Extragradient Method for Solutions of Variational Inequality Problems in Banach Spaces

Abstract

We introduce an iterative process which converges strongly to solutions of a certain variational inequity problem for 𝜂-inverse strongly accretive mappings in the set of common fixed points of finite family of strictly pseudocontractive mappings in Banach spaces. 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 real normed linear space with dual 𝐸^∗. For 1<𝑞<∞, we denote by 𝐽_𝑞 the generalized duality mapping from 𝐸 to 2^𝐸∗ defined by 𝐽_𝑞(𝑥)∶=𝑥^∗∈𝐸^∗∶𝑥,𝑥^∗=‖𝑥‖^𝑞,𝑥^∗=‖𝑥‖^𝑞−1, (1) where ⟨⋅,⋅⟩ denotes the duality pairing. In particular, 𝐽=𝐽₂ is called the normalized duality map. It is well known (see e.g., [1]) that 𝐽_𝑞 is single valued if 𝐸 is smooth and that 𝐽_𝑞(𝑥)=‖𝑥‖^𝑞−2𝐽(𝑥),𝑥≠0. (2) In the sequel, we will denote the single-valued generalized map by 𝑗_𝑞.

A mapping 𝐴 with domain 𝐷(𝐴)⊆𝐸 and range 𝑅(𝐴) in 𝐸 is called 𝛼-strongly accretive if there exist 𝛼∈(0,1) and 𝑗_𝑞(𝑥−𝑦)∈𝐽_𝑞(𝑥−𝑦) such that 𝐴𝑥−𝐴𝑦,𝑗_𝑞(𝑣−𝑢)≥𝛼𝑥−𝑦^𝑞. (3)𝐴 is called𝜂-inverse strongly accretive if there exist 𝜂∈(0,1) and 𝑗_𝑞(𝑥−𝑦)∈𝐽_𝑞(𝑥−𝑦) such that 𝐴𝑥−𝐴𝑦,𝑗_𝑞𝑥−𝑦≥𝜂𝐴𝑥−𝐴𝑦^𝑞,forevery𝑥,𝑦∈𝐷(𝐴). (4) Let 𝐶 be a nonempty, closed, and convex subset of 𝐸 and, let 𝐴∶𝐶→𝐸 be a nonlinear mapping. The variational inequality problem is to find𝑢∈𝐶suchthat𝐴𝑢,𝑗(𝑣−𝑢)≥0,∀𝑣∈𝐶, (5) for some 𝑗(𝑣−𝑢)∈𝐽(𝑣−𝑢). The set of solutions of variational inequality problem is denoted by 𝑉𝐼(𝐶,𝐴). If 𝐸∶=𝐻, a real Hilbert space, the variational inequality problem reduces to find𝑢∈𝐶suchthat⟨𝐴𝑢,𝑣−𝑢⟩≥0,∀𝑣∈𝐶, (6) which was introduced and studied by Stampacchia [2].

Variational inequality theory has emerged as an important tool in studying a wide class of related problems arising in mathematical, physical, regional, engineering, and nonlinear optimization sciences (see, for instance, [3–12]).

In 1976, Korpelevič [4] introduced the following well-known extragradient method: 𝑦_𝑛=𝑃_𝐶𝑥_𝑛−𝛾𝐴𝑥_𝑛,𝑥_𝑛+1=𝑃_𝐶𝑥_𝑛−𝛾𝐴𝑦_𝑛,𝑛≥0, (7) where 𝑃_𝐶 is the metric projection from ℝ^𝑛 onto its subset 𝐶, for some 𝛾>0, and 𝐴∶𝐶→ℝ^𝑛 is an accretive operator. He proved that the sequence {𝑥_𝑛} converges to a solution of the variational inequality (6).

Furthermore, Noor [6] proved that the iterative scheme, given by 𝑦_𝑛=𝑃_𝐶𝑥_𝑛−𝛾𝐴𝑥_𝑛,𝑥_𝑛+1=𝑃_𝐶𝑦_𝑛−𝛾𝐴𝑦_𝑛,𝑛≥0, (8) where 𝐴∶𝐶⊆ℝ^𝑛→ℝ^𝑛 is an accretive operator, converges to a solution of the variational inequality (6).

We note that the above algorithms give strong convergence to a solution of the variational inequality (6). However, both algorithms fail, in general, to converge strongly in the setting of infinite-dimensional Hilbert spaces.

In 2006, Aoyama et al. [13] introduced and studied the following iterative algorithm in a uniformly convex and 2-uniformly smooth Banach spaces possessing weakly sequentially continuous duality mapping: 𝑥_𝑛+1=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝑄_𝐶𝑥_𝑛−𝜆_𝑛𝐴𝑥_𝑛,𝑛≥0, (9) where 𝑄_𝐶 is a sunny nonexpansive retraction from 𝐸 onto a closed and convex 𝐶, 𝐴∶𝐶→𝐸 is an 𝜂-inverse strongly accretive mapping and {𝛼_𝑛} and {𝜆_𝑛} subsets of real numbers, satisfy certain conditions. They proved that the sequence in (9) converges weakly to a point 𝑧∈𝑉(𝐶,𝐴).

Recently, Yao et al. [8] introduced and considered the following iterative method for 𝜂-strongly accretive mappings in a uniformly convex and 2-uniformly smooth Banach space possessing weakly sequentially continuous duality mapping: 𝑦_𝑛=𝑄_𝐶𝑥_𝑛−𝜆_𝑛𝐴𝑥_𝑛,𝑥_𝑛+1=𝛼_𝑛𝑢+𝛽_𝑛𝑥_𝑛+𝛾_𝑛𝑄_𝐶𝑦_𝑛−𝜆_𝑛𝐴𝑦_𝑛,𝑛≥0, (10) where 𝑄_𝐶 is a sunny nonexpansive retraction from 𝐸 onto 𝐶. They proved that the sequence {𝑥_𝑛} defined by (10) converges strongly to 𝑄_{𝑉𝐼(𝐶,𝐴)}𝑢 provided that real sequences {𝛼_𝑛}, {𝛽_𝑛}, {𝛾_𝑛}, and {𝜆_𝑛} satisfy certain conditions.

Let 𝐶 be a nonempty subset of a real Banach space 𝐸. A mapping 𝑇∶𝐶→𝐸 is called 𝜆-strictly pseudocontractive of Browder-Petryshyn type [14] if for all 𝑥,𝑦∈𝐷(𝑇) there exist 𝜆>0 and 𝑗_𝑞(𝑥−𝑦)∈𝐽_𝑞(𝑥−𝑦) such that 𝑇𝑥−𝑇𝑦,𝑗_𝑞𝑥−𝑦≤𝑥−𝑦^𝑞−𝜆𝑥−𝑦−𝑇𝑥−𝑇𝑦^𝑞. (11)𝑇 is called Lipschitz if there exists 𝐿≥0 such that 𝑇𝑥−𝑇𝑦≤𝐿𝑥−𝑦∀𝑥,𝑦∈𝐷(𝑇). (12) If 𝐿<1 in (12), then 𝑇 is called contraction, while 𝑇 is said to be nonexpansive if 𝐿=1.

If 𝐸=𝐻, a real Hilbert space, then (11) is equivalent to the inequality 𝑇𝑥−𝑇𝑦²≤𝑥−𝑦²+𝑘𝑥−𝑦−𝑇𝑥−𝑇𝑦²,𝑘=(1−2𝜆), (13) and we can assume also that 𝑘≥0, so that 𝑘∈[0,1). A point 𝑥∈𝐶 is a fixed point of 𝑇 if 𝑇𝑥=𝑥, and we denote by 𝐹(𝑇) the set of fixed points of 𝑇; that is, 𝐹(𝑇)={𝑥∈𝐶∶𝑇𝑥=𝑥}.

In 2001, Yamada [7] introduced a hybrid steepest descent method which relates solutions of variational inequality problems with fixed point of mappings in Hilbert spaces. He proved that if 𝑇 is nonexpansive self-map on 𝐶 and 𝐴 is an 𝜂-strongly accretive mapping from 𝐶 into 𝐸 satisfying certain conditions, then the sequence defined by 𝑥_𝑛+1=𝑇𝑥_𝑛−𝜇𝜆_𝑛𝐴𝑇𝑥_𝑛,𝑛≥0, (14) converges strongly to the unique solution of the variational inequality Find𝑥^∗∈𝐹(𝑇)suchthat𝐴𝑥^∗,𝑥−𝑥^∗(𝑇).≥0,∀𝑥∈𝐹 (15) The above results naturally bring us to the following question.

Question. Could we produce an iterative scheme which approximates a solution of variational inequality (5) for 𝜂-inverse strongly accretive mappings in Banach spaces?

In this paper, motivated by Yao et al. [8] and Yamada [7], it is our purpose to introduce an iterative scheme which converges strongly to a solution of the variational inequality (5) for 𝜂-inverse strongly accretive mapping in the set of common fixed points of finite family of strictly pseudocontractive mappings in a uniformly convex and 𝑞-uniformly smooth Banach space 𝐸 possessing weakly sequentially continuous duality mapping. Our results complement or improve the results of Yao et al. [8], Aoyama et al. [13], and some authors.

2. Preliminaries

Let 𝐸 be a real Banach space. The modulus of smoothness of 𝐸 is the function 𝜌_𝐸∶[0,∞)→[0,∞) defined by 𝜌_𝐸(𝜏)∶=sup{(1/2)(‖𝑥+𝑦‖+‖𝑥−𝑦‖)−1∶‖𝑥‖=1,‖𝑦‖=𝜏}. If 𝜌_𝐸(𝜏)>0 for all 𝜏>0, then 𝐸 is said to be smooth. If there exists a constant 𝑐>0 and a real number 1<𝑞<∞, such that 𝜌_𝐸(𝜏)≤𝑐𝜏^𝑞, then 𝐸 is said to be 𝑞-uniformly smooth.

If 𝐸 is a real 𝑞-uniformly smooth Banach space, then by [15], the following geometric inequality holds: 𝑥+𝑦^𝑞≤‖𝑥‖^𝑞+𝑞𝑦,𝑗_𝑞(𝑥)+𝑐_𝑞𝑦^𝑞, (16) for all 𝑥,𝑦∈𝐸 and some real constant 𝑐_𝑞>0.

It is well known (see e.g., [16]) that 𝐿_𝑝𝑙_𝑝or𝑊^𝑝_𝑚is𝑝-uniformlysmoothif1<𝑝<2,2-uniformlysmoothif𝑝≥2. (17) The Banach space 𝐸 is said to be uniformly convex if, given 𝜀>0, there exists 𝛿>0, such that, for all 𝑥,𝑦∈𝐸 with ‖𝑥‖≤1, ‖𝑦‖≤1 and ‖𝑥−𝑦‖≥𝜀, ‖(1/2)(𝑥+𝑦)‖≤1−𝛿. It is well known that 𝐿_𝑝, ℓ_𝑝, and Sobolev spaces 𝑊^𝑝_𝑚, (1<𝑝<∞) are uniformly convex.

Let 𝐶⊆𝐸 be closed convex and 𝑄 a mapping of 𝐸 onto 𝐶. Then, 𝑄 is said to be sunny if 𝑄(𝑄(𝑥)+𝑡(𝑥−𝑄(𝑥)))=𝑄(𝑥) for all 𝑥∈𝐸 and 𝑡≥0. A mapping 𝑄 of 𝐸 into 𝐶 is said to be a retraction if 𝑄²=𝑄. If a mapping 𝑄 is a retraction, then 𝑄(𝑧)=(𝑧) for every 𝑧∈𝑅(𝑄), range of 𝑄. A subset 𝐶 of 𝐸 is said to be a sunny nonexpansive retract of 𝐸 if there exists a sunny nonexpansive retraction of 𝐸 onto 𝐶, and it is said to be a nonexpansive retract of 𝐸 if there exists a nonexpansive retraction of 𝐸 onto 𝐶. If 𝐸=𝐻, the metric projection 𝑃_𝐶 is a sunny nonexpansive retraction from 𝐻 to any closed convex subset of 𝐻. Moreover, if 𝐶 is a nonempty closed convex subset of a uniformly convex and uniformly smooth real Banach space 𝐸 and 𝑇 is a nonexpansive mapping of 𝐶 into itself with 𝐹(𝑇)≠∅, then the set 𝐹(𝑇) is a sunny nonexpansive retract of 𝐶.

In what follows, we will make use of the following lemmas.

Lemma 1 (see, e.g., [17]). Let 𝐸 be a smooth Banach space, and let 𝐾 be a nonempty subset of 𝐸. Let 𝑄∶𝐸→𝐾 be a retraction, and let 𝐽 be the normalized duality map on 𝐸. Then, the following are equivalent:(i)𝑄is sunny nonexpansive,(ii)⟨𝑥−𝑄(𝑥),𝐽(𝑦−𝑄(𝑥))⟩≤0 for all 𝑥∈𝐸 and 𝑦∈𝐾.

Lemma 2 (see [18]). Let {𝑎_𝑛} be a sequence of nonnegative real numbers satisfying the following relation: 𝑎_𝑛+1≤1−𝛼_𝑛𝑎_𝑛+𝛼_𝑛𝛿_𝑛,𝑛≥𝑛₀, (18) where {𝛼_𝑛}⊂(0,1) and {𝛿_𝑛}⊂ℝ satisfying the following conditions: lim_𝑛→∞𝛼_𝑛=0,∑^∞_𝑛=1𝛼_𝑛=∞, and limsup_𝑛→∞𝛿_𝑛≤0. Then, lim_𝑛→∞𝑎_𝑛=0.

Lemma 3 (see [13]). Let 𝐶 be a nonempty closed convex subset of a smooth Banach space 𝐸. Let 𝑄_𝐶 be a sunny nonexpansive retraction from 𝐸 onto 𝐶, and let 𝐴 be an accretive operator of 𝐶 into 𝐸. Then, for all 𝜆>0, 𝑉𝐼(𝐶,𝐴)=𝐹𝑄_𝐶(𝐼−𝜆𝐴), (19) where 𝑉𝐼(𝐶,𝐴)={𝑥^∗∈𝐶∶⟨𝐴𝑥^∗,𝐽(𝑥−𝑥^∗)⟩≥0,∀𝑥∈𝐶}.

Lemma 4 (see [17]). Let 𝐶 be a nonempty bounded closed convex subset of a uniformly convex Banach space 𝐸, 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 5 (see [19]). Let 𝐸 be a real Banach space. Then, for any given 𝑥,𝑦∈𝐸, the following inequality holds: 𝑥+𝑦²≤‖𝑥‖²+2𝑦,𝑗𝑥+𝑦,∀𝑗𝑥+𝑦∈𝐽𝑥+𝑦. (20)

Lemma 6 (see [20]). Let 𝐾 be a nonempty closed convex subset of a strictly convex Banach space 𝐸. Let 𝑇_𝑖∶𝐾→𝐸,𝑖=1,2,…,𝑟, be a family of nonexpansive mappings such that ⋂^𝑟_𝑖=1𝐹(𝑇_𝑖)≠∅. Let 𝛼₀,𝛼₁,𝛼₂,…,𝛼_𝑟 be real numbers in (0,1) such that ∑^𝑟_𝑖=0𝛼_𝑖=1, and let 𝑇∶=𝛼₀𝐼+𝛼₁𝑇₁+⋯+𝛼_𝑟𝑇_𝑟. Then, 𝑇 is nonexpansive, and 𝐹(𝑇)=⋂^𝑟_𝑖=1𝐹(𝑇_𝑖).

Lemma 7 (see [21]). Let 𝐶 be a nonempty, closed and convex subset of a real uniformly convex and smooth Banach space 𝐸. Let 𝑇_𝑖∶𝐶→𝐸, 𝑖=1,…,𝑁, be 𝜆_𝑖-strictly pseudocontractive mappings such that ⋂^𝑁_𝑖=1𝐹(𝑇_𝑖)≠∅. Let 𝑇∶=𝜃₁𝑇₁+𝜃₂𝑇₂+⋯+𝜃_𝑁𝑇_𝑁 with 𝜃₁+𝜃₂+⋯+𝜃_𝑁=1. Then 𝑇 is 𝜆-strictly pseudocontractive with 𝜆∶=min{𝜆_𝑖∶𝑖=1,…,𝑁} and 𝐹(𝑇)=⋂^𝑁_𝑖=1𝐹(𝑇_𝑖).

Lemma 8 (see [22]). Let 𝐶 be a nonempty closed and convex subset of a real 𝑞-uniformly smooth Banach space 𝐸 for 1<𝑞<∞. Let 𝑇∶𝐶→𝐸 be a 𝜆-strictly pseudocontractive mapping. Then, for 0<𝜇<𝜇₀=min{1,(𝑞𝜆/𝑐_𝑞)^1/(𝑞−1)}, where 𝐿 is the Lipchitz constant of 𝑇 and 𝑐_𝑞 is a constant in (16), the mapping 𝑇_𝜇𝑥∶=(1−𝜇)𝑥+𝜇𝑇𝑥 is nonexpansive, and 𝐹(𝑇_𝜇)=𝐹(𝑇).

Lemma 9. Let 𝐶 be a nonempty closed and convex subset of a a real 𝑞-uniformly smooth Banach space 𝐸 for 1<𝑞<∞. Let 𝐴∶𝐶→𝐸 be an 𝜂-inverse strongly accretive mapping. Then, for 0<𝛾<(𝑞𝜂/𝑐_𝑞)^1/(𝑞−1), the mapping 𝐴_𝜇𝑥∶=(𝑥−𝛾𝐴𝑥) is nonexpansive.

Proof. Now, using inequality (16), we get that 𝐴_𝛾𝑥−𝐴_𝛾𝑦^𝑞=𝑥−𝑦−𝛾𝐴𝑥−𝐴𝑦^𝑞≤𝑥−𝑦^𝑞−𝑞𝛾𝐴𝑥−𝐴𝑦,𝑗_𝑞𝑥−𝑦+𝛾^𝑞𝑐_𝑞𝐴𝑥−𝐴𝑦^𝑞≤𝑥−𝑦^𝑞−𝑞𝛾𝜂𝐴𝑥−𝐴𝑦²+𝛾^𝑞𝑐_𝑞𝐴𝑥−𝐴𝑦²≤𝑥−𝑦^𝑞−𝛾𝑞𝜂−𝛾^𝑞−1𝑐_𝑞𝐴𝑥−𝐴𝑦^𝑞,≤𝑥−𝑦^𝑞. (21) The proof is complete.

Lemma 10 (see [10]). Let 𝐸 be a uniformly convex Banach space, and let 𝐵_𝑅(0) be a closed ball of 𝐸. Then, there exists a continuous strictly increasing convex function 𝑔∶[0,∞)→[0,∞) with 𝑔(0)=0 such that 𝛼₀𝑥₀+𝛼₁𝑥₁+𝛼₂𝑥₂+⋯+𝛼_𝑘𝑥_𝑘²≤^𝑘_𝑖=0𝛼_𝑖𝑥_𝑖²−𝛼_𝑠𝛼_𝑡𝑔𝑥_𝑠−𝑥_𝑡, (22) for 𝑥_𝑖∈𝐵_𝑅(0)∶={𝑥∈𝐸∶‖𝑥‖≤𝑅}, 𝑖=0,1,2,…,𝑘 with ∑^𝑘_𝑖=0𝛼_𝑖=1.

Lemma 11 (see [5]). 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. (23) In fact, 𝑚_𝑘=max{𝑗≤𝑘∶𝑎_𝑗<𝑎_𝑗+1}.

3. Main Results

Let 𝐶 be a nonempty closed convex subset of a real uniformly convex and 𝑞-uniformly smooth Banach space 𝐸(1<𝑞<∞). Let 𝑇_𝑖∶𝐶→𝐶, for 𝑖=1,…,𝑁, be a 𝜆_𝑖-strictly pseudocontractive mappings, and let 𝐴∶𝐶→𝐸 be an 𝜂-inverse strongly accretive mapping. Then, in what follows, we will study the variational inequality Find𝑥^∗∈∩^𝑁_𝑖=1𝐹𝑇_𝑖suchthat𝐴𝑥^∗,𝐽𝑥−𝑥^∗≥0,∀𝑥∈∩^𝑁_𝑖=1𝐹𝑇_𝑖, (24) and the following iteration process: 𝑥₀𝑧∈𝐶,_𝑛=𝑐_𝑛𝑥_𝑛+1−𝑐_𝑛𝑆𝑥_𝑛,𝑥_𝑛+1=𝑄_𝐶1−𝛼_𝑛𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝑄_𝐶𝐼−𝛾𝐴𝑧_𝑛, (25) where 𝑆∶=[(1−𝜇)𝐼+𝜇𝑇], for 𝑇∶=𝜃₁𝑇₁+𝜃₂𝑇₂+⋯+𝜃_𝑛𝑇_𝑁, such that 𝜃₁+𝜃₂+⋯+𝜃_𝑁=1, 0<𝜇<𝜇₀=min{1,(𝑞𝜆/𝑐_𝑞)^1/(𝑞−1)}, for 𝜆∶=min{𝜆_𝑖∶𝑖=1,2,…,𝑁}, and 0<𝛾<(𝑞𝜂/𝑐_𝑞)^1/(𝑞−1), and 𝑐_𝑞 is the real number in (16). In addition, we assume {𝛼_𝑛}⊂(0,𝑐)⊂(0,1) and {𝛽_𝑛},{𝑐_𝑛}⊂[𝑎,𝑏]⊂(0,1) as real sequences satisfying the following control conditions: (i) lim_𝑛→∞𝛼_𝑛=0, (ii) ∑𝛼_𝑛=∞, lim_𝑛→∞(|𝛽_𝑛−𝛽_𝑛−1|/𝛼_𝑛−1)=0,lim_𝑛→∞(|𝛼_𝑛−𝛼_𝑛−1|/𝛼_𝑛−1)=0, andlim_𝑛→∞(|𝑐_𝑛−𝑐_𝑛−1|/𝛼_𝑛−1)=0.

We now prove our main theorem.

Theorem 12. Let 𝐶 be a nonempty closed convex subset of a real uniformly convex and 𝑞-uniformly smooth Banach space 𝐸 possessing weakly sequentially continuous duality mapping. Let 𝑇_𝑖∶𝐶→𝐶, for 𝑖=1,…,𝑁, be 𝜆_𝑖-strictly pseudocontractive mappings, and let 𝐴∶𝐶→𝐸 be 𝜂-inverse strongly accretive mapping. Let {𝑥_𝑛} be a sequence defined by (25). Assume that ℱ∶=𝐹∩𝑉𝐼(𝐶,𝐴)≠∅, where 𝐹=∩^𝑁_𝑖=1𝐹(𝑇_𝑖)=𝐹(𝑆). Then, {𝑥_𝑛} converges strongly to 𝑄_ℱ(0), where 𝑄_ℱ is a sunny nonexpansive retraction of 𝐸 onto ℱ, which is a solution of the variational inequality (24).

Proof. By Lemmas 7 and 8 we have that 𝑆 is nonexpansive. In addition, by Lemma 9 we get that 𝑄_𝐶[𝐼−𝛾𝐴] is nonexpansive. Let 𝑝∈ℱ and, let 𝑦_𝑛∶=𝛽_𝑛𝑥_𝑛+(1−𝛽_𝑛)𝑄_𝐶[𝐼−𝛾𝐴]𝑧_𝑛. Then from (25), Lemmas 8 and 9 we have that 𝑧_𝑛−𝑝≤𝑐_𝑛𝑥_𝑛−𝑝+1−𝑐_𝑛𝑆𝑥_𝑛−𝑝≤𝛽_𝑛𝑥_𝑛−𝑝+1−𝛽_𝑛𝑥_𝑛≤𝑥−𝑝_𝑛,𝑦−𝑝_𝑛−𝑝≤𝛽_𝑛𝑥_𝑛+−𝑝1−𝛽_𝑛𝑄_𝐶𝐼−𝛾𝐴𝑧_𝑛−𝑄_𝐶𝑝−𝛾𝐴𝑝≤𝛽_𝑛𝑥_𝑛−𝑝+1−𝛽_𝑛𝑧_𝑛−𝑝≤𝛽_𝑛𝑥_𝑛−𝑝+1−𝛽_𝑛𝑥_𝑛≤𝑥−𝑝_𝑛−𝑝. (26)(27)
Thus, from (25) and (27), we get that 𝑥_𝑛+1−𝑝=𝑄_𝐶1−𝛼_𝑛𝑦_𝑛−𝑄_𝐶𝑝≤1−𝛼_𝑛𝑦_𝑛−𝑝+𝛼_𝑛𝑝≤1−𝛼_𝑛𝑥_𝑛−𝑝+𝛼_𝑛𝑝. (28)
Therefore, by induction, 𝑥_𝑛+1−𝑝≤max𝑥₀−𝑝,𝑝,∀𝑛≥0, (29) which implies that {𝑥_𝑛} and hence {𝑦_𝑛}, {𝑧_𝑛}, and {𝐴𝑥_𝑛} are bounded. Furthermore, from (25), we obtain that 𝑧_𝑛+1−𝑧_𝑛=𝑐_𝑛+1𝑥_𝑛+1+1−𝑐_𝑛+1𝑆𝑥_𝑛+1−𝑐_𝑛𝑥_𝑛+1−𝑐_𝑛𝑆𝑥_𝑛=𝑐_𝑛+1𝑥_𝑛+1−𝑐_𝑛𝑥_𝑛+1−𝑐_𝑛+1𝑆𝑥_𝑛+1−1−𝑐_𝑛𝑆𝑥_𝑛≤𝑐_𝑛+1−𝑐_𝑛𝑥_𝑛+1+𝑐_𝑛𝑥_𝑛+1−𝑥_𝑛+𝑐_𝑛+1−𝑐_𝑛𝑆𝑥_𝑛+1+1−𝑐_𝑛𝑥_𝑛+1−𝑥_𝑛≤𝑥_𝑛+1−𝑥_𝑛+𝑐_𝑛+1−𝑐_𝑛𝑥_𝑛+1+𝑆𝑥_𝑛+1,𝑦_𝑛+1−𝑦_𝑛=𝛽_𝑛+1𝑥_𝑛+1+1−𝛽_𝑛+1𝑄_𝐶𝑧_𝑛+1−𝛾𝐴𝑧_𝑛+1−𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝑄_𝐶𝑧_𝑛−𝛾𝐴𝑧_𝑛=𝛽_𝑛+1𝑥_𝑛+1−𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛+1×𝑄_𝐶𝑧_𝑛+1−𝛾𝐴𝑧_𝑛+1−1−𝛽_𝑛𝑄_𝐶𝑧_𝑛−𝛾𝐴𝑧_𝑛≤𝛽_𝑛+1−𝛽_𝑛𝑥_𝑛+1+𝛽_𝑛𝑥_𝑛+1−𝑥_𝑛+𝛽_𝑛+1−𝛽_𝑛𝑄_𝐶𝑧_𝑛+1−𝛾𝐴𝑧_𝑛+1+1−𝛽_𝑛×𝑧_𝑛+1−𝑧_𝑛≤𝛽_𝑛𝑥_𝑛+1−𝑥_𝑛+𝛽_𝑛+1−𝛽_𝑛𝑥_𝑛+1+𝑄_𝐶𝑧_𝑛+1−𝛾𝐴𝑧_𝑛+1+1−𝛽_𝑛𝑧_𝑛+1−𝑧_𝑛≤𝛽_𝑛𝑥_𝑛+1−𝑥_𝑛+𝛽_𝑛+1−𝛽_𝑛𝑥_𝑛+1+𝑄_𝐶𝑧_𝑛+1−𝛾𝐴𝑧_𝑛+1+1−𝛽_𝑛𝑥_𝑛+1−𝑥_𝑛+𝑐_𝑛+1−𝑐_𝑛𝑥_𝑛+1+𝑆𝑥_𝑛+1≤𝑥_𝑛+1−𝑥_𝑛+𝛽_𝑛+1−𝛽_𝑛×𝑥_𝑛+1+𝑄_𝐶𝑧_𝑛+1−𝛾𝐴𝑧_𝑛+1+𝑐_𝑛+1−𝑐_𝑛𝑥_𝑛+1+𝑆𝑥_𝑛+1. (30)(31) And, hence, from (25) and (31), we have that 𝑥_𝑛+1−𝑥_𝑛=𝑄_𝐶1−𝛼_𝑛𝑦_𝑛−𝑄_𝐶1−𝛼_𝑛−1𝑦_𝑛−1≤1−𝛼_𝑛𝑦_𝑛−1−𝛼_𝑛−1𝑦_𝑛−1≤1−𝛼_𝑛𝑦_𝑛−1−𝛼_𝑛−1𝑦_𝑛+1−𝛼_𝑛−1𝑦_𝑛−1−𝛼_𝑛−1𝑦_𝑛−1≤𝛼_𝑛−𝛼_𝑛−1𝑦_𝑛+1−𝛼_𝑛−1𝑦_𝑛−1−𝑦_𝑛≤1−𝛼_𝑛−1𝑥_𝑛−𝑥_𝑛−1+𝛼_𝑛−𝛼_𝑛−1𝑀+𝛽_𝑛−𝛽_𝑛−1𝑀+𝑐_𝑛−𝑐_𝑛−1𝑀, (32) for some 𝑀>0. Thus, using the properties of {𝛼_𝑛}, {𝛽_𝑛}, {𝑐_𝑛}, (32), and Lemma 2, we obtain that ‖𝑥_𝑛+1−𝑥_𝑛‖→0, as 𝑛→∞, which implies from (31) that ‖𝑦_𝑛+1−𝑦_𝑛‖→0, as 𝑛→∞. Again from (25), we have that ‖𝑥_𝑛+1−𝑦_𝑛‖=‖𝑄_𝐶[(1−𝛼_𝑛)𝑦_𝑛]−𝑄_𝐶𝑦_𝑛‖=𝛼_𝑛‖𝑦_𝑛‖→0, as 𝑛→∞. Consequently, 𝑥_𝑛−𝑦_𝑛⟶0,as𝑛⟶∞. (33)
Now, we prove that {𝑥_𝑛} converges strongly to the point 𝑥^∗=𝑄_ℱ(0). Let 𝑡_𝑛=𝑄_𝐶[𝐼−𝛾𝐴]𝑧_𝑛, and let 𝑑_𝑛=(1−𝛼_𝑛)𝑦_𝑛. Then, since 𝛼_𝑛→0, we have that 𝑑_𝑛−𝑦_𝑛=𝛼_𝑛𝑦_𝑛⟶0,as𝑛⟶∞. (34) Furthermore, from (25), Lemma 5, and Lemma 10, we get that 𝑥_𝑛+1−𝑥^∗2=𝑄_𝐶1−𝛼_𝑛𝑦_𝑛−𝑄_𝐶𝑥^∗2≤𝛼_𝑛−𝑥^∗+1−𝛼_𝑛𝑦_𝑛−𝑥^∗2≤1−𝛼_𝑛𝑦_𝑛−𝑥^∗2−2𝛼_𝑛𝑥^∗,𝑗𝑑_𝑛−𝑥^∗≤1−𝛼_𝑛𝛽_𝑛𝑥_𝑛−𝑥^∗2+1−𝛽_𝑛𝑡_𝑛−𝑥^∗2−𝛽_𝑛1−𝛽_𝑛𝑔𝑡_𝑛−𝑥_𝑛−2𝛼_𝑛𝑥^∗,𝑗𝑑_𝑛−𝑥^∗≤1−𝛼_𝑛𝛽_𝑛𝑥_𝑛−𝑥^∗2+1−𝛼_𝑛1−𝛽_𝑛𝑧_𝑛−𝑥^∗2−𝛽_𝑛1−𝛽_𝑛1−𝛼_𝑛𝑔𝑡_𝑛−𝑥_𝑛−2𝛼_𝑛𝑥^∗,𝑗𝑑_𝑛−𝑥^∗≤1−𝛼_𝑛𝛽_𝑛𝑥_𝑛−𝑥^∗2+1−𝛼_𝑛1−𝛽_𝑛×𝑐_𝑛𝑥_𝑛−𝑝²+1−𝑐_𝑛𝑆𝑥_𝑛−𝑝²−𝑐_𝑛1−𝑐_𝑛𝑔𝑆𝑥_𝑛−𝑥_𝑛−𝛽_𝑛1−𝛽_𝑛1−𝛼_𝑛𝑔𝑡_𝑛−𝑥_𝑛−2𝛼_𝑛𝑥^∗,𝑗𝑑_𝑛−𝑥^∗≤1−𝛼_𝑛𝛽_𝑛𝑥_𝑛−𝑥^∗2+1−𝛼_𝑛1−𝛽_𝑛×𝑐_𝑛𝑥_𝑛−𝑝²+1−𝑐_𝑛𝑆𝑥_𝑛−𝑝²−𝑐_𝑛1−𝑐_𝑛𝑔𝑆𝑥_𝑛−𝑥_𝑛−𝛽_𝑛1−𝛽_𝑛1−𝛼_𝑛𝑔𝑡_𝑛−𝑥_𝑛−2𝛼_𝑛𝑥^∗,𝑗𝑑_𝑛−𝑥^∗, (35) which implies that 𝑥_𝑛+1−𝑥^∗2≤1−𝛼_𝑛𝑥_𝑛−𝑥^∗2−𝑐_𝑛1−𝑐_𝑛1−𝛼_𝑛1−𝛽_𝑛×𝑔𝑆𝑥_𝑛−𝑥_𝑛−𝛽_𝑛1−𝛽_𝑛1−𝛼_𝑛𝑔𝑡_𝑛−𝑥_𝑛−2𝛼_𝑛𝑥^∗,𝑗𝑑_𝑛−𝑥^∗≤1−𝛼_𝑛𝑥_𝑛−𝑥^∗2−2𝛼_𝑛𝑥^∗,𝑗𝑑_𝑛−𝑥^∗. (36)(37)
Now, following the method of proof of Lemma 3.2 of Maingé [5], we consider two cases.
Case 1. Suppose that there exists 𝑛₀∈ℕ such that {‖𝑥_𝑛−𝑥^∗‖} is decreasing for all 𝑛≥𝑛₀. Then, we get that {‖𝑥_𝑛−𝑥^∗‖} is convergent. Thus, from (36) and the fact that 𝛼_𝑛→0, as 𝑛→∞, we have that 𝑔𝑆𝑥_𝑛−𝑥_𝑛⟶0,𝑔𝑡_𝑛−𝑥_𝑛⟶0,as𝑛⟶∞, (38) which implies that 𝑆𝑥_𝑛−𝑥_𝑛𝑡⟶0,_𝑛−𝑥_𝑛=𝑄_𝐶𝐼−𝛾𝐴𝑥_𝑛−𝑥_𝑛⟶0,as𝑛⟶∞. (39)
In addition, since {𝑑_𝑛} is bounded subset of a reflexive space 𝐸, we can choose a subsequence {𝑑_𝑛𝑖} of {𝑑_𝑛} such that 𝑑_𝑛𝑖⇀𝑧 and limsup_𝑛→∞⟨𝑥^∗,𝑗(𝑑_𝑛−𝑥^∗)⟩=lim_𝑖→∞⟨𝑥^∗,𝑗(𝑑_𝑛𝑖−𝑥^∗)⟩. This implies from (34) and (33) that𝑥_𝑛𝑖⇀𝑧. Then, from (39) and Lemma 4, we have that 𝑧∈𝐹(𝑆)=∩^𝑁_𝑖=1𝐹(𝑇_𝑖). Moreover, from (39) and Lemma 4, we have that 𝑧∈𝐹(𝑄_𝐶[𝐼−𝛾𝐴]), and by Lemma 3, we get 𝑧∈𝑉𝐼(𝐶,𝐴), and hence 𝑧∈ℱ. Therefore, using the fact that 𝐸 has a weakly sequentially continuous duality mapping and Lemma 1, we immediately obtain that limsup_𝑛→∞𝑥^∗,𝑗𝑑_𝑛−𝑥^∗=lim_𝑖→∞𝑥^∗,𝑗𝑑_𝑛𝑖−𝑥^∗=𝑥^∗,𝑗𝑧−𝑥^∗≥0. (40) Then, it follows from (37), (40), and Lemma 2 that ‖𝑥_𝑛−𝑥^∗‖→0, as 𝑛→∞. Consequently, 𝑥_𝑛→𝑥^∗=𝑄_ℱ0.
Case 2. Suppose that there exists a subsequence {𝑛_𝑖} of {𝑛} such that 𝑥_𝑛𝑖−𝑥^∗<𝑥_𝑛𝑖+1−𝑥^∗, (41) for all 𝑖∈ℕ. Then, by Lemma 11, there exists a nondecreasing sequence {𝑚_𝑘}⊂ℕ such that 𝑚_𝑘→∞, and 𝑥_𝑚𝑘−𝑥^∗≤𝑥_𝑚𝑘+1−𝑥^∗,𝑥_𝑘−𝑥^∗≤𝑥_𝑚𝑘+1−𝑥^∗, (42) for all 𝑘∈ℕ. Now, from (36) and the fact that 𝛼_𝑛→0, we get that 𝑥_𝑚𝑘−𝑆𝑥_𝑚𝑘→0 and ‖𝑄_𝐶[𝐼−𝛾𝐴]𝑥_𝑚𝑘−𝑥_𝑚𝑘‖→0, as 𝑘→∞. Thus, like in Case 1, we obtain that limsup_𝑘→∞𝑥^∗,𝑗𝑑_𝑚𝑘−𝑥^∗≥0. (43) Moreover, from (37), we have that 𝑥_𝑚𝑘+1−𝑥^∗2≤1−𝛼_𝑚𝑘𝑥_𝑚𝑘−𝑥^∗2−2𝛼_𝑚𝑘𝑥^∗,𝑗𝑑_𝑚𝑘−𝑥^∗, (44) which implies from (42) and (44) that 𝛼_𝑚𝑘𝑥_𝑚𝑘−𝑥^∗2≤𝑥_𝑚𝑘−𝑥^∗2−𝑥_𝑚𝑘+1−𝑥^∗2−2𝛼_𝑚𝑘𝑥^∗,𝑗𝑑_𝑚𝑘−𝑥^∗≤−2𝛼_𝑚𝑘𝑥^∗,𝑗𝑑_𝑚𝑘−𝑥^∗. (45) Now, since 𝛼_𝑚𝑘>0, we obtain that 𝑥_𝑚𝑘−𝑥^∗2≤−2𝑥^∗,𝑗𝑑_𝑚𝑘−𝑥^∗, (46) and using (43), we get that ‖𝑥_𝑚𝑘−𝑥^∗‖→0. This together with (44) implies that ‖𝑥_𝑚𝑘+1−𝑥^∗‖→0, as 𝑘→∞. But ‖𝑥_𝑘−𝑥^∗‖≤‖𝑥_𝑚𝑘+1−𝑥^∗‖, for all 𝑘∈ℕ; thus, we obtain that 𝑥_𝑘→𝑥^∗. Therefore, from both cases, we can conclude that {𝑥_𝑛} converges strongly to 𝑥^∗=𝑃_ℱ(0), which is a solution of the variational inequality (24), and the proof is complete.

If in Theorem 12, we consider that 𝑁=1, we get the following corollary.

Corollary 13. Let 𝐶 be a nonempty closed convex subset of a real uniformly convex and 𝑞-uniformly smooth Banach space 𝐸 possessing weakly sequentially continuous duality mapping. Let 𝑇∶𝐶→𝐶 be a 𝜆-strictly pseudocontractive mapping, and let 𝐴∶𝐶→𝐸 be an 𝜂-inverse strongly accretive mapping. Let {𝑥_𝑛} be a sequence defined by (25), where 𝑆∶=[(1−𝜇)𝐼+𝜇𝑇]. Assume that 𝐹∶=𝐹(𝑇)∩𝑉𝐼(𝐶,𝐴)≠∅. Then, {𝑥_𝑛} converges strongly to 𝑄_𝐹(0) which is a solution of the variational inequality 𝐹𝑖𝑛𝑑𝑥^∗∈𝐹(𝑇)𝑠𝑢𝑐ℎ𝑡ℎ𝑎𝑡𝐴𝑥^∗,𝐽𝑥−𝑥^∗(𝑇).≥0,∀𝑥∈𝐹 (47)

If in Theorem 12, we assume that 𝑇_𝑖, for 𝑖=1,2,…,𝑁, are nonexpansive, we get the following corollary.

Corollary 14. Let 𝐶 be a nonempty closed convex subset of a real uniformly convex and 𝑞-uniformly smooth Banach space 𝐸 possessing weakly sequentially continuous duality mapping. Let 𝑇_𝑖∶𝐶→𝐶, for 𝑖=1,2,…,𝑁, be nonexpansive mappings, and let 𝐴∶𝐶→𝐸 be an 𝜂-inverse strongly accretive mapping. For 𝑥₀∈𝐶, let the sequence {𝑥_𝑛} be generated iteratively by 𝑧_𝑛=𝑐_𝑛𝑥_𝑛+1−𝑐_𝑛𝑇𝑥_𝑛,𝑥_𝑛+1=𝑄_𝐶1−𝛼_𝑛𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝑄_𝐶𝐼−𝛾𝐴𝑧_𝑛, (48) where 𝑇∶=𝜃₀𝐼+𝜃₁𝑇₁+⋯+𝜃_𝑁𝑇_𝑁 for {𝜃_𝑖}^𝑁_𝑖=1, {𝛼_𝑛}, {𝛽_𝑛}, 𝛾 are as in (24). Assume that ℱ∶=𝐹∩𝑉𝐼(𝐶,𝐴)≠∅, where 𝐹∶=∩^𝑁_𝑖=1𝐹(𝑇_𝑖)=𝐹(𝑇). Then, {𝑥_𝑛} converges strongly to 𝑄_ℱ(0), which is a solution of the variational inequality problem (24).

Proof. Lemma 6 and the method of proof of Theorem 12 provide the required assertion.

If in Corollary 14, we consider that 𝑁=1, we get the following corollary.

Corollary 15. Let 𝐶 be a nonempty closed convex subset of a real uniformly convex and 𝑞-uniformly smooth Banach space 𝐸 possessing weakly sequentially continuous duality mapping. Let 𝑇∶𝐶→𝐶 be a nonexpansive mapping, and let 𝐴∶𝐶→𝐸 be an 𝜂-inverse strongly accretive mapping. For 𝑥₀∈𝐶, let the sequence {𝑥_𝑛} be generated iteratively by 𝑧_𝑛=𝑐_𝑛𝑥_𝑛+1−𝑐_𝑛𝑇𝑥_𝑛,𝑥_𝑛+1=𝑄_𝐶1−𝛼_𝑛𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝑄_𝐶𝐼−𝛾𝐴𝑧_𝑛. (49) Assume that 𝐹∶=𝐹(𝑇)∩𝑉𝐼(𝐶,𝐴)≠∅. Then, {𝑥_𝑛} converges strongly to 𝑄_𝐹(0), which is a solution of the variational inequality problem 𝐹𝑖𝑛𝑑𝑥^∗∈𝐹(𝑇)𝑠𝑢𝑐ℎ𝑡ℎ𝑎𝑡𝐴𝑥^∗,𝐽𝑥−𝑥^∗(𝑇).≥0,∀𝑥∈𝐹 (50)

If in Corollary 14, we assume that 𝑇=𝑇₁=𝑇₂=⋯=𝑇_𝑁=𝐼, we obtain the following corollary.

Corollary 16. Let 𝐶 be a nonempty closed convex subset of a real uniformly convex and 𝑞-uniformly smooth Banach space 𝐸 possessing weakly sequentially continuous duality mapping. Let 𝐴∶𝐶→𝐸 be an 𝜂-inverse strongly accretive mapping. For 𝑥₀∈𝐶, let the sequence {𝑥_𝑛} be generated iteratively by 𝑥_𝑛+1=𝑄_𝐶1−𝛼_𝑛𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝑄_𝐶𝐼−𝛾𝐴𝑥_𝑛. (51) Assume that 𝑉𝐼(𝐶,𝐴)≠∅. Then, {𝑥_𝑛} converges strongly to 𝑄_{𝑉𝐼(𝐶,𝐴)}(0), where 𝑄_{𝑉𝐼(𝐶,𝐴)} is a sunny nonexpansive retraction of 𝐸 onto 𝑉𝐼(𝐶,𝐴).

If in Theorem 12, we assume that 𝐴 is an 𝛼-strongly accretive and 𝐿-Lipschitzian continuous mapping, we obtain the following corollary.

Corollary 17. Let 𝐶 be a nonempty closed convex subset of a real uniformly convex and 𝑞-uniformly smooth Banach space 𝐸 possessing weakly sequentially continuous duality mapping. Let 𝑇_𝑖∶𝐶→𝐶, for 𝑖=1,…,𝑁, be 𝜆_𝑖-strictly pseudocontractive mappings, and Let 𝐴∶𝐶→𝐸 be an 𝛼-strongly accretive and 𝐿-Lipschitzian continuous mapping. Let {𝑥_𝑛} be a sequence defined by (25) for 𝜂=𝛼/𝐿². Assume that ℱ∶=𝐹∩𝑉𝐼(𝐶,𝐴)≠∅, where 𝐹∶=∩^𝑁_𝑖=1𝐹(𝑇_𝑖)=𝐹(𝑆). Then, {𝑥_𝑛} converges strongly to 𝑄_ℱ(0), which is a solution of the variational inequality problem 𝐹𝑖𝑛𝑑𝑥^∗∈∩^𝑁_𝑖=1𝐹𝑇_𝑖𝑠𝑢𝑐ℎ𝑡ℎ𝑎𝑡𝐴𝑥^∗,𝐽𝑥−𝑥^∗≥0,∀𝑥∈∩^𝑁_𝑖=1𝐹𝑇_𝑖. (52)

Proof. We note that if 𝐴 is an 𝛼-strongly accretive and 𝐿-Lipschitzian continuous mapping of 𝐶 into 𝐸, then we have that ⟨𝐴𝑥−𝐴𝑦,𝑗𝑥−𝑦⟩≥𝛼𝑥−𝑦²≥𝛼𝐿²𝐴𝑥−𝐴𝑦²,∀𝑥,𝑦∈𝐶, (53) and hence, 𝐴 is an 𝜂-inverse strongly accretive mapping with 𝜂=𝛼/𝐿². Thus, the conclusion follows from Theorem 12.

If 𝐸=𝐻, a real Hilbert space, then 𝐸 is a uniformly convex and 𝑞-uniformly smooth Banach space 𝐸 for 1<𝑞<∞ possessing weakly sequentially continuous duality mapping. In this case, we have that 𝑄_𝐶=𝑃_𝐶, projection mapping from 𝐻 onto 𝐶. Thus, we have the following corollary.

Corollary 18. Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let 𝑇_𝑖∶𝐶→𝐶, for 𝑖=1,…,𝑁, be 𝜆_𝑖-strictly pseudocontractive mappings, and let 𝐴∶𝐶→𝐸 be an 𝜂-inverse strongly accretive mapping. For 𝑥₀∈𝐶, let the sequence {𝑥_𝑛} be generated iteratively by 𝑧_𝑛=𝑐_𝑛𝑥_𝑛+1−𝑐_𝑛𝑆𝑥_𝑛,𝑥_𝑛+1=𝑃_𝐶1−𝛼_𝑛𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝑃_𝐶𝐼−𝛾𝐴𝑧_𝑛, (54) where 𝑆∶=[(1−𝜇)𝐼+𝜇𝑇], for 𝑇∶=𝜃₁𝑇₁+𝜃₂𝑇₂+⋯+𝜃_𝑛𝑇_𝑁, such that 𝜃₁+𝜃₂+⋯+𝜃_𝑁=1, 0<𝜇<min{1,2𝜆}, for 𝜆∶=min{𝜆_𝑖∶𝑖=1,2,…,𝑁}, and 0<𝛾<2𝜂. Assume that ℱ∶=𝐹∩𝑉𝐼(𝐶,𝐴)≠∅, where 𝐹=∩^𝑁_𝑖=1𝐹(𝑇_𝑖)=𝐹(𝑆). Then, {𝑥_𝑛} converges strongly to 𝑃_ℱ(0), which is a solution of the variational inequality 𝐹𝑖𝑛𝑑𝑥^∗∈∩^𝑁_𝑖=1𝐹𝑇_𝑖𝑠𝑢𝑐ℎ𝑡ℎ𝑎𝑡𝐴𝑥^∗,𝑥−𝑥^∗≥0,∀𝑥∈∩^𝑁_𝑖=1𝐹𝑇_𝑖. (55)

Remark 19. Theorem 12 complements Theorem 3.2 of Yao et al. [8] in more general Banach spaces for 𝜂-inverse strongly accretive mappings. Moreover, Theorem 12 improves Theorem 3.1 of Aoyama et al. [13] and Theorem 3.7 of Saejung et al. [23] in the sense that our convergence is strong in the set of common fixed points of finite family of strictly pseudocontractive mappings.