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

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 𝜙∶𝐸×𝐸→ℝ the function defined by 𝜙(𝑦,𝑥)=‖𝑦‖²−2⟨𝑦,𝐽𝑥⟩+‖𝑥‖²,for𝑥,𝑦∈𝐸,(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 2^𝐸∗ defined by 𝐽𝑥∶=𝑓^∗∈𝐸^∗∶⟨𝑥,𝑓^∗⟩=‖𝑥‖²=‖𝑓^∗‖²,(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 𝐽⁻¹ is single valued, one-to-one, surjective, and it is the duality mapping from 𝐸^∗ into 𝐸, and thus 𝐽𝐽⁻¹=𝐼_𝐸^∗ and 𝐽⁻¹𝐽=𝐼_𝐸 (see [4]).

It is obvious from the definition of the function 𝜙 that (‖𝑥‖−‖𝑦‖)²≤𝜙(𝑥,𝑦)≤(‖𝑥‖+‖𝑦‖)²for𝑥,𝑦∈𝐸,(1.3) and, in a Hilbert space 𝐻, (1.1) reduces to 𝜙(𝑥,𝑦)=‖𝑥−𝑦‖², for 𝑥,𝑦∈𝐻.

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 Π_𝐶∶𝐸→𝐶 that assigns an arbitrary point 𝑥∈𝐸 to the minimizer, 𝑥, of 𝜙(⋅,𝑥) over 𝐶, that is, Π_𝐶𝑥=𝑥, where 𝑥 is the solution to the minimization problem 𝜙𝑥,𝑥=min{𝜙(𝑦,𝑥),𝑦∈𝐶}.(1.4) Let 𝐸 be a real Banach space with dual 𝐸^∗. A mapping 𝐴∶𝐷(𝐴)⊂𝐸→𝐸^∗ is said to be monotone if, for each 𝑥,𝑦∈𝐷(𝐴), the following inequality holds: ⟨𝑥−𝑦,𝐴𝑥−𝐴𝑦⟩≥0.(1.5)𝐴 is said to be 𝛾-inverse strongly monotone if there exists positive real number 𝛾 such that ⟨𝑥−𝑦,𝐴𝑥−𝐴𝑦⟩≥𝛾‖𝐴𝑥−𝐴𝑦‖²,∀𝑥,𝑦∈𝐾.(1.6) If 𝐴 is 𝛾-inverse strongly monotone, then it is Lipschitz continuous with constant 1/𝛾, that is, ‖𝐴𝑥−𝐴𝑦‖≤(1/𝛾)‖𝑥−𝑦‖, for all 𝑥,𝑦∈𝐷(𝐴), and it is called strongly monotone if there exists 𝑘>0 such that, for all 𝑥,𝑦∈𝐷(𝐴), ⟨𝑥−𝑦,𝐴𝑥−𝐴𝑦⟩≥𝑘‖𝑥−𝑦‖².(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 𝑢∈𝐶 such that ⟨𝑣−𝑢,𝐴𝑢⟩≥0, for all 𝑣∈𝐶. 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, 11–13]). Such a problem is connected with the convex minimization problem, the complementarity problem, the problem of finding point 𝑢∈𝐶 satisfying 0∈𝐴𝑢.

If 𝐸=𝐻, a Hilbert space, one method of solving a point 𝑢∈VI(𝐶,𝐴) is the projection algorithm which starts with any point 𝑥₁=𝑥∈𝐶 and updates iteratively as 𝑥_𝑛+1 according to the formula 𝑥_𝑛+1=𝑃_𝐶𝑥_𝑛−𝛼_𝑛𝐴𝑥_𝑛,forany𝑛≥1,(1.8) where 𝑃_𝐶 is the metric projection from 𝐻 onto 𝐶 and {𝛼_𝑛} 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 {𝑥_𝑛} 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: 𝑥_𝑛+1=Π_𝐶𝐽⁻¹𝐽𝑥_𝑛−𝛼_𝑛𝐴𝑥_𝑛,forany𝑛≥1,(1.9) where Π_𝐶 is the generalized projection from 𝐸 onto 𝐶, 𝐽 is the normalized duality mapping from 𝐸 into 𝐸^∗, and {𝛼_𝑛} is a sequence of positive real numbers. They proved that the sequence {𝑥_𝑛} generated by (1.9) converges weakly to some element of VI(𝐶,𝐴) provided that 𝐴 satisfies ‖𝐴𝑥‖≤‖𝐴𝑥−𝐴𝑝‖, for 𝑥∈𝐶 and 𝑝∈VI(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: 𝑥𝑦₀∈𝐶,chosenarbitrary,_𝑛𝐶_𝑛=𝑃_𝐶𝑥_𝑛−𝛼_𝑛𝐴𝑥_𝑛,𝑄=𝑧∈𝐶∶‖‖𝑦_𝑛−𝑧‖‖≤‖‖𝑥_𝑛−𝑧_𝑛‖‖,𝑥=𝑧∈𝐶∶⟨𝑥_𝑛−𝑧,𝑥₀−𝑥_𝑛⟩≥0_𝑛+1,=𝑃_{𝐶𝑛∩𝑄𝑛}𝑥₀,𝑛≥1,for𝑛≥1,(1.10) where {𝛼_𝑛} is a sequence in [0,2𝛾]. They proved that the sequence {𝑥_𝑛} generated by (1.10) converges strongly to 𝑃_{VI(𝐶,𝐴)}(𝑥₀), where 𝑃_{VI(𝐶,𝐴)} is the metric projection from 𝐻 onto VI(𝐶,𝐴) provided that 𝐴 satisfies ‖𝐴𝑥‖≤‖𝐴𝑥−𝐴𝑝‖, for 𝑥∈𝐶 and 𝑝∈VI(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: 𝑦𝑥₀∈𝐾,chosenarbitrary,_𝑛𝐶_𝑛=Π_𝐶𝐽⁻¹𝐽𝑥_𝑛−𝛼_𝑛𝐴𝑥_𝑛,𝑄=𝑧∈𝐸∶𝜙𝑧,𝑦_𝑛≤𝜙𝑧,𝑥_𝑛_𝑛,𝑥=𝑧∈𝐸∶⟨𝑥_𝑛−𝑧,𝐽𝑥₀−𝐽𝑥_𝑛⟩≥0_𝑛+1,=Π_{𝐶𝑛∩𝑄𝑛}𝑥₀,𝑛≥1,for𝑛≥1,(1.11) where Π_{𝐶𝑛∩𝑄𝑛} is the generalized projection from 𝐸 onto 𝐶_𝑛∩𝑄_𝑛,𝐽 is the normalized duality mapping from 𝐸 into 𝐸^∗, and {𝛼_𝑛} is a positive real sequence satisfying certain condition. Then, they proved that the sequence {𝑥_𝑛} converges strongly to an element of VI(𝐶,𝐴) provided that VI(𝐶,𝐴)≠∅ and 𝐴 satisfies ‖𝐴𝑥‖≤‖𝐴𝑥−𝐴𝑝‖ for all 𝑥∈𝐶 and 𝑝∈VI(𝐶,𝐴).

Remark 1.1. We remark that the computation of 𝑥_𝑛+1 in Algorithms (1.10) and (1.11) is not simple because of the involvement of computation of 𝐶_𝑛+1 from 𝐶_𝑛 and 𝑄_𝑛 for each 𝑛≥1.

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 {𝑥_𝑛} which converges weakly to 𝑝 such that lim_𝑛→∞‖𝑥_𝑛−𝑇𝑥_𝑛‖=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 ‖𝑇𝑥−𝑇𝑦‖≤‖𝑥−𝑦‖ for each 𝑥,𝑦∈𝐶 and is called relatively nonexpansive if (R1) 𝐹(𝑇)≠∅; (R2) 𝜙(𝑝,𝑇𝑥)≤𝜙(𝑝,𝑥) for 𝑥∈𝐶 and (R3) 𝐹(𝑇)=𝐹(𝑇). 𝑇 is called relatively quasi-nonexpansive if 𝐹(𝑇)≠∅ and 𝜙(𝑝,𝑇𝑥)≤𝜙(𝑝,𝑥) for all 𝑥∈𝐶, and 𝑝∈𝐹(𝑇).

A mapping 𝑇 from 𝐶 into itself is said to be asymptotically nonexpansive if there exists {𝑘_𝑛}⊂[1,∞) such that 𝑘_𝑛→1 and ‖𝑇^𝑛𝑥−𝑇^𝑛𝑦‖≤𝑘_𝑛‖𝑥−𝑦‖ for each 𝑥,𝑦∈𝐶 and is called relatively asymptotically nonexpansive if there exists {𝑘_𝑛}⊂[1,∞) such that (N1) 𝐹(𝑇)≠∅; (N2) 𝜙(𝑝,𝑇^𝑛𝑥)≤𝑘_𝑛𝜙(𝑝,𝑥) for 𝑥∈𝐶 and 𝑝∈𝐹(𝑇), and (N3) 𝐹(𝑇)=𝐹(𝑇), where 𝑘_𝑛→1 as 𝑛→∞. A-self mapping on 𝐶 is called uniformly 𝐿-Lipschitzian if there exists 𝐿>0 such that ‖𝑇^𝑛𝑥−𝑇^𝑛𝑦‖≤𝐿‖𝑥−𝑦‖ for all 𝑥,𝑦∈𝐶. 𝑇 is called closed if 𝑥_𝑛→𝑥 and 𝑇𝑥_𝑛→𝑦, 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=𝑙^𝑝, where 1<𝑝<∞, and 𝐶={𝑥=(𝑥₁,𝑥₂,…)∈𝑋;𝑥_𝑛≥0}. Then 𝐶 is closed and convex subset of 𝑋. Note that 𝐶 is not bounded. Obviously, 𝑋 is uniformly convex and uniformly smooth. Let {𝜆_𝑛} and{𝜆_𝑛} be sequences of real numbers satisfying the following properties:(i)0<𝜆_𝑛<1, 𝜆_𝑛>1, 𝜆_𝑛↑1, and 𝜆_𝑛↓1,(ii)𝜆_𝑛+1𝜆_𝑛=1 and 𝜆_𝑗+1𝜆_𝑛+𝑗<1 for all 𝑛 and 𝑗 (e.g., 𝜆_𝑛=1−1/(𝑛+1), 𝜆_𝑛=1+1/(𝑛+1)). Then, the map 𝑇∶𝐶→𝐶 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: 𝑦𝑥₀∈𝐶,chosenarbitrary,_𝑛𝐶_𝑛=𝐽⁻¹𝛼_𝑛𝐽𝑥_𝑛+1−𝛼_𝑛𝐽𝑇𝑥_𝑛,𝑄=𝑧∈𝐶∶𝜙𝑧,𝑦_𝑛≤𝜙𝑧,𝑥_𝑛_𝑛,=𝑥𝑧∈𝐶;⟨𝑥_𝑛−𝑧,𝐽𝑥₀−𝐽𝑥_𝑛⟩≥0_𝑛+1,=Π_{𝐶𝑛∩𝑄𝑛}𝑥₀,𝑛≥1.(1.13) They proved that, if the sequence {𝛼_𝑛} is bounded above from one, then the sequence {𝑥_𝑛} generated by (1.13) converges strongly to Π_𝐹(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, 18–20]).

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: 𝑥𝑥₁=𝑥∈𝐶,_𝑛+1=𝛼_𝑛𝑥+1−𝛼_𝑛𝑆𝑃_𝐶𝑥_𝑛−𝜆_𝑛𝐴𝑥_𝑛,𝑛≥1,(1.14) where {𝛼_𝑛} is sequences satisfying certain condition. They proved that the sequence {𝑥_𝑛} 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: 𝐶𝑧₁=𝐶,chosenarbitrary,_𝑛𝑦_𝑛=Π_𝐶𝑥_𝑛−𝜆_𝑛𝐴𝑥_𝑛,𝐶_𝑛+1=𝐽⁻¹𝛽𝐽𝑥_𝑛+(1−𝛽)𝐽𝑇𝑧_𝑛,=𝑥𝑧∈𝐶_𝑛∶𝜙𝑧,𝑦_𝑛≤𝜙𝑧,𝑥_𝑛_𝑛+1,=Π_𝐶𝑛+1𝑥₀,𝑛≥1,(1.15) where {𝜆_𝑛} is sequences satisfying certain condition. They proved that the sequence {𝑥_𝑛} converges strongly to an element of 𝐹∶=𝐹(𝑆)∩VI(𝐶,𝐴)≠∅ provided that 𝐹≠∅ and 𝐴 satisfies ‖𝐴𝑥‖≤‖𝐴𝑥−𝐴𝑝‖ for all 𝑥∈𝐶 and 𝑝∈𝐹.

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: 𝐶𝑧₁=𝐶,chosenarbitrary,_𝑛𝑦_𝑛=𝑃_𝐶𝑥_𝑛−𝜆_𝑛𝐴𝑥_𝑛,𝐶_𝑛+1=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝑆^𝑛𝑧_𝑛,=𝑧∈𝑥𝐶_𝑛𝑛+1∶‖‖𝑧−𝑢_𝑛‖‖²≤‖‖𝑧−𝑥_𝑛‖‖²+𝜃_𝑛,=𝑃_𝐶𝑛+1𝑥₀,𝑛≥1,(1.16) where 𝑃_𝐶𝑛 is the metric projection from 𝐻 into 𝐶_𝑛 and 𝜃_𝑛=(1−𝛼_𝑛)(𝑘^2𝑛−1)(diam(𝐶))² and {𝛼_𝑛},{𝜆_𝑛} are sequences satisfying certain condition. Then, they proved that the sequence {𝑥_𝑛} converges strongly to an element of 𝐹∶=𝐹(𝑆)∩VI(𝐶,𝐴)≠∅ provided that 𝐹≠∅ and 𝐴 satisfies ‖𝐴𝑥‖≤‖𝐴𝑥−𝐴𝑝‖ for all 𝑥∈𝐶 and 𝑝∈𝐹.

Remark 1.3. We again remark that the computation of 𝑥_𝑛+1 in Algorithms (1.13), (1.15), and (1.16) is not simple because of the involvement of computation of 𝐶_𝑛+1 from 𝐶_𝑛 for each 𝑛≥1.
It is our purpose in this paper to introduce an iterative scheme {𝑥_𝑛} which converges strongly to a common point of solution of variational inequality problem for a monotone operator 𝐴∶𝐶→𝐸^∗ 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 𝐶_𝑛+1 from 𝐶_𝑛 or 𝑄_𝑛, for each 𝑛≥1. 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 dim𝐸≥2. The modulus of smoothness of 𝐸 is the function 𝜌_𝐸∶[0,∞)→[0,∞) defined by 𝜌_𝐸(𝜏)∶=sup‖𝑥+𝑦‖+‖𝑥−𝑦‖2−1∶‖𝑥‖=1;‖𝑦‖=𝜏.(2.1) The space 𝐸 is said to be smooth if 𝜌_𝐸(𝜏)>0, for all 𝜏>0, and 𝐸 is called uniformly smooth if and only if lim_𝑡→0⁺(𝜌_𝐸(𝑡)/𝑡)=0.

The modulus of convexity of 𝐸 is the function 𝛿_𝐸∶(0,2]→[0,1] defined by 𝛿_𝐸(𝜖)∶=inf1−‖‖‖𝑥+𝑦2‖‖‖∶‖𝑥‖=‖𝑦‖=1;𝜖=‖𝑥−𝑦‖.(2.2)𝐸 is called uniformly convex if and only if 𝛿_𝐸(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 𝐴∶𝐶→𝐸^∗ 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 𝑥∈𝐸. Then, for all 𝑦∈𝐾, 𝜙𝑦,Π_𝐾𝑥+𝜙Π_𝐾𝑥,𝑥≤𝜙(𝑦,𝑥).(2.3)

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

We make use of the function 𝑉∶𝐸×𝐸^∗→ℝ defined by 𝑉𝑥,𝑥^∗=‖𝑥‖²−2⟨𝑥,𝑥^∗⟩+‖𝑥‖²,∀𝑥∈𝐸,𝑥^∗∈𝐸,(2.4) studied by Alber [1]. That is, 𝑉(𝑥,𝑦)=𝜙(𝑥,𝐽⁻¹𝑥^∗) for all 𝑥∈𝐸 and 𝑥^∗∈𝐸^∗. 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, 𝑉𝑥,𝑥^∗+2𝐽⁻¹𝑥^∗−𝑥,𝑦^∗≤𝑉𝑥,𝑥^∗+𝑦^∗,(2.5) for all 𝑥∈𝐸 and 𝑥^∗,𝑦^∗∈𝐸^∗.

Lemma 2.6 (see [1]). Let 𝐶 be a convex subset of a real smooth Banach space 𝐸. Let 𝑥∈𝐸. Then 𝑥₀=Π_𝐶𝑥 if and only if ⟨𝑧−𝑥₀,𝐽𝑥−𝐽𝑥₀⟩≤0,∀𝑧∈𝐶.(2.6)

Lemma 2.7 (see [12]). Let 𝐸 be a uniformly convex Banach space and 𝐵_𝑅(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+𝛼₃𝑥₃‖‖²≤𝛼_𝑖‖‖𝑥₁‖‖²+𝛼₂‖‖𝑥₂‖‖²+𝛼₃‖‖𝑥₃‖‖²−𝛼_𝑖𝛼_𝑗𝑔‖‖𝑥_𝑖−𝑥_𝑗‖‖,(2.7) for 𝛼_𝑖∈(0,1) such that 𝛼₁+𝛼₂+𝛼₃=1 and 𝑥_𝑖∈𝐵_𝑅(0)∶={𝑥∈𝐸∶‖𝑥‖≤𝑅}, for 𝑖=1,2,3.

Let 𝐸 be a smooth and strictly convex Banach space, 𝐶 a nonempty closed convex subset of 𝐸, and 𝐴∶𝐶→𝐸^∗ a monotone operator satisfying 𝐷(𝐴)⊆𝐶⊆𝐽⁻¹∩_𝑟>0𝑅(𝐽+𝑟𝐴),(2.8) for 𝑟>0. Then, we can define the resolvent 𝑄_𝑟∶𝐶→𝐷(𝐴) of 𝐴 by 𝑄_𝑟𝑥∶={𝑧∈𝐷(𝐴)∶𝐽𝑥∈𝐽𝑧+𝑟𝐴𝑧},∀𝑥∈𝐶.(2.9) In other words, 𝑄_𝑟𝑥=(𝐽+𝑟𝐴)⁻¹𝐽𝑥 for 𝑥∈𝐶. We know that 𝑄_𝑟𝑥 is single-valued mapping from 𝐶 into 𝐷(𝐴), for all 𝑥∈𝐶 and 𝑟>0 and 𝐹(𝑄_𝑟)=𝐴⁻¹(0), where 𝐹(𝑄_𝑟) is the set of fixed points of 𝑄_𝑟 (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 𝐴⊂𝐸×𝐸^∗ a monotone operator satisfying (2.8) and 𝐴⁻¹(0) is nonempty. Let 𝑄_𝑟 be the resolvent of 𝐴. Then, for each 𝑟>0, 𝜙𝑢,𝑄_𝑟𝑥+𝜙𝑄_𝑟𝑥,𝑥≤𝜙(𝑢,𝑥)(2.10) for all 𝑢∈𝐴⁻¹(0) and 𝑥∈𝐶, that is, 𝑄_𝑟 is relatively nonexpansive.

Lemma 2.9 (see [25]). Let {𝑎_𝑛} be a sequence of nonnegative real numbers satisfying the following relation: 𝑎_𝑛+1≤1−𝛽_𝑛𝑎_𝑛+𝛽_𝑛𝛿_𝑛,𝑛≥𝑛₀,(2.11) where {𝛽_𝑛}⊂(0,1) and {𝛿_𝑛}⊂𝑅 satisfying the following conditions: lim_𝑛→∞𝛽_𝑛=0,∑^∞𝑛=1𝛽_𝑛=∞, and limsup_𝑛→∞𝛿_𝑛≤0. Then, lim_𝑛→∞𝑎_𝑛=0.

Lemma 2.10 (see [26]). Let {𝑎_𝑛} be sequences of real numbers such that there exists a subsequence {𝑛_𝑖} of {𝑛} such that 𝑎_𝑛𝑖<𝑎_𝑛𝑖+1 for all 𝑖∈𝑁. Then, there exists a nondecreasing sequence {𝑚_𝑘}⊂𝑁 such that 𝑚_𝑘→∞, and the following properties are satisfied by all (sufficiently large) numbers 𝑘∈𝑁: 𝑎_𝑚𝑘≤𝑎_𝑚𝑘+1,𝑎_𝑘≤𝑎_𝑚𝑘+1.(2.12) In fact, 𝑚_𝑘=max{𝑗≤𝑘∶𝑎_𝑗<𝑎_𝑗+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 𝐴∶𝐶→𝐸^∗ is a mapping satisfying ‖𝐴𝑥‖≤‖𝐴𝑥−𝐴𝑝‖,for all 𝑥∈𝐶 and 𝑝∈VI(𝐶,𝐴), then VI(𝐶,𝐴)=𝐴⁻¹(0)={𝑝∈𝐶∶𝐴𝑝=0}.(3.1)

In fact, clearly, 𝐴⁻¹(0)⊆VI(𝐶,𝐴). Now, we show that VI(𝐶,𝐴)⊆𝐴⁻¹(0). Let 𝑝∈VI(𝐶,𝐴), then we have by hypothesis that ‖𝐴𝑝‖≤‖𝐴𝑝−𝐴𝑝‖=0 which implies that 𝑝∈𝐴⁻¹(0). Hence, VI(𝐶,𝐴)⊆𝐴⁻¹(0). Therefore, VI(𝐶,𝐴)=𝐴⁻¹(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 𝐴∶𝐶→𝐸^∗ be a monotone mapping satisfying (2.8) and ‖𝐴𝑥‖≤‖𝐴𝑥−𝐴𝑝‖,for all 𝑥∈𝐶 and 𝑝∈VI(𝐶,𝐴). Let 𝑇∶𝐶→𝐶 be a uniformly 𝐿-Lipschitzian relatively asymptotically nonexpansive mapping with sequence {𝑘_𝑛}. Assume that 𝐹∶=VI(𝐶,𝐴)∩𝐹(𝑆) is nonempty. Let 𝑄_𝑟 be the resolvent of 𝐴 and {𝑥_𝑛} a sequence generated by 𝑦𝑥₀=𝑤∈𝐶,chosenarbitrarily,_𝑛𝑥_𝑛+1=Π_𝐶𝐽⁻¹𝛼_𝑛𝐽𝑤+1−𝛼_𝑛𝐽𝑥_𝑛,=𝐽⁻¹𝛽_𝑛𝐽𝑥_𝑛+𝛾_𝑛𝐽𝑇^𝑛𝑦_𝑛+𝜃_𝑛𝐽𝑄_𝑟𝑦_𝑛,(3.2) where 𝛼_𝑛∈(0,1) such that lim_𝑛→∞𝛼_𝑛=0,lim_𝑛→∞((𝑘_𝑛−1)/𝛼_𝑛)=0,∑^∞𝑛=1𝛼_𝑛=∞, {𝛽_𝑛},{𝛾_𝑛},{𝜃_𝑛}⊂[𝑐,𝑑]⊂(0,1) such that 𝛽_𝑛+𝛾_𝑛+𝜃_𝑛=1. Then, {𝑥_𝑛} converges strongly to an element of 𝐹.

Proof. Let 𝑝∶=Π_𝐹𝑤. Then, from (3.2), Lemma 2.3, and property of 𝜙, we get that 𝜙𝑝,𝑦_𝑛=≤𝜙𝜙𝑝𝑝,,𝐽Π_𝐶−1𝐽⁻¹𝛼_𝑛𝐽𝑤+1−𝛼_𝑛𝐽𝑥_𝑛=‖𝑝‖₂𝛼_𝑛𝐽𝑤+1−𝛼_𝑛𝐽𝑥_𝑛−2𝑝,𝛼_𝑛𝐽𝑤+1−𝛼_𝑛𝐽𝑥_𝑛+‖‖𝛼_𝑛𝐽𝑤+1−𝛼_𝑛𝐽𝑥_𝑛‖‖²≤‖𝑝‖²−2𝛼_𝑛⟨𝑝,𝐽𝑤⟩−21−𝛼_𝑛⟨𝑝,𝐽𝑥_𝑛⟩+𝛼_𝑛‖𝐽𝑤‖²+1−𝛼_𝑛‖‖𝐽𝑥_𝑛‖‖²=𝛼_𝑛𝜙(𝑝,𝑤)+1−𝛼_𝑛𝜙𝑝,𝑥_𝑛.(3.3) Now, from (3.2) and relatively asymptotically nonexpansiveness of 𝑇, relatively nonexpansiveness of 𝑄_𝑟, property of 𝜙, and (3.3), we get that 𝜙𝑝,𝑥_𝑛+1=≤𝜙𝛽_𝑛𝑝,𝐽⁻¹𝛽_𝑛𝐽𝑥_𝑛+𝛾_𝑛𝐽𝑇^𝑛𝑦_𝑛+𝜃_𝑛𝐽𝑄_𝑟𝑦_𝑛≤𝛽_𝑛𝜙𝑝,𝑥_𝑛+𝛾_𝑛𝜙𝑝,𝑇^𝑛𝑦_𝑛+𝜃_𝑛𝜙𝑝,𝑄_𝑟𝑦_𝑛≤𝛽_𝑛𝜙𝑝,𝑥_𝑛+𝛾_𝑛𝑘_𝑛𝜙𝑝,𝑦_𝑛+𝜃_𝑛𝜙𝑝,𝑦_𝑛≤𝛽_𝑛𝜙𝑝,𝑥_𝑛+𝛾_𝑛𝑘_𝑛+𝜃_𝑛𝜙𝑝,𝑦_𝑛≤𝛾𝜙𝑝,𝑥_𝑛+𝛾_𝑛𝑘_𝑛+𝜃_𝑛𝛼_𝑛𝜙(𝑝,𝑤)+1−𝛼_𝑛𝜙𝑝,𝑥_𝑛_𝑛≤𝛾_𝑛𝑘_𝑛+𝜃_𝑛𝛼_𝑛𝜙(𝑝,𝑤)+𝛽_𝑛+𝛾_𝑛𝑘_𝑛+𝜃_𝑛1−𝛼_𝑛𝜙𝑝,𝑥_𝑛≤𝑐_𝑛𝑘_𝑛+𝜃_𝑛𝛼_𝑛𝜙(𝑝,𝑤)+1−𝛼_𝑛𝛾_𝑛𝑘_𝑛+𝜃_𝑛+𝛾_𝑛𝑘_𝑛−1×𝜙𝑝,𝑥_𝑛𝜙(𝑝,𝑤)+1−(1−𝜖)𝑐_𝑛𝜙𝑝,𝑥_𝑛,(3.4) where 𝑐_𝑛=𝛼_𝑛(𝛾_𝑛𝑘_𝑛+𝜃_𝑛), since there exists 𝑁₀>0 such that 𝛾_𝑛(𝑘_𝑛−1)/𝛼_𝑛≤𝜖(𝛾_𝑛𝑘_𝑛+𝜃_𝑛) for all 𝑛≥𝑁₀ and for some 𝜖>0 satisfying (1−𝜖)𝑐_𝑛≤1. Thus, by induction, 𝜙𝑝,𝑥_𝑛+1≤max𝜙𝑝,𝑥₀,(1−𝜖)⁻¹𝜙(𝑝,𝑤),∀𝑛≥𝑁₀(3.5) which implies that {𝑥_𝑛}, and hence {𝑦_𝑛} is bounded. Now, let 𝑧_𝑛=𝐽⁻¹(𝛼_𝑛𝐽𝑤+(1−𝛼_𝑛)𝐽𝑥_𝑛). Then we have that 𝑦_𝑛=Π_𝐶𝑧_𝑛. Using Lemmas 2.3, 2.5, and property of 𝜙, we obtain that 𝜙𝑝,𝑦_𝑛≤≤𝜙𝑉𝑝𝑝,,𝑧𝐽_𝑛𝑧_𝑛=𝑉𝑝,𝐽𝑧_𝑛=𝜙𝑝,𝐽₋₁−𝛼_𝑛(𝐽𝑤−𝐽𝑝)−2⟨𝑧_𝑛−𝑝,−𝛼_𝑛(𝐽𝑤−𝐽𝑝)⟩≤𝛼_𝑛𝛼_𝑛𝐽𝑝+1−𝛼_𝑛𝐽𝑤_𝑛+2𝛼_𝑛⟨𝑧_𝑛−𝑝,𝐽𝑤−𝐽𝑝⟩=1𝜙−(𝑝𝛼,_𝑛𝑝)+1−𝛼_𝑛𝜙𝑝,𝑤_𝑛+2𝛼_𝑛⟨𝑧_𝑛−𝑝,𝐽𝑤−𝐽𝑝⟩≤1−𝛼_𝑛𝜙𝑝,𝑤_𝑛+2𝛼_𝑛⟨𝑧_𝑛−𝑝,𝐽𝑤−𝐽𝑝⟩𝜙𝑝,𝑥_𝑛+2𝛼_𝑛⟨𝑧_𝑛−𝑝,𝐽𝑤−𝐽𝑝⟩.(3.6) Furthermore, from (3.2), Lemma 2.7, relatively asymptotically nonexpansiveness of 𝑇, relatively nonexpansiveness of 𝑄_𝑟, and (3.6), we have that 𝜙𝑝,𝑥_𝑛+1==‖𝜙𝑝‖𝑝₂,𝐽⁻¹𝛽_𝑛𝐽𝑥_𝑛+𝛾_𝑛𝐽𝑇^𝑛𝑦_𝑛+𝜃_𝑛𝐽𝑄_𝑟𝑦_𝑛−⟨𝑝,𝛽_𝑛𝐽𝑥_𝑛+𝛾_𝑛𝐽𝑇^𝑛𝑦_𝑛+𝜃_𝑛𝐽𝑄_𝑟𝑦_𝑛⟩+‖‖𝛽_𝑛𝐽𝑥_𝑛+𝛾_𝑛𝐽𝑇^𝑛𝑦_𝑛+𝜃_𝑛𝐽𝑄_𝑟𝑦_𝑛‖‖²≤‖𝑝‖₂(3.7)+𝛽_𝑛−2𝛽_𝑛⟨𝑝,𝐽𝑥_𝑛⟩−2𝛾_𝑛⟨𝑝,𝐽𝑇^𝑛𝑦_𝑛⟩−2𝜃_𝑛⟨𝑝,𝐽𝑄_𝑟𝑦_𝑛⟩≤𝛽_𝑛‖‖𝐽𝑥_𝑛‖‖²+𝛾_𝑛‖‖𝐽𝑇^𝑛𝑦_𝑛‖‖²+𝜃_𝑛‖‖𝐽𝑄_𝑟𝑦_𝑛‖‖²−𝛾_𝑛𝛽_𝑛𝑔‖‖𝐽𝑥_𝑛−𝐽𝑇^𝑛𝑦_𝑛‖‖≤𝛽_𝑛𝜙𝑝,𝑥_𝑛+𝛾_𝑛𝜙𝑝,𝑇^𝑛𝑦_𝑛+𝜃_𝑛𝜙𝑝,𝑄_𝑟𝑦_𝑛−𝛾_𝑛𝛽_𝑛𝑔‖‖𝐽𝑥_𝑛−𝐽𝑇^𝑛𝑦_𝑛‖‖≤𝛽_𝑛𝜙𝑝,𝑥_𝑛+𝛾_𝑛+𝜃_𝑛𝜙𝑝,𝑦_𝑛+𝛾_𝑛𝑘_𝑛−1𝜙𝑝,𝑦_𝑛−𝛾_𝑛𝛽_𝑛𝑔‖‖𝐽𝑥_𝑛−𝐽𝑇^𝑛𝑦_𝑛‖‖+𝛾𝜙_𝑛𝑝,𝑥_𝑛+𝛾_𝑛+𝜃_𝑛1−𝛼_𝑛𝜙𝑝,𝑥_𝑛+2𝛼_𝑛⟨𝑧_𝑛−𝑝,𝐽𝑤−𝐽𝑝⟩≤𝛽_𝑛𝑘_𝑛−1𝜙𝑝,𝑦_𝑛−𝛾_𝑛𝛽_𝑛𝑔‖‖𝐽𝑥_𝑛−𝐽𝑇^𝑛𝑦_𝑛‖‖+𝛾_𝑛+𝛾_𝑛+𝜃_𝑛1−𝛼_𝑛𝜙𝑝,𝑥_𝑛+2𝛼_𝑛𝛾_𝑛+𝜃_𝑛⟨𝑧_𝑛−𝑝,𝐽𝑤−𝐽𝑝⟩≤1−𝑘𝛿_𝑛𝑛−1𝜙𝑝,𝑦_𝑛−𝛾_𝑛𝛽_𝑛𝑔‖‖𝐽𝑥_𝑛−𝐽𝑇^𝑛𝑦_𝑛‖‖≤1−𝛿_𝑛𝜙𝑝,𝑥_𝑛+2𝛿_𝑛⟨𝑧_𝑛−𝑝,𝐽𝑤−𝐽𝑝⟩+𝑘_𝑛−1𝑀−𝛾_𝑛𝛽_𝑛𝑔‖‖𝐽𝑥_𝑛−𝐽𝑇^𝑛𝑦_𝑛‖‖(3.8)𝜙𝑝,𝑥_𝑛+2𝛿_𝑛⟨𝑧_𝑛−𝑝,𝐽𝑤−𝐽𝑝⟩+𝑘_𝑛−1𝑀,(3.9) for some 𝑀>0, where 𝛿_𝑛=(𝛾_𝑛+𝜃_𝑛)𝛼_𝑛.
Similarly, from (3.7), we obtain that 𝜙𝑝,𝑥_𝑛+1=≤𝜙1𝑝−,𝐽𝛿⁻¹_𝑛𝛽_𝑛𝐽𝑥_𝑛+𝛾_𝑛𝐽𝑇^𝑛𝑦_𝑛+𝜃_𝑛𝐽𝑄_𝑟𝑦_𝑛≤1−𝛿_𝑛𝜙𝑝,𝑥_𝑛+2𝛿⟨𝑧_𝑛−𝑝,𝐽𝑤−𝐽𝑝⟩+𝑘_𝑛−1𝑀−𝜃_𝑛𝛽_𝑛𝑔‖‖𝐽𝑥_𝑛−𝐽𝑄_𝑟𝑦_𝑛‖‖𝜙𝑝,𝑥_𝑛+2𝛿_𝑛⟨𝑧_𝑛−𝑝,𝐽𝑤−𝐽𝑝⟩+𝑘_𝑛−1𝑀,(∗) for some 𝑀>0. Note that {𝛿_𝑛} satisfies that lim_𝑛→∞𝛿_𝑛=0 and ∑𝛿_𝑛=∞.
Now, the rest of the proof is divided into two parts.
Case 1. Suppose that there exists 𝑛₀∈𝑁>𝑁₀ such that {𝜙(𝑝,𝑥_𝑛)} is nonincreasing for all 𝑛≥𝑛₀. In this situation, {𝜙(𝑝,𝑥_𝑛)} is then convergent. Then, from (3.8) and (*), we have that 𝛾_𝑛𝛽_𝑛𝑔‖‖𝐽𝑥_𝑛−𝐽𝑇^𝑛𝑦_𝑛‖‖⟶0,𝜃_𝑛𝛽_𝑛𝑔‖‖𝐽𝑥_𝑛−𝐽𝑄_𝑟𝑦_𝑛‖‖⟶0,(3.10) which implies, by the property of 𝑔, that 𝐽𝑥_𝑛−𝐽𝑇^𝑛𝑦_𝑛⟶0,𝐽𝑥_𝑛−𝐽𝑄_𝑟𝑦_𝑛⟶0,as𝑛⟶∞,(3.11) and, hence, since 𝐽⁻¹ is uniformly continuous on bounded sets, we obtain that 𝑥_𝑛−𝑇^𝑛𝑦_𝑛⟶0,𝑥_𝑛−𝑄_𝑟𝑦_𝑛⟶0,as𝑛⟶∞.(3.12) Furthermore, Lemma 2.3, property of 𝜙, and the fact that 𝛼_𝑛→0 as 𝑛→∞ imply that 𝜙𝑥_𝑛,𝑦_𝑛==𝜙𝜙(𝑥𝑥_𝑛𝑛,Π_𝐶𝑧_𝑛≤𝜙𝑥_𝑛,𝑧_𝑛≤𝛼_𝑛,𝐽⁻¹𝛼_𝑛𝐽𝑤+1−𝛼_𝑛𝐽𝑥_𝑛≤𝛼_𝑛𝜙𝑥_𝑛,𝑤+1−𝛼_𝑛𝜙𝑥_𝑛,𝑥_𝑛𝜙𝑥_𝑛,𝑤+1−𝛼_𝑛𝜙𝑥_𝑛,𝑥_𝑛⟶0,as𝑛⟶∞,(3.13) and hence 𝑥_𝑛−𝑦_𝑛⟶0,𝑥_𝑛−𝑧_𝑛⟶0,as𝑛⟶∞.(3.14) Therefore, from (3.12) and (3.14), we obtain that 𝑦_𝑛−𝑧_𝑛⟶0,𝑦_𝑛−𝑇^𝑛𝑦_𝑛⟶0,𝑦_𝑛−𝑄_𝑟𝑦_𝑛⟶0,as𝑛⟶∞.(3.15) But observe that from (3.2) and (3.11), we have ‖‖𝐽𝑥_𝑛+1−𝐽𝑥_𝑛‖‖≤𝛾_𝑛‖‖𝐽𝑇^𝑛𝑦_𝑛−𝐽𝑥_𝑛‖‖+𝜃_𝑛‖‖𝐽𝑄_𝑟𝑦_𝑛−𝐽𝑥_𝑛‖‖⟶0,(3.16) as 𝑛→∞. Thus, as 𝐽⁻¹ is uniformly continuous on bounded sets, we have that 𝑥_𝑛+1−𝑥_𝑛→0 which implies from (3.14) that 𝑥_𝑛+1−𝑦_𝑛→0, as 𝑛→∞, and that ‖‖𝑦_𝑛+1−𝑦_𝑛‖‖≤‖‖𝑦_𝑛+1−𝑥_𝑛+1‖‖+‖‖𝑥_𝑛+1−𝑦_𝑛‖‖⟶0,as𝑛⟶∞.(3.17) Furthermore, since ‖‖𝑦_𝑛−𝑇𝑦_𝑛‖‖≤≤‖‖‖‖𝑦𝑦_𝑛𝑛−𝑇^𝑛𝑦_𝑛‖‖+‖‖𝑇^𝑛𝑦_𝑛−𝑇^𝑛+1𝑦_𝑛‖‖+‖‖𝑇^𝑛+1𝑦_𝑛−𝑇𝑦_𝑛‖‖,+‖‖𝑦−_𝑛+1𝑇^𝑛𝑦_𝑛‖‖+‖‖𝑇^𝑛+1𝑦_𝑛−𝑇^𝑛+1𝑦_𝑛+1‖‖+‖‖𝑇^𝑛+1𝑦_𝑛+1−𝑦_𝑛+1‖‖≤‖‖𝑦_𝑛−𝑦_𝑛‖‖+‖‖𝑦_𝑛−𝑇^𝑛𝑦_𝑛‖‖+‖‖𝑇𝑇^𝑛𝑦_𝑛−𝑇𝑦_𝑛‖‖+‖‖𝑦−_𝑛+1𝑇^𝑛𝑦_𝑛‖‖+𝐿‖‖𝑦_𝑛−𝑦_𝑛+1‖‖+‖‖𝑇^𝑛+1𝑦_𝑛+1−𝑦_𝑛+1‖‖≤2‖‖𝑦_𝑛−𝑦_𝑛‖‖+‖‖𝑦_𝑛−𝑇^𝑛𝑦_𝑛‖‖+‖‖𝑇𝑇^𝑛𝑦_𝑛−𝑇𝑦_𝑛‖‖−𝑇^𝑛𝑦_𝑛‖‖+(1+𝐿)‖‖𝑦_𝑛−𝑦_𝑛+1‖‖+‖‖𝑇^𝑛+1𝑦_𝑛+1−𝑦_𝑛+1‖‖+‖‖𝑇𝑇^𝑛𝑦_𝑛−𝑇𝑦_𝑛‖‖,(3.18) we have from (3.17), (3.15), and uniform continuity of 𝑇 that ‖‖𝑦_𝑛−𝑇𝑦_𝑛‖‖⟶0,as𝑛⟶∞.(3.19) Since {𝑧_𝑛} is bounded and 𝐸 is reflexive, we choose a subsequence {𝑧_𝑛𝑖} of {𝑧_𝑛} such that 𝑧_𝑛𝑖⇀𝑧 and limsup_𝑛→∞⟨𝑧_𝑛−𝑝,𝐽𝑤−𝐽𝑝⟩=lim_𝑖→∞⟨𝑧_𝑛𝑖−𝑝,𝐽𝑤−𝐽𝑝⟩. Then, from (3.14) and (3.15) we get that 𝑦_𝑛𝑖⇀𝑧,𝑥_𝑛𝑖⇀𝑧,as𝑖⟶∞.(3.20) Thus, since 𝑇 satisfies condition (N3), we obtain from (3.19) that 𝑧∈𝐹(𝑇) and the fact that 𝑄_𝑟 is relatively nonexpansive and 𝑦_𝑛𝑖⇀𝑧 implies that 𝑧∈𝐹(𝑄_𝑟)=𝐴⁻¹(0), and, hence, using (3.1), we obtain that 𝑧∈VI(𝐶,𝐴).
Therefore, from the above discussions, we obtain that 𝑧∈𝐹∶=𝐹(𝑇)∩VI(𝐶,𝐴). Hence, by Lemma 2.6, we immediately obtain that limsup_𝑛→∞⟨𝑧_𝑛−𝑝,𝐽𝑤−𝐽𝑝⟩=lim_𝑖→∞⟨𝑧_𝑛𝑖−𝑝,𝐽𝑤−𝐽𝑝⟩=⟨𝑧−𝑝,𝐽𝑤−𝐽𝑝⟩≤0. It follows from Lemma 2.9 and (3.9) that 𝜙(𝑝,𝑥_𝑛)→0, as 𝑛→∞. Consequently, 𝑥_𝑛→𝑝.Case 2. Suppose that there exists a subsequence {𝑛_𝑖} of {𝑛} such that 𝜙𝑝,𝑥_𝑛𝑖<𝜙𝑝,𝑥_𝑛𝑖+1(3.21) for all 𝑖∈𝑁. Then, by Lemma 2.10, there exists a nondecreasing sequence {𝑚_𝑘}⊂𝑁 such that 𝑚_𝑘→∞, 𝜙(𝑝,𝑥_𝑚𝑘)≤𝜙(𝑝,𝑥_𝑚𝑘+1) and 𝜙(𝑝,𝑥_𝑘)≤𝜙(𝑝,𝑥_𝑚𝑘+1) for all 𝑘∈𝑁. Then, from (3.8), (*) and the fact 𝛿_𝑛→0, we have 𝑔‖‖𝐽𝑥_𝑚𝑘−𝐽𝑇^𝑚𝑘𝑦_𝑚𝑘‖‖⟶0,𝑔‖‖𝐽𝑥_𝑚𝑘−𝐽𝑄_𝑟𝑦_𝑚𝑘‖‖⟶0,as𝑘⟶∞.(3.22) Thus, using the same proof as in Case 1, we obtain that 𝑦_𝑚𝑘−𝑇𝑦_𝑚𝑘→0, 𝑦_𝑚𝑘−𝑄_𝑟𝑦_𝑚𝑘→0, as 𝑘→∞, and, hence, we obtain that limsup_𝑘→∞𝑧_𝑚𝑘−𝑝,𝐽𝑤−𝐽𝑝≤0.(3.23) Then, from (3.9), we have that 𝜙𝑝,𝑥_𝑚𝑘+1≤1−𝛿_𝑚𝑘𝜙𝑝,𝑥_𝑚𝑘+2𝛿_𝑚𝑘𝑧_𝑚𝑘−𝑝,𝐽𝑤−𝐽𝑝+𝑘_𝑚𝑘−1𝑀.(3.24) Since 𝜙(𝑝,𝑥_𝑚𝑘)≤𝜙(𝑝,𝑥_𝑚𝑘+1), (3.24) implies that 𝛿_𝑚𝑘𝜙𝑝,𝑥_𝑚𝑘≤≤2𝜙𝛿𝑝_𝑚𝑘,𝑥_𝑚𝑘−𝜙𝑝,𝑥_𝑚𝑘+1+2𝛿_𝑚𝑘𝑧_𝑚𝑘−𝑝,𝐽𝑤−𝐽𝑝+𝑘_𝑚𝑘−1𝑀𝑧_𝑚𝑘−𝑝,𝐽𝑤−𝐽𝑝+𝑘_𝑚𝑘−1𝑀.(3.25) In particular, since 𝛿_𝑚𝑘>0, we get 𝜙𝑝,𝑥_𝑚𝑘≤2𝑧_𝑚𝑘−𝑝,𝐽𝑤−𝐽𝑝+𝑘^𝑚𝑘−1𝛿_𝑚𝑘𝑀.(3.26) Then, from (3.23) and the fact that (𝑘_𝑚𝑘−1)/𝛿_𝑚𝑘→0, we obtain 𝜙(𝑝,𝑥_𝑚𝑘)→0, as 𝑘→∞. This together with (3.24) gives 𝜙(𝑝,𝑥_𝑚𝑘+1)→0, as 𝑘→∞. But 𝜙(𝑝,𝑥_𝑘)≤𝜙(𝑝,𝑥_𝑚𝑘+1), for all 𝑘∈𝑁, thus we obtain that 𝑥_𝑘→𝑝. Therefore, from the above two cases, we can conclude that {𝑥_𝑛} 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 𝐴∶𝐶→𝐸^∗ be a monotone mapping satisfying (2.8) and ‖𝐴𝑥‖≤‖𝐴𝑥−𝐴𝑝‖,for all 𝑥∈𝐶 and 𝑝∈VI(𝐶,𝐴). Let 𝑇∶𝐶→𝐶 be a relatively nonexpansive mapping. Assume that 𝐹∶=VI(𝐶,𝐴)∩𝐹(𝑆) is nonempty. Let {𝑥_𝑛} be a sequence generated by 𝑦𝑥₀=𝑤∈𝐶,chosenarbitrarily,_𝑛𝑥_𝑛+1=Π_𝐶𝐽⁻¹𝛼_𝑛𝐽𝑤+1−𝛼_𝑛𝐽𝑥_𝑛,=𝐽⁻¹𝛽_𝑛𝐽𝑥_𝑛+𝛾_𝑛𝐽𝑇𝑦_𝑛+𝜃_𝑛𝐽𝑄_𝑟𝑦_𝑛,(3.27) where 𝛼_𝑛∈(0,1) such that lim_𝑛→∞𝛼_𝑛=0, ∑^∞𝑛=1𝛼_𝑛=∞, {𝛽_𝑛},{𝛾_𝑛},{𝜃_𝑛}⊂[𝑐,𝑑]⊂(0,1) such that 𝛽_𝑛+𝛾_𝑛+𝜃_𝑛=1. Then, {𝑥_𝑛} 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 Π_𝐶=𝑃_𝐶, 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 𝐴∶𝐶→𝐻 be a monotone mapping satisfying (2.8) and ‖𝐴𝑥‖≤‖𝐴𝑥−𝐴𝑝‖,for all 𝑥∈𝐶 and 𝑝∈𝑉𝐼(𝐶,𝐴). Let 𝑇∶𝐶→𝐶 be a uniformly 𝐿-Lipschitzian relatively asymptotically nonexpansive mapping with sequence {𝑘_𝑛}. Assume that 𝐹∶=𝑉𝐼(𝐶,𝐴)∩𝐹(𝑆) is nonempty. Let {𝑥_𝑛} be a sequence generated by 𝑥𝑦₀=𝑤∈𝐶,chosenarbitrarily,_𝑛𝑥_𝑛+1=𝑃_𝐶𝛼_𝑛𝑤+1−𝛼_𝑛𝑥_𝑛,=𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝑇^𝑛𝑦_𝑛+𝜃_𝑛𝑄_𝑟𝑦_𝑛,(3.28) where 𝛼_𝑛∈(0,1) such that lim_𝑛→∞𝛼_𝑛=0,lim_𝑛→∞((𝑘_𝑛−1)/𝛼_𝑛)=0,∑^∞𝑛=1𝛼_𝑛=∞, {𝛽_𝑛},{𝛾_𝑛},{𝜃_𝑛}⊂[𝑐,𝑑]⊂(0,1) such that 𝛽_𝑛+𝛾_𝑛+𝜃_𝑛=1. Then, {𝑥_𝑛} 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 𝐴∶𝐶→𝐸^∗ be a monotone mapping satisfying (2.8). Let 𝑇∶𝐶→𝐶 be a uniformly 𝐿-Lipschitzian relatively asymptotically nonexpansive mapping with sequence {𝑘_𝑛}. Assume that 𝐹∶=𝐴⁻¹(0)∩𝐹(𝑆) is nonempty. Let 𝑄_𝑟 be the resolvent of 𝐴 and {𝑥_𝑛} a sequence generated by 𝑦𝑥₀=𝑤∈𝐶,chosenarbitrarily,_𝑛𝑥_𝑛+1=Π_𝐶𝐽⁻¹𝛼_𝑛𝐽𝑤+1−𝛼_𝑛𝐽𝑥_𝑛,=𝐽⁻¹𝛽_𝑛𝐽𝑥_𝑛+𝛾_𝑛𝐽𝑇^𝑛𝑦_𝑛+𝜃_𝑛𝐽𝑄_𝑟𝑦_𝑛,(3.29) where 𝛼_𝑛∈(0,1) such that lim_𝑛→∞𝛼_𝑛=0,lim_𝑛→∞((𝑘_𝑛−1)/𝛼_𝑛)=0, ∑^∞𝑛=1𝛼_𝑛=∞, {𝛽_𝑛},{𝛾_𝑛},{𝜃_𝑛}⊂[𝑐,𝑑]⊂(0,1) such that 𝛽_𝑛+𝛾_𝑛+𝜃_𝑛=1. Then, {𝑥_𝑛} 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 𝑇∶𝐶→𝐶 be a uniformly 𝐿-Lipschitzian relatively asymptotically nonexpansive mapping with sequence {𝑘_𝑛}. Assume that 𝐹∶=𝐹(𝑆) is nonempty. Let {𝑥_𝑛} be a sequence generated by 𝑦𝑥₀=𝑤∈𝐶,chosenarbitrarily,_𝑛𝑥_𝑛+1=Π_𝐶𝐽⁻¹𝛼_𝑛𝐽𝑤+1−𝛼_𝑛𝐽𝑥_𝑛,=𝐽⁻¹𝛽_𝑛𝐽𝑥_𝑛+1−𝛽_𝑛𝐽𝑇^𝑛𝑦_𝑛,(3.30) where 𝛼_𝑛∈(0,1) such that lim_𝑛→∞𝛼_𝑛=0,lim_𝑛→∞((𝑘_𝑛−1)/𝛼_𝑛)=0, ∑^∞𝑛=1𝛼_𝑛=∞, {𝛽_𝑛}⊂[𝑐,𝑑]⊂(0,1). Then, {𝑥_𝑛} 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 𝐴∶𝐶→𝐸^∗ be a monotone mapping satisfying (2.8). Assume that 𝐹∶=𝐴⁻¹(0) is nonempty. Let 𝑄_𝑟 be the resolvent of 𝐴 and {𝑥_𝑛} a sequence generated by 𝑦𝑥₀=𝑤∈𝐶,chosenarbitrarily,_𝑛𝑥_𝑛+1=Π_𝐶𝐽⁻¹𝛼_𝑛𝐽𝑤+1−𝛼_𝑛𝐽𝑥_𝑛,=𝐽⁻¹𝛽_𝑛𝐽𝑥_𝑛+1−𝛽_𝑛𝐽𝑄_𝑟𝑦_𝑛,(3.31) where 𝛼_𝑛∈(0,1) such that lim_𝑛→∞𝛼_𝑛=0, ∑^∞𝑛=1𝛼_𝑛=∞, {𝛽_𝑛}⊂[𝑐,𝑑]⊂(0,1). Then, {𝑥_𝑛} 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 𝐴∶𝐶→𝐸^∗ be a monotone mapping satisfying (2.8). Let 𝑇∶𝐶→𝐶 be a relatively nonexpansive mapping. Assume that 𝐹∶=𝐴⁻¹(0)∩𝐹(𝑆) is nonempty. Let {𝑥_𝑛} be a sequence generated by 𝑦𝑥₀=𝑤∈𝐶,chosenarbitrarily,_𝑛𝑥_𝑛+1=Π_𝐶𝐽⁻¹𝛼_𝑛𝐽𝑤+1−𝛼_𝑛𝐽𝑥_𝑛,=𝐽⁻¹𝛽_𝑛𝐽𝑥_𝑛+𝛾_𝑛𝐽𝑇𝑦_𝑛+𝜃_𝑛𝐽𝑄_𝑟𝑦_𝑛,(3.32) where 𝛼_𝑛∈(0,1) such that lim_𝑛→∞𝛼_𝑛=0, ∑^∞𝑛=1𝛼_𝑛=∞, {𝛽_𝑛},{𝛾_𝑛},{𝜃_𝑛}⊂[𝑐,𝑑]⊂(0,1) such that 𝛽_𝑛+𝛾_𝑛+𝜃_𝑛=1. Then, {𝑥_𝑛} 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,𝐵∶𝐶→𝐸^∗ be monotone mappings satisfying (2.8). Suppose that 𝑇₁=(𝐽+𝑟𝐴)⁻¹𝐽 and 𝑇₂=(𝐽+𝑟𝐵)⁻¹𝐽. Assume that 𝐹∶=𝐴⁻¹(0)∩𝐵⁻¹(0) is nonempty. Let {𝑥_𝑛} be a sequence generated by 𝑦𝑥₀=𝑤∈𝐶,chosenarbitrarily,_𝑛𝑥_𝑛+1=Π_𝐶𝐽⁻¹𝛼_𝑛𝐽𝑤+1−𝛼_𝑛𝐽𝑥_𝑛,=𝐽⁻¹𝛽_𝑛𝐽𝑥_𝑛+𝛾_𝑛𝐽𝑇₁𝑦_𝑛+𝜃_𝑛𝐽𝑇₂𝑦_𝑛,(3.33) where 𝛼_𝑛∈(0,1) such that lim_𝑛→∞𝛼_𝑛=0, ∑^∞𝑛=1𝛼_𝑛=∞, {𝛽_𝑛},{𝛾_𝑛},{𝜃_𝑛}⊂[𝑐,𝑑]⊂(0,1) such that 𝛽_𝑛+𝛾_𝑛+𝜃_𝑛=1. Then, {𝑥_𝑛} 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)=𝐴⁻¹(0) and 𝐹(𝑇₂)=𝐵⁻¹(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 𝐴_𝑖∶𝐶→𝐸^∗, 𝑖=1,2,…,𝑁 be monotone mappings satisfying (2.8). Suppose that 𝑇_𝑖=(𝐽+𝑟𝐴_𝑖)⁻¹𝐽, and assume that 𝐹∶=∩^𝑁𝑖=1𝐴^𝑖−1(0) is nonempty. Let {𝑥_𝑛} be a sequence generated by 𝑦𝑥₀=𝑤∈𝐶,chosenarbitrarily,_𝑛𝑥_𝑛+1=Π_𝐶𝐽⁻¹𝛼_𝑛𝐽𝑤+1−𝛼_𝑛𝐽𝑥_𝑛,=𝐽⁻¹𝛽_n,0𝐽𝑥_𝑛+𝛽_𝑛,1𝐽𝑇₁𝑦_𝑛+⋯+𝛽_𝑛,𝑁𝐽𝑇_𝑁𝑦_𝑛,(3.34) where 𝛼_𝑛∈(0,1) such that lim_𝑛→∞𝛼_𝑛=0, ∑^∞𝑛=1𝛼_𝑛=∞, {𝛽_𝑛,𝑖}⊂[𝑐,𝑑]⊂(0,1), for 𝑖=0,1,2,…,𝑁, such that ∑^𝑁𝑖=0𝛽_𝑛,𝑖=1. Then, {𝑥_𝑛} converges strongly to an element of 𝐹.

A monotone mapping 𝐴∶𝐶→𝐸^∗ 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 𝐴⁻¹(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 𝐴∶𝐶→𝐸^∗ 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 𝐴∶𝐶→𝐸^∗ be a maximal monotone mapping. Let 𝑇∶𝐶→𝐶 be a uniformly 𝐿-Lipschitzian relatively asymptotically nonexpansive mapping with sequence {𝑘_𝑛}. Assume that 𝐹∶=𝐴⁻¹(0)∩𝐹(𝑆) is nonempty. Let {𝑥_𝑛} be a sequence generated by 𝑦𝑥₀=𝑤∈𝐶,chosenarbitrarily,_𝑛𝑥_𝑛+1=Π_𝐶𝐽⁻¹𝛼_𝑛𝐽𝑤+1−𝛼_𝑛𝐽𝑥_𝑛,=𝐽⁻¹𝛽_𝑛𝐽𝑥_𝑛+𝛾_𝑛𝐽𝑇^𝑛𝑦_𝑛+𝜃_𝑛𝐽𝑄_𝑟𝑦_𝑛,(3.35) where 𝛼_𝑛∈(0,1) such that lim_𝑛→∞𝛼_𝑛=0,lim_𝑛→∞((𝑘_𝑛−1)/𝛼_𝑛)=0, ∑^∞𝑛=1𝛼_𝑛=∞, {𝛽_𝑛},{𝛾_𝑛},{𝜃_𝑛}⊂[𝑐,𝑑]⊂(0,1) such that 𝛽_𝑛+𝛾_𝑛+𝜃_𝑛=1. Then, {𝑥_𝑛} 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,𝑔∶𝐸→(−∞,∞) be a proper lower semicontinuous convex functions. Assume that 𝐹∶=(𝜕𝑓)⁻¹(0)∩(𝜕𝑔)⁻¹(0) is nonempty. Let {𝑥_𝑛} be a sequence generated by 𝑦𝑥₀=𝑤∈𝐶,chosenarbitrarily,_𝑛ℎ_𝑛=Π_𝐶𝐽⁻¹𝛼_𝑛𝐽𝑤+1−𝛼_𝑛𝐽𝑥_𝑛,=argmin𝑓(𝑧)+1𝑡2𝑟𝜙𝑧,𝑦^𝑛,𝑧∈𝐸_𝑛,=argmin𝑔(𝑧)+1𝑥2𝑟𝜙𝑧,𝑦^𝑛,𝑧∈𝐸_𝑛+1,=𝐽⁻¹𝛽_𝑛𝐽𝑥_𝑛+𝛾_𝑛𝐽ℎ_𝑛+𝜃_𝑛𝐽𝑡_𝑛,(4.1) where 𝛼_𝑛∈(0,1) such that lim_𝑛→∞𝛼_𝑛=0, ∑^∞𝑛=1𝛼_𝑛=∞, {𝛽_𝑛},{𝛾_𝑛},{𝜃_𝑛}⊂[𝑐,𝑑]⊂(0,1) such that 𝛽_𝑛+𝛾_𝑛+𝜃_𝑛=1. Then, {𝑥_𝑛} converges strongly to an element of 𝐹.

Proof. Let 𝐴 and 𝐵 be operators defined by 𝐴=𝜕𝑓 and 𝐵=𝜕𝑔 and 𝑄_𝑟=(𝐽+𝑟𝐴)⁻¹𝐽, 𝑄^𝐵𝑟=(𝐽+𝑟𝐵)⁻¹𝐽 for all 𝑟>0. Then, by Rockafellar [29], 𝐴 and 𝐵 are maximal monotone mappings. We also have that ℎ_𝑛=𝑄^𝐴𝑟𝑦=argmin𝑓(𝑧)+1𝑡2𝑟𝜙(𝑧,𝑦),𝑧∈𝐸_𝑛,=𝑄^𝐵𝑟𝑦=argmin𝑔(𝑧)+12𝑟𝜙(𝑧,𝑦),𝑧∈𝐸,(4.2) for all 𝑦∈𝐸 and 𝑟>0. Furthermore, we have that 𝐹(𝑄^𝐴𝑟)=𝐴⁻¹(0) and 𝐹(𝑄^𝐵𝑟)=𝐵⁻¹(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 𝐶_𝑛+1 for each 𝑛≥1 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 𝐶_𝑛+1 for each 𝑛≥1 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 𝐶_𝑛+1 for each 𝑛≥1 or the class of mappings considered in our corollary is more general.