Modified Relaxed Extragradient Method for a General System of Variational Inequalities and Nonexpansive Mappings in Banach Spaces

Abstract

The purpose of this paper is to introduce a new modified relaxed extragradient method and study for finding some common solutions for a general system of variational inequalities with inversestrongly monotone mappings and nonexpansive mappings in the framework of real Banach spaces. By using the demiclosedness principle, it is proved that the iterative sequence defined by the relaxed extragradient method converges strongly to a common solution for the system of variational inequalities and nonexpansive mappings under quite mild conditions.

1. Introduction

Let 𝐻 be a real Hilbert space with inner product ⟨⋅,⋅⟩ and norm ||⋅||, and 𝐶 be a nonempty closed convex subset of 𝐻. Let 𝑃_𝐶 be the projection of 𝐻 onto 𝐶, it is known that projection operator 𝑃_𝐶 is nonexpansive and satisfies the following: ⟨𝑥−𝑦,𝑃_𝐶𝑥−𝑃_𝐶𝑦⟩≥‖‖𝑃_𝐶𝑥−𝑃_𝐶𝑦‖‖²,∀𝑥,𝑦∈𝐻.(1.1) Moreover, 𝑃_𝐶𝑥 is characterized by the properties 𝑃_𝐶𝑥∈𝐶 and ⟨𝑥−𝑃_𝐶𝑥,𝑃_𝐶𝑥−𝑦⟩≥0 for all 𝑦∈𝐶.

Let 𝐴∶𝐶→𝐻 be a mapping. Recall that the classical variational inequality, denoted by VI(𝐶,𝐴), is to find 𝑢∈𝑉 such that ⟨𝐴𝑢,𝑣−𝑢⟩≥0,∀𝑣∈𝐶.(1.2)

One can see that the variational inequality (1.2) is equivalent to a fixed point problem. 𝐴 element 𝑥^∗∈𝐶 is a solution of the variational inequality (1.2) if and only if 𝑥^∗∈𝐶 is a fixed point of the mapping 𝑃_𝐶(𝐼−𝜆𝐴), where 𝐼 is the identity mapping and 𝜆>0 is a constant.

Variational inequalities introduced by Stampacchia [1] in the early sixties have had a great impact and influence in the development of almost all branches of pure and applied sciences and have witnessed an explosive growth in theoretical advances, algorithmic development, and so forth; see, for example, [1–18] and the references therein.

For a monotone mapping 𝐴∶𝐶→𝐻, Noor [2] studied the following problem of finding (𝑥^∗;𝑦^∗)∈𝐶×𝐶 such that: ⟨⟨𝜇𝜆𝐴𝐴𝑦𝑥^∗_∗+𝑥^∗−𝑦^∗,𝑥−𝑥^∗⟩≥0,∀𝑥∈𝐶,+𝑦^∗−𝑥^∗,𝑥−𝑦^∗⟩≥0,∀𝑥∈𝐶,(1.3) where 𝜆,𝜇>0 are constants. If we add up the requirement that 𝑥^∗=𝑦^∗, then the problem (1.3) is reduced to the classical variational inequality (1.2). The problem of finding solutions of (1.3) by using iterative methods has been studied by many authors; see [2–9] and the references therein.

Recently, some authors also studied the problem of finding a common element of the fixed point set of nonexpansive mappings and the solution set of variational inequalities for 𝛼-inversestrongly monotone mappings in the framework of real Hilbert spaces [10] and Banach spaces [11].

On the other hands, Ceng et al. [12] introduce the following general system of variational inequalities involving two different operators. For two given operators, consider the problem finding 𝑥^∗,𝑦^∗∈𝐶 such that ⟨⟨𝜇𝜆𝐴𝐵𝑦𝑥^∗_∗+𝑥^∗−𝑦^∗,𝑥−𝑥^∗⟩≥0,∀𝑥∈𝐶,+𝑦^∗−𝑥^∗,𝑥−𝑦^∗⟩≥0,∀𝑥∈𝐶,(1.4) where 𝜆,𝜇>0 are constants. To illustrate the applications of this system, we can refer to an example of related nonlinear optimization problem put forward by Zhu and Marcotte [13]. Very recently, Yao et al. [14] extend the system of variational inequality problems (1.4) to Banach spaces.

In the present paper, motivated and inspired by the methods of Ceng et al. [12], Iiduka and Takahashi [10], Qin et al. [11], and Yao et al. [14], we consider the following general system of variational inequalities in Banach spaces.

Let 𝐶 be a nonempty closed convex subset of a real smooth Banach space. Let 𝐴,𝐵∶𝐶→𝐸 be 𝛼-inversestrongly accretive mapping and 𝛽-inverse-strongly accretive mapping. Find (𝑥^∗,𝑦^∗)∈𝐶×𝐶 such that 𝜇𝜆𝐴𝐵𝑦𝑥^∗_∗+𝑥^∗−𝑦^∗,𝑗𝑥−𝑥^∗≥0,∀𝑥∈𝐶,+𝑦^∗−𝑥^∗,𝑗𝑥−𝑦^∗≥0,∀𝑥∈𝐶,(1.5) where 𝜆,𝜇>0 are constants, and 𝑗 is the normalized duality mapping. For more details of 𝑗, one may see Li [15]. In a real Hilbert space, 𝑗=𝐼 is the identity mapping, and the system (1.5) is reduced to (1.4). If we add up the requirement that 𝐴=𝐵, then the problem (1.4) is reduced to the generalized variational inequality (1.3), in particular.

And we consider the problem of finding a common element of the fixed point set of nonexpansive mappings and the solution set of the general system of variational inequalities for 𝛼-inversestrongly monotone mappings in the framework of real Banach spaces. By using the demiclosedness principle, we prove that under quite mild conditions the iterative sequence defined by the relaxed extragradient method converges strongly to a common solution of this system of variational inequalities and nonexpansive mappings. Our results improve and extend the corresponding results announced by other authors, such as [2–4, 6, 8, 10–12, 14].

2. Preliminaries

Let 𝐶 be a nonempty closed convex subset of a Banach space of 𝐸. Let 𝐸^∗ be the dual space of 𝐸, and let ⟨⋅,⋅⟩ denote the pairing between 𝐸 and 𝐸^∗. For 𝑞>1, the generalized duality mapping 𝐽_𝑞∶𝐸→2^𝐸∗ is defined by 𝐽_𝑞(𝑥)=𝑓∈𝐸^∗∶⟨𝑥,𝑓⟩=‖𝑥‖^𝑞,‖𝑓‖=‖𝑥‖^𝑞,(2.1) for all 𝑥∈𝐸. In particular, 𝐽=𝐽₂ is called the normalized duality mapping. It is known that 𝐽_𝑞(𝑥)=||𝑥||^𝑞−2𝐽(𝑥) for all 𝑥∈𝐸. If 𝐸 is a Hilbert space, then 𝐽=𝐼 is the identity mapping. Further, we have the following properties of the generalized duality mapping 𝐽_𝑞: (1)𝐽_𝑞(𝑥)=||𝑥||^𝑞−2𝐽₂(𝑥) for all 𝑥∈𝐸 with 𝑥≠0,(2)𝐽_𝑞(𝑡𝑥)=𝑡^𝑞−1𝐽_𝑞(𝑥) for all 𝑥∈𝐸 and 𝑡∈[0,∞),(3)𝐽_𝑞(−𝑥)=−𝐽_𝑞(𝑥) for all 𝑥∈𝐸.

Let 𝑈={𝑥∈𝐸∶||𝑥||=1}. 𝐸 is said to be uniformly convex if, for any 𝜖∈(0,2], there exists 𝛿>0 such that for any 𝑥,𝑦∈𝑈. ||𝑥−𝑦||≥𝜖 implies ||(𝑥+𝑦)/2||≤(1−𝛿).

It is known that a uniformly convex Banach space is reflexive and strictly convex, 𝐸 is said to be Gâteaux differentiable if the limit Lim_𝑡→0‖𝑥+𝑡𝑦‖−‖𝑥‖𝑡(2.2) exists for each 𝑥,𝑦∈𝑈. In this case, 𝐸 is said to be smooth. The norm of 𝐸 is said to be uniformly Gâteaux differentiable if, for each 𝑦∈𝑈, the limit (2.2) is attained uniformly for 𝑥∈𝑈. The norm of 𝐸 is said to be Frêchet differentiable, if, for each 𝑥∈𝑈, the limit (2.2) is attained uniformly for 𝑦∈𝑈. The norm of 𝐸 is said to be uniformly Frêchet differentiable if the limit (2.2) is attained uniformly for 𝑥,𝑦∈𝑈. It is well-known that (uniform) Frêchet differentiable of the norm of 𝐸 implies (uniform) Gâteaux differentiability of the norm of 𝐸.

The modulus of smoothness of 𝐸 is defined by 𝜌(𝜏)=sup12||||𝑥+𝑦||||+||||𝑥−𝑦||||−1∶𝑥,𝑦∈𝐸,|||𝑥|||=1,||||𝑦||||≤𝑡,(2.3) where 𝜌∶[0,∞)→[0,∞) is a function. It is known that a Banach space 𝐸 is uniformly smooth if and only if lim_(𝑛→∞)(𝜌(𝑡)/𝑡)=0. Let 𝑞 be a fixed real number with 1<𝑞≤2. A Banach space 𝐸 is said to be 𝑞 uniformly smooth if there exists a fixed constant 𝑐>0 such that 𝜌(𝑡)≤𝑐𝑡^𝑞, for all 𝑡>0.

Next, we always assume that 𝐸 is a smooth Banach space. Let 𝐶 be a nonempty closed convex subsets of 𝐸. Recall that an operator 𝐴 of 𝐶 into 𝐸 is said to be accretive if ⟨𝐴𝑥−𝐴𝑦,𝑗(𝑥−𝑦)⟩≥0,∀𝑥,𝑦∈𝐶.(2.4)𝐴 is said to be 𝛼-inversestrongly accretive if there exists a constant 𝛼>0 such that ⟨𝐴𝑥−𝐴𝑦,𝑗(𝑥−𝑦)⟩≥𝛼‖𝐴𝑥−𝐴𝑦‖²,∀𝑥,𝑦∈𝐶.(2.5)

Let 𝐷 be a subset of 𝐶 and 𝑄 be a mapping of 𝐶 into 𝐷. Then 𝑄 is said to be sunny if 𝑄(𝑄𝑥+𝑡(𝑥−𝑄𝑥))=𝑄𝑥,(2.6) whenever 𝑄𝑥+𝑡(𝑥−𝑄𝑥)∈𝐶 for 𝑥∈𝐶 and 𝑡≥0. A subset 𝐷 of 𝐶 is called a sunny nonexpansive retract of 𝐶 if there exists a sunny nonexpansive retraction from 𝐶 onto 𝐷.

In order to prove the main result, we also need the following lemmas. The following Lemma 2.2 describes characterization of sunny nonexpansive retraction on a smooth Banach space.

Lemma 2.1 (see [16]). Let 𝐸 be a real 2 uniformly smooth Banach space with the best smooth constant 𝐾. Then the following inequality holds: ‖𝑥+𝑦‖²≤‖𝑥‖²+2⟨𝑦,𝑗𝑥⟩+2‖𝐾𝑦‖²,∀𝑥,𝑦∈𝐸.(2.7)

Lemma 2.2 (see [17]). Let 𝐶 be a closed convex subset of a smooth Banach space 𝐸, let 𝐷 be a nonempty subset of 𝐶, and let 𝑄 be a retraction form 𝐶 onto 𝐷. Then 𝑄 is sunny and nonexpansive if and only if ⟨𝑢−𝑄𝑢,𝑗(𝑦−𝑄𝑢)⟩≤0,(2.8) for all 𝑢∈𝐶 and 𝑦∈𝐷.

Lemma 2.3 (see [18]). Let {𝑥_𝑛} and {𝑦_𝑛} be bounded sequences in a Banach space 𝐸 and a sequence {𝛽_𝑛} in [0,1] with 0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1.(2.9) Suppose that 𝑥_𝑛+1=(1−𝛽_𝑛)𝑦_𝑛+𝛽_𝑛𝑥_𝑛 for all integers 𝑛≥0 and limsup_𝑛→∞||||𝑦_𝑛+1−𝑦_𝑛||||−||||𝑥_𝑛+1−𝑥_𝑛||||≤0.(2.10) Then, lim_(𝑛→∞)||𝑦_𝑛−𝑥_𝑛||=0.

Lemma 2.4 (see [19]). Let 𝐶 be a nonempty closed convex subset of a real uniformly smooth Banach space. Let 𝑆₁ and 𝑆₂ be two nonexpansive mappings from 𝐶 into itself with a common fixed point. Define a mapping 𝑆∶𝐶→𝐶 by 𝑆𝑥=𝛿𝑆₁𝑥+(1−𝛿)𝑆₂𝑥,∀𝑥∈𝐶,(2.11) where 𝛿 is a constant in (0,1). Then 𝑆 is nonexpansive and 𝐹(𝑆)=𝐹(𝑆₁)⋂𝐹(𝑆₂).

Lemma 2.5 (see [20]). Assume that {𝛼_𝑛} is a sequence of nonnegative real numbers such that 𝛼_𝑛+1≤1−𝛾_𝑛𝛼_𝑛+𝛿_𝑛,(2.12) where {𝛾_𝑛} is a sequence in (0,1) and {𝛿_𝑛} is a sequence such that (a)∑^∞𝑛=1𝛾_𝑛=∞, (b)lim_(𝑛→∞)(𝛿_𝑛/𝛾_𝑛)≤0,or∑^∞𝑛=1|𝛿_𝑛|<∞. Then lim_(𝑛→∞)𝛼_𝑛=0.

Lemma 2.6. Let 𝐶 be a nonempty closed convex subset of a real 2 uniformly smooth Banach space 𝐸 with the best smooth constant 𝐾. Let the mappings 𝐴,𝐵∶𝐶→𝐸 be 𝛼-inverse strongly accretive and 𝛽-inverse strongly accretive, respectively, and then one has 𝐼−𝜆𝐴 and 𝐼−𝜇𝐵 are nonexpansive, where 𝜆∈(0,𝛼/𝐾²), 𝜇∈(0,𝛽/𝐾²).

Proof. Indeed, for all 𝑥,𝑦∈𝐶, from Lemma 2.1, we have ‖(𝐼−𝜆𝐴)𝑥−(𝐼−𝜆𝐴)𝑦‖²=‖(𝑥−𝑦)−𝜆(𝐴𝑥−𝐴𝑦)‖²≤‖𝑥−𝑦‖²−2𝜆⟨𝐴𝑥−𝐴𝑦,𝑗(𝑥−𝑦)⟩+2𝐾²𝜆²‖𝐴𝑥−𝐴𝑦‖²≤‖𝑥−𝑦‖²−2𝜆𝛼‖𝐴𝑥−𝐴𝑦‖²+2𝐾²𝜆²‖𝐴𝑥−𝐴𝑦‖²≤‖𝑥−𝑦‖²+2𝜆𝐾²𝜆−𝛼‖𝐴𝑥−𝐴𝑦‖²≤‖𝑥−𝑦‖².(2.13) This shows that 𝐼−𝜆𝐴 is nonexpansive mapping, so is 𝐼−𝜇𝐵.

Lemma 2.7. Let 𝐶 be a nonempty closed convex subset of a real 2 uniformly smooth Banach space 𝐸. Let 𝑄_𝐶 be the sunny nonexpansive retraction from 𝐸 onto 𝐶. Let 𝐴,𝐵∶𝐶→𝐸 be 𝛼-inverse strongly accretive mapping and 𝛽-inverse strongly accretive mapping, respectively. Let 𝐺∶𝐶→𝐶 be a mapping defined by 𝐺(𝑥)=𝑄_𝐶𝑄_𝐶(𝑥−𝜇𝐵𝑥)−𝜆𝐴𝑄_𝐶(𝑥−𝜇𝐵𝑥),∀𝑥∈𝐶.(2.14) If 𝜆∈(0,𝛼/𝐾²), 𝜇∈(0,𝛽/𝐾²), then 𝐺∶𝐶→𝐶 is nonexpansive.

Proof. For all 𝑥,𝑦∈𝐶, from Lemma 2.6, we have ||||𝐺(𝑥)−𝐺(𝑦)||||=‖‖𝑄−_𝐶𝑄_𝐶𝑄_𝐶(𝑥−𝜇𝐵𝑥)−𝜆𝐴𝑄_𝐶(𝑥−𝜇𝐵𝑥)≤‖‖𝑄𝑄_𝐶(𝑦−𝜇𝐵𝑦)−𝜆𝐴𝑄_𝐶(𝑦−𝜇𝐵𝑦)_𝐶‖‖=‖‖(𝐼−(𝑥𝜆−𝐴)𝜇𝑄𝐵_𝐶𝑥)−𝜆𝐴𝑄_𝐶(𝑥−𝜇𝐵𝑥)−𝑄_𝐶(𝑦−𝜇𝐵𝑦)−𝜆𝐴𝑄_𝐶(𝑦−𝜇𝐵𝑦)‖‖≤‖‖𝑄(𝐼−𝜇𝐵)𝑥−(𝐼−𝜆𝐴)𝑄_𝐶(𝐼−𝜇𝐵)𝑦_𝐶‖‖≤≤‖‖𝑥(𝐼−−(𝑦𝐼𝜇‖−𝐵.)𝜇𝑥𝐵−)𝑥(𝐼−−𝑄𝜇_𝐶𝐵(𝐼)𝑦−‖𝜇𝐵)𝑦‖‖(2.15) Therefore, from (2.15), we obtain immediately that the mapping 𝐺 is nonexpansive.

Lemma 2.8. Let 𝐶 be a nonempty closed convex subset of a real smooth Banach space 𝐸. Let 𝑄_𝐶 be the sunny nonexpansive retraction from 𝐸 onto 𝐶. Let 𝐴,𝐵∶𝐶→𝐸 be two possibly nonlinear mappings. For given 𝑥^∗,𝑦^∗∈𝐶, (𝑥^∗,𝑦^∗) is a solution of problem (1.5) if and only if 𝑥^∗=𝑄_𝐶(𝑦^∗−𝜆𝐴𝑦^∗), where 𝑦^∗=𝑄_𝐶(𝑥^∗−𝜇𝐵𝑥^∗).

Proof. We note that we can rewrite (1.5) as 𝑥𝑦^∗−𝑦^∗−𝜆𝐴𝑦^∗,𝑗𝑥−𝑥^∗≥0,∀𝑥∈𝐶,_∗−𝑥^∗−𝜆𝐵𝑥^∗,𝑗𝑥−𝑦^∗≥0,∀𝑥∈𝐶.(2.16) From Lemma 2.2, we can deduce that (2.16) is equivalent to 𝑦𝑥^∗_∗=𝑄_𝐶𝑦^∗−𝜆𝐴𝑦^∗,=𝑄_𝐶𝑥^∗−𝜇𝐵𝑥^∗.(2.17) This completes the proof.

Remark 2.9. From Lemma 2.8, we note that 𝑥^∗=𝑄_𝐶[𝑄_𝐶(𝑥^∗−𝜇𝐵𝑥^∗)−𝜆𝐴𝑄_𝐶(𝑥^∗−𝜇𝐵𝑥^∗)], which implies that 𝑥^∗ is a fixed point of the mappings 𝐺.

3. Main Result

To solve the general system of variation inequality problem (1.5), now we are in a position to state and prove the main result in this paper.

Theorem 3.1. Let 𝐸 be a uniformly convex and 2 uniformly smooth Banach space with the best smooth constant 𝐾, 𝐶 a nonempty closed convex subset of 𝐸, and 𝑄_𝐶 be the sunny nonexpansive retraction from 𝐸 onto 𝐶. Let 𝐴,𝐵∶𝐶→𝐸 be 𝛼-inverse strongly accretive mapping and 𝛽-inverse strongly accretive mapping, respectively, and 𝑆∶𝐶→𝐶 a nonexpansive mapping with a fixed point. Assume that 𝐹=𝐹(𝑆)⋂𝐹(𝐺)≠∅, where 𝐺 is defined as Lemma 2.7. Let {𝑥_𝑛} be a sequence generated in the following manner: 𝑦𝑥₁=𝑢∈𝐶,_𝑛𝑥_𝑛+1=𝑄_𝐶𝑥_𝑛−𝜇𝐵𝑥_𝑛,=𝛼_𝑛𝑢+𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝛿_𝑛𝑆𝑥_𝑛+1−𝛿_𝑛𝑄_𝐶𝑦_𝑛−𝜆𝐴𝑦_𝑛,𝑛≥1,(3.1) where 𝛿_𝑛⊂[0,1], 𝜆∈(0,𝛼/𝐾²), 𝜇∈(0,𝛽/𝐾²), and {𝛼_𝑛},{𝛽_𝑛}, and {𝛾_𝑛} are three sequences in (0,1), and the following conditions are satisfied (1)𝛼_𝑛+𝛽_𝑛+𝛾_𝑛=1,forall𝑛≥1, (2)lim_(𝑛→∞)𝛼_𝑛=0,∑^∞𝑛=0𝛼_𝑛=∞, (3)0<liminf_(𝑛→∞)𝛽_𝑛≤limsup_(𝑛→∞)𝛽_𝑛<1, (4)lim_(𝑛→∞)𝛿_𝑛=𝛿∈(0,1).
Then the sequence {𝑥_𝑛} defined by (3.1) converges strongly to 𝑥⁻=𝑄_𝐹𝑢, and (𝑥⁻,𝑦⁻) is a solution of the problem (1.5), where 𝑦⁻=𝑄_𝐶(𝑥⁻−𝜇𝐵𝑥⁻), 𝑄_𝐹 is a sunny nonexpansive retraction of 𝐶 onto 𝐹.

Proof. We divide the proof of Theorem 3.1 into six steps.
Step 1. First, we prove that 𝐹 is closed and convex. We know that 𝐹(𝑆) is closed and convex. Next, we show that 𝐹(𝐺) is closed and convex. From Lemma 3.1 and 3.2, we can see that 𝐼−𝜆𝐴,𝐼−𝜇𝐵,𝐺 are nonexpansive. This shows that 𝐹=𝐹(𝑆)⋂𝐹(𝐺) is closed and convex.
Step 2. Now we prove that the sequences {𝑥_𝑛},{𝑦_𝑛},{𝑡_𝑛},{𝐴𝑦_𝑛}, and {𝐵𝑥_𝑛} are bounded. Let 𝑥^∗∈𝐹(𝑆)⋂𝐹(𝐺), and from Remark 2.9, we obtain that 𝑥^∗=𝑄_𝐶𝑄_𝐶𝑥^∗−𝜇𝐵𝑥^∗−𝜆𝐴𝑄_𝐶𝑥^∗−𝜇𝐵𝑥^∗.(3.2) Putting 𝑦^∗=𝑄_𝐶(𝑥^∗−𝜇𝐵𝑥^∗), we see that 𝑥^∗=𝑄_𝐶𝑦^∗−𝜆𝐴𝑦^∗.(3.3) Putting 𝑡_𝑛=𝛿_𝑛𝑆𝑥_𝑛+(1−𝛿_𝑛)𝑄_𝐶(𝑦_𝑛−𝜆𝐴𝑦_𝑛) for each 𝑛≥1, we arrive at ‖‖𝑡_𝑛−𝑥^∗‖‖=≤𝛿‖‖_𝑛𝛿_𝑛𝑆𝑥_𝑛+1−𝛿_𝑛𝑄_𝐶𝑦_𝑛−𝜆𝐴𝑦_𝑛−𝑥^∗‖‖≤𝛿_𝑛‖‖𝑆𝑥_𝑛−𝑥^∗‖‖+1−𝛿_𝑛‖‖𝑄_𝐶𝑦_𝑛−𝜆𝐴𝑦_𝑛−𝑥^∗‖‖≤𝛿_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖+1−𝛿_𝑛‖‖𝑄_𝐶𝑦_𝑛−𝜆𝐴𝑦_𝑛−𝑄_𝐶𝑦^∗−𝜆𝐴𝑦^∗‖‖=𝛿_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖+1−𝛿_𝑛‖‖𝑦_𝑛−𝑦^∗‖‖≤𝛿_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖+1−𝛿_𝑛‖‖𝑄_𝐶𝑥_𝑛−𝜇𝐵𝑥_𝑛−𝑄_𝐶𝑥^∗−𝜇𝐵𝑥^∗‖‖=‖‖𝑥‖‖_𝑛𝑥_𝑛−𝑥^∗‖‖+1−𝛿_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖−𝑥^∗‖‖.(3.4) Hence, it follows that ‖‖𝑥_𝑛+1−𝑥^∗‖‖≤=𝛼‖‖_𝑛𝛼_𝑛𝑢+𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝑡_𝑛−𝑥^∗‖‖≤𝛼_𝑛‖𝑢−𝑥^∗‖+𝛽_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖+𝛾_𝑛‖‖𝑡_𝑛−𝑥^∗‖‖≤𝛼_𝑛‖𝑢−𝑥^∗‖+𝛽_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖+𝛾_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖≤ma‖x𝑢−‖‖𝑥𝑥^∗₁‖+1−𝛼_𝑛‖‖𝑥_𝑛−𝑥^∗‖‖=‖𝑢−𝑥_∗−𝑥^∗‖‖,‖𝑢−𝑥^∗‖‖.(3.5) Therefore, {𝑥_𝑛} is bounded. Hence {𝑥_𝑛},{𝑦_𝑛},{𝑡_𝑛},{𝐴𝑦_𝑛}, and {𝐵𝑥_𝑛} are bounded.
On the other hand, we have ‖‖𝑡_𝑛+1−𝑡_𝑛‖‖=‖‖𝛿+_𝑛+1𝑆1𝑥−𝛿_𝑛+1𝑛+1−𝛿_𝑛+1𝑄_𝐶𝑦_𝑛+1−𝜆_𝐴𝑦_𝑛+1−𝛿_𝑛𝑆𝑥_𝑛=‖‖𝛿_𝑛+1𝑄_𝐶𝑦_𝑛−𝜆𝐴𝑦_𝑛‖‖+𝛿𝑆𝑥_𝑛+1−𝑆𝑥_𝑛+1−𝛿_𝑛+1𝑄_𝐶𝑦_𝑛+1−𝜆𝐴𝑦_𝑛+1-𝑄_𝐶𝑦_𝑛−𝜆𝐴𝑦_𝑛_𝑛+1≤𝛿_𝑛+1−𝛿_𝑛𝑆𝑥_𝑛−𝑄_𝐶𝑦_𝑛−𝜆𝐴𝑦_𝑛‖‖+||𝛿‖‖_𝑛+1𝑆𝑥_𝑛+1−𝑆𝑥_𝑛‖‖+1−𝛿_𝑛+1‖‖𝑄_𝐶𝑦_𝑛+1−𝜆𝐴𝑦_𝑛+1−𝑄_𝐶𝑦_𝑛−𝜆𝐴𝑦_𝑛‖‖≤𝛿_𝑛+1−𝛿_𝑛||⋅‖‖𝑆𝑥_𝑛−𝑄_𝐶𝑦_𝑛−𝜆𝐴𝑦_𝑛‖‖=𝛿_𝑛+1‖‖𝑥_𝑛+1−𝑥_𝑛‖‖+1−𝛿_𝑛+1‖‖𝑦_𝑛+1−𝑦_𝑛‖‖+||𝛿_𝑛+1−𝛿_𝑛||⋅‖‖𝑆𝑥_𝑛−𝑄_𝐶𝑦_𝑛−𝜆𝐴𝑦_𝑛‖‖+||𝛿‖‖𝑥_𝑛+1𝑛+1−𝑥_𝑛‖‖+1−𝛿_𝑛+1+‖‖𝑄_𝐶𝑥_𝑛+1−𝜆𝐵𝑥_𝑛+1−𝑄_𝐶𝑥_𝑛−𝜆𝐵𝑥_𝑛‖‖≤𝛿_𝑛+1−𝛿_𝑛||⋅‖‖𝑆𝑥_𝑛−𝑄_𝐶𝑦_𝑛−𝜆𝐴𝑦_𝑛‖‖≤‖‖𝑥‖‖_𝑛+1𝑥_𝑛+1−𝑥_𝑛‖‖+1−𝛿_𝑛+1‖‖𝑥_𝑛+1−𝑥_𝑛‖‖+||𝛿_𝑛+1−𝛿_𝑛||⋅‖‖𝑆𝑥_𝑛−𝑄_𝐶𝑦_𝑛−𝜆_𝑛𝐴𝑦_𝑛‖‖−𝑥_𝑛‖‖+||𝛿_𝑛+1−𝛿_𝑛||𝑀,(3.6) where 𝑀 is an appropriate constant such that 𝑀≥sup_𝑛≥1‖‖𝑆𝑥_𝑛−𝑄_𝐶𝑦_𝑛−𝜆_𝑛𝐴𝑦_𝑛‖‖.(3.7)
Step 3. We prove that lim_(𝑛→∞)||𝑥_𝑛+1−𝑥_𝑛||=0.
Setting 𝑤_𝑛=(𝑥_(𝑛+1)−𝛽_𝑛𝑥_𝑛)/(1−𝛽_𝑛) for each 𝑛≥1, we see that 𝑥_𝑛+1=1−𝛽_𝑛𝑤_𝑛+𝛽_𝑛𝑥_𝑛,∀𝑛≥1.(3.8) Now, we compute ||𝑤_𝑛+1−𝑤_𝑛|| from ‖‖𝑤_𝑛+1−𝑤_𝑛‖‖=‖‖‖𝛼^𝑛+1𝑢+𝛾_𝑛+1𝑡_𝑛+11−𝛽_𝑛+1−𝛼^𝑛𝑢+𝛾_𝑛𝑡_𝑛1−𝛽_𝑛=‖‖‖𝛼_𝑛+1‖‖‖1−𝛽_𝑛+1𝑢+1−𝛽^𝑛+1−𝛼_𝑛+11−𝛽_𝑛+1𝑡_𝑛+1−𝛼^𝑛1−𝛽_𝑛𝑢−1−𝛽^𝑛−𝛼_𝑛1−𝛽_𝑛𝑡_𝑛=‖‖‖𝛼_𝑛+1‖‖‖1−𝛽_𝑛+1𝑢−𝑡_𝑛+1+1−𝛼^𝑛𝛽_𝑛𝑡_𝑛−𝑢+𝑡_𝑛+1−𝑡_𝑛≤𝛼_𝑛+1‖‖‖1−𝛽_𝑛+1‖‖𝑢−𝑡_𝑛+1‖‖+𝛼^𝑛1−𝛽_𝑛‖‖𝑡_𝑛−𝑢‖‖+‖‖𝑡_𝑛+1−𝑡_𝑛‖‖.(3.9) Combining (3.6) and (3.9), we arrive at ‖‖𝑤_𝑛+1−𝑤_𝑛‖‖−‖‖𝑥_𝑛+1−𝑥_𝑛‖‖≤𝛼^𝑛+11−𝛽_𝑛+1‖‖𝑢−𝑡_𝑛+1‖‖+𝛼^𝑛1−𝛽_𝑛‖‖𝑡_𝑛−𝑢‖‖+||𝛿_𝑛+1−𝛿_𝑛||𝑀₁.(3.10) It follows from the conditions (1.3), (1.4), and (1.5) that limsup_𝑛→∞‖‖𝑤_𝑛+1−𝑤_𝑛‖‖−‖‖𝑥_𝑛+1−𝑥_𝑛‖‖≤0.(3.11) Hence, by Lemma 2.3, we obtain that lim_(𝑛→∞)||𝑤_𝑛−𝑥_𝑛||=0. Consequently, lim_𝑛→∞‖‖𝑥_𝑛+1−𝑥_𝑛‖‖=lim_𝑛→∞1−𝛽_𝑛‖‖𝑤_𝑛−𝑥_𝑛‖‖=0.(3.12) On the other hand, it follows from the algorithm (3.1) that 𝑥_𝑛+1−𝑥_𝑛=𝛼_𝑛𝑢−𝑥_𝑛+𝛾_𝑛𝑡_𝑛−𝑥_𝑛.(3.13) From the condition (1.3) and formula (3.12), we see that lim_𝑛→∞‖‖𝑡_𝑛−𝑥_𝑛‖‖=0.(3.14)
Step 4. We prove that lim_(𝑛→∞)||𝑥_𝑛−𝑉𝑥_𝑛||=0. Define a mapping 𝑉∶𝐶→𝐶 by 𝑉𝑥=𝛿𝑆𝑥+(1−𝛿)𝑄_𝐶𝑄_𝐶(𝐼−𝜇𝐵)−𝜆𝐴𝑄_𝐶(𝐼−𝜇𝐵)𝑥,∀𝑥∈𝐶,(3.15) where lim_(𝑛→∞)𝛿_𝑛=𝛿∈(0,1). From Lemma 2.4, we see that 𝑉 is a nonexpansive mapping with 𝐹(𝑉)==𝐹𝐹((𝑆𝑆))𝐹𝐹𝑄𝑄_𝐶𝐶𝑄_𝐶(𝐼−𝜇𝐵)−𝜆𝐴𝑄_𝐶(𝐼−𝜇𝐵)==𝐹𝐹.(𝑆)𝐹(𝐺)(𝐼−𝜆𝐴)𝑄_𝐶(𝐼−𝜇𝐵)(3.16) On the other hand, we have ‖‖𝑥_𝑛−𝑉𝑥_𝑛‖‖≤≤‖‖‖‖𝑥𝑥_𝑛𝑛−𝑥_𝑛+1‖‖+‖‖𝑥_𝑛+1−𝑉𝑥_𝑛‖‖=‖‖𝑥_𝑛−𝑥_𝑛+1‖‖+𝛼_𝑛‖‖𝑢_𝑛−𝑉𝑥_𝑛‖‖+𝛽_𝑛‖‖𝑥_𝑛−𝑉𝑥_𝑛‖‖+𝛾_𝑛‖‖𝑡_𝑛−𝑉𝑥_𝑛‖‖+𝛾_𝑛−𝑥_𝑛+1‖‖+𝛼_𝑛‖‖𝑢_𝑛−𝑉𝑥_𝑛‖‖+𝛽_𝑛‖‖𝑥_𝑛−𝑉𝑥_𝑛‖‖≤‖‖𝑥_𝑛‖‖𝛿_𝑛𝑆𝑥_𝑛+1−𝛿_𝑛𝑄_𝐶𝑦_𝑛−𝜆𝐴𝑦_𝑛−𝛿𝑆𝑥_𝑛−(1−𝛿)𝑄_𝐶𝑦_𝑛−𝜆𝐴𝑦_𝑛‖‖+𝛾_𝑛−𝑥_𝑛+1‖‖+𝛼_𝑛‖‖𝑢_𝑛−𝑉𝑥_𝑛‖‖+𝛽_𝑛‖‖𝑥_𝑛−𝑉𝑥_𝑛‖‖≤‖‖𝑥_𝑛||𝛿_𝑛−𝛿||⋅‖‖𝑆𝑥_𝑛−𝑄_𝐶𝑦_𝑛−𝜆_𝑛𝑦_𝑛‖‖−𝑥_𝑛+1‖‖+𝛼_𝑛‖‖𝑢_𝑛−𝑉𝑥_𝑛‖‖+𝛽_𝑛‖‖𝑥_𝑛−𝑉𝑥_𝑛‖‖+𝛾_𝑛||𝛿_𝑛−𝛿||𝑀.(3.17) This implies that 1−𝛽_𝑛‖‖𝑥_𝑛−𝑉𝑥_𝑛‖‖≤‖‖𝑥_𝑛−𝑥_𝑛+1‖‖+𝛼_𝑛‖‖𝑢_𝑛−𝑉𝑥_𝑛‖‖+𝛾_𝑛||𝛿_𝑛−𝛿||𝑀.(3.18) It follows from the conditions (1.3), (1.4), (1.5), and (3.18) that lim_𝑛→∞‖‖𝑥_𝑛−𝑉𝑥_𝑛‖‖=0.(3.19)
Step 5. Next, we show that limsup_(𝑛→∞)⟨𝑢−𝑥⁻,𝑗(𝑥_𝑛−𝑥⁻)⟩≤0.
Let 𝑧_𝑡 be the fixed point of the contraction 𝑧↦𝑡𝑢+(1−𝑡)𝑉𝑧_𝑡, where 𝑡∈(0,1). That is, 𝑧_𝑡=𝑡𝑢+(1−𝑡)𝑉𝑧_𝑡. It follows that ‖‖𝑧_𝑡−𝑥_𝑛‖‖=‖‖(1−𝑡)𝑉𝑧_𝑡−𝑥_𝑛+𝑡𝑢−𝑥_𝑛‖‖.(3.20) On the other hand, for any 𝑡∈(0,1), we see that ‖‖𝑧_𝑡−𝑥_𝑛‖‖²==(1(−1−𝑡)𝑡)𝑉𝑧𝑉_𝑡𝑧_𝑡−𝑥_𝑛+𝑡𝑢−𝑥_𝑛,𝑗𝑧_𝑡−𝑥_𝑛=(1−𝑡)𝑉𝑧_𝑡−𝑥_𝑛,𝑗𝑧_𝑡−𝑥_𝑛+𝑡𝑢−𝑥_𝑛,𝑗𝑧_𝑡−𝑥_𝑛+𝑡𝑢−𝑧_𝑡−𝑉𝑥_𝑛,𝑗𝑧_𝑡−𝑥_𝑛+(1−𝑡)𝑉𝑥_𝑛−𝑥_𝑛,𝑗𝑧_𝑡−𝑥_𝑛≤(1−𝑡)‖‖𝑧_𝑡,𝑗𝑧_𝑡−𝑥_𝑛+𝑡𝑧_𝑡−𝑥_𝑛,𝑗𝑧_𝑡−𝑥_𝑛+𝑡𝑢−𝑧_𝑡−𝑥_𝑛‖‖²+(1−𝑡)‖‖𝑉𝑥_𝑛−𝑥_𝑛‖‖⋅‖‖𝑧_𝑡−𝑥_𝑛‖‖,𝑗𝑧_𝑡−𝑥_𝑛+𝑡‖‖𝑧_𝑡−𝑥_𝑛‖‖²≤‖‖𝑧_𝑡−𝑥_𝑛‖‖²+‖‖𝑉𝑥_𝑛−𝑥_𝑛‖‖⋅‖‖𝑧_𝑡−𝑥_𝑛‖‖+𝑡𝑢−𝑧_𝑡,𝑗𝑧_𝑡−𝑥_𝑛.(3.21) It follows that 𝑧_𝑡−𝑢,𝑗𝑧_𝑡−𝑥_𝑛≤1𝑡‖‖𝑉𝑥^𝑛−𝑥_𝑛‖‖⋅‖‖𝑧_𝑡−𝑥_𝑛‖‖,∀𝑡∈(0,1).(3.22) In view of (3.19), we see that limsup_𝑛→∞𝑧_𝑡−𝑢,𝑗𝑧_𝑡−𝑥_𝑛≤0.(3.23) On the other hand, we see that 𝑄_𝐹(𝑉)𝑢=lim_(𝑡→0)𝑧_𝑡 and 𝐹(𝑉)=𝐹. It follows that 𝑧_𝑡→𝑥⁻=𝑄_𝐹𝑢 as 𝑡→0. Owing to the fact that 𝑗 is strong to weak* uniformly continuous on bounded subsets of 𝐸, we see that lim_𝑡→0|||𝑢−⁻𝑥,𝑗𝑥_𝑛−𝑥⁻−𝑧_𝑡−𝑢,𝑗𝑧_𝑡−𝑥_𝑛|||≤|||+𝑢|||−𝑢−𝑥⁻,𝑗₋𝑥_𝑛−𝑥⁻−𝑢−𝑥⁻,𝑗𝑥_𝑛−𝑧_𝑡|||≤|||𝑢−₋𝑥,𝑗𝑥_𝑛−𝑧_𝑡−𝑧_𝑡−𝑥⁻,𝑗𝑧_𝑡−𝑥_𝑛|||+|||𝑧𝑥_𝑡,𝑗𝑥_𝑛−𝑥⁻−𝑗𝑥_𝑛−𝑧_𝑡|||≤‖‖𝑢−₋−𝑥⁻,𝑗𝑥_𝑛−𝑧_𝑡|||+‖‖𝑧𝑥_𝑡‖‖⋅‖‖𝑗𝑥_𝑛−𝑥⁻−𝑗𝑥_𝑛−𝑧_𝑡‖‖=0.−𝑥⁻‖‖⋅‖‖𝑥_𝑛−𝑧_𝑡‖‖(3.24) Hence, for any 𝜀>0, there exists 𝛿>0 such that forall𝑡∈(0,𝛿), the following inequality holds: 𝑢−𝑥⁻,𝑗𝑥_𝑛−𝑥⁻≤𝑧_𝑡−𝑢,𝑗𝑧_𝑡−𝑥_𝑛+𝜀.(3.25) Since 𝜀 is arbitrary and (3.23), we see that limsup_𝑛→∞𝑢−𝑥⁻,𝑗𝑥_𝑛−𝑥⁻≤0.(3.26)
Step 6. Finally, we show that 𝑥_𝑛→𝑥⁻ as 𝑛→∞. Observe that ‖‖𝑥_𝑛+1−𝑥⁻‖‖²==𝛼_𝑛𝛼_𝑛𝑢+𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝑡_𝑛−𝑥⁻,𝑗𝑥_𝑛+1−𝑥⁻≤𝛼_𝑛𝑢−𝑥⁻,𝑗𝑥_𝑛+1−𝑥⁻+𝛽_𝑛𝑥^𝑛−𝑥⁻,𝑗𝑥_𝑛+1−𝑥⁻+𝛾_𝑛𝑡^𝑛−𝑥⁻,𝑗𝑥_𝑛+1−𝑥⁻𝑢−𝑥⁻,𝑗𝑥_𝑛+1−𝑥⁻+𝛽_𝑛≤𝛼_𝑛‖‖𝑥_𝑛−𝑥⁻‖‖⋅‖‖𝑥_𝑛+1−𝑥⁻‖‖+𝛾_𝑛‖‖𝑡_𝑛−𝑥⁻‖‖⋅‖‖𝑥_𝑛+1−𝑥⁻‖‖𝑢−𝑥⁻,𝑗𝑥_𝑛+1−𝑥⁻+𝛽_𝑛=𝛼_𝑛‖‖𝑥_𝑛−𝑥⁻‖‖⋅‖‖𝑥_𝑛+1−𝑥⁻‖‖+𝛾_𝑛‖‖𝑥_𝑛−𝑥⁻‖‖⋅‖‖𝑥_𝑛+1−𝑥⁻‖‖≤𝛼_𝑛𝑢−𝑥⁻,𝑗𝑥_𝑛+1−𝑥⁻+1−𝛼_𝑛‖‖𝑥_𝑛−𝑥⁻‖‖⋅‖‖𝑥_𝑛+1−𝑥⁻‖‖𝑢−𝑥⁻,𝑗𝑥_𝑛+1−𝑥⁻+1−2𝛼^𝑛‖‖𝑥^𝑛−𝑥⁻‖‖²+‖‖𝑥_𝑛+1−𝑥⁻‖‖²(3.27) which implies that ‖‖𝑥_𝑛+1−𝑥_𝑛‖‖²≤1−𝛼_𝑛‖‖𝑥_𝑛−𝑥⁻‖‖²+2𝛼_𝑛𝑢−𝑥⁻,𝑗𝑥_𝑛+1−𝑥⁻.(3.28) From the conditions (1.3) and (3.26) and applying Lemma 2.5 to (3.28), we obtain that lim_𝑛→∞‖‖𝑥_𝑛−𝑥⁻‖‖=0.(3.29) This completes the proof.

Remark 3.2. Since 𝐿^𝑝 for all 𝑝≥2 is uniformly convex and 2 uniformly smooth, we see that Theorem 3.1 is applicable to 𝐿^𝑝 for all 𝑝≥2. There are a number of sequences satisfying the restrictions (C1)–(C3), for example, 𝛼_𝑛=1/(𝑛+1),𝛽_𝑛=𝑛/(2𝑛+1),𝛾_𝑛=𝑛²/(2𝑛²+3𝑛+1) for each 𝑛≥1.

Corollary 3.3. Let 𝐻 be a real Hilbert space and 𝐶 a nonempty closed convex subset of 𝐻. Let 𝐴,𝐵∶𝐶→𝐻 be 𝛼-inverse strongly monotone mapping and 𝛽-inverse strongly monotone mapping, respectively, and 𝑆∶𝐶→𝐶 nonexpansive mappings with a fixed point. Assume that 𝐹=𝐹(𝑆)⋂𝐹(𝐺^)≠∅, where 𝐺^=𝑃_𝐶[𝑃_𝐶(𝑥−𝜇𝐵𝑥)−𝜆𝐴𝑃_𝐶(𝑥−𝜇𝐵𝑥)]. Suppose that {𝑥_𝑛} is generated by 𝑦𝑥₁=𝑢∈𝐶,_𝑛𝑥_𝑛+1=𝑃_𝐶𝑥_𝑛−𝜇𝐵𝑥_𝑛,=𝛼_𝑛𝑢+𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝛿_𝑛𝑆𝑥_𝑛+1−𝛿_𝑛𝑃_𝐶𝑦_𝑛−𝜆𝐴𝑦_𝑛,𝑛≥1,(3.30) where 𝛿_𝑛⊂[0,1], 𝜆∈(0,2𝛼), 𝜇∈(0,2𝛽) and {𝛼_𝑛},{𝛽_𝑛}, and {𝛾_𝑛} are three sequences in (0,1), and the following conditions are satisfied: (1)𝛼_𝑛+𝛽_𝑛+𝛾_𝑛=1,forall𝑛≥1, (2)lim_(𝑛→∞)𝛼_𝑛=0,∑^∞𝑛=0𝛼_𝑛=∞, (3)0<liminf_(𝑛→∞)𝛽_𝑛≤limsup_(𝑛→∞)𝛽_𝑛<1, (4)lim_(𝑛→∞)𝛿_𝑛=𝛿∈(0,1).
Then the sequence {𝑥_𝑛} defined by (3.30) converges strongly to 𝑥⁻=𝑃_𝐹𝑢, and (𝑥⁻,𝑦⁻) is a solution of the problem (1.4), where 𝑦⁻=𝑃_𝐶(𝑥⁻−𝜇𝐵𝑥⁻), 𝑃_𝐶 is the projection of 𝐻 onto 𝐶, and 𝑃_𝐹 is the projection of 𝐶 onto 𝐹.

Remark 3.4. Theorem 3.1 and Corollary 3.3 improve and extend the corresponding results announced by other authors, such as [2–4, 6, 8, 10–12, 14].

Acknowledgments

The authors would like to thank editors and referees for many useful comments and suggestions for the improvement of the paper. This work was partially supported by the Natural Science Foundation of Zhejiang Province (Y6100696) and NSFC (11071169).