Extended Extragradient Methods for Generalized Variational Inequalities

Abstract

We suggest a modified extragradient method for solving the generalized variational inequalities in a Banach space. We prove some strong convergence results under some mild conditions on parameters. Some special cases are also discussed.

1. Introduction

The well-known variational inequality problem is to find 𝑥^∗∈𝐶 such that𝐴𝑥^∗,𝑥−𝑥^∗≥0,∀𝑥∈𝐶, (1.1) where 𝐶 is a nonempty closed convex subset of a real Hilbert space 𝐻 and 𝐴∶𝐶→𝐻 is a nonlinear operator. This problem has been researched extensively due to its applications in industry, finance, economics, optimization, medical sciences, and pure and applied sciences; see, for instance, [1–19] and the reference contained therein. For solving the above variational inequality, Korpelevičh [20] introduced the following so-called extragradient method:𝑥₀𝑦=𝑥∈𝐶,_𝑛=𝑃_𝐶𝑥_𝑛−𝜆𝐴𝑥_𝑛,𝑥_𝑛+1=𝑃_𝐶𝑥_𝑛−𝜆𝐴𝑦_𝑛 (1.2) for every 𝑛=0,1,2,…, where 𝑃_𝐶 is the metric projection from 𝑅^𝑛 onto 𝐶 and 𝜆∈(0,1/𝑘). He showed that the sequences {𝑥_𝑛} and {𝑦_𝑛} generated by (1.2) converge to the same point 𝑧∈VI(𝐶,𝐴). Since some methods related to extragradient methods have been considered in Hilbert spaces by many authors, please see, for example, [3, 5, 7, 14].

This naturally brings us to the following questions.

Question 1. Could we extend variational inequality from Hilbert spaces to Banach spaces?

Question 2. Could we extend the extragradient methods from Hilbert spaces to Banach spaces?

For solving Question 1, very recently, Aoyama et al. [21] first considered the following generalized variational inequality problem in a Banach space.

Problem 1. Let 𝑋 be a smooth Banach space and 𝐶 a nonempty closed convex subset of 𝑋. Let 𝐴 be an accretive operator of 𝐶 into 𝑋. Find a point 𝑥^∗∈𝐶 such that 𝐴𝑥^∗,𝐽𝑥−𝑥^∗≥0,∀𝑥∈𝐶. (1.3)
This problem is connected with the fixed point problem for nonlinear mapping, the problem of finding a zero point of an accretive operator, and so on. For the problem of finding a zero point of an accretive operator by the proximal point algorithm, please consult [22]. In order to find a solution of Problem 1, Aoyama et al. [21] introduced the following iterative scheme for an accretive operator 𝐴 in a Banach space 𝑋: 𝑥₁𝑦=𝑥∈𝐶,_𝑛=𝑄_𝐶𝑥_𝑛−𝜆_𝑛𝐴𝑥_𝑛,𝑥_𝑛+1=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝑦_𝑛, (1.4) for every 𝑛=1,2,…, where 𝑄_𝐶 is a sunny nonexpansive retraction from 𝑋 onto 𝐶. Then, they proved a weak convergence theorem in a Banach space which is generalized simultaneously by theorems of [4, 23] as follows.

Theorem 1.1. Let 𝑋 be a uniformly convex and 2-uniformly smooth Banach space, and let 𝐶 be a nonempty closed convex subset of 𝑋. Let 𝑄_𝐶 be a sunny nonexpansive retraction from 𝑋 onto 𝐶, let 𝛼>0, and let 𝐴 be an 𝛼-inverse-strongly accretive operator of 𝐶 into 𝑋 with 𝑆(𝐶,𝐴)≠∅. If {𝜆_𝑛} and {𝛼_𝑛} are chosen so that 𝜆_𝑛∈[𝑎,𝛼/𝐾²] for some 𝑎>0 and 𝛼_𝑛∈[𝑏,𝑐] for some 𝑏,𝑐 with 0<𝑏<𝑐<1, then {𝑥_𝑛} defined by (1.4) converges weakly to some element 𝑧 of 𝑆(𝐶,𝐴)∶={𝑥^∗∈𝐶∶⟨𝐴𝑥^∗,𝐽(𝑥−𝑥^∗)⟩≥0,forall𝑥∈𝐶}, where 𝐾 is the 2-uniformly smoothness constant of 𝑋.

In this paper, motivated by the ideas in the literature, we first introduce a new iterative method in a Banach space as follows.

For fixed 𝑢∈𝐶 and arbitrarily given 𝑥₀∈𝐶, define a sequence {𝑥_𝑛} iteratively by𝑦_𝑛=𝑄_𝐶𝑥_𝑛−𝜆_𝑛𝐴𝑥_𝑛,𝑥_𝑛+1=𝛼_𝑛𝑢+𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝑄_𝐶𝑦_𝑛−𝜆_𝑛𝐴𝑦_𝑛, (1.5) for every 𝑛=1,2,…, where 𝑄_𝐶 is a sunny nonexpansive retraction from 𝑋 onto 𝐶, {𝛼_𝑛}, {𝛽_𝑛}, and {𝛾_𝑛} are three sequences in (0,1), and {𝜆_𝑛} is a sequence of real numbers. We prove some strong convergence results under some mild conditions on parameters.

2. Preliminaries

Let 𝑋 be a real Banach space, and let 𝑋^∗ denote the dual of 𝑋. Let 𝐶 be a nonempty closed convex subset of 𝑋. A mapping 𝐴 of 𝐶 into 𝑋 is said to be accretive if there exists 𝑗(𝑥−𝑦)∈𝐽(𝑥−𝑦) such that 𝐴𝑥−𝐴𝑦,𝑗𝑥−𝑦≥0, (2.1) for all 𝑥,𝑦∈𝐶, where 𝐽 is called the duality mapping. A mapping 𝐴 of 𝐶 into 𝑋 is said to be 𝛼-strongly accretive if, for 𝛼>0, 𝐴𝑥−𝐴𝑦,𝑗𝑥−𝑦≥𝛼𝑥−𝑦², (2.2) for all 𝑥,𝑦∈𝐶. A mapping 𝐴 of 𝐶 into 𝑋 is said to be 𝛼-inverse-strongly accretive if, for 𝛼>0, 𝐴𝑥−𝐴𝑦,𝑗𝑥−𝑦≥𝛼𝐴𝑥−𝐴𝑦², (2.3) for all 𝑥,𝑦∈𝐶.

Remark 2.1. (1) Evidently, the definition of the inverse strongly accretive mapping is based on that of the inverse strongly monotone mapping.
(2) If 𝐴 is an 𝛼-strongly accretive and 𝐿-Lipschitz continuous mapping of 𝐶 into 𝑋, then 𝐴𝑥−𝐴𝑦,𝑗𝑥−𝑦≥𝛼𝑥−𝑦²≥𝛼𝐿²𝐴𝑥−𝐴𝑦²,∀𝑥,𝑦∈𝐶, (2.4) from which it follows that 𝐴 must be (𝛼/𝐿²)-inverse-strongly accretive mapping.

Let 𝑈={𝑥∈𝑋∶‖𝑥‖=1}. A Banach space 𝑋 is said to be uniformly convex if, for each 𝜖∈(0,2], there exists 𝛿>0 such that, for any 𝑥,𝑦∈𝑈, 𝑥−𝑦≥𝜖implies𝑥+𝑦2≤1−𝛿. (2.5) It is known that a uniformly convex Banach space is reflexive and strictly convex. A Banach space 𝑋 is said to be smooth if the limitlim_𝑡→0𝑥+𝑡𝑦−‖𝑥‖𝑡 (2.6) exists for all 𝑥,𝑦∈𝑈. It is also said to be uniformly smooth if the limit (2.6) is attained uniformly for 𝑥,𝑦∈𝑈. The norm of 𝑋 is said to be Frechet differentiable if, for each 𝑥∈𝑈, the limit (2.6) is attained uniformly for 𝑦∈U. And we define a function 𝜌∶[0,∞)→[0,∞) called the modulus of smoothness of 𝑋 as follows: 𝜌(𝜏)1=sup2𝑥+𝑦+𝑥−𝑦−1∶𝑥,𝑦∈𝑋,‖𝑥‖=1,𝑦=𝜏. (2.7) It is known that 𝑋 is uniformly smooth if and only if lim_𝜏→0𝜌(𝜏)/𝜏=0. Let 𝑞 be a fixed real number with 1<𝑞≤2. Then a Banach space 𝑋 is said to be 𝑞-uniformly smooth if there exists a constant 𝑐>0 such that 𝜌(𝜏)≤𝑐𝜏^𝑞 for all 𝜏>0.

Remark 2.2. Takahashi et al. [24] remind us of the following fact: no Banach space is 𝑞-uniformly smooth for 𝑞>2. So, in this paper, we study a strong convergence theorem in a 2-uniformly smooth Banach space.

We need the following lemmas for the proof of our main results.

Lemma 2.3 (see [25]). Let 𝑞 be a given real number with 1<𝑞≤2, and let 𝑋 be a 𝑞-uniformly smooth Banach space. Then, 𝑥+𝑦^𝑞≤‖𝑥‖^𝑞+𝑞𝑦,𝐽_𝑞(𝑥)+2𝐾𝑦^𝑞, (2.8) for all 𝑥,𝑦∈𝑋, where 𝐾 is the 𝑞-uniformly smoothness constant of 𝑋 and 𝐽_𝑞 is the generalized duality mapping from 𝑋 into 2^𝑋∗ defined by 𝐽_𝑞(𝑥)=𝑓∈𝑋^∗∶𝑥,𝑓=‖𝑥‖^𝑞,𝑓=‖𝑥‖^𝑞−1, (2.9) for all 𝑥∈𝑋.

Let 𝐷 be a subset of 𝐶, and let 𝑄 be a mapping of 𝐶 into 𝐷. Then, 𝑄 is said to be sunny if 𝑄(𝑄𝑥+𝑡(𝑥−𝑄𝑥))=𝑄𝑥, (2.10) whenever 𝑄𝑥+𝑡(𝑥−𝑄𝑥)∈𝐶 for 𝑥∈𝐶 and 𝑡≥0. A mapping 𝑄 of 𝐶 into itself is called a retraction if 𝑄²=𝑄. If a mapping 𝑄 of 𝐶 into itself is a retraction, then 𝑄𝑧=𝑧 for every 𝑧∈𝑅(𝑄), where 𝑅(𝑄) is the range of 𝑄. A subset 𝐷 of 𝐶 is called a sunny nonexpansive retract of 𝐶 if there exists a sunny nonexpansive retraction from 𝐶 onto 𝐷. We know the following lemma concerning sunny nonexpansive retraction.

Lemma 2.4 (see [26]). Let 𝐶 be a closed convex subset of a smooth Banach space 𝑋, 𝐷 a nonempty subset of 𝐶, and 𝑄 a retraction from 𝐶 onto 𝐷. Then, 𝑄 is sunny and nonexpansive if and only if 𝑢−𝑄𝑢,𝑗𝑦−𝑄𝑢≤0, (2.11) for all 𝑢∈𝐶 and 𝑦∈𝐷.

Remark 2.5. (1) It is well known that, if 𝑋 is a Hilbert space, then a sunny nonexpansive retraction 𝑄_𝐶 is coincident with the metric projection from 𝑋 onto 𝐶.
(2) Let 𝐶 be a nonempty closed convex subset of a uniformly convex and uniformly smooth Banach space 𝑋, and let 𝑇 be a nonexpansive mapping of 𝐶 into itself with 𝐹(𝑇)≠∅. Then, the set 𝐹(𝑇) is a sunny nonexpansive retract of 𝐶.

The following lemma is characterized by the set of solution Problem AIT by using sunny nonexpansive retractions.

Lemma 2.6 (see [21]). Let 𝐶 be a nonempty closed convex subset of a smooth Banach space 𝑋. Let 𝑄_𝐶 be a sunny nonexpansive retraction from 𝑋 onto 𝐶, and let 𝐴 be an accretive operator of 𝐶 into 𝑋. Then, for all 𝜆>0, 𝑆(𝐶,𝐴)=𝐹𝑄_𝐶(𝐼−𝜆𝐴), (2.12) where 𝑆(𝐶,𝐴)={𝑥^∗∈𝐶∶⟨𝐴𝑥^∗,𝐽(𝑥−𝑥^∗)⟩≥0,forall𝑥∈𝐶}.

Lemma 2.7 (see [27]). Let 𝐶 be a nonempty bounded closed convex subset of a uniformly convex Banach space 𝑋, and let 𝑇 be nonexpansive mapping of 𝐶 into itself. If {𝑥_𝑛} is a sequence of 𝐶 such that 𝑥_𝑛→𝑥 weakly and 𝑥_𝑛−𝑇𝑥_𝑛→0 strongly, then 𝑥 is a fixed point of 𝑇.

Lemma 2.8 (see [28]). Let {𝑥_𝑛} and {𝑧_𝑛} be bounded sequences in a Banach space 𝑋, and let {𝛼_𝑛} be a sequence in [0,1] which satisfies the following condition: 0<liminf_𝑛→∞𝛼_𝑛≤limsup_𝑛→∞𝛼_𝑛<1. (2.13) Suppose that 𝑥_𝑛+1=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝑧_𝑛,𝑛≥0,limsup_𝑛→∞𝑧_𝑛+1−𝑧_𝑛−𝑥_𝑛+1−𝑥_𝑛≤0. (2.14) Then, lim_𝑛→∞‖𝑧_𝑛−𝑥_𝑛‖=0.

Lemma 2.9 (see [26]). Assume that {𝑎_𝑛} is a sequence of nonnegative real numbers such that 𝑎_𝑛+1≤1−𝛾_𝑛𝑎_𝑛+𝛿_𝑛,𝑛≥0, (2.15) where {𝛾_𝑛} is a sequence in (0,1) and {𝛿_𝑛} is a sequence in 𝑅 such that (i)∑^∞_𝑛=0𝛾_𝑛=∞, (ii)limsup_𝑛→∞𝛿_𝑛/𝛾_𝑛≤0 or ∑^∞_𝑛=0|𝛿_𝑛|<∞. Then, lim_𝑛→∞𝑎_𝑛=0.

3. Main Results

In this section, we obtain a strong convergence theorem for finding a solution of Problem AIT for an 𝛼-strongly accretive and 𝐿-Lipschitz continuous mapping in a uniformly convex and 2-uniformly smooth Banach space. First, we assume that 𝛼>0 is a constant, 𝐿>0 a Lipschitz constant of 𝐴, and 𝐾>0 the 2-uniformly smoothness constant of 𝑋 appearing in the following.

In order to obtain our main result, we need the following lemma concerning (𝛼/𝐿²)-inverse-strongly accretive mapping.

Lemma 3.1. Let 𝑋 be a uniformly convex and 2-uniformly smooth Banach space, and let 𝐶 be a nonempty closed convex subset of 𝑋. Let 𝑄_𝐶 be a sunny nonexpansive retraction from 𝑋 onto 𝐶, and let 𝐴 be an (𝛼/𝐿²)-inverse-strongly accretive mapping of 𝐶 into 𝑋 with 𝑆(𝐶,𝐴)≠∅. For given 𝑥₀∈𝐶, let the sequence {𝑥_𝑛} be generated iteratively by (1.5), where {𝛼_𝑛}, {𝛽_𝑛}, and {𝛾_𝑛} are three sequences in (0,1) and {𝜆_𝑛} is a real number sequence in [𝑎,𝑏] for some 𝑎,𝑏 with 0<𝑎<𝑏<𝛼/𝐾²𝐿² satisfying the following conditions: (i)𝛼_𝑛+𝛽_𝑛+𝛾_𝑛=1,forall𝑛≥0; (ii)lim_𝑛→∞𝛼_𝑛=0; (iii)0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1; (iv)lim_𝑛→∞(𝜆_𝑛+1−𝜆_𝑛)=0. Then we have lim_𝑛→∞‖𝑥_𝑛+1−𝑥_𝑛‖=0 and lim_𝑛→∞‖𝐴𝑦_𝑛−𝐴𝑥_𝑛‖=0.

Proof. First, we observe that 𝐼−𝜆_𝑛𝐴 is nonexpansive. Indeed, for all 𝑥,𝑦∈𝐶, from Lemma 2.3, we have 𝐼−𝜆_𝑛𝐴𝑥−𝐼−𝜆_𝑛𝐴𝑦²=𝑥−𝑦−𝜆_𝑛𝐴𝑥−𝐴𝑦²≤𝑥−𝑦²−2𝜆_𝑛𝐴𝑥−𝐴𝑦,𝑗𝑥−𝑦+2𝐾²𝜆²_𝑛𝐴𝑥−𝐴𝑦²≤𝑥−𝑦²−2𝜆_𝑛𝛼𝐿²𝐴𝑥−𝐴𝑦²+2𝐾²𝜆²_𝑛𝐴𝑥−𝐴𝑦²=𝑥−𝑦²+2𝜆_𝑛𝐾²𝜆_𝑛−𝛼𝐿²𝐴𝑥−𝐴𝑦². (3.1) If 0<𝑎<𝜆_𝑛<𝑏<𝛼/(𝐾²𝐿²), then 𝐼−𝜆_𝑛𝐴 is a nonexpansive mapping.
Letting 𝑝∈𝑆(𝐶,𝐴), it follows from Lemma 2.6 that 𝑝=𝑄_𝐶(𝑝−𝜆_𝑛𝐴𝑝). Setting 𝑧_𝑛=𝑄_𝐶(𝑦_𝑛−𝜆_𝑛𝐴𝑦_𝑛), from (3.1), we have 𝑧_𝑛−𝑝=𝑄_𝐶𝑦_𝑛−𝜆_𝑛𝐴𝑦_𝑛−𝑄_𝐶𝑝−𝜆_𝑛≤𝑦𝐴𝑝_𝑛−𝜆_𝑛𝐴𝑦_𝑛−𝑝−𝜆_𝑛≤𝑦𝐴𝑝_𝑛=𝑄−𝑝_𝐶𝑥_𝑛−𝜆_𝑛𝐴𝑥_𝑛−𝑄_𝐶𝑝−𝜆_𝑛≤𝑥𝐴𝑝_𝑛−𝜆_𝑛𝐴𝑥_𝑛−𝑝−𝜆_𝑛≤𝑥𝐴𝑝_𝑛−𝑝. (3.2) By (1.5) and (3.2), we have 𝑥_𝑛+1−𝑝=𝛼_𝑛𝑢+𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝑧_𝑛−𝑝≤𝛼_𝑛𝑢−𝑝+𝛽_𝑛𝑥_𝑛−𝑝+𝛾_𝑛𝑧_𝑛−𝑝=𝛼_𝑛𝑢−𝑝+1−𝛼_𝑛𝑥_𝑛,𝑥−𝑝≤max𝑢−𝑝₀−𝑝. (3.3) Therefore, {𝑥_𝑛} is bounded. Hence {𝑧_𝑛}, {𝐴𝑥_𝑛}, and {𝐴𝑦_𝑛} are also bounded. We observe that 𝑧_𝑛+1−𝑧_𝑛=𝑄_𝐶𝑦_𝑛+1−𝜆_𝑛+1𝐴𝑦_𝑛+1−𝑄_𝐶𝑦_𝑛−𝜆_𝑛𝐴𝑦_𝑛≤𝑦_𝑛+1−𝜆_𝑛+1𝐴𝑦_𝑛+1−𝑦_𝑛−𝜆_𝑛𝐴𝑦_𝑛=𝑦_𝑛+1−𝜆_𝑛+1𝐴𝑦_𝑛+1−𝑦_𝑛−𝜆_𝑛+1𝐴𝑦_𝑛+𝜆_𝑛−𝜆_𝑛+1𝐴𝑦_𝑛≤𝑦_𝑛+1−𝜆_𝑛+1𝐴𝑦_𝑛+1−𝑦_𝑛−𝜆_𝑛+1𝐴𝑦_𝑛+𝜆_𝑛−𝜆_𝑛+1𝐴𝑦_𝑛≤𝑦_𝑛+1−𝑦_𝑛+𝜆_𝑛−𝜆_𝑛+1𝐴𝑦_𝑛=𝑄_𝐶𝑥_𝑛+1−𝜆_𝑛+1𝐴𝑥_𝑛+1−𝑄_𝐶𝑥_𝑛−𝜆_𝑛𝐴𝑥_𝑛+𝜆_𝑛−𝜆_𝑛+1𝐴𝑦_𝑛≤𝑥_𝑛+1−𝑥_𝑛+𝜆_𝑛−𝜆_𝑛+1𝐴𝑥_𝑛+𝐴𝑦_𝑛. (3.4) Setting 𝑥_𝑛+1=𝛽_𝑛𝑥_𝑛+(1−𝛽_𝑛)𝑤_𝑛 for all 𝑛≥0 we obtain 𝑤_𝑛+1−𝑤_𝑛=𝛼_𝑛+1𝑢+𝛾_𝑛+1𝑧_𝑛+11−𝛽_𝑛+1−𝛼_𝑛𝑢+𝛾_𝑛𝑧_𝑛1−𝛽_𝑛=𝛼_𝑛+11−𝛽_𝑛+1−𝛼_𝑛1−𝛽_𝑛𝛾𝑢+^𝑛+11−𝛽_𝑛+1𝑧_𝑛+1−𝑧_𝑛+𝛾_𝑛+11−𝛽_𝑛+1−𝛾_𝑛1−𝛽_𝑛𝑧_𝑛. (3.5) Combining (3.4) and (3.5), we have 𝑤_𝑛+1−𝑤_𝑛−𝑥_𝑛+1−𝑥_𝑛≤𝛼_𝑛+11−𝛽_𝑛+1−𝛼_𝑛1−𝛽_𝑛𝑢‖+𝛾‖^𝑛+11−𝛽_𝑛+1𝑥_𝑛+1−𝑥_𝑛+𝛾_𝑛+11−𝛽_𝑛+1𝜆_𝑛+1−𝜆_𝑛𝐴𝑥_𝑛+𝐴𝑦_𝑛+𝛾_𝑛+11−𝛽_𝑛+1−𝛾_𝑛1−𝛽_𝑛𝑧_𝑛−𝑥_𝑛+1−𝑥_𝑛≤𝛼_𝑛+11−𝛽_𝑛+1−𝛼_𝑛1−𝛽_𝑛‖𝑢‖+𝑧_𝑛+𝛾_𝑛+11−𝛽_𝑛+1𝜆_𝑛+1−𝜆_𝑛𝐴𝑥_𝑛+𝐴𝑦_𝑛; (3.6) this together with (ii) and (iv) implies that limsup_𝑛→∞𝑤_𝑛+1−𝑤_𝑛−𝑥_𝑛+1−𝑥_𝑛≤0. (3.7) Hence, by Lemma 2.8, we obtain ‖𝑤_𝑛−𝑥_𝑛‖→0 as 𝑛→∞. Consequently, lim_𝑛→∞𝑥_𝑛+1−𝑥_𝑛=lim_𝑛→∞1−𝛽_𝑛𝑤_𝑛−𝑥_𝑛=0. (3.8) From (1.5), we can write 𝑥_𝑛+1−𝑥_𝑛=𝛼_𝑛(𝑢−𝑥_𝑛)+𝛾_𝑛(𝑧_𝑛−𝑥_𝑛) and note that 0<liminf_𝑛→∞𝛾_𝑛≤limsup_𝑛→∞𝛾_𝑛<1 and lim_𝑛→∞𝛼_𝑛=0. It follows from (3.8) that lim_𝑛→∞𝑥_𝑛−𝑧_𝑛=0. (3.9) For 𝑝∈𝑆(𝐶,𝐴), from (3.1) and (3.2), we obtain 𝑥_𝑛+1−𝑝²=𝛼_𝑛𝑢+𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝑧_𝑛−𝑝²≤𝛼_𝑛𝑢−𝑝²+𝛽_𝑛𝑥_𝑛−𝑝²+𝛾_𝑛𝑧_𝑛−𝑝²≤𝛼_𝑛𝑢−𝑝²+𝛽_𝑛𝑥_𝑛−𝑝²+𝛾_𝑛𝑦_𝑛−𝑝²≤𝛼_𝑛𝑢−𝑝²+𝛽_𝑛𝑥_𝑛−𝑝²+𝛾_𝑛𝑥_𝑛−𝜆_𝑛𝐴𝑥_𝑛−𝑝−𝜆_𝑛𝐴𝑝²≤𝛼_𝑛𝑢−𝑝²+𝛽_𝑛𝑥_𝑛−𝑝²+𝛾_𝑛𝑥_𝑛−𝑝²+2𝜆_𝑛𝐾²𝜆_𝑛−𝛼𝐿²𝐴𝑥_𝑛−𝐴𝑝²≤𝛼_𝑛𝑢−𝑝²+𝑥_𝑛−𝑝²+2𝛾_𝑛𝑎𝐾²𝛼𝑏−𝐿²𝐴𝑥_𝑛−𝐴𝑝². (3.10) Therefore, we have 0≤−2𝛾_𝑛𝑎𝐾²𝛼𝑏−𝐿²𝐴𝑥_𝑛−𝐴p²≤𝛼_𝑛𝑢−𝑝²+𝑥_𝑛−𝑝²−𝑥_𝑛+1−𝑝²=𝛼_𝑛𝑢−𝑝²+𝑥_𝑛−𝑝+𝑥_𝑛+1−𝑝𝑥_𝑛−𝑝−𝑥_𝑛+1−𝑝≤𝛼_𝑛𝑢−𝑝²+𝑥_𝑛−𝑝+𝑥_𝑛+1−𝑝𝑥_𝑛−𝑥_𝑛+1. (3.11) Since 𝛼_𝑛→0 and ‖𝑥_𝑛−𝑥_𝑛+1‖→0 as 𝑛→∞, from (3.11), we obtain 𝐴𝑥_𝑛−𝐴𝑝⟶0as𝑛⟶∞. (3.12) From the definition of 𝑧_𝑛 and (3.1), we also have 𝑧_𝑛−𝑝²=𝑄_𝐶𝑦_𝑛−𝜆_𝑛𝐴𝑦_𝑛−𝑄_𝐶𝑝−𝜆_𝑛𝐴𝑝²≤𝑦_𝑛−𝜆_𝑛𝐴𝑦_𝑛−𝑝−𝜆_𝑛𝐴𝑝²≤𝑦_𝑛−𝑝²+2𝜆_𝑛𝐾²𝜆_𝑛−𝛼𝐿²𝐴𝑦_𝑛−𝐴𝑝². (3.13) From the above results and assumptions, we note that ‖𝑦_𝑛−𝑝‖≤‖𝑥_𝑛−𝑝‖, 0<𝑎<𝑏<𝛼/(𝐾²𝐿²), ‖𝑥_𝑛−𝑝‖,‖𝑧_𝑛−𝑝‖ are bounded, and ‖𝑥_𝑛−𝑧_𝑛‖→0 as 𝑛→∞. Therefore, from (3.13), we have 0≤−2𝑎𝐾²𝛼𝑏−𝐿²𝐴𝑦_𝑛−𝐴𝑝²≤𝑦_𝑛−𝑝²−𝑧_𝑛−𝑝²≤𝑥_𝑛−𝑝²−𝑧_𝑛−𝑝²=𝑥_𝑛−𝑝+𝑧_𝑛−𝑝𝑥_𝑛−𝑝−𝑧_𝑛≤𝑥−𝑝_𝑛−𝑝+𝑧_𝑛−𝑝𝑥_𝑛−𝑧_𝑛⟶0, (3.14) which implies that 𝐴𝑦_𝑛−𝐴𝑝⟶0as𝑛⟶∞. (3.15) It follows from (3.12) and (3.15) that 𝐴𝑦_𝑛−𝐴𝑥_𝑛⟶0as𝑛⟶∞. (3.16) This completes the proof.

Now we state and study our main result.

Theorem 3.2. Let 𝑋 be a uniformly convex and 2-uniformly smooth Banach space with weakly sequentially continuous duality mapping, and let 𝐶 be a nonempty closed convex subset of 𝑋. Let 𝑄_𝐶 be a sunny nonexpansive retraction from 𝑋 onto 𝐶, and let 𝐴 be an 𝛼-strongly accretive and L-Lipschitz continuous mapping of 𝐶 into 𝑋 with 𝑆(𝐶,𝐴)≠∅. Let {𝛼_𝑛}, {𝛽_𝑛}, and {𝛾_𝑛} be three sequences in (0,1) and {𝜆_𝑛} a real number sequence in [𝑎,𝑏] for some 𝑎,𝑏 with 0<𝑎<𝑏<𝛼/(𝐾²𝐿²) satisfying the following conditions: (i)𝛼_𝑛+𝛽_𝑛+𝛾_𝑛=1,forall𝑛≥0; (ii)lim_𝑛→∞𝛼_𝑛=0 and ∑^∞_𝑛=0𝛼_𝑛=∞; (iii)0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1; (iv)lim_𝑛→∞(𝜆_𝑛+1−𝜆_𝑛)=0. Then {𝑥_𝑛} defined by (1.5) converges strongly to 𝑄′𝑢, where 𝑄′ is a sunny nonexpansive retraction of 𝐶 onto 𝑆(𝐶,𝐴).

Proof. From Remark 2.1(2), we have that 𝐴 is an (𝛼/𝐿²)-inverse-strongly accretive mapping. Then, from Lemma 3.1, we have lim_𝑛→∞𝐴𝑦_𝑛−𝐴𝑥_𝑛=0. (3.17) On the other hand, we note that 𝐴𝑦_𝑛−𝐴𝑥_𝑛≥𝛼𝑦_𝑛−𝑥_𝑛, (3.18) which implies that lim_𝑛→∞𝑦_𝑛−𝑥_𝑛=0, (3.19) that is, lim_𝑛→∞𝑄_𝐶𝑥_𝑛−𝜆_𝑛𝐴𝑥_𝑛−𝑥_𝑛=0. (3.20) Next, we show that limsup_𝑛→∞𝑢−𝑄^′𝑢,𝑗𝑥_𝑛−𝑄^′𝑢≤0. (3.21) To show (3.21), since {𝑥_𝑛} is bounded, we can choose a sequence {𝑥_𝑛𝑖} of {𝑥_𝑛} that converges weakly to 𝑧 such that limsup_𝑛→∞𝑢−𝑄^′𝑢,𝑗𝑥_𝑛−𝑄^′𝑢=limsup_𝑖→∞𝑢−𝑄^′𝑢,𝑗𝑥_𝑛𝑖−𝑄^′𝑢. (3.22) We first prove 𝑧∈𝑆(𝐶,𝐴). Since 𝜆_𝑛 is in [𝑎,𝑏], it follows that {𝜆_𝑛𝑖} is bounded, and so there exists a subsequence {𝜆_𝑛𝑖𝑗} of {𝜆_𝑛𝑖} which converges to 𝜆₀∈[𝑎,𝑏]. We may assume, without loss of generality, that 𝜆_𝑛𝑖→𝜆₀ as 𝑖→∞. Since 𝑄_𝐶 is nonexpansive, it follows that 𝑄_𝐶𝑥_𝑛𝑖−𝜆₀𝐴𝑥_𝑛𝑖−𝑥_𝑛𝑖≤𝑄_𝐶𝑥_𝑛𝑖−𝜆₀𝐴𝑥_𝑛𝑖−𝑄_𝐶𝑥_𝑛𝑖−𝜆_𝑛𝑖𝐴𝑥_𝑛𝑖+𝑄_𝐶𝑥_𝑛𝑖−𝜆_𝑛𝑖𝐴𝑥_𝑛𝑖−𝑥_𝑛𝑖≤𝑥_𝑛𝑖−𝜆₀𝐴𝑥_𝑛𝑖−𝑥_𝑛𝑖−𝜆_𝑛𝑖𝐴𝑥_𝑛𝑖+𝑄_𝐶𝑥_𝑛𝑖−𝜆_𝑛𝑖𝐴𝑥_𝑛𝑖−𝑥_𝑛𝑖≤𝜆_𝑛𝑖−𝜆₀𝐴𝑥_𝑛𝑖+𝑄_𝐶𝑥_𝑛𝑖−𝜆_𝑛𝑖𝐴𝑥_𝑛𝑖−𝑥_𝑛𝑖, (3.23) which implies that (noting that (3.20)) lim_𝑖→∞𝑄_𝐶𝐼−𝜆₀𝐴𝑥_𝑛𝑖−𝑥_𝑛𝑖=0. (3.24) By Lemma 2.7 and (3.24), we have 𝑧∈𝐹(𝑄_𝐶(𝐼−𝜆₀𝐴)), and it follows from Lemma 2.6 that 𝑧∈𝑆(𝐶,𝐴).
Now, from (3.22) and Lemma 2.4, we have limsup_𝑛→∞𝑢−𝑄^′𝑢,𝑗𝑥_𝑛−𝑄^′𝑢=limsup_𝑖→∞𝑢−𝑄^′𝑢,𝑗𝑥_𝑛𝑖−𝑄^′𝑢=𝑢−𝑄^′𝑢,𝑗𝑧−𝑄^′𝑢≤0. (3.25)
Finally, from (1.5) and (3.2), we have 𝑥_𝑛+1−𝑧²=𝛼_𝑛𝑢+𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝑧_𝑛−𝑧,𝑗𝑥_𝑛+1−𝑧=𝛼_𝑛𝑢−𝑧,𝑗𝑥_𝑛+1−𝑧+𝛽_𝑛𝑥_𝑛−𝑧,𝑗𝑥_𝑛+1−𝑧+𝛾_𝑛𝑧_𝑛−𝑧,𝑗𝑥_𝑛+1≤1−𝑧2𝛽_𝑛𝑥_𝑛−𝑧²+𝑥_𝑛+1−𝑧²+𝛼_𝑛𝑢−𝑧,𝑗𝑥_𝑛+1+1−𝑧2𝛾_𝑛𝑧_𝑛−𝑧²+𝑥_𝑛+1−𝑧²≤121−𝛼_𝑛𝑥_𝑛−𝑧²+𝑥_𝑛+1−𝑧²+𝛼_𝑛𝑢−𝑧,𝑗𝑥_𝑛+1−𝑧, (3.26) which implies that 𝑥_𝑛+1−𝑧²≤1−𝛼_𝑛𝑥_𝑛−𝑧²+2𝛼_𝑛𝑢−𝑧,𝑗𝑥_𝑛+1−𝑧. (3.27) Finally, by Lemma 2.9 and (3.27), we conclude that 𝑥_𝑛 converges strongly to 𝑄^′𝑢. This completes the proof.

Remark 3.3. From (3.1), we know that 𝑄(𝐼−𝜆_𝑛𝐴) is nonexpansive. If 𝑆(𝐶,𝐴)≠∅, it follows that there exists a sunny nonexpansive retraction 𝑄^′ of 𝐶 onto 𝐹(𝑄(𝐼−𝜆_𝑛𝐴))=𝑆(𝐶,𝐴).

4. Application

In this section, we prove a strong convergence theorem in a uniformly convex and 2-uniformly smooth Banach space by using Theorem 3.2. We study the problem of finding a fixed point of a strictly pseudocontractive mapping.

A mapping 𝑇 of 𝐶 into itself is said to be strictly pseudocontractive if there exists 0≤𝜎<1 such that for all 𝑥,𝑦∈𝐶 there exists 𝑗(𝑥−𝑦)∈𝐽(𝑥−𝑦) such that 𝑇𝑥−𝑇𝑦,𝑗𝑥−𝑦≤𝜎𝑥−𝑦². (4.1) This inequality can be written in the following form(𝐼−𝑇)𝑥−(𝐼−𝑇)𝑦,𝑗𝑥−𝑦≥(1−𝜎)𝑥−𝑦². (4.2) Now we give an application concerning a strictly pseudocontractive mapping.

Theorem 4.1. Let 𝑋 be a uniformly convex and 2-uniformly smooth Banach space with weakly sequentially continuous duality mapping, and let 𝐶 be a nonempty closed convex subset and a sunny nonexpansive retract of 𝑋. Let 𝑇 be a strictly pseudocontractive and L-Lipschitz continuous mapping of 𝐶 into itself with 𝐹(𝑇)≠∅. Let {𝛼_𝑛}, {𝛽_𝑛}, and {𝛾_𝑛} be three sequences in (0,1) and {𝜆_𝑛} a real number sequence in [𝑎,𝑏] for some 𝑎,𝑏 with 0<𝑎<𝑏<1−𝜎/𝐾²(𝐿+1)² satisfying the following conditions: (i)𝛼_𝑛+𝛽_𝑛+𝛾_𝑛=1,forall𝑛≥0; (ii)lim_𝑛→∞𝛼_𝑛=0 and ∑^∞_𝑛=0𝛼_𝑛=∞; (iii)0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1; (iv)lim_𝑛→∞(𝜆_𝑛+1−𝜆_𝑛)=0. For fixed 𝑢∈𝐶 and arbitrarily given 𝑥₀∈𝐶, define a sequence {𝑥_𝑛} iteratively by 𝑦_𝑛=1−𝜆_𝑛𝑥_𝑛+𝜆_𝑛𝑇𝑥_𝑛,𝑧_𝑛=1−𝜆_𝑛𝑦_𝑛+𝜆_𝑛𝑇𝑦_𝑛,𝑥_𝑛+1=𝛼_𝑛𝑢+𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝑧_𝑛, (4.3) for every 𝑛=1,2,…. Then {𝑥_𝑛} converges strongly to a fixed point of 𝑇.

Proof. Putting 𝐴=𝐼−𝑇, we have from (4.2) that 𝐴 is (1−𝜎)-strongly accretive. At the same time, since 𝑇 is 𝐿-Lipschitz continuous, then we have 𝐴𝑥−𝐴𝑦=(𝐼−𝑇)𝑥−(𝐼−𝑇)𝑦≤(𝐿+1)𝑥−𝑦, (4.4) for all 𝑥,𝑦∈𝐶, that is, 𝐴 is (𝐿+1)-Lipschitz continuous mapping. It follows from Remark 2.1 (2) that 𝐴 is (1−𝜎)/(𝐿+1)²-inverse-strongly accretive mapping. It is easy to show that 𝑆(𝐶,𝐴)=𝑆(𝐶,𝐼−𝑇)=𝐹(𝑇)≠∅. Therefore, using Theorem 3.2, we can obtain the desired conclusion. This completes the proof.

Acknowledgment

The paper is partially supported by the Program TH-1-3, Optimization Lean Cycle, of Sub-Projects TH-1 of Spindle Plan Four in Excellence Teaching and Learning Plan of Cheng Shiu University. The second author was partially supported by the Program TH-1-3, Optimization Lean Cycle, of Sub-Projects TH-1 of Spindle Plan Four inExcellence Teaching and Learning Plan of Cheng Shiu University and wassupported in part by NSC 100-2221-E-230-012.