Viscosity Approximation Methods with Errors and Strong Convergence Theorems for a Common Point of Pseudocontractive and Monotone Mappings: Solutions of Variational Inequality Problems

We introduce two proximal iterative algorithms with errors which converge strongly to the common solution of certain variational inequality problems for a finite family of pseudocontractive mappings and a finite family of monotone mappings. The strong convergence theorems are obtained under some mild conditions. Our theorems extend and unify some of the results that have been proposed for this class of nonlinear mappings.


Introduction
In many problems, for example, convex optimization, linear programming, monotone inclusions, elliptic differential equations, and variational inequalities, it is quite often to seek a proximal point of a given nonlinear problem. The proximal point algorithm is recognized as a powerful and successful algorithm in finding a common point of the fixed points of pseudocontractive mappings and the solutions of monotone mappings. Let be a closed convex subset of a real Hilbert space with inner product ⟨⋅, ⋅⟩ and norm ‖ ⋅ ‖. We recall that a mapping : → is called monotone if and only if ⟨ − , − ⟩ ≥ 0, ∀ , ∈ .
A mapping : → is called -inverse strongly monotone if there exists a positive real number > 0 such that Obviously, the class of monotone mappings includes the class of the -inverse strongly monotone mappings. The class of monotone mappings is one of the most important classes of mappings among nonlinear mappings. The classical variational inequality problem is formulated as finding a point ∈ such that ⟨V − , ⟩ ≥ 0, for all V ∈ . The set of solutions of variational inequality problems is denoted by VI( , ).
A mapping : → is called pseudocontractive if, for all , ∈ , we have A mapping : → is called -strict pseudocontractive if there exists a constant 0 ≤ ≤ 1 such that A mapping : → is called nonexpansive if − ≤ − , ∀ , ∈ .
For finding an element of the set of fixed points of the nonexpansive mappings, Halpern [1] was the first to study the convergence of the scheme in 1967: In 2000, Moudafi [2] introduced the viscosity approximation methods and proved the strong convergence of the following iterative algorithm under some suitable conditions: Takahashi et al. [19,20] introduced the following scheme and studied the weak and strong convergence theorems of the elements of ( ) ∩ VI( , ), respectively, under different conditions: where is a nonexpansive mapping and is an -inverse strong monotone operator. Recently, Zegeye and Shahzad [21] introduced the mappings as follows: Very recently, Tang [22] introduced the following sequence and obtained the strong convergence theorems: For other related results, see [11][12][13][23][24][25]. On the other side, there are perturbations always occurring in the iterative processes because the manipulations are inaccurate. There is no doubt that researching the convergent problems of iterative methods with perturbation members is a significant job. Starting from any initial guess, 0 ∈ , the proximal point algorithm generates a sequence { } according to the inclusion: where is a maximal monotone operator and > 0 is a parameter. For solving the original problem of finding a solution to the inclusion 0 ∈ , Rockafellar [23] introduced the following algorithm: where { } is a sequence of errors. Rockafellar [23] obtained the weak convergence of the algorithm. Very recently Yao and Shahzad [24] proved that sequences generated from the method of resolvent are given by where { } is a sequence in [0, 1], the sequence { } ⊂ is a small perturbation, and is a nonexpansive mapping.
The following is our concern now: Is it possible to construct a new sequence with general errors that converges strongly to a common element of fixed points of pseudocontractive mappings and the solution set of monotone mappings and converges strongly to the unique solution of certain variational inequality?
In this paper, motivated and inspired by the above results, we introduce two iterations with perturbations which converge strongly to a common element of the set of fixed points of a finite family of pseudocontractive mappings more general than nonexpansive mappings and the solution set of a finite family of monotone mappings more general thaninverse strongly monotone mappings or maximal monotone mappings. Our theorems presented in this paper improve and extend the corresponding results of Yao and Shahzad [24], Zegeye and Shahzad [21], and Tang [22] and some other results in this direction.

Preliminaries
In the sequel, we will use the following lemmas.
Lemma 1 (see [6]). 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 It is well known that is a nonexpansive mapping.
Let +1 = + (1 − ) . Hence we have that Then we have from (46), (45), and (38) that Notice conditions (ii), (iii), and (iv); we have that Hence we have from Lemma 4 that lim sup Therefore we have that Hence we have from (37), (38), and (45) that In addition, since +1 = ( ) + + Σ =1 , , = ( , for all ∈ , we have from 6 Abstract and Applied Analysis the monotonicity of , the nonexpansivity of , and the convexity of ‖ ⋅ ‖ 2 that (52) So we have that Since → 0, → 0, we have from (50) that In a similar way, we have that Consequently, we have that Since the sequence { } is bounded, there exists a subsequence { } of { } and ∈ such that → weakly. And because → V , , V , → weakly. Next we show that ∈ .
By using the weakly lower semicontinuity of the norm on , we get that which implies that Thus, from Lemma 1, we have that Putting := in (68) and := * in (69), we get that Adding (70) and (71) we get that ⟨ * − , * − ⟩ ≤ 0; that is, ‖ − * ‖ 2 ≤ 0; thus = * . Furthermore, from (67), we get that the sequence → = ( ) strongly and is the solution of the following variational inequality: Now we show that is the unique solution of the variational inequality ⟨ − , ( − ) ⟩ ≤ 0, for all ∈ . Suppose that ∈ is another solution of the variational inequality; that is, Let := in (72) and let := in (73); we have that Adding (74) and (75), we have that Hence Because ∈ (0, 1), we conclude that = ; the uniqueness of the solution is obtained. The proof is complete. = Ø, and let : → be a contraction with a contraction coefficient ∈ (0, 1). and are defined as (21) and (22), respectively. Let { } be a sequence generated by 0 ∈ , (iii) 0 < lim inf → ∞ < lim sup → ∞ < 1; For ≥ 0, because and are nonexpansive and is contractive, we have from (28) that This implies that Notice condition (iv); therefore, { } is bounded. Consequently, we get that { }, { } and { }, { ( )}, and ( ; then we get that Repeating equations from (29) to (38), we have that Therefore, Similar to the rest of the proof of Theorem 6, we obtain the result.
Then the sequence { } converges strongly to an element = Π and also is the unique solution of the variational inequality ⟨ − , − ⟩ ≤ 0, ∀ ∈ . (89) The conclusions of Theorems 6 and 7 are true under the same conditions.
Remark 11. Our theorems extend and unify some of the results that have been proved for these important classes of nonlinear operators. In particular, Theorem 6 extends Theorem 6 of Yao and Shahzad [24] in the sense that our convergence is for the more general class of continuous pseudocontractive and continuous monotone mappings. Theorem 6 also extends Theorem 3.2 of Tang [22] in the sense that our convergence is for the more general algorithm with perturbations.

Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.