Weak and Strong Convergence Theorems for Zeroes of Accretive Operators in Banach Spaces

Abstract

The purpose of this paper is to present two new forward-backward splitting schemes with relaxations and errors for finding a common element of the set of solutions to the variational inclusion problem with two accretive operators and the set of fixed points of nonexpansive mappings in infinite-dimensional Banach spaces. Under mild conditions, some weak and strong convergence theorems for approximating this common elements are proved. 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 the very recent literature.

1. Introduction

Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let 𝐴∶𝐶→𝐻 be a single-valued nonlinear mapping and let 𝐵∶𝐻→2^𝐻 be a multivalued mapping. The so-called quasivariational inclusion problem is to find a 𝑧∈𝐻 such that 0∈(𝐴+𝐵)𝑧. (1) The set of solutions of (1) is denoted by (𝐴+𝐵)⁻¹(0). A number of problems arising in structural analysis, mechanics, and economics can be studied in the framework of this kind of variational inclusions; see, for instance, [1–4]. The problem (1) includes many problems as special cases as follows.(1)If 𝐵=𝜕𝜙∶𝐻→2^𝐻, where 𝜙∶𝐻→ℝ∪{+∞} is a proper convex lower semicontinuous function and 𝜕𝜙 is the subdifferential of 𝜙, then the variational inclusion problem (1) is equivalent to find 𝑢∈𝐻 such that 𝐴𝑢,𝑦−𝑢+𝜙𝑦−𝜙(𝑢)≥0,∀𝑦∈𝐻, (2) which is called the mixed quasivariational inequality (see, Noor [5]).(2)If 𝐵=𝜕𝛿_𝐶, where 𝐶 is a nonempty closed convex subset of 𝐻 and 𝛿_𝐶∶𝐻→[0,∞] is the indicator function of 𝐶, that is, 𝛿_𝐶=0,𝑥∈𝐶,+∞,𝑥∉𝐶, (3)then the variational inclusion problem (1) is equivalent to find 𝑢∈𝐶 such that ⟨𝐴𝑢,𝑣−𝑢⟩≥0,∀𝑣∈𝐶. (4) This problem is called Hartman-Stampacchia variational inequality (see, e.g., [6]).

In [7], Zhang et al. investigated the problem of finding a common element of the set of solutions to the inclusion problem and the set of fixed points of nonexpansive mappings by considering the following iterative algorithm: 𝑦_𝑛=𝐽_𝑀,𝜆𝑥_𝑛−𝜆𝐴𝑥_𝑛,𝑥_𝑛+1=𝛼_𝑛𝑥+1−𝛼_𝑛𝑆𝑦_𝑛, (5) 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.

In [8], Manaka and Takahashi introduced the following iteration: 𝑥₁∈𝐶,𝑥_𝑛+1=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝑆𝐽_𝜆𝑛𝐼−𝜆_𝑛𝐴𝑥_𝑛,𝑛≥1, (6) where {𝛼_𝑛} is a sequence in (0, 1), {𝜆_𝑛} is a positive sequence, 𝑆∶𝐶→𝐶 is a nonexpansive mapping, 𝐴∶𝐶→𝐻 is an inversely strongly monotone mapping, 𝐵∶𝐷(𝐵)⊂𝐶→2^𝐻 is a maximal monotone operator, and 𝐽_𝜆𝑛=(𝐼+𝜆_𝑛𝐵)⁻¹ is the resolvent of 𝐵. They showed that the sequence {𝑥_𝑛} generated in (6) converges weakly to some 𝑧∈(𝐴+𝐵)⁻¹(0)∩𝐹(𝑆) provided that the control sequence satisfies some restrictions.

It is well known that, the quasivariational inclusion problem in the setting of Hilbert spaces has been extensively studied in the literature; see, for instance, [3–14]. However, there is little work in the existing literature on this problem in the setting of Banach spaces (though there were some work on finding a common zero of a finite family of accretive operators [15–17]). The main difficulties are due to the fact that the inner product structure of a Hilbert space fails to be true in a Banach space. To overcome these difficulties, López et al. [18] use the new technique to carry out certain initiative investigations on splitting methods for accretive operators in Banach spaces. They considered the following algorithms with errors in Banach spaces: 𝑥_𝑛+1=1−𝛼_𝑛𝑥_𝑛+𝛼_𝑛𝐽^𝐵_𝑟𝑛𝑥_𝑛−𝑟_𝑛𝐴𝑥_𝑛+𝑎_𝑛+𝑏_𝑛𝑥_𝑛+1,=𝛼_𝑛𝑢+1−𝛼_𝑛𝐽^𝐵_𝑟𝑛𝑥_𝑛−𝑟_𝑛𝐴𝑥_𝑛+𝑎_𝑛+𝑏_𝑛, (7)(8) where 𝑢∈𝐸,{𝑎_𝑛},{𝑏_𝑛}⊂𝐸, and 𝐽^𝐵_𝑟𝑛=(𝐼+𝑟_𝑛𝐵)⁻¹ are the resolvent of 𝐵. Then they studied the weak and strong convergence of the algorithms (7) and (8), respectively.

Motivated and inspired by Zhang et al. [7], Manaka and Takahashi [8], López et al. [18], and Cho et al. [19], the purpose of this paper is to introduce two iterative forward-backward splitting methods for finding a common element of the set of solutions of the variational inclusion problem (1) with 𝑚-accretive operators and inverse strongly accretive operators and the set of fixed points of nonexpansive mappings in the setting of Banach space. Under suitable conditions, some weak and strong convergence theorems for approximating to this common elements are proved. The results presented in the paper not only improve and extend the main result in Zhang et al. [7], but also replenish and extend the corresponding results in Manaka and Takahashi [8], López et al. [18], and Cho et al. [19].

2. Preliminaries

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 be 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 (9) 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 (10) exists for all 𝑥,𝑦 on the unit sphere 𝑆(𝐸)={𝑥∈𝐸∶‖𝑥‖=1}. If, for each 𝑦∈𝑆(𝐸), the limit (10) 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 (10) 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=sup2+(𝐸),𝑦.𝑥+𝑦𝑥−𝑦−1∶𝑥∈𝑆≤𝑡 (11)

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.

A Banach space is said to have the Kadec-Klee property if for every sequence {𝑥_𝑛} in 𝐸, 𝑥_𝑛⇀𝑥 and ‖𝑥_𝑛‖→‖𝑥‖ together imply ‖𝑥_𝑛−𝑥‖→0.

A Banach space 𝐸 is said to satisfy Opial’s condition if for any sequence {𝑥_𝑛} in 𝐸 the condition that {𝑥_𝑛} converges weakly to 𝑥∈𝐸 implies that the inequality liminf_𝑛→∞𝑥_𝑛−𝑥≤liminf_𝑛→∞𝑥_𝑛−𝑦 (12) holds for every 𝑦∈𝐸 with 𝑦≠𝑥.

Definition 1. A mapping 𝑇∶𝐶→𝐸 is said to be (1)nonexpansive if 𝑇𝑥−𝑇𝑦≤𝑥−𝑦∀𝑥,𝑦∈𝐶; (13)(2)𝑟-contractive if for all 𝑥,𝑦∈𝐶, there exists 𝑟∈(0,1) such that 𝑇𝑥−𝑇𝑦≤𝑟𝑥−𝑦; (14)(3)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; (15)(4)𝜂-strongly accretive if for all 𝑥,𝑦∈𝐶, there exists 𝜂>0 and 𝑗_𝑞(𝑥−𝑦)∈𝐽_𝑞(𝑥−𝑦) such that 𝑇𝑥−𝑇𝑦,𝑗_𝑞𝑥−𝑦≥𝜂𝑥−𝑦^𝑞; (16)(5)𝜇-inverse strongly accretive if for all 𝑥,𝑦∈𝐶, there exists 𝜇>0 and 𝑗_𝑞(𝑥−𝑦)∈𝐽_𝑞(𝑥−𝑦) such that 𝑇𝑥−𝑇𝑦,𝑗_𝑞𝑥−𝑦≥𝜇𝑇𝑥−𝑇𝑦^𝑞. (17)

Definition 2. A set-valued mapping 𝑇∶𝐷(𝑇)⊆𝐸→2^𝐸 is said to be(1)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; (18)(2)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, (19) 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.

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

Lemma 3 (see [20]). Let 𝐸 be a Banach space and 𝐽_𝑞 be a generalize duality mapping. Then for any given 𝑥,𝑦∈𝐸, the following inequality holds: 𝑥+𝑦^𝑞≤‖𝑥‖^𝑞+𝑞𝑦,𝑗_𝑞,𝑗𝑥+𝑦_𝑞𝑥+𝑦∈𝐽_𝑞𝑥+𝑦. (20) In particular, we have, for any given 𝑥,𝑦∈𝐸, 𝑥+𝑦²≤‖𝑥‖²+2𝑦,𝑗𝑥+𝑦,𝑗𝑥+𝑦∈𝐽𝑥+𝑦. (21)

Lemma 4 (see [21]). 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∈ℕ, (22) where {𝛾_𝑛}, {𝑏_𝑛}, and {𝑐_𝑛} satisfy the restrictions:(i)∑^∞_𝑛=1𝛾_𝑛=∞,(ii)𝑏_𝑛≥0, ∑^∞_𝑛=1𝑏_𝑛<∞,(iii)limsup_𝑛→∞𝑐_𝑛≤0.Then, lim_𝑛→∞𝛼_𝑛=0.

Lemma 5 (see [22]). Let {𝑎_𝑛}, {𝑏_𝑛}, and {𝛿_𝑛} be sequences of nonnegative real numbers satisfying the inequality 𝑎_𝑛+1≤1+𝛿_𝑛𝑎_𝑛+𝑏_𝑛,∀𝑛=1,2,…. (23) If ∑^∞_𝑛=0𝛿_𝑛<∞ and ∑^∞_𝑛=0𝑏_𝑛<∞, then lim_𝑛→∞𝑎_𝑛 exists. In particular, lim_𝑛→∞𝑎_𝑛=0 whenever there exists a subsequence {𝑎_𝑛𝑘} in {𝑎_𝑛} which strongly converges to zero.

Lemma 6 (see [20]). Let 1<𝑝<∞, 𝑞∈(1,2], 𝑟>0 be given.(i)If 𝐸 is uniformly convex, then there exists a continuous, strictly increasing, and convex function 𝜑∶[0,∞)→[0,∞) with 𝜑(0)=0 such that 𝜆𝑥+(1−𝜆)𝑦^𝑝≤𝜆‖𝑥‖^𝑝+𝜆𝑦^𝑝−𝑊_𝑝(𝜆)𝜑𝑥−𝑦,𝑥,𝑦∈𝐵_𝑟,0≤𝜆≤1, (24) where 𝑊_𝑝(𝜆)=𝜆^𝑝(1−𝜆)+(1−𝜆)^𝑝𝜆, 𝐵_𝑟={𝑧∈𝐸∶‖𝑧‖≤𝑟}.(ii)If 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. (25)

Lemma 7 (see [23]). Let 𝐸 be a uniformly convex Banach space, 𝐶 a closed convex subset of 𝐸, and 𝑇∶𝐶→𝐸 a nonexpansive mapping. Then, 𝐼−𝑇 is demiclosed at zero.

Lemma 8 (see [24]). If 𝐸 is a uniformly convex Banach space and 𝐶 is a closed convex bounded subset of 𝐸, there is a continuous strictly increasing function 𝑔∶[0,∞)→[0,∞) with 𝑔(0)=0 such that 𝑔𝑆()𝑦−()≤−𝑡𝑥+1−𝑡𝛼𝑆𝑥+1−𝛼𝑆𝑦𝑥−𝑦𝑆𝑥−𝑆𝑦 (26) for all 𝑥,𝑦∈𝐶,𝑡∈[0,1] and nonexpansive mappings 𝑆∶𝐶→𝐸.

Lemma 9 (see [25]). Let 𝐸 be a real reflexive Banach space such that its dual 𝐸^∗ has the Kadec-Klee property. Let {𝑥_𝑛} be a bounded sequence in 𝐸 and 𝑥^∗,𝑦^∗∈𝜔_𝑤(𝑥_𝑛); here 𝜔_𝑤(𝑥_𝑛) denotes the weak 𝑤-limit set of {𝑥_𝑛}. Suppose lim_𝑛→∞‖𝑡𝑥_𝑛+(1−𝑡)𝑥^∗−𝑦^∗‖ exists for all 𝑡∈[0,1]. Then 𝑥^∗=𝑦^∗.

Lemma 10. 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. (27) 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 6 that (𝐼−𝜆𝐴)𝑥−(𝐼−𝜆𝐴)𝑦^𝑞=𝑥−𝑦−𝜆𝐴𝑥−𝐴𝑦^𝑞≤𝑥−𝑦^𝑞−𝑞𝜆𝐴𝑥−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. (28) It is clear that if 0<𝜆≤(𝑞𝛼/𝐶_𝑞)^1/(𝑞−1), then 𝐼−𝜆𝐴 is nonexpansive. This completes the proof.

Lemma 11. Assume that 𝐶 is a nonempty closed subset of a real uniformly convex and 𝑞-uniformly smooth Banach space 𝐸. Suppose that 𝐴 is a single-valued and 𝛼-inverse strongly accretive operator for some 𝛼>0 and 𝐵 is an 𝑚-accretive operator in 𝐸, with 𝐶⊂𝐷(𝐴) and 𝐶⊂𝐷(𝐵). Moreover, denote 𝐽_𝑟 by 𝐽_𝑟≡𝐽^𝐵_𝑟=(𝐼+𝑟𝐵)⁻¹ (29) and 𝑇_𝑟 by 𝑇_𝑟=𝐽_𝑟(𝐼−𝑟𝐴)=(𝐼+𝑟𝐵)⁻¹(𝐼−𝑟𝐴). (30) Then, it is holds for all 𝑟>0 that 𝐹(𝑇_𝑟)=(𝐴+𝐵)⁻¹(0).

Proof. From the definition of 𝑇_𝑟, it follows that 𝑥=𝑇_𝑟𝑥⟺𝑥=(𝐼+𝑟𝐵)⁻¹()𝑥⟺()()𝑥()𝐼−𝑟𝐴𝐼−𝑟𝐴𝑥∈𝐼+𝑟𝐵⟺0∈𝐴+𝐵𝑥. (31)
This lemma alludes to the fact that in order to solve the inclusion problem (1), it suffices to find a fixed point of 𝑇_𝑟. Since 𝑇_𝑟 is already split, an iterative algorithm for 𝑇_𝑟 corresponds to a splitting algorithm for (1). However, to guarantee convergence (weak or strong) of an iterative algorithm for 𝑇_𝑟, we need good metric properties of 𝑇_𝑟 such as nonexpansivity. To this end, some relate geometric conditions on the underlying space 𝐸 are very necessary (see Lemmas 12 and 13 below).

Lemma 12 (see [18]). Assume that 𝐶 is a nonempty closed subset of a real uniformly convex and 𝑞-uniformly smooth Banach space 𝐸. Suppose that 𝐴 is a single-valued and 𝛼-inverse strongly accretive operator for some 𝛼>0 and 𝐵 is an 𝑚-accretive operator in 𝐸, with 𝐶⊂𝐷(𝐴) and 𝐶⊂𝐷(𝐵). Then, the following relations hold.(i)Given 0<𝑠≤𝑟 and 𝑥∈𝐸, 𝑇_𝑠𝑥−𝑇_𝑟𝑥≤𝑠1−𝑟𝑥−𝑇_𝑟𝑥,𝑥−𝑇_𝑠𝑥≤2𝑥−𝑇_𝑟𝑥. (32)(ii)Given 𝑠>0, there exists a continuous, strictly increasing, and convex function 𝜙_𝑞∶[0,∞)→[0,∞) with 𝜙_𝑞(0)=0 such that for all 𝑥,𝑦∈𝐵_𝑠, 𝑇_𝑟𝑥−𝑇_𝑟𝑦^𝑞≤𝑥−𝑦^𝑞−𝑟𝛼𝑞−𝑟^𝑞−1𝐶_𝑞𝐴𝑥−𝐴𝑦^𝑞−𝜙_𝑞𝐼−𝐽_𝑟(𝐼−𝑟𝐴)𝑥−𝐼−𝐽_𝑟×(𝐼−𝑟𝐴)𝑦. (33)

Lemma 13. Assume that 𝐶 is a nonempty closed subset of a real uniformly convex and 𝑞-uniformly smooth Banach space 𝐸. Suppose that 𝑆 is a nonexpansive mapping and 𝐴 is a single-valued and 𝛼-inverse strongly accretive operator for some 𝛼>0 and 𝐵 is a 𝑚-accretive operator in 𝐸, with 𝐶⊂𝐷(𝐴) and 𝐶⊂𝐷(𝐵). Assume 0<𝑟<(𝑞𝛼/𝐶_𝑞)^1/(𝑞−1). Then 𝐹(𝑆𝑇_𝑟)=𝐹(𝑆)∩𝐹(𝑇_𝑟).

Proof. Suppose that 𝑥₁∈𝐹(𝑆𝑇_𝑟), it is sufficient to show that 𝑥₁∈𝐹(𝑆)∩𝐹(𝑇_𝑟). Indeed, for 𝑥₂∈𝐹(𝑆)∩𝐹(𝑇_𝑟), we have by Lemma 12 that 𝑥₁−𝑥₂^𝑞=𝑆𝑇_𝑟𝑥₁−𝑆𝑇_𝑟𝑥₂^𝑞≤𝑇_𝑟𝑥₁−𝑇_𝑟𝑥₂^𝑞≤𝑥₁−𝑥₂^𝑞−𝑟𝛼𝑞−𝑟^𝑞−1𝐶_𝑞×𝐴𝑥₁−𝐴𝑥₂^𝑞−𝜙_𝑞𝐼−𝐽_𝑟(𝐼−𝑟𝐴)𝑥₁−𝐼−𝐽_𝑟(𝐼−𝑟𝐴)𝑥₂. (34) The property of 𝜙 and the condition 0<𝑟<(𝑞𝛼/𝐶_𝑞)^1/(𝑞−1) together imply that 𝐴𝑥₁−𝐴𝑥₂=𝐼−𝐽_𝑟(𝐼−𝑟𝐴)𝑥₁−𝐼−𝐽_𝑟(𝐼−𝑟𝐴)𝑥₂=0. (35) It turns out that 𝑥₁−𝑇_𝑟𝑥₁−𝑥₂+𝑇_𝑟𝑥₂=0, (36) which imply 𝑇_𝑟𝑥₁=𝑥₁. (37) Noticing the assumption of 𝑥₁=𝑆𝑇_𝑟𝑥₁, we can deduce 𝑥₁=𝑆𝑥₁. This means that 𝑥₁∈𝐹(𝑆)∩𝐹(𝑇_𝑟).
Next we give a weak convergence theorem in a Banach space 𝐸.

3. Main Results

Theorem 14. Let 𝐸 be a uniformly convex and 𝑞-uniformly smooth Banach space. Let 𝐴∶𝐸→𝐸 be 𝛼-inverse strongly accretive, let 𝐵∶𝐸→2^𝐸 be 𝑚-accretive, and let 𝑆∶𝐸→𝐸 be nonexpansive. Assume that 𝐹(𝑆)∩(𝐴+𝐵)⁻¹(0)≠∅. We define a sequence {𝑥_𝑛} by the perturbed iterative scheme: 𝑥_𝑛+1=1−𝛼_𝑛𝑥_𝑛+𝛼_𝑛𝑆𝐽_𝑟𝑛𝑥_𝑛−𝑟_𝑛𝐴𝑥_𝑛+𝑎_𝑛+𝑏_𝑛,∀𝑛≥1, (38) where 𝐽_𝑟𝑛=(𝐼+𝑟_𝑛𝐵)⁻¹, {𝑎_𝑛},{𝑏_𝑛}⊂𝐸, {𝛼_𝑛}⊂(0,1], and {𝑟_𝑛}⊂(0,+∞). Assume that(i)∑^∞_𝑛=1‖𝑎_𝑛‖<∞ and ∑^∞_𝑛=1‖𝑏_𝑛‖<∞;(ii)0<liminf_𝑛→∞𝛼_𝑛≤limsup_𝑛→∞𝛼_𝑛<1; (iii)0<liminf_𝑛→∞𝑟_𝑛≤limsup_𝑛→∞𝑟_𝑛<(𝑞𝛼/𝐶_𝑞)^1/(𝑞−1).Then {𝑥_𝑛} converges weakly to some point 𝑧∈𝐹(𝑆)∩(𝐴+𝐵)⁻¹(0).

Proof. We divide the proof into several steps.
First we prove that lim_𝑛→∞‖𝑥_𝑛−𝑧‖ exist for any point 𝑧∈𝐹(𝑆)∩(𝐴+𝐵)⁻¹(0).
Putting 𝑇_𝑛=𝐽_𝑟𝑛(𝐼−𝑟_𝑛𝐴)=(𝐼+𝑟_𝑛𝐵)⁻¹(𝐼−𝑟_𝑛𝐴), one has 𝑆𝐽_𝑟𝑛𝑥_𝑛−𝑟_𝑛𝐴𝑥_𝑛+𝑎_𝑛+𝑏_𝑛=𝑆𝑇_𝑛𝑥_𝑛+𝑔_𝑛, (39) where 𝑔_𝑛=𝑆𝐽_𝑟𝑛𝑥_𝑛−𝑟_𝑛𝐴𝑥_𝑛+𝑎_𝑛+𝑏_𝑛−𝑆𝑇_𝑛𝑥_𝑛. (40) Then the iterative formula (38) turns into the form 𝑥_𝑛+1=1−𝛼_𝑛𝑥_𝑛+𝛼_𝑛𝑆𝑇_𝑛𝑥_𝑛+𝑔_𝑛. (41) Thus, by virtue of nonexpansivity of 𝑆 and 𝐽_𝑟𝑛, it follows that 𝑔_𝑛=𝑆𝐽_𝑟𝑛𝑥_𝑛−𝑟_𝑛𝐴𝑥_𝑛+𝑎_𝑛+𝑏_𝑛−𝑆𝑇_𝑛𝑥_𝑛≤𝐽_𝑟𝑛𝑥_𝑛−𝑟_𝑛𝐴𝑥_𝑛+𝑎_𝑛−𝑇_𝑛𝑥_𝑛+𝑏_𝑛≤𝑟_𝑛𝑎_𝑛+𝑏_𝑛. (42) It follows from (42) and the condition (i) that ^∞_𝑛=1𝑔_𝑛<∞. (43) Since 𝑧∈𝐹(𝑆)∩(𝐴+𝐵)⁻¹(0), we can deduce 𝑧∈𝐹(𝑆)∩𝐹(𝑇_𝑛) due to Lemma 13. And 𝑇_𝑛 is nonexpansive due to Lemma 12 and condition (iii). Therefore, we can get from (41) that 𝑥_𝑛+1−𝑧=1−𝛼_𝑛𝑥_𝑛−𝑧+𝛼_𝑛𝑆𝑇_𝑛𝑥_𝑛+𝑔_𝑛≤−𝑧1−𝛼_𝑛𝑥_𝑛−𝑧+𝛼_𝑛𝑆𝑇_𝑛𝑥_𝑛−𝑧+𝛼_𝑛𝑔_𝑛≤1−𝛼_𝑛𝑥_𝑛−𝑧+𝛼_𝑛𝑥_𝑛−𝑧+𝛼_𝑛𝑔_𝑛≤𝑥_𝑛−𝑧+𝛼_𝑛𝑔_𝑛. (44) In view of (43), (44), and Lemma 5, we can get that lim_𝑛→∞‖𝑥_𝑛−𝑧‖ exists. Therefor {𝑥_𝑛} is bounded.
Next, we show lim_𝑛→∞‖𝑇_𝑛𝑥_𝑛−𝑥_𝑛‖=0.
Let 𝑀₁>0 be such that ‖𝑥_𝑛‖<𝑀₁, for all 𝑛∈ℕ, and let 𝑠=𝑞(𝑀₁+‖𝑧‖)^𝑞−1. By (41), Lemmas 3, and 12, we have 𝑥_𝑛+1−𝑧^𝑞=(1−𝛼_𝑛)(𝑥_𝑛−𝑧)+𝛼_𝑛(𝑆𝑇_𝑛𝑥_𝑛+𝑔_𝑛−𝑧)^𝑞≤(1−𝛼_𝑛)(𝑥_𝑛−𝑧)+𝛼_𝑛(𝑆𝑇_𝑛𝑥_𝑛−𝑧)^𝑞+𝛼_𝑛𝑞⟨𝑔_𝑛,𝑗_𝑞𝑥_𝑛+1⟩≤−𝑧1−𝛼_𝑛𝑥_𝑛−𝑧^𝑞+𝛼_𝑛𝑆𝑇_𝑛𝑥_𝑛−𝑧^𝑞+𝛼_𝑛𝑞𝑔_𝑛𝑥_𝑛+1−𝑧^𝑞−1≤1−𝛼_𝑛𝑥_𝑛−𝑧^𝑞+𝛼_𝑛𝑇_𝑛𝑥_𝑛−𝑧^𝑞+𝛼_𝑛𝑞𝑔_𝑛𝑥_𝑛+1−𝑧^𝑞−1≤𝑥_𝑛−𝑧^𝑞−𝛼_𝑛𝑟_𝑛𝛼𝑞−𝑟_𝑛^𝑞−1𝐶_𝑞×𝐴𝑥_𝑛−𝐴𝑧^𝑞−𝛼_𝑛𝜙_𝑞×𝐼−𝐽_𝑟𝑛𝐼−𝑟_𝑛𝐴𝑥_𝑛−𝐼−𝐽_𝑟𝑛𝐼−𝑟_𝑛𝐴𝑧+𝛼_𝑛𝑠𝑔_𝑛≤𝑥_𝑛−𝑧^𝑞−𝛼_𝑛𝑟_𝑛𝛼𝑞−𝑟_𝑛^𝑞−1𝐶_𝑞×𝐴𝑥_𝑛−𝐴𝑧^𝑞−𝛼_𝑛𝜙_𝑞×𝑥_𝑛−𝑟_𝑛𝐴𝑥_𝑛−𝑇_𝑛𝑥_𝑛+𝑟_𝑛𝐴𝑧+𝛼_𝑛𝑠𝑔_𝑛. (45) Meanwhile, by the fact that 𝑎^𝑟−𝑏^𝑟≤𝑟𝑎^𝑟−1(𝑎−𝑏), ∀𝑟≥1, and (45), we can get that 𝛼_𝑛𝑟_𝑛𝛼𝑞−𝑟_𝑛^𝑞−1𝐶_𝑞𝐴𝑥_𝑛−𝐴𝑧^𝑞+𝛼_𝑛𝜙_𝑞𝑥_𝑛−𝑟_𝑛𝐴𝑥_𝑛−𝑇_𝑛𝑥_𝑛+𝑟_𝑛≤𝑥𝐴𝑧_𝑛−𝑧^𝑞−𝑥_𝑛+1−𝑧^𝑞+𝛼_𝑛𝑠𝑔_𝑛≤𝑞𝑥_𝑛−𝑧^𝑞−1𝑥_𝑛−𝑧−𝑥_𝑛+1−𝑧+𝛼_𝑛𝑠𝑔_𝑛. (46) Thanks to the conditions of (ii),(iii), and (43) and existence of lim_𝑛→∞‖𝑥_𝑛−𝑧‖, one has lim_𝑛→∞𝐴𝑥_𝑛−𝐴𝑧=lim_𝑛→∞𝑥_𝑛−𝑟_𝑛𝐴𝑥_𝑛−𝑇_𝑛𝑥_𝑛+𝑟_𝑛𝐴𝑧=0. (47) Consequently, lim_𝑛→∞𝑇_𝑛𝑥_𝑛−𝑥_𝑛=0. (48)
Step  1. We prove lim_𝑛→∞‖𝑆𝑇_𝑛𝑥_𝑛−𝑇_𝑛𝑥_𝑛‖=0.
Noticing (43) and Lemma 6, we have 𝑥_𝑛+1−𝑧²=(1−𝛼_𝑛)(𝑥_𝑛−𝑧)+𝛼_𝑛(𝑆𝑇_𝑛𝑥_𝑛+𝑔_𝑛−𝑧)²≤1−𝛼_𝑛𝑥_𝑛−𝑧²+𝛼_𝑛𝑆𝑇_𝑛𝑥_𝑛+𝑔_𝑛−𝑧²−𝑊₂𝛼_𝑛𝜑𝑆𝑇_𝑛𝑥_𝑛+𝑔_𝑛−𝑥_𝑛≤1−𝛼_𝑛𝑥_𝑛−𝑧²+𝛼_𝑛𝑆𝑇_𝑛𝑥_𝑛−𝑧²+2𝑆𝑇_𝑛𝑥_𝑛−𝑧𝑔_𝑛+𝑔_𝑛²−𝑊₂𝛼_𝑛𝜑𝑆𝑇_𝑛𝑥_𝑛+𝑔_𝑛−𝑥_𝑛≤1−𝛼_𝑛𝑥_𝑛−𝑧²+𝛼_𝑛𝑥_𝑛−𝑧²+2𝑥_𝑛−𝑧𝑔_𝑛+𝑔_𝑛²−𝑊₂𝛼_𝑛𝜑𝑆𝑇_𝑛𝑥_𝑛+𝑔_𝑛−𝑥_𝑛≤𝑥_𝑛−𝑧²+2𝛼_𝑛𝑥_𝑛−𝑧𝑔_𝑛+𝛼_𝑛𝑔_𝑛²−𝑊₂𝛼_𝑛𝜑𝑆𝑇_𝑛𝑥_𝑛+𝑔_𝑛−𝑥_𝑛≤𝑥_𝑛−𝑧²+2𝑥_𝑛−𝑧𝑔_𝑛+𝑔_𝑛²−𝑊₂𝛼_𝑛𝜑𝑆𝑇_𝑛𝑥_𝑛+𝑔_𝑛−𝑥_𝑛, (49) which implies 𝑊₂𝛼_𝑛𝜑𝑆𝑇_𝑛𝑥_𝑛+𝑔_𝑛−𝑥_𝑛≤𝑥_𝑛−𝑧²−𝑥_𝑛+1−𝑧²+2𝑥_𝑛−𝑧𝑔_𝑛+𝑔_𝑛², (50) where 𝑊₂(𝛼_𝑛)=𝛼_𝑛(1−𝛼_𝑛). From (50), assumptions (ii), (43), and existence of lim_𝑛→∞‖𝑥_𝑛−𝑧‖, it turns out that lim_𝑛→∞𝜑𝑆𝑇_𝑛𝑥_𝑛+𝑔_𝑛−𝑥_𝑛=0. (51) It follows from the property of 𝜑 that lim_𝑛→∞𝑆𝑇_𝑛𝑥_𝑛+𝑔_𝑛−𝑥_𝑛=0. (52) Noticing (43), we have lim_𝑛→∞𝑆𝑇_𝑛𝑥_𝑛−𝑥_𝑛=0. (53) By (48) and (53), we get lim_𝑛→∞𝑆𝑇_𝑛𝑥_𝑛−𝑇_𝑛𝑥_𝑛=0. (54)
Step  2. We prove 𝜔_𝑤(𝑥_𝑛)⊂𝐹(𝑆)∩(𝐴+𝐵)⁻¹(0).
Since liminf_𝑛→∞𝑟_𝑛>0, there exists 𝜀>0 such that 𝑟_𝑛≥𝜀 for all 𝑛≥1. Then, by Lemma 12, we have lim_𝑛→∞𝑇_𝜀𝑥_𝑛−𝑥_𝑛≤2lim_𝑛→∞𝑇_𝑛𝑥_𝑛−𝑥_𝑛=0. (55) It follows from (48), (53), and (54) that 𝑆𝑇_𝜀𝑥_𝑛−𝑥_𝑛≤𝑆𝑇_𝜀𝑥_𝑛−𝑆𝑇_𝑛𝑥_𝑛+𝑆𝑇_𝑛𝑥_𝑛−𝑥_𝑛≤𝑇_𝜀𝑥_𝑛−𝑇_𝑛𝑥_𝑛+𝑆𝑇_𝑛𝑥_𝑛−𝑥_𝑛≤𝑇_𝜀𝑥_𝑛−𝑥_𝑛+𝑥_𝑛−𝑇_𝑛𝑥_𝑛+𝑆𝑇_𝑛𝑥_𝑛−𝑥_𝑛⟶0. (56) By Lemmas 7 and 13, we get 𝜔_𝑤𝑥_𝑛⊂𝐹𝑆𝑇_𝜀=𝐹(𝑆)(𝐴+𝐵)⁻¹(0). (57) Finally, we show that {𝑥_𝑛} converges weakly to a fixed point of 𝑧∈𝐹(𝑆)∩(𝐴+𝐵)⁻¹(0).
Indeed, it suffices to show that 𝜔_𝑤(𝑥_𝑛) consists of exactly only one point. To this end, we suppose that two different points 𝑧 and 𝑧 are in 𝜔_𝑤(𝑥_𝑛). Then there exist two different subsequences {𝑛_𝑖} and {𝑛_𝑗} such that 𝑥_𝑛𝑖⇀𝑧 and 𝑥_𝑛𝑗⇀𝑧 as 𝑖→∞ and 𝑗→∞. Define 𝑆_𝑛,𝑚∶𝐷→𝐸 by 𝑆_𝑛,𝑚=𝑉_{𝑛+𝑚−1}𝑉_{𝑛+𝑚−2}⋯𝑉_𝑛,𝑉_𝑛=1−𝛼_𝑛𝐼+𝛼_𝑛𝑆𝑇_𝑛. (58) Then 𝑥_𝑛 can be written 𝑥_𝑛+𝑚=𝑆_𝑛,𝑚𝑥_𝑛+𝑐_𝑛,𝑚, (59) where 𝑐_𝑛,𝑚=𝑉_{𝑛+𝑚−1}𝑉_{𝑛+𝑚−2}×⋯𝑉_𝑛+1𝑉_𝑛𝑥_𝑛+𝛼_𝑛𝑔_𝑛+𝛼_𝑛+1𝑔_𝑛+1⋯+𝛼_{𝑛+𝑚−2}𝑔_{𝑛+𝑚−2}+𝛼_{𝑛+𝑚−1}𝑔_{𝑛+𝑚−1}−𝑆_𝑛,𝑚𝑥_𝑛. (60) Thanks to the nonexpansivity of 𝑉_𝑛, we have 𝑐_𝑛,𝑚≤^{𝑛+𝑚−1}_𝑘=𝑛𝛼_𝑘𝑔_𝑘≤^{𝑛+𝑚−1}_𝑘=𝑛𝑔_𝑘. (61) It follows from (43) that lim_{𝑚,𝑛→∞}𝑐_𝑛,𝑚⟶0. (62) Let 𝑓_𝑛(𝑡)=𝑡𝑥_𝑛+()𝑧,𝑑1−𝑡𝑧−_𝑛,𝑚=𝑆_𝑛,𝑚𝑡𝑥_𝑛+(1−𝑡)𝑧−𝑡𝑆_𝑛,𝑚𝑥_𝑛+(1−𝑡)𝑧. (63) Apply Lemma 8 to the closed convex bounded subset 𝐾∶=𝑐𝑜({𝑥_𝑛}∪{𝑧}) to obtain 𝑔𝑑_𝑛,𝑚≤𝑥_𝑛−𝑧−𝑆_𝑛,𝑚𝑥_𝑛−𝑆_𝑛,𝑚𝑧≤𝑥_𝑛−𝑧−𝑥_𝑛+𝑚−𝑧−𝑐_𝑛,𝑚≤𝑥_𝑛−𝑧−𝑥_𝑛+𝑚−𝑧+𝑐_𝑛,𝑚. (64) Since lim_𝑛→∞‖𝑥_𝑛−𝑧‖ exists, (62) and (64) together imply that lim_{𝑚,𝑛→∞}𝑑_𝑛,𝑚⟶0. (65) Furthermore, we have 𝑓_𝑛+𝑚(𝑡)=𝑡𝑥_𝑛+𝑚+()𝑧≤𝑑1−𝑡𝑧−_𝑛,𝑚+𝑆_𝑛,𝑚𝑡𝑥_𝑛+()𝑧−𝑧𝑐1−𝑡+𝑡_𝑚,𝑛≤𝑑_𝑛,𝑚+𝑡𝑥_𝑛+(1−𝑡)𝑧−𝑧+𝑡𝑐_𝑚,𝑛=𝑑_𝑛,𝑚+𝑓_𝑛(𝑡)+𝑡𝑐_𝑚,𝑛. (66) After taking first limsup_𝑚→∞ and then liminf_𝑛→∞ in (66) and using (62) and (65), we get limsup_𝑚→∞𝑓_𝑚(𝑡)≤liminf_𝑛→∞𝑓_𝑛(𝑡)+lim_{𝑚,𝑛→∞}𝑑_𝑚,𝑛+𝑐_𝑛,𝑚=liminf_𝑛→∞𝑓_𝑛(𝑡). (67) So that lim_𝑛→∞‖𝑡𝑥_𝑛+(1−𝑡)𝑧−𝑧‖ exists for all 𝑡∈[0,1]. It follows from Lemma 9 that 𝑧=𝑧. This completes the proof.

Remark 15. Compared with the known results in the literature, our results are very different from those in the following aspects. (i)Theorem 14 improves and extends Theorem 3.1 of Manaka and Takahashi [8] and Theorem 3 of Kamimura and Takahashi [9] from Hilbert spaces to uniformly convex and 𝑞-uniformly smooth Banach spaces.(ii)Theorem 14 also improves and extends Theorem 3.6 of López et al. [18] from the problem of finding an element of (𝐴+𝐵)⁻¹(0) to the problem of finding an element of (𝐴+𝐵)⁻¹(0)∩𝐹(𝑆).

Theorem 16. Let 𝐸 be a uniformly convex and 𝑞-uniformly smooth Banach space. Let 𝐴∶𝐸→𝐸 be 𝛼-inverse strongly accretive, let 𝐵∶𝐸→2^𝐸 be 𝑚-accretive, let 𝑓∶𝐸→𝐸 be 𝑟-contractive, and let 𝑆∶𝐸→𝐸 be nonexpansive. Assume that 𝐹(𝑆)∩(𝐴+𝐵)⁻¹(0)≠∅. We define a sequence {𝑥_𝑛} by the perturbed iterative scheme: 𝑥_𝑛+1=𝛼_𝑛𝑓𝑥_𝑛+1−𝛼_𝑛×𝑆𝐽_𝑟𝑛𝑥_𝑛−𝑟_𝑛𝐴𝑥_𝑛+𝑎_𝑛+𝑏_𝑛,∀𝑛≥1, (68) where 𝐽_𝑟𝑛=(𝐼+𝑟_𝑛𝐵)⁻¹. Assume that {𝛼_𝑛}⊂[0,1], {𝑟_𝑛}⊂(0,+∞), and {𝑎_𝑛},{𝑏_𝑛}⊂𝐸 satisfying the following conditions:(i)∑^∞_𝑛=1‖𝑎_𝑛‖<∞ and ∑^∞_𝑛=1‖𝑏_𝑛‖<∞;(ii)∑^∞_𝑛=1𝛼_𝑛=∞lim_𝑛→∞𝛼_𝑛=0 and ∑^∞_𝑛=1|𝛼_𝑛+1−𝛼_𝑛|<∞;(iii)0<liminf_𝑛→∞𝑟_𝑛≤limsup_𝑛→∞𝑟_𝑛<(𝑞𝛼/𝐶_𝑞)^1/(𝑞−1) and ∑^∞_𝑛=1|𝑟_𝑛+1−𝑟_𝑛|<∞.Then {𝑥_𝑛} converges strongly to some point 𝑧∈𝐹(𝑆)∩(𝐴+𝐵)⁻¹(0), which is the unique solution of the variational inequality ⟨𝑓(𝑧)−𝑧,𝑗(𝑝−𝑧)⟩≤0, ∀𝑝∈𝐹(𝑆)∩(𝐴+𝐵)⁻¹(0).

Proof. Let {𝑦_𝑛} be a sequence generated by 𝑦_𝑛+1=𝛼_𝑛𝑓𝑦_𝑛+1−𝛼_𝑛𝑆𝑇_𝑛𝑦_𝑛, (69) where 𝑇_𝑛∶=𝐽_𝑟𝑛(𝐼−𝑟_𝑛𝐴). Hence to show the desired result, it suffices to prove that 𝑦_𝑛→𝑧. Indeed, since 𝐽_𝑟𝑛 and (𝐼−𝑟_𝑛𝐴) are both nonexpansive under the condition of (iii), it follows that 𝑦_𝑛+1−𝑥_𝑛+1≤𝛼_𝑛𝑓𝑦_𝑛+1−𝛼_𝑛𝑆𝐽_𝑟𝑛𝑦_𝑛−𝑟_𝑛𝐴𝑦_𝑛−𝛼_𝑛𝑓𝑥_𝑛−1−𝛼_𝑛×𝑆𝐽_𝑟𝑛𝑥_𝑛−𝑟_𝑛𝐴𝑥_𝑛+𝑎_𝑛+𝑏_𝑛≤1−𝛼_𝑛𝑆𝐽_𝑟𝑛𝑦_𝑛−𝑟_𝑛𝐴𝑦_𝑛−𝑆𝐽_𝑟𝑛𝑥_𝑛−𝑟_𝑛𝐴𝑥_𝑛+𝑎_𝑛+𝑏_𝑛+𝛼_𝑛𝑟𝑦_𝑛−𝑥_𝑛≤1−𝛼_𝑛𝐽_𝑟𝑛𝑦_𝑛−𝑟_𝑛𝐴𝑦_𝑛−𝐽_𝑟𝑛𝑥_𝑛−𝑟_𝑛𝐴𝑥_𝑛+𝑎_𝑛+𝑏_𝑛+𝛼_𝑛𝑟𝑦_𝑛−𝑥_𝑛≤1−𝛼_𝑛(1−𝑟)𝑦_𝑛−𝑥_𝑛+𝑟_𝑛𝑎_𝑛+𝑏_𝑛. (70) By virtue of Lemma 4 and (70), one has lim_𝑛→∞‖𝑦_𝑛−𝑥_𝑛‖=0.
We first prove that the sequences {𝑦_𝑛} is bounded.
Thanks to (69) and Lemma 13, we have 𝑦_𝑛+1−𝑧=𝛼_𝑛𝑓𝑦_𝑛+1−𝛼_𝑛𝑆𝑇_𝑛𝑦_𝑛−𝑧≤𝛼_𝑛𝑓𝑦_𝑛−𝑧+1−𝛼_𝑛𝑆𝑇_𝑛𝑦_𝑛−𝑧≤𝛼_𝑛𝑓𝑦_𝑛−𝑓(𝑧)+𝛼_𝑛𝑓(𝑧)+−𝑧1−𝛼_𝑛𝑆𝑇_𝑛𝑦_𝑛−𝑧≤𝛼_𝑛𝑟𝑦_𝑛−𝑧+𝛼_𝑛𝑓(𝑧)+−𝑧1−𝛼_𝑛𝑦_𝑛=−𝑧1−𝛼_𝑛(1−𝑟)𝑦_𝑛−𝑧+𝛼_𝑛𝑓(𝑧)𝑓(𝑧)−𝑧≤max−𝑧,𝑦1−𝑟^𝑛−𝑧. (71) By induction, we have 𝑦_𝑛𝑓(𝑧)−𝑧≤max−𝑧,𝑦1−𝑟¹−𝑧,∀𝑛≥1. (72) Hence, {𝑦_𝑛} is bounded, so are {𝑓(𝑦_𝑛)} and {𝑇_𝑛(𝑦_𝑛)}.
Next we prove that lim_𝑛→∞𝑦_𝑛+1−𝑦_𝑛⟶0. (73) Putting 𝑧_𝑛=𝑇_𝑛𝑦_𝑛, it follows from Lemma 12 that 𝑧_𝑛+1−𝑧_𝑛=𝑇_𝑛+1𝑦_𝑛+1−𝑇_𝑛𝑦_𝑛≤𝑇_𝑛+1𝑦_𝑛+1−𝑇_𝑛𝑦_𝑛+1+𝑇_𝑛𝑦_𝑛+1−𝑇_𝑛𝑦_𝑛≤𝑟1−^𝛼𝑛𝑟_𝛽𝑛𝑦_𝑛+1−𝐽_𝑟𝛽𝑛1−𝑟_𝛽𝑛𝐴𝑦_𝑛+1+𝑦_𝑛+1−𝑦_𝑛≤𝑟_𝛽𝑛−𝑟_𝛼𝑛𝑦_𝑛+1−𝐽_𝑟𝛽𝑛1−𝑟_𝛽𝑛𝐴𝑦_𝑛+1𝑟_𝛽𝑛+𝑦_𝑛+1−𝑦_𝑛≤𝑟_𝑛+1−𝑟_𝑛𝑀₂+𝑦_𝑛+1−𝑦_𝑛, (74) where 𝑀₂>sup_𝑛≥1{‖𝑦_𝑛+1−𝐽_𝑟𝛽𝑛(1−𝑟_𝛽𝑛𝐴)𝑦_𝑛+1‖/𝑟_𝛽𝑛}, 𝑟_𝛼𝑛=min{𝑟_𝑛+1,𝑟_𝑛}, and 𝑟_𝛽𝑛=max{𝑟_𝑛+1,𝑟_𝑛}. Hence from (69) and (74) we have 𝑦_𝑛+1−𝑦_𝑛=𝛼_𝑛𝑓𝑦_𝑛+1−𝛼_𝑛𝑆𝑧_𝑛−𝛼_𝑛−1𝑓𝑦_𝑛−1+1−𝛼_𝑛−1𝑆𝑧_𝑛−1=𝛼_𝑛−𝛼_𝑛−1𝑓𝑦_𝑛−1−𝑆𝑧_𝑛−1+1−𝛼_𝑛𝑆𝑧_𝑛−𝑆𝑧_𝑛−1+𝛼_𝑛𝑓𝑦_𝑛−𝑓𝑦_𝑛−1≤𝛼_𝑛−𝛼_𝑛−1𝑓𝑦_𝑛−1−𝑆𝑧_𝑛−1+1−𝛼_𝑛𝑆𝑧_𝑛−𝑆𝑧_𝑛−1+𝛼_𝑛𝑟𝑦_𝑛−𝑦_𝑛−1≤𝛼_𝑛−𝛼_𝑛−1𝑀₃+1−𝛼_𝑛𝑧_𝑛−𝑧_𝑛−1+𝛼_𝑛𝑟𝑦_𝑛−𝑦_𝑛−1≤𝛼_𝑛−𝛼_𝑛−1𝑀₃+𝑟_𝑛−𝑟_𝑛−1𝑀₂+1−𝛼_𝑛(1−𝑟)𝑦_𝑛−𝑦_𝑛−1, (75) where 𝑀₃>sup_𝑛≥1{‖𝑓(𝑦_𝑛)−𝑆𝑧_𝑛‖}. It follows from Lemma 4, (ii), and (iii) that ‖𝑦_𝑛+1−𝑦_𝑛‖→0.
Again from Lemmas 3 and 12, we obtain 𝑦_𝑛+1−𝑧^𝑞=𝛼_𝑛𝑓𝑦_𝑛+−𝑧1−𝛼_𝑛𝑆𝑇_𝑛𝑦_𝑛−𝑧^𝑞≤1−𝛼_𝑛𝑇_𝑛𝑦_𝑛−𝑧^𝑞+𝑞𝛼_𝑛×𝑓𝑦_𝑛−𝑧,𝑗_𝑞𝑦_𝑛+1.≤𝑇−𝑧_𝑛𝑦_𝑛−𝑧^𝑞+𝑞𝛼_𝑛𝑀₄≤𝑦_𝑛−𝑧^𝑞−𝑟_𝑛𝛼𝑞−𝑟_𝑛^𝑞−1𝐶_𝑞𝐴𝑦_𝑛−𝐴𝑧^𝑞−𝜙_𝑞𝑦_𝑛−𝑟_𝑛𝐴𝑦_𝑛−𝑇_𝑛𝑦_𝑛+𝑟_𝑛𝐴𝑧+𝑞𝛼_𝑛𝑀₄, (76) where 𝑀₄>sup_𝑛≥1{⟨𝑓(𝑦_𝑛)−𝑧,𝑗_𝑞(𝑦_𝑛+1−𝑧)⟩}. It follows immediately from (76) that lim_𝑛→∞𝐴𝑦_𝑛−𝐴𝑧=lim_𝑛→∞𝑦_𝑛−𝑟_𝑛𝐴𝑦_𝑛−𝑇_𝑛𝑦_𝑛+𝑟_𝑛𝐴𝑧=0. (77) Hence we obtain that lim_𝑛→∞𝑇_𝑛𝑦_𝑛−𝑦_𝑛=0. (78) By condition (iii), there exists 𝜀>0 such that 𝑟_𝑛≥𝜀 for all 𝑛≥1. Then, by Lemma 12, we get lim_𝑛→∞𝑇_𝜀𝑦_𝑛−𝑦_𝑛≤lim_𝑛→∞2𝑇_𝑛𝑦_𝑛−𝑦_𝑛=0. (79)
Step  1. We show lim_𝑛→∞‖𝑆𝑇_𝜀𝑦_𝑛−𝑦_𝑛‖=0.
From (73), (78), (79), and (ii), we have 𝑆𝑇_𝜀𝑦_𝑛−𝑦_𝑛≤𝑆𝑇_𝜀𝑦_𝑛−𝑆𝑇_𝑛𝑦_𝑛+𝑆𝑇_𝑛𝑦_𝑛−𝑦_𝑛≤𝑇_𝜀𝑦_𝑛−𝑇_𝑛𝑦_𝑛+𝑆𝑇_𝑛𝑦_𝑛−𝑦_𝑛≤𝑇_𝜀𝑦_𝑛−𝑦_𝑛+𝑦_𝑛−𝑇_𝑛𝑦_𝑛+𝑆𝑇_𝑛𝑦_𝑛−𝑦_𝑛+1+𝑦_𝑛+1−𝑦_𝑛≤𝑇_𝜀𝑦_𝑛−𝑦_𝑛+𝑦_𝑛−𝑇_𝑛𝑦_𝑛+𝛼_𝑛𝑓𝑦_𝑛−𝑆𝑇_𝑛𝑦_𝑛+𝑦_𝑛+1−𝑦_𝑛⟶0. (80) Lemmas 7 and 13 imply that 𝜔_𝑤𝑦_𝑛⊂𝐹𝑆𝑇_𝜀=𝐹(𝑆)𝐹𝑇_𝜀. (81) Next we prove that limsup_𝑛→∞𝑓(𝑧)−𝑧,𝑗_𝑞𝑦_𝑛−𝑧≤0. (82) Equivalently (should ‖𝑦_𝑛−𝑧‖≠0), we need to prove that limsup_𝑛→∞𝑓(𝑧)−𝑧,𝑗𝑦_𝑛−𝑧≤0. (83) To this end, let 𝑧_𝑡 satisfy 𝑧_𝑡=𝑡𝑓(𝑧_𝑡)+(1−𝑡)𝑆𝑇_𝜀𝑧_𝑡. By Xu’s Theorem 4.1 [26], we get 𝑧_𝑡→𝑧∈𝐹(𝑆𝑇_𝜀) as 𝑡→0, which 𝑧 is the unique solution of the variational inequality: 𝑓(𝑧)−𝑧,𝑗(𝑥−𝑧)≤0,∀𝑥∈𝐹𝑆𝑇_𝜀. (84) Using subdifferential inequality, we deduce that 𝑧_𝑡−𝑦_𝑛²=𝑡𝑓𝑧_𝑡−𝑦_𝑛,𝑗𝑧_𝑡−𝑦_𝑛+(1−𝑡)𝑆𝑇_𝜀𝑧_𝑡−𝑦_𝑛,𝑗𝑧_𝑡−𝑦_𝑛=𝑡𝑓𝑧_𝑡−𝑧_𝑡,𝑗𝑧_𝑡−𝑦_𝑛+𝑡𝑧_𝑡−𝑦_𝑛,𝑗𝑧_𝑡−𝑦_𝑛+(1−𝑡)𝑆𝑇_𝜀𝑧_𝑡−𝑆𝑇_𝜀𝑦_𝑛,𝑗𝑧_𝑡−𝑦_𝑛+(1−𝑡)𝑆𝑇_𝜀𝑦_𝑛−𝑦_𝑛,𝑗𝑧_𝑡−𝑦_𝑛≤𝑡𝑓𝑧_𝑡−𝑧_𝑡,𝑗𝑧_𝑡−𝑦_𝑛+𝑡𝑧_𝑡−𝑦_𝑛²+(1−𝑡)𝑧_𝑡−𝑦_𝑛²+()×1−𝑡𝑆𝑇_𝜀𝑦_𝑛−𝑦_𝑛𝑧_𝑡−𝑦_𝑛≤𝑡𝑓𝑧_𝑡−𝑧_𝑡,𝑗𝑧_𝑡−𝑦_𝑛+𝑧_𝑡−𝑦_𝑛²+𝑆𝑇_𝜀𝑦_𝑛−𝑦_𝑛𝑧_𝑡−𝑦_𝑛, (85) which implies that 𝑓𝑧_𝑡−𝑧_𝑡,𝑗𝑦_𝑛−𝑧_𝑡≤𝑆𝑇_𝜀𝑦_𝑛−𝑦_𝑛𝑡𝑧_𝑡−𝑦_𝑛. (86) Using (80), taking the upper limit as 𝑛→∞ firstly and then as 𝑡→0 in (86), we have limsup_𝑡→0limsup_𝑛→∞𝑓𝑧_𝑡−𝑧_𝑡,𝑗𝑦_𝑛−𝑧_𝑡≤0. (87) Since 𝐸 is a uniformly smooth Banach space, we have the duality mapping 𝑗 is norm-to-norm uniformly on any bounded subset of 𝐸, which ensures that the limits limsup_𝑡→0 and limsup_𝑛→∞ are interchangeable, and we have limsup_𝑛→∞𝑓(𝑧)−𝑧,𝑗𝑦_𝑛−𝑧≤0. (88) Finally, we show ‖𝑦_𝑛−𝑧‖→0.
By Lemma 3 and the fact that 𝑎𝑏≤(1/𝑞)𝑎^𝑞+((𝑞−1)/𝑞)𝑏^{𝑞/(𝑞−1)}, we get 𝑦_𝑛+1−𝑧^𝑞=𝛼_𝑛𝑓𝑦_𝑛+1−𝛼_𝑛𝑆𝑇_𝑛𝑦_𝑛−𝑧^𝑞=𝛼_𝑛𝑓𝑦_𝑛+1−𝛼_𝑛𝑆𝑇_𝑛𝑦_𝑛−𝑧,𝑗_𝑞𝑦_𝑛+1−𝑧=𝛼_𝑛𝑓𝑦_𝑛−𝑓(𝑧),𝑗_𝑞𝑦_𝑛+1−𝑧+𝛼_𝑛𝑓(𝑧)−𝑧,𝑗_𝑞𝑦_𝑛+1+−𝑧1−𝛼_𝑛𝑆𝑇_𝑛𝑦_𝑛−𝑧,𝑗_𝑞𝑦_𝑛+1−𝑧≤𝛼_𝑛𝑓𝑦_𝑛−𝑓(𝑧)𝑦_𝑛+1−𝑧^𝑞−1+𝛼_𝑛𝑓(𝑧)−𝑧,𝑗_𝑞𝑦_𝑛+1+−𝑧1−𝛼_𝑛𝑦_𝑛−𝑧𝑦_𝑛+1−𝑧^𝑞−1≤𝛼_𝑛𝑟𝑦_𝑛−𝑧𝑦_𝑛+1−𝑧^𝑞−1+𝛼_𝑛𝑓(𝑧)−𝑧,𝑗_𝑞𝑦_𝑛+1−𝑧+1−𝛼_𝑛×𝑦_𝑛−𝑧𝑦_𝑛+1−𝑧^𝑞−1≤1−𝛼_𝑛(1−𝑟)𝑦_𝑛−𝑧𝑦_𝑛+1−𝑧^𝑞−1+𝛼_𝑛𝑓(𝑧)−𝑧,𝑗_𝑞𝑦_𝑛+1≤−𝑧1−𝛼_𝑛(11−𝑟)𝑞𝑦_𝑛−𝑧^𝑞+𝑞−1𝑞×𝑦_𝑛+1−𝑧^𝑞+𝛼_𝑛𝑓(𝑧)−𝑧,𝑗_𝑞𝑦_𝑛+1−𝑧, (89) which implies that 𝑦_𝑛+1−𝑧^𝑞≤1−𝛼_𝑛(1−𝑟)𝑦_𝑛−𝑧^𝑞+𝑞𝛼_𝑛𝑓(𝑧)−𝑧,𝑗_𝑞𝑦_𝑛+1−𝑧. (90) Apply Lemma 4 to (90) to conclude 𝑦_𝑛→𝑧 as 𝑛→∞. This completes the proof.

Remark 17. Theorem 16 improves and extends Theorem 3.7 of López et al. [18] in the following ways: (i)from the problem of finding an element of (𝐴+𝐵)⁻¹(0) to the problem of finding an element of (𝐴+𝐵)⁻¹(0)∩𝐹(𝑆),(ii)from a fixed element 𝑢 in 𝐸 to a contractive mapping 𝑓.

Remark 18. Theorem 16 improves and extends Theorem 2.1 of Zhang et al. [7] in the following ways: (i)from Hilbert spaces to uniformly convex and 𝑞-uniformly smooth Banach spaces,(ii)from a fixed element 𝑢 in 𝐸 to a contractive mapping 𝑓,(iii)from a fixed positive number 𝜆 to a positive sequence {𝑟_𝑛}.

As a direct consequence of Theorem 16, we obtain the following result.

Corollary 19. Let 𝐻 be a Hilbert space. Let 𝐴∶𝐻→𝐻 be 𝛼-inverse strongly monotone, let 𝐵∶𝐻→2^𝐻 be maximal monotone, let 𝑓∶𝐸→𝐸 be 𝑟-contractive, and let 𝑆∶𝐻→𝐻 be nonexpansive. Assume that 𝐹(𝑆)∩(𝐴+𝐵)⁻¹(0)≠∅. We define a sequence {𝑥_𝑛} by the perturbed iterative scheme: 𝑥_𝑛+1=𝛼_𝑛𝑓𝑥_𝑛+1−𝛼_𝑛×𝑆𝐽_𝑟𝑛𝑥_𝑛−𝑟_𝑛𝐴𝑥_𝑛+𝑎_𝑛+𝑏_𝑛,∀𝑛≥1, (91) where 𝐽_𝑟𝑛=(𝐼+𝑟_𝑛𝐵)⁻¹. Assume that {𝛼_𝑛}⊂[0,1], {𝑟_𝑛}⊂(0,+∞), and {𝑎_𝑛},{𝑏_𝑛}⊂𝐻 satisfying the following conditions:(i)∑^∞_𝑛=1‖𝑎_𝑛‖<∞ and ∑^∞_𝑛=1‖𝑏_𝑛‖<∞;(ii)∑^∞_𝑛=1𝛼_𝑛=∞lim_𝑛→∞𝛼_𝑛=0 and ∑^∞_𝑛=1|𝛼_𝑛+1−𝛼_𝑛|<∞;(iii)0<liminf_𝑛→∞𝑟_𝑛≤limsup_𝑛→∞𝑟_𝑛<2𝛼 and ∑^∞_𝑛=1|𝑟_𝑛+1−𝑟_𝑛|<∞.Then {𝑥_𝑛} converges strongly to some point 𝑧∈𝐹(𝑆)∩(𝐴+𝐵)⁻¹(0), which is the unique solution of the variational inequality ⟨𝑓(𝑧)−𝑧,𝑝−𝑧⟩≤0, ∀𝑝∈𝐹(𝑆)∩(𝐴+𝐵)⁻¹(0).

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

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