On Variational Inclusion and Common Fixed Point Problems in q-Uniformly Smooth Banach Spaces

Abstract

We introduce a general iterative algorithm for finding a common element of the common fixed-point set of an infinite family of 𝜆_𝑖-strict pseudocontractions and the solution set of a general system of variational inclusions for two inverse strongly accretive operators in a q-uniformly smooth Banach space. Then, we prove a strong convergence theorem for the iterative sequence generated by the proposed iterative algorithm under very mild conditions. The methods in the paper are novel and different from those in the early and recent literature. Our results can be viewed as the improvement, supplementation, development, and extension of the corresponding results in some references to a great extent.

1. Introduction

Throughout this paper, we denote by 𝐸 and 𝐸^∗ a real Banach space and the dual space of 𝐸, respectively. Let 𝐶 be a subset of 𝐸 and 𝑇 a mapping on 𝐶. We use 𝐹(𝑇) to denote the set of fixed points of 𝑇. Let 𝑞>1 be a real number. The (generalized) duality mapping 𝐽_𝑞∶𝐸→2^𝐸∗ is defined by 𝐽_𝑞(𝑥)=𝑥^∗∈𝐸^∗∶𝑥,𝑥^∗=‖𝑥‖^𝑞,𝑥^∗=‖𝑥‖^𝑞−1 (1.1) for all 𝑥∈𝐸, where ⟨⋅,⋅⟩ denotes the generalized duality pairing between 𝐸 and 𝐸^∗. In particular, 𝐽=𝐽₂ is called the normalized duality mapping and 𝐽_𝑞(𝑥)=‖𝑥‖^𝑞−2𝐽₂(𝑥) for 𝑥≠0. If 𝐸 is a Hilbert space, then 𝐽=𝐼, where 𝐼 is the identity mapping. It is well known that if 𝐸 is smooth, then 𝐽_𝑞 is single-valued, which is denoted by 𝑗_𝑞.

The norm of a Banach space 𝐸 is said to be Gâteaux differentiable if the limit lim_𝑡→0𝑥+𝑡𝑦−‖𝑥‖𝑡 (1.2) exists for all 𝑥, 𝑦 on the unit sphere 𝑆(𝐸)={𝑥∈𝐸∶‖𝑥‖=1}. If, for each 𝑦∈𝑆(𝐸), the limit (1.2) is uniformly attained for 𝑥∈𝑆(𝐸), then the norm of 𝐸 is said to be uniformly Gâteaux differentiable. The norm of 𝐸 is said to be Fréchet differentiable if, for each 𝑥∈𝑆(𝐸), the limit (1.2) is attained uniformly for 𝑦∈𝑆(𝐸).

Let 𝜌_𝐸∶[0,1)→[0,1) be the modulus of smoothness of 𝐸 defined by 𝜌_𝐸(𝑡)1=sup2𝑥+𝑦+𝑥−𝑦−1∶𝑥∈𝑆(𝐸),𝑦≤𝑡. (1.3)

A Banach space 𝐸 is said to be uniformly smooth if 𝜌_𝐸(𝑡)/𝑡→0 as 𝑡→0. Let 𝑞>1. A Banach space 𝐸 is said to be 𝑞-uniformly smooth, if there exists a fixed constant 𝑐>0 such that 𝜌_𝐸(𝑡)≤𝑐𝑡^𝑞. It is well known that 𝐸 is uniformly smooth if and only if the norm of 𝐸 is uniformly Fréchet differentiable. If 𝐸 is 𝑞-uniformly smooth, then 𝑞≤2 and 𝐸 is uniformly smooth, and hence the norm of 𝐸 is uniformly Fréchet differentiable; in particular, the norm of 𝐸 is Fréchet differentiable. Typical examples of both uniformly convex and uniformly smooth Banach spaces are 𝐿^𝑝, where 𝑝>1. More precisely, 𝐿^𝑝 is min{𝑝,2}-uniformly smooth for every 𝑝>1.

A Banach space 𝐸 is said to be uniformly convex if, for any 𝜀∈(0,2], there exists 𝛿>0 such that, for any 𝑥, 𝑦∈𝑆(𝐸), ‖𝑥−𝑦‖≥𝜀 implies ‖(𝑥+𝑦)/2‖≤1−𝛿. It is known that a uniformly convex Banach space is reflexive and strictly convex.

Recall that if 𝐶 and 𝐷 are nonempty subsets of a Banach space 𝐸 such that 𝐶 is nonempty closed convex and 𝐷⊂𝐶, then a mapping 𝑄∶𝐶→𝐷 is sunny (see [1]) provided that 𝑄(𝑥+𝑡(𝑥−𝑄(𝑥)))=𝑄(𝑥) (1.4) for all 𝑥∈𝐶 and 𝑡≥0, whenever 𝑄𝑥+𝑡(𝑥−𝑄(𝑥))∈𝐶. A mapping 𝑄∶𝐶→𝐷 is called a retraction if 𝑄𝑥=𝑥 for all 𝑥∈𝐷. Furthermore, 𝑄 is a sunny nonexpansive retraction from 𝐶 onto 𝐷 if 𝑄 is retraction from 𝐶 onto 𝐷 which is also sunny and nonexpansive. A subset 𝐷 of 𝐶 is called a sunny nonexpansive retraction of 𝐶 if there exists a sunny nonexpansive retraction from 𝐶 onto 𝐷. The following proposition concerns the sunny nonexpansive retraction.

Proposition 1.1 (see [1]). Let 𝐶 be a closed convex subset of a smooth Banach space 𝐸. Let 𝐷 be a nonempty subset of 𝐶. Let 𝑄∶𝐶→𝐷 be a retraction and let 𝐽 be the normalized duality mapping on 𝐸. Then the following are equivalent:(a)𝑄 is sunny and nonexpansive,(b)‖𝑄𝑥−𝑄𝑦‖²≤⟨𝑥−𝑦,𝐽(𝑄𝑥−𝑄𝑦)⟩, for all 𝑥, 𝑦∈𝐶,(c)⟨𝑥−𝑄𝑥,𝐽(𝑦−𝑄𝑥)⟩≤0, for all 𝑥∈𝐶, 𝑦∈𝐷.

Among nonlinear mappings, the classes of nonexpansive mappings and strict pseudocontractions are two kinds of the most important nonlinear mappings. The studies on them have a very long history (see, e.g., [1–29] and the references therein). Recall that a mapping 𝑇∶𝐶→𝐸 is said to be nonexpansive, if 𝑇𝑥−𝑇𝑦≤𝑥−𝑦forall𝑥,𝑦∈𝐶. (1.5)

A mapping 𝑇∶𝐶→𝐸 is said to be 𝜆-strict pseudocontractive in the terminology of Browder and Petryshyn (see [2–4]), if there exists a constant 𝜆>0 such that 𝑇𝑥−𝑇𝑦,𝑗_𝑞𝑥−𝑦≤𝑥−𝑦^𝑞−𝜆(𝐼−𝑇)𝑥−(𝐼−𝑇)𝑦^𝑞, (1.6) for every 𝑥, 𝑦∈𝐶 and for some 𝑗_𝑞(𝑥−𝑦)∈𝐽_𝑞(𝑥−𝑦). It is clear that (1.6) is equivalent to the following: (𝐼−𝑇)𝑥−(𝐼−𝑇)𝑦,𝑗_𝑞𝑥−𝑦≥𝜆(𝐼−𝑇)𝑥−(𝐼−𝑇)𝑦^𝑞. (1.7)

A mapping 𝑇∶𝐶→𝐸 is said to be 𝐿-Lipschitz if for all 𝑥,𝑦∈𝐶 there exists a constant 𝐿>0 such that 𝑇𝑥−𝑇𝑦≤𝐿𝑥−𝑦forall𝑥,𝑦∈𝐶. (1.8) In particular, if 0<𝐿<1, then 𝑇 is called contractive and if 𝐿=1, then 𝑇 reduces to a nonexpansive mapping.

A mapping 𝑇∶𝐶→𝐸 is said to be accretive if for all 𝑥, 𝑦∈𝐶 there exists 𝑗_𝑞(𝑥−𝑦)∈𝐽_𝑞(𝑥−𝑦) such that 𝑇𝑥−𝑇𝑦,𝑗_𝑞𝑥−𝑦≥0. (1.9)

For some 𝜂>0, 𝑇∶𝐶→𝐸 is said to be 𝜂-strongly accretive if for all 𝑥, 𝑦∈𝐶, there exists 𝑗_𝑞(𝑥−𝑦)∈𝐽_𝑞(𝑥−𝑦) such that 𝑇𝑥−𝑇𝑦,𝑗_𝑞𝑥−𝑦≥𝜂𝑥−𝑦^𝑞. (1.10)

For some 𝜇>0,𝑇∶𝐶→𝐸 is said to be 𝜇-inverse strongly accretive if for all 𝑥, 𝑦∈𝐶 there exists 𝑗_𝑞(𝑥−𝑦)∈𝐽_𝑞(𝑥−𝑦) such that ⟨𝑇𝑥−𝑇𝑦,𝑗_𝑞𝑥−𝑦⟩≥𝜇𝑇𝑥−𝑇𝑦^𝑞. (1.11)

A set-valued mapping 𝑇∶𝐷(𝑇)⊆𝐸→2^𝐸 is said to be accretive if for any 𝑥, 𝑦∈𝐷(𝑇), there exists 𝑗(𝑥−𝑦)∈𝐽(𝑥−𝑦), such that for all 𝑢∈𝑇(𝑥) and 𝑣∈𝑇(𝑦)⟨𝑢−𝑣,𝑗𝑥−𝑦⟩≥0. (1.12)

A set-valued mapping 𝑇∶𝐷(𝑇)⊆𝐸→2^𝐸 is said to be 𝑚-accretive if 𝑇 is accretive and (𝐼+𝜌𝑇)(𝐷(𝑇))=𝐸 for every (equivalently, for some) 𝜌>0, where 𝐼 is the identity mapping.

Let 𝑀∶𝐷(𝑀)→2^𝐸 be 𝑚-accretive. The mapping 𝐽_𝑀,𝜌∶𝐸→𝐷(𝑀) defined by 𝐽_𝑀,𝜌(𝑢)=𝐼+𝜌𝑀⁻¹(𝑢),∀𝑢∈𝐸 (1.13) is called the resolvent operator associated with 𝑀, where 𝜌 is any positive number and 𝐼 is the identity mapping. It is well known that 𝐽_𝑀,𝜌 is single valued and nonexpansive (see [5]).

In order to find the common element of the solutions set of a variational inclusion and the set of fixed points of a nonexpansive mapping 𝑆, Zhang et al. [6] introduced the following new iterative scheme in a Hilbert space 𝐻. Starting with an arbitrary point 𝑥₁=𝑥∈𝐻, define sequences {𝑥_𝑛} by 𝑦_𝑛=𝐽_𝑀,𝜆𝑥_𝑛−𝜆𝐴𝑥_𝑛,𝑥_𝑛+1=𝛼_𝑛𝑥+1−𝛼_𝑛𝑆𝑦_𝑛, (1.14) where 𝐴∶𝐻→𝐻 is an 𝛼-cocoercive mapping, 𝑀∶𝐻→2^𝐻 is a maximal monotone mapping, 𝑆∶𝐻→𝐻 is a nonexpansive mapping, and {𝛼_𝑛} is a sequence in [0,1]. Under mild conditions, they obtained a strong convergence theorem.

Let 𝐶 be a nonempty closed convex subset of a real reflexive, strictly convex, and 𝑞-uniformly smooth Banach space 𝐸. In this paper, we consider the general system of finding (𝑥^∗,𝑦^∗)∈𝐶×𝐶 such that 𝜃∈𝑥^∗−𝑦^∗+𝜌₁𝐴𝑦^∗+𝑀₁𝑥^∗,𝜃∈𝑦^∗−𝑥^∗+𝜌₂𝐵𝑥^∗+𝑀₂𝑦^∗, (1.15) where 𝐴,𝐵∶𝐶→𝐸, 𝑀₁∶𝐷(𝑀₁)→2^𝐸 and 𝑀₂∶𝐷(𝑀₂)→2^𝐸 are nonlinear mappings.

In the case where 𝐶=𝐸, a uniformly convex and 2-uniformly smooth Banach space, Qin et al. [8] introduced the following scheme for finding a common element of the solution set of the variational inclusions and the fixed-point set of a 𝜆-strict pseudocontraction. Starting with an arbitrary point 𝑥₁=𝑢∈𝐸, define sequences {𝑥_𝑛} by 𝑧_𝑛=𝐽_𝑀2,𝜌2𝑥_𝑛−𝜌₂𝐴₂𝑥_𝑛,𝑦_𝑛=𝐽_𝑀1,𝜌1𝑧_𝑛−𝜌₁𝐴₁𝑧_𝑛,𝑥_𝑛+1=𝛼_𝑛𝑢+𝛽_𝑛𝑥_𝑛+1−𝛼_𝑛−𝛽_𝑛𝜇𝑆𝑥_𝑛+1−𝜇𝑦_𝑛,𝑛≥1, (1.16) where 𝐴₁,𝐴₂∶𝐸→𝐸 are two inverse strongly accretive operators, 𝑀₁, 𝑀₂∶𝐸→2^𝐸 are two maximal monotone mappings, 𝑇∶𝐸→𝐸 is a 𝜆-strict pseudocontraction, and 𝑆∶𝐸→𝐸 is defined as 𝑆𝑥=(1−𝜆/𝐾²)𝑥+(𝜆/𝐾²)𝑇𝑥, for all 𝑥∈𝐸. Then they proved a strong convergence theorem under mild conditions.

In this paper, motivated by Zhang et al. [6], Qin et al. [8], Yao et al. [9], Hao [10], Yao and Yao [11], and Takahashi and Toyoda [12], we consider a relaxed extragradient-type method for finding a common element of the solution set of a general system of variational inclusions for inverse strongly accretive mappings and the common fixed-point set of an infinite family of 𝜆_𝑖-strict pseudocontractions. Furthermore, we obtain strong convergence theorems under mild conditions. The results presented by us improve and extend the corresponding results announced by many others.

2. Preliminaries

In order to prove our main results, we need the following lemmas.

Lemma 2.1 (see [16]). Let 𝐶 be a closed convex subset of a strictly convex Banach space 𝐸. Let 𝑇₁ and 𝑇₂ be two nonexpansive mappings from 𝐶 into itself with 𝐹(𝑇₁)⋂𝐹(𝑇₂)≠∅. Define a mapping 𝑆 by 𝑆𝑥=𝜆𝑇₁𝑥+(1−𝜆)𝑇₂𝑥,∀𝑥∈𝐶, (2.1) where 𝜆 is a constant in (0,1). Then 𝑆 is nonexpansive and 𝐹(𝑆)=𝐹(𝑇₁)⋂𝐹(𝑇₂).

Lemma 2.2 (see [30]). Let {𝛼_𝑛} be a sequence of nonnegative numbers satisfying the property: 𝛼_𝑛+1≤1−𝛾_𝑛𝛼_𝑛+𝑏_𝑛+𝛾_𝑛𝑐_𝑛,𝑛≥0, (2.2) where {𝛾_𝑛}, {𝑏_𝑛}, and {𝑐_𝑛} satisfy the restrictions: (i)lim_𝑛→∞𝛾_𝑛=0, ∑^∞_𝑛=1𝛾_𝑛=∞, (ii)𝑏_𝑛≥0,∑^∞_𝑛=1𝑏_𝑛<∞, (iii)limsup_𝑛→∞𝑐_𝑛≤0.
Then, lim_𝑛→∞𝛼_𝑛=0.

Lemma 2.3 (see [31, page 63]). Let 𝑞>1. Then the following inequality holds: 1𝑎𝑏≤𝑞𝑎^𝑞+𝑞−1𝑞𝑏^{𝑞/(𝑞−1)} (2.3) for arbitrary positive real numbers 𝑎, 𝑏.

Lemma 2.4 (see [17]). Let 𝐸 be a real 𝑞-uniformly smooth Banach space, then there exists a constant 𝐶_𝑞>0 such that 𝑥+𝑦^𝑞≤‖𝑥‖^𝑞+𝑞𝑦,𝐽_𝑞(𝑥)+𝐶_𝑞𝑦^𝑞,∀𝑥,𝑦∈𝐸. (2.4) In particular, if 𝐸 is a real 2-uniformly smooth Banach space, then there exists a best smooth constant 𝐾>0 such that 𝑥+𝑦²≤‖𝑥‖²+2⟨𝑦,𝐽(𝑥)⟩+2𝐾𝑦²,∀𝑥,𝑦∈𝐸. (2.5)

Lemma 2.5 (see [20]). Let 𝐶 be a nonempty convex subset of a real 𝑞-uniformly smooth Banach space 𝐸 and let 𝑇∶𝐶→𝐶 be a 𝜆-strict pseudocontraction. For 𝛼∈(0,1), one defines 𝑇_𝛼𝑥=(1−𝛼)𝑥+𝛼𝑇𝑥. Then, as 𝛼∈(0,𝜇], 𝜇=min{1,{𝑞𝜆/𝐶_𝑞}^1/(𝑞−1)}, 𝑇_𝛼∶𝐶→𝐶 is nonexpansive such that 𝐹(𝑇_𝛼)=𝐹(𝑇).

Lemma 2.6 (see [21]). Let 𝐶 be a nonempty, closed, and convex subset of a real 𝑞-uniformly smooth Banach space 𝐸 which admits weakly sequentially continuous generalized duality mapping 𝑗_𝑞 from 𝐸 into 𝐸^∗. Let 𝑇∶𝐶→𝐶 be a nonexpansive mapping. Then, for all {𝑥_𝑛}⊂𝐶, if 𝑥_𝑛⇀𝑥 and 𝑥_𝑛−𝑇𝑥_𝑛→0, then 𝑥=𝑇𝑥.

Lemma 2.7 (see [21]). Let 𝐶 be a nonempty, closed, and convex subset of a real 𝑞-uniformly smooth Banach space 𝐸. Let 𝑉∶𝐶→𝐸 be a 𝑘-Lipschitzian and 𝜂-strongly accretive operator with constants 𝑘,𝜂>0. Let 0<𝜇<(𝑞𝜂/𝐶_𝑞𝑘^𝑞)^1/(𝑞−1) and 𝜏=𝜇(𝜂−𝐶_𝑞𝜇^𝑞−1𝑘^𝑞/𝑞). Then for each 𝑡∈(0,min{1,1/𝜏}), the mapping 𝑆∶𝐶→𝐸 defined by 𝑆∶=(𝐼−𝑡𝜇𝑉) is a contraction with a constant 1−𝑡𝜏.

Lemma 2.8 (see [21]). Let 𝐶 be a nonempty, closed and convex subset of a real 𝑞-uniformly smooth Banach space 𝐸. Let 𝑄_𝐶 be a sunny nonexpansive retraction from 𝐸 onto 𝐶. Let 𝑉∶𝐶→𝐸 be a 𝑘-Lipschitzian and 𝜂-strongly accretive operator with constants 𝑘,𝜂>0, 𝑓∶𝐶→𝐸 a 𝐿-Lipschitzian mapping with constant 𝐿≥0, and 𝑇∶𝐶→𝐶 a nonexpansive mapping such that 𝐹(𝑇)≠∅. Let 0<𝜇<(𝑞𝜂/𝐶_𝑞𝑘^𝑞)^1/(𝑞−1) and 0≤𝛾𝐿<𝜏, where 𝜏=𝜇(𝜂−𝐶_𝑞𝜇^𝑞−1𝑘^𝑞/𝑞). Then {𝑥_𝑡} defined by 𝑥_𝑡=𝑄_𝐶𝑡𝛾𝑓𝑥_𝑡+𝐼−𝑡𝜇𝑉𝑇𝑥_𝑡. (2.6) Has the following properties:(i){𝑥_𝑡} is bounded for each 𝑡∈(0,min{1,1/𝜏}),(ii)lim_𝑡→0‖𝑥_𝑡−𝑇𝑥_𝑡‖=0,(iii){𝑥_𝑡} defines a continuous curve from (0,min{1,1/𝜏}) into 𝐶.

Lemma 2.9. Let 𝐶 be a closed convex subset of a smooth Banach space 𝐸. Let 𝐷 be a nonempty subset of 𝐶. Let 𝑄∶𝐶→𝐷 be a retraction and let 𝑗, 𝑗_𝑞 be the normalized duality mapping and generalized duality mapping on 𝐸, respectively. Then the following are equivalent:(a)𝑄 is sunny and nonexpansive,(b)‖𝑄𝑥−𝑄𝑦‖²≤⟨𝑥−𝑦,𝑗(𝑄𝑥−𝑄𝑦)⟩, for all 𝑥, 𝑦∈𝐶,(c)⟨𝑥−𝑄𝑥,𝑗(𝑦−𝑄𝑥)⟩≤0, for all 𝑥∈𝐶, 𝑦∈𝐷,(d)⟨𝑥−𝑄𝑥,𝑗_𝑞(𝑦−𝑄𝑥)⟩≤0, for all 𝑥∈𝐶, 𝑦∈𝐷.

Proof. From Proposition 1.1, we have a⇔b⇔c. We need only to prove c⇔d.
Indeed, if 𝑦−𝑄𝑥≠0, then ⟨𝑥−𝑄𝑥,𝑗(𝑦−𝑄𝑥)⟩≤0⇔⟨𝑥−𝑄𝑥,𝑗_𝑞(𝑦−𝑄𝑥)⟩≤0, for all 𝑥∈𝐶, 𝑦∈𝐷 (by the fact that 𝑗_𝑞(𝑥)=‖𝑥‖^𝑞−2𝑗(𝑥),∀𝑥≠0).
If 𝑦−𝑄𝑥=0, then ⟨𝑥−𝑄𝑥,𝑗(𝑦−𝑄𝑥)⟩=⟨𝑥−𝑄𝑥,𝑗_𝑞(𝑦−𝑄𝑥)⟩=0, for all 𝑥∈𝐶, 𝑦∈𝐷. This completes the proof.

Lemma 2.10. Let 𝐶 be a nonempty, closed, and convex subset of a 𝑞-uniformly smooth Banach space 𝐸 which admits a weakly sequentially continuous generalized duality mapping 𝑗_𝑞 from 𝐸 into 𝐸^∗. Let 𝑄_𝐶 be a sunny nonexpansive retraction from 𝐸 onto 𝐶. Let 𝑉∶𝐶→𝐸 be a 𝑘-Lipschitzian and 𝜂-strongly accretive operator with constants 𝑘,𝜂>0, 𝑓∶𝐶→𝐸 a 𝐿-Lipschitzian with constant 𝐿≥0, and 𝑇∶𝐶→𝐶 a nonexpansive mapping such that 𝐹(𝑇)≠∅. Let 0<𝜇<(𝑞𝜂/𝐶_𝑞𝑘^𝑞)^1/(𝑞−1) and 0≤𝛾𝐿<𝜏, where 𝜏=𝜇(𝜂−𝐶_𝑞𝜇^𝑞−1𝑘^𝑞/𝑞). For each 𝑡∈(0,min{1,1/𝜏}), let {𝑥_𝑡} be defined by (2.6), then {𝑥_𝑡} converges strongly to 𝑥^∗∈𝐹(𝑇) as 𝑡→0, which is the unique solution of the following variational inequality: 𝛾𝑓𝑥^∗−𝜇𝑉𝑥^∗,𝑗_𝑞𝑝−𝑥^∗≤0,∀𝑝∈𝐹(𝑇). (2.7)

Proof. We first show the uniqueness of a solution of the variational inequality (2.7). Suppose both 𝑥∈𝐹(𝑇) and 𝑥^∗∈𝐹(𝑇) are solutions of (2.7). It follows that 𝛾𝑓𝑥^∗−𝜇𝑉𝑥^∗,𝑗_𝑞𝑥−𝑥^∗≤0,𝛾𝑓𝑥−𝜇𝑉𝑥,𝑗_𝑞𝑥^∗−𝑥≤0. (2.8) Adding up (2.8), we have 𝛾𝑓−𝜇𝑉𝑥−𝛾𝑓−𝜇𝑉𝑥^∗,𝑗_𝑞𝑥^∗−𝑥≤0. (2.9) On the other hand, we have that 𝛾𝑓−𝜇𝑉𝑥−𝛾𝑓−𝜇𝑉𝑥^∗,𝑗_𝑞𝑥^∗−𝑥=𝜇𝑉𝑥^∗−𝑉𝑥,𝑗_𝑞𝑥^∗−𝑥−𝛾𝑓𝑥^∗−𝑓𝑥,𝑗_𝑞𝑥^∗−𝑥≥𝜇𝜂𝑥^∗−𝑥^𝑞−𝛾𝐿𝑥^∗−𝑥^𝑞≥𝜇𝜂−𝛾𝐿𝑥^∗−𝑥^𝑞≥𝜏−𝛾𝐿𝑥^∗−𝑥^𝑞>0. (2.10) It is a contradiction. Therefore, 𝑥^∗=𝑥 and the uniqueness is proved. Below we use 𝑥^∗ to denote the unique solution of (2.7).
Next, we prove that 𝑥_𝑡→𝑥^∗ as 𝑡→0.
Since 𝐸 is reflexive and {𝑥_𝑡} is bounded due to Lemma 2.8 (i), there exists a subsequence {𝑥_𝑡𝑛} of {𝑥_𝑡} and some point 𝑥∈𝐶 such that 𝑥_𝑡𝑛⇀𝑥. By Lemma 2.8(ii), we have lim_𝑡→0‖𝑥_𝑡𝑛−𝑇𝑥_𝑡𝑛‖=0. Taken together with Lemma 2.6, we can get that 𝑥∈𝐹(𝑇). Setting 𝑦_𝑡=𝑡𝛾𝑓𝑥_𝑡+(𝐼−𝑡𝜇𝑉)𝑇𝑥_𝑡, where 𝑡∈(0,min{1,1/𝜏}), then we can rewrite (2.6) as 𝑥_𝑡=𝑄_𝐶𝑦_𝑡.
We claim ‖𝑥_𝑡𝑛−𝑥‖→0.
From Lemma 2.9, we have 𝑦_𝑡−𝑄_𝐶𝑦_𝑡,𝑗_𝑞𝑥−𝑄_𝐶𝑦_𝑡≤0. (2.11) It follows from (2.11) and Lemma 2.7 that 𝑥_𝑡𝑛−𝑥^𝑞=𝑄_𝐶𝑦_𝑡𝑛−𝑦_𝑡𝑛,𝑗_𝑞𝑥_𝑡𝑛−𝑥+𝑦_𝑡𝑛−𝑥,𝑗_𝑞𝑥_𝑡𝑛−𝑥≤𝑦_𝑡𝑛−𝑥,𝑗_𝑞𝑥_𝑡𝑛−𝑥=⟨𝐼−𝑡_𝑛𝜇𝑉𝑇𝑥_𝑡𝑛−𝐼−𝑡_𝑛𝜇𝑉𝑥,𝑗_𝑞𝑥_𝑡𝑛−𝑥⟩+𝑡_𝑛𝛾𝑓𝑥_𝑡𝑛−𝜇𝑉𝑥,𝑗_𝑞𝑥_𝑡𝑛−𝑥≤1−𝑡_𝑛𝜏𝑥_𝑡𝑛−𝑥^𝑞+𝑡_𝑛𝛾𝑓𝑥_𝑡𝑛−𝜇𝑉𝑥,𝑗_𝑞𝑥_𝑡𝑛−𝑥. (2.12) It follows that 𝑥_𝑡𝑛−𝑥^𝑞≤1𝜏𝛾𝑓𝑥_𝑡𝑛−𝜇𝑉𝑥,𝑗_𝑞𝑥_𝑡𝑛−𝑥=1𝜏𝛾𝑓𝑥_𝑡𝑛−𝛾𝑓𝑥,𝑗_𝑞𝑥_𝑡𝑛−𝑥+𝛾𝑓𝑥−𝜇𝑉𝑥,𝑗_𝑞𝑥_𝑡𝑛−𝑥≤1𝜏𝛾𝐿𝑥_𝑡𝑛−𝑥^𝑞+𝛾𝑓𝑥−𝜇𝑉𝑥,𝑗_𝑞𝑥_𝑡𝑛−𝑥. (2.13) Therefore, we get 𝑥_𝑡𝑛−𝑥^𝑞≤𝛾𝑓𝑥−𝜇𝑉𝑥,𝑗_𝑞𝑥_𝑡𝑛−𝑥.𝜏−𝛾𝐿 (2.14) Using that the duality map 𝑗_𝑞 is weakly sequentially continuous from 𝐸 to 𝐸^∗ and noticing (2.14), we get that lim_𝑛→∞𝑥_𝑡𝑛−𝑥=0. (2.15)
We prove that 𝑥 solves the variational inequality (2.7). Since 𝑥_𝑡=𝑄_𝐶𝑦_𝑡=𝑄_𝐶𝑦_𝑡−𝑦_𝑡+𝑡𝛾𝑓𝑥_𝑡+𝐼−𝑡𝜇𝑉𝑇𝑥_𝑡, (2.16) we derive that 𝜇𝑉−𝛾𝑓𝑥_𝑡=1𝑡𝑄_𝐶𝑦_𝑡−𝑦_𝑡−1𝑡(𝐼−𝑇)𝑥_𝑡+𝜇𝑉𝑥_𝑡−𝑉𝑇𝑥_𝑡. (2.17) For all 𝑧∈𝐹(𝑇), note that (𝐼−𝑇)𝑥_𝑡−(𝐼−𝑇)𝑧,𝑗_𝑞𝑥_𝑡−𝑧≥𝑥_𝑡−𝑧^𝑞−𝑇𝑥_𝑡−𝑇𝑧𝑥_𝑡−𝑧^𝑞−1≥𝑥_𝑡−𝑧^𝑞−𝑥_𝑡−𝑧^𝑞=0. (2.18) It follows from Lemma 2.9 and (2.18) that 𝜇𝑉−𝛾𝑓𝑥_𝑡,𝑗_𝑞𝑥_𝑡=1−𝑧𝑡𝑄_𝐶𝑦_𝑡−𝑦_𝑡,𝑗_𝑞𝑥_𝑡−1−𝑧𝑡(𝐼−𝑇)𝑥_𝑡,𝑗_𝑞𝑥_𝑡−𝑧+⟨𝜇𝑉𝑥_𝑡−𝑉𝑇𝑥_𝑡,𝑗_𝑞𝑥_𝑡⟩=1−𝑧𝑡⟨𝑄_𝐶𝑦_𝑡−𝑦_𝑡,𝑗_𝑞𝑥_𝑡1−𝑧⟩−𝑡⟨(𝐼−𝑇)𝑥_𝑡−(𝐼−𝑇)𝑧,𝑗_𝑞𝑥_𝑡−𝑧⟩+⟨𝜇𝑉𝑥_𝑡−𝑉𝑇𝑥_𝑡,𝑗_𝑞𝑥_𝑡−𝑧⟩≤𝜇⟨𝑉𝑥_𝑡−𝑉𝑇𝑥_𝑡,𝑗_𝑞𝑥_𝑡⟩≤𝑥−𝑧_𝑡−𝑇𝑥_𝑡𝑀, (2.19) where 𝑀=sup_𝑛≥0{𝜇𝑘‖𝑥_𝑡−𝑧‖^𝑞−1}<∞.
Now replacing 𝑡 in (2.19) with 𝑡_𝑛 and letting 𝑛→∞, from (2.15) and Lemma 2.8 (ii), we obtain ⟨(𝜇𝑉−𝛾𝑓)𝑥,𝑗_𝑞(𝑥−𝑧)⟩≤0, that is, 𝑥∈𝐹(𝑇) is a solution of (2.7). Hence 𝑥=𝑥^∗ by uniqueness. Therefore, 𝑥_𝑡𝑛→𝑥^∗ as 𝑛→∞. And consequently, 𝑥_𝑡→𝑥^∗ as 𝑡→0.

Lemma 2.11. Let 𝐶 be a nonempty closed convex subset of a real 𝑞-uniformly smooth Banach space 𝐸. Let the mapping 𝐴∶𝐶→𝐸 be a 𝛼-inverse-strongly accretive operator. Then the following inequality holds: (𝐼−𝜆𝐴)𝑥−(𝐼−𝜆𝐴)𝑦^𝑞≤𝑥−𝑦^𝑞−𝜆𝑞𝛼−𝐶_𝑞𝜆^𝑞−1𝐴𝑥−𝐴𝑦^𝑞. (2.20) In particular, if 0<𝜆≤(𝑞𝛼/𝐶_𝑞)^1/(𝑞−1), then 𝐼−𝜆𝐴 is nonexpansive.

Proof. Indeed, for all 𝑥,𝑦∈𝐶, it follows from Lemma 2.4 that (𝐼−𝜆𝐴)𝑥−(𝐼−𝜆𝐴)𝑦^𝑞=(𝑥−𝑦)−𝜆(𝐴𝑥−𝐴𝑦)^𝑞≤𝑥−𝑦^𝑞−𝑞𝜆⟨𝐴𝑥−𝐴𝑦,𝑗_𝑞𝑥−𝑦⟩+𝐶_𝑞𝜆^𝑞𝐴𝑥−𝐴𝑦^𝑞≤𝑥−𝑦^𝑞−𝑞𝛼𝜆𝐴𝑥−𝐴𝑦^𝑞+𝐶_𝑞𝜆^𝑞𝐴𝑥−𝐴𝑦^𝑞≤𝑥−𝑦^𝑞−𝜆𝑞𝛼−𝐶_𝑞𝜆^𝑞−1𝐴𝑥−𝐴𝑦^𝑞. (2.21) It is clear that if 0<𝜆≤(𝑞𝛼/𝐶_𝑞)^1/(𝑞−1), then 𝐼−𝜆𝐴 is nonexpansive. This completes the proof.

Lemma 2.12. Let 𝐶 be a nonempty closed convex subset of a real 𝑞-uniformly smooth Banach space 𝐸. Suppose 𝑀₁,𝑀₂∶𝐶→2^𝐸 are two 𝑚-accretive mappings and 𝜌₁, 𝜌₂ are two arbitrary positive constants. Let 𝐴,𝐵∶𝐶→𝐸 be 𝛼-inverse strongly accretive and 𝛽-inverse strongly accretive, respectively. Let 𝐺∶𝐶→𝐶 be a mapping defined by 𝐺(𝑥)=𝐽_𝑀1,𝜌1𝐽_𝑀2,𝜌2𝑥−𝜌₂𝐵𝑥−𝜌₁𝐴𝐽_𝑀2,𝜌2𝑥−𝜌₂𝐵𝑥,∀𝑥∈𝐶. (2.22) If 0<𝜌₁≤(𝑞𝛼/𝐶_𝑞)^1/(𝑞−1) and 0<𝜌₂≤(𝑞𝛽/𝐶_𝑞)^1/(𝑞−1), then 𝐺∶𝐶→𝐶 is nonexpansive.

Proof. For all 𝑥, 𝑦∈𝐶, by Lemma 2.11, we have 𝐺𝑥−𝐺𝑦=𝐽_𝑀1,𝜌1𝐽_𝑀2,𝜌2𝑥−𝜌₂𝐵𝑥−𝜌₁𝐴𝐽_𝑀2,𝜌2𝑥−𝜌₂𝐵𝑥−𝐽_𝑀1,𝜌1𝐽_𝑀2,𝜌2𝑦−𝜌₂𝐵𝑦−𝜌₁𝐴𝐽_𝑀2,𝜌2𝑦−𝜌₂≤𝐽𝐵𝑦_𝑀2,𝜌2𝑥−𝜌₂𝐵𝑥−𝜌₁𝐴𝐽_𝑀2,𝜌2𝑥−𝜌₂−𝐽𝐵𝑥_𝑀2,𝜌2𝑦−𝜌₂𝐵𝑦−𝜌₁𝐴𝐽_𝑀2,𝜌2𝑦−𝜌₂≤𝐵𝑦𝐼−𝜌₁𝐴𝐽_𝑀2,𝜌2𝑥−𝜌₂𝐵𝑥−𝐼−𝜌₁𝐴𝐽_𝑀2,𝜌2𝑦−𝜌₂≤𝐽𝐵𝑦_𝑀2,𝜌2𝑥−𝜌₂𝐵𝑥−𝐽_𝑀2,𝜌2𝑦−𝜌₂≤𝐵𝑦𝑥−𝜌₂𝐵𝑥−𝑦−𝜌₂≤,𝐵𝑦𝑥−𝑦 (2.23) which implies that 𝐺∶𝐶→𝐶 is nonexpansive. This completes the proof.

Lemma 2.13. Let 𝐶 be a nonempty closed convex subset of a real 𝑞-uniformly smooth Banach space 𝐸. Suppose 𝐴,𝐵∶𝐶→𝐸 are two inverse strongly accretive operators, 𝑀₁, 𝑀₂∶𝐶→2^𝐸 are two 𝑚-accretive mappings, and 𝜌₁, 𝜌₂ are two arbitrary positive constants. Then (𝑥^∗,𝑦^∗)∈𝐶×𝐶 is a solution of general system (1.15) if and only if 𝑥^∗=𝐺𝑥^∗, where 𝐺 is defined by Lemma 2.12.

Proof. Note that 𝜃∈𝑥^∗−𝑦^∗+𝜌₁𝐴𝑦^∗+𝑀₁𝑥^∗𝜃∈𝑦^∗−𝑥^∗+𝜌₂𝐵𝑥^∗+𝑀₂𝑦^∗⇕𝑥^∗=𝐽_𝑀1,𝜌1𝑦^∗−𝜌₁𝐴𝑦^∗𝑦^∗=𝐽_𝑀2,𝜌2𝑥^∗−𝜌₂𝐵𝑥^∗⇕𝐺𝑥^∗=𝐽_𝑀1,𝜌1𝐽_𝑀2,𝜌2𝑥^∗−𝜌₂𝐵𝑥^∗−𝜌₁𝐴𝐽_𝑀2,𝜌2𝑥^∗−𝜌₂𝐵𝑥^∗=𝑥^∗. (2.24) This completes the proof.

Lemma 2.14 (see [18]). Let 𝐸 be a 𝑞-uniformly smooth Banach space and 𝐶 a nonempty convex subset of 𝐸. Assume for each 𝑖≥0, 𝑇_𝑖∶𝐶→𝐸 is a 𝜆_𝑖-strict pseudocontraction with 𝜆_𝑖∈(0,1). Assume inf{𝜆_𝑖∶𝑖≥1}=𝜆>0 and 𝐹=⋂^∞_𝑖=1𝐹(𝑇_𝑖)≠∅. Let {𝜉_𝑖}^∞_𝑖=1 be a positive sequence such that ∑^∞_𝑖=1𝜉_𝑖=1, then ∑^∞_𝑖=1𝜉_𝑖𝑇_𝑖∶𝐶→𝐸 is a 𝜆-strict pseudocontraction and 𝐹(∑^∞_𝑖=1𝜉_𝑖𝑇_𝑖)=𝐹.

Remark 2.15. Under the assumptions of Lemma 2.14, if for each 𝑖≥1 the mapping 𝑇_𝑖∶𝐶→𝐸 is replaced by 𝑇_𝑖∶𝐶→𝐶, respectively, where 𝐶 is a nonempty closed convex subset of 𝐸, then noticing the fact ^∞_𝑖=1𝜉_𝑖𝑇_𝑖𝑥=lim^𝑛_{𝑛→∞𝑖=1}𝜉_𝑖𝑇_𝑖𝑥=lim_𝑛→∞1∑^𝑛_𝑖=1𝜉_𝑖^𝑛_𝑖=1𝜉_𝑖𝑇_𝑖𝑥∈𝐶, (2.25) by Lemma 2.14, we deduce that ∑^∞_𝑖=1𝜉_𝑖𝑇_𝑖∶𝐶→𝐶 is a 𝜆-strict pseudocontraction with 𝜆=inf{𝜆_𝑖∶𝑖≥1} and 𝐹(∑^∞_𝑖=1𝜉_𝑖𝑇_𝑖)=𝐹.

3. Main Results

Theorem 3.1. Let 𝐶 be a nonempty closed convex subset of a strictly convex, and uniformly smooth Banach space 𝐸 which admits a weakly sequentially continuous generalized duality mapping 𝑗_𝑞∶𝐸→𝐸^∗. Let 𝑄_𝐶 be a sunny nonexpansive retraction from 𝐸 onto 𝐶. Assume the mappings 𝐴, 𝐵∶𝐶→𝐸 are 𝛼-inverse strongly accretive and 𝛽-inverse strongly accretive, respectively. Let 𝑀₁, 𝑀₂∶𝐶→2^𝐸 two 𝑚-accretive mappings and 𝜌₁, 𝜌₂ be two arbitrary positive constants. Suppose 𝑉∶𝐶→𝐸 is 𝑘-Lipschitz and 𝜂-strongly accretive with constants 𝑘,𝜂>0,𝑓∶𝐶→𝐸 being 𝐿-Lipschitz with constant 𝐿≥0. Let {𝑆_𝑖∶𝐶→𝐶}^∞_𝑖=0 be an infinite family of 𝜆_𝑖-strict pseudocontractions with {𝜆_𝑖}⊂(0,1) and inf{𝜆_𝑖∶𝑖≥0}=𝜆>0. Let 0<𝜇<(𝑞𝜂/𝐶_𝑞𝑘^𝑞)^1/(𝑞−1), 0<𝜌₁<(𝑞𝛼/𝐶_𝑞)^1/(𝑞−1), 0<𝜌₂<(𝑞𝛽/𝐶_𝑞)^1/(𝑞−1), 0≤𝛾𝐿<𝜏,  0<𝜎≤𝑑, where 𝜏=𝜇(𝜂−𝐶_𝑞𝜇^𝑞−1𝑘^𝑞/𝑞) and 𝑑=min{1,{𝑞𝜆/𝐶_𝑞}^1/(𝑞−1)}. Assume {𝜉_𝑖}⊂(0,1) and ∑^∞_𝑖=0𝜉_𝑖=1. Define a mapping 𝑇𝑥∶=(1−𝜎)𝑥+𝜎∑^∞_𝑖=0𝜉_𝑖𝑆_𝑖𝑥, for all 𝑥∈𝐶. For arbitrarily given 𝑥₀∈𝐶 and 𝛿∈(0,1), let {𝑥_𝑛} be the sequence generated iteratively by 𝑥_𝑛+1=𝑄_𝐶𝛼_𝑛𝛾𝑓𝑥_𝑛+𝛾_𝑛𝑥_𝑛+1−𝛾_𝑛𝐼−𝛼_𝑛𝜇𝑉𝛿𝑇𝑥_𝑛+(1−𝛿)𝑦_𝑛,𝑦_𝑛=1−𝛽_𝑛𝑥_𝑛+𝛽_𝑛𝑘_𝑛,𝑘_𝑛=𝐽_𝑀1,𝜌1𝑧_𝑛−𝜌₁𝐴𝑧_𝑛,𝑧_𝑛=𝐽_𝑀2,𝜌2𝑥_𝑛−𝜌₂𝐵𝑥_𝑛,𝑛≥0. (3.1) Assume that {𝛼_𝑛}, {𝛽_𝑛}, and {𝛾_𝑛} are three sequences in (0,1) satisfying the following conditions: (i)∑^∞_𝑖=0𝛼_𝑛=∞,lim_𝑛→∞𝛼_𝑛=0,∑^∞_𝑛=0|𝛼_𝑛+1−𝛼_𝑛|<∞, (ii)0<liminf_𝑛→∞𝛾_𝑛≤limsup_𝑛→∞𝛾_𝑛<1,∑^∞_𝑛=0|𝛾_𝑛+1−𝛾_𝑛|<∞, (iii)∑^∞_𝑛=0|𝛽_𝑛+1−𝛽_𝑛|<∞,lim_𝑛→∞𝛽_𝑛=𝛽>0.
Suppose in addition that 𝐹∶=⋂^∞_𝑖=0𝐹(𝑆_𝑖)⋂𝐹(𝐺)≠∅. Then {𝑥_𝑛} converges strongly to some point 𝑥^∗∈𝐹, which is the unique solution of the following variational inequality: ⟨𝛾𝑓𝑥^∗−𝜇𝑉𝑥^∗,𝑗_𝑞𝑝−𝑥^∗⟩≤0,∀𝑝∈𝐹. (3.2)

Proof. We divide the proof into several steps.
Step  1. First, we show that sequences {𝑥_𝑛} are bounded. From lim_𝑛→∞𝛼_𝑛=0 and 0<liminf_𝑛→∞𝛾_𝑛≤limsup_𝑛→∞𝛾_𝑛<1, there exist some 𝑎,𝑏∈(0,1) such that {𝛾_𝑛}⊂[𝑎,𝑏]. We may assume, without loss of generality, that {𝛼_𝑛}⊂(0,(1−𝑏)min{1,1/𝜏}). From Lemma 2.7, we deduce that 1−𝛾_𝑛𝐼−𝛼_𝑛𝜇𝑉≤1−𝛾_𝑛−𝛼_𝑛𝜏. (3.3) Taking 𝑥^∗∈𝐹, it follows from Lemma 2.13 that 𝑥^∗=𝐽_𝑀1,𝜌1𝐽_𝑀2,𝜌2𝑥^∗−𝜌₂𝐵𝑥^∗−𝜌₁𝐴𝐽_𝑀2,𝜌2𝑥^∗−𝜌₂𝐵𝑥^∗. (3.4) Putting 𝑦^∗=𝐽_𝑀2,𝜌2(𝑥^∗−𝜌₂𝐵𝑥^∗), then we can deduce that 𝑥^∗=𝐽_𝑀1,𝜌1(𝑦^∗−𝜌₁𝐴𝑦^∗). By Lemma 2.11, we obtain 𝑘_𝑛−𝑥^∗=𝐽_𝑀1,𝜌1𝑧_𝑛−𝜌₁𝐴𝑧_𝑛−𝐽_𝑀1,𝜌1𝑦^∗−𝜌₁𝐴𝑦^∗≤𝐼−𝜌₁𝐴𝑧_𝑛−𝐼−𝜌₁𝐴𝑦^∗≤𝑧_𝑛−𝑦^∗=𝐽_𝑀2,𝜌2𝑥_𝑛−𝜌₂𝐵𝑥_𝑛−𝐽_𝑀2,𝜌2𝑥^∗−𝜌₂𝐵𝑥^∗≤𝐼−𝜌₂𝐵𝑥_𝑛−𝐼−𝜌₂𝐵𝑥^∗≤𝑥_𝑛−𝑥^∗. (3.5) It follows from (3.5) that 𝑦_𝑛−𝑥^∗=1−𝛽_𝑛𝑥_𝑛+𝛽_𝑛𝑘_𝑛−𝑥^∗≤1−𝛽_𝑛𝑥_𝑛−𝑥^∗+𝛽_𝑛𝑘_𝑛−𝑥^∗≤1−𝛽_𝑛𝑥_𝑛−𝑥^∗+𝛽_𝑛𝑥_𝑛−𝑥^∗≤𝑥_𝑛−𝑥^∗. (3.6) In view of Remark 2.15, let 𝑆∶𝐶→𝐶 be the mapping defined by 𝑆𝑥=∑^∞_𝑖=0𝜉_𝑖𝑆_𝑖𝑥 for all 𝑥∈𝐶, then we can deduce that 𝑆∶𝐶→𝐶 is a 𝜆-strict pseudocontraction and 𝐹(𝑆)=⋂^∞_𝑖=0𝐹(𝑆_𝑖). By virtue of Lemma 2.5 and 0<𝜎≤𝑑, where 𝑑=min{1,{𝑞𝜆/𝐶_𝑞}^1/(𝑞−1)}, we can get that 𝑇∶𝐶→𝐶 is nonexpansive and 𝐹(𝑇)=𝐹(𝑆)=⋂^∞_𝑖=0𝐹(𝑆_𝑖). Putting 𝑙_𝑛=𝛿𝑇𝑥_𝑛+(1−𝛿)𝑦_𝑛, it follows that 𝑙_𝑛−𝑥^∗=𝛿𝑇𝑥_𝑛+(1−𝛿)𝑦_𝑛−𝑥^∗≤𝛿𝑇𝑥_𝑛−𝑥^∗+(1−𝛿)𝑦_𝑛−𝑥^∗≤𝛿𝑇𝑥_𝑛−𝑇𝑥^∗+(1−𝛿)𝑦_𝑛−𝑥^∗≤𝛿𝑥_𝑛−𝑥^∗+(1−𝛿)𝑥_𝑛−𝑥^∗=𝑥_𝑛−𝑥^∗. (3.7) It follows from (3.7) that 𝑥_𝑛+1−𝑥^∗=𝑄_𝐶𝛼_𝑛𝛾𝑓𝑥_𝑛+𝛾_𝑛𝑥_𝑛+1−𝛾_𝑛𝐼−𝛼_𝑛𝜇𝑉𝑙_𝑛−𝑥^∗≤𝛼_𝑛𝛾𝑓𝑥_𝑛+𝛾_𝑛𝑥_𝑛+1−𝛾_𝑛𝐼−𝛼_𝑛𝜇𝑉𝑙_𝑛−𝑥^∗=1−𝛾_𝑛𝐼−𝛼_𝑛𝜇𝑉𝑙_𝑛−𝑥^∗+𝛼_𝑛𝛾𝑓𝑥_𝑛−𝜇𝑉𝑥^∗+𝛾_𝑛𝑥_𝑛−𝑥^∗≤1−𝛾_𝑛−𝛼_𝑛𝜏𝑙_𝑛−𝑥^∗+𝛼_𝑛𝛾𝑓𝑥_𝑛−𝜇𝑉𝑥^∗+𝛾_𝑛𝑥_𝑛−𝑥^∗≤1−𝛾_𝑛−𝛼_𝑛𝜏𝑙_𝑛−𝑥^∗+𝛼_𝑛𝛾𝑓𝑥_𝑛−𝑓𝑥^∗+𝛼_𝑛𝛾𝑓𝑥^∗−𝜇𝑉𝑥^∗+𝛾_𝑛𝑥_𝑛−𝑥^∗≤1−𝛾_𝑛−𝛼_𝑛𝜏𝑥_𝑛−𝑥^∗+𝛼_𝑛𝛾𝐿𝑥_𝑛−𝑥^∗+𝛼_𝑛𝛾𝑓𝑥^∗−𝜇𝑉𝑥^∗+𝛾_𝑛𝑥_𝑛−𝑥^∗≤1−𝛼_𝑛𝜏−𝛾𝐿𝑥_𝑛−𝑥^∗+𝛼_𝑛𝛾𝑓𝑥^∗−𝜇𝑉𝑥^∗≤max𝑥₀−𝑥^∗,𝛾𝑓𝑥^∗−𝜇𝑉𝑥^∗.𝜏−𝛾𝐿 (3.8) Hence, {𝑥_𝑛} is bounded, so are {𝑦_𝑛}, {𝑘_𝑛}, {𝑧_𝑛}, and {𝑙_𝑛}.
Step  2. In this part, we will claim that ‖𝑥_𝑛+1−𝑥_𝑛‖→0, as 𝑛→∞.
We observe that 𝑘_𝑛+1−𝑘_𝑛=𝐽_𝑀1,𝜌1𝑧_𝑛+1−𝜌₁𝐴𝑧_𝑛+1−𝐽_𝑀1,𝜌1𝑧_𝑛−𝜌₁𝐴𝑧_𝑛≤𝐼−𝜌₁𝐴𝑧_𝑛+1−𝐼−𝜌₁𝐴𝑧_𝑛≤𝑧_𝑛+1−𝑧_𝑛=𝐽_𝑀2,𝜌2𝑥_𝑛+1−𝜌₂𝐵𝑥_𝑛+1−𝐽_𝑀2,𝜌2𝑥_𝑛−𝜌₂𝐵𝑥_𝑛≤𝐼−𝜌₂𝐵𝑥_𝑛+1−𝐼−𝜌₂𝐵𝑥_𝑛≤𝑥_𝑛+1−𝑥_𝑛. (3.9) It follows from (3.9) that 𝑦_𝑛+1−𝑦_𝑛=1−𝛽_𝑛+1𝑥_𝑛+1+𝛽_𝑛+1𝑘_𝑛+1−1−𝛽_𝑛𝑥_𝑛+𝛽_𝑛𝑘_𝑛=1−𝛽_𝑛+1𝑥_𝑛+1−𝑥_𝑛+𝛽_𝑛+1𝑘_𝑛+1−𝑘_𝑛+𝛽_𝑛+1−𝛽_𝑛𝑘_𝑛−𝑥_𝑛≤1−𝛽_𝑛+1𝑥_𝑛+1−𝑥_𝑛+𝛽_𝑛+1𝑘_𝑛+1−𝑘_𝑛+𝛽_𝑛+1−𝛽_𝑛𝑘_𝑛−𝑥_𝑛≤1−𝛽_𝑛+1𝑥_𝑛+1−𝑥_𝑛+𝛽_𝑛+1𝑥_𝑛+1−𝑥_𝑛+𝛽_𝑛+1−𝛽_𝑛𝑘_𝑛−𝑥_𝑛≤𝑥_𝑛+1−𝑥_𝑛+𝛽_𝑛+1−𝛽_𝑛𝑘_𝑛−𝑥_𝑛. (3.10) Again from (3.1), we have 𝑥_𝑛+2−𝑥_𝑛+1≤𝛼_𝑛+1𝛾𝑓𝑥_𝑛+1+𝛾_𝑛+1𝑥_𝑛+1+1−𝛾_𝑛+1𝐼−𝛼_𝑛+1𝜇𝑉𝑙_𝑛+1−𝛼_𝑛𝛾𝑓𝑥_𝑛+𝛾_𝑛𝑥_𝑛+1−𝛾_𝑛𝐼−𝛼_𝑛𝜇𝑉𝑙_𝑛≤𝛼_𝑛+1𝛾𝑓𝑥_𝑛+1−𝑓𝑥_𝑛+𝛾_𝑛+1𝑥_𝑛+1−𝑥_𝑛+1−𝛾_𝑛+1𝐼−𝛼_𝑛+1𝜇𝑉𝑙_𝑛+1−𝑙_𝑛+𝛼_𝑛+1−𝛼_𝑛𝛾𝑓𝑥_𝑛+𝛼_𝑛+1−𝛼_𝑛𝜇𝑉𝑙_𝑛+𝛾_𝑛+1−𝛾_𝑛𝑙_𝑛−𝑥_𝑛≤𝛼_𝑛+1𝛾𝐿𝑥_𝑛+1−𝑥_𝑛+𝛾_𝑛+1𝑥_𝑛+1−𝑥_𝑛+1−𝛾_𝑛+1−𝛼_𝑛+1𝜏𝑙_𝑛+1−𝑙_𝑛+𝛼_𝑛+1−𝛼_𝑛𝛾𝑓𝑥_𝑛+𝛼_𝑛+1−𝛼_𝑛𝜇𝑉𝑙_𝑛+𝛾_𝑛+1−𝛾_𝑛𝑙_𝑛−𝑥_𝑛. (3.11) It follows from (3.10) that 𝑙_𝑛+1−𝑙_𝑛=𝛿𝑇𝑥_𝑛+1+(1−𝛿)𝑦_𝑛+1−𝛿𝑇𝑥_𝑛−(1−𝛿)𝑦_𝑛≤𝛿𝑇𝑥_𝑛+1−𝑇𝑥_𝑛+(1−𝛿)𝑦_𝑛+1−𝑦_𝑛≤𝛿𝑥_𝑛+1−𝑥_𝑛+(1−𝛿)𝑦_𝑛+1−𝑦_𝑛≤𝑥_𝑛+1−𝑥_𝑛+𝛽_𝑛+1−𝛽_𝑛𝑘_𝑛−𝑥_𝑛. (3.12) Substituting (3.12) into (3.11), we have 𝑥_𝑛+2−𝑥_𝑛+1≤1−𝛾_𝑛+1−𝛼_𝑛+1𝜏𝑥_𝑛+1−𝑥_𝑛+𝛽_𝑛+1−𝛽_𝑛𝑘_𝑛−𝑥_𝑛+𝛼_𝑛+1−𝛼_𝑛𝛾𝑓𝑥_𝑛+𝛼_𝑛+1𝛾𝐿𝑥_𝑛+1−𝑥_𝑛+𝛾_𝑛+1𝑥_𝑛+1−𝑥_𝑛+𝛼_𝑛+1−𝛼_𝑛𝜇𝑉𝑙_𝑛+𝛾_𝑛+1−𝛾_𝑛𝑙_𝑛−𝑥_𝑛≤1−𝛼_𝑛+1𝜏−𝛾𝐿𝑥_𝑛+1−𝑥_𝑛+𝛼_𝑛+1−𝛼_𝑛+𝛾_𝑛+1−𝛾_𝑛+𝛽_𝑛+1−𝛽_𝑛𝑀^′, (3.13) where 𝑀^′=sup_𝑛≥0{𝜇‖𝑉𝑙_𝑛‖+𝛾‖𝑓𝑥_𝑛‖,‖𝑙_𝑛−𝑥_𝑛‖,‖𝑘_𝑛−𝑥_𝑛‖}<∞. From (i), (ii), (iii), (3.13), and Lemma 2.2, we deduce that lim_𝑛→∞𝑥_𝑛+1−𝑥_𝑛=0. (3.14) We observe that 𝑙_𝑛−𝑥_𝑛≤𝑥_𝑛+1−𝑥_𝑛+𝑥_𝑛+1−𝑙_𝑛=𝑥_𝑛+1−𝑥_𝑛+𝑄_𝐶𝛼_𝑛𝛾𝑓𝑥_𝑛+𝛾_𝑛𝑥_𝑛+1−𝛾_𝑛𝐼−𝛼_𝑛𝜇𝑉𝑙_𝑛−𝑙_𝑛≤𝑥_𝑛+1−𝑥_𝑛+𝛼_𝑛𝛾𝑓𝑥_𝑛+𝛾_𝑛𝑥_𝑛+1−𝛾_𝑛𝐼−𝛼_𝑛𝜇𝑉𝑙_𝑛−𝑙_𝑛≤𝑥_𝑛+1−𝑥_𝑛+𝛼_𝑛𝛾𝑓𝑥_𝑛−𝜇𝑉𝑙_𝑛+𝛾_𝑛𝑥_𝑛−𝑙_𝑛≤𝑥_𝑛+1−𝑥_𝑛+𝛼_𝑛𝛾𝑓𝑥_𝑛−𝜇𝑉𝑙_𝑛+𝛾_𝑛𝑥_𝑛−𝑙_𝑛, (3.15) which implies that 𝑙_𝑛−𝑥_𝑛≤11−𝛾_𝑛𝑥_𝑛+1−𝑥_𝑛+𝛼_𝑛𝛾𝑓𝑥_𝑛−𝜇𝑉𝑙_𝑛. (3.16) Noticing conditions (i) and (ii) and (3.14), we have lim_𝑛→∞𝑙_𝑛−𝑥_𝑛=0. (3.17) Let 𝑊𝑥=𝛿𝑇𝑥+(1−𝛿)1−𝛽𝑥+𝛽𝐽_𝑀1,𝜌1𝐼−𝜌₁𝐴𝐽_𝑀2,𝜌2𝐼−𝜌₂𝐵𝑥,∀𝑥∈𝐶. (3.18) In view of Lemma 2.1, we see that 𝑊∶𝐶→𝐶 is nonexpansive such that 𝐹(𝑊)=𝐹(𝑇)𝐹𝐽_𝑀1,𝜌1𝐼−𝜌₁𝐴𝐽_𝑀2,𝜌2𝐼−𝜌₂𝐵=^∞_𝑖=0𝑆_𝑖𝐹(𝐺)=𝐹. (3.19) Noticing that 𝑊𝑥_𝑛−𝑙_𝑛=𝛿𝑇𝑥_𝑛+(1−𝛿)1−𝛽𝑥_𝑛+𝛽𝐽_𝑀1,𝜌1𝐼−𝜌₁𝐴𝐽_𝑀2,𝜌2𝐼−𝜌₂𝐵𝑥_𝑛−𝛿𝑇𝑥_𝑛−(1−𝛿)1−𝛽_𝑛𝑥_𝑛+𝛽_𝑛𝐽_𝑀1,𝜌1𝐼−𝜌₁𝐴𝐽_𝑀2,𝜌2𝐼−𝜌₂𝐵𝑥_𝑛=(1−𝛿)𝛽−𝛽_𝑛𝐽_𝑀1,𝜌1𝐼−𝜌₁𝐴𝐽_𝑀2,𝜌2𝐼−𝜌₂𝐵𝑥_𝑛−𝑥_𝑛, (3.20) one has 𝑊𝑥_𝑛−𝑥_𝑛≤𝑊𝑥_𝑛−𝑙_𝑛+𝑙_𝑛−𝑥_𝑛≤(1−𝛿)𝛽−𝛽_𝑛𝐽_𝑀1,𝜌1𝐼−𝜌₁𝐴𝐽_𝑀2,𝜌2𝐼−𝜌₂𝐵𝑥_𝑛−𝑥_𝑛+𝑙_𝑛−𝑥_𝑛. (3.21) In view of (3.17), (iii) and (3.21), we deduce that 𝑊𝑥_𝑛−𝑥_𝑛→0as𝑛→∞. (3.22) We define 𝑥_𝑡=𝑄_𝐶[𝑡𝛾𝑓𝑥_𝑡+(𝐼−𝑡𝜇𝑉)𝑊𝑥_𝑡], then it follows from Lemma 2.10 that {𝑥_𝑡} converges strongly to some point 𝑥^∗∈𝐹(𝑊)=𝐹, which is the unique solution of the variational inequality (3.2).
Step  3. We show that limsup_𝑛→∞⟨𝛾𝑓𝑥^∗−𝜇𝑉𝑥^∗,𝑗_𝑞𝑥_𝑛−𝑥^∗⟩≤0, (3.23) where 𝑥^∗ is the solution of the variational inequality of (3.2). To show this, we take a subsequence {𝑥_𝑛𝑖} of {𝑥_𝑛} such that limsup_𝑛→∞𝛾𝑓𝑥^∗−𝜇𝑉𝑥^∗,𝑗_𝑞𝑥_𝑛−𝑥^∗=lim_𝑖→∞⟨𝛾𝑓𝑥^∗−𝜇𝑉𝑥^∗,𝑗_𝑞𝑥_𝑛𝑖−𝑥^∗⟩. (3.24) Without loss of generality, we may further assume that 𝑥_𝑛𝑖⇀𝑧 for some point 𝑧∈𝐶 due to reflexivity of the Banach space 𝐸 and boundness of {𝑥_𝑛}, it follows from (3.22) and Lemma 2.6 that 𝑧∈𝐹(𝑊)=𝐹. Since the Banach space 𝐸 has a weakly sequentially continuous generalized duality mapping 𝑗_𝑝∶𝐸→𝐸^∗, we obtain that limsup_𝑛→∞𝛾𝑓𝑥^∗−𝜇𝑉𝑥^∗,𝑗_𝑞𝑥_𝑛−𝑥^∗=lim_𝑖→∞𝛾𝑓𝑥^∗−𝜇𝑉𝑥^∗,𝑗_𝑞𝑥_𝑛𝑖−𝑥^∗=⟨𝛾𝑓𝑥^∗−𝜇𝑉𝑥^∗,𝑗_𝑞𝑧−𝑥^∗⟩≤0. (3.25)Step  4. We prove that lim_𝑛→∞‖𝑥_𝑛−𝑥^∗‖. Setting ℎ_𝑛=𝛼_𝑛𝛾𝑓𝑥_𝑛+𝛾_𝑛𝑥_𝑛+[(1−𝛾_𝑛)𝐼−𝛼_𝑛𝜇𝑉]𝑙_𝑛, for all 𝑛≥0. It follows from (3.1) that 𝑥_𝑛+1=𝑄_𝐶ℎ_𝑛. In view of Lemmas 2.3, 2.7, and 2.9, we have 𝑥_𝑛+1−𝑥^∗𝑞=𝑄_𝐶ℎ_𝑛−ℎ_𝑛,𝑗_𝑞𝑥_𝑛+1−𝑥^∗+ℎ_𝑛−𝑥^∗,𝑗_𝑞𝑥_𝑛+1−𝑥^∗≤ℎ_𝑛−𝑥^∗,𝑗_𝑞𝑥_𝑛+1−𝑥^∗=1−𝛾_𝑛𝐼−𝛼_𝑛𝜇𝑉𝑙_𝑛−𝑥^∗,𝑗_𝑞𝑥_𝑛+1−𝑥^∗+𝛾_𝑛𝑥_𝑛−𝑥^∗,𝑗_𝑞𝑥_𝑛+1−𝑥^∗+𝛼_𝑛𝛾𝑓𝑥_𝑛−𝜇𝑉𝑥^∗,𝑗_𝑞𝑥_𝑛+1−𝑥^∗≤1−𝛾_𝑛−𝛼_𝑛𝜏𝑙_𝑛−𝑥^∗𝑥_𝑛+1−𝑥^∗𝑞−1+𝛾_𝑛𝑥_𝑛−𝑥^∗𝑥_𝑛+1−𝑥^∗𝑞−1+𝛼_𝑛𝛾𝑓𝑥_𝑛−𝛾𝑓𝑥^∗,𝑗_𝑞𝑥_𝑛+1−𝑥^∗+𝛼_𝑛𝛾𝑓𝑥^∗−𝜇𝑉𝑥^∗,𝑗_𝑞𝑥_𝑛+1−𝑥^∗≤1−𝛾_𝑛−𝛼_𝑛𝜏𝑥_𝑛−𝑥^∗𝑥_𝑛+1−𝑥^∗𝑞−1+𝛾_𝑛𝑥_𝑛−𝑥^∗𝑥_𝑛+1−𝑥^∗𝑞−1+𝛼_𝑛𝛾𝐿𝑥_𝑛−𝑥^∗𝑥_𝑛+1−𝑥^∗𝑞−1+𝛼_𝑛⟨𝛾𝑓𝑥^∗−𝜇𝑉𝑥^∗,𝑗_𝑞𝑥_𝑛+1−𝑥^∗⟩≤1−𝛼_𝑛𝜏−𝛾𝐿𝑥_𝑛−𝑥^∗𝑥_𝑛+1−𝑥^∗𝑞−1+𝛼_𝑛⟨𝛾𝑓𝑥^∗−𝜇𝑉𝑥^∗,𝑗_𝑞𝑥_𝑛+1−𝑥^∗⟩≤1−𝛼_𝑛1𝜏−𝛾𝐿𝑞𝑥_𝑛−𝑥^∗𝑞+𝑞−1𝑞𝑥_𝑛+1−𝑥^∗𝑞+𝛼_𝑛𝛾𝑓𝑥^∗−𝜇𝑉𝑥^∗,𝑗_𝑞𝑥_𝑛+1−𝑥^∗ (3.26) which implies 𝑥_𝑛+1−𝑥^∗𝑞≤1−𝛼_𝑛𝜏−𝛾𝐿1+𝑞−1𝜏−𝛾𝐿𝛼_𝑛𝑥_𝑛−𝑥^∗𝑞+𝑞𝛼_𝑛1+𝑞−1𝜏−𝛾𝐿𝛼_𝑛×𝛾𝑓𝑥^∗−𝜇𝑉𝑥^∗,𝑗_𝑞𝑥_𝑛+1−𝑥^∗≤1−𝛼_𝑛𝜏−𝛾𝐿𝑥_𝑛−𝑥^∗𝑞+𝑞𝛼_𝑛1+𝑞−1𝜏−𝛾𝐿𝛼_𝑛𝛾𝑓𝑥^∗−𝜇𝑉𝑥^∗,𝑗_𝑞𝑥_𝑛+1−𝑥^∗. (3.27) Put 𝑎_𝑛=𝛼_𝑛(𝜏−𝛾𝐿) and 𝑐_𝑛=𝑞⟨𝛾𝑓𝑥^∗−𝜇𝑉𝑥^∗,𝑗_𝑞(𝑥_𝑛+1−𝑥^∗)⟩/[1+(𝑞−1)(𝜏−𝛾𝐿)𝛼_𝑛](𝜏−𝛾𝐿). Apply Lemma 2.2 to (3.27) to obtain 𝑥_𝑛→𝑥^∗∈𝐹 as 𝑛→∞. This completes the proof.

Remark 3.2. Compared with the known results in the literature, our results are very different from those in the following aspects. (i)The results in this paper improve and extend corresponding results in [6–13]. Especially, our result extends their results from 2-uniformly smooth Banach space or Hilbert space to more general 𝑞-uniformly smooth Banach space.(ii)Our Theorem 3.1 extends one nonexpansive mapping in [6, Theorem 2.1], one 𝜆-strict pseudocontraction in [8, Theorem 3.1], and an infinite family of nonexpansive mappings in [10, Theorem 3.1] to an infinite family of 𝜆_𝑖-strict pseudocontractions. And our Theorem 3.1 gets a common element of the common fixed-point set of an infinite family of 𝜆_𝑖-strict pseudocontractions and the solution set of the general system of variational inclusions for two inverse strongly accretive mappings in a 𝑞-uniformly smooth Banach space.(iii)We by 𝑓(𝑥_𝑛) replace the 𝑢 which is a fixed element in iterative scheme (1.16), where 𝑓 is a 𝐿-Lipschitzian. And we also add a Lipschitz and strong accretive operator 𝑉 in our scheme (3.1). In particular, whenever 𝐶=𝐸, 𝑓=𝑢, 𝑉=𝐼, {𝑇_𝑛}^∞_𝑛=0={𝑇} and 𝑞=2, our scheme (3.1) reduces to (1.16). (iv)It is worth noting that the Banach space 𝐸 does not have to be uniformly convex in our Theorem 3.1. However, it is very necessary in Theorem 3.1 of Qin et al. [8] and many other literature.

Corollary 3.3. Let 𝐶 be a nonempty closed convex subset of a strictly convex, and 2-reflexive 𝐸 which admits a weakly sequentially continuous normalized duality mapping 𝑗∶𝐸→𝐸^∗. Let 𝑄_𝐶 be a sunny nonexpansive retraction from 𝐸 onto 𝐶. Assume the mappings 𝐴, 𝐵∶𝐶→𝐸 are 𝛼-inverse strongly accretive and 𝛽-inverse strongly accretive, respectively. Let 𝑀₁, 𝑀₂∶𝐶→2^𝐸 be two 𝑚-accretive operators and 𝜌₁, 𝜌₂ two arbitrary positive constants. Suppose 𝑉∶𝐶→𝐸 is a 𝑘-Lipschitzian and 𝜂-strongly accretive operator with constants 𝑘, 𝜂>0, 𝑓∶𝐶→𝐸 being a 𝐿-Lipschitzian with constant 𝐿≥0. Let 0<𝜇<𝜂/𝐾²𝑘², 0<𝜌₁<𝛼/𝐾², 0<𝜌₂<𝛽/𝐾² and 0≤𝛾𝐿<𝜏, where 𝜏=𝜇(𝜂−𝐾²𝜇𝑘²). Let 𝑇∶𝐶→𝐶 be a nonexpansive with 𝐹=𝐹(𝑇)⋂𝐹(𝐺)≠∅. For arbitrarily given 𝛿∈(0,1) and 𝑥₀∈𝐶, let {𝑥_𝑛} be the sequence generated iteratively by 𝑥_𝑛+1=𝑄_𝐶𝛼_𝑛𝛾𝑓𝑥_𝑛+𝛾_𝑛𝑥_𝑛+1−𝛾_𝑛𝐼−𝛼_𝑛𝜇𝑉𝛿𝑇𝑥_𝑛+(1−𝛿)𝑦_𝑛,𝑦_𝑛=1−𝛽_𝑛𝑥_𝑛+𝛽_𝑛𝑘_𝑛,𝑘_𝑛=𝐽_𝑀1,𝜌1𝑧_𝑛−𝜌₁𝐴𝑧_𝑛,𝑧_𝑛=𝐽_𝑀2,𝜌2𝑥_𝑛−𝜌₂𝐵𝑥_𝑛,𝑛≥0. (3.28) Assume that {𝛼_𝑛}, {𝛽_𝑛}, and {𝛾_𝑛} are three sequences in (0,1) satisfying the following conditions: (i)∑^∞_𝑖=0𝛼_𝑛=∞, lim_𝑛→∞𝛼_𝑛=0, ∑^∞_𝑛=0|𝛼_𝑛+1−𝛼_𝑛|<∞,(ii)0<liminf_𝑛→∞𝛾_𝑛≤limsup_𝑛→∞𝛾_𝑛<1, ∑^∞_𝑛=0|𝛾_𝑛+1−𝛾_𝑛|<∞,(iii)∑^∞_𝑛=0|𝛽_𝑛+1−𝛽_𝑛|<∞, lim_𝑛→∞𝛽_𝑛=𝛽>0.
Then {𝑥_𝑛} converges strongly to 𝑥^∗∈𝐹, which is the unique solution of the following variational inequality: 𝛾𝑓𝑥^∗−𝜇𝑉𝑥^∗,𝑗𝑝−𝑥^∗≤0,∀𝑝∈𝐹. (3.29)

Acknowledgments

This research was supported by the National Science Foundation of China (11071169) and the Innovation Program of Shanghai Municipal Education Commission (09ZZ133).