Modified Hybrid Steepest-Descent Methods for General Systems of Variational Inequalities with Solutions to Zeros of 𝑚-Accretive Operators in Banach Spaces

Abstract

The purpose of this paper is to introduce and analyze modified hybrid steepest-descent methods for a general system of variational inequalities (GSVI), with solutions being also zeros of an 𝑚-accretive operator 𝐴 in the setting of real uniformly convex and 2-uniformly smooth Banach space 𝑋. Here the modified hybrid steepest-descent methods are based on Korpelevich's extragradient method, hybrid steepest-descent method, and viscosity approximation method. We propose and consider modified implicit and explicit hybrid steepest-descent algorithms for finding a common element of the solution set of the GSVI and the set 𝐴⁻¹(0) of zeros of 𝐴 in 𝑋. Under suitable assumptions, we derive some strong convergence theorems. The results presented in this paper improve, extend, supplement, and develop the corresponding results announced in the earlier and very recent literature.

1. Introduction

Let 𝑋 be a real Banach space whose dual space is denoted by 𝑋^∗. The normalized duality mapping 𝐽∶𝑋→2^𝑋∗ is defined by 𝐽(𝑥)=𝑥^∗∈𝑋^∗∶𝑥,𝑥^∗=‖𝑥‖²=𝑥^∗2,∀𝑥∈𝑋, (1) where ⟨⋅,⋅⟩ denotes the generalized duality pairing. It is an immediate consequence of the Hahn-Banach theorem that 𝐽(𝑥) is nonempty for each 𝑥∈𝑋. Let 𝐶 be a nonempty closed convex subset of 𝑋. A mapping 𝑇∶𝐶→𝐶 is called nonexpansive if ‖𝑇𝑥−𝑇𝑦‖≤‖𝑥−𝑦‖ for every 𝑥,𝑦∈𝐶. The set of fixed points of 𝑇 is denoted by Fix(𝑇). We use the notation ⇀ to indicate the weak convergence and → to indicate the strong convergence. A mapping 𝐴∶𝐶→𝑋 is said to be

(i) accretive if for each 𝑥,𝑦∈𝐶 there exists 𝑗(𝑥−𝑦)∈𝐽(𝑥−𝑦) such that 𝐴𝑥−𝐴𝑦,𝑗𝑥−𝑦≥0; (2)

(ii) 𝛼-strongly accretive if for each 𝑥,𝑦∈𝐶 there exists 𝑗(𝑥−𝑦)∈𝐽(𝑥−𝑦) such that 𝐴𝑥−𝐴𝑦,𝑗𝑥−𝑦≥𝛼𝑥−𝑦², (3) for some 𝛼∈(0,1);

(iii) 𝛽-inverse strongly accretive if for each 𝑥,𝑦∈𝐶 there exists 𝑗(𝑥−𝑦)∈𝐽(𝑥−𝑦) such that 𝐴𝑥−𝐴𝑦,𝑗𝑥−𝑦≥𝛽𝐴𝑥−𝐴𝑦², (4) for some𝛽>0;

(iv) 𝜆-strictly pseudocontractive [1] (see also [2]) if for each 𝑥,𝑦∈𝐶 there exists 𝑗(𝑥−𝑦)∈𝐽(𝑥−y) such that 𝐴𝑥−𝐴𝑦,𝑗𝑥−𝑦≤𝑥−𝑦²−𝜆𝑥−𝑦−(𝐴𝑥−𝐴𝑦)² (5) for some 𝜆∈(0,1).

It is worth emphasizing that the definition of the inverse strongly accretive mapping is based on that of the inverse strongly monotone mapping, which was studied by so many authors; see, for example, [3–5]. Let 𝑈={𝑥∈𝑋∶‖𝑥‖=1} denote the unite sphere of 𝑋. A Banach space 𝑋 is said to be uniformly convex if for each 𝜖∈(0,2] there exists 𝛿>0 such that for all 𝑥,𝑦∈𝑈𝑥−𝑦≥𝜖⟹𝑥+𝑦2≤1−𝛿. (6) 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 limit lim_𝑡→0𝑥+𝑡𝑦−‖𝑥‖𝑡 (7) exists for all 𝑥,𝑦∈𝑈; in this case, 𝑋 is also said to have a Gateaux differentiable norm. Moreover, it is said to be uniformly smooth if this limit is attained uniformly for 𝑥,𝑦∈𝑈; in this case, 𝑋 is also said to have a uniformly Frechet differentiable norm. The norm of 𝑋 is said to be the Frechet differential if for each 𝑥∈𝑈 this limit is attained uniformly for 𝑦∈𝑈. In the meantime, we define a function 𝜌∶[0,∞)→[0,∞) called the modulus of smoothness of 𝑋 as follows: 𝜌(𝜏)1=sup2+‖𝑥‖𝑦.𝑥+𝑦𝑥−𝑦−1∶𝑥,𝑦∈𝑋,=1,=𝜏 (8) 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. As pointed out in [6], no Banach space is 𝑞-uniformly smooth for 𝑞>2. In addition, it is also known that 𝐽 is single-valued if and only if 𝑋 is smooth, whereas if 𝑋 is uniformly smooth, then the mapping 𝐽 is norm-to-norm uniformly continuous on bounded subsets of 𝑋.

Let 𝐴 be an operator with domain 𝐷(𝐴) and range 𝑅(𝐴) in 𝑋 is said to be accretive if for each 𝑥_𝑖∈𝐷(𝐴) and 𝑦_𝑖∈𝐴𝑥_𝑖(𝑖=1,2) there exists 𝑗(𝑥₂−𝑥₁)∈𝐽(𝑥₂−𝑥₁) such that 𝑦₂−𝑦₁,𝑗𝑥₂−𝑥₁≥0. (9) An accretive operator 𝐴 is said to be 𝑚-accretive if 𝑅(𝐼+𝜆𝐴)=𝑋 for all 𝜆>0. Denote by 𝐽_𝑟 the resolvent of 𝐴; that is, for each 𝑟>0, 𝐽_𝑟=(𝐼+𝑟𝐴)⁻¹. (10) It is known that 𝐽_𝑟 is a nonexpansive mapping from 𝑋 to 𝐶∶=𝐷(𝐴) which will be assumed convex. In 2008, Chen and Zhu [7] derived the following strong convergence theorems for viscosity approximation methods for accretive operators in a uniformly smooth Banach space 𝑋.

Theorem  CZ1. Let 𝑋 be a uniformly smooth Banach space. Suppose that 𝐴 is an 𝑚-accretive operator in 𝑋 such that 𝐶=𝐷(𝐴) is convex and 𝑓∶𝐶→𝐶 is a fixed contractive map. For each 𝑡∈(0,1), {𝑥_𝑡,𝑛} is defined by𝑥_𝑡,𝑛=𝑡𝑓𝑥_𝑡,𝑛+(1−𝑡)𝐽_𝑟𝑛𝑥_𝑡,𝑛. (11)Then as 𝑡→0, {𝑥_𝑡,𝑛} converges strongly to a zero of 𝐴.

Theorem  CZ2. Let 𝑋 be a uniformly smooth Banach space. Suppose that 𝐴 is an 𝑚-accretive operator in 𝑋 such that 𝐶=𝐷(𝐴) is convex and 𝑓∶𝐶→𝐶 is a fixed contractive map. The sequence {𝑥_𝑛} is defined by𝑥_𝑛+1=𝛼_𝑛𝑓𝑥_𝑛+1−𝛼_𝑛𝐽_𝑟𝑛𝑥_𝑛,∀𝑛≥0, (12)where {𝛼_𝑛} and {𝑟_𝑛} satisfy the following conditions: (i)𝛼_𝑛→0,∑^∞_𝑛=0𝛼_𝑛=∞, and ∑^∞_𝑛=0|𝛼_𝑛+1−𝛼_𝑛|<∞,(ii)𝑟_𝑛≥𝜀 for all 𝑛 and ∑^∞_𝑛=0|𝑟_𝑛+1−𝑟_𝑛|<∞.

Then as 𝑛→∞, {𝑥_𝑛} converges strongly to a zero of 𝐴.

In the meantime, Ceng et al. [8] derived some strong convergence theorems of composite iterative schemes for zeros of 𝑚-accretive operators in uniformly smooth Banach spaces. Furthermore, motivated by strong convergence results for hybrid steepest-descent methods in [9, 10], Ceng et al. [11] established some strong convergence theorems for hybrid steepest-descent methods for nonexpansive and 𝑚-accretive operators in a uniformly smooth Banach space 𝑋. Subsequently, Ceng et al. [12] introduced hybrid viscosity approximation method for finding zeros of 𝑚-accretive operators, which combine viscosity approximation method with hybrid steepest-descent method, and obtained the following strong convergence theorems.

Theorem  CASY1. Let 𝑋 be a uniformly smooth Banach space, let 𝐴 be an 𝑚-accretive operator in 𝑋 with 𝐴⁻¹(0)≠∅, and let 𝑓∶𝑋→𝐶(=𝐷(𝐴)) be a contractive map. Assume that 𝐹∶𝑋→𝑋 is 𝛿-strongly accretive and 𝜆-strictly pseudocontractive with 𝛿+𝜆>1. For each 𝑡∈(0,1) and each integer 𝑛≥0, let {𝑥_𝑡,𝑛} be defined by𝑥_𝑡,𝑛=𝑡𝑓𝑥_𝑡,𝑛+(1−𝑡)𝐽_𝑟𝑛𝑥_𝑡,𝑛−𝜃_𝑡𝐹𝑥_𝑡,𝑛, (13)where {𝑟_𝑛}⊂[𝜀,∞) for some 𝜀>0 and {𝜃_𝑡∶𝑡∈(0,1)}⊂[0,1) with lim_𝑡→0(𝜃_𝑡/𝑡)=0. Then as 𝑡→0,{𝑥_𝑡,𝑛} converges strongly to a zero 𝑝 of 𝐴, which is a unique solution of the variational inequality problem (VIP)I−𝑓𝑝,𝐽𝑝−𝑢≤0,∀𝑢∈𝐴⁻¹(0). (14)

Theorem  CASY2. Let 𝑋 be a uniformly smooth Banach space, let 𝐴 be an 𝑚-accretive operator in 𝑋 with 𝐴⁻¹(0)≠∅, and let 𝑓∶𝑋→𝐶(=𝐷(𝐴)) be a contractive map. Assume that 𝐹∶𝑋→𝑋 is 𝛿-strongly accretive and 𝜆-strictly pseudocontractive with 𝛿+𝜆>1. Given sequences {𝜆_𝑛}^∞_𝑛=0,{𝜇_𝑛}^∞_𝑛=0 in [0,1],{𝛼_𝑛}^∞_𝑛=0,{𝛽_𝑛}^∞_𝑛=0 in (0,1], and {𝑟_𝑛}^∞_𝑛=0 in [𝜀,∞) for some 𝜀>0, suppose that there hold the following conditions: (i)lim_𝑛→∞𝛽_𝑛=0 and ∑^∞_𝑛=0𝛽_𝑛=∞,(ii)lim_𝑛→∞(𝜆_𝑛𝜇_𝑛)/𝛽_𝑛=0,(iii){𝛼_𝑛}⊂[𝑎,𝑏] for some 𝑎,𝑏∈(0,1),(iv)∑^∞_𝑛=0|𝛼_𝑛+1−𝛼_𝑛|<∞, ∑^∞_𝑛=0|𝛽_𝑛+1−𝛽_𝑛|<∞, ∑^∞_𝑛=0|𝜆_𝑛+1−𝜆_𝑛|<∞, ∑^∞_𝑛=0|𝜇_𝑛+1−𝜇_𝑛|<∞ and ∑^∞_𝑛=0|𝑟_𝑛+1−𝑟_𝑛|<∞.

Then for any given point 𝑥₀∈𝑋, the sequence {𝑥_𝑛} generated by𝑦_𝑛=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝐽_𝑟𝑛𝑥_𝑛,𝑥_𝑛+1=𝛽_𝑛𝑓𝑥_𝑛+1−𝛽_𝑛𝐽_𝑟𝑛𝑦_𝑛−𝜆_𝑛𝜇_𝑛𝐹𝐽_𝑟𝑛𝑦_𝑛,∀𝑛≥0, (15)converges strongly to a zero 𝑝 of 𝐴, which is a unique solution of the VIP as above.

On the other hand, Cai and Bu [13] considered the following general system of variational inequalities (GSVI) in a real smooth Banach space 𝑋, which involves finding (𝑥^∗,𝑦^∗)∈𝐶×𝐶 such that 𝜇₁𝐵₁𝑦^∗+𝑥^∗−𝑦^∗,𝐽𝑥−𝑥^∗𝜇≥0,∀𝑥∈𝐶,₂𝐵₂𝑥^∗+𝑦^∗−𝑥^∗,𝐽𝑥−𝑦^∗≥0,∀𝑥∈𝐶, (16) where 𝐶 is a nonempty, closed and convex subset of 𝑋,𝐵₁,𝐵₂∶𝐶→𝑋 are two nonlinear mappings, and 𝜇₁ and 𝜇₂ are two positive constants. Here the set of solutions of GSVI (16) is denoted by GSVI (𝐶,𝐵₁,𝐵₂). In particular, if 𝑋=𝐻, a real Hilbert space, then GSVI (16) reduces to the following GSVI of finding (𝑥^∗,𝑦^∗)∈𝐶×𝐶 such that 𝜇₁𝐵₁𝑦^∗+𝑥^∗−𝑦^∗,𝑥−𝑥^∗𝜇≥0,∀𝑥∈𝐶,₂𝐵₂𝑥^∗+𝑦^∗−𝑥^∗,𝑥−𝑦^∗≥0,∀𝑥∈𝐶, (17) where 𝜇₁ and 𝜇₂ are two positive constants. The set of solutions of problem (17) is still denoted by GSVI (𝐶,𝐵₁,𝐵₂). In particular, if 𝐵₁=𝐵₂=𝐴, then problem (17) reduces to the new system of variational inequalities (NSVI), introduced and studied by Verma [14]. Further, if 𝑥^∗=𝑦^∗ additionally, then the NSVI reduces to the classical variational inequality problem (VIP) of finding 𝑥^∗∈𝐶 such that 𝐴𝑥^∗,𝑥−𝑥^∗≥0,∀𝑥∈𝐶. (18) The solution set of the VIP (18) is denoted by VI (𝐶,𝐴). Variational inequality theory has been studied quite extensively and has emerged as an important tool in the study of a wide class of obstacle, unilateral, free, moving, equilibrium problems. It is now well known that the variational inequalities are equivalent to the fixed point problems, the origin of which can be traced back to Lions and Stampacchia [15]. This alternative formulation has been used to suggest and analyze projection iterative method for solving variational inequalities under the condition that the involved operator must be strongly monotone and Lipschitz continuous.

Recently, Ceng et al. [16] transformed problem (17) into a fixed point problem in the following way.

Lemma 1 (see [16]). For given 𝑥,𝑦∈𝐶,(𝑥,𝑦) is a solution of problem (17) if and only if 𝑥 is a fixed point of the mapping 𝐺∶𝐶→𝐶 defined by 𝐺(𝑥)=𝑃_𝐶𝑃_𝐶𝑥−𝜇₂𝐵₂𝑥−𝜇₁𝐵₁𝑃_𝐶𝑥−𝜇₂𝐵₂𝑥,∀𝑥∈𝐶, (19) where 𝑦=𝑃_𝐶(𝑥−𝜇₂𝐵₂𝑥) and 𝑃_𝐶 is the the projection of 𝐻 onto 𝐶.
In particular, if the mapping 𝐵_𝑖∶𝐶→𝐻 is 𝛽_𝑖-inverse strongly monotone for 𝑖=1,2, then the mapping 𝐺 is nonexpansive provided 𝜇_𝑖∈(0,2𝛽_𝑖) for 𝑖=1,2.

In 1976, Korpelevič [17] proposed an iterative algorithm for solving the VIP (18) in Euclidean space 𝐑^𝑛: 𝑦_𝑛=𝑃_𝐶𝑥_𝑛−𝜏𝐴𝑥_𝑛,𝑥_𝑛+1=𝑃_𝐶𝑥_𝑛−𝜏𝐴𝑦_𝑛,∀𝑛≥0, (20) with 𝜏>0 being a given number, which is known as the extragradient method (see also [18]). The literature on the VIP is vast, and Korpelevich’s extragradient method has received great attention given by many authors, who improved it in various ways; see, for example, [3, 13, 19–32] and references therein, to name but a few.

In particular, whenever 𝑋 is still a real smooth Banach space, 𝐵₁=𝐵₂=𝐴, and 𝑥^∗=𝑦^∗, then GSVI (16) reduces to the variational inequality problem (VIP) of finding 𝑥^∗∈𝐶 such that 𝐴𝑥^∗,𝐽𝑥−𝑥^∗≥0,∀𝑥∈𝐶, (21) which was considered by Aoyama et al. [33]. Note that VIP (21) is connected with the fixed point problem for nonlinear mapping (see, e.g., [34]), the problem of finding a zero point of a nonlinear operator (see, e.g., [35]), and so on. It is clear that VIP (21) extends VIP (18) from Hilbert spaces to Banach spaces.

In order to find a solution of VIP (21), Aoyama et al. [33] introduced the following iterative scheme for an accretive operator 𝐴: 𝑥_𝑛+1=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛Π_𝐶𝑥_𝑛−𝜆_𝑛𝐴𝑥_𝑛,∀𝑛≥1, (22) where Π_𝐶 is a sunny nonexpansive retraction from 𝑋 onto 𝐶. Then they proved a weak convergence theorem.

Beyond doubt, it is an interesting and valuable problem of constructing some algorithms with strong convergence for solving GSVI (16) which contains VIP (21) as a special case. Very recently, Cai and Bu [13] constructed an iterative algorithm for solving GSVI (16) and a common fixed point problem of an infinite family of nonexpansive mappings in a uniformly convex and 2-uniformly smooth Banach space. They proved the strong convergence of the proposed algorithm by virtue of the following inequality in a 2-uniformly smooth Banach space 𝑋.

Lemma 2 (see [36]). Let 𝑋 be a 2-uniformly smooth Banach space. Then 𝑥+𝑦²≤‖𝑥‖²+2𝑦,𝐽(𝑥)+2𝜅𝑦²,∀𝑥,𝑦∈𝑋, (23) where 𝜅 is the 2-uniformly smooth constant of 𝑋 and 𝐽 is the normalized duality mapping from 𝑋 into 𝑋^∗.

Define the mapping 𝐺∶𝐶→𝐶 as follows: 𝐺(𝑥)∶=Π_𝐶𝐼−𝜇₁𝐵₁Π_𝐶𝐼−𝜇₂𝐵₂𝑥,∀𝑥∈𝐶. (24) The fixed point set of 𝐺 is denoted by Ω. Then their strong convergence theorem on the proposed method is stated as follows.

Theorem  CB (see [13, Theorem 3.1]). Let 𝐶 be a nonempty closed convex subset of a uniformly convex and 2-uniformly smooth Banach space 𝑋. Let Π_𝐶 be a sunny nonexpansive retraction from 𝑋 onto 𝐶. Let the mapping 𝐵_𝑖∶𝐶→𝑋 be 𝛼_𝑖-inverse strongly accretive with 0<𝜇_𝑖<𝛼_𝑖/𝜅² for 𝑖=1,2. Let 𝑓 be a contraction of 𝐶 into itself with coefficient 𝜌∈(0,1). Let {𝑆_𝑛}^∞_𝑛=1 be an infinite family of nonexpansive mappings of 𝐶 into itself such that Δ=⋂^∞_𝑖=1Fix(𝑆_𝑖)∩Ω≠∅, where Ω is the fixed point set of the mapping 𝐺 defined by (24). For arbitrarily given 𝑥₁∈𝐶, let {𝑥_𝑛} be the sequence generated by𝑥_𝑛+1=𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝑆_𝑛𝑦_𝑛,𝑦_𝑛=𝛼_𝑛𝑓𝑥_𝑛+1−𝛼_𝑛𝑧_𝑛,𝑧_𝑛=Π_𝐶𝑢_𝑛−𝜇₁𝐵₁𝑢_𝑛,𝑢_𝑛=Π_𝐶𝑥_𝑛−𝜇₂𝐵₂𝑥_𝑛,∀𝑛≥1. (25)Suppose that {𝛼_𝑛} and {𝛽_𝑛} are two sequences in (0,1) satisfying the following conditions: (i)lim_𝑛→∞𝛼_𝑛=0 and ∑^∞_𝑛=1𝛼_𝑛=∞,(ii)0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1.

Assume that ∑^∞_𝑛=1sup_𝑥∈𝐷‖𝑆_𝑛+1𝑥−𝑆_𝑛𝑥‖<∞ for any bounded subset 𝐷 of 𝐶, let 𝑆 be a mapping of 𝐶 into 𝑋 defined by 𝑆𝑥=lim_𝑛→∞𝑆_𝑛𝑥 for all 𝑥∈𝐶, and suppose that Fix(𝑆)=⋂^∞_𝑛=1Fix(𝑆_𝑛). Then {𝑥_𝑛} converges strongly to 𝑝∈Δ, which solves the VIP:𝐼−𝑓𝑝,𝐽𝑝−𝑢≤0,∀𝑢∈Δ. (26)

It is easy to see that the iterative scheme in Theorem CB is essentially equivalent to the following two-step iterative scheme:𝑦_𝑛=𝛼_𝑛𝑓𝑥_𝑛+1−𝛼_𝑛𝐺𝑥_𝑛,𝑥_𝑛+1=𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝑆_𝑛𝑦_𝑛,∀𝑛≥1. (27)

For the convenience of implementing the argument techniques in [16], the authors [13] have used the following inequality in a real smooth and uniform convex Banach space 𝑋.

Proposition 3 (see [37]). Let 𝑋 be a real smooth and uniform convex Banach space and let 𝑟>0. Then there exists a strictly increasing, continuous and convex function 𝑔∶[0,2𝑟]→𝐑,𝑔(0)=0, such that 𝑔𝑥−𝑦≤‖𝑥‖²−2𝑥,𝐽𝑦+𝑦²,∀𝑥,𝑦∈𝐵_𝑟, (28) where 𝐵_𝑟={𝑥∈𝑋∶‖𝑥‖≤𝑟}.

Let 𝐶 be a nonempty closed convex subset of a real uniformly convex and 2-uniformly smooth Banach space 𝑋. Let Π_𝐶 be a sunny nonexpansive retraction from 𝑋 onto 𝐶 and let 𝑓∶𝐶→𝐶 be a contraction with coefficient 𝜌∈(0,1). Motivated and inspired by the research going on in this area, we introduce and analyze modified hybrid steepest-descent methods for the GSVI (16), with solutions being also zeros of an 𝑚-accretive operator 𝐴 in 𝑋. Here the modified hybrid steepest-descent methods are based on Korpelevich’s extragradient method, hybrid steepest-descent method, and viscosity approximation method. We propose and consider modified implicit and explicit hybrid steepest-descent algorithms for finding a common element of the solution set of the GSVI (16) and the set 𝐴⁻¹(0) of zeros of 𝐴 in 𝑋. Under suitable assumptions, we derive some strong convergence theorems. The results presented in this paper improve, extend, supplement, and develop the corresponding results announced in the earlier and very recent literature [12, 13, 16, 32].

2. Preliminaries

We list some lemmas that will be used in the sequel. Lemma 4 can be found in [38]. Lemma 5 is an immediate consequence of the subdifferential inequality of the function (1/2)‖⋅‖².

Lemma 4. Let{𝑎_𝑛}be a sequence of nonnegative real numbers such that 𝑎_𝑛+1≤1−𝑏_𝑛𝑎_𝑛+𝑏_𝑛𝑐_𝑛,∀𝑛≥0, (29) where {𝑏_𝑛} and {𝑐_𝑛} are sequences of real numbers satisfying the following conditions:(i){𝑏_𝑛}⊂[0,1] and ∑^∞_𝑛=0𝑏_𝑛=∞,(ii)either limsup_𝑛→∞𝑐_𝑛≤0 or ∑^∞_𝑛=0|𝑏_𝑛𝑐_𝑛|<∞.
Then, lim_𝑛→∞𝑎_𝑛=0.

Lemma 5. In a real smooth Banach space 𝑋, there holds the inequality ‖𝑥‖²+2𝑦,𝐽(𝑥)≤𝑥+𝑦²≤‖𝑥‖²+2𝑦,𝐽𝑥+𝑦,∀𝑥,𝑦∈𝑋, (30) where 𝐽∶𝑋→𝑋^∗ is the normalized duality mapping.

Let 𝐷 be a subset of 𝐶 and let Π be a mapping of 𝐶 into 𝐷. Then Π is said to be sunny if Π[Π(𝑥)+𝑡(𝑥−Π(𝑥))]=Π(𝑥), (31) whenever Π(𝑥)+𝑡(𝑥−Π(𝑥))∈𝐶 for 𝑥∈𝐶 and 𝑡≥0. A mapping Π of 𝐶 into itself is called a retraction if Π²=Π. If a mapping Π of𝐶into itself is a retraction, then Π(𝑧)=𝑧 for every 𝑧∈𝑅(Π), where 𝑅(Π) is the range of Π. A subset 𝐷 of 𝐶 is called a sunny nonexpansive retract of 𝐶 if there exists a sunny nonexpansive retraction from 𝐶 onto 𝐷. The following lemma concerns the sunny nonexpansive retraction.

Lemma 6 (see [39]). Let 𝐶 be a nonempty closed convex subset of a real smooth Banach space 𝑋. Let 𝐷 be a nonempty subset of 𝐶. Let Π be a retraction of 𝐶 onto 𝐷. Then the following are equivalent:(i)Π is sunny and nonexpansive,(ii)‖Π(𝑥)−Π(𝑦)‖²≤⟨𝑥−𝑦,𝐽(Π(𝑥)−Π(𝑦))⟩,∀𝑥,𝑦∈𝐶,(iii)⟨𝑥−Π(𝑥),𝐽(𝑦−Π(𝑥))⟩≤0,∀𝑥∈𝐶,𝑦∈𝐷.

It is well known that if 𝑋=𝐻, a Hilbert space, then a sunny nonexpansive retraction Π_𝐶 is coincident with the metric projection from 𝑋 onto 𝐶; that is, Π_𝐶=𝑃_𝐶. If 𝐶 is a nonempty closed convex subset of a strictly convex and uniformly smooth Banach space 𝑋 and if 𝑇∶𝐶→𝐶 is a nonexpansive mapping with the fixed point set Fix(𝑇)≠∅, then the set Fix(𝑇)is a sunny nonexpansive retract of 𝐶. The following lemma follows easily from Lemma 6.

Lemma 7. Let 𝐶 be a nonempty closed convex subset of a smooth Banach space 𝑋. Let Π_𝐶 be a sunny nonexpansive retraction from 𝑋 onto 𝐶 and let 𝐵₁,𝐵₂∶𝐶→𝑋 be nonlinear mappings. For given 𝑥^∗,𝑦^∗∈𝐶,(𝑥^∗,𝑦^∗) is a solution of GSVI (16) if and only if 𝑥^∗=Π_𝐶(𝑦^∗−𝜇₁𝐵₁𝑦^∗), where 𝑦^∗=Π_𝐶(𝑥^∗−𝜇₂𝐵₂𝑥^∗).

In terms of Lemma 7, we observe that 𝑥^∗=Π_𝐶Π_𝐶𝑥^∗−𝜇₂𝐵₂𝑥^∗−𝜇₁𝐵₁Π_𝐶𝑥^∗−𝜇₂𝐵₂𝑥^∗, (32) which implies that 𝑥^∗ is a fixed point of the mapping 𝐺. Throughout this paper, the set of fixed points of the mapping 𝐺 is denoted by Ω.

Lemma 8 is the resolvent identity which can be found in [40], and Lemma 9 can be found in [41].

Lemma 8. For 𝜆,𝜇>0, there holds the identity 𝐽_𝜆𝑥=𝐽_𝜇𝜇𝜆𝜇𝑥+1−𝜆𝐽_𝜆𝑥,∀𝑥∈𝑋. (33)

Lemma 9. Assume that 𝑐₂≥𝑐₁>0. Then ‖𝐽_𝑐1𝑥−𝑥‖≤2‖𝐽_𝑐2𝑥−𝑥‖ for all 𝑥∈𝑋.

Lemma 10 (see [42]). Let 𝐶 be a nonempty closed convex subset of a strictly convex Banach space 𝑋. Let {𝑇_𝑛}^∞_𝑛=0 be a sequence of nonexpansive mappings on 𝐶. Suppose that ⋂^∞_𝑛=0Fix(𝑇_𝑛) is nonempty. Let {𝜆_𝑛} be a sequence of positive numbers with ∑^∞_𝑛=0𝜆_𝑛=1. Then a mapping 𝑆 on 𝐶 defined by 𝑆𝑥=∑^∞_𝑛=0𝜆_𝑛𝑇_𝑛𝑥 for 𝑥∈𝐶 is defined well, nonexpansive, and Fix(𝑆)=⋂^∞_𝑛=0Fix(𝑇_𝑛) holds.

Let 𝜇 be a mean if 𝜇 is a continuous linear functional on 𝑙^∞ satisfying ‖𝜇‖=1=𝜇(1). Then we know that 𝜇 is a mean on 𝐍 if and only if inf𝑎_𝑛∶𝑛∈𝐍≤𝜇(𝑎)≤sup𝑎_𝑛∶𝑛∈𝐍 (34) for every 𝑎=(𝑎₁,𝑎₂,…)∈𝑙^∞. According to time and circumstances, we use 𝜇_𝑛(𝑎_𝑛) instead of 𝜇(𝑎). A mean 𝜇 on 𝐍 is called a Banach limit if and only if 𝜇_𝑛𝑎_𝑛=𝜇_𝑛𝑎_𝑛+1 (35) for every 𝑎=(𝑎₁,𝑎₂,…)∈𝑙^∞. We know that if 𝜇 is a Banach limit, then liminf_𝑛→∞𝑎_𝑛≤𝜇_𝑛𝑎_𝑛≤limsup_𝑛→∞𝑎_𝑛 (36) for every 𝑎=(𝑎₁,𝑎₂,…)∈𝑙^∞. So if 𝑎=(𝑎₁,𝑎₂,…),𝑏=(𝑏₁,𝑏₂,…)∈𝑙^∞, and 𝑎_𝑛→𝑐 (resp., 𝑎_𝑛−𝑏_𝑛→0), as 𝑛→∞, we have 𝜇_𝑛𝑎_𝑛=𝜇(𝑎)=𝑐resp.,𝜇_𝑛𝑎_𝑛=𝜇_𝑛𝑏_𝑛. (37)

Further, it is well known that there holds the following result.

Lemma 11 (see [43]). Let 𝐶 be a nonempty closed convex subset of a uniformly smooth Banach space 𝑋. Let {𝑥_𝑛} be a bounded sequence of 𝑋, let 𝜇 be a mean on 𝐍, and let 𝑧∈𝐶. Then 𝜇_𝑛𝑥_𝑛−𝑧²=min_𝑦∈𝐶𝜇_𝑛𝑥_𝑛−𝑦² (38) if and only if 𝜇_𝑛𝑦−𝑧,𝐽𝑥_𝑛−𝑧≤0,∀𝑦∈𝐶, (39) where 𝐽 is the normalized duality mapping of 𝑋.

Let 𝐶 be a nonempty closed convex subset of a Banach space 𝑋 and let 𝑇∶𝐶→𝐶 be a nonexpansive mapping with Fix(𝑇)≠∅. Let Ξ_𝐶 be the set of all contractive self-mappings on 𝐶. For 𝑡∈(0,1) and 𝑓∈Ξ_𝐶, let 𝑥_𝑡∈𝐶 be the unique fixed point of the contraction 𝑥↦𝑡𝑓(𝑥)+(1−𝑡)𝑇𝑥 on 𝐶; that is, 𝑥_𝑡=𝑡𝑓𝑥_𝑡+(1−𝑡)𝑇𝑥_𝑡. (40)

Lemma 12 (see [34]). Let 𝑋 be a uniformly smooth Banach space. Let 𝐶 be a nonempty closed convex subset of 𝑋, let 𝑇∶𝐶→𝐶 be a nonexpansive mapping with Fix(𝑇)≠∅, and let 𝑓∈Ξ_𝐶. Then the net {𝑥_𝑡} defined by 𝑥_𝑡=𝑡𝑓(𝑥_𝑡)+(1−𝑡)𝑇𝑥_𝑡 converges strongly to a point in Fix(𝑇). If one defines a mapping 𝑄∶Ξ_𝐶→Fix(𝑇) by 𝑄(𝑓)∶=𝑠−lim_𝑡→0𝑥_𝑡,for all 𝑓∈Ξ_𝐶, then 𝑄(𝑓) solves the VIP: 𝐼−𝑓𝑄𝑓,𝐽𝑄𝑓−𝑢≤0,∀𝑓∈Ξ_𝐶,𝑢∈Fix(𝑇). (41)

The following proposition will be used frequently throughout this paper.

Proposition 13 (see [11]). Let 𝑋 be a real smooth Banach space and let 𝐹∶𝑋→𝑋 be a mapping.(i)If 𝐹 is 𝜆-strictly pseudocontractive, then 𝐹 is Lipschitzian with constant 1+1/𝜆.(ii)If 𝐹 is 𝛿-strongly accretive and 𝜆-strictly pseudocontractive with 𝛿+𝜆>1, then 𝐼−𝐹 is contractive with constant (1−𝛿)/𝜆.(iii)If 𝐹 is 𝛿-strongly accretive and 𝜆-strictly pseudocontractive with 𝛿+𝜆>1, then, for any fixed number 𝜏∈(0,1), 𝐼−𝜏𝐹 is contractive with constant 1−𝜏(1−(1−𝛿)/𝜆).

3. Main Results

In this section, we introduce our modified hybrid steepest-descent schemes and show the strong convergence theorems. We will need the following useful lemmas in the sequel.

Lemma 14 (see [13, Lemma 2.8]). Let 𝐶 be a nonempty closed convex subset of a real 2-uniformly smooth Banach space 𝑋. Let the mapping 𝐵_𝑖∶𝐶→𝑋 be 𝛼_𝑖-inverse strongly accretive. Then, one has (𝐼−𝜇_𝑖𝐵_𝑖)𝑥−(𝐼−𝜇_𝑖𝐵_𝑖)𝑦²≤𝑥−𝑦²+2𝜇_𝑖𝜇_𝑖𝜅²−𝛼_𝑖𝐵_𝑖𝑥−𝐵_𝑖𝑦²,∀𝑥,𝑦∈𝐶, (42) for 𝑖=1,2, where 𝜇_𝑖>0. In particular, if 0<𝜇_𝑖≤𝛼_𝑖/𝜅², then 𝐼−𝜇_𝑖𝐵_𝑖 is nonexpansive for 𝑖=1,2.

Lemma 15 (see [13, Lemma 2.9]). Let 𝐶 be a nonempty closed convex subset of a real 2-uniformly smooth Banach space 𝑋. Let Π_𝐶 be a sunny nonexpansive retraction from 𝑋 onto 𝐶. Let the mapping 𝐵_𝑖∶𝐶→𝑋 be 𝛼_𝑖-inverse strongly accretive for 𝑖=1,2. Let 𝐺∶𝐶→𝐶 be the mapping defined by 𝐺𝑥=Π_𝐶Π_𝐶𝑥−𝜇₂𝐵₂𝑥−𝜇₁𝐵₁Π_𝐶𝑥−𝜇₂𝐵₂𝑥,∀𝑥∈𝐶. (43) If  0<𝜇_𝑖≤𝛼_𝑖/𝜅² for 𝑖=1,2, then 𝐺∶𝐶→𝐶 is nonexpansive.

Let 𝑋 be a real smooth Banach space and let 𝐴 be an 𝑚-accretive operator in 𝑋such that 𝐶=𝐷(𝐴) is convex. Let Π_𝐶 be a sunny nonexpansive retraction from 𝑋 onto 𝐶. Let 𝐹∶𝑋→𝑋 be 𝛿-strongly accretive and 𝜆-strictly pseudocontractive with 𝛿+𝜆>1. Let the mapping 𝐵_𝑖∶𝐶→𝑋 be 𝛼_𝑖-inverse strongly accretive for 𝑖=1,2 and let 𝑓∶𝑋→𝑋 be a contractive map with coefficient 𝜌∈(0,1). In this section, we will consider the problem of finding a point 𝑝∈Δ=𝐴⁻¹(0)∩Ω(≠∅), which is a unique solution of the VIP: 𝐼−𝑓𝑝,𝐽𝑝−𝑢≤0,∀𝑢∈Δ, (44) where Ω is the fixed point set of the mapping 𝐺=Π_𝐶(𝐼−𝜇₁𝐵₁)Π_𝐶(𝐼−𝜇₂𝐵₂) with 0<𝜇_𝑖≤𝛼_𝑖/𝜅² for 𝑖=1,2. For each 𝑡∈(0,1) and each integer 𝑛≥0, we choose a number 𝜃_𝑡∈[0,1) arbitrarily and then consider the following mapping Γ_𝑡,𝑛∶𝑋→𝑋 defined as Γ_𝑡,𝑛𝑥=𝑡𝑓(𝑥)+(1−𝑡)𝐺𝐽_𝑟𝑛𝑥−𝜃_𝑡𝐹𝐺𝐽_𝑟𝑛𝑥,∀𝑥∈𝑋. (45) Then, Γ_𝑡,𝑛∶𝑋→𝑋 is a contractive map. Indeed, utilizing Proposition 13(iii) and Lemma 15, we have for all 𝑥,𝑦∈𝑋Γ_𝑡,𝑛𝑥−Γ_𝑡,𝑛𝑦=𝑡𝑓(𝑥)+(1−𝑡)𝐺𝐽_𝑟𝑛𝑥−𝜃_𝑡𝐹𝐺𝐽_𝑟𝑛𝑥−𝑡𝑓𝑦−(1−𝑡)𝐺𝐽_𝑟𝑛𝑦−𝜃_𝑡𝐹𝐺𝐽_𝑟𝑛𝑦𝑓(𝑥)𝑦+(≤𝑡−𝑓1−𝑡)𝐼−𝜃_𝑡𝐹𝐺𝐽_𝑟𝑛𝑥−𝐼−𝜃_𝑡𝐹𝐺𝐽_𝑟𝑛𝑦≤𝑡𝜌𝑥−𝑦+(1−𝑡)1−𝜃_𝑡1−1−𝛿𝜆×𝐺𝐽_𝑟𝑛𝑥−𝐺𝐽_𝑟𝑛𝑦≤𝑡𝜌𝑥−𝑦+(1−𝑡)1−𝜃_t1−1−𝛿𝜆×𝐽_𝑟𝑛𝑥−𝐽_𝑟𝑛𝑦≤𝑡𝜌𝑥−𝑦+(1−𝑡)1−𝜃_𝑡1−1−𝛿𝜆+(=𝑡,𝑥−𝑦≤𝑡𝜌𝑥−𝑦1−𝑡)𝑥−𝑦1−1−𝜌𝑥−𝑦 (46) and hence Γ_𝑡,𝑛∶𝑋→𝑋 is contractive due to (1−𝜌)𝑡∈(0,1). By Banach’s Contraction Mapping Principle, there exists a unique fixed point 𝑥_𝑡,𝑛 of  Γ_𝑡,𝑛 in 𝑋; that is, 𝑥_𝑡,𝑛=𝑡𝑓𝑥_𝑡,𝑛+(1−𝑡)𝐺𝐽_𝑟𝑛𝑥_𝑡,𝑛−𝜃_𝑡𝐹𝐺𝐽_𝑟𝑛𝑥_𝑡,𝑛. (47)

Theorem 16. Let 𝑋 be a uniformly convex and 2-uniformly smooth Banach space and let 𝐴 be an 𝑚-accretive operator in 𝑋 such that 𝐶=𝐷(𝐴) is convex. Let Π_𝐶 be a sunny nonexpansive retraction from 𝑋 onto 𝐶. Let the mapping 𝐵_𝑖∶𝐶→𝑋 be 𝛼_𝑖-inverse strongly accretive for 𝑖=1,2, let 𝑓∶𝑋→𝐶 be a contractive map with coefficient 𝜌∈(0,1), and let 𝐹∶𝑋→𝑋 be 𝛿-strongly accretive and 𝜆-strictly pseudocontractive with 𝛿+𝜆>1. Assume that Δ=𝐴⁻¹(0)∩Ω≠∅, where Ω is the fixed point set of the mapping 𝐺=Π_𝐶(𝐼−𝜇₁𝐵₁)Π_𝐶(𝐼−𝜇₂𝐵₂) with 0<𝜇_𝑖<𝛼_𝑖/𝜅² for 𝑖=1,2. For each 𝑡∈(0,1) and each integer 𝑛≥0, let {𝑥_𝑡,𝑛} be defined by (47), where {𝑟_𝑛}⊂[𝜀,∞) for some 𝜀>0 and {𝜃_𝑡∶𝑡∈(0,1)}⊂[0,1) with lim_𝑡→0(𝜃_𝑡/𝑡)=0. Then as 𝑡→0,{𝑥_𝑡,𝑛} converges strongly to 𝑝∈Δ, which is a unique solution of the VIP (44).

Proof. First let us show that for some 𝑎∈(0,1),{𝑥_𝑡,𝑛∶𝑡∈(0,𝑎],𝑛≥0} is bounded. Indeed, since {𝜃_𝑡∶𝑡∈(0,1)}⊂[0,1) with lim_𝑡→0(𝜃_𝑡/𝑡)=0, there exists some 𝑎∈(0,1) such that0≤𝜃_𝑡/𝑡<1 for all 𝑡∈(0,𝑎].
Take 𝑝∈Δ. Then utilizing Proposition 13, we have 𝑥_𝑡,𝑛𝑓𝑥−𝑝≤𝑡_𝑡,𝑛+()×𝐺𝐽−𝑝1−𝑡_𝑟𝑛𝑥_𝑡,𝑛−𝜃_𝑡𝐹G𝐽_𝑟𝑛𝑥_𝑡,𝑛𝑥−𝑝≤𝑡𝜌_𝑡,𝑛𝑓𝑝+()×−𝑝+𝑡−𝑝1−𝑡𝐼−𝜃_𝑡𝐹𝐺𝐽_𝑟𝑛𝑥_𝑡,𝑛−𝐼−𝜃_𝑡𝐹𝐺𝐽_𝑟𝑛𝑝+𝐼−𝜃_𝑡𝐹𝐺𝐽_𝑟𝑛𝑝𝑥−𝑝≤𝑡𝜌_𝑡,𝑛𝑓𝑝+()×−𝑝+𝑡−𝑝1−𝑡1−𝜃_𝑡1−1−𝛿𝜆𝐺𝐽_𝑟𝑛𝑥_𝑡,𝑛−𝐺𝐽_𝑟𝑛𝑝+(1−𝑡)𝜃_𝑡𝐹𝑝≤𝑡𝜌𝑥_𝑡,𝑛𝑓𝑝+()×−𝑝+𝑡−𝑝1−𝑡1−𝜃_𝑡1−1−𝛿𝜆𝐽_𝑟𝑛𝑥_𝑡,𝑛−𝐽_𝑟𝑛𝑝+𝜃_𝑡𝐹𝑝≤𝑡𝜌𝑥_𝑡,𝑛𝑓𝑝+()×−𝑝+𝑡−𝑝1−𝑡1−𝜃_𝑡1−1−𝛿𝜆𝑥_𝑡,𝑛−𝑝+𝜃_𝑡𝐹𝑝≤𝑡𝜌𝑥_𝑡,𝑛−𝑝+𝑡𝑓𝑝−𝑝+(1−𝑡)𝑥_𝑡,𝑛−𝑝+𝜃_𝑡𝐹𝑝, (48) and, hence, for all 𝑡∈(0,𝑎]𝑥_𝑡,𝑛≤1−𝑝𝑓𝑝+𝜃1−𝜌−𝑝^𝑡𝑡𝐹𝑝≤1𝑓𝑝+𝐹𝑝.1−𝜌−𝑝 (49) Thus, this implies that {𝑥_𝑡,𝑛∶𝑡∈(0,𝑎],𝑛≥0} is bounded and so are {𝑓(𝑥_𝑡,𝑛)∶𝑡∈(0,𝑎],𝑛≥0}, {𝐽_𝑟𝑛𝑥_𝑡,𝑛∶𝑡∈(0,𝑎],𝑛≥0}, and {𝐺(𝐽_𝑟𝑛𝑥_𝑡,𝑛)∶𝑡∈(0,𝑎],𝑛≥0}.
Let us show that ‖𝐽_𝑟𝑛𝑥_𝑡,𝑛−𝐺(𝐽_𝑟𝑛𝑥_𝑡,𝑛)‖→0 as 𝑡→0.
Indeed, for simplicity we put 𝑞=Π_𝐶(𝐼−𝜇₂𝐵₂)𝑝, 𝑥_𝑡,𝑛=𝐽_𝑟𝑛𝑥_𝑡,𝑛, 𝑢_𝑡,𝑛=Π_𝐶(𝐼−𝜇₂𝐵₂)𝑥_𝑡,𝑛, and 𝑣_𝑡,𝑛=Π_𝐶(𝐼−𝜇₁𝐵₁)𝑢_𝑡,𝑛. Then it is clear that 𝑝=Π_𝐶(𝐼−𝜇₁𝐵₁)𝑞 and 𝑣_𝑡,𝑛=𝐺(𝑥_𝑡,𝑛)=𝐺(𝐽_𝑟𝑛𝑥_𝑡,𝑛). Taking into account 𝑥_𝑡,𝑛−𝑝=𝑡(𝑓(𝑥_𝑡,𝑛)−𝑝)+(1−𝑡)[𝐺(𝐽_𝑟𝑛𝑥_𝑡,𝑛)−𝜃_𝑡𝐹𝐺(𝐽_𝑟𝑛𝑥_𝑡,𝑛)−𝑝], we get 𝑥_𝑡,𝑛−𝑝²=𝑡𝑓𝑥_𝑡,𝑛−𝑝,𝐽𝑥_𝑡,𝑛+(𝐺𝐽−𝑝1−𝑡)_𝑟𝑛𝑥_𝑡,𝑛−𝑝,𝐽𝑥_𝑡,𝑛−()𝜃−𝑝1−𝑡_𝑡𝐹𝐺𝐽_𝑟𝑛𝑥_𝑡,𝑛,𝐽𝑥_𝑡,𝑛𝑓𝑥−𝑝≤𝑡_𝑡,𝑛−𝑓𝑝,𝐽𝑥_𝑡,𝑛𝑓𝑝𝑥−𝑝+𝑡−𝑝,𝐽_𝑡,𝑛+(𝐺𝐽−𝑝1−𝑡)_𝑟𝑛𝑥_𝑡,𝑛−𝑝𝐽𝑥_𝑡,𝑛+()𝜃−𝑝1−𝑡_𝑡𝐹𝐺𝐽_𝑟𝑛𝑥_𝑡,𝑛𝐽𝑥_𝑡,𝑛𝑥−𝑝≤𝜌𝑡_𝑡,𝑛−𝑝²+𝑡𝑓𝑝−𝑝𝑥_𝑡,𝑛+(𝑣−𝑝1−𝑡)_𝑡,𝑛−𝑝𝑥_𝑡,𝑛−𝑝+𝜃_𝑡𝐹𝑣_𝑡,𝑛𝑥_𝑡,𝑛−𝑝. (50) From Lemma 14 we have 𝑢_𝑡,𝑛−𝑞²=Π_𝐶(𝑥_𝑡,𝑛−𝜇₂𝐵₂𝑥_𝑡,𝑛)−Π_𝐶(𝑝−𝜇₂𝐵₂𝑝)²≤𝑥_𝑡,𝑛−𝑝−𝜇₂(𝐵₂𝑥_𝑡,𝑛−𝐵₂𝑝)²≤𝑥_𝑡,𝑛−𝑝²−2𝜇₂𝛼₂−𝜅²𝜇₂𝐵₂𝑥_𝑡,𝑛−𝐵₂𝑝²,𝑣_𝑡,𝑛−𝑝²=Π_𝐶(𝑢_𝑡,𝑛−𝜇₁𝐵₁𝑢_𝑡,𝑛)−Π_𝐶(𝑞−𝜇₁𝐵₁𝑞)²≤𝑢_𝑡,𝑛−𝑞−𝜇₁(𝐵₁𝑢_𝑡,𝑛−𝐵₁𝑞)²≤𝑢_𝑡,𝑛−𝑞²−2𝜇₁𝛼₁−𝜅²𝜇₁𝐵₁𝑢_𝑡,𝑛−𝐵₁𝑞². (51) From the last two inequalities, we obtain 𝑣_𝑡,𝑛−𝑝²≤𝑥_𝑡,𝑛−𝑝²−2𝜇₂𝛼₂−𝜅²𝜇₂𝐵₂𝑥_𝑡,𝑛−𝐵₂𝑝²−2𝜇₁𝛼₁−𝜅²𝜇₁𝐵₁𝑢_𝑡,𝑛−𝐵₁𝑞²≤𝑥_𝑡,𝑛−𝑝²−2𝜇₂𝛼₂−𝜅²𝜇₂𝐵₂𝑥_𝑡,𝑛−𝐵₂𝑝²−2𝜇₁𝛼₁−𝜅²𝜇₁𝐵₁𝑢_𝑡,𝑛−𝐵₁𝑞², (52) which, together with (50), implies that 𝑥_𝑡,𝑛−𝑝²≤𝜌𝑡𝑥_𝑡,𝑛−𝑝²+𝑡𝑓𝑝−𝑝𝑥_𝑡,𝑛+(𝑣−𝑝1−𝑡)_𝑡,𝑛−𝑝𝑥_𝑡,𝑛−𝑝+𝜃_𝑡𝐹𝑣_𝑡,𝑛𝑥_𝑡,𝑛𝑥−𝑝≤𝜌𝑡_𝑡,𝑛−𝑝²+𝑡𝑓𝑝−𝑝𝑥_𝑡,𝑛+()1−𝑝1−𝑡2𝑥_𝑡,𝑛−𝑝²+𝑣_𝑡,𝑛−𝑝²+𝜃_𝑡𝐹𝑣_𝑡,𝑛𝑥_𝑡,𝑛𝑥−𝑝≤𝜌𝑡_𝑡,𝑛−𝑝²+𝑡𝑓𝑝−𝑝𝑥_𝑡,𝑛+()1−𝑝1−𝑡2𝑥_𝑡,𝑛−𝑝²+𝑥_𝑡,𝑛−𝑝²−2𝜇₂𝛼₂−𝜅²𝜇₂𝐵₂𝑥_𝑡,𝑛−𝐵₂𝑝²−2𝜇₁𝛼₁−𝜅²𝜇₁𝐵₁𝑢_𝑡,𝑛−𝐵₁𝑞²+𝜃_𝑡𝐹𝑣_𝑡,𝑛𝑥_𝑡,𝑛=𝑥−𝑝1−𝑡1−𝜌_𝑡,𝑛−𝑝²+𝑡𝑓𝑝−𝑝𝑥_𝑡,𝑛−(𝜇−𝑝1−𝑡)₂𝛼₂−𝜅²𝜇₂𝐵₂𝑥_𝑡,𝑛−𝐵₂𝑝²+𝜇₁𝛼₁−𝜅²𝜇₁𝐵₁𝑢_𝑡,𝑛−𝐵₁𝑞²+𝜃_𝑡𝐹𝑣_𝑡,𝑛𝑥_𝑡,𝑛≤𝑥−𝑝_𝑡,𝑛−𝑝²+𝑡𝑓𝑝−𝑝𝑥_𝑡,𝑛−(𝜇−𝑝1−𝑡)₂𝛼₂−𝜅²𝜇₂𝐵₂𝑥_𝑡,𝑛−𝐵₂𝑝²+𝜇₁𝛼₁−𝜅²𝜇₁𝐵₁𝑢_𝑡,𝑛−𝐵₁𝑞²+𝜃_𝑡𝐹𝑣_𝑡,𝑛𝑥_𝑡,𝑛−𝑝. (53) So it immediately follows that (1−𝑡)𝜇₂𝛼₂−𝜅²𝜇₂𝐵₂𝑥_𝑡,𝑛−𝐵₂𝑝²+𝜇₁𝛼₁−𝜅²𝜇₁𝐵₁𝑢_𝑡,𝑛−𝐵₁𝑞²≤𝑡𝑓𝑝−𝑝𝑥_𝑡,𝑛−𝑝+𝜃_𝑡𝐹𝑣_𝑡,𝑛𝑥_𝑡,𝑛−𝑝. (54) Since 0<𝜇_𝑖<𝛼_𝑖/𝜅² for 𝑖=1,2, we have lim_𝑡→0𝐵₂𝑥_𝑡,𝑛−𝐵₂𝑝=0,lim_𝑡→0𝐵₁𝑢_𝑡,𝑛−𝐵₁𝑞=0. (55) Utilizing Proposition 3 and Lemma 6, we have 𝑢_𝑡,𝑛−𝑞²=Π_𝐶(𝑥_𝑡,𝑛−𝜇₂𝐵₂𝑥_𝑡,𝑛)−Π_𝐶𝑝−𝜇₂𝐵₂𝑝²≤𝑥_𝑡,𝑛−𝜇₂𝐵₂𝑥_𝑡,𝑛−𝑝−𝜇₂𝐵₂𝑝,𝐽𝑢_𝑡,𝑛=𝑥−𝑞_𝑡,𝑛−𝑝,𝐽𝑢_𝑡,𝑛−𝑞+𝜇₂𝐵₂𝑝−𝐵₂𝑥_𝑡,𝑛,𝐽𝑢_𝑡,𝑛≤1−𝑞2𝑥_𝑡,𝑛−𝑝²+𝑢_𝑡,𝑛−𝑞²−𝑔₁𝑥_𝑡,𝑛−𝑢_𝑡,𝑛−𝑝−𝑞+𝜇₂𝐵₂𝑝−𝐵₂𝑥_𝑡,𝑛𝑢_𝑡,𝑛−𝑞, (56) which implies that 𝑢_𝑡,𝑛−𝑞²≤𝑥_𝑡,𝑛−𝑝²−𝑔₁𝑥_𝑡,𝑛−𝑢_𝑡,𝑛−𝑝−𝑞+2𝜇₂𝐵₂𝑝−𝐵₂𝑥_𝑡,𝑛𝑢_𝑡,𝑛−𝑞. (57) In the same way, we derive 𝑣_𝑡,𝑛−𝑝²=Π_𝐶(𝑢_𝑡,𝑛−𝜇₁𝐵₁𝑢_𝑡,𝑛)−Π_𝐶(𝑞−𝜇₁𝐵₁𝑞)²≤𝑢_𝑡,𝑛−𝜇₁𝐵₁𝑢_𝑡,𝑛−𝑞−𝜇₁𝐵₁𝑞,𝐽𝑣_𝑡,𝑛=𝑢−𝑝_𝑡,𝑛−𝑞,𝐽𝑣_𝑡,𝑛−𝑝+𝜇₁𝐵₁𝑞−𝐵₁𝑢_𝑡,𝑛,𝐽𝑣_𝑡,𝑛≤1−𝑝2𝑢_𝑡,𝑛−𝑞²+𝑣_𝑡,𝑛−𝑝²−𝑔₂𝑢_𝑡,𝑛−𝑣_𝑡,𝑛+𝑝−𝑞+𝜇₁𝐵₁𝑞−𝐵₁𝑢_𝑡,𝑛𝑣_𝑡,𝑛−𝑝, (58) which implies that 𝑣_𝑡,𝑛−𝑝²≤𝑢_𝑡,𝑛−𝑞²−𝑔₂𝑢_𝑡,𝑛−𝑣_𝑡,𝑛+𝑝−𝑞+2𝜇₁𝐵₁𝑞−𝐵₁𝑢_𝑡,𝑛𝑣_𝑡,𝑛−𝑝. (59) Substituting (57) for (59), we get 𝑣_𝑡,𝑛−𝑝²≤𝑥_𝑡,𝑛−𝑝²−𝑔₁𝑥_𝑡,𝑛−𝑢_𝑡,𝑛−𝑝−𝑞−𝑔₂𝑢_𝑡,𝑛−𝑣_𝑡,𝑛+𝑝−𝑞+2𝜇₂𝐵₂𝑝−𝐵₂𝑥_𝑡,𝑛𝑢_𝑡,𝑛−𝑞+2𝜇₁𝐵₁𝑞−𝐵₁𝑢_𝑡,𝑛𝑣_𝑡,𝑛≤𝑥−𝑝_𝑡,𝑛−𝑝²−𝑔₁𝑥_𝑡,𝑛−𝑢_𝑡,𝑛−𝑝−𝑞−𝑔₂𝑢_𝑡,𝑛−𝑣_𝑡,𝑛+𝑝−𝑞+2𝜇₂𝐵₂𝑝−𝐵₂𝑥_𝑡,𝑛𝑢_𝑡,𝑛−𝑞+2𝜇₁𝐵₁𝑞−𝐵₁𝑢_𝑡,𝑛𝑣_𝑡,𝑛−𝑝, (60) which, together with (50), implies that 𝑥_𝑡,𝑛−𝑝²≤𝜌𝑡𝑥_𝑡,𝑛−𝑝²+𝑡𝑓𝑝−𝑝𝑥_t,𝑛+()1−𝑝1−𝑡2𝑥_𝑡,𝑛−𝑝²+𝑣_𝑡,𝑛−𝑝²+𝜃_𝑡𝐹𝑣_𝑡,𝑛𝑥_𝑡,𝑛𝑥−𝑝≤𝜌𝑡_𝑡,𝑛−𝑝²+𝑡𝑓𝑝−𝑝𝑥_𝑡,𝑛+()1−𝑝1−𝑡2𝑥_𝑡,𝑛−𝑝²+𝑥_𝑡,𝑛−𝑝²−𝑔₁𝑥_𝑡,𝑛−𝑢_𝑡,𝑛−𝑝−𝑞−𝑔₂𝑢_𝑡,𝑛−𝑣_𝑡,𝑛+𝑝−𝑞+2𝜇₂𝐵₂𝑝−𝐵₂𝑥_𝑡,𝑛𝑢_𝑡,𝑛−𝑞+2𝜇₁𝐵₁𝑞−𝐵₁𝑢_𝑡,𝑛𝑣_𝑡,𝑛−𝑝+𝜃_𝑡𝐹𝑣_𝑡,𝑛𝑥_𝑡,𝑛=𝑥−𝑝1−𝑡1−𝜌_𝑡,𝑛−𝑝²+𝑡𝑓𝑝−𝑝𝑥_𝑡,𝑛−()1−𝑝1−𝑡2𝑔₁𝑥_𝑡,𝑛−𝑢_𝑡,𝑛−𝑝−𝑞+𝑔₂𝑢_𝑡,𝑛−𝑣_𝑡,𝑛++(𝜇𝑝−𝑞1−𝑡)₂𝐵₂𝑝−𝐵₂𝑥_𝑡,𝑛𝑢_𝑡,𝑛−𝑞+𝜇₁𝐵₁𝑞−𝐵₁𝑢_𝑡,𝑛𝑣_𝑡,𝑛−𝑝+𝜃_𝑡𝐹𝑣_𝑡,𝑛𝑥_𝑡,𝑛≤𝑥−𝑝_𝑡,𝑛−𝑝²+𝑡𝑓𝑝−𝑝𝑥_𝑡,𝑛−()1−𝑝1−𝑡2𝑔₁𝑥_𝑡,𝑛−𝑢_𝑡,𝑛−𝑝−𝑞+𝑔₂𝑢_𝑡,𝑛−𝑣_𝑡,𝑛+𝑝−𝑞+𝜇₂𝐵₂𝑝−𝐵₂𝑥_𝑡,𝑛𝑢_𝑡,𝑛−𝑞+𝜇₁𝐵₁𝑞−𝐵₁𝑢_𝑡,𝑛𝑣_𝑡,𝑛−𝑝+𝜃_𝑡𝐹𝑣_𝑡,𝑛𝑥_𝑡,𝑛−𝑝. (61) So it immediately follows that ()11−𝑡2𝑔₁𝑥_𝑡,𝑛−𝑢_𝑡,𝑛−𝑝−𝑞+𝑔₂𝑢_𝑡,𝑛−𝑣_𝑡,𝑛+𝑓𝑝𝑥𝑝−𝑞≤𝑡−𝑝_𝑡,𝑛−𝑝+𝜇₂𝐵₂𝑝−𝐵₂𝑥_𝑡,𝑛𝑢_𝑡,𝑛−𝑞+𝜇₁𝐵₁𝑞−𝐵₁𝑢_𝑡,𝑛𝑣_𝑡,𝑛−𝑝+𝜃_𝑡𝐹𝑣_𝑡,𝑛𝑥_𝑡,𝑛−𝑝. (62) Hence, from (55) we conclude that lim_𝑡→0𝑔₁𝑥_𝑡,𝑛−𝑢_𝑡,𝑛−𝑝−𝑞=0,lim_𝑡→0𝑔₂𝑢_𝑡,𝑛−𝑣_𝑡,𝑛+𝑝−𝑞=0. (63) Utilizing the properties of 𝑔₁ and 𝑔₂, we get lim_𝑡→0𝑥_𝑡,𝑛−𝑢_𝑡,𝑛−𝑝−𝑞=0,lim_𝑡→0𝑢_𝑡,𝑛−𝑣_𝑡,𝑛+𝑝−𝑞=0, (64) which leads to 𝑥_𝑡,𝑛−𝑣_𝑡,𝑛≤𝑥_𝑡,𝑛−𝑢_𝑡,𝑛−+𝑢𝑝−𝑞_𝑡,𝑛−𝑣_𝑡,𝑛+𝑝−𝑞⟶0as𝑡⟶0. (65) That is, lim_𝑡→0𝐽_𝑟𝑛𝑥_𝑡,𝑛−𝐺𝐽_𝑟𝑛𝑥_𝑡,𝑛=lim_𝑡→0𝑥_𝑡,𝑛−𝑣_𝑡,𝑛=0. (66)
Note that Fix(𝐽_𝑟𝑛)=𝐴⁻¹(0) for all 𝑛≥0 and that {𝑥_𝑡,𝑛∶𝑡∈(0,𝑎],𝑛≥0} is bounded and so are {𝑓(𝑥_𝑡,𝑛)∶𝑡∈(0,𝑎],𝑛≥0}, {𝐽_𝑟𝑛𝑥_𝑡,𝑛∶𝑡∈(0,𝑎],𝑛≥0}, and {𝐺(𝐽_𝑟𝑛𝑥_𝑡,𝑛)∶𝑡∈(0,𝑎],𝑛≥0}. Hence, we have 𝑥_𝑡,𝑛−𝐺𝐽_𝑟𝑛𝑥_𝑡,𝑛=𝑡𝑓𝑥_𝑡,𝑛−𝐺𝐽_𝑟𝑛𝑥_𝑡,𝑛−()𝜃1−𝑡^𝑡𝑡𝐹𝐺𝐽_𝑟𝑛𝑥_𝑡,𝑛⟶0, (67) as 𝑡→0. Also, observe that 𝑥_𝑡,𝑛−𝐽_𝑟𝑛𝑥_𝑡,𝑛≤𝑥_𝑡,𝑛−𝐺𝐽_𝑟𝑛𝑥_𝑡,𝑛+𝐺𝐽_𝑟𝑛𝑥_𝑡,𝑛−𝐽_𝑟𝑛𝑥_𝑡,𝑛. (68) This, together with (66) and (67), implies that lim_𝑡→0𝑥_𝑡,𝑛−𝐽_𝑟𝑛𝑥_𝑡,𝑛=0. (69) Utilizing the nonexpansivity of 𝐺, we obtain from (67) and (69) that 𝑥_𝑡,𝑛−𝐺𝑥_𝑡,𝑛≤𝑥_𝑡,𝑛−𝐺𝐽_𝑟𝑛𝑥_𝑡,𝑛+𝐺𝐽_𝑟𝑛𝑥_𝑡,𝑛−𝐺𝑥_𝑡,𝑛≤𝑥_𝑡,𝑛−𝐺𝐽_𝑟𝑛𝑥_𝑡,𝑛+𝐽_𝑟𝑛𝑥_𝑡,𝑛−𝑥_𝑡,𝑛⟶0,as𝑡⟶0. (70) Since 𝑟_𝑛≥𝜀 for all 𝑛, utilizing Lemma 9 we have 𝑥_𝑡,𝑛−𝐽_𝜀𝑥_𝑡,𝑛≤2𝑥_𝑡,𝑛−𝐽_𝑟𝑛𝑥_𝑡,𝑛⟶0,as𝑡⟶0. (71)
For any integer 𝑛≥0, for simplicity put 𝑤_𝑡=𝑥_𝑡,𝑛 for all 𝑡∈(0,𝑎]. Now let {𝑡_𝑘} be a sequence in (0,𝑎] that converges to 0 as 𝑘→∞ and define a function 𝑔 on 𝐶 by 𝑔(𝑤)=𝜇_𝑘12𝑤_𝑡𝑘−𝑤²,∀𝑤∈𝐶, (72) where 𝜇 is a Banach limit. Define the set 𝐾∶=𝑤∈𝐶∶𝑔(𝑤)=min𝑔𝑦∶𝑦∈𝐶 (73) and the mapping 𝑊𝑥=(1−𝜃)𝐽_𝜀𝑥+𝜃𝐺𝑥,∀𝑥∈𝐶, (74) where 𝜃 is a constant in (0,1). Then by Lemma 10, we know that Fix(𝑊)=Fix(𝐽_𝜀)∩Fix(𝐺)=Δ. We observe that 𝑤_𝑡−𝑊𝑤_𝑡=(1−𝜃)𝑤_𝑡−𝐽_𝜀𝑤_𝑡+𝜃𝑤_𝑡−𝐺𝑤_𝑡≤(1−𝜃)𝑤_𝑡−𝐽_𝜀𝑤_𝑡+𝜃𝑤_𝑡−𝐺𝑤_𝑡. (75) So from (70) and (71) we obtain lim_𝑛→∞𝑤_𝑡−𝑊𝑤_𝑡=0. (76) Since 𝑋 is a uniformly smooth Banach space, 𝐾 is a nonempty bounded closed convex subset of 𝐶; for more details, see [43]. We claim that 𝐾 is also invariant under the nonexpansive mapping 𝑊. Indeed, noticing (76), we have for 𝑤∈𝐾𝑔(𝑊𝑤)=𝜇_𝑘12𝑤_𝑡𝑘−𝑊𝑤²=𝜇_𝑘12𝑊𝑤_𝑡𝑘−𝑊𝑤²≤𝜇_𝑘12𝑤_𝑡𝑘−𝑤²=𝑔(𝑤). (77) Since every nonempty closed bounded convex subset of uniformly smooth Banach space 𝑋 has the fixed point property for nonexpansive mappings and 𝑊 is a nonexpansive mapping of 𝐾,𝑊 has a fixed point in 𝐾, say 𝑝. Utilizing Lemma 11, we get 𝜇_𝑘𝑥−𝑝,𝐽𝑤_𝑡𝑘−𝑝≤0,∀𝑥∈𝐶. (78) Putting 𝑥=𝑓(𝑝) we have 𝜇_𝑘𝑓𝑝−𝑝,𝐽𝑤_𝑡𝑘−𝑝≤0,∀𝑥∈𝐶. (79) Since 𝑤_𝑡𝑘−𝑝=𝑡_𝑘(𝑓(𝑤_𝑡𝑘)−𝑝)+(1−𝑡_𝑘)[𝐺(𝐽_𝑟𝑛𝑤_𝑡𝑘)−𝜃_𝑡𝑘𝐹𝐺(𝐽_𝑟𝑛𝑤_𝑡𝑘)−𝑝], we get 𝑤_𝑡𝑘−𝑝²=𝑡_𝑘𝑓𝑤_𝑡𝑘−𝑝,𝐽𝑤_𝑡𝑘+−𝑝1−𝑡_𝑘𝐺𝐽_𝑟𝑛𝑤_𝑡𝑘−𝑝,𝐽𝑤_𝑡𝑘−−𝑝1−𝑡_𝑘𝜃_𝑡𝑘𝐹𝐺𝐽_𝑟𝑛𝑤_𝑡𝑘,𝐽𝑤_𝑡𝑘−𝑝≤𝑡_𝑘𝑓𝑤_𝑡𝑘−𝑓𝑝,𝐽𝑤_𝑡𝑘−𝑝+𝑡_𝑘𝑓𝑝−𝑝,𝐽𝑤_𝑡𝑘+−𝑝1−𝑡_𝑘𝐺𝐽_𝑟𝑛𝑤_𝑡𝑘−𝑝𝐽𝑤_𝑡𝑘+−𝑝1−𝑡_𝑘𝜃_𝑡𝑘𝐹𝐺𝐽_𝑟𝑛𝑤_𝑡𝑘𝐽𝑤_𝑡𝑘−𝑝≤𝜌𝑡_𝑘𝑤_𝑡𝑘−𝑝²+𝑡_𝑘𝑓𝑝−𝑝,𝐽𝑤_𝑡𝑘+−𝑝1−𝑡_𝑘𝑤_𝑡𝑘−𝑝²+𝜃_𝑡𝑘𝐹𝐺𝐽_𝑟𝑛𝑤_𝑡𝑘𝑤_𝑡𝑘−𝑝. (80) It follows that 𝑤_𝑡𝑘−𝑝²≤1𝑓𝑝𝑤1−𝜌−𝑝,𝐽_𝑡𝑘+𝜃−𝑝_𝑡𝑘𝑡_𝑘𝐹𝐺𝐽_𝑟𝑛𝑤_𝑡𝑘𝑤_𝑡𝑘−𝑝. (81) Since lim_𝑘→∞(𝜃_𝑡𝑘/𝑡_𝑘)=0, from (79) and the boundedness of sequences {𝐹𝐺(𝐽_𝑟𝑛𝑤_𝑡𝑘)},{𝑤_𝑡𝑘}, it follows that 𝜇_𝑘𝑤_𝑡𝑘−𝑝²≤1𝜇1−𝜌^𝑘𝑓𝑝−𝑝,𝐽𝑤_𝑡𝑘+𝜃−𝑝_𝑡𝑘𝑡_𝑘𝐹𝐺𝐽_𝑟𝑛𝑤_𝑡𝑘𝑤_𝑡𝑘=1−𝑝𝜇1−𝜌^𝑘𝑓𝑝−𝑝,𝐽𝑤_𝑡𝑘−𝑝+𝜇_𝑘𝜃_𝑡𝑘𝑡_𝑘𝐹𝐺𝐽_𝑟𝑛𝑤_𝑡𝑘𝑤_𝑡𝑘−𝑝≤0. (82) Therefore, it is known that for any sequence {𝑤_𝑡𝑘} in {𝑤_𝑡∶𝑡∈(0,𝑎]} there exists a subsequence which is still denoted by {𝑤_𝑡𝑘} that converges strongly to some fixed point 𝑝 of 𝑊. To prove that the net {𝑤_𝑡∶𝑡∈(0,𝑎]} converges strongly to 𝑝 as 𝑡→0, suppose that there exists another subsequence {𝑤_𝑠𝑘}⊂{𝑤_𝑡} such that 𝑤_𝑠𝑘→𝑞 as 𝑠_𝑘→0, and then we also have 𝑞∈Fix(𝑊)=𝐴⁻¹(0)∩Ω=∶Δ due to (76).
Since the sets {𝑤_𝑡−𝑢∶𝑡∈(0,𝑎]} and {𝑤_𝑡−𝑓(𝑤_𝑡)∶𝑡∈(0,𝑎]} are bounded and the duality map 𝐽 is single-valued and norm-to-norm uniformly continuous on bounded sets of uniformly smooth Banach space 𝑋, for any 𝑢∈Δ=𝐴⁻¹(0)∩Ω, from 𝑤_𝑠𝑘→𝑞  (𝑠_𝑘→0) we obtain 𝐼−𝑓𝑤_𝑠𝑘−𝐼−𝑓𝑞⟶0𝑠_𝑘,𝑤→0_𝑠𝑘−𝑓𝑤_𝑠𝑘,𝐽𝑤_𝑠𝑘−=𝑤−𝑢𝐼−𝑓𝑞,𝐽𝑞−𝑢𝐼−𝑓_𝑠𝑘−𝐼−𝑓𝑞,𝐽𝑤_𝑠𝑘+𝑤−𝑢𝐼−𝑓𝑞,𝐽_𝑠𝑘≤𝑤−𝑢−𝐽𝑞−𝑢𝐼−𝑓_𝑠𝑘−𝐼−𝑓𝑞𝑤_𝑠𝑘+𝑤−𝑢𝐼−𝑓𝑞,𝐽_𝑠𝑘−𝑢−𝐽𝑞−𝑢⟶0, (83) as 𝑠_𝑘→0. Therefore, 𝐼−𝑓𝑞,𝐽𝑞−𝑢=lim_𝑠𝑘→0𝑤_𝑠𝑘−𝑓𝑤_𝑠𝑘,J𝑤_𝑠𝑘−𝑢. (84) Since 𝑢∈Δ=𝐴⁻¹(0)∩Ω, 𝑥_𝑡,𝑛−[(1−𝑡)𝑢+𝑡𝑓(𝑥_𝑡,𝑛)]²=(1−𝑡)𝐺𝐽_𝑟𝑛𝑥_𝑡,𝑛+𝑡𝑓𝑥_𝑡,𝑛−(1−𝑡)𝑢−𝑡𝑓𝑥_𝑡,𝑛−(1−𝑡)𝜃_𝑡𝐹𝐺(𝐽_𝑟𝑛𝑥_𝑡,𝑛)²≤(1−𝑡)𝐺𝐽_𝑟𝑛𝑥_𝑡,𝑛−𝑢+(1−𝑡)𝜃_𝑡𝐹𝐺𝐽_𝑟𝑛𝑥_𝑡,𝑛²≤(1−𝑡)𝑥_𝑡,𝑛−𝑢+(1−𝑡)𝜃_𝑡𝐹𝐺𝐽_𝑟𝑛𝑥_𝑡,𝑛²=(1−𝑡)²𝑥_𝑡,𝑛−𝑢²+2(1−𝑡)²𝜃_𝑡𝑥_𝑡,𝑛−𝑢𝐹𝐺𝐽_𝑟𝑛𝑥_𝑡,𝑛+(1−𝑡)²𝜃²_𝑡𝐹𝐺(𝐽_𝑟𝑛𝑥_𝑡,𝑛)². (85) Utilizing Lemma 5, we have 𝑥_𝑡,𝑛−[(1−𝑡)𝑢+𝑡𝑓(𝑥_𝑡,𝑛)]²=(1−𝑡)(𝑥_𝑡,𝑛−𝑢)+𝑡(𝑥_𝑡,𝑛−𝑓(𝑥_𝑡,𝑛))²≥(1−𝑡)²𝑥_𝑡,𝑛−𝑢²+2𝑡(1−𝑡)𝑥_𝑡,𝑛−𝑓𝑥_𝑡,𝑛,𝐽𝑥_𝑡,𝑛−𝑢. (86) Consequently, from the last two inequalities we deduce that 2𝑡(1−𝑡)𝑥_𝑡,𝑛−𝑓𝑥_𝑡,𝑛,𝐽𝑥_𝑡,𝑛()−𝑢≤21−𝑡²𝜃_𝑡𝑥_𝑡,𝑛−𝑢𝐹𝐺𝐽_𝑟𝑛𝑥_𝑡,𝑛+(1−𝑡)²𝜃²_𝑡𝐹𝐺(𝐽_𝑟𝑛𝑥_𝑡,𝑛)², (87) and hence 𝑥_𝑡,𝑛−𝑓𝑥_𝑡,𝑛,𝐽𝑥_𝑡,𝑛≤()𝜃−𝑢1−𝑡^𝑡𝑡𝑥_𝑡,𝑛−𝑢𝐹𝐺𝐽_𝑟𝑛𝑥_𝑡,𝑛+()𝜃1−𝑡^2𝑡2𝑡𝐹𝐺(𝐽_𝑟𝑛𝑥_𝑡,𝑛)². (88)
Noticing (84), from lim_𝑡→0(𝜃_𝑡/𝑡)=0 and the boundedness of sequences {𝐹𝐺(𝐽_𝑟𝑛𝑥_𝑡,𝑛)},{𝑥_𝑡,𝑛}, we conclude that 𝐼−𝑓𝑞,𝐽𝑞−𝑢≤0. (89) Interchanging 𝑝 and 𝑢 leads to 𝐼−𝑓𝑞,𝐽𝑞−𝑝≤0. (90) Interchanging 𝑞 and 𝑝 leads to 𝐼−𝑓𝑝,𝐽𝑝−𝑞≤0. (91) This implies that 𝑞−𝑝−𝑓𝑞−𝑓𝑝,𝐽𝑝−𝑞≤0, (92) and hence 𝑞−𝑝²≤𝜌𝑞−𝑝². (93) Taking into account 𝜌∈(0,1), we obtain 𝑝=𝑞. Furthermore, by the careful analysis of the above proof, we can readily see that 𝑝 is also a unique solution of the VIP: 𝐼−𝑓𝑝,𝐽𝑝−𝑢≤0,∀𝑢∈Δ. (94) This completes the proof.

Remark 17. In the assertion of Theorem 16, “as 𝑡→0, {𝑥_𝑡,𝑛} converges strongly to 𝑝∈Δ;” this 𝑝 does not depend on 𝑛. Indeed, it is known that there holds the condition that {𝑟_𝑛}⊂[𝜀,∞) for some 𝜀>0. Moreover, in the proof of Theorem 16 it can be readily seen that 𝑝 is first found out as a fixed point of the nonexpansive self-mapping 𝑊 of 𝐾. This shows that 𝑝 depends on neither 𝑛 nor 𝑡.

Theorem 18. Let 𝑋 be a uniformly convex and 2-uniformly smooth Banach space and let 𝐴 be an 𝑚-accretive operator in 𝑋 such that 𝐶=𝐷(𝐴) is convex. Let Π_𝐶 be a sunny nonexpansive retraction from 𝑋 onto 𝐶. Let the mapping 𝐵_𝑖∶𝐶→𝑋 be 𝛼_𝑖-inverse strongly accretive for 𝑖=1,2, 𝑓∶𝑋→𝐶 be a contractive map with coefficient 𝜌∈(0,1), and let 𝐹∶𝑋→𝑋 be 𝛿-strongly accretive and 𝜆-strictly pseudocontractive with 𝛿+𝜆>1. Assume that Δ=𝐴⁻¹(0)∩Ω≠∅, where Ω is the fixed point set of the mapping 𝐺=Π_𝐶(𝐼−𝜇₁𝐵₁)Π_𝐶(𝐼−𝜇₂𝐵₂) with 0<𝜇_𝑖<𝛼_𝑖/𝜅² for 𝑖=1,2. Given sequences {𝜆_𝑛}^∞_𝑛=0,{𝜇_𝑛}^∞_𝑛=0 in [0,1], {𝛼_𝑛}^∞_𝑛=0,{𝛽_𝑛}^∞_𝑛=0 in (0,1], and {𝑟_𝑛}^∞_𝑛=0 in [𝜀,∞) for some 𝜀>0, suppose that there hold the following conditions:(i)lim_𝑛→∞𝛽_𝑛=0 and ∑^∞_𝑛=0𝛽_𝑛=∞,(ii)lim_𝑛→∞(𝜆_𝑛𝜇_𝑛)/𝛽_𝑛=0,(iii){𝛼_𝑛}⊂[𝑎,𝑏] for some 𝑎,𝑏∈(0,1),(iv)∑^∞_𝑛=0|𝛼_𝑛+1−𝛼_𝑛|<∞, ∑^∞_𝑛=0|𝛽_𝑛+1−𝛽_𝑛|<∞, ∑^∞_𝑛=0|𝜆_𝑛+1−𝜆_𝑛|<∞, ∑^∞_𝑛=0|𝜇_𝑛+1−𝜇_𝑛|<∞ and ∑^∞_𝑛=0|𝑟_𝑛+1−𝑟_𝑛|<∞.
Then for any given point 𝑥₀∈𝑋, the sequence {𝑥_𝑛} generated by 𝑦_𝑛=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝐺𝐽_𝑟𝑛𝑥_𝑛,𝑥_𝑛+1=𝛽_𝑛𝑓𝑥_𝑛+1−𝛽_𝑛𝐺𝐽_𝑟𝑛𝑦_𝑛−𝜆_𝑛𝜇_𝑛𝐹𝐺𝐽_𝑟𝑛𝑦_𝑛,∀𝑛≥0, (95) converges strongly to 𝑝∈Δ, which is a unique solution of the VIP (44).

Proof. First, let us show that {𝑥_𝑛} is bounded. Indeed, taking a fixed 𝑢∈Δ arbitrarily, we have 𝑦_𝑛−𝑢=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝐺𝐽_𝑟𝑛𝑥_𝑛−𝑢≤𝛼_𝑛𝑥_𝑛−𝑢+1−𝛼_𝑛𝐺𝐽_𝑟𝑛𝑥_𝑛−𝑢≤𝛼_𝑛𝑥_𝑛−𝑢+1−𝛼_𝑛𝐽_𝑟𝑛𝑥_𝑛−𝑢≤𝛼_𝑛𝑥_𝑛−𝑢+1−𝛼_𝑛𝑥_𝑛=𝑥−𝑢_𝑛−𝑢. (96) So ‖𝑦_𝑛−𝑢‖≤‖𝑥_𝑛−𝑢‖ for all 𝑛≥0. Thus, by Proposition 13(iii), we have𝑥_𝑛+1=𝛽−𝑢_𝑛𝑓𝑥_𝑛+1−𝛽_𝑛𝐺𝐽_𝑟𝑛𝑦_𝑛−𝜆_𝑛𝜇_𝑛𝐹𝐺𝐽_𝑟𝑛𝑦_𝑛−𝑢≤𝛽_𝑛𝑓𝑥_𝑛−𝑢+1−𝛽_𝑛×𝜆_𝑛𝐼−𝜇_𝑛𝐹𝐺𝐽_𝑟𝑛𝑦_𝑛+1−𝜆_𝑛𝐺𝐽_𝑟𝑛𝑦_𝑛−𝑢≤𝛽_𝑛𝜌𝑥_𝑛−𝑢+𝛽_𝑛𝑓(𝑢)+−𝑢1−𝛽_𝑛𝜆_𝑛𝐼−𝜇_𝑛𝐹𝐺𝐽_𝑟𝑛𝑦_𝑛+−𝑢1−𝜆_𝑛𝐺𝐽_𝑟𝑛𝑦_𝑛−𝑢≤𝛽_𝑛𝜌𝑥_𝑛−𝑢+𝛽_𝑛𝑓(𝑢)+−𝑢1−𝛽_𝑛𝜆_𝑛𝐼−𝜇_𝑛𝐹𝐺𝐽_𝑟𝑛𝑦_𝑛−𝐼−𝜇_𝑛𝐹𝑢+𝜆_𝑛𝐼−𝜇_𝑛𝐹+𝑢−𝑢1−𝜆_𝑛𝐺𝐽_𝑟𝑛𝑦_𝑛−𝑢≤𝛽_𝑛𝜌𝑥_𝑛−𝑢+𝛽_𝑛𝑓(𝑢)+−𝑢1−𝛽_𝑛𝜆_𝑛1−𝜇_𝑛1−1−𝛿𝜆𝐺𝐽_𝑟𝑛𝑦_𝑛−𝑢+𝜆_𝑛𝜇_𝑛‖𝐹𝑢‖+1−𝜆_𝑛𝐺𝐽_𝑟𝑛𝑦_𝑛−𝑢=𝛽_𝑛𝜌𝑥_𝑛−𝑢+𝛽_𝑛𝑓(𝑢)+−𝑢1−𝛽_𝑛1−𝜆_𝑛𝜇_𝑛1−1−𝛿𝜆𝐺𝐽_𝑟𝑛𝑦_𝑛−𝑢+𝜆_𝑛𝜇_𝑛‖𝐹𝑢‖=𝛽_𝑛𝜌𝑥_𝑛−𝑢+𝛽_𝑛𝑓(𝑢)+−𝑢1−𝛽_𝑛1−𝜆_𝑛𝜇_𝑛1−1−𝛿𝜆𝐺𝐽_𝑟𝑛𝑦_𝑛−𝑢+𝜆_𝑛𝜇_𝑛1−1−𝛿𝜆1−1−𝛿𝜆⁻¹‖𝐹𝑢‖≤𝛽_𝑛𝜌𝑥_𝑛−𝑢+𝛽_𝑛𝑓(𝑢)+−𝑢1−𝛽_𝑛max𝐺𝐽_𝑟𝑛𝑦_𝑛−𝑢,1−1−𝛿𝜆⁻¹‖𝐹𝑢‖≤𝛽_𝑛𝜌𝑥_𝑛−𝑢+𝛽_𝑛𝑓(𝑢)+−𝑢1−𝛽_𝑛max𝑦_𝑛−𝑢,1−1−𝛿𝜆⁻¹‖𝐹𝑢‖≤𝛽_𝑛𝜌𝑥_𝑛−𝑢+𝛽_𝑛𝑓(𝑢)+−𝑢1−𝛽_𝑛max𝑥_𝑛−𝑢,1−1−𝛿𝜆⁻¹‖𝐹𝑢‖≤𝛽_𝑛𝑓(𝑢)−𝑢+1−1−𝜌𝛽_𝑛×max𝑥_𝑛−𝑢,1−1−𝛿𝜆⁻¹‖=𝛽𝐹𝑢‖1−𝜌_𝑛𝑓(𝑢)−𝑢+𝛽1−𝜌1−1−𝜌_𝑛×max𝑥_𝑛−𝑢,1−1−𝛿𝜆⁻¹‖𝑥𝐹𝑢‖≤max_𝑛,𝑓(𝑢)−𝑢−𝑢,1−𝜌1−1−𝛿𝜆⁻¹‖𝐹𝑢‖. (97)By induction, 𝑥_𝑛𝑥−𝑢≤max₀,𝑓(𝑢)−𝑢−𝑢,1−𝜌1−1−𝛿𝜆⁻¹‖𝐹𝑢‖,∀𝑛≥0. (98) Thus, {𝑥_𝑛} is bounded and so is {𝑦_𝑛}. Because 𝐺 and 𝐽_𝑟𝑛 are nonexpansive for all 𝑛≥0, 𝑓 is contractive, and 𝐹 is Lipschitzian, {𝐽_𝑟𝑛𝑥_𝑛}, {𝐽_𝑟𝑛𝑦_𝑛}, {𝐺(𝐽_𝑟𝑛𝑥_𝑛)}, {𝐺(𝐽_𝑟𝑛𝑦_𝑛)}, {𝑓(𝑥_𝑛)}, and {𝐹𝐺(𝐽_𝑟𝑛𝑦_𝑛)} are bounded. From conditions (i), (ii) we have 𝑥_𝑛+1−𝐺𝐽_𝑟𝑛𝑦_𝑛≤𝛽_𝑛𝑓𝑥_𝑛−𝐺𝐽_𝑟𝑛𝑦_𝑛+1−𝛽_𝑛𝐺𝐽_𝑟𝑛𝑦_𝑛−𝜆_𝑛𝜇_𝑛𝐹𝐺𝐽_𝑟𝑛𝑦_𝑛−𝐺𝐽_𝑟𝑛𝑦_𝑛=𝛽_𝑛𝑓𝑥_𝑛−𝐺𝐽_𝑟𝑛𝑦_𝑛+1−𝛽_𝑛𝜆_𝑛𝜇_𝑛𝐹𝐺𝐽_𝑟𝑛𝑦_𝑛⟶0as𝑛⟶∞. (99)
Now, we claim that 𝑥_𝑛+1−𝑥_𝑛⟶0as𝑛⟶∞. (100) In order to prove (100), we estimate ‖𝑥_𝑛+1−𝑥_𝑛‖ first. From (95) we have 𝑦_𝑛=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝐺𝐽_𝑟𝑛𝑥_𝑛,𝑦_𝑛−1=𝛼_𝑛−1𝑥_𝑛−1+1−𝛼_𝑛−1𝐺𝐽_𝑟𝑛−1𝑥_𝑛−1. (101) Simple calculations show that 𝑦_𝑛−𝑦_𝑛−1=1−𝛼_𝑛𝐺𝐽_𝑟𝑛𝑥_𝑛−𝐺𝐽_𝑟𝑛−1𝑥_𝑛−1+𝛼_𝑛𝑥_𝑛−𝑥_𝑛−1+𝑥_𝑛−1−𝐺𝐽_𝑟𝑛−1𝑥_𝑛−1𝛼_𝑛−𝛼_𝑛−1. (102) It follows that 𝑦_𝑛−𝑦_𝑛−1≤1−𝛼_𝑛𝐺𝐽_𝑟𝑛𝑥_𝑛−𝐺𝐽_𝑟𝑛−1𝑥_𝑛−1+𝛼_𝑛𝑥_𝑛−𝑥_𝑛−1+𝑥_𝑛−1−𝐺𝐽_𝑟𝑛−1𝑥_𝑛−1𝛼_𝑛−𝛼_𝑛−1≤1−𝛼_𝑛𝐽_𝑟𝑛𝑥_𝑛−𝐽_𝑟𝑛−1𝑥_𝑛−1+𝛼_𝑛𝑥_𝑛−𝑥_𝑛−1+𝑥_𝑛−1−𝐺𝐽_𝑟𝑛−1𝑥_𝑛−1𝛼_𝑛−𝛼_𝑛−1. (103)
On the other hand, if 𝑟_𝑛−1≤𝑟_𝑛, using the resolvent identity in Lemma 8𝐽_𝑟𝑛𝑥_𝑛=𝐽_𝑟𝑛−1𝑟_𝑛−1𝑟_𝑛𝑥_𝑛+𝑟1−^𝑛−1𝑟_𝑛𝐽_𝑟𝑛𝑥_𝑛, (104) we get 𝐽_𝑟𝑛𝑥_𝑛−𝐽_𝑟𝑛−1𝐺_𝑛−1=𝐽_𝑟𝑛−1𝑟_𝑛−1𝑟_𝑛𝑥_𝑛+𝑟1−^𝑛−1𝑟_𝑛𝐽_𝑟𝑛𝑥_𝑛−𝐽_𝑟𝑛−1𝑥_𝑛−1≤𝑟_𝑛−1𝑟_𝑛𝑥_𝑛−𝑥_𝑛−1+𝑟1−^𝑛−1𝑟_𝑛𝐽_𝑟𝑛𝑥_𝑛−𝑥_𝑛−1≤𝑥_𝑛−𝑥_𝑛−1+𝑟_𝑛−𝑟_𝑛−1𝑟_𝑛𝐽_𝑟𝑛𝑥_𝑛−𝑥_𝑛−1≤𝑥_𝑛−𝑥_𝑛−1+1𝜀𝑟_𝑛−𝑟_𝑛−1𝐽_𝑟𝑛𝑥_𝑛−𝑥_𝑛−1. (105) If 𝑟_𝑛≤𝑟_𝑛−1, it is easy to see that 𝐽_𝑟𝑛𝑥_𝑛−𝐽_𝑟𝑛−1𝑥_𝑛−1≤𝑥_𝑛−1−𝑥_𝑛+1𝜀𝑟_𝑛−1−𝑟_𝑛𝐽_𝑟𝑛−1𝑥_𝑛−1−𝑥_𝑛. (106) So combining the above cases we obtain 𝐽_𝑟𝑛𝑥_𝑛−𝐽_𝑟𝑛−1𝑥_𝑛−1≤𝑥_𝑛−1−𝑥_𝑛+𝑟_𝑛−1−𝑟_𝑛𝜀×sup_𝑛≥1𝐽_𝑟𝑛𝑥_𝑛−𝑥_𝑛−1+𝐽_𝑟𝑛−1𝑥_𝑛−1−𝑥_𝑛,∀𝑛≥1. (107) In the similar way we can derive 𝐽_𝑟𝑛𝑦_𝑛−𝐽_𝑟𝑛−1𝑦_𝑛−1≤𝑦_𝑛−1−𝑦_𝑛+𝑟_𝑛−1−𝑟_𝑛𝜀×sup_𝑛≥1𝐽_𝑟𝑛𝑦_𝑛−𝑦_𝑛−1+𝐽_𝑟𝑛−1𝑦_𝑛−1−𝑦_𝑛,∀𝑛≥1. (108) Therefore, we have 𝐽_𝑟𝑛𝑥_n−𝐽_𝑟𝑛−1𝑥_𝑛−1≤𝑥_𝑛−1−𝑥_𝑛+𝑟_𝑛−1−𝑟_𝑛𝑀₀,𝐽_𝑟𝑛𝑦_𝑛−𝐽_𝑟𝑛−1𝑦_𝑛−1≤𝑦_𝑛−1−𝑦_𝑛+𝑟_𝑛−1−𝑟_𝑛𝑀₀, (109) for all 𝑛≥1, where sup_𝑛≥1{(1/𝜀)(‖𝐽_𝑟𝑛𝑥_𝑛−𝑥_𝑛−1‖+‖𝐽_𝑟𝑛−1𝑥_𝑛−1−𝑥_𝑛‖)}≤𝑀₀ and sup_𝑛≥1{(1/𝜀)(‖𝐽_𝑟𝑛𝑦_𝑛−𝑦_𝑛−1‖+‖𝐽_𝑟𝑛−1𝑦_𝑛−1−𝑦_𝑛‖)}≤M₀ for some 𝑀₀>0.
Substituting (109) for (103), we obtain 𝑦_𝑛−𝑦_𝑛−1≤1−𝛼_𝑛𝐽_𝑟𝑛𝑥_𝑛−𝐽_𝑟𝑛−1𝑥_𝑛−1+𝛼_𝑛𝑥_𝑛−𝑥_𝑛−1+𝑥_𝑛−1−𝐺𝐽_𝑟𝑛−1𝑥_𝑛−1𝛼_𝑛−𝛼_𝑛−1≤1−𝛼_𝑛𝑥_𝑛−𝑥_𝑛−1+𝑟_𝑛−𝑟_𝑛−1𝑀₀+𝛼_𝑛𝑥_𝑛−𝑥_𝑛−1+𝑥_𝑛−1−𝐺𝐽_𝑟𝑛−1𝑥_𝑛−1𝛼_𝑛−𝛼_𝑛−1≤𝑥_𝑛−𝑥_𝑛−1+𝑀₁𝑟_𝑛−𝑟_𝑛−1+𝛼_𝑛−𝛼_𝑛−1, (110) where sup_𝑛≥0{𝑀₀,‖𝑥_𝑛−𝐺(𝐽_𝑟𝑛𝑥_𝑛)‖}≤𝑀₁ for some constant 𝑀₁>0.
In the meantime, it follows from (95) that 𝑥_𝑛+1=𝛽_𝑛𝑓𝑥_𝑛+1−𝛽_𝑛𝐺𝐽_𝑟𝑛𝑦_𝑛−𝜆_𝑛𝜇_𝑛𝐹𝐺𝐽_𝑟𝑛𝑦_𝑛,𝑥_𝑛=𝛽_𝑛−1𝑓𝑥_𝑛−1+1−𝛽_𝑛−1×𝐺𝐽_𝑟𝑛−1𝑦_𝑛−1−𝜆_𝑛−1𝜇_𝑛−1𝐹𝐺𝐽_𝑟𝑛−1𝑦_𝑛−1. (111) Simple calculations show that 𝑥_𝑛+1−𝑥_𝑛=𝛽_𝑛−𝛽_𝑛−1𝑓𝑥_𝑛−1+𝛽_𝑛𝑓𝑥_𝑛−𝑓𝑥_𝑛−1+𝛽_𝑛−1−𝛽_𝑛𝐼−𝜆_𝑛−1𝜇_𝑛−1𝐹𝐺𝐽_𝑟𝑛−1𝑦_𝑛−1+1−𝛽_𝑛𝐼−𝜆_𝑛𝜇_𝑛𝐹𝐺𝐽_𝑟𝑛𝑦_𝑛−𝐼−𝜆_𝑛−1𝜇_𝑛−1𝐹𝐺𝐽_𝑟𝑛−1𝑦_𝑛−1=𝛽_𝑛−𝛽_𝑛−1𝑓𝑥_𝑛−1+𝛽_𝑛𝑓𝑥_𝑛−𝑓𝑥_𝑛−1+𝛽_𝑛−1−𝛽_𝑛𝐼−𝜆_𝑛−1𝜇_𝑛−1𝐹𝐺𝐽_𝑟𝑛−1𝑦_𝑛−1+1−𝛽_𝑛𝐼−𝜆_𝑛𝜇_𝑛𝐹𝐺𝐽_𝑟𝑛𝑦_𝑛−𝐼−𝜆_𝑛𝜇_𝑛𝐹𝐺𝐽_𝑟𝑛−1𝑦_𝑛−1+𝜆_𝑛−1𝜇_𝑛−1−𝜆_𝑛𝜇_𝑛𝐹𝐺𝐽_𝑟𝑛−1𝑦_𝑛−1. (112) It follows from Proposition 13 (iii) and (109) that 𝑥_𝑛+1−𝑥_𝑛≤𝛽_𝑛−𝛽_𝑛−1𝑓𝑥_𝑛−1+𝛽_𝑛𝑓𝑥_𝑛−𝑓𝑥_𝑛−1+𝛽_𝑛−1−𝛽_𝑛𝐼−𝜆_𝑛−1𝜇_𝑛−1𝐹𝐺𝐽_𝑟𝑛−1𝑦_𝑛−1+1−𝛽_𝑛×𝐼−𝜆_𝑛𝜇_𝑛𝐹𝐺𝐽_𝑟n𝑦_𝑛−𝐼−𝜆_𝑛𝜇_𝑛𝐹𝐺𝐽_𝑟𝑛−1𝑦_𝑛−1+𝜆_𝑛−1𝜇_𝑛−1−𝜆_𝑛𝜇_𝑛𝐹𝐺𝐽_𝑟𝑛−1𝑦_𝑛−1≤𝛽_𝑛−𝛽_𝑛−1𝑓𝑥_𝑛−1+𝛽_𝑛𝜌𝑥_𝑛−𝑥_𝑛−1+𝛽_𝑛−1−𝛽_𝑛𝐼−𝜆_𝑛−1𝜇_𝑛−1𝐹𝐺𝐽_𝑟𝑛−1𝑦_𝑛−1+1−𝛽_𝑛1−𝜆_𝑛𝜇_𝑛1−1−𝛿𝜆×𝐺𝐽_𝑟𝑛𝑦_𝑛−𝐺𝐽_𝑟𝑛−1𝑦_𝑛−1+𝜆_𝑛𝜇_𝑛−𝜆_𝑛−1𝜇_𝑛−1𝐹𝐺𝐽_𝑟𝑛−1𝑦_𝑛−1≤𝛽_𝑛−𝛽_𝑛−1𝑓𝑥_𝑛−1+𝛽_𝑛𝜌𝑥_𝑛−𝑥_𝑛−1+𝛽_𝑛−1−𝛽_𝑛𝐼−𝜆_𝑛−1𝜇_𝑛−1𝐹𝐺𝐽_𝑟𝑛−1𝑦_𝑛−1+1−𝛽_𝑛1−𝜆_𝑛𝜇_𝑛1−1−𝛿𝜆×𝐽_𝑟𝑛𝑦_𝑛−𝐽_𝑟𝑛−1𝑦_𝑛−1+𝜆_𝑛𝜇_𝑛−𝜆_𝑛−1𝜇_𝑛−1𝐹𝐺𝐽_𝑟𝑛−1𝑦_𝑛−1≤𝛽_𝑛−𝛽_𝑛−1𝑓𝑥_𝑛−1+𝛽_𝑛𝜌𝑥_𝑛−𝑥_𝑛−1+𝛽_𝑛−1−𝛽_𝑛𝐼−𝜆_𝑛−1𝜇_𝑛−1𝐹𝐺𝐽_𝑟𝑛−1𝑦_𝑛−1+1−𝛽_𝑛1−𝜆_𝑛𝜇_𝑛1−1−𝛿𝜆×𝑦_𝑛−1−𝑦_𝑛+𝑟_𝑛−1−𝑟_𝑛𝑀₀+𝜆_𝑛𝜇_𝑛−𝜆_𝑛−1𝜇_𝑛−1𝐹𝐺𝐽_𝑟𝑛−1𝑦_𝑛−1≤𝛽_𝑛𝜌𝑥_𝑛−𝑥_𝑛−1+𝛽_𝑛−𝛽_𝑛−1×𝑓𝑥_𝑛−1+𝐼−𝜆_𝑛−1𝜇_𝑛−1𝐹𝐺𝐽_𝑟𝑛−1𝑦_𝑛−1+1−𝛽_𝑛𝑦_𝑛−1−𝑦_𝑛+𝑟_𝑛−1−𝑟_𝑛𝑀₀+𝜆_𝑛𝜇_𝑛−𝜆_𝑛−1𝜇_𝑛−1𝐹𝐺𝐽_𝑟𝑛−1𝑦_𝑛−1≤𝛽_𝑛𝜌𝑥_𝑛−𝑥_𝑛−1+𝛽_𝑛−𝛽_𝑛−1𝑀₂+1−𝛽_𝑛𝑦_𝑛−1−𝑦_𝑛+𝑟_𝑛−1−𝑟_𝑛𝑀₂+𝜆_𝑛𝜇_𝑛−𝜆_𝑛−1𝜇_𝑛−1𝑀₂=𝛽_𝑛𝜌𝑥_𝑛−𝑥_𝑛−1+1−𝛽_𝑛𝑦_𝑛−𝑦_𝑛−1+𝑀₂𝛽_𝑛−𝛽_𝑛−1+𝜆_𝑛𝜇_𝑛−𝜆_𝑛−1𝜇_𝑛−1+𝑟_𝑛−𝑟_𝑛−1, (113) where sup_𝑛≥0{‖𝑓(𝑥_𝑛)‖+‖(𝐼−𝜆_𝑛𝜇_𝑛𝐹)𝐺(𝐽_𝑟𝑛𝑦_𝑛)‖, ‖𝐹𝐺(𝐽_𝑟𝑛𝑦_𝑛)‖,𝑀₀,𝑀₁}≤𝑀₂ for some 𝑀₂>0.
Substituting (110) for (113), we get 𝑥_𝑛+1−𝑥_𝑛≤𝛽_𝑛𝜌𝑥_𝑛−𝑥_𝑛−1+1−𝛽_𝑛𝑦_𝑛−𝑦_𝑛−1+𝑀₂𝛽_𝑛−𝛽_𝑛−1+𝜆_𝑛𝜇_𝑛−𝜆_𝑛−1𝜇_𝑛−1+𝑟_𝑛−𝑟_𝑛−1≤𝛽_𝑛𝜌𝑥_𝑛−𝑥_𝑛−1+1−𝛽_𝑛×𝑥_𝑛−𝑥_𝑛−1+𝑀₁𝑟_𝑛−𝑟_𝑛−1+𝛼_𝑛−𝛼_𝑛−1+𝑀₂𝛽_𝑛−𝛽_𝑛−1+𝜆_𝑛𝜇_𝑛−𝜆_𝑛−1𝜇_𝑛−1+𝑟_𝑛−𝑟_𝑛−1≤1−1−𝜌𝛽_𝑛𝑥_𝑛−𝑥_𝑛−1+2𝑀₂𝛼_𝑛−𝛼_𝑛−1+𝛽_𝑛−𝛽_𝑛−1+𝜆_𝑛−𝜆_𝑛−1+𝜇_𝑛−𝜇_𝑛−1+𝑟_𝑛−𝑟_𝑛−1. (114) Since it follows from conditions (i) and (iv) that ∑^∞_𝑛=0(1−𝜌)𝛽_𝑛=∞ and ^∞_𝑛=02𝑀₂𝛼_𝑛−𝛼_𝑛−1+𝛽_𝑛−𝛽_𝑛−1+𝜆_𝑛−𝜆_𝑛−1+𝜇_𝑛−𝜇_𝑛−1+𝑟_𝑛−𝑟_𝑛−1<∞, (115) Lemma 4 is applicable to (114) and we obtain lim_𝑛→∞𝑥_𝑛+1−𝑥_𝑛=0. (116) By condition (iii) and (95), we have 𝑦_𝑛−𝑥_𝑛=1−𝛼_𝑛𝐺𝐽_𝑟𝑛𝑥_𝑛−𝑥_𝑛≤(1−𝑎)𝐺𝐽_𝑟𝑛𝑥_𝑛−𝐺𝐽_𝑟𝑛𝑦_𝑛+𝐺𝐽_𝑟𝑛𝑦_𝑛−𝑥_𝑛+1+𝑥_𝑛+1−𝑥_𝑛≤(1−𝑎)𝑥_𝑛−𝑦_𝑛+𝐺𝐽_𝑟𝑛𝑦_𝑛−𝑥_𝑛+1+𝑥_𝑛+1−𝑥_𝑛, (117) which implies that 𝑦_𝑛−𝑥_𝑛≤1−𝑎𝑎𝐺𝐽_𝑟𝑛𝑦_𝑛−𝑥_𝑛+1+𝑥_𝑛+1−𝑥_𝑛. (118) This together with (99)-(100) implies that lim_𝑛→∞𝑥_𝑛−𝑦_𝑛=0. (119) So we obtain 𝑥_𝑛−𝐺𝐽_𝑟𝑛𝑥_𝑛≤𝑥_𝑛−𝑦_𝑛+𝑦_𝑛−𝐺𝐽_𝑟𝑛𝑥_𝑛≤𝑥_𝑛−𝑦_𝑛+𝛼_𝑛𝑥_𝑛−𝐺𝐽_𝑟𝑛𝑥_𝑛≤𝑥_𝑛−𝑦_𝑛+𝑏𝑥_𝑛−𝐺𝐽_𝑟𝑛𝑥_𝑛, (120) which implies that 𝑥_𝑛−𝐺𝐽_𝑟𝑛𝑥_𝑛≤1𝑥1−𝑏^𝑛−𝑦_𝑛, (121) and hence lim_𝑛→∞𝑥_𝑛−𝐺𝐽_𝑟𝑛𝑥_𝑛=0. (122)
Next let us show that lim_𝑛→∞‖𝑥_𝑛−𝐽_𝑟𝑛𝑥_𝑛‖=0 and lim_𝑛→∞‖𝑥_𝑛−𝐺𝑥_𝑛‖=0.
Indeed, for simplicity, put 𝑣=Π_𝐶(𝑢−𝜇₂𝐵₂𝑢), 𝑥_𝑛=𝐽_𝑟𝑛𝑥_𝑛, 𝑢_𝑛=Π_𝐶(𝑥_𝑛−𝜇₂𝐵₂𝑥_𝑛) and 𝑣_𝑛=Π_𝐶(𝑢_𝑛−𝜇₁𝐵₁𝑢_𝑛). Then 𝑢=Π_𝐶(𝑣−𝜇₁𝐵₁𝑣), and 𝑣_𝑛=𝐺𝑥_𝑛=𝐺(𝐽_𝑟𝑛𝑥_𝑛) for all 𝑛≥0. It is clear from (95) that 𝑦_𝑛−𝑢²≤𝛼_𝑛𝑥_𝑛−𝑢²+1−𝛼_𝑛𝐺(𝐽_𝑟𝑛𝑥_𝑛)−𝑢²=𝛼_𝑛𝑥_𝑛−𝑢²+1−𝛼_𝑛𝑣_𝑛−𝑢². (123) Utilizing Lemma 14, we have 𝑢_𝑛−𝑣²=Π_𝐶(𝑥_𝑛−𝜇₂𝐵₂𝑥_𝑛)−Π_𝐶(𝑢−𝜇₂𝐵₂𝑢)²≤𝑥_𝑛−𝑢−𝜇₂(𝐵₂𝑥_𝑛−𝐵₂𝑢)²≤𝑥_𝑛−𝑢²−2𝜇₂𝛼₂−𝜅²𝜇₂𝐵₂𝑥_𝑛−𝐵₂𝑢²,𝑣_𝑛−𝑢²=Π_𝐶(𝑢_𝑛−𝜇₁𝐵₁𝑢_𝑛)−Π_𝐶(𝑣−𝜇₁𝐵₁𝑣)²≤𝑢_𝑛−𝑣−𝜇₁(𝐵₁𝑢_𝑛−𝐵₁𝑣)²≤𝑢_𝑛−𝑣²−2𝜇₁𝛼₁−𝜅²𝜇₁𝐵₁𝑢_𝑛−𝐵₁𝑣². (124)(125) Substituting (124) for (125), we obtain 𝑣_𝑛−𝑢²≤𝑥_𝑛−𝑢²−2𝜇₂𝛼₂−𝜅²𝜇₂𝐵₂𝑥_𝑛−𝐵₂𝑢²−2𝜇₁𝛼₁−𝜅²𝜇₁𝐵₁𝑢_𝑛−𝐵₁𝑣²≤𝑥_𝑛−𝑢²−2𝜇₂𝛼₂−𝜅²𝜇₂𝐵₂𝑥_𝑛−𝐵₂𝑢²−2𝜇₁𝛼₁−𝜅²𝜇₁𝐵₁𝑢_𝑛−𝐵₁𝑣², (126) which, together with (123), implies that 𝑦_𝑛−𝑢²≤𝛼_𝑛𝑥_𝑛−𝑢²+1−𝛼_𝑛𝑣_𝑛−𝑢²≤𝛼_𝑛𝑥_𝑛−𝑢²+1−𝛼_𝑛×𝑥_𝑛−𝑢²−2𝜇₂𝛼₂−𝜅²𝜇₂𝐵₂𝑥_𝑛−𝐵₂𝑢²−2𝜇₁𝛼₁−𝜅²𝜇₁𝐵₁𝑢_𝑛−𝐵₁𝑣²=𝑥_𝑛−𝑢²−21−𝛼_𝑛×𝜇₂𝛼₂−𝜅²𝜇₂𝐵₂𝑥_𝑛−𝐵₂𝑢²+𝜇₁𝛼₁−𝜅²𝜇₁𝐵₁𝑢_𝑛−𝐵₁𝑣². (127) It immediately follows that 21−𝛼_𝑛𝜇₂𝛼₂−𝜅²𝜇₂𝐵₂𝑥_𝑛−𝐵₂𝑢²+𝜇₁𝛼₁−𝜅²𝜇₁𝐵₁𝑢_𝑛−𝐵₁𝑣²≤𝑥_𝑛−𝑢²−𝑦_𝑛−𝑢²≤𝑥_𝑛−𝑢+𝑦_𝑛−𝑢𝑥_𝑛−𝑦_𝑛. (128) Since {𝑥_𝑛} and {𝑦_𝑛} are bounded and 0<𝜇_𝑖<𝛼_𝑖/𝜅² for 𝑖=1,2, we deduce from (119) and condition (iii) that lim_𝑛→∞𝐵₂𝑥_𝑛−𝐵₂𝑢=0,lim_𝑛→∞𝐵₁𝑢_𝑛−𝐵₁𝑣=0. (129) Utilizing Proposition 3 and Lemma 6, we have 𝑢_𝑛−𝑣²=Π_𝐶(𝑥_𝑛−𝜇₂𝐵₂𝑥_𝑛)−Π_𝐶(𝑢−𝜇₂𝐵₂𝑢)²≤𝑥_𝑛−𝜇₂𝐵₂𝑥_𝑛−𝑢−𝜇₂𝐵₂𝑢,𝐽𝑢_𝑛=𝑥−𝑣_𝑛−𝑢,𝐽𝑢_𝑛−𝑣+𝜇₂𝐵₂𝑝−𝐵₂𝑥_𝑛,𝐽𝑢_𝑛≤1−𝑣2𝑥_𝑛−𝑢²+𝑢_𝑛−𝑣²−𝑔₁𝑥_𝑛−𝑢_𝑛−(𝑢−𝑣)+𝜇₂𝐵₂𝑢−𝐵₂𝑥_𝑛𝑢_𝑛−𝑣, (130) which implies that 𝑢_𝑛−𝑣²≤𝑥_𝑛−𝑢²−𝑔₁𝑥_𝑛−𝑢_𝑛−(𝑢−𝑣)+2𝜇₂𝐵₂𝑢−𝐵₂𝑥_𝑛𝑢_𝑛−𝑣. (131) In the same way, we derive 𝑣_𝑛−𝑢²=Π_𝐶(𝑢_𝑛−𝜇₁𝐵₁𝑢_𝑛)−Π_𝐶(𝑣−𝜇₁𝐵₁𝑣)²≤𝑢_𝑛−𝜇₁𝐵₁𝑢_𝑛−𝑣−𝜇₁𝐵₁𝑣,𝐽𝑣_𝑛=𝑢−𝑢_𝑛−𝑣,𝐽𝑣_𝑛−𝑢+𝜇₁𝐵₁𝑣−𝐵₁𝑢_𝑛,𝐽𝑣_𝑛≤1−𝑢2𝑢_𝑛−𝑣²+𝑣_𝑛−𝑢²−𝑔₂𝑢_𝑛−𝑣_𝑛+(𝑢−𝑣)+𝜇₁𝐵₁𝑣−𝐵₁𝑢_𝑛𝑣_𝑛−𝑢, (132) which implies that 𝑣_𝑛−𝑢²≤𝑢_𝑛−𝑣²−𝑔₂𝑢_𝑛−𝑣_𝑛+(𝑢−𝑣)+2𝜇₁𝐵₁𝑣−𝐵₁𝑢_𝑛𝑣_𝑛−𝑢. (133) Substituting (131) for (133), we get 𝑣_𝑛−𝑢²≤𝑥_𝑛−𝑢²−𝑔₁𝑥_𝑛−𝑢_𝑛−(𝑢−𝑣)−𝑔₂𝑢_𝑛−𝑣_𝑛+(𝑢−𝑣)+2𝜇₂𝐵₂𝑢−𝐵₂𝑥_𝑛𝑢_𝑛−𝑣+2𝜇₁𝐵₁𝑣−𝐵₁𝑢_𝑛𝑣_𝑛≤𝑥−𝑢_𝑛−𝑢²−𝑔₁𝑥_𝑛−𝑢_𝑛−(𝑢−𝑣)−𝑔₂𝑢_𝑛−𝑣_𝑛+(𝑢−𝑣)+2𝜇₂𝐵₂𝑢−𝐵₂𝑥_𝑛𝑢_𝑛−𝑣+2𝜇₁𝐵₁𝑣−𝐵₁𝑢_𝑛𝑣_𝑛−𝑢, (134) which, together with (123), implies that 𝑦_𝑛−𝑢²≤𝛼_𝑛𝑥_𝑛−𝑢²+1−𝛼_𝑛𝑣_𝑛−𝑢²≤𝛼_𝑛𝑥_𝑛−𝑢²+1−𝛼_𝑛×𝑥_𝑛−𝑢²−𝑔₁𝑥_𝑛−𝑢_𝑛−(𝑢−𝑣)−𝑔₂𝑢_𝑛−𝑣_𝑛+(𝑢−𝑣)+2𝜇₂𝐵₂𝑢−𝐵₂𝑥_𝑛𝑢_𝑛−𝑣+2𝜇₁𝐵₁𝑣−𝐵₁𝑢_𝑛𝑣_𝑛=𝑥−𝑢_𝑛−𝑢²−1−𝛼_𝑛×𝑔₁𝑥_𝑛−𝑢_𝑛−(𝑢−𝑣)+𝑔₂𝑢_𝑛−𝑣_𝑛+(𝑢−𝑣)+21−𝛼_𝑛𝜇₂𝐵₂𝑢−𝐵₂𝑥_𝑛𝑢_𝑛−𝑣+𝜇₁𝐵₁𝑣−𝐵₁𝑢_𝑛𝑣_𝑛−𝑢. (135) It immediately follows that 1−𝛼_𝑛𝑔₁𝑥_𝑛−𝑢_𝑛−(𝑢−𝑣)+𝑔₂𝑢_𝑛−𝑣_𝑛+(≤𝑥𝑢−𝑣)_𝑛−𝑢²−𝑦_𝑛−𝑢²+21−𝛼_𝑛𝜇₂𝐵₂𝑢−𝐵₂𝑥_𝑛𝑢_𝑛−𝑣+𝜇₁𝐵₁𝑣−𝐵₁𝑢_𝑛𝑣_𝑛≤𝑥−𝑢_𝑛−𝑢+𝑦_𝑛−𝑢𝑥_𝑛−𝑦_𝑛+2𝜇₂𝐵₂𝑢−𝐵₂𝑥_𝑛𝑢_𝑛−𝑣+2𝜇₁𝐵₁𝑣−𝐵₁𝑢_𝑛𝑣_𝑛−𝑢. (136) Since {𝑥_𝑛}, {𝑦_𝑛}, {𝑢_𝑛}, and {𝑣_𝑛} are bounded, we deduce from (119), (129), and condition (iii) that lim_𝑛→∞𝑔₁𝑥_𝑛−𝑢_𝑛−(𝑢−𝑣)=0,lim_𝑛→∞𝑔₂𝑢_𝑛−𝑣_𝑛+(𝑢−𝑣)=0. (137) Utilizing the properties of 𝑔₁ and 𝑔₂, we get lim_𝑛→∞𝑥_𝑛−𝑢_𝑛−(𝑢−𝑣)=0,lim_𝑛→∞𝑢_𝑛−𝑣_𝑛+(𝑢−𝑣)=0, (138) which hence yields 𝑥_𝑛−𝑣_𝑛≤𝑥_𝑛−𝑢_𝑛−(+𝑢𝑢−𝑣)_𝑛−𝑣_𝑛+(𝑢−𝑣)⟶0as𝑛⟶∞. (139) That is, lim_𝑛→∞𝐽_𝑟𝑛𝑥_𝑛−𝐺𝐽_𝑟𝑛𝑥_𝑛=lim_𝑛→∞𝑥_𝑛−𝑣_𝑛=0. (140) Note that 𝑥_𝑛−𝐽_𝑟𝑛𝑥_𝑛≤𝑥_𝑛−𝐺𝐽_𝑟𝑛𝑥_𝑛+𝐺𝐽_𝑟𝑛𝑥_𝑛−𝐽_𝑟𝑛𝑥_𝑛. (141) So from (122) and (140) we have lim_𝑛→∞𝑥_𝑛−𝐽_𝑟𝑛𝑥_𝑛=0, (142) which, together with (122), leads to 𝑥_𝑛−𝐺𝑥_𝑛≤𝑥_𝑛−𝐺𝐽_𝑟𝑛𝑥_𝑛+𝐺𝐽_𝑟𝑛𝑥_𝑛−𝐺𝑥_𝑛≤𝑥_𝑛−𝐺𝐽_𝑟𝑛𝑥_𝑛+𝐽_𝑟𝑛𝑥_𝑛−𝑥_𝑛⟶0as𝑛⟶∞. (143) That is, lim_𝑛→∞𝑥_𝑛−𝐺𝑥_𝑛=0. (144) In addition, utilizing Lemma 9 we obtain from {𝑟_𝑛}⊂[𝜀,∞) that 𝑥_𝑛−𝐽_𝜀𝑥_𝑛≤2𝑥_𝑛−𝐽_𝑟𝑛𝑥_𝑛, (145) which, together with (142), implies that lim_𝑛→∞𝑥_𝑛−𝐽_𝜀𝑥_𝑛=0. (146)
Define a mapping 𝑊𝑥=(1−𝜃)𝐽_𝜀𝑥+𝜃𝐺𝑥,∀𝑥∈𝐶, (147) where 𝜃 is a constant in (0,1). Then by Lemma 10, we know that Fix(𝑊)=Fix(𝐽_𝜀)∩Fix(𝐺)=Δ. We observe that 𝑥_𝑛−𝑊𝑥_𝑛=(1−𝜃)𝑥_𝑛−𝐽_𝜀𝑥_𝑛+𝜃𝑥_𝑛−𝐺𝑥_𝑛≤(1−𝜃)𝑥_𝑛−𝐽_𝜀𝑥_𝑛+𝜃𝑥_𝑛−𝐺𝑥_𝑛. (148) So from (144) and (146) we obtain lim_𝑛→∞𝑥_𝑛−𝑊𝑥_𝑛=0. (149)
Now, we claim that limsup_𝑛→∞𝑓𝑝−𝑝,𝐽𝑥_𝑛−𝑝≤0, (150) where 𝑝=𝑠−lim_𝑡→0𝑥_𝑡, with 𝑥_𝑡 being the fixed point of the contraction 𝑥↦𝑡𝑓(𝑥)+(1−𝑡)𝑊𝑥 of 𝐶 into itself (due to Lemma 12). Then 𝑥_𝑡 solves the fixed point equation 𝑥_𝑡=𝑡𝑓(𝑥_𝑡)+(1−𝑡)𝑊𝑥_𝑡. Thus we have 𝑥_𝑡−𝑥_𝑛=(1−𝑡)𝑊𝑥_𝑡−𝑥_𝑛+𝑡𝑓𝑥_𝑡−𝑥_𝑛. (151) By Lemma 5 we conclude that 𝑥_𝑡−𝑥_𝑛²=(1−𝑡)(𝑊𝑥_𝑡−𝑥_𝑛)+𝑡(𝑓(𝑥_𝑡)−𝑥_𝑛)²≤(1−𝑡)²𝑊𝑥_𝑡−𝑥_𝑛²+2𝑡𝑓𝑥_𝑡−𝑥_𝑛,𝐽𝑥_𝑡−𝑥_𝑛≤(1−𝑡)²𝑊𝑥_𝑡−𝑊𝑥_𝑛+𝑊𝑥_𝑛−𝑥_𝑛²+2𝑡𝑓𝑥_𝑡−𝑥_𝑛,𝐽𝑥_𝑡−𝑥_𝑛≤(1−𝑡)²𝑥_𝑡−𝑥_𝑛+𝑊𝑥_𝑛−𝑥_𝑛²+2𝑡𝑓𝑥_𝑡−𝑥_𝑛,𝐽𝑥_𝑡−𝑥_𝑛=(1−𝑡)²𝑥_𝑡−𝑥_𝑛²+2𝑥_𝑡−𝑥_𝑛𝑊𝑥_𝑛−𝑥_𝑛+𝑊𝑥_𝑛−𝑥_𝑛²+2𝑡𝑓𝑥_𝑡−𝑥_𝑡,𝐽𝑥_𝑡−𝑥_𝑛+2𝑡𝑥_𝑡−𝑥_𝑛,𝐽𝑥_t−𝑥_𝑛=1−2𝑡+𝑡²𝑥_𝑡−𝑥_𝑛²+𝑓_𝑛(𝑡)+2𝑡𝑓𝑥_𝑡−𝑥_𝑡,𝐽𝑥_𝑡−𝑥_𝑛+2𝑡𝑥_𝑡−𝑥_𝑛², (152) where 𝑓_𝑛(𝑡)=(1−𝑡)²2𝑥_𝑡−𝑥_𝑛+𝑥_𝑛−𝑊𝑥_𝑛×𝑥_𝑛−𝑊𝑥_𝑛⟶0,as𝑛⟶∞. (153) It follows from (152) that 𝑥_𝑡−𝑓𝑥_𝑡,𝐽𝑥_𝑡−𝑥_𝑛≤𝑡2𝑥_𝑡−𝑥_𝑛²+1𝑓2𝑡^𝑛(𝑡). (154) Letting 𝑛→∞ in (154) and noticing (153), we derive limsup_𝑛→∞𝑥_𝑡−𝑓𝑥_𝑡,𝐽𝑥_𝑡−𝑥_𝑛≤𝑡2𝑀₃, (155) where 𝑀₃>0 is a constant such that ‖𝑥_𝑡−𝑥_𝑛‖²≤𝑀₃ for all 𝑡∈(0,1) and 𝑛≥0. Taking 𝑡→0 in (155), we have limsup_𝑡→0limsup_𝑛→∞𝑥_𝑡−𝑓𝑥_𝑡,𝐽𝑥_𝑡−𝑥_𝑛≤0. (156) On the other hand, we have 𝑓𝑝−𝑝,𝐽𝑥_𝑛=𝑓𝑝𝑥−𝑝−𝑝,𝐽_𝑛−𝑝−𝑓𝑝−𝑝,𝐽𝑥_𝑛−𝑥_𝑡+𝑓𝑝−𝑝,𝐽𝑥_𝑛−𝑥_𝑡−𝑓𝑝−𝑥_𝑡,𝐽𝑥_𝑛−𝑥_𝑡+𝑓𝑝−𝑥_𝑡,𝐽𝑥_𝑛−𝑥_𝑡−𝑓𝑥_𝑡−𝑥_𝑡,𝐽𝑥_𝑛−𝑥_𝑡+𝑓𝑥_𝑡−𝑥_𝑡,𝐽𝑥_𝑛−𝑥_𝑡=𝑓𝑝−𝑝,𝐽𝑥_𝑛−𝑝−𝐽𝑥_𝑛−𝑥_𝑡+𝑥_𝑡−𝑝,𝐽𝑥_𝑛−𝑥_𝑡+𝑓𝑝−𝑓𝑥_𝑡,𝐽𝑥_𝑛−𝑥_𝑡+𝑓𝑥_𝑡−𝑥_𝑡,𝐽𝑥_𝑛−𝑥_𝑡. (157) It follows that limsup_𝑛→∞𝑓𝑝−𝑝,𝐽𝑥_𝑛−𝑝≤limsup_𝑛→∞𝑓𝑝−𝑝,𝐽𝑥_𝑛−𝑝−𝐽𝑥_𝑛−𝑥_𝑡+𝑥_𝑡−𝑝limsup_𝑛→∞𝑥_𝑛−𝑥_𝑡+𝜌𝑝−𝑥_𝑡limsup_𝑛→∞𝑥_𝑛−𝑥_𝑡+limsup_𝑛→∞𝑓𝑥_𝑡−𝑥_𝑡,𝐽𝑥_𝑛−𝑥_𝑡. (158) Taking into account that 𝑥_𝑡→𝑝 as 𝑡→0, we have from (156) limsup_𝑛→∞𝑓𝑝−𝑝,𝐽𝑥_𝑛−𝑝=limsup_𝑡→0limsup_𝑛→∞𝑓𝑝−𝑝,𝐽𝑥_𝑛−𝑝≤limsup_𝑡→0limsup_𝑛→∞𝑓𝑝−𝑝,𝐽𝑥_𝑛−𝑝−𝐽𝑥_𝑛−𝑥_𝑡. (159) Since 𝑋 has a uniformly Frechet differentiable norm, the duality mapping 𝐽 is norm-to-norm uniformly continuous on bounded subsets of 𝑋. Consequently, the two limits are interchangeable and hence (150) holds. It is clear from (150) that limsup_𝑛→∞𝑓𝑝−𝑝,𝐽𝑥_𝑛+1−𝑝≤0. (160)
Finally, let us show that 𝑥_𝑛→𝑝 as 𝑛→∞. We observe that 𝑦_𝑛−𝑝²≤𝛼_𝑛𝑥_𝑛−𝑝²+1−𝛼_𝑛𝐺(𝐽_𝑟𝑛𝑥_𝑛)−𝑝²≤𝛼_𝑛𝑥_𝑛−𝑝²+1−𝛼_𝑛𝑥_𝑛−𝑝²=𝑥_𝑛−𝑝², (161) and hence 𝑥_𝑛+1−𝑝²=𝛽_𝑛𝑓𝑥_𝑛𝑝+−𝑓1−𝛽_𝑛𝐺𝐽_𝑟𝑛𝑦_𝑛−𝜆_𝑛𝜇_𝑛𝐹𝐺𝐽_𝑟𝑛𝑦_𝑛−𝑝+𝛽_𝑛𝑓𝑝−𝑝²≤𝛽_𝑛𝑓𝑥_𝑛𝑝+−𝑓1−𝛽_𝑛𝐺𝐽_𝑟𝑛𝑦_𝑛−𝜆_𝑛𝜇_𝑛𝐹𝐺𝐽_𝑟𝑛𝑦_𝑛−𝑝²+2𝛽_𝑛𝑓𝑝−𝑝,𝐽𝑥_𝑛+1−𝑝≤𝛽_𝑛𝑓(𝑥_𝑛)−𝑓(𝑝)²+1−𝛽_𝑛𝐺(𝐽_𝑟𝑛𝑦_𝑛)−𝜆_𝑛𝜇_𝑛𝐹𝐺(𝐽_𝑟𝑛𝑦_𝑛)−𝑝²+2𝛽_𝑛𝑓𝑝−𝑝,𝐽𝑥_𝑛+1−𝑝=𝛽_𝑛𝑓(𝑥_𝑛)−𝑓(𝑝)²+1−𝛽_𝑛𝐼−𝜆_𝑛𝜇_𝑛𝐹𝐺𝐽_𝑟𝑛𝑦_𝑛−(𝐼−𝜆_𝑛𝜇_𝑛𝐹)𝑝−𝜆_𝑛𝜇_𝑛𝐹𝑝²+2𝛽_𝑛𝑓𝑝−𝑝,𝐽𝑥_𝑛+1−𝑝≤𝛽_𝑛𝜌𝑥_𝑛−𝑝²+1−𝛽_𝑛𝐼−𝜆_𝑛𝜇_𝑛𝐹𝐺𝐽_𝑟𝑛𝑦_𝑛−𝐼−𝜆_𝑛𝜇_𝑛𝐹𝑝+𝜆_𝑛𝜇_𝑛𝐹𝑝²+2𝛽_𝑛𝑓𝑝−𝑝,𝐽𝑥_𝑛+1−𝑝≤𝛽_𝑛𝜌𝑥_𝑛−𝑝²+1−𝛽_𝑛×1−𝜆_𝑛𝜇_𝑛1−1−𝛿𝜆𝐺𝐽_𝑟𝑛𝑦_𝑛−𝑝+𝜆_𝑛𝜇_𝑛𝐹𝑝²+2𝛽_𝑛𝑓𝑝−𝑝,𝐽𝑥_𝑛+1−𝑝≤𝛽_𝑛𝜌𝑥_𝑛−𝑝²+1−𝛽_𝑛𝑦_𝑛−𝑝+𝜆_𝑛𝜇_𝑛𝐹𝑝²+2𝛽_𝑛𝑓𝑝−𝑝,𝐽𝑥_𝑛+1−𝑝≤𝛽_𝑛𝜌𝑥_𝑛−𝑝²+1−𝛽_𝑛𝑥_𝑛−𝑝+𝜆_𝑛𝜇_𝑛𝐹𝑝²+2𝛽_𝑛𝑓𝑝−𝑝,𝐽𝑥_𝑛+1−𝑝=𝛽_𝑛𝜌𝑥_𝑛−𝑝²+1−𝛽_𝑛𝑥_𝑛−𝑝²+𝜆_𝑛𝜇_𝑛×2𝑥𝐹𝑝_𝑛−𝑝+𝜆_𝑛𝜇_𝑛𝐹𝑝+2𝛽_𝑛𝑓𝑝−𝑝,𝐽𝑥_𝑛+1≤𝛽−𝑝1−1−𝜌_𝑛𝑥_𝑛−𝑝²+𝜆_𝑛𝜇_𝑛𝐹𝑝2𝑥_𝑛−𝑝+𝜆_𝑛𝜇_𝑛𝐹𝑝+2𝛽_𝑛𝑓𝑝−𝑝,𝐽𝑥_𝑛+1=𝛽−𝑝1−1−𝜌_𝑛𝑥_𝑛−𝑝²+1−𝜌𝛽_𝑛𝜆_𝑛𝜇_𝑛𝛽_𝑛𝐹𝑝2𝑥_𝑛−𝑝+𝜆_𝑛𝜇_𝑛𝐹𝑝+2𝑓𝑝𝑥1−𝜌−𝑝,𝐽_𝑛+1−𝑝.1−𝜌 (162) Taking into account (160) and conditions (i), (ii), we obtain that ∑^∞_𝑛=0(1−𝜌)𝛽_𝑛=∞ and limsup_𝑛→∞𝜆_𝑛𝜇_𝑛𝛽_𝑛𝐹𝑝2𝑥_𝑛−𝑝+𝜆_𝑛𝜇_𝑛𝐹𝑝+2𝑓𝑝𝑥1−𝜌−𝑝,𝐽_𝑛+1−𝑝1−𝜌≤0. (163) Therefore, applying Lemma 4 to (162), we infer that lim_𝑛→∞𝑥_𝑛−𝑝=0. (164) This completes the proof.

Remark 19. As pointed out in [12, Remark 3.2], the sequences {𝜆_𝑛}, {𝜇_𝑛}, and {𝛽_𝑛} can be taken, which satisfy the conditions in Theorem 18. As a matter of fact, put 𝜆_𝑛=(1+𝑛)^−5/6, 𝜇_𝑛=1, and 𝛽_𝑛=(1+𝑛)^−2/3 for all 𝑛≥0. Then there hold the following statements:(i)lim_𝑛→∞𝛽_𝑛=0 and ∑^∞_𝑛=0𝛽_𝑛=∞,(ii)lim_𝑛→∞(𝜆_𝑛𝜇_𝑛)/𝛽_𝑛=0,(iii)∑^∞_𝑛=0|𝛽_𝑛+1−𝛽_𝑛|<∞, ∑^∞_𝑛=0|𝜆_𝑛+1−𝜆_𝑛|<∞, and∑^∞_𝑛=0|𝜇_𝑛+1−𝜇_𝑛|<∞.

By the careful analysis of the proof of Theorem 18, we can obtain the following result. Because its proof is much simpler than that of Theorem 18, we omit its proof.

Theorem 20. Let 𝑋 be a uniformly convex and 2-uniformly smooth Banach space and let 𝐴 be an 𝑚-accretive operator in 𝑋 such that 𝐶=𝐷(𝐴) is convex. Let Π_𝐶 be a sunny nonexpansive retraction from 𝑋 onto 𝐶. Let the mapping 𝐵_𝑖∶𝐶→𝑋 be 𝛼_𝑖-inverse strongly accretive for 𝑖=1,2, let 𝑓∶𝑋→𝐶 be a contractive map with coefficient 𝜌∈(0,1), and let 𝐹∶𝑋→𝑋 be 𝛿-strongly accretive and 𝜆-strictly pseudocontractive with 𝛿+𝜆>1. Assume that Δ=𝐴⁻¹(0)∩Ω≠∅, where Ω is the fixed point set of the mapping 𝐺=Π_𝐶(𝐼−𝜇₁𝐵₁)Π_𝐶(𝐼−𝜇₂𝐵₂) with 0<𝜇_𝑖<𝛼_𝑖/𝜅² for 𝑖=1,2. Given sequences {𝜆_𝑛}^∞_𝑛=0 in [0,1], {𝛼_𝑛}^∞_𝑛=0,{𝛽_𝑛}^∞_𝑛=0 in (0,1], and {𝑟_𝑛}^∞_𝑛=0 in [𝜀,∞) for some 𝜀>0, suppose that there hold the following conditions:(i)lim_𝑛→∞𝛽_𝑛=0 and ∑^∞_𝑛=0𝛽_𝑛=∞,(ii)lim_𝑛→∞𝜆_𝑛/𝛽_𝑛=0 and ∑^∞_𝑛=0|𝜆_𝑛+1−𝜆_𝑛|<∞,(iii){𝛼_𝑛}⊂[𝑎,𝑏] for some 𝑎,𝑏∈(0,1),(iv)∑^∞_𝑛=0|𝛼_𝑛+1−𝛼_𝑛|<∞, ∑^∞_𝑛=0|𝛽_𝑛+1−𝛽_𝑛|<∞, and ∑^∞_𝑛=0|𝑟_𝑛+1−𝑟_𝑛|<∞.
Then for any given point 𝑥₀∈𝑋, the sequence {𝑥_𝑛} generated by 𝑦_𝑛=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝐺𝐽_𝑟𝑛𝑥_𝑛,𝑥_𝑛+1=𝛽_𝑛𝑓𝑥_𝑛+1−𝛽_𝑛𝑦_𝑛−𝜆_𝑛𝐹𝑦_𝑛,∀𝑛≥0, (165) converges strongly to 𝑝∈Δ, which is a unique solution of the VIP (44).

Remark 21. Our Theorems 16–20 improve, extend, supplement, and develop Cai and Bu [13, Theorem 3.1] and Ceng et al. [12, Theorems 3.1–3.3] in the following aspects.(i)The problem of finding a point 𝑝∈𝐴⁻¹(0)∩Ω in our Theorems 16–20 is very different from everyone of both the problem of finding a point 𝑝∈⋂_𝑛Fix(𝑆_𝑛)∩Ω in Cai and Bu [13, Theorem 3.1] and the problem of finding a point 𝑝∈𝐴⁻¹(0) in Ceng et al. [12, Theorems 3.1–3.3]. There is no doubt that our problem of finding a point 𝑝∈𝐴⁻¹(0)∩Ω is more general than the problem of finding a point 𝑝∈𝐴⁻¹(0) in [12, Theorems 3.1–3.3].(ii)Compared with the choice of iterative parameters in [12, Theorems 3.1–3.3], the choice of iterative parameters in our Theorems 16–20 is the same as that in [12, Theorems 3.1–3.3].(iii)The iterative schemes in [12, Theorems 3.1–3.3] are extended to develop the iterative schemes in our Theorems 16–20 by virtue of the iterative scheme of [13, Theorems 3.1]. The iterative schemes in our Theorems 16–20 are more advantageous and more flexible than the iterative schemes of [12, Theorems 3.1–3.3] because they involve solving two problems: the GSVI (16) and the problem of finding zeros of an 𝑚-accretive operator.(iv)The iterative schemes in our Theorems 16–20 are very different from everyone in both [13, Theorem 3.1] and [12, Theorems 3.1–3.3] because the iterative scheme in our Theorem 16 is implicit and because the mapping 𝐺 in [13, Theorem 3.1] and the mapping 𝐽_𝑟𝑛 in [12, Theorems 3.1–3.3] are replaced by the same composite mapping 𝐺∘𝐽_𝑟𝑛 in the iterative schemes of our Theorems 16–20.(v)Cai and Bu’s proof in [13, Theorem 3.1] depends on the argument techniques in [16], the inequality in 2-uniformly smooth Banach spaces (see Lemma 2), and the inequality in smooth and uniform convex Banach spaces (see Proposition 3). Because the composite mapping 𝐺∘𝐽_𝑟𝑛 appears in the iterative schemes in our Theorems 16–20, the proof of our Theorems 16–20 depends on the argument techniques in [16], the inequality in 2-uniformly smooth Banach spaces (see Lemma 2), the inequality in smooth and uniform convex Banach spaces (see Proposition 3), and the properties of the resolvent of an𝑚-accretive operator (see Lemmas 8 and 9), the Banach limit (see Lemma 11) and the strongly accretive and strictly pseudocontractive mapping (see Proposition 13).

Acknowledgments

This research was partially supported by the National Science Foundation of China (11071169), Innovation Program of Shanghai Municipal Education Commission (09ZZ133), and Ph.D. Program Foundation of Ministry of Education of China (20123127110002). This research was partially supported by a Grant from NSC 102-2115-M-037-001.