Implicit and Explicit Iterative Methods for Systems of Variational Inequalities and Zeros of Accretive Operators

Abstract

Based on Korpelevich's extragradient method, hybrid steepest-descent method, and viscosity approximation method, we propose implicit and explicit iterative schemes for computing a common element of the solution set of a system of variational inequalities and the set of zeros of an accretive operator, which is also a unique solution of a variational inequality. Under suitable assumptions, we study the strong convergence of the sequences generated by the proposed algorithms. The results of this paper improve and extend several known results in the literature.

1. Introduction and Preliminaries

Throughout the paper, we use the notation ⇀ to indicate the weak convergence and → to indicate the strong convergence.

Let 𝑋 be a real Banach space whose dual space is denoted by 𝑋^∗. Let 𝑈={𝑥∈𝑋∶‖𝑥‖=1} denote the unit 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∈𝑈, 𝑥−𝑦≥𝜖⇒𝑥+𝑦2≤1−𝛿. (1) 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 (2) exists for all 𝑥,𝑦∈𝑈; in this case, 𝑋 is also said to have a Gâteaux differentiable norm. Moreover, it is said to be uniformly smooth if this limit is attained uniformly for all 𝑥,𝑦∈𝑈; in this case, 𝑋 is also said to have a uniformly Fréchet differentiable norm. The norm of 𝑋 is said to be Fréchet differentiable if for each 𝑥∈𝑈, the limit is attained uniformly for all 𝑦∈𝑈. A function 𝜌∶[0,∞)→[0,∞) defined by 𝜌(𝜏)1=sup2+‖𝑥‖𝑦𝑥+𝑦𝑥−𝑦−1∶𝑥,𝑦∈𝑋,=1,=𝜏 (3) is called the modulus of smoothness of 𝑋. It is known that 𝑋 is uniformly smooth if and only if lim_𝜏→0𝜌(𝜏)/𝜏=0. If 𝑞∈(1,2] is a fixed number, 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 [1], no Banach space is 𝑞-uniformly smooth for 𝑞>2.

The normalized duality mapping 𝐽∶𝑋→2^𝑋∗ is defined by 𝐽(𝑥)=𝑥^∗∈𝑋^∗∶𝑥,𝑥^∗=‖𝑥‖²=𝑥^∗2,∀𝑥∈𝑋, (4) where ⟨⋅,⋅⟩ denotes the generalized duality pairing. It is an immediate consequence of the Hahn-Banach theorem that 𝐽(𝑥) is nonempty for all 𝑥∈𝑋. 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. For further details on the geometry of Banach spaces, we refer to [2, 3] and the references therein.

Lemma 1 (see [4]). Let 𝑋 be a 2-uniformly smooth Banach space. Then 𝑥+𝑦²≤‖𝑥‖²+2𝑦,𝐽(𝑥)+2𝜅𝑦²,∀𝑥,𝑦∈𝑋, (5) where 𝜅 is the 2-uniformly smooth constant of 𝑋 and 𝐽 is the normalized duality mapping from 𝑋 into 𝑋^∗.

The following lemma is an immediate consequence of the subdifferential inequality of the function (1/2)‖⋅‖².

Lemma 2. Let 𝑋 be a smooth Banach space. Then, ‖𝑥‖²+2𝑦,𝐽(𝑥)≤𝑥+𝑦²≤‖𝑥‖²+2𝑦,𝐽𝑥+𝑦,∀𝑥,𝑦∈𝑋, (6) where 𝐽 is the normalized duality mapping of 𝑋.

Proposition 3 (see [5]). Let 𝑋 be a real smooth and uniform convex Banach space and 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, (7) where 𝐵_𝑟={𝑥∈𝑋∶‖𝑥‖≤𝑟}.

Let 𝐶 be a nonempty closed convex subset of a real Banach space 𝑋. A mapping 𝑇∶𝐶→𝐶 is said to be 𝐿-Lipschitzian (or Lipschtiz continuous) if there exists a constant 𝐿>0 such that ‖𝑇𝑥−𝑇𝑦‖≤𝐿‖𝑥−𝑦‖ for all 𝑥,𝑦∈𝐶. If 𝐿∈(0,1), then 𝑇 is called 𝐿-contraction. If 𝐿=1, then 𝑇 is called nonexpansive.

The set of fixed points of 𝑇 is denoted by Fix(𝑇).

Lemma 4 (see [6]). Let 𝐶 be a nonempty closed convex subset of a strictly convex Banach space 𝑋, and 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 𝑇 on 𝐶 defined by 𝑇𝑥=∑^∞_𝑛=0𝜆_𝑛𝑇_𝑛𝑥 for 𝑥∈𝐶 is defined well, nonexpansive, and Fix(𝑇)=⋂^∞_𝑛=0Fix(𝑇_𝑛).

Lemma 5 (see [7]). Let 𝐶 be a nonempty closed and convex subset of a Banach space 𝑋 and 𝑇∶𝐶→𝐶 be a continuous and strong pseudocontraction mapping. Then, 𝑇 has a unique fixed point in 𝐶.

Let 𝐶 be a nonempty subset of a Banach space 𝑋. A mapping 𝐴∶𝐶→𝑋 is said to be(a) accretive if for each 𝑥,𝑦∈𝐶, there exists 𝑗(𝑥−𝑦)∈𝐽(𝑥−𝑦) such that 𝐴𝑥−𝐴𝑦,𝑗𝑥−𝑦≥0, (8) where 𝐽 is the normalized duality mapping of 𝑋;(b)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,forsome𝛼∈(0,1); (9)(c) pseudocontractive if for each 𝑥,𝑦∈𝐶, there exists 𝑗(𝑥−𝑦)∈𝐽(𝑥−𝑦) such that 𝐴𝑥−𝐴𝑦,𝑗𝑥−𝑦≤𝑥−𝑦²; (10) (d)𝛽-strongly pseudocontractive if for each 𝑥,𝑦∈𝐶, there exists 𝑗(𝑥−𝑦)∈𝐽(𝑥−𝑦) such that 𝐴𝑥−𝐴𝑦,𝑗𝑥−𝑦≤𝛽𝑥−𝑦²,forsome𝛽∈(0,1); (11)(e)𝜆-strictly pseudocontractive 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−𝐴𝑦²,forsome𝜆∈(0,1). (12)

It is worth to mention 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, [8, 9].

Lemma 6. Let 𝐶 be a nonempty closed convex subset of a real smooth Banach space 𝑋, and let 𝐹∶𝐶→𝑋 be a mapping.(a) If 𝐹∶𝐶→𝑋 is 𝛼-strongly accretive and 𝜆-strictly pseudocontractive with 𝛼+𝜆≥1, then 𝐼−𝐹 is nonexpansive and 𝐹 is Lipschitz continuous with constant 1+1/𝜆.(b) If 𝐹∶𝐶→𝑋 is 𝛼-strongly accretive and 𝜆-strictly pseudocontractive with 𝛼+𝜆>1, then for any fixed 𝜏∈(0,1),𝐼−𝜏𝐹 is contractive with coefficient 1−𝜏(1−(1−𝛼)/𝜆).

Proof. (a) From the 𝜆-strictly pseudocontractivity of 𝐹, we have for all 𝑥,𝑦∈𝐶, 𝜆(𝐼−𝐹)𝑥−(𝐼−𝐹)𝑦²≤)()≤)()𝑦,(𝐼−𝐹𝑥−𝐼−𝐹𝑦,𝐽𝑥−𝑦(𝐼−𝐹𝑥−𝐼−𝐹𝑥−𝑦 (13) which implies that )()𝑦≤1(𝐼−𝐹𝑥−𝐼−𝐹𝜆𝑥−𝑦. (14) Hence, 𝐹(𝑥)𝑦≤)()𝑦+≤1−𝐹(𝐼−𝐹𝑥−𝐼−𝐹𝑥−𝑦1+𝜆𝑥−𝑦. (15) From the 𝜆-strictly pseudocontractiveness and 𝛼-strongly accretiveness of 𝐹, we have for all 𝑥,𝑦∈𝐶, 𝜆(𝐼−𝐹)𝑥−(𝐼−𝐹)𝑦²≤𝑥−𝑦²−𝐹(𝑥)𝑦≤(−𝐹,𝐽𝑥−𝑦1−𝛼)𝑥−𝑦², (16) which implies that (𝐼−𝐹)𝑥−(𝐼−𝐹)𝑦≤1−𝛼𝜆𝑥−𝑦. (17) Since 𝛼+𝜆≥1⇔(1−𝛼)/𝜆≤1, 𝐼−𝐹 is nonexpansive.
(b) Take a fixed 𝜏∈(0,1) arbitrarily. Observe that for all 𝑥,𝑦∈𝐶, )()𝑦=))()𝑦≤()()𝑦≤((𝐼−𝜏𝐹𝑥−𝐼−𝜏𝐹(1−𝜏𝑥−𝑦+𝜏(𝐼−𝐹𝑥−𝐼−𝐹1−𝜏)𝑥−𝑦+𝜏(𝐼−𝐹𝑥−𝐼−𝐹1−𝜏)𝑥−𝑦+𝜏1−𝛼𝜆=𝑥−𝑦1−𝜏1−1−𝛼𝜆𝑥−𝑦. (18) Since 𝛼+𝜆>1⇔(1−𝛼)/𝜆<1, 𝐼−𝜏𝐹 is contractive with coefficient 1−𝜏(1−(1−𝛼)/𝜆).

Let 𝑋 be a real smooth Banach space 𝑋. An operator 𝑉∶𝑋→𝑋^∗ is said to be strongly positive [10] if there exists a constant 𝛾>0 such that ⟨𝑉𝑥,𝐽(𝑥)⟩≥𝛾‖𝑥‖²,‖𝑎𝐼−𝑏𝑉‖=sup_{‖𝑥‖≤1})(𝑥,[][],|⟨(𝑎𝐼−𝑏𝑉𝑥,𝐽)⟩|𝑎∈0,1,𝑏∈−1,1 (19) where 𝐼 is the identity mapping.

Lemma 7 (see [7]). Let 𝑉 be a strongly positive linear bounded operator on a smooth Banach space 𝑋 with coefficient 𝛾>0 and 0<𝜌≤‖𝑉‖⁻¹. Then, ‖𝐼−𝜌𝐴‖≤1−𝜌𝛾.

Recall that a possibly multivalued operator 𝐴⊂𝑋×𝑋 with domain 𝐷(𝐴) and range 𝑅(𝐴) in 𝑋 is accretive if for each 𝑥_𝑖∈𝐷(𝐴) and 𝑦_𝑖∈𝐴𝑥_𝑖 (𝑖=1,2), there exists 𝑗(𝑥₁−𝑥₂)∈𝐽(𝑥₁−𝑥₂) such that ⟨𝑦₁−𝑦₂,𝑗(𝑥₁−𝑥₂)⟩≥0. An accretive operator 𝐴 is said to satisfy the range condition if 𝐷(𝐴)⊂𝑅(𝐼+𝑟𝐴) for all 𝑟>0. An accretive operator 𝐴 is 𝑚-accretive if 𝑅(𝐼+𝑟𝐴)=𝑋 for each 𝑟>0. If 𝐴 is an accretive operator which satisfies the range condition, then we can define, for each 𝑟>0, a mapping 𝐽_𝑟∶𝑅(𝐼+𝑟𝐴)→𝐷(𝐴) by 𝐽_𝑟=(𝐼+𝑟𝐴)⁻¹, which is called the resolvent of 𝐴. It is well known that 𝐽_𝑟 is nonexpansive and Fix(𝐽_𝑟)=𝐴⁻¹𝟎 for all 𝑟>0. Hence, Fix𝐽_𝑟=𝐴⁻¹𝟎={𝑧∈𝐷(𝐴)∶𝟎∈𝐴𝑧}. (20) If 𝐴⁻¹𝟎≠∅, then the inclusion 𝟎∈𝐴𝑧 is solvable.

The following resolvent identity is well-known.

Lemma 8 (resolvent identity). For 𝜆>0, 𝜇>0, and 𝑥∈𝑋, 𝐽_𝜆𝑥=𝐽_𝜇𝜇𝜆𝜇𝑥+1−𝜆𝐽_𝜆𝑥. (21)

Recently, Aoyama et al. [11] studied an iterative scheme in the setting of a uniformly convex Banach space 𝑋 having a uniformly Gâteaux differentiable norm to compute the approximate zero of an accretive operator 𝐴. They proved that the sequence generated by their scheme converges strongly to a zero of 𝐴 under some appropriate assumptions. Subsequently, Ceng et al. [12] introduced and analyzed a composite iterative scheme in the setting of a uniformly smooth Banach space or a reflexive Banach space having a weakly sequentially continuous duality mapping. Further, Jung [13] proposed and analyzed a composite iterative scheme by the viscosity approximation method for finding a zero of an accretive operator 𝐴 and established the strong convergence of the sequence generated by the proposed scheme.

Let 𝐶 be a closed and convex subset of a Banach space 𝑋, and let 𝐷 be a nonempty subset of 𝐶. A mapping Π∶𝐶→𝐷 is said to be sunny if Π[Π(𝑥)+𝑡(𝑥−Π(𝑥))]=Π(𝑥), (22) whenever Π(𝑥)+𝑡(𝑥−Π(𝑥))∈𝐶 for all 𝑥∈𝐶 and 𝑡≥0. A mapping Π∶𝐶→𝐶 is called a retraction if Π²=Π. If a mapping Π∶𝐶→𝐶 is a retraction, then Π(𝑧)=𝑧 for all 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 9 (see [14]). Let 𝐶 be a nonempty closed convex subset of a real smooth Banach space 𝑋, let 𝐷 be a nonempty subset of 𝐶, and let Π be a retraction of 𝐶 onto 𝐷. Then, the following statements are equivalent:(a)Π is sunny and nonexpansive;(b)‖Π(𝑥)−Π(𝑦)‖²≤⟨𝑥−𝑦,𝐽(Π(𝑥)−Π(𝑦))⟩,∀𝑥,𝑦∈𝐶;(c)⟨𝑥−Π(𝑥),𝐽(𝑦−Π(𝑥))⟩≤0,∀𝑥∈𝐶,𝑦∈𝐷.

It is well known that if 𝑋=𝐻 is 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.

Let 𝑋 be a smooth Banach space, let 𝐶⊆𝑋 be a nonempty closed convex set, and let 𝐵₁,𝐵₂∶𝐶→𝑋 be two nonlinear mappings. We consider the following system of variational inequalities (in short, SVI) which is earlier studied by Cai and Bu [15]. Find (𝑥^∗,𝑦^∗)∈𝐶×𝐶 such that 𝜇₁𝐵₁𝑦^∗+𝑥^∗−𝑦^∗,𝐽𝑥−𝑥^∗𝜇≥0,∀𝑥∈𝐶,₂𝐵₂𝑥^∗+𝑦^∗−𝑥^∗,𝐽𝑥−𝑦^∗≥0,∀𝑥∈𝐶, (23) where 𝜇₁ and 𝜇₂ are two positive constants. When 𝑋=𝐻 is a real Hilbert space, then SVI (23) is considered and studied by Ceng et al. [16]. In addition, if 𝐵₁=𝐵₂=𝐴 and 𝑥^∗=𝑦^∗, then the SVI reduces to the classical variational inequality problem (VIP) of finding 𝑥^∗∈𝐶 such that 𝐴𝑥^∗,𝑥−𝑥^∗≥0,∀𝑥∈𝐶. (24) It is worth to mention that the system of variational inequalities is used as a tool to study the Nash equilibrium problem; see, for example, [17–19] and the references therein. We believe that the problem (23) could be used to study Nash equilibrium problem for two-player game. The theory of variational inequalities is well established because it has a wide range of applications in science, engineering, management, and social sciences. In 1976, Korpelevič [20] proposed an iterative algorithm for solving VIP (24) in Euclidean space, which is known as the extragradient method or Korpelevich’s extragradient method. For different generalizations of extragradient method for variational inequalities and fixed point problems, we refer to [21–23] and the references therein. For further details on variational inequalities, we refer to [2, 19, 24, 25] and the references therein.

Whenever 𝑋 is a real smooth Banach space, 𝐵₁=𝐵₂=𝐴, and 𝑥^∗=𝑦^∗, then SVI (23) reduces to the variational inequality problem (VIP) of finding 𝑥^∗∈𝐶 such that 𝐴𝑥^∗,𝐽𝑥−𝑥^∗≥0,∀𝑥∈𝐶, (25) which was considered by Aoyama et al. [26]. Very recently, Cai and Bu [15] constructed an iterative algorithm for solving SVI (23) 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 sequence generated by the proposed algorithm.

Lemma 10. 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 SVI (23) if and only if 𝑥^∗=Π_𝐶(𝑦^∗−𝜇₁𝐵₁𝑦^∗), where 𝑦^∗=Π_𝐶(𝑥^∗−𝜇₂𝐵₂𝑥^∗).

Proof. We can rewrite SVI (23) as 𝑥^∗−𝑦^∗−𝜇₁𝐵₁𝑦^∗,𝐽𝑥−𝑥^∗𝑦≥0,∀𝑥∈𝐶,_∗−𝑥^∗−𝜇₂𝐵₂𝑥^∗,𝐽𝑥−𝑦^∗≥0,∀𝑥∈𝐶, (26) which is equivalent to 𝑥^∗=Π_𝐶𝑦^∗−𝜇₁𝐵₁𝑦^∗,𝑦^∗=Π_𝐶𝑥^∗−𝜇₂𝐵₂𝑥^∗. (27) From Lemma 9, we get the conclusion.

Remark 11. In view of Lemma 10, we observe that 𝑥^∗=Π_𝐶Π_𝐶𝑥^∗−𝜇₂𝐵₂𝑥^∗−𝜇₁𝐵₁Π_𝐶𝑥^∗−𝜇₂𝐵₂𝑥^∗, (28) which implies that 𝑥^∗ is a fixed point of the mapping 𝐺=Π_𝐶(𝐼−𝜇₁𝐵₁)Π_𝐶(𝐼−𝜇₂𝐵₂).

Lemma 12 (see [15, Lemma 2.8]). Let 𝐶 be a nonempty closed convex subset of a real 2-uniformly smooth Banach space 𝑋, and for each 𝑖=1,2, let 𝐵_𝑖∶𝐶→𝑋 be an 𝛼_𝑖-inverse-strongly accretive. Then, (𝐼−𝜇_𝑖𝐵_𝑖)𝑥−(𝐼−𝜇_𝑖𝐵_𝑖)𝑦²≤𝑥−𝑦²+2𝜇_𝑖𝜇_𝑖𝜅²−𝛼_𝑖𝐵_𝑖𝑥−𝐵_𝑖𝑦²,∀𝑥,𝑦∈𝐶,𝑖=1,2, (29) 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 13 (see [15, Lemma 2.9]). Let 𝐶 be a nonempty closed convex subset of a real 2-uniformly smooth Banach space 𝑋, and let Π_𝐶 be a sunny nonexpansive retraction from 𝑋 onto 𝐶. For each 𝑖=1,2, let 𝐵_𝑖∶𝐶→𝑋 be a 𝛼_𝑖-inverse-strongly accretive mapping and let 𝐺∶𝐶→𝐶 be the mapping defined by 𝐺𝑥=Π_𝐶Π_𝐶𝑥−𝜇₂𝐵₂𝑥−𝜇₁𝐵₁Π_𝐶𝑥−𝜇₂𝐵₂𝑥,∀𝑥∈𝐶. (30) If 0<𝜇_𝑖≤𝛼_𝑖/𝜅² for 𝑖=1,2, then 𝐺∶𝐶→𝐶 is nonexpansive.

Throughout this paper, the set of fixed points of the mapping 𝐺=Π_𝐶(𝐼−𝜇₁𝐵₁)Π_𝐶(𝐼−𝜇₂𝐵₂) is denoted by Ω.

Let 𝑋 be a real uniformly convex and 2-uniformly smooth Banach space, and let 𝐶 be a nonempty closed convex subset of 𝑋 such that 𝐶±𝐶⊂𝐶. Let 𝑉 be a 𝛾-strongly positive linear bounded operator with 𝛾∈(1,2). The purpose of this paper is to introduce and analyze implicit and explicit iterative schemes for computing a common element of the solution set of system of variational inequalities and set of zeros of an accretive operator 𝐴 with domain 𝐷(𝐴) and range 𝑅(𝐴) in 𝑋 such that 𝐷(𝐴)⊂𝐶⊂⋂_𝑟>0𝑅(𝐼+𝑟𝐴). Our methods are based on Korpelevich’s extragradient method, hybrid steepest-descent method, and viscosity approximation method. In particular, we propose the iterative methods for computing a point 𝑝∈Ω∩𝐴⁻¹𝟎, which is also a unique solution of the following variational inequality: (𝑉−𝐼)𝑝,𝐽𝑝−𝑢≤0,∀𝑢∈Ω∩𝐴⁻¹𝟎. (31) Under suitable assumptions, the strong convergence of the sequences generated by the proposed schemes is studied. Our results extend and improve several results presented in the recent past in the literature; see, for example, [10, 13, 15, 27–29] and the references therein.

Now we present some known results and definitions which will be used in the sequel.

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 (32) 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 (33) 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 (34) 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. (35)

Lemma 14 (see [30]). Let 𝐶 be a nonempty closed convex subset of a uniformly smooth Banach space 𝑋, and let {𝑥_𝑛} be a bounded sequence in 𝑋. Let 𝜇 be a mean on 𝐍 and 𝑧∈𝐶. Then, 𝜇_𝑛𝑥_𝑛−𝑧²=min_𝑦∈𝐶𝜇_𝑛𝑥_𝑛−𝑦² (36) if and only if 𝜇_𝑛𝑦−𝑧,𝐽𝑥_𝑛−𝑧≤0,∀𝑦∈𝐶, (37) where 𝐽 is the normalized duality mapping of 𝑋.

Recall that a Banach space 𝑋 is said to satisfy Opial’s condition if whenever {𝑥_𝑛} is a sequence in 𝑋 which converges weakly to 𝑥 as 𝑛→∞, then limsup_𝑛→∞𝑥_𝑛−𝑥<limsup_𝑛→∞𝑥_𝑛−𝑦,∀𝑦∈𝑋,𝑦≠𝑥. (38)

Lemma 15 (demiclosedness principle, see [31, Theorem 10.3]). Let 𝑋 be a reflexive Banach space satisfying Opial’s condition, let 𝐶 be a nonempty closed convex subset of 𝑋, and let 𝑇∶𝐶→𝐶 be a nonexpansive mapping. Then the mapping 𝐼−𝑇 is demiclosed on 𝐶, where 𝐼 is the identity mapping; that is, if {𝑥_𝑛} is a sequence of 𝐶 such that 𝑥_𝑛⇀𝑥 and (𝐼−𝑇)𝑥_𝑛→𝑦, then (𝐼−𝑇)𝑥=𝑦.

Remark 16. If the duality mapping 𝐽∶𝑥↦{𝑥^∗∈𝑋^∗∶⟨𝑥,𝑥^∗⟩=‖𝑥‖²=‖𝑥^∗‖²} from 𝑋 into 𝑋^∗ is single-valued and weakly sequentially continuous, then 𝑋 satisfies Opial’s condition; see [32, Theorem 1]. Each Hilbert space and the sequence spaces 𝑙^𝑝 with 1<𝑝<∞ satisfy Opial’s condition; see [32]. Though an 𝐿^𝑝-space with 𝑝≠2 does not usually satisfy Opial’s condition, each separable Banach space can be equivalently renormed, so that it satisfies Opial’s condition; see [33].

We close this section by mentioning a result on the sequence which will be used in the proof of the main results of this paper.

Lemma 17 (see [34]). Let {𝑠_𝑛} be a sequence of nonnegative real numbers satisfying 𝑠_𝑛+1≤1−𝛼_𝑛𝑠_𝑛+𝛼_𝑛𝛽_𝑛+𝛾_𝑛,∀𝑛≥0, (39) where {𝛼_𝑛},{𝛽_𝑛}, and {𝛾_𝑛} satisfy the following conditions:(i){𝛼_𝑛}⊂[0,1] and ∑^∞_𝑛=0𝛼_𝑛=∞;(ii)limsup_𝑛→∞𝛽_𝑛≤0;(iii)𝛾_𝑛≥0,∀𝑛≥0, and ∑^∞_𝑛=0𝛾_𝑛<∞. Then lim_𝑛→∞𝑠_𝑛=0.

2. Implicit Iterative Scheme and Its Convergence Criteria

In this section, we propose an implicit hybrid steepest-descent viscosity scheme for finding the solution of SVI (23) such that such solution is also a zero of an accretive operator. We also study the strong convergence of the sequence generated by the proposed algorithm.

Theorem 18. Let 𝐶 be a nonempty closed convex subset of a uniformly convex and 2-uniformly smooth Banach space 𝑋 such that 𝐶±𝐶⊂𝐶. Let Π_𝐶 be a sunny nonexpansive retraction from 𝑋 onto 𝐶, and let 𝐴⊂𝑋×𝑋 be an accretive operator in 𝑋 such that 𝐷(𝐴)⊂𝐶⊂⋂_𝑟>0𝑅(𝐼+𝑟𝐴). For each 𝑖=1,2, let 𝐵_𝑖∶𝐶→𝑋 be an 𝛼_𝑖-inverse-strongly accretive mapping such that Δ=Ω∩𝐴⁻¹𝟎≠∅, 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. Let 𝐹∶𝐶→𝐶 be 𝛼-strongly accretive and 𝜆-strictly pseudocontractive with 𝛼+𝜆>1, and let 𝑓∶𝐶→𝐶 be a fixed Lipschitzian strongly pseudocontractive mapping with pseudocontractive coefficient 𝛽∈(0,𝛾₀), 𝛾₀=1−(1−𝛼)/𝜆, and Lipschitzian constant 𝐿>0. Let 𝑉∶𝐶→𝐶 be a 𝛾-strongly positive linear bounded operator with 𝛾∈(1,2). For each integer 𝑛≥0, let {𝑥_𝑡,𝑛} be defined by 𝑥_𝑡,𝑛=𝐼−𝜃_𝑡𝑉𝐽_𝑟𝑛𝐺𝑥_𝑡,𝑛+𝜃_𝑡𝐽_𝑟𝑛𝐺𝑥_𝑡,𝑛−𝑡𝐹𝐽_𝑟𝑛𝐺𝑥_𝑡,𝑛−𝑓𝑥_𝑡,𝑛, (40) where {𝑟_𝑛}⊂[𝜀,∞) for some 𝜀>0 and 0<𝜃_𝑡≤‖𝑉‖⁻¹ for all 𝑡∈(0,(2−𝛾)/(𝛾₀−𝛽)) with lim_𝑡→0𝜃_𝑡=0. Then, as 𝑡→0,{𝑥_𝑡,𝑛} converges strongly to 𝑝∈Δ, which is a unique solution of the following variational inequality: (𝑉−𝐼)𝑝,𝐽𝑝−𝑢≤0,∀𝑢∈Δ. (41)

Proof. We first show that {𝑥_𝑡,𝑛} is defined well. Since 0<𝜃_𝑡≤‖𝑉‖⁻¹, for all 𝑡∈(0,(2−𝛾)/(𝛾₀−𝛽)) and ‖𝑉‖≥𝛾>1, we have 0<𝜃_𝑡≤‖𝑉‖⁻¹<1, for all 𝑡∈(0,(2−𝛾)/(𝛾₀−𝛽)). Define a mapping Γ_𝑡,𝑛∶𝐶→𝐶 by Γ_𝑡,𝑛𝑥=𝐼−𝜃_𝑡𝑉𝐽_𝑟𝑛𝐺𝑥+𝜃_𝑡𝐽_𝑟𝑛𝐺𝑥−𝑡𝐹𝐽_𝑟𝑛𝐺𝑥−𝑓(𝑥),∀𝑥∈𝐶. (42) Observe that Γ_𝑡,𝑛𝑥−Γ_𝑡,𝑛=𝑦,𝐽𝑥−𝑦𝐼−𝜃_𝑡𝑉𝐽_𝑟𝑛𝐺𝑥−𝐼−𝜃_𝑡𝑉𝐽_𝑟𝑛𝐺𝑦,𝐽𝑥−𝑦+𝜃_𝑡(𝐼−𝑡𝐹)𝐽_𝑟𝑛𝐺𝑥−(𝐼−𝑡𝐹)𝐽_𝑟𝑛𝐺𝑦,𝐽𝑥−𝑦+𝜃_𝑡𝑡𝑓(𝑥)𝑦≤−𝑓,𝐽𝑥−𝑦𝐼−𝜃_𝑡𝑉𝐽_𝑟𝑛𝐺𝑥−𝐽_𝑟𝑛𝐺𝑦𝑥−𝑦+𝜃_𝑡(𝐼−𝑡𝐹)𝐽_𝑟𝑛𝐺𝑥−(𝐼−𝑡𝐹)𝐽_𝑟𝑛𝐺𝑦𝑥−𝑦+𝑡𝜃_𝑡𝛽𝑥−𝑦²≤1−𝜃_𝑡𝛾𝐽_𝑟𝑛𝐺𝑥−𝐽_𝑟𝑛𝐺𝑦𝑥−𝑦+𝜃_𝑡1−𝑡1−1−𝛼𝜆𝐽_𝑟𝑛𝐺𝑥−𝐽_𝑟𝑛𝐺𝑦𝑥−𝑦+𝑡𝜃_𝑡𝛽𝑥−𝑦²≤1−𝜃_𝑡𝛾𝐺𝑥−𝐺𝑦𝑥−𝑦+𝜃_𝑡1−𝑡1−1−𝛼𝜆𝐺𝑥−𝐺𝑦𝑥−𝑦+𝑡𝜃_𝑡𝛽𝑥−𝑦²≤1−𝜃_𝑡𝛾𝑥−𝑦²+𝜃_𝑡1−𝑡1−1−𝛼𝜆𝑥−𝑦²+𝑡𝜃_𝑡𝛽𝑥−𝑦²=1−𝜃_𝑡𝛾−1+𝑡𝛾₀−𝛽𝑥−𝑦². (43) Whenever 𝑛≥0 and 𝑡∈(0,(2−𝛾)/(𝛾₀−𝛽)), Γ_𝑡,𝑛∶𝐶→𝐶 is a continuous and strongly pseudocontractive mapping with pseudocontractive coefficient 1−𝜃_𝑡(𝛾−1+𝑡(𝛾₀−𝛽))∈(0,1). By Lemma 5, we deduce that there exists a unique fixed point in 𝐶, denoted by 𝑥_𝑡,𝑛, which uniquely solves the fixed point equation 𝑥_𝑡,𝑛=𝐼−𝜃_𝑡𝑉𝐽_𝑟𝑛𝐺𝑥_𝑡,𝑛+𝜃_𝑡𝐽_𝑟𝑛𝐺𝑥_𝑡,𝑛−𝑡𝐹𝐽_𝑟𝑛𝐺𝑥_𝑡,𝑛−𝑓𝑥_𝑡,𝑛. (44)
Let us show the uniqueness of a solution of VIP (41). Suppose that both 𝑝₁∈Δ and 𝑝₂∈Δ are solutions to VIP (41). Then, we have (𝑉−𝐼)𝑝₁,𝐽𝑝₁−𝑝₂)𝑝≤0,(𝑉−𝐼₂,𝐽𝑝₂−𝑝₁≤0. (45) Adding up the above two inequalities, we obtain (𝑉−𝐼)𝑝₁−(𝑉−𝐼)𝑝₂,𝐽𝑝₁−𝑝₂≤0. (46) Note that (𝑉−𝐼)𝑝₁−(𝑉−𝐼)𝑝₂,𝐽𝑝₁−𝑝₂=𝑉𝑝₁−𝑝₂,𝐽𝑝₁−𝑝₂−𝑝₁−𝑝₂,𝐽𝑝₁−𝑝₂≥𝛾𝑝₁−𝑝₂²−𝑝₁−𝑝₂²=𝛾−1𝑝₁−𝑝₂²≥0. (47) Taking into account 0<𝛾−1<1, we have 𝑝₁=𝑝₂, and hence the uniqueness is proved.
Next, we show that {𝑥_𝑡,𝑛} is bounded. Indeed, we note that 0<𝜃_𝑡≤‖𝑉‖⁻¹, for all 𝑡∈(0,(2−𝛾)/(𝛾₀−𝛽)). Take a fixed 𝑝∈Δ arbitrarily. Utilizing Lemmas 6 and 7, we obtain 𝑥_𝑡,𝑛−𝑝²=𝐼−𝜃_𝑡𝑉𝐽_𝑟𝑛𝐺𝑥_𝑡,𝑛+𝜃_𝑡𝐽_𝑟𝑛𝐺𝑥_𝑡,𝑛−𝑡𝐹𝐽_𝑟𝑛𝐺𝑥_𝑡,𝑛−𝑓𝑥_𝑡,𝑛−𝑝,𝐽𝑥_𝑡,𝑛=−𝑝𝐼−𝜃_𝑡𝑉𝐽_𝑟𝑛𝐺𝑥_𝑡,𝑛−𝐼−𝜃_𝑡𝑉𝐽_𝑟𝑛𝐺𝑝,𝐽𝑥_𝑡,𝑛−𝑝+𝜃_𝑡𝐽_𝑟𝑛𝐺𝑥_𝑡,𝑛−𝑡𝐹𝐽_𝑟𝑛𝐺𝑥_𝑡,𝑛−𝑓𝑥_𝑡,𝑛−𝑝,𝐽𝑥_𝑡,𝑛−𝑝+𝜃_𝑡(𝐼−𝑉)𝑝,𝐽𝑥_𝑡,𝑛=−𝑝𝐼−𝜃_𝑡𝑉𝐽_𝑟𝑛𝐺𝑥_𝑡,𝑛−𝐼−𝜃_𝑡𝑉𝐽_𝑟𝑛𝐺𝑝,𝐽𝑥_𝑡,𝑛−𝑝+𝜃_𝑡(𝐼−𝑡𝐹)𝐽_𝑟𝑛𝐺𝑥_𝑡,𝑛−(𝐼−𝑡𝐹)𝐽_𝑟𝑛𝐺𝑝,𝐽𝑥_𝑡,𝑛𝑓𝑥−𝑝+𝑡_𝑡,𝑛−𝑓𝑝,𝐽𝑥_𝑡,𝑛+𝑓𝑝𝑝𝑥−𝑝−𝐹,𝐽_𝑡,𝑛−𝑝+𝜃_𝑡(𝐼−𝑉)𝑝,𝐽𝑥_𝑡,𝑛≤−𝑝𝐼−𝜃_𝑡𝑉𝐽_𝑟𝑛𝐺𝑥_𝑡,𝑛−𝐼−𝜃_𝑡𝑉𝐽_𝑟𝑛𝐺𝑝𝑥_𝑡,𝑛−𝑝+𝜃_𝑡(𝐼−𝑡𝐹)𝐽_𝑟𝑛𝐺𝑥_𝑡,𝑛−(𝐼−𝑡𝐹)𝐽_𝑟𝑛𝐺𝑝𝑥_𝑡,𝑛𝑓𝑥−𝑝+𝑡_𝑡,𝑛−𝑓𝑝,𝐽𝑥_𝑡,𝑛+𝑓𝑝𝑝𝑥−𝑝−𝐹_𝑡,𝑛−𝑝+𝜃_𝑡‖𝐼−𝑉‖𝑝𝑥_𝑡,𝑛≤−𝑝1−𝜃_𝑡𝛾𝐽_𝑟𝑛𝐺𝑥_𝑡,𝑛−𝐽_𝑟𝑛𝐺𝑝𝑥_𝑡,𝑛−𝑝+𝜃_𝑡1−𝑡𝛾₀𝐽_𝑟𝑛𝐺𝑥_𝑡,𝑛−𝐽_𝑟𝑛𝐺𝑝𝑥_𝑡,𝑛𝛽𝑥−𝑝+𝑡_𝑡,𝑛−𝑝²+𝑓𝑝−𝐹𝑝𝑥_𝑡,𝑛−𝑝+𝜃_𝑡‖𝐼−𝑉‖𝑝𝑥_𝑡,𝑛≤−𝑝1−𝜃_𝑡𝛾𝐺𝑥_𝑡,𝑛−𝐺𝑝𝑥_𝑡,𝑛−𝑝+𝜃_𝑡1−𝑡𝛾₀𝐺𝑥_𝑡,𝑛−𝐺𝑝𝑥_𝑡,𝑛𝛽𝑥−𝑝+𝑡_𝑡,𝑛−𝑝²+𝑓𝑝−𝐹𝑝𝑥_𝑡,𝑛−𝑝+𝜃_𝑡‖𝐼−𝑉‖𝑝𝑥_𝑡,𝑛≤−𝑝1−𝜃_𝑡𝛾𝑥_𝑡,𝑛−𝑝²+𝜃_𝑡1−𝑡𝛾₀𝑥_𝑡,𝑛−𝑝²+𝑡𝛽𝑥_𝑡,𝑛−𝑝²+𝑓𝑝−𝐹𝑝𝑥_𝑡,𝑛−𝑝+𝜃_𝑡‖𝐼−𝑉‖𝑝𝑥_𝑡,𝑛−𝑝, (48) which immediately yields 𝑥_𝑡,𝑛−𝑝≤1−𝜃_𝑡𝛾𝑥_𝑡,𝑛−𝑝+𝜃_𝑡1−𝑡𝛾₀𝑥_𝑡,𝑛𝛽𝑥−𝑝+𝑡_𝑡,𝑛−𝑝+𝑓𝑝−𝐹𝑝+𝜃_𝑡‖𝐼−𝑉‖𝑝. (49) It follows that 𝑥_𝑡,𝑛≤‖𝑝𝑓𝑝𝑝−𝑝𝐼−𝑉‖+𝑡−𝐹𝛾−1+𝑡𝛾₀≤‖𝑝+𝑓𝑝𝑝−𝛽𝐼−𝑉‖−𝐹.𝛾−1 (50) Thus, {𝑥_𝑡,𝑛} is bounded and so are {𝑓(𝑥_𝑡,𝑛)}, {𝐺𝑥_𝑡,𝑛}, and {𝐽_𝑟𝑛𝐺𝑥_𝑡,𝑛}.
Let us show that ‖𝑥_𝑡,𝑛−𝐺𝑥_𝑡,𝑛‖→0 as 𝑡→0.
Indeed, for simplicity we put 𝑞=Π_𝐶(𝐼−𝜇₂𝐵₂)𝑝,𝑢_𝑡,𝑛=Π_𝐶(𝐼−𝜇₂𝐵₂)𝑥_𝑡,𝑛 and 𝑣_𝑡,𝑛=Π_𝐶(𝐼−𝜇₁𝐵₁)𝑢_𝑡,𝑛. Then, 𝑝=Π_𝐶(𝐼−𝜇₁𝐵₁)𝑞 and 𝑣_𝑡,𝑛=𝐺𝑥_𝑡,𝑛. Hence from (48), we get 𝑥_𝑡,𝑛−𝑝²≤1−𝜃_𝑡𝛾𝑣_𝑡,𝑛−𝑝𝑥_𝑡,𝑛−𝑝+𝜃_𝑡1−𝑡𝛾₀𝑣_𝑡,𝑛−𝑝𝑥_𝑡,𝑛𝛽𝑥−𝑝+𝑡_𝑡,𝑛−𝑝²+𝑓𝑝−𝐹𝑝𝑥_𝑡,𝑛−𝑝+𝜃_𝑡‖𝐼−𝑉‖𝑝𝑥_𝑡,𝑛=−𝑝1−𝜃_𝑡𝛾−1+𝑡𝛾₀𝑣_𝑡,𝑛−𝑝𝑥_𝑡,𝑛−𝑝+𝜃_𝑡𝑡𝛽𝑥_𝑡,𝑛−𝑝²+𝑓𝑝−𝐹𝑝𝑥_𝑡,𝑛−𝑝+𝜃_𝑡‖𝐼−𝑉‖𝑝𝑥_𝑡,𝑛≤−𝑝1−𝜃_𝑡𝛾−1+𝑡𝛾₀12𝑣_𝑡,𝑛−𝑝²+𝑥_𝑡,𝑛−𝑝²+𝜃_𝑡𝑡𝛽𝑥_𝑡,𝑛−𝑝²+𝑓𝑝−𝐹𝑝𝑥_𝑡,𝑛−𝑝+𝜃_𝑡‖𝐼−𝑉‖𝑝𝑥_𝑡,𝑛−𝑝. (51) From Lemma 12, we have 𝑢_𝑡,𝑛−𝑞²=Π_𝐶(𝑥_𝑡,𝑛−𝜇₂𝐵₂𝑥_𝑡,𝑛)−Π_𝐶(𝑝−𝜇₂𝐵₂𝑝)²≤𝑥_𝑡,𝑛−𝑝−𝜇₂𝐵₂𝑥_𝑡,𝑛−𝐵₂𝑝²≤𝑥_𝑡,𝑛−𝑝²−2𝜇₂𝛼₂−𝜅²𝜇₂𝐵₂𝑥_𝑡,𝑛−𝐵₂𝑝²,𝑣_𝑡,𝑛−𝑝²=Π_𝐶𝑢_𝑡,𝑛−𝜇₁𝐵₁𝑢_𝑡,𝑛−Π_𝐶𝑞−𝜇₁𝐵₁𝑞²≤𝑢_𝑡,𝑛−𝑞−𝜇₁𝐵₁𝑢_𝑡,𝑛−𝐵₁𝑞²≤𝑢_𝑡,𝑛−𝑞²−2𝜇₁𝛼₁−𝜅²𝜇₁𝐵₁𝑢_𝑡,𝑛−𝐵₁𝑞². (52) 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, (53) which together with (51) implies that 𝑥_𝑡,𝑛−𝑝²≤1−𝜃_𝑡𝛾−1+𝑡𝛾₀12𝑥_𝑡,𝑛−𝑝²+𝑣_𝑡,𝑛−𝑝²+𝜃_𝑡𝑡𝛽𝑥_𝑡,𝑛−𝑝²+𝑓𝑝−𝐹𝑝𝑥_𝑡,𝑛−𝑝+𝜃_𝑡‖𝐼−𝑉‖𝑝𝑥_𝑡,𝑛≤−𝑝1−𝜃_𝑡𝛾−1+𝑡𝛾₀×12𝑥_𝑡,𝑛−𝑝²+𝑥_𝑡,𝑛−𝑝²−2𝜇₂𝛼₂−𝜅²𝜇₂𝐵₂𝑥_𝑡,𝑛−𝐵₂𝑝²−2𝜇₁𝛼₁−𝜅²𝜇₁𝐵₁𝑢_𝑡,𝑛−𝐵₁𝑞²+𝜃_𝑡𝑡𝛽𝑥_𝑡,𝑛−𝑝²+𝑓𝑝−𝐹𝑝𝑥_𝑡,𝑛−𝑝+𝜃_𝑡‖𝐼−𝑉‖𝑝𝑥_𝑡,𝑛=−𝑝1−𝜃_𝑡𝛾−1+𝑡𝛾₀−𝛽𝑥_𝑡,𝑛−𝑝²+𝜃_𝑡𝑡𝑓𝑝−𝐹𝑝𝑥_𝑡,𝑛−𝑝+𝜃_𝑡‖𝐼−𝑉‖𝑝𝑥_𝑡,𝑛−−𝑝1−𝜃_𝑡𝛾−1+𝑡𝛾₀×𝜇₂𝛼₂−𝜅²𝜇₂𝐵₂𝑥_𝑡,𝑛−𝐵₂𝑝²+𝜇₁𝛼₁−𝜅²𝜇₁𝐵₁𝑢_𝑡,𝑛−𝐵₁𝑞²≤𝑥_𝑡,𝑛−𝑝²+𝜃_𝑡𝑡𝑓𝑝−𝐹𝑝𝑥_𝑡,𝑛−𝑝+𝜃_𝑡‖𝐼−𝑉‖𝑝𝑥_𝑡,𝑛−−𝑝1−𝜃_𝑡𝛾−1+𝑡𝛾₀×𝜇₂𝛼₂−𝜅²𝜇₂𝐵₂𝑥_𝑡,𝑛−𝐵₂𝑝²+𝜇₁𝛼₁−𝜅²𝜇₁𝐵₁𝑢_𝑡,𝑛−𝐵₁𝑞². (54) Therefore, 1−𝜃_𝑡𝛾−1+𝑡𝛾₀𝜇₂𝛼₂−𝜅²𝜇₂𝐵₂𝑥_𝑡,𝑛−𝐵₂𝑝²+𝜇₁𝛼₁−𝜅²𝜇₁𝐵₁𝑢_𝑡,𝑛−𝐵₁𝑞²≤𝜃_𝑡𝑡𝑓𝑝−𝐹𝑝𝑥_𝑡,𝑛−𝑝+𝜃_𝑡‖𝐼−𝑉‖𝑝𝑥_𝑡,𝑛−𝑝. (55) Since 0<𝜇_𝑖<𝛼_𝑖/𝜅² for 𝑖=1,2, it follows from lim_𝑡→0𝜃_𝑡=0 that lim_𝑡→0𝐵₂𝑥_𝑡,𝑛−𝐵₂𝑝=0,lim_𝑡→0𝐵₁𝑢_𝑡,𝑛−𝐵₁𝑞=0. (56) Utilizing Proposition 3 and Lemma 9, we have 𝑢_𝑡,𝑛−𝑞²=Π_𝐶𝑥_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, (57) 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. (58) Similarly, 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+𝜇₁⟨𝐵₁𝑞−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, (59) 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. (60) Combining (58) and (60), 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−𝑝, (61) which together with (51) implies that 𝑥_𝑡,𝑛−𝑝²≤1−𝜃_𝑡𝛾−1+𝑡𝛾₀12𝑥_𝑡,𝑛−𝑝²+𝑣_𝑡,𝑛−𝑝²+𝜃_𝑡𝑡𝛽𝑥_𝑡,𝑛−𝑝²+𝑓𝑝−𝐹𝑝𝑥_𝑡,𝑛−𝑝+𝜃_𝑡‖𝐼−𝑉‖𝑝𝑥_𝑡,𝑛≤−𝑝1−𝜃_𝑡𝛾−1+𝑡𝛾₀×12𝑥_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_𝑡𝑡𝛽𝑥_𝑡,𝑛−𝑝²+𝑓𝑝−𝐹𝑝𝑥_𝑡,𝑛−𝑝+𝜃_𝑡‖𝐼−𝑉‖𝑝𝑥_𝑡,𝑛=−𝑝1−𝜃_𝑡𝛾−1+𝑡𝛾₀−𝛽𝑥_𝑡,𝑛−𝑝²−121−𝜃_𝑡𝛾−1+𝑡𝛾₀×𝑔₁𝑥_𝑡,𝑛−𝑢_𝑡,𝑛−𝑝−𝑞+𝑔₂𝑢_𝑡,𝑛−𝑣_𝑡,𝑛++𝑝−𝑞1−𝜃_𝑡𝛾−1+𝑡𝛾₀×𝜇₂𝐵₂𝑝−𝐵₂𝑥_𝑡,𝑛𝑢_𝑡,𝑛−𝑞+𝜇₁𝐵₁𝑞−𝐵₁𝑢_𝑡,𝑛𝑣_𝑡,𝑛−𝑝+𝜃_𝑡𝑡𝛽𝑥_𝑡,𝑛−𝑝²+𝑓𝑝−𝐹𝑝𝑥_𝑡,𝑛−𝑝+𝜃_𝑡‖𝐼−𝑉‖𝑝𝑥_𝑡,𝑛≤𝑥−𝑝_𝑡,𝑛−𝑝²−121−𝜃_𝑡𝛾−1+𝑡𝛾₀×𝑔₁𝑥_𝑡,𝑛−𝑢_𝑡,𝑛−𝑝−𝑞+𝑔₂𝑢_𝑡,𝑛−𝑣_𝑡,𝑛++𝑝−𝑞1−𝜃_𝑡𝛾−1+𝑡𝛾₀×𝜇₂𝐵₂𝑝−𝐵₂𝑥_𝑡,𝑛𝑢_𝑡,𝑛−𝑞+𝜇₁𝐵₁𝑞−𝐵₁𝑢_𝑡,𝑛𝑣_𝑡,𝑛−𝑝+𝜃_𝑡𝑡𝛽𝑥_𝑡,𝑛−𝑝²+𝑓𝑝−𝐹𝑝𝑥_𝑡,𝑛−𝑝+𝜃_𝑡‖𝐼−𝑉‖𝑝𝑥_𝑡,𝑛−𝑝. (62) Therefore, 121−𝜃_𝑡𝛾−1+𝑡𝛾₀𝑔₁𝑥_𝑡,𝑛−𝑢_𝑡,𝑛−𝑝−𝑞+𝑔₂𝑢_𝑡,𝑛−𝑣_𝑡,𝑛+≤𝑝−𝑞1−𝜃_𝑡𝛾−1+𝑡𝛾₀𝜇₂𝐵₂𝑝−𝐵₂𝑥_𝑡,𝑛𝑢_𝑡,𝑛−𝑞+𝜇₁𝐵₁𝑞−𝐵₁𝑢_𝑡,𝑛𝑣_𝑡,𝑛−𝑝+𝜃_𝑡𝑡𝛽𝑥_𝑡,𝑛−𝑝²+𝑓𝑝−𝐹𝑝𝑥_𝑡,𝑛−𝑝+𝜃_𝑡‖𝐼−𝑉‖𝑝𝑥_𝑡,𝑛−𝑝. (63) Hence, from (56) and lim_𝑡→0𝜃_𝑡=0, 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. (64) 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, (65) 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; (66) that is, lim_𝑡→0𝑥_𝑡,𝑛−𝐺𝑥_𝑡,𝑛=lim_𝑡→0𝑥_𝑡,𝑛−𝑣_𝑡,𝑛=0. (67) Note that Fix(𝐽_𝑟𝑛)=𝐴⁻¹0 for all 𝑛≥0 and that {𝑥_𝑡,𝑛∶𝑡∈(0,𝑏),𝑛≥0} is bounded and so are {𝑓(𝑥_𝑡,𝑛)∶𝑡∈(0,𝑏),𝑛≥0}, {𝐽_𝑟𝑛𝑥_𝑡,𝑛∶𝑡∈(0,𝑏),𝑛≥0} and {𝐽_𝑟𝑛𝐺𝑥_𝑡,𝑛∶𝑡∈(0,𝑏),𝑛≥0}, where 𝑏=(2−𝛾)/(𝛾₀−𝛽). Hence, we have 𝑥_𝑡,𝑛−𝐽_𝑟𝑛𝐺𝑥_𝑡,𝑛=𝜃_𝑡(𝐴−𝐼)𝐽_𝑟𝑛𝐺𝑥_𝑡,𝑛−𝑡𝑓𝑥_𝑡,𝑛−𝐹𝐽_𝑟𝑛𝐺𝑥_𝑡,𝑛⟶0,as𝑡⟶0. (68) Also, observe that 𝑥_𝑡,𝑛−𝐽_𝑟𝑛𝑥_𝑡,𝑛≤𝑥_𝑡,𝑛−𝐽_𝑟𝑛𝐺𝑥_𝑡,𝑛+𝐽_𝑟𝑛𝐺𝑥_𝑡,𝑛−𝐽_𝑟𝑛𝑥_𝑡,𝑛≤𝑥_𝑡,𝑛−𝐽_𝑟𝑛𝐺𝑥_𝑡,𝑛+𝐺𝑥_𝑡,𝑛−𝑥_𝑡,𝑛. (69) This together with (67) and (68) implies that lim_𝑡→0𝑥_𝑡,𝑛−𝐽_𝑟𝑛𝑥_𝑡,𝑛=0. (70) Since 𝑟_𝑛≥𝜀 for all 𝑛≥0, utilizing Lemma 8, we have 𝑥_𝑡,𝑛−𝐽_𝜀𝑥_𝑡,𝑛≤𝑥_𝑡,𝑛−𝐽_𝑟𝑛𝑥_𝑡,𝑛+𝐽_𝑟𝑛𝑥_𝑡,𝑛−𝐽_𝜀𝑥_𝑡,𝑛=𝑥_𝑡,𝑛−𝐽_𝑟𝑛𝑥_𝑡,𝑛+𝐽_𝜀𝜀𝑟_𝑛𝑥_𝑡,𝑛+𝜀1−𝑟_𝑛𝐽_𝑟𝑛𝑥_𝑡,𝑛−𝐽_𝜀𝑥_𝑡,𝑛≤2𝑥_𝑡,𝑛−𝐽_𝑟𝑛𝑥_𝑡,𝑛⟶0,as𝑡⟶0. (71) For any integer 𝑛≥0, for simplicity we put 𝑤_𝑡=𝑥_𝑡,𝑛 for all 𝑡∈(0,𝑏). Now let {𝑡_𝑘} be a sequence in (0,𝑏) that converges to 0 as 𝑘→∞, and define a function 𝑔∶𝐶→ℝ by 𝑔(𝑤)=𝜇_𝑘12𝑤_𝑡𝑘−𝑤²,∀𝑤∈𝐶, (72) where 𝜇 is a Banach limit. Define a set 𝐾∶=𝑤∈𝐶∶𝑔(𝑤)=min𝑔𝑦∶𝑦∈𝐶 (73) and a mapping 𝑊𝑥=(1−𝜃)𝐽_𝜀𝑥+𝜃𝐺𝑥,∀𝑥∈𝐶, (74) where 𝜃 is a constant in (0,1). Then by Lemma 4, we have 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 (67) 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 (see [30]). We claim that 𝐾 is also invariant under the nonexpansive mapping 𝑊. Indeed, noticing (76), we have for all 𝑤∈𝐾, 𝑔(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 15, we get 𝜇_𝑘𝑥−𝑝,𝐽𝑤_𝑡𝑘−𝑝≤0,∀𝑥∈𝐶. (78) Putting 𝑥=(2𝐼−𝑉)𝑝, we have 𝜇_𝑘(𝐼−𝑉)𝑝,𝐽𝑤_𝑡𝑘−𝑝≤0. (79) Noticing 𝑤_𝑡𝑘=𝑥_{𝑡𝑘,𝑛}, we conclude from (48) that 𝑤_𝑡𝑘−𝑝²=𝐼−𝜃_𝑡𝑘𝑉𝐽_𝑟𝑛𝐺𝑤_𝑡𝑘−𝐼−𝜃_𝑡𝑘𝑉𝐽_𝑟𝑛𝐺𝑝,𝐽𝑤_𝑡𝑘−𝑝+𝜃_𝑡𝑘𝐼−𝑡_𝑘𝐹𝐽_𝑟𝑛𝐺𝑤_𝑡𝑘−𝐼−𝑡_𝑘𝐹𝐽_𝑟𝑛𝐺𝑝,𝐽𝑤_𝑡𝑘−𝑝+𝑡_𝑘𝑓𝑤_𝑡𝑘−𝑓𝑝,𝐽𝑤_𝑡𝑘+𝑓𝑝𝑝𝑤−𝑝−𝐹,𝐽_𝑡𝑘−𝑝+𝜃_𝑡𝑘(𝐼−𝑉)𝑝,𝐽𝑤_𝑡𝑘≤−𝑝1−𝜃_𝑡𝑘𝛾𝐽_𝑟𝑛𝐺𝑤_𝑡𝑘−𝐽_𝑟𝑛𝐺𝑝𝑤_𝑡𝑘−𝑝+𝜃_𝑡𝑘1−𝑡_𝑘𝛾₀𝐽_𝑟𝑛𝐺𝑤_𝑡𝑘−𝐽_𝑟𝑛𝐺𝑝𝑤_𝑡𝑘−𝑝+𝑡_𝑘𝛽𝑤_𝑡𝑘−𝑝²+𝑓𝑝−𝐹𝑝𝑤_𝑡𝑘−𝑝+𝜃_𝑡𝑘(𝐼−𝑉)𝑝,𝐽𝑤_𝑡𝑘≤−𝑝1−𝜃_𝑡𝑘𝛾𝑤_𝑡𝑘−𝑝²+𝜃_𝑡𝑘1−𝑡_𝑘𝛾₀𝑤_𝑡𝑘−𝑝²+𝑡_𝑘𝛽𝑤_𝑡𝑘−𝑝²+𝑓𝑝−𝐹𝑝𝑤_𝑡𝑘−𝑝+𝜃_𝑡𝑘(𝐼−𝑉)𝑝,𝐽𝑤_𝑡𝑘=−𝑝1−𝜃_𝑡𝑘𝛾−1+𝑡_𝑘𝛾₀−𝛽𝑤_𝑡𝑘−𝑝²+𝜃_𝑡𝑘𝑡_𝑘𝑓𝑝−𝐹𝑝𝑤_𝑡𝑘−𝑝+𝜃_𝑡𝑘(𝐼−𝑉)𝑝,𝐽𝑤_𝑡𝑘−𝑝. (80) It follows that 𝑤_𝑡𝑘−𝑝²≤1𝛾−1+𝑡_𝑘𝛾₀)𝑤−𝛽(𝐼−𝑉𝑝,𝐽_𝑡𝑘−𝑝+𝑡_𝑘𝑓𝑝−𝐹𝑝𝑤_𝑡𝑘≤1−𝑝)𝑤𝛾−1(𝐼−𝑉𝑝,𝐽_𝑡𝑘−𝑝+𝑡_𝑘𝑓𝑝−𝐹𝑝𝑤_𝑡𝑘−𝑝. (81) Since lim_𝑘→∞𝑡_𝑘=0, from (79) and the boundedness of {𝑤_𝑡𝑘}, it follows that 𝜇_𝑘𝑤_𝑡𝑘−𝑝²≤1𝜇𝛾−1^𝑘(𝐼−𝑉)𝑝,𝐽𝑤_𝑡𝑘−𝑝+𝑡_𝑘𝑓𝑝−𝐹𝑝𝑤_𝑡𝑘=1−𝑝𝜇𝛾−1^𝑘(𝐼−𝑉)𝑝,𝐽𝑤_𝑡𝑘−𝑝+𝜇_𝑘𝑡_𝑘𝑓𝑝−𝐹𝑝𝑤_𝑡𝑘−𝑝≤0. (82) Therefore, it is known that for any sequence {𝑤_𝑡𝑘} in {𝑤_𝑡∶𝑡∈(0,𝑏)}, there exists a subsequence which is still denoted by {𝑤_𝑡𝑘} that converges strongly to some fixed point 𝑝 of 𝑊.
Now we claim that such a 𝑝∈Δ is a unique solution of variational inequality (41).
Indeed, from (48), it follows that for all 𝑢∈Δ=Ω∩𝐴⁻¹𝟎𝑤_𝑡−𝑢²=𝐼−𝜃_𝑡𝑉𝐽_𝑟𝑛𝐺𝑤_𝑡−𝐼−𝜃_𝑡𝑉𝐽_𝑟𝑛𝐺𝑢,𝐽𝑤_𝑡−𝑢+𝜃_𝑡(𝐼−𝑡𝐹)𝐽_𝑟𝑛𝐺𝑤_𝑡−(𝐼−𝑡𝐹)𝐽_𝑟𝑛𝐺𝑢,𝐽𝑤_𝑡𝑓𝑤−𝑢+𝑡_𝑡−𝑓(𝑢),𝐽𝑤_𝑡+𝑓(𝑢)(𝑢)𝑤−𝑢−𝐹,𝐽_𝑡−𝑢+𝜃_𝑡(𝐼−𝑉)𝑢,𝐽𝑤_𝑡≤−𝑢1−𝜃_𝑡𝛾𝐽_𝑟𝑛𝐺𝑤_𝑡−𝐽_𝑟𝑛𝐺𝑢𝑤_𝑡−𝑢+𝜃_𝑡1−𝑡𝛾₀𝐽_𝑟𝑛𝐺𝑤_𝑡−𝐽_𝑟𝑛𝐺𝑢𝑤_𝑡𝛽𝑤−𝑢+𝑡_𝑡−𝑢²+𝑓(𝑢)−𝐹(𝑢)𝑤_𝑡−𝑢+𝜃_𝑡(𝐼−𝑉)𝑢,𝐽𝑤_𝑡≤−𝑢1−𝜃_𝑡𝛾𝑤_𝑡−𝑢²+𝜃_𝑡1−𝑡𝛾₀𝑤_𝑡−𝑢²+𝑡𝛽𝑤_𝑡−𝑢²+𝑓(𝑢)−𝐹(𝑢)𝑤_𝑡−𝑢+𝜃_𝑡(𝐼−𝑉)𝑢,𝐽𝑤_𝑡=−𝑢1−𝜃_𝑡𝛾−1+𝑡𝛾₀−𝛽𝑤_𝑡−𝑢²+𝜃_𝑡𝑡𝑓(𝑢)−𝐹(𝑢)𝑤_𝑡−𝑢+𝜃_𝑡(𝐼−𝑉)𝑢,𝐽𝑤_𝑡≤𝑤−𝑢_𝑡−𝑢²+𝜃_𝑡𝑡𝑓(𝑢)−𝐹(𝑢)𝑤_𝑡−𝑢+𝜃_𝑡(𝐼−𝑉)𝑢,𝐽𝑤_𝑡−𝑢, (83) which implies that (𝑉−𝐼)𝑢,𝐽𝑤_𝑡−𝑢≤𝑡𝑓(𝑢)−𝐹(𝑢)𝑤_𝑡−𝑢,∀𝑢∈Δ. (84) Since 𝑤_𝑡𝑘→𝑝 as 𝑡_𝑘→0, we obtain from the last inequality that (𝑉−𝐼)𝑢,𝐽𝑝−𝑢≤0,∀𝑢∈Δ. (85) Utilizing the well-known Minty-type Lemma [25], we get (𝑉−𝐼)𝑝,𝐽𝑝−𝑢≤0,∀𝑢∈Δ. (86) So, 𝑝∈Δ is a solution of variational inequality (41).
In order to prove that the net {𝑥_𝑡∶𝑡∈(0,𝑏)} converges strongly to 𝑝 as 𝑡→0, suppose that there exists another subsequence {𝑤_𝑠𝑘}⊂{𝑤_𝑡∶𝑡∈(0,𝑏)} such that 𝑤_𝑠𝑘→𝑞 as 𝑠_𝑘→0; then we also have 𝑞∈Fix(𝑊)=Ω∩𝐴⁻¹𝟎=∶Δ due to (76). Repeating the same argument as above, we know that 𝑞∈Δ is another solution of variational inequality (41). In terms of the uniqueness of solutions in Δ to the variational inequality (41), we get 𝑝=𝑞.

Remark 19. In the assertion of Theorem 18, “as 𝑡→0,{𝑥_𝑡,𝑛} converges strongly to 𝑝∈Δ”, 𝑝 does not depend on 𝑛. Indeed, it is known that {𝑟_𝑛}⊂[𝜀,∞) for some 𝜀>0. Moreover, in the proof of Theorem 18, it can be readily seen that 𝑝 is a fixed point of the nonexpansive self-mapping 𝑊 of 𝐾. This shows that 𝑝 depends on neither 𝑛 nor 𝑡.

Remark 20. Theorem 18 improves and extends [15, Theorem 3.1], [10, Lemma 2.5], and [13, Theorem 3.1] in the following aspects.(a)The problem and iterative scheme in Theorem 18 are different from those considered in [15, Theorem 3.1], [10, Lemma 2.5], and [13, Theorem 3.1].(b)Theorem 18 drops the assumption of the asymptotical regularity for the iterative scheme in [13, Theorem 3.1].

3. Explicit Iterative Scheme and Its Convergence Criteria

In this section, we propose an explicit iterative scheme to compute a common element of the solution set of SVI (23) and the solution set of zeros of an accretive operator. We study the strong convergence of the sequence generated by the proposed scheme.

Theorem 21. Let 𝑋 be an uniformly convex and 2-uniformly smooth Banach space which has the weakly sequentially continuous duality mapping 𝐽. Let 𝐶 be a nonempty closed convex subset of 𝑋 such that 𝐶±𝐶⊂𝐶, and let Π_𝐶 be a sunny nonexpansive retraction from 𝑋 onto 𝐶. Let 𝐴⊂𝑋×𝑋 be an accretive operator in 𝑋 such that 𝐷(𝐴)⊂𝐶⊂⋂_𝑟>0𝑅(𝐼+𝑟𝐴). For each 𝑖=1,2, let 𝐵_𝑖∶𝐶→𝑋 be an 𝛼_𝑖-inverse-strongly accretive mapping such that Δ=Ω∩𝐴⁻¹𝟎≠∅, 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. Let 𝐹∶𝐶→𝐶 be 𝛼-strongly accretive and 𝜆-strictly pseudocontractive with 𝛼+𝜆>1, and let 𝑓∶𝐶→𝐶 be a contractive mapping with coefficient 𝛽∈(0,𝛾₀), where 𝛾₀=1−(1−𝛼)/𝜆. Let 𝑉∶𝐶→𝐶 be a 𝛾-strongly positive linear bounded operator with 𝛾∈(1,2). For any given point 𝑥₀∈𝐶, the sequence {𝑥_𝑛} is generated by 𝑦_𝑛=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝐽_𝑟𝑛𝐺𝑥_𝑛,𝑥_𝑛+1=𝐼−𝛽_𝑛𝑉𝐽_𝑟𝑛𝐺𝑦_𝑛+𝛽_𝑛𝐽_𝑟𝑛𝐺𝑦_𝑛−𝜆_𝑛𝐹𝐽_𝑟𝑛𝐺𝑦_𝑛−𝑓𝑦_𝑛,∀𝑛≥0, (87) where {𝑟_𝑛}^∞_𝑛=0⊂[𝜀,∞) for some 𝜀>0 and {𝛼_𝑛}^∞_𝑛=0, {𝛽_𝑛}^∞_𝑛=0, and {𝜆_𝑛}^∞_𝑛=0 are the sequences in (0,1] satisfying the following conditions:(i)lim_𝑛→∞𝜆_𝑛=0;(ii)lim_𝑛→∞𝛽_𝑛=0 and ∑^∞_𝑛=0𝛽_𝑛=∞;(iii){𝛼_𝑛}⊂[𝑎,𝑏] for some 𝑎,𝑏∈(0,1);(iv)∑^∞_𝑛=0(|𝛼_𝑛+1−𝛼_𝑛|+|𝛽_𝑛+1−𝛽_𝑛|+|𝑟_𝑛+1−𝑟_𝑛|)<∞. Then, {𝑥_𝑛} converges strongly to 𝑝∈Δ, which is a unique solution of the following variational inequality: (𝑉−𝐼)𝑝,𝐽𝑝−𝑢≤0,∀𝑢∈Δ. (88)

Proof. We first show that {𝑥_𝑛} is bounded. Indeed, since lim_𝑛→∞𝛽_𝑛=0 and ‖𝑉‖≥𝛾>1, without loss of generality, we may assume that 0<𝛽_𝑛≤‖𝑉‖⁻¹<1, for all 𝑛≥0. Take a fixed 𝑝∈Δ arbitrarily. Then, 𝑝=𝐺𝑝, 𝑝=𝐽_𝑟𝑛𝑝, for all 𝑛≥0, and 𝑦_𝑛−𝑝≤𝛼_𝑛𝑥_𝑛−𝑝+1−𝛼_𝑛𝐽_𝑟𝑛𝐺𝑥_𝑛−𝑝≤𝛼_𝑛𝑥_𝑛−𝑝+1−𝛼_𝑛𝐺𝑥_𝑛−𝑝≤𝛼_𝑛𝑥_𝑛−𝑝+1−𝛼_𝑛𝑥_𝑛=𝑥−𝑝_𝑛−𝑝. (89) Note that 𝑥_𝑛+1=−𝑝𝐼−𝛽_𝑛𝑉𝐽_𝑟𝑛𝐺𝑥_𝑛−𝐼−𝛽_𝑛𝑉𝐽_𝑟𝑛𝐺𝑝+𝛽_𝑛𝐽_𝑟𝑛𝐺𝑦_𝑛−𝜆_𝑛𝐹𝐽_𝑟𝑛𝐺𝑦_𝑛−𝑓𝑦_𝑛−𝑝+𝛽_𝑛()𝑝=𝐼−𝑉𝐼−𝛽_𝑛𝑉𝐽_𝑟𝑛𝐺𝑥_𝑛−𝐽_𝑟𝑛𝐺𝑝+𝛽_𝑛𝜆_𝑛𝑓𝑦_𝑛−𝐹𝑝+𝐼−𝜆_𝑛𝐹𝐽_𝑟𝑛𝐺𝑦_𝑛−𝐼−𝜆_𝑛𝐹𝐽_𝑟𝑛𝐺𝑝+𝛽_𝑛(𝐼−𝑉)𝑝. (90) Therefore, we have 𝑥_𝑛+1=−𝑝𝐼−𝛽_𝑛𝑉𝐽_𝑟𝑛𝐺𝑦_𝑛−𝐽_𝑟𝑛𝐺𝑝+𝛽_𝑛𝜆_𝑛𝑓𝑦_𝑛𝑝+−𝐹𝐼−𝜆_𝑛𝐹𝐽_𝑟𝑛𝐺𝑦_𝑛−𝐼−𝛼_𝑛𝐹𝐽_𝑟𝑛𝐺𝑝+𝛽_𝑛()𝑝≤𝐼−𝑉𝐼−𝛽_𝑛𝑉𝐽_𝑟𝑛𝐺𝑦_𝑛−𝐽_𝑟𝑛𝐺𝑝+𝛽_𝑛𝜆_𝑛𝑓𝑦_𝑛𝑝+−𝐹𝐼−𝜆_𝑛𝐹𝐽_𝑟𝑛𝐺𝑦_𝑛−𝐼−𝜆_𝑛𝐹𝐽_𝑟𝑛𝐺𝑝+𝛽_𝑛)𝑝≤(𝐼−𝑉1−𝛽_𝑛𝛾𝑦_𝑛−𝑝+𝛽_𝑛𝜆_𝑛𝑓𝑦_𝑛𝑝+𝑓𝑝𝑝+−𝑓−𝐹1−𝜆_𝑛𝛾₀𝐽_𝑟𝑛𝐺𝑦_𝑛−𝐽_𝑟𝑛𝐺𝑝+𝛽_𝑛‖𝑝≤𝐼−𝑉‖1−𝛽_𝑛𝛾𝑦_𝑛−𝑝+𝛽_𝑛𝜆_𝑛𝛽𝑦_𝑛+𝑓𝑝𝑝+−𝑝−𝐹1−𝜆_𝑛𝛾₀𝑦_𝑛−𝑝+𝛽_𝑛‖𝑝≤𝐼−𝑉‖1−𝛽_𝑛𝛾𝑦_𝑛−𝑝+𝛽_𝑛1−𝜆_𝑛𝛾₀−𝛽𝑦_𝑛−𝑝+𝛼_𝑛𝑓𝑝−𝐹𝑝+𝛽_𝑛‖𝑝≤𝐼−𝑉‖1−𝛽_𝑛𝛾𝑦_𝑛−𝑝+𝛽_𝑛𝑦_𝑛−𝑝+𝑓𝑝−𝐹𝑝+𝛽_𝑛‖𝑝=𝐼−𝑉‖1−𝛽_𝑛𝛾−1𝑦_𝑛−𝑝+𝛽_𝑛𝑓𝑝𝑝+‖𝑝=−𝐹𝐼−𝑉‖1−𝛽_𝑛𝛾−1𝑦_𝑛−𝑝+𝛽_𝑛𝑓𝑝𝑝+‖𝑝𝛾−1−𝐹𝐼−𝑉‖𝑦𝛾−1≤max_𝑛,𝑓𝑝𝑝+‖𝑝−𝑝−𝐹𝐼−𝑉‖𝑥𝛾−1≤max_𝑛,𝑓𝑝𝑝+‖𝑝−𝑝−𝐹𝐼−𝑉‖.𝛾−1 (91) By induction, we obtain 𝑥_𝑛−𝑝≤max𝑥₀,𝑓𝑝𝑝+‖𝑝−𝑝−𝐹𝐼−𝑉‖,𝛾−1∀𝑛≥0, (92) and hence {𝑥_𝑛} is bounded and so are {𝑦_𝑛}, {𝐽_𝑟𝑛𝐺𝑥_𝑛}, {𝐽_𝑟𝑛𝐺𝑦_𝑛}, {𝑓(𝑥_𝑛)}, and {𝐹(𝐽_𝑟𝑛𝐺𝑦_𝑛)}. From conditions (i), (ii), we have 𝑥_𝑛+1−𝐽_𝑟𝑛𝐺𝑦_𝑛=𝛽_𝑛(𝐼−𝑉)𝐽_𝑟𝑛𝐺𝑦_𝑛−𝜆_𝑛𝐹𝐽_𝑟𝑛𝐺𝑦_𝑛−𝑓𝑦_𝑛⟶0,as𝑛⟶∞. (93)
Now we claim that lim_𝑛→∞𝑥_𝑛+1−𝑥_𝑛=0. (94) Indeed, (88) can be rewritten as 𝑦_𝑛=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝐽_𝑟𝑛𝐺𝑥_𝑛,𝑧_𝑛=𝐽_𝑟𝑛𝐺𝑦_𝑛−𝜆_𝑛𝐹𝐽_𝑟𝑛𝐺𝑦_𝑛−𝑓𝑦_𝑛,𝑥_𝑛+1=𝐼−𝛽_𝑛𝑉𝐽_𝑟𝑛𝐺𝑦_𝑛+𝛽_𝑛𝑧_𝑛,∀𝑛≥0. (95) If 𝑟_𝑛−1≤𝑟_𝑛, using the resolvent identity in Lemma 8, 𝐽_𝑟𝑛𝐺𝑥_𝑛=𝐽_𝑟𝑛−1𝑟_𝑛−1𝑟_𝑛𝐺𝑥_𝑛+𝑟1−^𝑛−1𝑟_𝑛𝐽_𝑟𝑛𝐺𝑥_𝑛. (96) We get 𝐽_𝑟𝑛𝐺𝑥_𝑛−𝐽_𝑟𝑛−1𝐺𝑥_𝑛−1=𝐽_𝑟𝑛−1𝑟_𝑛−1𝑟_𝑛𝐺𝑥_𝑛+𝑟1−^𝑛−1𝑟_𝑛𝐽_𝑟𝑛𝐺𝑥_𝑛−𝐽_𝑟𝑛−1𝐺𝑥_𝑛−1≤𝑟_𝑛−1𝑟_𝑛𝐺𝑥_𝑛−𝐺𝑥_𝑛−1+𝑟1−^𝑛−1𝑟_𝑛𝐽_𝑟𝑛𝐺𝑥_𝑛−𝐺𝑥_𝑛−1≤𝑥_𝑛−𝑥_𝑛−1+𝑟_𝑛−𝑟_𝑛−1𝑟_𝑛𝐽_𝑟𝑛𝐺𝑥_𝑛−𝐺𝑥_𝑛−1≤𝑥_𝑛−𝑥_𝑛−1+1𝜀𝑟_𝑛−𝑟_𝑛−1𝐽_𝑟𝑛𝐺𝑥_𝑛−𝐺𝑥_𝑛−1. (97) If 𝑟_𝑛≤𝑟_𝑛−1, then it can be easily seen that 𝐽_𝑟𝑛𝐺𝑥_𝑛−𝐽_𝑟𝑛−1𝐺𝑥_𝑛−1≤𝑥_𝑛−1−𝑥_𝑛+1𝜀𝑟_𝑛−1−𝑟_𝑛𝐽_𝑟𝑛−1𝐺𝑥_𝑛−1−𝐺𝑥_𝑛. (98) By combining the above two cases, we obtain 𝐽_𝑟𝑛𝐺𝑥_𝑛−𝐽_𝑟𝑛−1𝐺𝑥_𝑛−1≤𝑥_𝑛−1−𝑥_𝑛+𝑟_𝑛−1−𝑟_𝑛𝜀×sup_𝑛≥1𝐽_𝑟𝑛𝐺𝑥_𝑛−𝐺𝑥_𝑛−1+𝐽_𝑟𝑛−1𝐺𝑥_𝑛−1−𝐺𝑥_𝑛,∀𝑛≥1. (99) Similarly, we obtain 𝐽_𝑟𝑛𝐺𝑦_𝑛−𝐽_𝑟𝑛−1𝐺𝑦_𝑛−1≤𝑦_𝑛−1−𝑦_𝑛+𝑟_𝑛−1−𝑟_𝑛𝜀×sup_𝑛≥1𝐽_𝑟𝑛𝐺𝑦_𝑛−𝐺𝑦_𝑛−1+𝐽_𝑟𝑛−1𝐺𝑦_𝑛−1−𝐺𝑦_𝑛,∀𝑛≥1. (100) Thus, 𝐽_𝑟𝑛𝐺𝑥_𝑛−𝐽_𝑟𝑛−1𝐺𝑥_𝑛−1≤𝑥_𝑛−1−𝑥_𝑛+𝑟_𝑛−1−𝑟_𝑛𝑀₀,𝐽_𝑟𝑛𝐺𝑦_𝑛−𝐽_𝑟𝑛−1𝐺𝑦_𝑛−1≤𝑦_𝑛−1−𝑦_𝑛+𝑟_𝑛−1−𝑟_𝑛𝑀₀,∀𝑛≥1, (101) where sup_𝑛≥1{(1/𝜀)(‖𝐽_𝑟𝑛𝐺𝑥_𝑛−𝐺𝑥_𝑛−1‖+‖𝐽_𝑟𝑛−1𝐺𝑥_𝑛−1−𝐺𝑥_𝑛‖)}≤𝑀₀ and sup_𝑛≥1{(1/𝜀)(‖𝐽_𝑟𝑛𝐺𝑦_𝑛−𝐺𝑦_𝑛−1‖+‖𝐽_𝑟𝑛−1𝐺𝑦_𝑛−1−𝐺𝑦_𝑛‖)}≤𝑀₀ for some 𝑀₀>0. From (95) and (101), we have 𝑦_𝑛−𝑦_𝑛−1=1−𝛼_𝑛𝐽_𝑟𝑛𝐺𝑥_𝑛−𝐽_𝑟𝑛−1𝐺𝑥_𝑛−1+𝛼_𝑛𝑥_𝑛−𝑥_𝑛−1+𝑥_𝑛−1−𝐽_𝑟𝑛−1𝐺𝑥_𝑛−1𝛼_𝑛−𝛼_𝑛−1≤1−𝛼_𝑛𝐽_𝑟𝑛𝐺𝑥_𝑛−𝐽_𝑟𝑛−1𝐺𝑥_𝑛−1+𝛼_𝑛𝑥_𝑛−𝑥_𝑛−1+𝑥_𝑛−1−𝐽_𝑟𝑛−1𝐺𝑥_𝑛−1𝛼_𝑛−𝛼_𝑛−1≤1−𝛼_𝑛𝐽_𝑟𝑛𝐺𝑥_𝑛−𝐽_𝑟𝑛−1𝐺𝑥_𝑛−1+𝛼_𝑛𝑥_𝑛−𝑥_𝑛−1+𝑥_𝑛−1−𝐽_𝑟𝑛−1𝐺𝑥_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−𝐽_𝑟𝑛−1𝐺𝑥_𝑛−1𝛼_𝑛−𝛼_𝑛−1≤𝑥_𝑛−𝑥_𝑛−1+𝑟_𝑛−𝑟_𝑛−1𝑀₀+𝑥_𝑛−1−𝐽_𝑟𝑛−1𝐺𝑥_𝑛−1𝛼_𝑛−𝛼_𝑛−1,𝑧_𝑛−𝑧_𝑛−1=𝐼−𝜆_𝑛𝐹𝐽_𝑟𝑛𝐺𝑦_𝑛+𝜆_𝑛𝑓𝑦_𝑛−𝐼−𝜆_𝑛−1𝐹𝐽_𝑟𝑛−1𝐺𝑦_𝑛−1−𝜆_𝑛−1𝑓𝑦_𝑛−1=𝜆_𝑛𝑓𝑦_𝑛−𝑓𝑦_𝑛−1+𝜆_𝑛−𝜆_𝑛−1×𝑓𝑦_𝑛−1−𝐹𝐽_𝑟𝑛−1𝐺𝑦_𝑛−1+𝐼−𝜆_𝑛𝐹𝐽_𝑟𝑛𝐺𝑦_𝑛−𝐼−𝜆_𝑛𝐹𝐽_𝑟𝑛−1𝐺𝑦_𝑛−1≤𝜆_𝑛𝑓𝑦_𝑛−𝑓𝑦_𝑛−1+𝜆_𝑛−𝜆_𝑛−1×𝑓𝑦_𝑛−1−𝐹𝐽_𝑟𝑛−1𝐺𝑦_𝑛−1+𝐼−𝜆_𝑛𝐹𝐽_𝑟𝑛𝐺𝑦_𝑛−𝐼−𝜆_𝑛𝐹𝐽_𝑟𝑛−1𝐺𝑦_𝑛−1≤𝜆_𝑛𝛽𝑦_𝑛−𝑦_𝑛−1+𝜆_𝑛−𝜆_𝑛−1×𝑓𝑦_𝑛−1−𝐹𝐽_𝑟𝑛−1𝐺𝑦_𝑛−1+1−𝜆_𝑛𝛾₀𝐽_𝑟𝑛𝐺𝑦_𝑛−𝐽_𝑟𝑛−1𝐺𝑦_𝑛−1≤𝜆_𝑛𝛽𝑦_𝑛−𝑦_𝑛−1+𝜆_𝑛−𝜆_𝑛−1×𝑓𝑦_𝑛−1−𝐹𝐽_𝑟𝑛−1𝐺𝑦_𝑛−1+1−𝜆_𝑛𝛾₀𝑦_𝑛−𝑦_𝑛−1+𝑟_𝑛−𝑟_𝑛−1𝑀₀=1−𝜆_𝑛𝛾₀−𝛽𝑦_𝑛−𝑦_𝑛−1+1−𝜆_𝑛𝛾₀𝑟_𝑛−𝑟_𝑛−1𝑀₀+𝜆_𝑛−𝜆_𝑛−1𝑓𝑦_𝑛−1−𝐹𝐽_𝑟𝑛−1𝐺𝑦_𝑛−1≤𝑦_𝑛−𝑦_𝑛−1+𝑟_𝑛−𝑟_𝑛−1𝑀₀+𝜆_𝑛−𝜆_𝑛−1𝑓𝑦_𝑛−1−𝐹𝐽_𝑟𝑛−1𝐺𝑦_𝑛−1, (102) and hence, 𝑥_𝑛+1−𝑥_𝑛=𝐼−𝛽_𝑛𝑉𝐽_𝑟𝑛𝐺𝑦_𝑛+𝛽_𝑛𝑧_𝑛−𝐼−𝛽_𝑛−1𝑉𝐽_𝑟𝑛−1𝐺𝑦_𝑛−1−𝛽_𝑛−1𝑧_𝑛−1=𝛽_𝑛𝑧_𝑛−𝑧_𝑛−1+𝛽_𝑛−𝛽_𝑛−1𝑧_𝑛−1−𝑉𝐽_𝑟𝑛−1𝐺𝑦_𝑛−1+𝐼−𝛽_𝑛𝑉𝐽_𝑟𝑛𝐺𝑦_𝑛−𝐽_𝑟𝑛−1𝐺𝑦_𝑛−1≤𝛽_𝑛𝑧_𝑛−𝑧_𝑛−1+𝛽_𝑛−𝛽_𝑛−1𝑧_𝑛−1−𝑉𝐽_𝑟𝑛−1𝐺𝑦_𝑛−1+1−𝛽_𝑛𝛾𝐽_𝑟𝑛𝐺𝑦_𝑛−𝐽_𝑟𝑛−1𝐺𝑦_𝑛−1≤𝛽_𝑛𝑦_𝑛−𝑦_𝑛−1+𝑟_𝑛−𝑟_𝑛−1𝑀₀+𝜆_𝑛−𝜆_𝑛−1𝑓𝑦_𝑛−1−𝐹𝐽_𝑟𝑛−1𝐺𝑦_𝑛−1+𝛽_𝑛−𝛽_𝑛−1𝑧_𝑛−1−𝑉𝐽_𝑟𝑛−1𝐺𝑦_𝑛−1+1−𝛽_𝑛𝛾𝑦_𝑛−𝑦_𝑛−1+𝑟_𝑛−𝑟_𝑛−1𝑀₀=1−𝛽_𝑛𝛾−1𝑦_𝑛−𝑦_𝑛−1+𝑟_𝑛−𝑟_𝑛−1𝑀₀+𝛽_𝑛𝜆_𝑛−𝜆_𝑛−1𝑓𝑦_𝑛−1−𝐹𝐽_𝑟𝑛−1𝐺𝑦_𝑛−1+𝛽_𝑛−𝛽_𝑛−1𝑧_𝑛−1−𝑉𝐽_𝑟𝑛−1𝐺𝑦_𝑛−1≤1−𝛽_𝑛×𝑥𝛾−1_𝑛−𝑥_𝑛−1+2𝑟_𝑛−𝑟_𝑛−1𝑀₀+𝑥_𝑛−1−𝐽_𝑟𝑛−1𝐺𝑥_𝑛−1𝛼_𝑛−𝛼_𝑛−1+𝛽_𝑛𝜆_𝑛−𝜆_𝑛−1𝑓𝑦_𝑛−1−𝐹𝐽_𝑟𝑛−1𝐺𝑦_𝑛−1+𝛽_𝑛−𝛽_𝑛−1𝑧_𝑛−1−𝑉𝐽_𝑟𝑛−1𝐺𝑦_𝑛−1≤1−𝛽_𝑛𝛾−1𝑥_𝑛−𝑥_𝑛−1+𝛽_𝑛𝜆_𝑛−𝜆_𝑛−1𝑀₁+𝛼_𝑛−𝛼_𝑛−1+𝛽_𝑛−𝛽_𝑛−1+𝑟_𝑛−𝑟_𝑛−1𝑀₁, (103) where sup_𝑛≥0{‖𝑥_𝑛−𝐽_𝑟𝑛𝐺𝑥_𝑛‖+‖𝑧_𝑛−𝑉𝐽_𝑟𝑛𝐺𝑦_𝑛‖+‖𝑓(𝑦_𝑛)−𝐹(𝐽_𝑟𝑛𝐺𝑦_𝑛)‖+2𝑀₀}≤𝑀₁ for some 𝑀₁>0 (it is easy to see that {𝑧_𝑛} is bounded due to the boundedness of {𝑦_𝑛}). Utilizing Lemma 17, we conclude from conditions (i), (ii), and (iv) that lim_𝑛→∞𝑥_𝑛+1−𝑥_𝑛=0. (104) By condition (iii) and (95), we have 𝑦_𝑛−𝑥_𝑛=1−𝛼_𝑛𝐽_𝑟𝑛𝐺𝑥_𝑛−𝑥_𝑛≤(1−𝑎)𝐽_𝑟𝑛𝐺𝑥_𝑛−𝐽_𝑟𝑛𝐺𝑦_𝑛+𝐽_𝑟𝑛𝐺𝑦_𝑛−𝑥_𝑛+1+𝑥_𝑛+1−𝑥_𝑛≤(1−𝑎)𝑥_𝑛−𝑦_𝑛+𝐽_𝑟𝑛𝐺𝑦_𝑛−𝑥_𝑛+1+𝑥_𝑛+1−𝑥_𝑛, (105) which implies that 𝑦_𝑛−𝑥_𝑛≤1−𝑎𝑎𝐽_𝑟𝑛𝐺𝑦_𝑛−𝑥_𝑛+1+𝑥_𝑛+1−𝑥_𝑛. (106) This together with (93)-(94) implies that lim_𝑛→∞𝑥_𝑛−𝑦_𝑛=0. (107) Therefore, 𝑥_𝑛−𝐽_𝑟𝑛𝐺𝑥_𝑛≤𝑥_𝑛−𝑦_𝑛+𝑦_𝑛−𝐽_𝑟𝑛𝐺𝑥_𝑛≤𝑥_𝑛−𝑦_𝑛+𝛼_𝑛𝑥_𝑛−𝐽_𝑟𝑛𝐺𝑥_𝑛≤𝑥_𝑛−𝑦_𝑛+𝑏𝑥_𝑛−𝐽_𝑟𝑛𝐺𝑥_𝑛, (108) which implies that 𝑥_𝑛−𝐽_𝑟𝑛𝐺𝑥_𝑛≤1𝑥1−𝑏^𝑛−𝑦_𝑛, (109) and hence, lim_𝑛→∞𝑥_𝑛−𝐽_𝑟𝑛𝐺𝑥_𝑛=0. (110)
Next, we show that lim_𝑛→∞‖𝑥_𝑛−𝐽_𝑟𝑛𝑥_𝑛‖=0 and lim_𝑛→∞‖𝑥_𝑛−𝐺𝑥_𝑛‖=0.
Indeed, for simplicity, put 𝑞=Π_𝐶(𝑝−𝜇₂𝐵₂𝑝), 𝑢_𝑛=Π_𝐶(𝑥_𝑛−𝜇₂𝐵₂𝑥_𝑛), and 𝑣_𝑛=Π_𝐶(𝑢_𝑛−𝜇₁𝐵₁𝑢_𝑛). Then, 𝑝=Π_𝐶(𝑞−𝜇₁𝐵₁𝑞) and 𝑣_𝑛=𝐺𝑥_𝑛 for all 𝑛≥0. From (95), we have 𝑦_𝑛−𝑝²≤𝛼_𝑛𝑥_𝑛−𝑝²+1−𝛼_𝑛𝐽_𝑟𝑛𝐺𝑥_𝑛−𝑝²=𝛼_𝑛𝑥_𝑛−𝑝²+1−𝛼_𝑛𝑣_𝑛−𝑝². (111) Utilizing Lemma 12, we have 𝑢_𝑛−𝑞²=Π_𝐶𝑥_𝑛−𝜇₂𝐵₂𝑥_𝑛−Π_𝐶𝑝−𝜇₂𝐵₂𝑝²≤𝑥_𝑛−𝑝−𝜇₂𝐵₂𝑥_𝑛−𝐵₂𝑝²≤𝑥_𝑛−𝑝²−2𝜇₂𝛼₂−𝜅²𝜇₂𝐵₂𝑥_𝑛−𝐵₂𝑝²,𝑣_𝑛−𝑝²=Π_𝐶𝑢_𝑛−𝜇₁𝐵₁𝑢_𝑛−Π_𝐶𝑞−𝜇₁𝐵₁𝑞²≤𝑢_𝑛−𝑞−𝜇₁𝐵₁𝑢_𝑛−𝐵₁𝑞²≤𝑢_𝑛−𝑞²−2𝜇₁𝛼₁−𝜅²𝜇₁𝐵₁𝑢_𝑛−𝐵₁𝑞². (112)(113) Combining (112) and (113), we obtain 𝑣_𝑛−𝑝²≤𝑥_𝑛−𝑝²−2𝜇₂𝛼₂−𝜅²𝜇₂𝐵₂𝑥_𝑛−𝐵₂𝑝²−2𝜇₁𝛼₁−𝜅²𝜇₁𝐵₁𝑢_𝑛−𝐵₁𝑞², (114) which together with (111) implies that 𝑦_𝑛−𝑝²≤𝛼_𝑛𝑥_𝑛−𝑝²+1−𝛼_𝑛𝑣_𝑛−𝑝²≤𝛼_𝑛𝑥_𝑛−𝑝²+1−𝛼_𝑛×𝑥_𝑛−𝑝²−2𝜇₂𝛼₂−𝜅²𝜇₂𝐵₂𝑥_𝑛−𝐵₂𝑝²−2𝜇₁𝛼₁−𝜅²𝜇₁𝐵₁𝑢_𝑛−𝐵₁𝑞²=𝑥_𝑛−𝑝²−21−𝛼_𝑛𝜇₂𝛼₂−𝜅²𝜇₂𝐵₂𝑥_𝑛−𝐵₂𝑝²+𝜇₁𝛼₁−𝜅²𝜇₁𝐵₁𝑢_𝑛−𝐵₁𝑞². (115) It immediately follows that 21−𝛼_𝑛𝜇₂𝛼₂−𝜅²𝜇₂𝐵₂𝑥_𝑛−𝐵₂𝑝²+𝜇₁𝛼₁−𝜅²𝜇₁𝐵₁𝑢_𝑛−𝐵₁𝑞²≤𝑥_𝑛−𝑝²−𝑦_𝑛−𝑝²≤𝑥_𝑛−𝑝+𝑦_𝑛−𝑝𝑥_𝑛−𝑦_𝑛. (116) Since {𝑥_𝑛} and {𝑦_𝑛} are bounded and 0<𝜇_𝑖<𝛼_𝑖/𝜅² for 𝑖=1,2, we deduce from (107) and condition (iii) that lim_𝑛→∞𝐵₂𝑥_𝑛−𝐵₂𝑝=0,lim_𝑛→∞𝐵₁𝑢_𝑛−𝐵₁𝑞=0. (117) Utilizing Proposition 3 and Lemma 9, we have 𝑢_𝑛−𝑞²=Π_𝐶𝑥_𝑛−𝜇₂𝐵₂𝑥_𝑛−Π_𝐶𝑝−𝜇₂𝐵₂𝑝²≤𝑥_𝑛−𝜇₂𝐵₂𝑥_𝑛−𝑝−𝜇₂𝐵₂𝑝,𝐽𝑢_𝑛=𝑥−𝑞_𝑛−𝑝,𝐽𝑢_𝑛−𝑞+𝜇₂𝐵₂𝑝−𝐵₂𝑥_𝑛,𝐽𝑢_𝑛≤1−𝑞2𝑥_𝑛−𝑝²+𝑢_𝑛−𝑞²−𝑔₁𝑥_𝑛−𝑢_𝑛−𝑝−𝑞+𝜇₂𝐵₂𝑝−𝐵₂𝑥_𝑛𝑢_𝑛−𝑞, (118) which implies that 𝑢_𝑛−𝑞²≤𝑥_𝑛−𝑝²−𝑔₁𝑥_𝑛−𝑢_𝑛−𝑝−𝑞+2𝜇₂𝐵₂𝑝−𝐵₂𝑥_𝑛𝑢_𝑛−𝑞. (119) Similarly, we derive 𝑣_𝑛−𝑝²=Π_𝐶(𝑢_𝑛−𝜇₁𝐵₁𝑢_𝑛)−Π_𝐶(𝑞−𝜇₁𝐵₁𝑞)²≤𝑢_𝑛−𝜇₁𝐵₁𝑢_𝑛−𝑞−𝜇₁𝐵₁𝑞,𝐽𝑣_𝑛=𝑢−𝑝_𝑛−𝑞,𝐽𝑣_𝑛−𝑝+𝜇₁𝐵₁𝑞−𝐵₁𝑢_𝑛,𝐽𝑣_𝑛≤1−𝑝2𝑢_𝑛−𝑞²+𝑣_𝑛−𝑝²−𝑔₂𝑢_𝑛−𝑣_𝑛+𝑝−𝑞+𝜇₁𝐵₁𝑞−𝐵₁𝑢_𝑛𝑣_𝑛−𝑝, (120) which implies that 𝑣_𝑛−𝑝²≤𝑢_𝑛−𝑞²−𝑔₂𝑢_𝑛−𝑣_𝑛+𝑝−𝑞+2𝜇₁𝐵₁𝑞−𝐵₁𝑢_𝑛𝑣_𝑛−𝑝. (121) Combining (119) and (121), we get 𝑣_𝑛−𝑝²≤𝑥_𝑛−𝑝²−𝑔₁𝑥_𝑛−𝑢_𝑛−𝑝−𝑞−𝑔₂𝑢_𝑛−𝑣_𝑛+𝑝−𝑞+2𝜇₂𝐵₂𝑝−𝐵₂𝑥_𝑛𝑢_𝑛−𝑞+2𝜇₁𝐵₁𝑞−𝐵₁𝑢_𝑛𝑣_𝑛−𝑝, (122) which together with (111) implies that 𝑦_𝑛−𝑝²≤𝛼_𝑛𝑥_𝑛−𝑝²+1−𝛼_𝑛𝑣_𝑛−𝑝²≤𝛼_𝑛𝑥_𝑛−𝑝²+1−𝛼_𝑛×𝑥_𝑛−𝑝²−𝑔₁𝑥_𝑛−𝑢_𝑛−𝑝−𝑞−𝑔₂𝑢_𝑛−𝑣_𝑛+𝑝−𝑞+2𝜇₂𝐵₂𝑝−𝐵₂𝑥_𝑛𝑢_𝑛−𝑞+2𝜇₁𝐵₁𝑞−𝐵₁𝑢_𝑛𝑣_𝑛=𝑥−𝑝_𝑛−𝑝²−1−𝛼_𝑛×𝑔₁𝑥_𝑛−𝑢_𝑛−𝑝−𝑞+𝑔₂𝑢_𝑛−𝑣_𝑛+𝑝−𝑞+21−𝛼_𝑛𝜇₂𝐵₂𝑝−𝐵₂𝑥_𝑛𝑢_𝑛−𝑞+𝜇₁𝐵₁𝑞−𝐵₁𝑢_𝑛𝑣_𝑛−𝑝. (123) It immediately follows that 1−𝛼_𝑛𝑔₁𝑥_𝑛−𝑢_𝑛−𝑝−𝑞+𝑔₂𝑢_𝑛−𝑣_𝑛+≤𝑥𝑝−𝑞_𝑛−𝑝²−𝑦_𝑛−𝑝²+21−𝛼_𝑛×𝜇₂𝐵₂𝑝−𝐵₂𝑥_𝑛𝑢_𝑛−𝑞+𝜇₁𝐵₁𝑞−𝐵₁𝑢_𝑛𝑣_𝑛≤𝑥−𝑝_𝑛−𝑝+𝑦_𝑛−𝑝𝑥_𝑛−𝑦_𝑛+2𝜇₂𝐵₂𝑝−𝐵₂𝑥_𝑛𝑢_𝑛−𝑞+2𝜇₁𝐵₁𝑞−𝐵₁𝑢_𝑛𝑣_𝑛−𝑝. (124) Since {𝑥_𝑛},{𝑦_𝑛},{𝑢_𝑛}, and {𝑣_𝑛} are bounded, we deduce from (107), (117) and condition (iii) that lim_𝑛→∞𝑔₁𝑥_𝑛−𝑢_𝑛−𝑝−𝑞=0,lim_𝑛→∞𝑔₂𝑢_𝑛−𝑣_𝑛+𝑝−𝑞=0. (125) Utilizing the properties of 𝑔₁ and 𝑔₂, we get lim_𝑛→∞𝑥_𝑛−𝑢_𝑛−𝑝−𝑞=0,lim_𝑛→∞𝑢_𝑛−𝑣_𝑛+𝑝−𝑞=0, (126) which yields 𝑥_𝑛−𝑣_𝑛≤𝑥_𝑛−𝑢_𝑛−+𝑢𝑝−𝑞_𝑛−𝑣_𝑛+𝑝−𝑞⟶0as𝑛⟶∞; (127) that is, lim_𝑛→∞𝑥_𝑛−𝐺𝑥_𝑛=0. (128) Note that 𝑥_𝑛−𝐽_𝑟𝑛𝑥_𝑛≤𝑥_𝑛−𝐽_𝑟𝑛𝐺𝑥_𝑛+𝐽_𝑟𝑛𝐺𝑥_𝑛−𝐽_𝑟𝑛𝑥_𝑛≤𝑥_𝑛−𝐽_𝑟𝑛𝐺𝑥_𝑛+𝐺𝑥_𝑛−𝑥_𝑛. (129) From (110) and (128), we have lim_𝑛→∞𝑥_𝑛−𝐽_𝑟𝑛𝑥_𝑛=0. (130) By utilizing the resolvent identity in Lemma 8, we obtain from {𝑟_𝑛}⊂[𝜀,∞) that 𝑥_𝑛−𝐽_𝜀𝑥_𝑛≤𝑥_𝑛−𝐽_𝑟𝑛𝑥_𝑛+𝐽_𝑟𝑛𝑥_𝑛−𝐽_𝜀𝑥_𝑛=𝑥_𝑛−𝐽_𝑟𝑛𝑥_𝑛+𝐽_𝑟𝑛𝑥_𝑛−𝐽_𝑟𝑛𝑟_𝑛𝜀𝑥_𝑛+𝑟1−^𝑛𝜀𝐽_𝑟𝑛𝑥_𝑛≤2𝑥_𝑛−𝐽_𝑟𝑛𝑥_𝑛, (131) which together with (130) implies that lim_𝑛→∞𝑥_𝑛−𝐽_𝜀𝑥_𝑛=0. (132)
For every 𝑛≥0 and 𝑡∈(0,(2−𝛾)/(𝛾₀−𝛽)), let 𝑥_𝑡,𝑛=𝐼−𝜃_𝑡𝑉𝐽_𝑟𝑛𝐺𝑥_𝑡,𝑛+𝜃_𝑡𝐽_𝑟𝑛𝐺𝑥_𝑡,𝑛−𝑡𝐹𝐽_𝑟𝑛𝐺𝑥_𝑡,𝑛−𝑓𝑥_𝑡,𝑛, (133) where {𝑟_𝑛}⊂[𝜀,∞) and 0<𝜃_𝑡≤‖𝑉‖⁻¹, for all 𝑡∈(0,(2−𝛾)/(𝛾₀−𝛽)) with lim_𝑡→0𝜃_𝑡=0. Then, by Theorem 18, {𝑥_𝑡,𝑛} converges strongly to 𝑝∈Δ as 𝑡→0, which is the unique solution of the variational inequality (88).
Define a mapping 𝑊𝑥=(1−𝜃)𝐽_𝜀𝑥+𝜃𝐺𝑥,∀𝑥∈𝐶, (134) where 𝜃 is a constant in (0,1). Then by Lemma 4, we have Fix(𝑊)=Fix(𝐽_𝜀)∩Fix(𝐺)=Δ. 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. (135) From (128) and (132), we obtain lim_𝑛→∞𝑥_𝑛−𝑊𝑥_𝑛=0. (136)
Further, we show that limsup_𝑛→∞(𝐼−𝑉)𝑝,𝐽𝑥_𝑛−𝑝≤0. (137) Indeed, take a subsequence {𝑥_𝑛𝑖} of {𝑥_𝑛} such that limsup_𝑛→∞(𝐼−𝑉)𝑝,𝐽𝑥_𝑛−𝑝=lim_𝑖→∞(𝐼−𝑉)𝑝,𝐽𝑥_𝑛𝑖−𝑝. (138) Without loss of generality, we may assume that 𝑥_𝑛𝑖⇀𝑥. Utilizing Lemma 15, we obtain from (136) that 𝑥∈Fix(𝑊)=Fix(𝐽_𝜀)∩Fix(𝐺)=Δ. Hence from (88) and (138), we get limsup_𝑛→∞(𝐼−𝑉)𝑝,𝐽𝑥_𝑛−𝑝=(𝐼−𝑉)𝑝,𝐽𝑥−𝑝≤0. (139)
Finally, we show that 𝑥_𝑛→𝑝 as 𝑛→∞.
Indeed, 𝑦_𝑛−𝑝≤𝛼_𝑛𝑥_𝑛−𝑝+1−𝛼_𝑛𝐽_𝑟𝑛𝐺𝑥_𝑛−𝑝≤𝛼_𝑛𝑥_𝑛−𝑝+1−𝛼_𝑛𝑥_𝑛−𝑝=𝑥_𝑛,𝑥−𝑝_𝑛+1−𝑝²=𝐼−𝛽_𝑛𝑉𝐽_𝑟𝑛𝐺𝑦_𝑛−𝐼−𝛽_𝑛𝑉𝐽_𝑟𝑛𝐺𝑝,𝐽𝑥_𝑛+1−𝑝+𝛽_𝑛𝐼−𝜆_𝑛𝐹𝐽_𝑟𝑛𝐺𝑦_𝑛−𝐼−𝜆_𝑛𝐹𝐽_𝑟𝑛𝐺𝑝,𝐽𝑥_𝑛+1−𝑝+𝜆_𝑛𝑓𝑦_𝑛−𝑓𝑝,𝐽𝑥_𝑛+1𝑝𝑝𝑥−𝑝+⟨𝑓−𝐹,𝐽_𝑛+1−𝑝⟩+𝛽_𝑛(𝐼−𝑉)𝑝,𝐽𝑥_𝑛+1≤−𝑝1−𝛽_𝑛𝛾𝐽_𝑟𝑛𝐺𝑦_𝑛−𝐽_𝑟𝑛𝐺𝑝𝑥_𝑛+1−𝑝+𝛽_𝑛1−𝜆_𝑛𝛾₀𝐽_𝑟𝑛𝐺𝑦_𝑛−𝐽_𝑟𝑛𝐺𝑝𝑥_𝑛+1−𝑝+𝜆_𝑛𝛽𝑦_𝑛−𝑝𝑥_𝑛+1+𝑓𝑝𝑝𝑥−𝑝−𝐹_𝑛+1−𝑝+𝛽_𝑛(𝐼−𝑉)𝑝,𝐽𝑥_𝑛+1≤−𝑝1−𝛽_𝑛𝛾𝑦_𝑛−𝑝𝑥_𝑛+1−𝑝+𝛽_𝑛1−𝜆_𝑛𝛾₀𝑦_𝑛−𝑝𝑥_𝑛+1−𝑝+𝜆_𝑛𝛽𝑦_𝑛−𝑝𝑥_𝑛+1+𝑓𝑝𝑝𝑥−𝑝−𝐹_𝑛+1−𝑝+𝛽_𝑛(𝐼−𝑉)𝑝,𝐽𝑥_𝑛+1=−𝑝1−𝛽_𝑛𝛾𝑦_𝑛−𝑝𝑥_𝑛+1−𝑝+𝛽_𝑛1−𝜆_𝑛𝛾₀−𝛽𝑦_𝑛−𝑝𝑥_𝑛+1−𝑝+𝜆_𝑛𝑓𝑝−𝐹𝑝𝑥_𝑛+1−𝑝+𝛽_𝑛(𝐼−𝑉)𝑝,𝐽𝑥_𝑛+1≤−𝑝1−𝛽_𝑛𝛾𝑦_𝑛−𝑝𝑥_𝑛+1−𝑝+𝛽_𝑛𝑦_𝑛−𝑝𝑥_𝑛+1−𝑝+𝜆_𝑛𝑓𝑝−𝐹𝑝𝑥_𝑛+1−𝑝+𝛽_𝑛(𝐼−𝑉)𝑝,𝐽𝑥_𝑛+1=−𝑝1−𝛽_𝑛𝛾−1𝑦_𝑛−𝑝𝑥_𝑛+1−𝑝+𝛽_𝑛𝜆_𝑛𝑓𝑝−𝐹𝑝𝑥_𝑛+1−𝑝+𝛽_𝑛(𝐼−𝑉)𝑝,𝐽𝑥_𝑛+1≤−𝑝1−𝛽_𝑛1𝛾−12𝑦_𝑛−𝑝²+𝑥_𝑛+1−𝑝²+𝛽_𝑛𝜆_𝑛𝑓𝑝−𝐹𝑝𝑥_𝑛+1−𝑝+𝛽_𝑛(𝐼−𝑉)𝑝,𝐽𝑥_𝑛+1−𝑝, (140) which yields 𝑥_𝑛+1−𝑝²≤1−𝛽_𝑛𝛾−11+𝛽_𝑛𝑦𝛾−1^𝑛−𝑝²+𝛽_𝑛11+𝛽_𝑛×𝛾−12𝜆_𝑛𝑓𝑝−𝐹𝑝𝑥_𝑛+1)𝑥−𝑝+2(𝐼−𝑉𝑝,𝐽_𝑛+1≤−𝑝1−𝛽_𝑛𝛾−1𝑥_𝑛−𝑝²+𝛽_𝑛11+𝛽_𝑛×𝛾−12𝜆_𝑛𝑓𝑝−𝐹𝑝𝑥_𝑛+1)𝑥−𝑝+2(𝐼−𝑉𝑝,𝐽_𝑛+1=−𝑝1−𝛾_𝑛𝑥_𝑛−𝑝²+𝛾_𝑛𝛿_𝑛, (141) where 𝛾_𝑛=𝛽_𝑛(𝛾−1) and 𝛿_𝑛=11+𝛽_𝑛𝛾−1×𝛾−12𝜆_𝑛𝑓𝑝−𝐹𝑝𝑥_𝑛+1)𝑥−𝑝+2(𝐼−𝑉𝑝,𝐽_𝑛+1−𝑝. (142) It can be easily seen from (137) and conditions (i), (ii) that ^∞_𝑛=0𝛾_𝑛=∞,limsup_𝑛→∞𝛿_𝑛≤0. (143) Applying Lemma 17 to (141), we infer that 𝑥_𝑛→𝑝 as 𝑛→∞. This completes the proof.

Remark 22. By a careful analysis of the proof of Theorem 21, we reached to the conclusion that the iterative method (87) in Theorem 21 can be replaced by the following iterative method: 𝑦_𝑛=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝐽_𝑟𝑛𝐺𝑥_𝑛,𝑥_𝑛+1=𝐼−𝛽_𝑛𝑉𝑦_𝑛+𝛽_𝑛𝑦_𝑛−𝜆_𝑛𝐹𝑦_𝑛−𝑓𝑦_𝑛,∀𝑛≥0. (144)

Remark 23. Theorem 21 improves and extends [15, Theorem 3.1], [10, Theorem 3.1], and [13, Theorem 3.1] in the following aspects.(a)The problem in Theorem 21 is different from the one considered in [15, Theorem 3.1], [10, Theorem 3.1], and [13, Theorem 3.1]. Also, the iterative scheme in Theorem 21 is different from the one considered in [15, Theorem 3.1], [10, Theorem 3.1] and [13, Theorem 3.1].(b)The iterative scheme in Theorem 21 is more advantageous and more flexible than the iterative scheme in [13, Theorem 3.1].(c)Theorem 21 drops the assumption of the asymptotical regularity for the iterative scheme in [13, Theorem 3.1].

Remark 24. Theorem 21 extends and improves [28, Theorem 3.1] to a great extent in the following aspects.(a)The 𝑢 is replaced by a fixed contractive mapping.(b)One continuous pseudocontractive mapping (including nonexpansive mapping) is replaced by the mapping 𝐽_𝑟𝑛∘𝐺.(c)We add a strongly positive linear bounded operator 𝐴 and a strongly accretive and strictly pseudocontractive mapping 𝐹 in our iterative algorithm.

Acknowledgments

In this research, first author 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).