Abstract

Suppose that is a nonempty closed convex subset of a real reflexive Banach space which has a uniformly Gateaux differentiable norm. A viscosity iterative process is constructed in this paper. A strong convergence theorem is proved for a common element of the set of fixed points of a finite family of pseudocontractive mappings and the set of solutions of a finite family of monotone mappings. And the common element is the unique solution of certain variational inequality. The results presented in this paper extend most of the results that have been proposed for this class of nonlinear mappings.

1. Introduction

Let be a real Banach space with dual . A normalized duality mapping is defined by where denotes the generalized duality pairing between and . It is well known that is smooth if and only if is single valued, and if is uniformly smooth, then is uniformly continuous on bounded subsets of . Moreover, if is reflexive and strictly convex Banach space with a strictly convex dual then is single valued, one-to-one surjective, and it is the duality mapping from into , and then and .

A mapping is said to be monotone if, for each , the following inequality holds:

A mapping is said to be strongly monotone, if there exists a positive real number such that for all

Obviously, the class of monotone mappings includes the class of the -inverse strongly monotone mappings.

Let be closed convex subset of Banach space . A mapping is said to be pseudocontractive if, for any , there exists such that

A mapping is said to be -strictly pseudo-contractive, if, for any , there exist and a constant such that

A mapping is called nonexpansive if

Clearly, the class of pseudo-contractive mappings includes the class of strict pseudo-contractive mappings and non-expansive mappings. We denote by the set of fixed points of ; that is, .

A mapping is called contractive with a contraction coefficient if there exists a constant such that

Let be a real Banach space with dual . The norm on is said to be uniformly Gateaux differentiable if for each the limit exists uniformly for .

For finding an element of the set of fixed points of the non-expansive mappings, Halpern [1] was the first to study the convergence of the scheme in 1967:

Viscosity approximation methods are very important because they are applied to convex optimization, linear programming, monotone inclusions, and elliptic differential equations. In Hilbert spaces, many authors have studied the fixed points problems of the fixed points for the non-expansive mappings and monotone mappings by the viscosity approximation methods, and obtained a series of good results (see [2–17]).

In 2000, Moudifi [18] introduced the viscosity approximation methods and proved the strong convergence of the following iterative algorithm in Hilbert spaces under some suitable conditions:

Suppose that is monotone mapping from into . The classical variational inequality problem is formulated as finding a point such that , for all . The set of solutions of variational inequality problems is denoted by .

Takahashi and Toyoda [19, 20] introduced the following scheme in Hilbert spaces and studied the weak and strong convergence theorem of the elements of , respectively, under different conditions: where is non-expansive mapping and is -inverse strong monotone operator.

Recently, Zegeye and Shahzad [21] introduced the following algorithm and obtained the strong convergence theorem but still in Hilbert spaces: where , are nonexpansive mappings.

Our concern now is the following: is it possible to construct a new sequence in Banach spaces which converges strongly to a common element of fixed points of a finite family of pseudocontractive mappings and the solution set of a variational inequality problems for finite family of monotone mappings?

2. Preliminaries

In the sequel, we will use the following lemmas.

Lemma 1 (see, e.g., [5]). Let be a sequence of nonnegative real numbers satisfying the following relation: where is a sequence in (0,1) and is a real sequence such that (i); (ii) or . Then .

Lemma 2 (see, e.g., [10]). Let be a nonempty, closed, and convex subset of uniformly smooth strictly convex real Banach space with dual . Let be a continuous monotone mapping; define mapping as follows: Then the following hold:(i) is well defined and single valued; (ii) is a firmly non-expansive mapping; that is, ;(iii);(iv) is closed and convex.

Lemma 3 (see, e.g., [22]). Let be a nonempty closed convex subset of uniformly smooth strictly convex real Banach space . Let , be a continuous strictly pseudocontractive mapping; define mapping as follows: Then the following hold:(i) is single valued; (ii) is a firmly nonexpansive mapping; that is, ;(iii); (iv) is closed and convex.

Lemma 4 (see, e.g., [23]). Let and be bounded sequence in a Banach space, and let be a sequence in which satisfies the following condition: Suppose Then .

Lemma 5 (see, e.g., [16]). Let be a nonempty closed and convex subset of a real smooth Banach space . A mapping is a generalized projection. Let ; then if and only if

Lemma 6. Let be a real Banach space with dual . is the generalized duality pairing; then, for all ,

3. Main Results

Let be a nonempty closed convex and bounded subset of a smooth, strictly convex and reflexive real Banach space with dual . Let be a finite family of continuous monotone mappings, and let , be a finite family of continuous strictly pseudo-contractive mappings. For the rest of this paper, and are mappings defined as follows: for , , Consider Denote , .

Lemma 7. Let be a nonempty closed convex and bounded subset of a smooth Banach space , and let , be a finite family of non-expansive mappings such that . Suppose that ; then there exists non-expansive mapping such that .

Proof. Let be any sequence of positive real numbers satisfying and set . Since each is non-expansive for any , we have that is well-defined nonexpansive mapping (see, e.g., [20]), and Next, we claim that .
Clearly . Now, we prove that . Let and . Then and non-expansivity of each implies that for each , which implies for each . Therefore, , and hence . The proof is complete.

Theorem 8. Let be a nonempty closed convex subset of a real reflexive Banach space which has a uniformly Gateaux differentiable norm. Let , be a finite family of continuous strictly pseudocontractive mappings, let , be a finite family of continuous monotone mappings such that , and let be a contraction with a contraction coefficient . and are defined as (19) and (20), respectively. Let be a sequence generated by where and , , and are sequences of nonnegative real numbers in , and (i), , , , , ; (ii), ;(iii);(iv), .
Then the sequence converges strongly to an element , and also is the unique solution of the variational inequality

Proof. First we prove that is bounded. Take , because is non-expansive; then we have that For , because and are nonexpansive and is contractive, we have from (25) that
Therefore, is bounded. Consequently, we get that , and , are bounded.
Next, we show that .
Consider
Let , ; by the definition of mapping , we have that
Putting in (28) and leting in (29), we have that
Adding (30), we have that
Since , are monotone mappings, which implies that therefore we have that That is,
Without loss of generality, let be a real number such that ; then we have that where .
Then we have from (35) and (27) that
On the other hand, let , ; we have that
Let in (37), and let in (38); we have that
Adding (39) and because is pseudo-contractive, we have that
Therefore we have
Hence we have that where .
Let . Hence we have that
Hence we have from (43), (42), and (36) that
Noticing the conditions (ii) and (iv), we have that
Hence we have from Lemma 4 that
Therefore we have that
Hence we have from (35), (36), and (42) that
In addition, since , , for all , we have from the monotonicity of , the non-expansivity of , and the convexity of that
So we have that
Since , so we have from (47) that
In a similar way, we have that
Consequently, we have that
Since is a reflexive real Banach space and the sequence is bounded, so there exists the subsequence of and such that . And because , , therefore . Next we show that .
Because , by the definition of mapping , we have that
Let , , for all ; we have that
Because , so , and because are monotone, we have that
Consequently we have that
If , by the continuity of , we have that ; that is, , and then .
Similarly, because , by the definition of mapping , we have that
Let , for all . Because are pseudocontractive mappings, we have that
Because , so ; we have that
Consequently we have that
If , by the continuity of , we have that , for all ; we conclude that ; that is, , and then . Consequently .
Because , we have from Lemma 5 that
Next we show that . Since , from formula (23) and Lemma 6 we have that where , . Let ; according to Lemma 1 and formula (62), we have that ; that is, the sequence converges strongly to .
According to formula (62) we conclude that is the solution of the variational inequality (24). Now we show that is the unique solution of the variational inequality (24). Suppose that is another solution of the variational inequality (24). Because is the solution of the variational inequality (24), that is, , for all . And because that , then we have
On the other hand, to the solution , since , so
Adding (65) and (64), we have that Hence
Because , hence we conclude that , and the uniqueness of the solution is obtained.