptotically nonexpansive mapping, but the converse is not true;
  1. (3)

    The class of quasi-ϕ-asymptotically nonexpansive mappings contains properly the class of quasi-ϕ-nonexpansive mappings as a subclass, but the converse is not true;

     
  2. (4)

    The class of quasi-ϕ-nonexpansive mappings contains properly the class of weak relatively nonexpansive mappings as a subclass, but the converse is not true;

     
  3. (5)

    The class of weak relatively nonexpansive contains properly the class of relatively nonexpansive mappings as a subclass, but the converse is not true.

     

Iterative approximation of fixed points for relatively nonexpansive mappings, weak relatively nonexpansive mappings, quasi-ϕ nonexpansive mappings, quasi-ϕ-asymptotically non-expansive mappings in the setting of Banach spaces has been studied extensively by many authors (see [513]).

Motivated by the above, the purpose of this article is to introduce the concept of total quasi-ϕ-asymptotically nonexpansive mapping which contains many kinds of mappings as its special cases and we prove a strong convergence theorem by using a hybrid method for finding a common element of the set of solutions for a generalized mixed equilibrium problems, the set of fixed points of a family of total quasi- ϕ-asymptotically nonexpansive mappings in uniformly smooth and strictly convex Banach space with the Kadec-Klee property. The results presented in the paper improve and extend some recent results.

2 Preliminaries

Throughout this article, we assume that all the Banach spaces are real. We denote by and the sets of positive integers and real numbers, respectively. Let E be a Banach space and let E* be the topological dual of E. For all x E and x* E* , we denote by 〈x, x*〉 the value of x* at x. The mapping J: E → 2E*defined by
J ( x ) = { x * E * : x , x * = | | x | | 2 = | | x * | | 2 } , x E ,
(2.1)

is normalized duality mapping. We denote the weak convergence and the strong convergence of a sequence {x n } to x by x n x and x n x, respectively.

A Banach spaces E is said to be strictly convex if | | x + y | | 2 < 1 for x, y S(E) = {z E: ||z|| = 1} with xy. It is said to be uniformly convex if for any given ε (0, 2], there exists δ > 0 such that | | x + y | | 2 < 1 - δ for x, y S(E) with ||x - y|| ≥ ε. E is said to have the Kadec-Klee property, if for any sequence {x n } E such that x n x E and ||x n || → ||x||, then x n x.

Define f: S(E) × S(E) × \ {0} by
f ( x , y , t ) = x + t y - x t

for x, y S(E) and t \ {0}. A norm of E is said to be Gâteaux differentiable if limt→ 0f(x, y, t) has a limit for each x, y S(E). In this case, E is said to be smooth. We know that if E is smooth, strictly convex, and reflexive, then the duality mapping J is single valued, one to one, and onto. In this case, the inverse mapping J-1 coincides with the duality mapping J* on E*. See [14] for more details.

Remark 2.1 If E is a reflexive and strictly convex Banach space, then J-1 is hemi-continuous, i.e., J-1 is norm-weak-continuous.

Let {C n } be a sequence of nonempty closed convex subset of a reflexive Banach space E. We define two subsets s - Li n C n and w - Ls n C n as follows: x s - Li n C n if and only if there exists {x n } E such that {x n } converges strongly to x and that x n C n for all n . Similarly, y w - Ls n C n if and only if there exists a subsequence { C n i } of {C n } and a sequence {y i } E such that {y i } converges weakly to y and y i C n i for all i . We define the Mosco convergence [15] of {C n } as follows: If C0 = s - Li n C n = w - Ls n C n , then {C n } is said to be convergent to C0 in the sense of Mosco and we write C0 = M - limn→∞C n .

For more details, see [16].

Let C be a nonempty closed convex subset of a smooth, strictly convex, and reflexive Banach space E. Then, for arbitrarily fixed x E, the function y α ||x - y||2: C → R+ has a unique minimizer y x C. Using such a point, we define the metric projection P C by P C x = y x = arg minyC||x - y||2 for every x E. In a similar fashion, we can see that the function y α ϕ(x, y): C → R+ has a unique minimizer z x C. The generalized projection Π C of E onto C is defined by Π C = z x = arg minyCϕ(x, y) for every x E; see [17].

The generalized projection Π C from E onto C is well defined, single valued and satisfies
( | | x | | - | | y | | ) 2 ϕ ( y , x ) ( | | x | | + | | y | | ) 2 , x , y E
(2.2)

If E is a Hilbert space, then ϕ(y, x) = ||y - x||2 and Π C is the metric projection P C of E onto C.

It is well-known that the following conclusions hold:

Lemma 2.1[17, 18]. Let C be a nonempty closed convex subsets of a smooth, strictly convex, and reflexive Banach spaces. Then
ϕ ( x , Π C y ) + ϕ ( Π C y , y ) ϕ ( x , y ) , x C , y E .
(2.3)
Lemma 2.2. Let C be a nonempty closed convex subsets of a smooth, strictly convex, and reflexive Banach spaces E, let x E and z C. Then the following conclusions hold:
  1. (a)

    z = ∏ C xy - z, Jx - jz〉 ≤ 0, y C.

     
  2. (b)

    For x, y E, ϕ (x, y) = 0 if and only if x = y.

     

The following theorem proved by Tsukada [19] plays an important role in our results.

Theorem 2.1. Let E be a smooth, reflexive, and strictly convex Banach spaces having the Kadec-Klee property. Let {K n } be a sequence of nonempty closed convex subsets of E. If K0 = M - limn→K n exists and is nonempty, then P K n x converges strongly to P K 0 x for each x C.

Theorem 2.1 is still valid if we replace the metric projections with the generalized pro-jections as follows:

Theorem 2.2 Let E be a smooth, reflexive, and strictly convex Banach spaces having the Kadec-Klee property. Let {K n } be a sequence of nonempty closed convex subsets of E. If K0 = M - limn→∞K n exists and is nonempty, then K n x converges strongly to K 0 x for each x C.

For solving the equilibrium problem for bifunction F: C × C, let us 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 α F (x, y) is a convex and lower semicontinuous.

If an equilibrium bifunction F: C × CR satisfies conditions (A1)-(A4), then we have the following results.

Lemma 2.3[20]. Let C be a nonempty closed convex subset of a smooth, strictly convex, and reflexive Banach spaces E, let F be an equilibrium bifunction from C × C to satisfying conditions (A1)-(A4), let r > 0 and let x E. Then, there exists z C such that
F ( z , y ) + 1 r y - z , J z - J x 0 , y C .
Lemma 2.4[21]. Let C be a nonempty closed convex subset of a smooth, strictly convex, and reflexive Banach spaces E, let F: C × C be an equilibrium bifunction satisfying conditions (A1)-(A4). For 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 .
for all x E. Then, the following hold:
  1. (1)

    T r is single-valued;

     
  2. (2)
    T r is a firmly nonexpansive-type mapping, i.e., for any 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 ) = F ^ ( T r ) = E P ( F ) ;

     
  4. (4)

    EP (F) is a closed and convex set.

     
Lemma 2.5[21]. Let C be a nonempty closed convex subset of a smooth, strictly convex, and reflexive Banach spaces E, let F: C × C be an equilibrium bifunction satisfying conditions (A1)-(A4). For r > 0 and x E and q F (T r ),
ϕ ( q , T r x ) + ϕ ( T r x , x ) ϕ ( q , x )
Lemma 2.6[22]. Let E be a uniformly convex Banach space and let r > 0. Then there exists a strictly increasing, continuous, and convex function g: [0, 2r] → 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 ) .
(2.4)

3 The main results

Theorem 3.1. Let E be a uniformly smooth and strictly convex Banach space with Kadec-Klee property and C be a nonempty closed convex subset of E. A: CE* be a continuous and monotone mapping, ψ: C → be a lower semi-continuous and convex function and F be a bifunction from C ×C to which satisfies the conditions (A1)-(A4). Let {T λ }(λ Λ): CC be a family of uniformly L λ -Lipschitzian continuous and uniformly total quasi-ϕ-asymptotically nonexpansive mappings such that = λ Λ F T λ G M E P . Assume that K = sup{||u||: u F} < ∞. Let {x n } be the sequence generated by x1 = x C, C1 = C and
y λ , n = J - 1 ( α n J x n + ( 1 - α n ) J T λ n x n ) , λ Λ , u λ , n C s u c h t h a t F ( u λ , n , y ) + A u λ , n , y - u λ , n + ψ ( y ) - ψ ( u λ , n ) + 1 r λ , n y - u λ , n , J u λ , n - J y λ , n 0 , y C , λ Λ C n + 1 = { z C n : sup λ Λ ϕ ( z , u λ , n ) ϕ ( z , x n ) + ξ n } , x n + 1 = Π C n + 1 x , n 0 .
(3.1)

where ξ n = (1 - α n )(ν n supuFζ(ϕ(u, x n )) + μ n ),{α n } is a sequence in [0, 1] such that lim infn→∞α n (1 - α n ) > 0, lim inf n→∞ α n < 1 and {r λ , n } [a, ∞) for some a > 0, then {x n } converge strongly to some point x* in F.

Proof. We define a bifunction G: C × CR by
G ( z , y ) = F ( z , y ) + A z , y - z + ψ ( y ) - ψ ( z ) , z , y C .

It is easy to prove that the bifunction G satisfies conditions(A1)-(A4).

Therefore, the generalized mixed equilibrium problem (1.1) is equivalent to the following equilibrium problem: find z C such that
G ( z , y ) 0 , y C ,
and GMEP = EP (G), F = GMEPλΛ=EP(G) ∩λΛF(T λ ). So, (3.1) can be written as:
y λ , n = J - 1 ( α n J x n + ( 1 - α n ) J T λ n x n ) for all λ Λ , u λ , n C such that G ( u λ , n , y ) + 1 r λ , n y - u λ , n , J u λ , n - J y λ , n 0 , y C , λ Λ C n + 1 = { z C n : sup λ Λ ϕ ( z , u λ , n ) ϕ ( z , x n ) + ξ n } , x n + 1 = Π C n + 1 x , n 0 .
(3.2)
Since the bifunction G satisfies conditions (A1)-(A4), from Lemma 2.4, for given r > 0 and x E, the mapping W r : EC defined by
W r ( x ) = z C : G ( z , y ) + 1 r y - z , J z - J x 0 , y C .

has the same properties as in Lemma 2.4.

Putting u λ , n = W r λ , n y λ , n for all n , we have from Lemmas 2.4 and 2.5 that W r λ , n is relatively nonexpansive.

We divide the proof of Theorem 3.1 into five steps:

Step 1. We first show that C n is closed and convex for every n .

From the definition of ϕ, we may show that
C n + 1 = { z C n : sup λ Λ ϕ ( z , u λ , n ) ϕ ( z , x n ) + ξ n } = λ Λ { z C n : ϕ ( z , u λ , n ) ϕ ( z , x n ) + ξ n } = λ Λ { z C : 2 z , J x n - J u λ , n + u λ , n 2 - x n 2 - ξ n 0 } C n ,

and thus C n is closed and convex for every n .

Step 2. Sequence {x n } is bounded.

In fact, since x n = Π C n x , for any p F, from Lemma 2.1, we have
ϕ ( x n , x ) = ϕ ( Π C n x , x ) ϕ ( p , x ) - ϕ ( p , x n ) ϕ ( p , x ) .

This implies that the sequence {ϕ (x n , x)} is bounded, and so {x n } is bounded.

Step 3. Next we show that C n for each n .

For any u , since W r λ , n is relatively nonexpansive, {T λ }, λ Λ is uniformly total quasi-ϕ-asymptotically nonexpansive and E* is uniformly convex, it follows from Lemma 2.6 that
ϕ ( u , u λ , n ) = ϕ ( u , W r λ , n y λ , n ) ϕ ( u , y λ , n ) = ϕ ( u , J - 1 ( α n J x n + ( 1 - α n ) J T λ n x n ) = | | u | | 2 - 2 u , α n J x n + ( 1 - α n ) J T λ n x n + | | α n J x n + ( 1 - α n ) J T λ n x n | | 2 | | u | | 2 - 2 α n u , J x n - 2 ( 1 - α n ) u , J T λ n x n + α n | | J x n | | 2 + ( 1 - α n ) | | J T λ n x n | | 2 - α n ( 1 - α n ) g ( | | J x n - J T λ n x n | | ) = α n ϕ ( u , x n ) + ( 1 - α n ) ϕ ( u , T λ n x n ) - α n ( 1 - α n ) g ( | | J x n - J T λ n x n | | ) α n ϕ ( u , x n ) + ( 1 - α n ) [ ϕ ( u , x n ) + v n ζ ( ϕ ( u , x n ) )  +  μ n ] - α n ( 1 - α n ) g ( | | J x n J T λ n x n | | ) = ϕ ( u , x n ) + ( 1 - α n ) [ v n ζ ( ϕ ( u , x n ) )  +  μ n ] - α n ( 1 - α n ) g ( | | J x n J T λ n x n | | ) ϕ ( u , x n ) + ξ n - α n ( 1 - α n ) g ( | | J x n - J T λ n x n | | ) ϕ ( u , x n ) + ξ n .
(3.3)
This shows that {y λ, n } is bounded and supλΛϕ (u, u λ, n ) ≤ ϕ(u, x n ) + ξ n , i.e., u C n , this implies that
C n , n .

Step 4. Now we prove that the limit lim n→∞ x n exists.

Since is nonempty, C n is a nonempty closed convex subset of E and thus Π C n exists for every n . Hence {x n } is well defined. Also, since {C n } is a decreasing sequence of closed convex subsets of C such that C 0 = n = 1 C n is nonempty. Therefore, we have
M - lim n C n = C 0 = n = 1 C n .
By Theorem 2.2, {x n } = { Π C n x } converges strongly to x* = Π C 0 x . Therefore, we have
x n + 1 - x n 0 .
(3.4)
Step 5. Next we prove x * .
  1. (a)

    First, we prove x*λ Λ F(T λ ).

     
In fact, since x n x*, we have
ϕ ( x n + 1 , x n ) 0 .
(3.5)
In view of xn+1 Cn+1, from the definition of Cn+1, we have
sup λ Λ ϕ ( x n + 1 , u λ , n ) ϕ ( x n + 1 , x n ) + ξ n .
From (3.5) and ξ n → 0, we have
sup λ Λ ϕ ( x n + 1 , u λ , n ) 0 .
From (2.2) it yields supλΛ(||xn+1|| - ||u λ, n ||)2 0. Since ||xn+1|| ||x*||, we have
u λ , n x * ( n ) , λ Λ
(3.6)
Hence we have
| | J u λ , n | | | | J x * | | n , λ Λ
(3.7)
This implies that {Ju λ, n } is uniformly bounded in E*. Since E is reflexive, and so is E*. We can assume that Ju λ, n f0 E*. In view of the reflexive of E, we see that J(E) = E*. Hence there exists p E such that Jp = f0. Since
ϕ x n + 1 , u λ , n = | | x n + 1 | | 2 - 2 x n + 1 , J u λ , n + | | u λ , n | | 2 = | | x n + 1 | | 2 - 2 x n + 1 , J u λ , n + | | u λ , n | | 2
Taking lim inf n→∞ on the both sides of equality above and in view of the weak lower semicontinuity of norm || · ||, it yields that
0 | | x * | | 2 - 2 x * , f 0 + | | f 0 | | 2 = | | x * | | 2 - 2 x * , J p + | | J p | | 2 = | | x * | | 2 - 2 x * , J p + | | p | | 2 = ϕ x * , p
i.e., x* = p. This implies that f0 = Jx*, and so Ju λ, n Jx*, λ Λ. It follows from (3.7) and the Kadec-Klee property of E* that Ju λ, n → Jx*(n → ∞). Note that J-1: E* → E is hemi-continuous, it yields that u λ, n x*. In view of (3.6) and the Kadec-Klee property of E, we have
lim n u λ , n = x * . λ Λ
(3.8)
From (3.8), we have
lim n | | x n - u λ , n | | = 0 , λ Λ
(3.9)
Since J is uniformly continuous, we have that
| | J x n - J u λ , n | | 0 , λ Λ
(3.10)
From (3.9) and (3.10), we have
ϕ u , x n - ϕ u , u λ , n = | | x n | | 2 - | | u λ , n | | 2 - 2 u , J x n - J u λ , n | | | x n | | 2 - | | u λ , n | | 2 | + 2 | u , J x n - J u λ , n | | | | x n | | - | | u λ , n | | | | | x n | | + | | u λ , n | | + 2 | | u | | | | J x n - J u λ , n | | | | x n - u λ , n | | | | x n | | + | | u λ , n | | + 2 | | u | | | | J x n - J u λ , n | | . 0
(3.11)
It follows from (3.3), (3.11) and ξ n 0 that
α n 1 - α n g ( | | J x n - J T λ n x n | | ) ϕ u , x n - ϕ ( u , u λ , n + ξ n 0 .
(3.12)
In view of condition lim inf n→∞ α n (1 - α n ) > 0, we see that
g ( | | J x n - J T λ n x n | | ) 0 a s n .
It follows from the property of g that
| | J x n - J T λ n x n | | 0 a s n .
(3.13)
Since x n → x* and so Jx n → Jx*. From (3.13) we have
J T λ n x n J x * a s n .
Since J-1: E* → E is hemi-continuous, it follows that
T λ n x n x * , λ Λ .
(3.14)
On the other hand, for each λ Λ we have
| | | T λ n x n | | - | | x * | | | = | | | J ( T λ n x n ) | | - | | J x * | | | | | J ( T λ n x n ) - J x * | | 0 a s n .
This together with (3.14) shows that
T λ n x n x * , λ Λ .
(3.15)
Furthermore, by the assumption that for each λ Λ, T λ is uniformly L λ -Lipschitz continuous, hence from (3.4) and (3.15), we have
| | T λ n + 1 x n - T λ n x n | | | | T λ n + 1 x n - T λ n + 1 x n + 1 | | + | | T λ n + 1 x n + 1 - x n + 1 | | + | | x n + 1 - x n | | + | | x n - T λ n x n | | L λ + 1 | | x n + 1 - x n | | + | | T λ n + 1 x n + 1 - x n + 1 | | + | | x n - T λ n x n | | 0
This implies that T λ n + 1 x n x * , i.e., T λ T λ n x n x * . In view of (3.15) and T λ is uniformly Lipschitzian continuous, it yields that T λ x* = x*, λ Λ. This implies that x*λ ΛF(T λ ).
  1. (b)

    Next, we prove x* EP(G).

     
Since
ϕ ( u λ , n , y λ , n ) = ϕ ( W r λ , n y λ , n , y λ , n ) ϕ ( u , y λ , n ) - ϕ ( u , W r λ , n y λ , n ) ϕ u , x n + ξ n - ϕ ( u , W r λ , n y λ , n ) = ϕ u , x n + ξ n - ϕ u , u λ , n .
(3.16)
Hence it follows from (3.11) and (3.16) that
lim n ϕ u λ , n , y λ , n = 0 .
(3.17)
From (2.2) and (3.17) it yields (||u λ, n || - ||y λ, n ||) 2 →0. Since ||u λ, n ||→ ||x*||, we have
| | y λ , n | | | | x * | | n .
(3.18)
Hence we have
| | J y λ , n | | | | J x * | | ( n ) .
(3.19)
This implies that {Jy λ, n } is bounded in E*. Since E is reflexive, and so is E*. we can assume that Jy λ, n g0 E*. In view of the reflexive of E, we see that J(E) = E*. Hence there exists y E such that Jy = g0. Since
ϕ u λ , n , y λ , n = | | u λ , n | | 2 - 2 u λ , n , J y λ , n + | | y λ , n | | 2 = | | u λ , n | | 2 - 2 u λ , n , J y λ , n + | | J y λ , n | | 2
Taking lim inf n→∞ on the both sides of equality above and in view of the weak lower semicontinuity of norm || · ||, it yields that
0 | | x * | | 2 - 2 x * , g 0 + | | g 0 | | 2 = | | x * | | 2 - 2 x * , J y + | | J y | | 2 = | | x * | | 2 - 2 x * , J y + | | y | | 2 = ϕ x * , y
i.e., x* = y. This implies that g0 = Jx*, and so Jy λ, n Jx*. It follows from (3.19) and the Kadec-Klee property of E* that Jy n (λ) → Jx*(n). Note that J-1: E* → E is hemicontinuous, it yields that y λ, n x*. It follows from (3.18) and the Kadec-Klee property of E that
lim n y λ , n = x * .
(3.20)
Since u λ, n x*, from (3.20), we have
lim n | | u λ , n - y λ , n | | = 0 .
(3.21)
Since J is uniformly norm-to-norm continuous on bounded sets, from (3.21), we have
lim n | | J u λ , n - J y λ , n | | = 0 .
(3.22)
From r λ, n ≥ a, we have
lim n | | J u λ , n - J y λ , n | | r λ , n = 0 .
(3.23)
By u λ , n = W r λ , n y λ , n , we have
G ( u λ , n , y ) + 1 r λ , n y - u λ , n , J u λ , n - J y λ , n 0 , y C .
(3.24)
From condition (A2), we have
1 r λ , n y - u λ , n , J u λ , n - J y λ , n - G ( u λ , n , y ) G ( y , u λ , n ) , y C .
(3.25)
Since G(x,·) is convex and lower semicontinuous, it is also weakly lower semicontinuous, letting n, we have from (3.25) and (A4) that
G ( y , x * ) 0 , y C .
(3.26)
For any t with 0 < t ≤ 1 and y C, let y t = ty + (1 - t)x*. Since y C and hence G(y t , x*) ≤ 0, from conditions (A1) and (A4), we have
0 = G ( y t , y t ) t G ( y t , y ) + ( 1 - t ) G ( y t , x * ) t G ( y t , y )

This implies that G(y t , y) ≥ 0. Hence from condition (A3), we have G(x*, y) ≥ 0 for all y C, and hence x* EP (G).

This completes the proof of Theorem 3.1.

The proof of Theorem 3.1 shows that the properties of generalized projections used in the iterative scheme do not interact with the properties of mappings {T λ }.

Theorem 3.2. Let E be a uniformly smooth and strictly convex Banach space with Kadec-Klee property and C be a nonempty closed convex subset of E. A: CE* be a continuous and monotone mapping, ψ: C be a lower semi-continuous and convex function and F be a bifunction from C ×C to which satisfies the conditions (A1)-(A4). Let {T λ }(λ Λ): CC be a family of uniformly L λ -Lipschitzian continuous and uniformly quasi-ϕ-asymptotically nonexpansive mappings such that = λ Λ F ( T λ ) G M E P . Assume that K = sup { | | u | | : u } < . Let {x n } be the sequence generated by x1 = x C, C1 = C and
y λ , n = J - 1 ( α n J x n + ( 1 - α n ) J T λ n x n ) , λ Λ , u λ , n C such that F ( u λ , n , y ) + A u λ , n , y - u λ , n + ψ ( y ) - ψ ( u λ , n ) + 1 r λ , n y - u λ , n , J u λ , n - J y λ , n 0 , y C , λ Λ C n + 1 = z C n : sup λ Λ ϕ ( z , u λ , n ) ϕ ( z , x n ) + ξ n , x n + 1 = Π C n + 1 x , n 0 .
(3.27)

where ξ n = ( 1 - α n ) ( v n sup u ζ ( ϕ ( u , x n ) ) + μ n ) , {α n } is a sequence in 0[1] such that lim inf n→∞ α n (1 - α n ) > 0, lim inf n→∞ α n < 1 and {r λ, n } [a, ∞) for some a > 0, then {x n } converge strongly to some point x* in F.

Proof. In Theorem 3.1 take ζ(t) = t, ν n = k n - 1, μ n = 0. Therefore the conclusion of Theorem 3.2 can be obtained form Theorem 3.1.

Declarations

Acknowledgements

This study was supported by Scientific Research Fund of Sichuan Provincial Education Department (11ZB146)(2012), the natural Science Foundation of Yibin University (2011Z08) and Scientific Research Fund of Science Technology Department of Sichuan Province (2011JYZ011).

Authors’ Affiliations

(1)
Department of Mathematics, Yibin university
(2)
Department of Mathematics, College of Statistics and Mathematics, Yunnan University of Finance and Economics

References

  1. Ceng LC, Yao JC: A hybrid iterative scheme for mixed equilibrium problems and fixed point problems. J Comput Appl Math 2008, 214: 186–201.MathSciNetView ArticleGoogle Scholar
  2. Takahashi S, Takahashi W: Strong convergence theorem for a generalized equilibrium problem and a nonexpansive mapping in a Hilbert space. Nonlinear Anal 2008, 69(3):1025–1033.MathSciNetView ArticleGoogle Scholar
  3. Tada A, Takahashi W: Strong convergence theorem for an equilibrium problem and a nonexpansive mapping. JOptim Theory Appl 2007, 133: 359–370.MathSciNetView ArticleGoogle Scholar
  4. Takahashi S, Takahashi W: Viscosity approximation methods for equilibrium problem and fixed point problems in Hilbert spaces. J Math Anal Appl 2007, 331: 506–515.MathSciNetView ArticleGoogle Scholar
  5. Kimura Y, Takahashi W: On a hybrid method for a family of relatively nonexpansive mappings in a Banach space. J Math Anal Appl 2009, 357: 356–363.MathSciNetView ArticleGoogle Scholar
  6. Takahashi W: Nonlinear Functional Analysis. Yokohama Publishers, Yokohama; 2000.Google Scholar
  7. Liu M, Chang SS, Zuo P: On a hybrid method for generalized mixed equilibrium problem and fixed point problem of a family of quasi- φ -asymptotically nonexpansive mappings in Banach space. Fixed Point Theory and Applications 2010, 2010: 18. Article ID 157278MathSciNetGoogle Scholar
  8. Liu M, Chang SS, Zuo P: Shrinking projection method of common solutions for generalized equilibrium quasi- φ -nonexpansive mapping and relatively nonexpansive mapping. J Inequal Appl 2010, 2010: 15. Article ID 101690Google Scholar
  9. Chang SS: The generalized mixed equilibrium problem in Banach. Appl Math Mech 2009, 30: 1105–1112.View ArticleGoogle Scholar
  10. Su YF, Zhang HK: Strong convergence theorems for two countable families of weak relatively nonexpansive mappings and applications. Nonlinear Anal 2010, 73: 3890–3906.MathSciNetView ArticleGoogle Scholar
  11. Zegeye H, Shahzad N: Strong convergence theorems for monotone mappings and relatively weak nonexpansive mappings. Nonlinear Anal Theory Methods Appl 2009, 70(7):2707–2716.MathSciNetView ArticleGoogle Scholar
  12. Zhang HC, Su YF: Strong convergence of modified hybrid algorithm for quasi- φ -asymptotically nonexpansive mappings. Commun Korean Math Soc 2009, 24(4):539–551.MathSciNetView ArticleGoogle Scholar
  13. Qin XL, Cho YS, Kang SM: On hybrid projection methods for asymptotically quasi- φ -nonexpansive mappings. Appl Math Comput 2010, 215(11):3874–3883.MathSciNetView ArticleGoogle Scholar
  14. Goebel K, Reich S: Uniform convexity, hyperbolic geometry, and nonexpansive mappings. In Monogr Textbooks Pure Appl Math. Volume 83. Marcel Dekker Inc., New York; 1984.Google Scholar
  15. Mosco U: Convergence of convex sets and of solutions of variational inequalities. Adv Math 1969, 3: 510–585.MathSciNetView ArticleGoogle Scholar
  16. Beer G: Topologies on Closed and Closed Convex Sets. Kluwer Academic Publishers Group, Dordrecht; 1993.View ArticleGoogle Scholar
  17. Alber YI: Metric and generalized projection operators in Banach spaces: properties and applications. In Theory and Applications of Nonlinear Operators of Accretive and Monotone Type. Volume 178. Edited by: Kartsators AG. Dekker, New York, NY, USA; 1996:15–50. of Lecture Notes in Pure and Applied MathematicsGoogle Scholar
  18. Kamimura S, Takahashi W: Strong convergence of a proximal-type algorithm in a Banach space. SLAM J Optim 2002, 13(3):938–945.MathSciNetView ArticleGoogle Scholar
  19. Tsukada M: Convergence of best approximations in a smooth Banach space. J Approx Theory 1984, 40: 301–309.MathSciNetView ArticleGoogle Scholar
  20. Blum E, Oettli W: From optimization and varitional inequalities to equilibrium problems. Math Student 1994, 63: 123–145.MathSciNetGoogle Scholar
  21. Takahashi W, Zembayashi K: Strong convergence theorems by a new hybrid method for equilibrium problems and relatively nonexpansive mappings. Fixed Point Theory and Applications 2008, 2008: 11. Article ID 528476. doi:10.1155/2008/528476MathSciNetView ArticleGoogle Scholar
  22. Xu HK: Inequalities in Banach spaces with applications. Nonlinear Anal: Theory Methods Appl 1991, 16(12):1127–1138.View ArticleGoogle Scholar

Copyright

© Zuo et al; licensee Springer. 2012

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.