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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 504503, 19 pages
http://dx.doi.org/10.1155/2012/504503
Research Article

An Iteration to a Common Point of Solution of Variational Inequality and Fixed Point-Problems in Banach Spaces

1Department of Mathematics, University of Botswana, Private Bag 00704, Gaborone, Botswana
2Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

Received 24 February 2012; Accepted 5 April 2012

Academic Editor: Yonghong Yao

Copyright © 2012 H. Zegeye and N. Shahzad. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We introduce an iterative process which converges strongly to a common point of solution of variational inequality problem for a monotone mapping and fixed point of uniformly Lipschitzian relatively asymptotically nonexpansive mapping in Banach spaces. As a consequence, we provide a scheme that converges strongly to a common zero of finite family of monotone mappings under suitable conditions. Our theorems improve and unify most of the results that have been proved for this important class of nonlinear operators.

1. Introduction

Let 𝐸 be a smooth Banach space. Throughout this paper, we denote by 𝜙𝐸×𝐸 the function defined by 𝜙(𝑦,𝑥)=𝑦22𝑦,𝐽𝑥+𝑥2,for𝑥,𝑦𝐸,(1.1) which was studied by Alber [1], Kamimura and Takahashi [2], and Reich [3], where 𝐽 is the normalized duality mapping from 𝐸 to 2𝐸 defined by 𝑓𝐽𝑥=𝐸𝑥,𝑓=𝑥2=𝑓2,(1.2) where , denotes the duality pairing. It is well known that if 𝐸 is smooth, then 𝐽 is single-valued, and, if 𝐸 has uniformly Gâteaux differentiable norm, then 𝐽 is uniformly continuous on bounded subsets of 𝐸. Moreover, if 𝐸 is a reflexive and strictly convex Banach space with a strictly convex dual, then 𝐽1 is single valued, one-to-one, surjective, and it is the duality mapping from 𝐸 into 𝐸, and thus 𝐽𝐽1=𝐼𝐸 and 𝐽1𝐽=𝐼𝐸 (see [4]).

It is obvious from the definition of the function 𝜙 that ()𝑥𝑦2)𝜙(𝑥,𝑦)(𝑥+𝑦2for𝑥,𝑦𝐸,(1.3) and, in a Hilbert space 𝐻, (1.1) reduces to 𝜙(𝑥,𝑦)=𝑥𝑦2, for 𝑥,𝑦𝐻.

Let 𝐸 be a reflexive, strictly convex, and smooth Banach space, and let 𝐶 be a nonempty closed and convex subset of 𝐸. The generalized projection mapping, introduced by Alber [1], is a mapping Π𝐶𝐸𝐶 that assigns an arbitrary point 𝑥𝐸 to the minimizer, 𝑥, of 𝜙(,𝑥) over 𝐶, that is, Π𝐶𝑥=𝑥, where 𝑥 is the solution to the minimization problem 𝜙𝑥,𝑥=min{𝜙(𝑦,𝑥),𝑦𝐶}.(1.4) Let 𝐸 be a real Banach space with dual 𝐸. A mapping 𝐴𝐷(𝐴)𝐸𝐸 is said to be monotone if, for each 𝑥,𝑦𝐷(𝐴), the following inequality holds: 𝑥𝑦,𝐴𝑥𝐴𝑦0.(1.5)𝐴 is said to be 𝛾-inverse strongly monotone if there exists positive real number 𝛾 such that 𝑥𝑦,𝐴𝑥𝐴𝑦𝛾𝐴𝑥𝐴𝑦2,𝑥,𝑦𝐾.(1.6) If 𝐴 is 𝛾-inverse strongly monotone, then it is Lipschitz continuous with constant 1/𝛾, that is, 𝐴𝑥𝐴𝑦(1/𝛾)𝑥𝑦, for all 𝑥,𝑦𝐷(𝐴), and it is called strongly monotone if there exists 𝑘>0 such that, for all 𝑥,𝑦𝐷(𝐴), 𝑥𝑦,𝐴𝑥𝐴𝑦𝑘𝑥𝑦2.(1.7) Clearly, the class of monotone mappings include the class of strongly monotone and 𝛾-inverse strongly monotone mappings.

Suppose that 𝐴 is monotone mapping from 𝐶 into 𝐸. The variational inequality problem is formulated as finding a point 𝑢𝐶 such that 𝑣𝑢,𝐴𝑢0, for all 𝑣𝐶. The set of solutions of the variational inequality problems is denoted by VI(𝐶,𝐴).

The notion of monotone mappings was introduced by Zarantonello [5], Minty [6], and Kacurovskii [7] in Hilbert spaces. Monotonicity conditions in the context of variational methods for nonlinear operator equations were also used by Vainberg and Kachurovisky [8]. Variational inequalities were initially studied by Stampacchia [9, 10] and ever since have been widely studied in general Banach spaces (see, e.g., [2, 1113]). Such a problem is connected with the convex minimization problem, the complementarity problem, the problem of finding point 𝑢𝐶 satisfying 0𝐴𝑢.

If 𝐸=𝐻, a Hilbert space, one method of solving a point 𝑢VI(𝐶,𝐴) is the projection algorithm which starts with any point 𝑥1=𝑥𝐶 and updates iteratively as 𝑥𝑛+1 according to the formula 𝑥𝑛+1=𝑃𝐶𝑥𝑛𝛼𝑛𝐴𝑥𝑛,forany𝑛1,(1.8) where 𝑃𝐶 is the metric projection from 𝐻 onto 𝐶 and {𝛼𝑛} is a sequence of positive real numbers. In the case that 𝐴 is 𝛾-inverse strongly monotone, Iiduka et al. [14] proved that the sequence {𝑥𝑛} generated by (3.35) converges weakly to some element of VI(𝐶,𝐴).

In the case that 𝐸 is a 2-uniformly convex and uniformly smooth Banach space, Iiduka and Takahashi [15] introduced the following iteration scheme for finding a solution of the variational inequality problem for an inverse strongly monotone operator 𝐴: 𝑥𝑛+1=Π𝐶𝐽1𝐽𝑥𝑛𝛼𝑛𝐴𝑥𝑛,forany𝑛1,(1.9) where Π𝐶 is the generalized projection from 𝐸 onto 𝐶, 𝐽 is the normalized duality mapping from 𝐸 into 𝐸, and {𝛼𝑛} is a sequence of positive real numbers. They proved that the sequence {𝑥𝑛} generated by (1.9) converges weakly to some element of VI(𝐶,𝐴) provided that 𝐴 satisfies 𝐴𝑥𝐴𝑥𝐴𝑝, for 𝑥𝐶 and 𝑝VI(𝐶,𝐴).

It is worth to mention that the convergence is weak convergence.

To obtain strong convergence, when 𝐸=𝐻, a Hilbert space and 𝐴 is 𝛾-inverse strongly monotone; Iiduka et al. [14] studied the following iterative scheme: 𝑥0𝑦𝐶,chosenarbitrary,𝑛=𝑃𝐶𝑥𝑛𝛼𝑛𝐴𝑥𝑛,𝐶𝑛=𝑦𝑧𝐶𝑛𝑥𝑧𝑛,𝑄𝑧𝑛=𝑧𝐶𝑥𝑛𝑧,𝑥0𝑥𝑛,𝑥0𝑛+1=𝑃𝐶𝑛𝑄𝑛𝑥0,𝑛1,for𝑛1,(1.10) where {𝛼𝑛} is a sequence in [0,2𝛾]. They proved that the sequence {𝑥𝑛} generated by (1.10) converges strongly to 𝑃VI(𝐶,𝐴)(𝑥0), where 𝑃VI(𝐶,𝐴) is the metric projection from 𝐻 onto VI(𝐶,𝐴) provided that 𝐴 satisfies 𝐴𝑥𝐴𝑥𝐴𝑝, for 𝑥𝐶 and 𝑝VI(𝐶,𝐴).

In the case that 𝐸 is 2-uniformly convex and uniformly smooth Banach space, Iiduka and Takahashi [11] studied the following iterative scheme for a variational inequality problem for 𝛾-inverse strongly monotone mapping: 𝑥0𝑦𝐾,chosenarbitrary,𝑛=Π𝐶𝐽1𝐽𝑥𝑛𝛼𝑛𝐴𝑥𝑛,𝐶𝑛=𝑧𝐸𝜙𝑧,𝑦𝑛𝜙𝑧,𝑥𝑛,𝑄𝑛=𝑧𝐸𝑥𝑛𝑧,𝐽𝑥0𝐽𝑥𝑛,𝑥0𝑛+1=Π𝐶𝑛𝑄𝑛𝑥0,𝑛1,for𝑛1,(1.11) where Π𝐶𝑛𝑄𝑛 is the generalized projection from 𝐸 onto 𝐶𝑛𝑄𝑛,𝐽 is the normalized duality mapping from 𝐸 into 𝐸, and {𝛼𝑛} is a positive real sequence satisfying certain condition. Then, they proved that the sequence {𝑥𝑛} converges strongly to an element of VI(𝐶,𝐴) provided that VI(𝐶,𝐴) and 𝐴 satisfies 𝐴𝑥𝐴𝑥𝐴𝑝 for all 𝑥𝐶 and 𝑝VI(𝐶,𝐴).

Remark 1.1. We remark that the computation of 𝑥𝑛+1 in Algorithms (1.10) and (1.11) is not simple because of the involvement of computation of 𝐶𝑛+1 from 𝐶𝑛 and 𝑄𝑛 for each 𝑛1.

Let 𝑇 be a mapping from 𝐶 into itself. We denote by 𝐹(𝑇) the fixed points set of 𝑇. A point 𝑝 in 𝐶 is said to be an asymptotic fixed point of 𝑇 (see [3]) if 𝐶 contains a sequence {𝑥𝑛} which converges weakly to 𝑝 such that lim𝑛𝑥𝑛𝑇𝑥𝑛=0. The set of asymptotic fixed points of 𝑇 will be denoted by 𝐹(𝑇). A mapping 𝑇 from 𝐶 into itself is said to be nonexpansive if 𝑇𝑥𝑇𝑦𝑥𝑦 for each 𝑥,𝑦𝐶 and is called relatively nonexpansive if (R1) 𝐹(𝑇); (R2) 𝜙(𝑝,𝑇𝑥)𝜙(𝑝,𝑥) for 𝑥𝐶 and (R3) 𝐹(𝑇)=𝐹(𝑇). 𝑇 is called relatively quasi-nonexpansive if 𝐹(𝑇) and 𝜙(𝑝,𝑇𝑥)𝜙(𝑝,𝑥) for all 𝑥𝐶, and 𝑝𝐹(𝑇).

A mapping 𝑇 from 𝐶 into itself is said to be asymptotically nonexpansive if there exists {𝑘𝑛}[1,) such that 𝑘𝑛1 and 𝑇𝑛𝑥𝑇𝑛𝑦𝑘𝑛𝑥𝑦 for each 𝑥,𝑦𝐶 and is called relatively asymptotically nonexpansive if there exists {𝑘𝑛}[1,) such that (N1) 𝐹(𝑇); (N2) 𝜙(𝑝,𝑇𝑛𝑥)𝑘𝑛𝜙(𝑝,𝑥) for 𝑥𝐶 and 𝑝𝐹(𝑇), and (N3) 𝐹(𝑇)=𝐹(𝑇), where 𝑘𝑛1 as 𝑛. A-self mapping on 𝐶 is called uniformly 𝐿-Lipschitzian if there exists 𝐿>0 such that 𝑇𝑛𝑥𝑇𝑛𝑦𝐿𝑥𝑦 for all 𝑥,𝑦𝐶. 𝑇 is called closed if 𝑥𝑛𝑥 and 𝑇𝑥𝑛𝑦, then 𝑇𝑥=𝑦.

Clearly, we note that the class of relatively nonexpansive mappings is contained in a class of relatively asymptotically nonexpansive mappings but the converse is not true. Now, we give an example of relatively asymptotically nonexpansive mapping which is not relatively nonexpansive.

Example 1.2 (see [16]). Let 𝑋=𝑙𝑝, where 1<𝑝<, and 𝐶={𝑥=(𝑥1,𝑥2,)𝑋;𝑥𝑛0}. Then 𝐶 is closed and convex subset of 𝑋. Note that 𝐶 is not bounded. Obviously, 𝑋 is uniformly convex and uniformly smooth. Let {𝜆𝑛} and{𝜆𝑛} be sequences of real numbers satisfying the following properties:(i)0<𝜆𝑛<1, 𝜆𝑛>1, 𝜆𝑛1, and 𝜆𝑛1,(ii)𝜆𝑛+1𝜆𝑛=1 and 𝜆𝑗+1𝜆𝑛+𝑗<1 for all 𝑛 and 𝑗 (e.g., 𝜆𝑛=11/(𝑛+1), 𝜆𝑛=1+1/(𝑛+1)). Then, the map 𝑇𝐶𝐶 defined by 𝑇𝑥=0,𝜆1||sin𝑥1||,𝜆2𝑥2,𝜆2𝑥3,𝜆3𝑥4,𝜆3𝑥5,,(1.12)for all 𝑥=(𝑥1,𝑥2,)𝐶, is uniformly Lipschitzian which is relatively asymptotically nonexpansive but not relatively nonexpansive (see [16] for the details). Note also that 𝐹(𝑇)={0}.

In 2005, Matsushita and Takahashi [17] proposed the following hybrid iteration method with generalized projection for relatively nonexpansive mapping 𝑇 in a Banach space 𝐸: 𝑥0𝑦𝐶,chosenarbitrary,𝑛=𝐽1𝛼𝑛𝐽𝑥𝑛+1𝛼𝑛𝐽𝑇𝑥𝑛,𝐶𝑛=𝑧𝐶𝜙𝑧,𝑦𝑛𝜙𝑧,𝑥𝑛,𝑄𝑛=𝑧𝐶;𝑥𝑛𝑧,𝐽𝑥0𝐽𝑥𝑛,𝑥0𝑛+1=Π𝐶𝑛𝑄𝑛𝑥0,𝑛1.(1.13) They proved that, if the sequence {𝛼𝑛} is bounded above from one, then the sequence {𝑥𝑛} generated by (1.13) converges strongly to Π𝐹(𝑇)𝑥0.

Recently, many authors have considered the problem of finding a common element of the fixed-point set of relatively nonexpansive mapping and the solution set of variational inequality problem for 𝛾-inverse monotone mapping (see, e.g., [12, 13, 1820]).

In [21], Iiduka and Takahashi studied the following iterative scheme for a common point of solution of a variational inequality problem for 𝛾-inverse strongly monotone mapping 𝐴 and fixed point of nonexpansive mapping 𝑇 in a Hilbert space 𝐻: 𝑥1𝑥=𝑥𝐶,𝑛+1=𝛼𝑛𝑥+1𝛼𝑛𝑆𝑃𝐶𝑥𝑛𝜆𝑛𝐴𝑥𝑛,𝑛1,(1.14) where {𝛼𝑛} is sequences satisfying certain condition. They proved that the sequence {𝑥𝑛} converges strongly to an element of 𝐹=𝐹(𝑆)VI(𝐶,𝐴) provided that 𝐹.

In the case that 𝐸 is a Banach space more general than Hilbert spaces, Zegeye et al. [12] studied the following iterative scheme for a common point of solution of a variational inequality problem for 𝛾-inverse strongly monotone mapping 𝐴 and fixed point of a closed relatively quasi-nonexpansive mapping 𝑇 in a 2-uniformly convex and uniformly smooth Banach space 𝐸: 𝐶1𝑧=𝐶,chosenarbitrary,𝑛=Π𝐶𝑥𝑛𝜆𝑛𝐴𝑥𝑛,𝑦𝑛=𝐽1𝛽𝐽𝑥𝑛+(1𝛽)𝐽𝑇𝑧𝑛,𝐶𝑛+1=𝑧𝐶𝑛𝜙𝑧,𝑦𝑛𝜙𝑧,𝑥𝑛,𝑥𝑛+1=Π𝐶𝑛+1𝑥0,𝑛1,(1.15) where {𝜆𝑛} is sequences satisfying certain condition. They proved that the sequence {𝑥𝑛} converges strongly to an element of 𝐹=𝐹(𝑆)VI(𝐶,𝐴) provided that 𝐹 and 𝐴 satisfies 𝐴𝑥𝐴𝑥𝐴𝑝 for all 𝑥𝐶 and 𝑝𝐹.

Furthermore, Zegeye and Shahzad [22] studied the following iterative scheme for common point of solution of a variational inequality problem for 𝛾-inverse strongly monotone mapping 𝐴 and fixed point of a relatively asymptotically nonexpansive mapping on a closed convex and bounded set 𝐶 which is a subset of a real Hilbert space 𝐻: 𝐶1𝑧=𝐶,chosenarbitrary,𝑛=𝑃𝐶𝑥𝑛𝜆𝑛𝐴𝑥𝑛,𝑦𝑛=𝛼𝑛𝑥𝑛+1𝛼𝑛𝑆𝑛𝑧𝑛,𝐶𝑛+1=𝑧𝐶𝑛𝑧𝑢𝑛2𝑧𝑥𝑛2+𝜃𝑛,𝑥𝑛+1=𝑃𝐶𝑛+1𝑥0,𝑛1,(1.16) where 𝑃𝐶𝑛 is the metric projection from 𝐻 into 𝐶𝑛 and 𝜃𝑛=(1𝛼𝑛)(𝑘2𝑛1)(diam(𝐶))2 and {𝛼𝑛},{𝜆𝑛} are sequences satisfying certain condition. Then, they proved that the sequence {𝑥𝑛} converges strongly to an element of 𝐹=𝐹(𝑆)VI(𝐶,𝐴) provided that 𝐹 and 𝐴 satisfies 𝐴𝑥𝐴𝑥𝐴𝑝 for all 𝑥𝐶 and 𝑝𝐹.

Remark 1.3. We again remark that the computation of 𝑥𝑛+1 in Algorithms (1.13), (1.15), and (1.16) is not simple because of the involvement of computation of 𝐶𝑛+1 from 𝐶𝑛 for each 𝑛1.
It is our purpose in this paper to introduce an iterative scheme {𝑥𝑛} which converges strongly to a common point of solution of variational inequality problem for a monotone operator 𝐴𝐶𝐸 satisfying appropriate conditions, for some nonempty closed convex subset 𝐶 of a Banach space 𝐸 and fixed points of uniformly 𝐿-Lipschitzian relatively asymptotically nonexpansive mapping in Banach spaces. As a consequence, we provide a scheme which converges strongly to a common zero of finite family of monotone mappings. Our scheme does not involve computation of 𝐶𝑛+1 from 𝐶𝑛 or 𝑄𝑛, for each 𝑛1. Our theorems improve and unify most of the results that have been proved for this important class of nonlinear operators.

2. Preliminaries

Let 𝐸 be a normed linear space with dim𝐸2. The modulus of smoothness of 𝐸 is the function 𝜌𝐸[0,)[0,) defined by 𝜌𝐸(𝜏)=sup𝑥+𝑦+𝑥𝑦21𝑥=1;𝑦=𝜏.(2.1) The space 𝐸 is said to be smooth if 𝜌𝐸(𝜏)>0, for all 𝜏>0, and 𝐸 is called uniformly smooth if and only if lim𝑡0+(𝜌𝐸(𝑡)/𝑡)=0.

The modulus of convexity of 𝐸 is the function 𝛿𝐸(0,2][0,1] defined by 𝛿𝐸(𝜖)=inf1𝑥+𝑦2𝑥=𝑦=1;𝜖=𝑥𝑦.(2.2)𝐸 is called uniformly convex if and only if 𝛿𝐸(𝜖)>0, for every 𝜖(0,2].

In the sequel, we will need the following results.

Lemma 2.1 (see [23]). Let 𝐶 be a nonempty closed and convex subset of a real reflexive, strictly convex, and smooth Banach space 𝐸. If 𝐴𝐶𝐸 is continuous monotone mapping, then VI(𝐶,𝐴) is closed and convex.

Lemma 2.2. Let 𝐶 be a closed convex subset of a uniformly convex and smooth Banach space 𝐸, and let 𝑆 be continuous relatively asymptotically nonexpansive mapping from 𝐶 into itself. Then, 𝐹(𝑆) is closed and convex.

Lemma 2.3 (see [1]). Let 𝐾 be a nonempty closed and convex subset of a real reflexive, strictly convex, and smooth Banach space 𝐸, and let 𝑥𝐸. Then, for all 𝑦𝐾, 𝜙𝑦,Π𝐾𝑥Π+𝜙𝐾𝑥,𝑥𝜙(𝑦,𝑥).(2.3)

Lemma 2.4 (see [2]). Let 𝐸 be a real smooth and uniformly convex Banach space, and let {𝑥𝑛} and {𝑦𝑛} be two sequences of 𝐸. If either {𝑥𝑛} or {𝑦𝑛} is bounded and 𝜙(𝑥𝑛,𝑦𝑛)0 as 𝑛, then 𝑥𝑛𝑦𝑛0, as 𝑛.

We make use of the function 𝑉𝐸×𝐸 defined by 𝑉𝑥,𝑥=𝑥22𝑥,𝑥+𝑥2,𝑥𝐸,𝑥𝐸,(2.4) studied by Alber [1]. That is, 𝑉(𝑥,𝑦)=𝜙(𝑥,𝐽1𝑥) for all 𝑥𝐸 and 𝑥𝐸. We know the following lemma.

Lemma 2.5 (see [1]). Let 𝐸 be reflexive strictly convex and smooth Banach space with 𝐸 as its dual. Then, 𝑉𝑥,𝑥𝐽+21𝑥𝑥,𝑦𝑉𝑥,𝑥+𝑦,(2.5) for all 𝑥𝐸 and 𝑥,𝑦𝐸.

Lemma 2.6 (see [1]). Let 𝐶 be a convex subset of a real smooth Banach space 𝐸. Let 𝑥𝐸. Then 𝑥0=Π𝐶𝑥 if and only if 𝑧𝑥0,𝐽𝑥𝐽𝑥00,𝑧𝐶.(2.6)

Lemma 2.7 (see [12]). Let 𝐸 be a uniformly convex Banach space and 𝐵𝑅(0) a closed ball of 𝐸. Then, there exists a continuous strictly increasing convex function 𝑔[0,)[0,) with 𝑔(0)=0 such that 𝛼1𝑥1+𝛼2𝑥2+𝛼3𝑥32𝛼𝑖𝑥12+𝛼2𝑥22+𝛼3𝑥32𝛼𝑖𝛼𝑗𝑔𝑥𝑖𝑥𝑗,(2.7) for 𝛼𝑖(0,1) such that 𝛼1+𝛼2+𝛼3=1 and 𝑥𝑖𝐵𝑅(0)={𝑥𝐸𝑥𝑅}, for 𝑖=1,2,3.

Let 𝐸 be a smooth and strictly convex Banach space, 𝐶 a nonempty closed convex subset of 𝐸, and 𝐴𝐶𝐸 a monotone operator satisfying 𝐷(𝐴)𝐶𝐽1𝑟>0𝑅(𝐽+𝑟𝐴),(2.8) for 𝑟>0. Then, we can define the resolvent 𝑄𝑟𝐶𝐷(𝐴) of 𝐴 by 𝑄𝑟𝑥={𝑧𝐷(𝐴)𝐽𝑥𝐽𝑧+𝑟𝐴𝑧},𝑥𝐶.(2.9) In other words, 𝑄𝑟𝑥=(𝐽+𝑟𝐴)1𝐽𝑥 for 𝑥𝐶. We know that 𝑄𝑟𝑥 is single-valued mapping from 𝐶 into 𝐷(𝐴), for all 𝑥𝐶 and 𝑟>0 and 𝐹(𝑄𝑟)=𝐴1(0), where 𝐹(𝑄𝑟) is the set of fixed points of 𝑄𝑟 (see, [4]).

Lemma 2.8 (see [24]). Let 𝐸 be a smooth and strictly convex Banach space, 𝐶 a nonempty closed convex subset of 𝐸, and 𝐴𝐸×𝐸 a monotone operator satisfying (2.8) and 𝐴1(0) is nonempty. Let 𝑄𝑟 be the resolvent of 𝐴. Then, for each 𝑟>0, 𝜙𝑢,𝑄𝑟𝑥𝑄+𝜙𝑟𝑥,𝑥𝜙(𝑢,𝑥)(2.10) for all 𝑢𝐴1(0) and 𝑥𝐶, that is, 𝑄𝑟 is relatively nonexpansive.

Lemma 2.9 (see [25]). Let {𝑎𝑛} be a sequence of nonnegative real numbers satisfying the following relation: 𝑎𝑛+11𝛽𝑛𝑎𝑛+𝛽𝑛𝛿𝑛,𝑛𝑛0,(2.11) where {𝛽𝑛}(0,1) and {𝛿𝑛}𝑅 satisfying the following conditions: lim𝑛𝛽𝑛=0,𝑛=1𝛽𝑛=, and limsup𝑛𝛿𝑛0. Then, lim𝑛𝑎𝑛=0.

Lemma 2.10 (see [26]). Let {𝑎𝑛} be sequences of real numbers such that there exists a subsequence {𝑛𝑖} of {𝑛} such that 𝑎𝑛𝑖<𝑎𝑛𝑖+1 for all 𝑖𝑁. Then, there exists a nondecreasing sequence {𝑚𝑘}𝑁 such that 𝑚𝑘, and the following properties are satisfied by all (sufficiently large) numbers 𝑘𝑁: 𝑎𝑚𝑘𝑎𝑚𝑘+1,𝑎𝑘𝑎𝑚𝑘+1.(2.12) In fact, 𝑚𝑘=max{𝑗𝑘𝑎𝑗<𝑎𝑗+1}.

3. Main Result

We note that, as it is mentioned in [27], if 𝐶 is a subset of a real Banach space 𝐸 and 𝐴𝐶𝐸 is a mapping satisfying 𝐴𝑥𝐴𝑥𝐴𝑝,for all 𝑥𝐶 and 𝑝VI(𝐶,𝐴), then VI(𝐶,𝐴)=𝐴1(0)={𝑝𝐶𝐴𝑝=0}.(3.1)

In fact, clearly, 𝐴1(0)VI(𝐶,𝐴). Now, we show that VI(𝐶,𝐴)𝐴1(0). Let 𝑝VI(𝐶,𝐴), then we have by hypothesis that 𝐴𝑝𝐴𝑝𝐴𝑝=0 which implies that 𝑝𝐴1(0). Hence, VI(𝐶,𝐴)𝐴1(0). Therefore, VI(𝐶,𝐴)=𝐴1(0). Now we prove the main theorem of our paper.

Theorem 3.1. Let 𝐶 be a nonempty, closed, and convex subset of a uniformly convex and uniformly smooth real Banach space 𝐸. Let 𝐴𝐶𝐸 be a monotone mapping satisfying (2.8) and 𝐴𝑥𝐴𝑥𝐴𝑝,for all 𝑥𝐶 and 𝑝VI(𝐶,𝐴). Let 𝑇𝐶𝐶 be a uniformly 𝐿-Lipschitzian relatively asymptotically nonexpansive mapping with sequence {𝑘𝑛}. Assume that 𝐹=VI(𝐶,𝐴)𝐹(𝑆) is nonempty. Let 𝑄𝑟 be the resolvent of 𝐴 and {𝑥𝑛} a sequence generated by 𝑥0𝑦=𝑤𝐶,chosenarbitrarily,𝑛=Π𝐶𝐽1𝛼𝑛𝐽𝑤+1𝛼𝑛𝐽𝑥𝑛,𝑥𝑛+1=𝐽1𝛽𝑛𝐽𝑥𝑛+𝛾𝑛𝐽𝑇𝑛𝑦𝑛+𝜃𝑛𝐽𝑄𝑟𝑦𝑛,(3.2) where 𝛼𝑛(0,1) such that lim𝑛𝛼𝑛=0,lim𝑛((𝑘𝑛1)/𝛼𝑛)=0,𝑛=1𝛼𝑛=, {𝛽𝑛},{𝛾𝑛},{𝜃𝑛}[𝑐,𝑑](0,1) such that 𝛽𝑛+𝛾𝑛+𝜃𝑛=1. Then, {𝑥𝑛} converges strongly to an element of 𝐹.

Proof. Let 𝑝=Π𝐹𝑤. Then, from (3.2), Lemma 2.3, and property of 𝜙, we get that 𝜙𝑝,𝑦𝑛=𝜙𝑝,Π𝐶𝐽1𝛼𝑛𝐽𝑤+1𝛼𝑛𝐽𝑥𝑛𝜙𝑝,𝐽1𝛼𝑛𝐽𝑤+1𝛼𝑛𝐽𝑥𝑛=𝑝22𝑝,𝛼𝑛𝐽𝑤+1𝛼𝑛𝐽𝑥𝑛+𝛼𝑛𝐽𝑤+1𝛼𝑛𝐽𝑥𝑛2𝑝22𝛼𝑛𝑝,𝐽𝑤21𝛼𝑛𝑝,𝐽𝑥𝑛+𝛼𝑛𝐽𝑤2+1𝛼𝑛𝐽𝑥𝑛2=𝛼𝑛𝜙(𝑝,𝑤)+1𝛼𝑛𝜙𝑝,𝑥𝑛.(3.3) Now, from (3.2) and relatively asymptotically nonexpansiveness of 𝑇, relatively nonexpansiveness of 𝑄𝑟, property of 𝜙, and (3.3), we get that 𝜙𝑝,𝑥𝑛+1=𝜙𝑝,𝐽1𝛽𝑛𝐽𝑥𝑛+𝛾𝑛𝐽𝑇𝑛𝑦𝑛+𝜃𝑛𝐽𝑄𝑟𝑦𝑛𝛽𝑛𝜙𝑝,𝑥𝑛+𝛾𝑛𝜙𝑝,𝑇𝑛𝑦𝑛+𝜃𝑛𝜙𝑝,𝑄𝑟𝑦𝑛𝛽𝑛𝜙𝑝,𝑥𝑛+𝛾𝑛𝑘𝑛𝜙𝑝,𝑦𝑛+𝜃𝑛𝜙𝑝,𝑦𝑛𝛽𝑛𝜙𝑝,𝑥𝑛+𝛾𝑛𝑘𝑛+𝜃𝑛𝜙𝑝,𝑦𝑛𝛽𝑛𝜙𝑝,𝑥𝑛+𝛾𝑛𝑘𝑛+𝜃𝑛𝛼𝑛𝜙(𝑝,𝑤)+1𝛼𝑛𝜙𝑝,𝑥𝑛𝛾𝑛𝑘𝑛+𝜃𝑛𝛼𝑛𝛽𝜙(𝑝,𝑤)+𝑛+𝛾𝑛𝑘𝑛+𝜃𝑛1𝛼𝑛𝜙𝑝,𝑥𝑛𝛾𝑛𝑘𝑛+𝜃𝑛𝛼𝑛𝜙(𝑝,𝑤)+1𝛼𝑛𝛾𝑛𝑘𝑛+𝜃𝑛+𝛾𝑛𝑘𝑛1×𝜙𝑝,𝑥𝑛𝑐𝑛𝜙(𝑝,𝑤)+1(1𝜖)𝑐𝑛𝜙𝑝,𝑥𝑛,(3.4) where 𝑐𝑛=𝛼𝑛(𝛾𝑛𝑘𝑛+𝜃𝑛), since there exists 𝑁0>0 such that 𝛾𝑛(𝑘𝑛1)/𝛼𝑛𝜖(𝛾𝑛𝑘𝑛+𝜃𝑛) for all 𝑛𝑁0 and for some 𝜖>0 satisfying (1𝜖)𝑐𝑛1. Thus, by induction, 𝜙𝑝,𝑥𝑛+1𝜙max𝑝,𝑥0,(1𝜖)1𝜙(𝑝,𝑤),𝑛𝑁0(3.5) which implies that {𝑥𝑛}, and hence {𝑦𝑛} is bounded. Now, let 𝑧𝑛=𝐽1(𝛼𝑛𝐽𝑤+(1𝛼𝑛)𝐽𝑥𝑛). Then we have that 𝑦𝑛=Π𝐶𝑧𝑛. Using Lemmas 2.3, 2.5, and property of 𝜙, we obtain that 𝜙𝑝,𝑦𝑛𝜙𝑝,𝑧𝑛=𝑉𝑝,𝐽𝑧𝑛𝑉𝑝,𝐽𝑧𝑛𝛼𝑛(𝐽𝑤𝐽𝑝)2𝑧𝑛𝑝,𝛼𝑛(𝐽𝑤𝐽𝑝)=𝜙𝑝,𝐽1𝛼𝑛𝐽𝑝+1𝛼𝑛𝐽𝑤𝑛+2𝛼𝑛𝑧𝑛𝑝,𝐽𝑤𝐽𝑝𝛼𝑛𝜙(𝑝,𝑝)+1𝛼𝑛𝜙𝑝,𝑤𝑛+2𝛼𝑛𝑧𝑛=𝑝,𝐽𝑤𝐽𝑝1𝛼𝑛𝜙𝑝,𝑤𝑛+2𝛼𝑛𝑧𝑛𝑝,𝐽𝑤𝐽𝑝1𝛼𝑛𝜙𝑝,𝑥𝑛+2𝛼𝑛𝑧𝑛𝑝,𝐽𝑤𝐽𝑝.(3.6) Furthermore, from (3.2), Lemma 2.7, relatively asymptotically nonexpansiveness of 𝑇, relatively nonexpansiveness of 𝑄𝑟, and (3.6), we have that 𝜙𝑝,𝑥𝑛+1=𝜙𝑝,𝐽1𝛽𝑛𝐽𝑥𝑛+𝛾𝑛𝐽𝑇𝑛𝑦𝑛+𝜃𝑛𝐽𝑄𝑟𝑦𝑛=𝑝2𝑝,𝛽𝑛𝐽𝑥𝑛+𝛾𝑛𝐽𝑇𝑛𝑦𝑛+𝜃𝑛𝐽𝑄𝑟𝑦𝑛𝛽+𝑛𝐽𝑥𝑛+𝛾𝑛𝐽𝑇𝑛𝑦𝑛+𝜃𝑛𝐽𝑄𝑟𝑦𝑛2(3.7)𝑝22𝛽𝑛𝑝,𝐽𝑥𝑛2𝛾𝑛𝑝,𝐽𝑇𝑛𝑦𝑛2𝜃𝑛𝑝,𝐽𝑄𝑟𝑦𝑛+𝛽𝑛𝐽𝑥𝑛2+𝛾𝑛𝐽𝑇𝑛𝑦𝑛2+𝜃𝑛𝐽𝑄𝑟𝑦𝑛2𝛾𝑛𝛽𝑛𝑔𝐽𝑥𝑛𝐽𝑇𝑛𝑦𝑛𝛽𝑛𝜙𝑝,𝑥𝑛+𝛾𝑛𝜙𝑝,𝑇𝑛𝑦𝑛+𝜃𝑛𝜙𝑝,𝑄𝑟𝑦𝑛𝛾𝑛𝛽𝑛𝑔𝐽𝑥𝑛𝐽𝑇𝑛𝑦𝑛𝛽𝑛𝜙𝑝,𝑥𝑛+𝛾𝑛+𝜃𝑛𝜙𝑝,𝑦𝑛+𝛾𝑛𝑘𝑛𝜙1𝑝,𝑦𝑛𝛾𝑛𝛽𝑛𝑔𝐽𝑥𝑛𝐽𝑇𝑛𝑦𝑛𝛽𝑛𝜙𝑝,𝑥𝑛+𝛾𝑛+𝜃𝑛1𝛼𝑛𝜙𝑝,𝑥𝑛+2𝛼𝑛𝑧𝑛𝑝,𝐽𝑤𝐽𝑝+𝛾𝑛𝑘𝑛𝜙1𝑝,𝑦𝑛𝛾𝑛𝛽𝑛𝑔𝐽𝑥𝑛𝐽𝑇𝑛𝑦𝑛𝛽𝑛+𝛾𝑛+𝜃𝑛1𝛼𝑛𝜙𝑝,𝑥𝑛+2𝛼𝑛𝛾𝑛+𝜃𝑛𝑧𝑛𝑝,𝐽𝑤𝐽𝑝+𝛾𝑛𝑘𝑛𝜙1𝑝,𝑦𝑛𝛾𝑛𝛽𝑛𝑔𝐽𝑥𝑛𝐽𝑇𝑛𝑦𝑛1𝛿𝑛𝜙𝑝,𝑥𝑛+2𝛿𝑛𝑧𝑛𝑘𝑝,𝐽𝑤𝐽𝑝+𝑛1𝑀𝛾𝑛𝛽𝑛𝑔𝐽𝑥𝑛𝐽𝑇𝑛𝑦𝑛(3.8)1𝛿𝑛𝜙𝑝,𝑥𝑛+2𝛿𝑛𝑧𝑛𝑘𝑝,𝐽𝑤𝐽𝑝+𝑛1𝑀,(3.9) for some 𝑀>0, where 𝛿𝑛=(𝛾𝑛+𝜃𝑛)𝛼𝑛.
Similarly, from (3.7), we obtain that 𝜙𝑝,𝑥𝑛+1=𝜙𝑝,𝐽1𝛽𝑛𝐽𝑥𝑛+𝛾𝑛𝐽𝑇𝑛𝑦𝑛+𝜃𝑛𝐽𝑄𝑟𝑦𝑛1𝛿𝑛𝜙𝑝,𝑥𝑛+2𝛿𝑧𝑛𝑘𝑝,𝐽𝑤𝐽𝑝+𝑛1𝑀𝜃𝑛𝛽𝑛𝑔𝐽𝑥𝑛𝐽𝑄𝑟𝑦𝑛1𝛿𝑛𝜙𝑝,𝑥𝑛+2𝛿𝑛𝑧𝑛𝑘𝑝,𝐽𝑤𝐽𝑝+𝑛1𝑀,() for some 𝑀>0. Note that {𝛿𝑛} satisfies that lim𝑛𝛿𝑛=0 and 𝛿𝑛=.
Now, the rest of the proof is divided into two parts.
Case 1. Suppose that there exists 𝑛0𝑁>𝑁0 such that {𝜙(𝑝,𝑥𝑛)} is nonincreasing for all 𝑛𝑛0. In this situation, {𝜙(𝑝,𝑥𝑛)} is then convergent. Then, from (3.8) and (*), we have that 𝛾𝑛𝛽𝑛𝑔𝐽𝑥𝑛𝐽𝑇𝑛𝑦𝑛0,𝜃𝑛𝛽𝑛𝑔𝐽𝑥𝑛𝐽𝑄𝑟𝑦𝑛0,(3.10) which implies, by the property of 𝑔, that 𝐽𝑥𝑛𝐽𝑇𝑛𝑦𝑛0,𝐽𝑥𝑛𝐽𝑄𝑟𝑦𝑛0,as𝑛,(3.11) and, hence, since 𝐽1 is uniformly continuous on bounded sets, we obtain that 𝑥𝑛𝑇𝑛𝑦𝑛0,𝑥𝑛𝑄𝑟𝑦𝑛0,as𝑛.(3.12) Furthermore, Lemma 2.3, property of 𝜙, and the fact that 𝛼𝑛0 as 𝑛 imply that 𝜙𝑥𝑛,𝑦𝑛𝑥=𝜙𝑛,Π𝐶𝑧𝑛𝑥𝜙𝑛,𝑧𝑛=𝜙(𝑥𝑛,𝐽1𝛼𝑛𝐽𝑤+1𝛼𝑛𝐽𝑥𝑛𝛼𝑛𝜙𝑥𝑛+,𝑤1𝛼𝑛𝜙𝑥𝑛,𝑥𝑛𝛼𝑛𝜙𝑥𝑛+,𝑤1𝛼𝑛𝜙𝑥𝑛,𝑥𝑛0,as𝑛,(3.13) and hence 𝑥𝑛𝑦𝑛0,𝑥𝑛𝑧𝑛0,as𝑛.(3.14) Therefore, from (3.12) and (3.14), we obtain that 𝑦𝑛𝑧𝑛0,𝑦𝑛𝑇𝑛𝑦𝑛0,𝑦𝑛𝑄𝑟𝑦𝑛0,as𝑛.(3.15) But observe that from (3.2) and (3.11), we have 𝐽𝑥𝑛+1𝐽𝑥𝑛𝛾𝑛𝐽𝑇𝑛𝑦𝑛𝐽𝑥𝑛+𝜃𝑛𝐽𝑄𝑟𝑦𝑛𝐽𝑥𝑛0,(3.16) as 𝑛. Thus, as 𝐽1 is uniformly continuous on bounded sets, we have that 𝑥𝑛+1𝑥𝑛0 which implies from (3.14) that 𝑥𝑛+1𝑦𝑛0, as 𝑛, and that 𝑦𝑛+1𝑦𝑛𝑦𝑛+1𝑥𝑛+1+𝑥𝑛+1𝑦𝑛0,as𝑛.(3.17) Furthermore, since 𝑦𝑛𝑇𝑦𝑛𝑦𝑛𝑇𝑛𝑦𝑛+𝑇𝑛𝑦𝑛𝑇𝑛+1𝑦𝑛+𝑇𝑛+1𝑦𝑛𝑇𝑦𝑛,𝑦𝑛𝑇𝑛𝑦𝑛+𝑇𝑛+1𝑦𝑛𝑇𝑛+1𝑦𝑛+1+𝑇𝑛+1𝑦𝑛+1𝑦𝑛+1+𝑦𝑛+1𝑦𝑛+𝑦𝑛𝑇𝑛𝑦𝑛+𝑇𝑇𝑛𝑦𝑛𝑇𝑦𝑛𝑦𝑛𝑇𝑛𝑦𝑛𝑦+𝐿𝑛𝑦𝑛+1+𝑇𝑛+1𝑦𝑛+1𝑦𝑛+1+𝑦𝑛+1𝑦𝑛+𝑦𝑛𝑇𝑛𝑦𝑛+𝑇𝑇𝑛𝑦𝑛𝑇𝑦𝑛𝑦2𝑛𝑇𝑛𝑦𝑛𝑦+(1+𝐿)𝑛𝑦𝑛+1+𝑇𝑛+1𝑦𝑛+1𝑦𝑛+1+𝑇𝑇𝑛𝑦𝑛𝑇𝑦𝑛,(3.18) we have from (3.17), (3.15), and uniform continuity of 𝑇 that 𝑦𝑛𝑇𝑦𝑛0,as𝑛.(3.19) Since {𝑧𝑛} is bounded and 𝐸 is reflexive, we choose a subsequence {𝑧𝑛𝑖} of {𝑧𝑛} such that 𝑧𝑛𝑖𝑧 and limsup𝑛𝑧𝑛𝑝,𝐽𝑤𝐽𝑝=lim𝑖𝑧𝑛𝑖𝑝,𝐽𝑤𝐽𝑝. Then, from (3.14) and (3.15) we get that 𝑦𝑛𝑖𝑧,𝑥𝑛𝑖𝑧,as𝑖.(3.20) Thus, since 𝑇 satisfies condition (N3), we obtain from (3.19) that 𝑧𝐹(𝑇) and the fact that 𝑄𝑟 is relatively nonexpansive and 𝑦𝑛𝑖𝑧 implies that 𝑧𝐹(𝑄𝑟)=𝐴1(0), and, hence, using (3.1), we obtain that 𝑧VI(𝐶,𝐴).
Therefore, from the above discussions, we obtain that 𝑧𝐹=𝐹(𝑇)VI(𝐶,𝐴). Hence, by Lemma 2.6, we immediately obtain that limsup𝑛𝑧𝑛𝑝,𝐽𝑤𝐽𝑝=lim𝑖𝑧𝑛𝑖𝑝,𝐽𝑤𝐽𝑝=𝑧𝑝,𝐽𝑤𝐽𝑝0. It follows from Lemma 2.9 and (3.9) that 𝜙(𝑝,𝑥𝑛)0, as 𝑛. Consequently, 𝑥𝑛𝑝.
Case 2. Suppose that there exists a subsequence {𝑛𝑖} of {𝑛} such that 𝜙𝑝,𝑥𝑛𝑖<𝜙𝑝,𝑥𝑛𝑖+1(3.21) for all 𝑖𝑁. Then, by Lemma 2.10, there exists a nondecreasing sequence {𝑚𝑘}𝑁 such that 𝑚𝑘, 𝜙(𝑝,𝑥𝑚𝑘)𝜙(𝑝,𝑥𝑚𝑘+1) and 𝜙(𝑝,𝑥𝑘)𝜙(𝑝,𝑥𝑚𝑘+1) for all 𝑘𝑁. Then, from (3.8), (*) and the fact 𝛿𝑛0, we have 𝑔𝐽𝑥𝑚𝑘𝐽𝑇𝑚𝑘𝑦𝑚𝑘0,𝑔𝐽𝑥𝑚𝑘𝐽𝑄𝑟𝑦𝑚𝑘0,as𝑘.(3.22) Thus, using the same proof as in Case 1, we obtain that 𝑦𝑚𝑘𝑇𝑦𝑚𝑘0, 𝑦𝑚𝑘𝑄𝑟𝑦𝑚𝑘0, as 𝑘, and, hence, we obtain that limsup𝑘𝑧𝑚𝑘𝑝,𝐽𝑤𝐽𝑝0.(3.23) Then, from (3.9), we have that 𝜙𝑝,𝑥𝑚𝑘+11𝛿𝑚𝑘𝜙𝑝,𝑥𝑚𝑘+2𝛿𝑚𝑘𝑧𝑚𝑘+𝑘𝑝,𝐽𝑤𝐽𝑝𝑚𝑘1𝑀.(3.24) Since 𝜙(𝑝,𝑥𝑚𝑘)𝜙(𝑝,𝑥𝑚𝑘+1), (3.24) implies that 𝛿𝑚𝑘𝜙𝑝,𝑥𝑚𝑘𝜙𝑝,𝑥𝑚𝑘𝜙𝑝,𝑥𝑚𝑘+1+2𝛿𝑚𝑘𝑧𝑚𝑘+𝑘𝑝,𝐽𝑤𝐽𝑝𝑚𝑘𝑀12𝛿𝑚𝑘𝑧𝑚𝑘+𝑘𝑝,𝐽𝑤𝐽𝑝𝑚𝑘1𝑀.(3.25) In particular, since 𝛿𝑚𝑘>0, we get 𝜙𝑝,𝑥𝑚𝑘𝑧2𝑚𝑘+𝑘𝑝,𝐽𝑤𝐽𝑝𝑚𝑘1𝛿𝑚𝑘𝑀.(3.26) Then, from (3.23) and the fact that (𝑘𝑚𝑘1)/𝛿𝑚𝑘0, we obtain 𝜙(𝑝,𝑥𝑚𝑘)0, as 𝑘. This together with (3.24) gives 𝜙(𝑝,𝑥𝑚𝑘+1)0, as 𝑘. But 𝜙(𝑝,𝑥𝑘)𝜙(𝑝,𝑥𝑚𝑘+1), for all 𝑘𝑁, thus we obtain that 𝑥𝑘𝑝. Therefore, from the above two cases, we can conclude that {𝑥𝑛} converges strongly to 𝑝 and the proof is complete.

If, in Theorem 3.1, we assume that 𝑇 is relatively nonexpansive, we get the following corollary.

Corollary 3.2. Let 𝐶 be a nonempty, closed, and convex subset of a uniformly convex and uniformly smooth real Banach space 𝐸. Let 𝐴𝐶𝐸 be a monotone mapping satisfying (2.8) and 𝐴𝑥𝐴𝑥𝐴𝑝,for all 𝑥𝐶 and 𝑝VI(𝐶,𝐴). Let 𝑇𝐶𝐶 be a relatively nonexpansive mapping. Assume that 𝐹=VI(𝐶,𝐴)𝐹(𝑆) is nonempty. Let {𝑥𝑛} be a sequence generated by 𝑥0𝑦=𝑤𝐶,chosenarbitrarily,𝑛=Π𝐶𝐽1𝛼𝑛𝐽𝑤+1𝛼𝑛𝐽𝑥𝑛,𝑥𝑛+1=𝐽1𝛽𝑛𝐽𝑥𝑛+𝛾𝑛𝐽𝑇𝑦𝑛+𝜃𝑛𝐽𝑄𝑟𝑦𝑛,(3.27) where 𝛼𝑛(0,1) such that lim𝑛𝛼𝑛=0, 𝑛=1𝛼𝑛=, {𝛽𝑛},{𝛾𝑛},{𝜃𝑛}[𝑐,𝑑](0,1) such that 𝛽𝑛+𝛾𝑛+𝜃𝑛=1. Then, {𝑥𝑛} converges strongly to an element of 𝐹.

Proof. We note that the method of proof of Theorem 3.1 provides the required assertion.

If 𝐸=𝐻, a real Hilbert space, then 𝐸 is uniformly convex and uniformly smooth real Banach space. In this case, 𝐽=𝐼, identity map on 𝐻 and Π𝐶=𝑃𝐶, projection mapping from 𝐻 onto 𝐶. Thus, the following corollary holds.

Corollary 3.3. Let 𝐶 be a nonempty, closed, and convex subset of a real Hilbert space 𝐻. Let 𝐴𝐶𝐻 be a monotone mapping satisfying (2.8) and 𝐴𝑥𝐴𝑥𝐴𝑝,for all 𝑥𝐶 and 𝑝𝑉𝐼(𝐶,𝐴). Let 𝑇𝐶𝐶 be a uniformly 𝐿-Lipschitzian relatively asymptotically nonexpansive mapping with sequence {𝑘𝑛}. Assume that 𝐹=𝑉𝐼(𝐶,𝐴)𝐹(𝑆) is nonempty. Let {𝑥𝑛} be a sequence generated by 𝑥0𝑦=𝑤𝐶,chosenarbitrarily,𝑛=𝑃𝐶𝛼𝑛𝑤+1𝛼𝑛𝑥𝑛,𝑥𝑛+1=𝛽𝑛𝑥𝑛+𝛾𝑛𝑇𝑛𝑦𝑛+𝜃𝑛𝑄𝑟𝑦𝑛,(3.28) where 𝛼𝑛(0,1) such that lim𝑛𝛼𝑛=0,lim𝑛((𝑘𝑛1)/𝛼𝑛)=0,𝑛=1𝛼𝑛=, {𝛽𝑛},{𝛾𝑛},{𝜃𝑛}[𝑐,𝑑](0,1) such that 𝛽𝑛+𝛾𝑛+𝜃𝑛=1. Then, {𝑥𝑛} converges strongly to an element of 𝐹.

Now, we state the second main theorem of our paper.

Theorem 3.4. Let 𝐶 be a nonempty, closed, and convex subset of a uniformly convex and uniformly smooth real Banach space 𝐸. Let 𝐴𝐶𝐸 be a monotone mapping satisfying (2.8). Let 𝑇𝐶𝐶 be a uniformly 𝐿-Lipschitzian relatively asymptotically nonexpansive mapping with sequence {𝑘𝑛}. Assume that 𝐹=𝐴1(0)𝐹(𝑆) is nonempty. Let 𝑄𝑟 be the resolvent of 𝐴 and {𝑥𝑛} a sequence generated by 𝑥0𝑦=𝑤𝐶,chosenarbitrarily,𝑛=Π𝐶𝐽1𝛼𝑛𝐽𝑤+1𝛼𝑛𝐽𝑥𝑛,𝑥𝑛+1=𝐽1𝛽𝑛𝐽𝑥𝑛+𝛾𝑛𝐽𝑇𝑛𝑦𝑛+𝜃𝑛𝐽𝑄𝑟𝑦𝑛,(3.29) where 𝛼𝑛(0,1) such that lim𝑛𝛼𝑛=0,lim𝑛((𝑘𝑛1)/𝛼𝑛)=0, 𝑛=1𝛼𝑛=, {𝛽𝑛},{𝛾𝑛},{𝜃𝑛}[𝑐,𝑑](0,1) such that 𝛽𝑛+𝛾𝑛+𝜃𝑛=1. Then, {𝑥𝑛} converges strongly to an element of 𝐹.

Proof. Similar method of proof of Theorem 3.1 provides the required assertion.

If, in Theorem 3.4, 𝐴=0, then we have the following corollary. Similar proof of Theorem 3.1 provides the assertion.

Corollary 3.5. Let 𝐶 be a nonempty, closed, and convex subset of a uniformly convex and uniformly smooth real Banach space 𝐸. Let 𝑇𝐶𝐶 be a uniformly 𝐿-Lipschitzian relatively asymptotically nonexpansive mapping with sequence {𝑘𝑛}. Assume that 𝐹=𝐹(𝑆) is nonempty. Let {𝑥𝑛} be a sequence generated by 𝑥0𝑦=𝑤𝐶,chosenarbitrarily,𝑛=Π𝐶𝐽1𝛼𝑛𝐽𝑤+1𝛼𝑛𝐽𝑥𝑛,𝑥𝑛+1=𝐽1𝛽𝑛𝐽𝑥𝑛+1𝛽𝑛𝐽𝑇𝑛𝑦𝑛,(3.30) where 𝛼𝑛(0,1) such that lim𝑛𝛼𝑛=0,lim𝑛((𝑘𝑛1)/𝛼𝑛)=0, 𝑛=1𝛼𝑛=, {𝛽𝑛}[𝑐,𝑑](0,1). Then, {𝑥𝑛} converges strongly to an element of 𝐹.

If, in Theorem 3.4, 𝑇=𝐼, identity mapping on 𝐶, then we have the following corollary.

Corollary 3.6. Let 𝐶 be a nonempty, closed, and convex subset of a uniformly smooth and uniformly convex real Banach space 𝐸. Let 𝐴𝐶𝐸 be a monotone mapping satisfying (2.8). Assume that 𝐹=𝐴1(0) is nonempty. Let 𝑄𝑟 be the resolvent of 𝐴 and {𝑥𝑛} a sequence generated by 𝑥0𝑦=𝑤𝐶,chosenarbitrarily,𝑛=Π𝐶𝐽1𝛼𝑛𝐽𝑤+1𝛼𝑛𝐽𝑥𝑛,𝑥𝑛+1=𝐽1𝛽𝑛𝐽𝑥𝑛+1𝛽𝑛𝐽𝑄𝑟𝑦𝑛,(3.31) where 𝛼𝑛(0,1) such that lim𝑛𝛼𝑛=0, 𝑛=1𝛼𝑛=, {𝛽𝑛}[𝑐,𝑑](0,1). Then, {𝑥𝑛} converges strongly to an element of 𝐹.

If, in Theorem 3.4, we assume that 𝑇 is relatively nonexpansive, we get the following corollary.

Corollary 3.7. Let 𝐶 be a nonempty, closed, and convex subset of a uniformly convex and uniformly smooth real Banach space 𝐸. Let 𝐴𝐶𝐸 be a monotone mapping satisfying (2.8). Let 𝑇𝐶𝐶 be a relatively nonexpansive mapping. Assume that 𝐹=𝐴1(0)𝐹(𝑆) is nonempty. Let {𝑥𝑛} be a sequence generated by 𝑥0𝑦=𝑤𝐶,chosenarbitrarily,𝑛=Π𝐶𝐽1𝛼𝑛𝐽𝑤+1𝛼𝑛𝐽𝑥𝑛,𝑥𝑛+1=𝐽1𝛽𝑛𝐽𝑥𝑛+𝛾𝑛𝐽𝑇𝑦𝑛+𝜃𝑛𝐽𝑄𝑟𝑦𝑛,(3.32) where 𝛼𝑛(0,1) such that lim𝑛𝛼𝑛=0, 𝑛=1𝛼𝑛=, {𝛽𝑛},{𝛾𝑛},{𝜃𝑛}[𝑐,𝑑](0,1) such that 𝛽𝑛+𝛾𝑛+𝜃𝑛=1. Then, {𝑥𝑛} converges strongly to an element of 𝐹.

We may also get the following corollary for a common zero of monotone mappings.

Corollary 3.8. Let 𝐶 be a nonempty, closed, and convex subset of a uniformly convex and uniformly smooth real Banach space 𝐸. Let 𝐴,𝐵𝐶𝐸 be monotone mappings satisfying (2.8). Suppose that 𝑇1=(𝐽+𝑟𝐴)1𝐽 and 𝑇2=(𝐽+𝑟𝐵)1𝐽. Assume that 𝐹=𝐴1(0)𝐵1(0) is nonempty. Let {𝑥𝑛} be a sequence generated by 𝑥0𝑦=𝑤𝐶,chosenarbitrarily,𝑛=Π𝐶𝐽1𝛼𝑛𝐽𝑤+1𝛼𝑛𝐽𝑥𝑛,𝑥𝑛+1=𝐽1𝛽𝑛𝐽𝑥𝑛+𝛾𝑛𝐽𝑇1𝑦𝑛+𝜃𝑛𝐽𝑇2𝑦𝑛,(3.33) where 𝛼𝑛(0,1) such that lim𝑛𝛼𝑛=0, 𝑛=1𝛼𝑛=, {𝛽𝑛},{𝛾𝑛},{𝜃𝑛}[𝑐,𝑑](0,1) such that 𝛽𝑛+𝛾𝑛+𝜃𝑛=1. Then, {𝑥𝑛} converges strongly to an element of 𝐹.

Proof. Clearly, from Lemma 2.8, we know that 𝑇1 and 𝑇2 are relatively nonexpansive mappings. We also have that 𝐹(𝑇1)=𝐴1(0) and 𝐹(𝑇2)=𝐵1(0). Thus, the conclusion follows from Corollary 3.7.

Remark 3.9. We remark that from Corollary 3.8 the scheme converges strongly to a common zero of two monotone operators. We may also have the following theorem for a common zero of finite family of monotone mappings.

Theorem 3.10. Let 𝐶 be a nonempty, closed, and convex subset of a uniformly convex and uniformly smooth real Banach space 𝐸. Let 𝐴𝑖𝐶𝐸, 𝑖=1,2,,𝑁 be monotone mappings satisfying (2.8). Suppose that 𝑇𝑖=(𝐽+𝑟𝐴𝑖)1𝐽, and assume that 𝐹=𝑁𝑖=1𝐴𝑖1(0) is nonempty. Let {𝑥𝑛} be a sequence generated by 𝑥0𝑦=𝑤𝐶,chosenarbitrarily,𝑛=Π𝐶𝐽1𝛼𝑛𝐽𝑤+1𝛼𝑛𝐽𝑥𝑛,𝑥𝑛+1=𝐽1𝛽n,0𝐽𝑥𝑛+𝛽𝑛,1𝐽𝑇1𝑦𝑛++𝛽𝑛,𝑁𝐽𝑇𝑁𝑦𝑛,(3.34) where 𝛼𝑛(0,1) such that lim𝑛𝛼𝑛=0, 𝑛=1𝛼𝑛=, {𝛽𝑛,𝑖}[𝑐,𝑑](0,1), for 𝑖=0,1,2,,𝑁, such that 𝑁𝑖=0𝛽𝑛,𝑖=1. Then, {𝑥𝑛} converges strongly to an element of 𝐹.

A monotone mapping 𝐴𝐶𝐸 is said to be maximal monotone if its graph is not properly contained in the graph of any monotone operator. We know that if 𝐴 is maximal monotone operator, then 𝐴1(0) is closed and convex: see [4] for more details. The following Lemma is well known.

Lemma 3.11 (see [28]). Let 𝐸 be a smooth and strictly convex and reflexive Banach space, let 𝐶 be a nonempty closed convex subset of 𝐸, and let 𝐴𝐶𝐸 be a monotone operator. Then 𝐴 is maximal if and only if 𝑅(𝐽+𝑟𝐴)=𝐸 for all 𝑟>0.

We note from the above lemma that if 𝐴 is maximal then it satisfies condition (2.8) and hence we have the following corollary.

Corollary 3.12. Let 𝐶 be a nonempty, closed, and convex subset of a uniformly convex and uniformly smooth real Banach space 𝐸. Let 𝐴𝐶𝐸 be a maximal monotone mapping. Let 𝑇𝐶𝐶 be a uniformly 𝐿-Lipschitzian relatively asymptotically nonexpansive mapping with sequence {𝑘𝑛}. Assume that 𝐹=𝐴1(0)𝐹(𝑆) is nonempty. Let {𝑥𝑛} be a sequence generated by 𝑥0𝑦=𝑤𝐶,chosenarbitrarily,𝑛=Π𝐶𝐽1𝛼𝑛𝐽𝑤+1𝛼𝑛𝐽𝑥𝑛,𝑥𝑛+1=𝐽1𝛽𝑛𝐽𝑥𝑛+𝛾𝑛𝐽𝑇𝑛𝑦𝑛+𝜃𝑛𝐽𝑄𝑟𝑦𝑛,(3.35) where 𝛼𝑛(0,1) such that lim𝑛𝛼𝑛=0,lim𝑛((𝑘𝑛1)/𝛼𝑛)=0, 𝑛=1𝛼𝑛=, {𝛽𝑛},{𝛾𝑛},{𝜃𝑛}[𝑐,𝑑](0,1) such that 𝛽𝑛+𝛾𝑛+𝜃𝑛=1. Then, {𝑥𝑛} converges strongly to an element of 𝐹.

4. Application

In this section, we study the problem of finding a minimizer of a lower semicontinuous continuously convex functional in Banach spaces.

Theorem 4.1. Let 𝐸 be a uniformly convex and uniformly smooth real Banach space. Let 𝑓,𝑔𝐸(,) be a proper lower semicontinuous convex functions. Assume that 𝐹=(𝜕𝑓)1(0)(𝜕𝑔)1(0) is nonempty. Let {𝑥𝑛} be a sequence generated by 𝑥0𝑦=𝑤𝐶,chosenarbitrarily,𝑛=Π𝐶𝐽1𝛼𝑛𝐽𝑤+1𝛼𝑛𝐽𝑥𝑛,𝑛1=argmin𝑓(𝑧)+𝜙2𝑟𝑧,𝑦𝑛,𝑡,𝑧𝐸𝑛1=argmin𝑔(𝑧)+𝜙2𝑟𝑧,𝑦𝑛,𝑥,𝑧𝐸𝑛+1=𝐽1𝛽𝑛𝐽𝑥𝑛+𝛾𝑛𝐽𝑛+𝜃𝑛𝐽𝑡𝑛,(4.1) where 𝛼𝑛(0,1) such that lim𝑛𝛼𝑛=0, 𝑛=1𝛼𝑛=, {𝛽𝑛},{𝛾𝑛},{𝜃𝑛}[𝑐,𝑑](0,1) such that 𝛽𝑛+𝛾𝑛+𝜃𝑛=1. Then, {𝑥𝑛} converges strongly to an element of 𝐹.

Proof. Let 𝐴 and 𝐵 be operators defined by 𝐴=𝜕𝑓 and 𝐵=𝜕𝑔 and 𝑄𝑟=(𝐽+𝑟𝐴)1𝐽, 𝑄𝐵𝑟=(𝐽+𝑟𝐵)1𝐽 for all 𝑟>0. Then, by Rockafellar [29], 𝐴 and 𝐵 are maximal monotone mappings. We also have that 𝑛=𝑄𝐴𝑟1𝑦=argmin𝑓(𝑧)+,𝑡2𝑟𝜙(𝑧,𝑦),𝑧𝐸𝑛=𝑄𝐵𝑟1𝑦=argmin𝑔(𝑧)+,2𝑟𝜙(𝑧,𝑦),𝑧𝐸(4.2) for all 𝑦𝐸 and 𝑟>0. Furthermore, we have that 𝐹(𝑄𝐴𝑟)=𝐴1(0) and 𝐹(𝑄𝐵𝑟)=𝐵1(0). Thus, by Corollary 3.8, we obtain the desired result.

Remark 4.2. Consider the following.(1) Theorem 3.1 improves and extends the corresponding results of Zegeye et al. [12] and Zegeye and Shahzad [22] in the sense that either our scheme does not require computation of 𝐶𝑛+1 for each 𝑛1 or the space considered is more general.(2) Corollary 3.5 improves the corresponding results of Nakajo and Takahashi [30] and Matsushita and Takahashi [17] in the sense that either our scheme does not require computation of 𝐶𝑛+1 for each 𝑛1 or the class of mappings considered in our corollary is more general.(3) Corollary 3.6 improves the corresponding results of Iiduka and Takahashi [11], Iiduka et al. [14], and Alber [1] in the sense that our scheme does not require computation of 𝐶𝑛+1 for each 𝑛1 or the class of mappings considered in our corollary is more general.

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