A Strong Convergence Theorem for Relatively Nonexpansive Mappings and Equilibrium Problems in Banach Spaces

Abstract

Relatively nonexpansive mappings and equilibrium problems are considered based on a shrinking projection method. Using properties of the generalized f-projection operator, a strong convergence theorem for relatively nonexpansive mappings and equilibrium problems is proved in Banach spaces under some suitable conditions.

1. Introduction

It is well known that metric projection operator in Hilbert and Banach spaces is widely used in different areas of mathematics such as functional analysis and numerical analysis, theory of optimization and approximation, and also for the problems of optimal control and operations research, nonlinear and stochastic programming and game theory.

Let 𝑋 be a real Banach space with its dual 𝑋^∗, and let 𝐾 be a nonempty, closed, and convex subset of 𝑋. In 1994, Alber [1] introduced the generalized projections 𝜋_𝐾∶𝑋^∗→𝐾 and Π_𝐾∶𝑋→𝐾 in uniformly convex and uniformly smooth Banach spaces based on the function 𝜙(𝑦,𝑥) defined on p.3 and studied their properties in detail. In 2005, Li [2] extended the definition of the generalized projection operator from uniformly convex and uniformly smooth Banach spaces to reflexive Banach spaces and studied some properties of the generalized projection operator. Recently, Wu and Huang [3] introduced a new generalized 𝑓-projection operator in a Banach space. By making use of (2.5), they extended the definition of the generalized projection operators introduced by Abler [1] and proved some properties of the generalized 𝑓-projection operator. Wu and Huang [4] studied a relation between the generalized projection operator and the resolvent operator for the subdifferential of a proper convex and lower semicontinuous functional in reflexive and smooth Banach spaces (see [5–9]). Very recently, Li et al. [10] studied some properties of the generalized 𝑓-projection operator, and proved the strong convergence theorems for relatively nonexpansive mappings in Banach spaces.

On the other hand, equilibrium problem was introduced by Blum and Oettli [11], in 1994. It is a hot topic of intensive research efforts, because it has a great impact and influence in the development of several branches of pure and applied sciences. It has been shown that equilibrium problem theory provides a novel and unified treatment of a wide class of problems arisen in economics, finance, physics, image reconstruction, ecology, transportation, network, elasticity, and optimization problems. Numerous issues in physics, optimization, and economics reduce to finding a solution of equilibrium problem. Some methods have been proposed to solve the equilibrium problems (see, e.g, [12–14] and the references therein).

In this paper, motivated and inspired by the work mentioned above, we introduce a new hybrid projection algorithm based on the shrinking projection method for relatively nonexpansive mapping and equilibrium problem. Using the new algorithm, we prove a strong convergence theorem for relatively nonexpansive mappings and equilibrium problems in Banach spaces. The result presented in this paper extends and improves the main result of Li et al. [10].

2. Preliminaries

Let 𝑋 be a real Banach space with its dual 𝑋^∗ and 𝑅=(−∞,+∞). We denote the duality between 𝑋 and 𝑋^∗ by ⟨⋅,⋅⟩, and the norms of Banach space 𝑋 and 𝑋^∗ by ‖⋅‖_𝑋 and ‖⋅‖_𝑋^∗, respectively. A Banach space 𝑋 is said to be strictly convex if ‖(𝑥+𝑦)/2‖<1 for all 𝑥,𝑦∈𝑋 with ‖𝑥‖=‖𝑦‖=1 and 𝑥≠𝑦. It is also said to be uniformly convex if lim_𝑛→∞‖𝑥_𝑛−𝑦_𝑛‖=0 for any two sequences {𝑥_𝑛}, {𝑦_𝑛} in 𝑋, such that ‖𝑥_𝑛‖=‖𝑦_𝑛‖=1 and lim_𝑛→∞‖(𝑥_𝑛+𝑦_𝑛)/2‖=1. The function 𝛿_𝑋(𝜀)=inf1−‖𝑥+𝑦‖2∶‖𝑥‖=1,‖𝑦‖=1,‖𝑥−𝑦‖≥𝜀(2.1) is called the modulus of convexity of 𝑋.

A Banach space 𝑋 is said to be smooth provided that lim_𝑡→0(‖𝑥+𝑡𝑦‖−‖𝑥‖)/𝑡 exists for all 𝑥,𝑦∈𝑋 with ‖𝑥‖=‖𝑦‖=1. It is also said to be uniformly smooth if the limit is attained uniformly for ‖𝑥‖=‖𝑦‖=1. The function 𝜌_𝑋(𝑡)=sup‖𝑥+𝑦‖+‖𝑥−𝑦‖2−1∶‖𝑥‖=1,‖𝑦‖≤𝑡(2.2) is called the modulus of smoothness of 𝑋.

When {𝑥_𝑛} is a sequence in 𝑋, we denote the strong convergence of {𝑥_𝑛} with a cluster 𝑥∈𝑋 by 𝑥_𝑛→𝑥 and the weak convergence of {𝑥_𝑛} with a weak cluster 𝑥∈𝑋 by 𝑥_𝑛⇀𝑥. A Banach space 𝑋 is said to have the Kadec-Klee property if a sequence {𝑥_𝑛} of 𝑋 satisfies that 𝑥_𝑛⇀𝑥∈𝑋 and ‖𝑥_𝑛‖→‖𝑥‖, then 𝑥_𝑛→𝑥. It is known that if 𝑋 is uniformly convex, then 𝑋 has the Kadec-Klee property.

The normalized duality mapping 𝐽 from 𝑋 to 𝑋^∗ is defined by 𝐽𝑥=𝑥^∗∈𝑋^∗∶⟨𝑥,𝑥^∗⟩=‖𝑥‖²=‖𝑥^∗‖²(2.3) for any 𝑥∈𝑋. We list some properties of mapping 𝐽 as follows. (i) If 𝑋 is a smooth Banach space (with Gâteaux differential norm), then 𝐽 is single-valued and demicontinuous. If 𝑋 is a smooth reflexive Banach space, then 𝐽 is single-valued and hemicontinuous. If 𝑋 is a strongly smooth Banach space (with Fréchet differential norm), then 𝐽 is single-valued and continuous. (ii)𝐽 is uniformly continuous on every bounded set of a uniformly smooth Banach space. (iii) If 𝑋 is a reflexive, smooth and strictly convex Banach space, 𝐽^∗∶𝑋^∗→𝑋 is the duality mapping of 𝑋^∗, then 𝐽⁻¹=𝐽^∗,𝐽𝐽^∗=𝐼_𝑋^∗,𝐽^∗𝐽=𝐼_𝑋.

Let 𝑋 be a smooth Banach space and 𝐾 be a nonempty, closed and convex subset of 𝑋. The function 𝜙∶𝑋×𝑋→𝑅 is defined by 𝜙(𝑦,𝑥)=‖𝑦‖²−2⟨𝑦,𝐽𝑥⟩+‖𝑥‖²(2.4) for all 𝑥,𝑦∈𝑋.

Next, we recall the concept of the generalized 𝑓-projector operator, together with its properties. Let 𝐺∶𝐾×𝑋^∗→𝑅∪{+∞} be a functional defined as follows: 𝐺(𝜉,𝜑)=‖𝜉‖²−2⟨𝜉,𝜑⟩+‖𝜑‖²+2𝜌𝑓(𝜉),(2.5) where 𝜉∈𝐾,𝜑∈𝑋^∗, 𝜌 is a positive number and 𝑓∶𝐾→𝑅∪{+∞} is proper, convex, and lower semicontinuous.

From the definitions of 𝐺 and 𝑓, it is easy to have the following properties: (i)𝐺(𝜉,𝜑) is convex and continuous with respect to 𝜑 when 𝜉 is fixed; (ii)𝐺(𝜉,𝜑) is convex and lower semicontinuous with respect to 𝜉 when 𝜑 is fixed.

Definition 2.1. Let 𝑋 be a real smooth Banach space and 𝐾 be a nonempty, closed and convex subset of 𝑋. We say that Π^𝑓𝐾∶𝑋→2^𝐾 is a generalized 𝑓-projection operator if Π^𝑓𝐾𝑥=𝑢∈𝐾∶𝐺(𝑢,𝐽𝑥)=inf_𝜉∈𝐾𝐺(𝜉,𝐽𝑥),∀𝑥∈𝑋.(2.6)

In order to obtain our results, the following lemmas are crucial to us.

Lemma 2.2 (see [15]). Let 𝑋 be a real Banach space and 𝑓∶𝑋→𝑅∪{+∞} be a lower semicontinuous convex functional. Then there exist 𝑥^∗∈𝑋^∗ and 𝛼∈𝑅 such that 𝑓(𝑥)≥⟨𝑥,𝑥^∗⟩+𝛼,∀𝑥∈𝑋.(2.7)

Lemma 2.3 (see [16]). Let 𝑋 be a uniformly convex and smooth Banach space and let {𝑦_𝑛}, {𝑧_𝑛} be two sequences of 𝑋. If 𝜙(𝑦_𝑛,𝑧_𝑛)→0 and either {𝑦_𝑛} or {𝑧_𝑛} is bounded, then 𝑦_𝑛−𝑧_𝑛→0.

Let 𝐾 be a closed subset of a real Banach space 𝑋, and let 𝑇 be a mapping from 𝐾 to 𝐾. We denote by 𝐹(𝑇) the set of all fixed points of 𝑇. A point 𝑝 in 𝐾 is said to be an asymptotic fixed point of 𝑇, if 𝐾 contains a sequence {𝑥_𝑛} which converges weakly to 𝑝 such that lim_𝑛→∞(𝑥_𝑛−𝑇𝑥_𝑛)=0. The set of all asymptotic fixed points of 𝑇 will be denoted by 𝐹(𝑇). 𝑇 is called nonexpansive if ‖𝑇𝑥−𝑇𝑦‖≤‖𝑥−𝑦‖ for all 𝑥,𝑦∈𝐾, and relatively nonexpansive if 𝐹(𝑇)=𝐹(𝑇) and 𝜙(𝑝,𝑇𝑥)≤𝜙(𝑝,𝑥) for all 𝑥∈𝐾 and 𝑝∈𝐹(𝑇). Obviously, the definition of relatively nonexpansive mapping 𝑇 is equivalent to 𝐹(𝑇)=𝐹(𝑇) and 𝐺(𝑝,𝐽𝑇𝑥)≤𝐺(𝑝,𝐽𝑥) for all 𝑥∈𝐾 and 𝑝∈𝐹(𝑇).

Lemma 2.4 (see [17]). Let 𝑋 be a strictly convex and smooth Banach space, let 𝐾 be a closed, and convex subset of 𝑋, and let 𝑇 be a relatively nonexpansive mapping from 𝐾 into itself. Then 𝐹(𝑇) is closed, and convex.

Lemma 2.5 (see [10]). Let 𝑋 be a real reflexive and smooth Banach space and let 𝐾 be a nonempty, closed, and convex subset of 𝑋. The following statements hold: (i)Π^𝑓𝐾𝑥 is a nonempty, closed, and convex subset of 𝐾 for all 𝑥∈𝑋;(ii) for all 𝑥∈𝑋,𝑥̂∈Π^𝑓𝐾𝑥 if and only if ⟨𝑥̂−𝑦,𝐽𝑥−𝐽𝑥̂⟩+𝜌𝑓(𝑦)−𝜌𝑓(𝑥̂)≥0,∀𝑦∈𝐾;(2.8)(iii) if 𝑋 is strictly convex, then Π^𝑓𝐾 is a single-valued mapping.

Lemma 2.6 (see [10]). Let 𝑋 be a real reflexive and smooth Banach space, let 𝐾 be a nonempty, closed, and convex subset of 𝑋, and let 𝑥∈𝑋, 𝑥̂∈Π^𝑓𝐾𝑥. Then 𝜙(𝑦,𝑥̂)+𝐺(𝑥̂,𝐽𝑥)≤𝐺(𝑦,𝐽𝑥),∀𝑦∈𝐾.(2.9)

Lemma 2.7 (see [10]). Let 𝑋 be a Banach space and 𝑦∈𝑋. Let 𝑓∶𝑋→𝑅∪{+∞} be a proper, convex and lower semicontinuous functional with convex domain 𝐷(𝑓). If {𝑥_𝑛} is a sequence in 𝐷(𝑓) such that 𝑥_𝑛⇀𝑥̂∈int(𝐷(𝑓)) and lim_n→∞𝐺(𝑥_𝑛,𝐽𝑦)=𝐺(𝑥̂,𝐽𝑦), then lim_𝑛→∞‖𝑥_𝑛‖=‖𝑥̂‖.

Let 𝑀 be a closed and convex subset of a real Banach space 𝑋 and 𝑔∶𝑀×𝑀→𝑅 be a bifunction. The equilibrium problem for 𝑔 is as follows. Find 𝑥̂∈𝑀 such that 𝑔(𝑥̂,𝑦)≥0,∀𝑦∈𝑀.(2.10) The set of all solutions for the above equilibrium problem is denoted by EP(𝑔). For solving the equilibrium problem, one always assumes that the bifunction 𝑔 satisfies the following conditions:(A1)𝑔(𝑥,𝑥)=0, for all 𝑥∈𝑀; (A2)𝑔 is monotone, that is, 𝑔(𝑥,𝑦)+𝑔(𝑦,𝑥)≤0, for all 𝑥,𝑦∈𝑀; (A3) for all 𝑥,𝑦,𝑧∈𝑀,limsup_𝑡↓0𝑔(𝑡𝑧+(1−𝑡)𝑥,𝑦)⩽𝑔(𝑥,𝑦); (A4) for all 𝑥∈𝑀, 𝑔(𝑥,⋅) is convex and lower semicontinuous.

In order to prove our results, we present several necessary lemmas.

Lemma 2.8 (see [14]). Let 𝑀 be a closed and convex subset of a uniformly smooth, strictly convex and reflexive Banach space 𝑋, and 𝑔(⋅,⋅) be a bifunction from 𝑀×𝑀→𝑅 satisfying the conditions (A1)–(A4). For all 𝑟>0 and 𝑥∈𝑋, define the mappingas follows. 𝑇_𝑟𝑥=𝑧∈𝑀∶𝑔(𝑧,𝑦)+1𝑟⟨𝐽𝑧−𝐽𝑥,𝑦−𝑧⟩≥0,∀𝑦∈𝑀.(2.11) Then, the following statements hold: (B1)𝑇_𝑟 is single-valued; (B2)𝑇_𝑟 is a firmly nonexpansive-type mapping, that is, for all 𝑥,𝑦∈𝑋, ⟨𝐽𝑇_𝑟𝑥−𝐽𝑇_𝑟𝑦,𝑇_𝑟𝑥−𝑇_𝑟𝑦⟩≤⟨𝐽𝑥−𝐽𝑦,𝑇_𝑟𝑥−𝑇_𝑟𝑦⟩;(2.12)(B3)𝐹(𝑇_𝑟)=𝐹(𝑇_𝑟)=EP(𝑔); (B4)EP(𝑔) is closed and convex.

Lemma 2.9 (see [14]). Let 𝑀 be a closed and convex subset of a smooth, strictly convex, and reflexive Banach space 𝑋, 𝑔 be a bifunction from 𝑀×𝑀 to 𝑅 satisfying the conditions (A1)–(A4), and 𝑟>0. Then, for any 𝑥∈𝑋 and 𝑞∈𝐹(𝑇_𝑟), 𝜙𝑞,𝑇_𝑟𝑥+𝜙𝑇_𝑟𝑥,𝑥≤𝜙(𝑞,𝑥).(2.13)

3. The Main Result

In this section, we prove a strong convergence theorem for relatively nonexpansive mappings and equilibrium problems in Banach spaces.

Theorem 3.1. Let 𝑋 be a uniformly convex and uniformly smooth Banach space, 𝐾 and 𝑀 be two nonempty, closed and convex subsets of 𝑋 such that 𝐾∩𝑀≠∅. Let 𝑇∶𝐾→𝐾 be a relatively nonexpansive mapping and 𝑓∶𝑋→𝑅 a convex and lower semicontinuous mapping with 𝐾⊂int(𝐷(𝑓)). Let 𝑔(⋅,⋅) be a bifunction from 𝑀×𝑀→𝑅, which satisfies the conditions (A1)–(A4). Assume that {𝛼_𝑛}^∞𝑛=0 is a sequence in [0,1) such that limsup_𝑛→∞𝛼_𝑛<1, and {𝑟_𝑛}⊂[𝑎,∞) for some 𝑎>0. Define a sequence {𝑥_𝑛} in 𝐾∩𝑀 by the following algorithm: 𝑥𝑦₀=𝑥∈𝐾∩𝑀,𝐻₀=𝐾∩𝑀,_𝑛𝑢_𝑛=𝐽⁻¹𝛼_𝑛𝐽𝑥_𝑛+1−𝛼_𝑛𝐽𝑇𝑥_𝑛,∈𝑀𝑠𝑢𝑐ℎ𝑡ℎ𝑎𝑡𝑔𝑢_𝑛,𝑦+1𝑟_𝑛𝐻⟨𝐽𝑢_𝑛−𝐽𝑦_𝑛,𝑦−𝑢_𝑛⟩≥0,∀𝑦∈𝑀,_𝑛+1=𝑧∈𝑥𝐻_𝑛∶𝐺𝑧,𝐽𝑢_𝑛≤𝐺𝑧,𝐽𝑦_𝑛≤𝐺𝑧,𝐽𝑥_𝑛_n+1,=Π^𝑓𝐻_𝑛+1𝑥,𝑛=0,1,2,….(3.1) If 𝐹=𝐹(𝑇)∩EP(𝑔) is nonempty, then {𝑥_𝑛} converges strongly to Π^𝑓𝐹𝑥.

Proof . The proof is divided into the following four steps.
(I) First, we prove the following conclusion: 𝐻_𝑛 is a closed convex set and 𝐹⊂𝐻_𝑛 for all 𝑛≥0.
It is obvious that 𝐻₀ is a closed convex set and 𝐹⊂𝐻₀. Thus, we only need to show that 𝐻_𝑛 is a closed convex set and 𝐹⊂𝐻_𝑛 for all 𝑛≥1.
Since 𝐺(𝑧,𝐽𝑢_𝑛)≤𝐺(𝑧,𝐽𝑦_𝑛) and 𝐺(𝑧,𝐽𝑦_𝑛)≤𝐺(𝑧,𝐽𝑥_𝑛) are respectively equivalent to 22⟨⟨𝑧𝑧,,𝐽𝐽𝑥𝑦_𝑛𝑛−𝐽𝑢_𝑛⟩+‖‖𝑢_𝑛‖‖²−‖‖𝑦_𝑛‖‖²≤0,−𝐽𝑦_𝑛⟩+‖‖𝑦_𝑛‖‖²−‖‖𝑥_𝑛‖‖²≤0,(3.2) it follows that 𝐻_𝑛+1 is closed and convex for all 𝑛≥0. Thus, we know that {𝑥_𝑛} is well defined. Further, for any 𝑢∈𝐹 and 𝑛≥0, we have 𝐺𝑢,𝐽=𝑦_𝑛‖𝑢‖₂≤‖𝑢‖₂−2𝑢,𝛼_𝑛𝐽𝑥_𝑛+1−𝛼_𝑛𝐽𝑇𝑥_𝑛+‖‖𝛼_𝑛𝐽𝑥_𝑛+1−𝛼_𝑛𝐽𝑇𝑥_𝑛‖‖²+2𝜌𝑓(𝑢)=𝛼_𝑛−2𝛼_𝑛⟨𝑢,𝐽𝑥_𝑛⟩−21−𝛼_𝑛⟨𝑢,𝐽𝑇𝑥_𝑛⟩+𝛼_𝑛‖‖𝑥_𝑛‖‖²+1−𝛼_𝑛‖‖𝑇𝑥_𝑛‖‖²+2𝜌𝑓(𝑢)+1‖−𝑢‖𝛼²_𝑛−2⟨𝑢,𝐽𝑥_𝑛⟩+‖‖𝑥_𝑛‖‖²+2𝜌𝑓(𝑢)=𝛼_𝑛‖𝑢‖²−2⟨𝑢,𝐽𝑇𝑥_𝑛⟩+‖‖𝑇𝑥_𝑛‖‖²+2𝜌𝑓(𝑢)≤𝐺𝐺𝑢,𝐽𝑢,𝑥𝐽_𝑛𝑥_𝑛+1−𝛼_𝑛𝐺𝑢,𝐽𝑇𝑥_𝑛.(3.3) On the other hand, it follows from the definition of {𝑢_𝑛} and Lemma 2.8 that 𝑢_𝑛=𝑇_𝑟𝑛𝑦_𝑛. From Lemma 2.9, we obtain 𝜙𝑢,𝑢_𝑛=𝜙𝑢,𝑇_𝑟𝑛𝑦_𝑛≤𝜙𝑢,𝑦_𝑛,(3.4) which implies that 𝐺𝑢,𝐽𝑢_𝑛≤𝐺𝑢,𝐽𝑦_𝑛.(3.5) Therefore, 𝑢∈𝐻_𝑛+1 for all 𝑛≥0.
(II) Second, we show that {𝑥_𝑛} is bounded and lim_𝑛→∞𝐺(𝑥_𝑛,𝐽𝑥) exists.
Since 𝑓∶𝑋→𝑅 is a convex and lower semicontinuous mapping, a direct application of Lemma 2.2 yields that there exist 𝑥^∗∈𝑋^∗ and 𝛼∈𝑅 such that 𝑓(𝑦)≥⟨𝑦,𝑥^∗⟩+𝛼,∀𝑦∈𝑋.(3.6) It follows that 𝐺𝑥_𝑛,𝐽𝑥=≥‖‖‖‖𝑥𝑥_𝑛𝑛‖‖²−2⟨𝑥_𝑛,𝐽𝑥⟩+‖𝑥‖²+2𝜌𝑓𝑥_𝑛=‖‖𝑥‖‖²−2⟨𝑥_𝑛,𝐽𝑥⟩+‖𝑥‖²+2𝜌⟨𝑥_𝑛,𝑥^∗⟩+2𝜌𝛼_𝑛≥‖‖𝑥‖‖²−2⟨𝑥_𝑛,𝐽𝑥−𝜌𝑥^∗⟩+‖𝑥‖²+2𝜌𝛼_𝑛=‖‖𝑥‖‖²−2‖𝐽𝑥−𝜌𝑥^∗‖‖‖𝑥_𝑛‖‖+‖𝑥‖²+2𝜌𝛼_𝑛‖‖−‖𝐽𝑥−𝜌𝑥^∗‖²+‖𝑥‖²−‖𝐽𝑥−𝜌𝑥^∗‖²+2𝜌𝛼.(3.7) Since 𝑥_𝑛=Π^𝑓𝐻_𝑛𝑥, it follows from (3.7) that 𝐺(𝑢,𝐽𝑥)≥≥𝐺‖‖𝑥𝑥_𝑛,𝐽𝑥_𝑛‖‖−‖𝐽𝑥−𝜌𝑥^∗‖²+‖𝑥‖²−‖𝐽𝑥−𝜌𝑥^∗‖²+2𝜌𝛼,∀𝑢∈𝐹,(3.8) which implies that {𝑥_𝑛} is bounded and so is {𝐺(𝑥_𝑛,𝐽𝑥)}. By the fact that 𝑥_𝑛+1∈𝐻_𝑛+1⊂𝐻_𝑛 and Lemma 2.6, we obtain 𝜙𝑥_𝑛+1,𝑥_𝑛+𝐺𝑥_𝑛,𝐽𝑥≤𝐺𝑥_𝑛+1,𝐽𝑥.(3.9) It is obvious that 𝜙𝑥_𝑛+1,𝑥_𝑛≥‖‖𝑥_𝑛+1‖‖−‖‖𝑥_𝑛‖‖²≥0,(3.10) and so {𝐺(𝑥_𝑛,𝐽𝑥)} is nondecreasing. Therefore, we know that lim_𝑛→∞𝐺(𝑥_𝑛,𝐽𝑥) exists.
(III) Third, we prove that, if 𝑥_𝑛𝑘⇀𝑥̂, then 𝑥̂∈𝐹, where {𝑥_𝑛𝑘} is an arbitrarily weakly convergent subsequence of {𝑥_𝑛}.
It follows from the definition of 𝐻_𝑛+1 and 𝑥_𝑛+1∈𝐻_𝑛 that 𝜙𝑥_𝑛+1,𝑢_𝑛≤𝜙𝑥_𝑛+1,𝑦_𝑛≤𝜙𝑥_𝑛+1,𝑥_𝑛≤𝐺𝑥_𝑛+1,𝐽𝑥−𝐺𝑥_𝑛,𝐽𝑥.(3.11) Taking lim_𝑛→∞ in (3.11), we get lim_𝑛→∞𝜙𝑥_𝑛+1,𝑢_𝑛=lim_𝑛→∞𝜙𝑥_𝑛+1,𝑦_𝑛=lim_𝑛→∞𝜙𝑥_𝑛+1,𝑥_𝑛=0.(3.12) Applying Lemma 2.3, we obtain lim_𝑛→∞‖‖𝑥_𝑛+1−𝑢_𝑛‖‖=lim_𝑛→∞‖‖𝑥_𝑛+1−𝑦_𝑛‖‖=lim_𝑛→∞‖‖𝑥_𝑛+1−𝑥_𝑛‖‖=0.(3.13)
Next, we show that 𝑥̂∈𝐹(𝑇)=𝐹(𝑇).
From the fact that 𝐽 is uniformly norm-to-norm continuous on bounded sets, we have lim_𝑛→∞‖‖𝐽𝑥_𝑛+1−𝐽𝑦_𝑛‖‖=lim_𝑛→∞‖‖𝐽𝑥_𝑛+1−𝐽𝑥_𝑛‖‖=0.(3.14) Note that ‖‖𝐽𝑥_𝑛+1−𝐽𝑦_𝑛‖‖==‖‖‖‖𝐽𝑥1−_𝑛+1𝛼−_𝑛𝛼_𝑛𝐽𝑥_𝑛−1−𝛼_𝑛𝐽𝑇𝑥_𝑛‖‖≥1−𝛼_𝑛𝐽𝑥_𝑛+1−1−𝛼_𝑛𝐽𝑇𝑥_𝑛+𝛼_𝑛𝐽𝑥_𝑛+1−𝛼_𝑛𝐽𝑥_𝑛‖‖‖‖𝐽𝑥_𝑛+1−𝐽𝑇𝑥_𝑛‖‖−𝛼_𝑛‖‖𝐽𝑥_𝑛+1−𝐽𝑥_𝑛‖‖(3.15) and thus ‖‖𝐽𝑥_𝑛+1−𝐽𝑇𝑥_𝑛‖‖≤11−𝛼_𝑛‖‖𝐽𝑥_𝑛+1−𝐽𝑦_𝑛‖‖+𝛼^𝑛1−𝛼_𝑛≤1‖‖𝐽𝑥_𝑛+1−𝐽𝑥_𝑛‖‖1−𝛼_𝑛‖‖𝐽𝑥_𝑛+1−𝐽𝑦_𝑛‖‖+‖‖𝐽𝑥_𝑛+1−𝐽𝑥_𝑛‖‖.(3.16) From (3.14) and limsup_𝑛→∞𝛼_𝑛<1, we get lim_𝑛→∞‖‖𝐽𝑥_𝑛+1−𝐽𝑇𝑥_𝑛‖‖=0.(3.17) Since 𝐽⁻¹ is uniformly norm-to-norm continuous on bounded sets, we have lim_𝑛→∞‖‖𝑥_𝑛+1−𝑇𝑥_𝑛‖‖=0.(3.18) Since ‖‖𝑥_𝑛−𝑇𝑥_𝑛‖‖≤=‖‖‖‖𝑥𝑥_𝑛𝑛−𝑥_𝑛+1+𝑥_𝑛+1−𝑇𝑥_𝑛‖‖−𝑥_𝑛+1‖‖+‖‖𝑥_𝑛+1−𝑇𝑥_𝑛‖‖,(3.19) we have lim_𝑛→∞‖‖𝑥_𝑛−𝑇𝑥_𝑛‖‖=lim_𝑘→∞‖‖𝑥_𝑛𝑘−𝑇𝑥_𝑛𝑘‖‖=0(3.20) and so 𝑥̂∈𝐹(𝑇)=𝐹(𝑇).(3.21) Now, we show 𝑥̂∈EP(𝑔). Since ‖‖𝑢_𝑛−𝑦_𝑛‖‖≤‖‖𝑥_𝑛+1−𝑢_𝑛‖‖+‖‖𝑥_𝑛+1−𝑦_𝑛‖‖,(3.22) we get lim_𝑛→∞‖‖𝑢_𝑛−𝑦_𝑛‖‖=0.(3.23) Since 𝐽 is a uniformly norm-to-norm continuous on bounded sets, we have lim_𝑛→∞‖‖𝐽𝑢_𝑛−𝐽𝑦_𝑛‖‖=0.(3.24) From the assumption that 𝑟_𝑛≥𝑎, we get lim_𝑛→∞‖‖𝐽𝑢_𝑛−𝐽𝑦_𝑛‖‖𝑟_𝑛=0.(3.25) It follows from 𝑢_𝑛=𝑇_𝑟𝑛𝑦_𝑛 that 𝑔𝑢_𝑛,𝑦+1𝑟_𝑛⟨𝐽𝑢_𝑛−𝐽𝑦_𝑛,𝑦−𝑢_𝑛⟩≥0,∀𝑦∈𝑀.(3.26) From (A2), we obtain ‖‖𝑦−𝑢_𝑛‖‖⋅‖‖𝐽𝑢^𝑛−𝐽𝑦_𝑛‖‖𝑟_𝑛≥1𝑟_𝑛⟨𝐽𝑢_𝑛−𝐽𝑦_𝑛,𝑦−𝑢_𝑛⟩≥−𝑔𝑢_𝑛,𝑦≥𝑔𝑦,𝑢_𝑛,∀𝑦∈𝑀.(3.27) Since lim_𝑛→∞‖𝑥_𝑛+1−𝑢_𝑛‖=lim_𝑛→∞‖𝑥_𝑛+1−𝑥_𝑛‖=0, we obtain lim_𝑛→∞‖‖𝑥_𝑛−𝑢_𝑛‖‖=lim_𝑘→∞‖‖𝑥_𝑛𝑘−𝑢_𝑛𝑘‖‖=0.(3.28) For any ℎ∈𝑋^∗, it follows that lim_𝑘→∞ℎ𝑢_𝑛𝑘−ℎ(𝑥̂)=lim_𝑘→∞ℎ𝑢_𝑛𝑘−𝑥_𝑛𝑘+ℎ𝑥_𝑛k−𝑥̂=0(3.29) and so 𝑢_𝑛𝑘⇀𝑥̂. From (3.27) and (A4), we know that 𝑔(𝑦,𝑥̂)≤liminf_𝑘→∞𝑔𝑦,𝑢_𝑛𝑘≤lim_𝑘→∞‖‖𝑦−𝑢_𝑛𝑘‖‖⋅‖‖𝐽𝑢^𝑛𝑘−𝐽𝑦_𝑛𝑘‖‖𝑟_𝑛𝑘=0,∀𝑦∈𝑀.(3.30) Letting 𝑦_𝑡=𝑡𝑦+(1−𝑡)𝑥̂∈𝑀,∀0<𝑡<1,𝑦∈𝑀,(3.31) we have 𝑔𝑦_𝑡,𝑥̂≤0.(3.32) It follows from (A1) that 0=𝑔𝑦_𝑡,𝑦_𝑡≤𝑡𝑔𝑦_𝑡,𝑦+(1−𝑡)𝑔𝑦_𝑡,𝑥̂≤𝑡𝑔𝑦_𝑡,𝑦(3.33) and thus 𝑔𝑦_𝑡,𝑦≥0.(3.34) Taking the limit as 𝑡↓0 in (3.34) and from (A3), we have 𝑔(𝑥̂,𝑦)≥0,∀𝑦∈𝑀(3.35) and so 𝑥̂∈EP(𝑔).
(IV) Last, we prove that 𝑥_𝑛→Π^𝑓𝐹𝑥.
Since 𝐹 is a closed convex set, from Lemma 2.5, we know that Π^𝑓𝐹𝑥 is single-valued and denote that 𝑤=Π^𝑓𝐹𝑥. Since 𝑥_𝑛=Π^𝑓𝐻_𝑛𝑥 and 𝑤∈𝐹⊂𝐻_𝑛, we have 𝐺𝑥_𝑛,𝐽𝑥≤𝐺(𝑤,𝐽𝑥),∀𝑛≥1.(3.36) For each given 𝑥, 𝐺(𝜉,𝐽𝑥) is convex and lower semicontinuous with respect to 𝜉, it is easy to see that 𝐺(𝜉,𝐽𝑥) is weakly lower semicontinuous with respect to 𝜉 and so 𝐺(𝑥̂,𝐽𝑥)≤liminf_𝑘→∞𝐺𝑥_𝑛𝑘,𝐽𝑥≤limsup_𝑘→∞𝐺𝑥_𝑛𝑘,𝐽𝑥≤𝐺(𝑤,𝐽𝑥).(3.37) From the definition of Π^𝑓𝐹𝑥 and 𝑥̂∈𝐹, we know that 𝑥̂=𝑤 and so lim_𝑘→∞𝐺(𝑥_𝑛𝑘,𝐽𝑥)=𝐺(𝑥̂,𝐽𝑥). It follows from Lemma 2.7 that lim_𝑘→∞‖𝑥_𝑛𝑘‖=‖𝑥̂‖. The Kadec-Klee property of 𝑋 implies that {𝑥_𝑛𝑘} converges strongly to Π^𝑓𝐹𝑥. Since {𝑥_𝑛𝑘} is an arbitrarily weakly convergent sequence of {𝑥_𝑛}, we conclude that {𝑥_𝑛} converges strongly to Π^𝑓𝐹𝑥. This completes the proof.

Remark 3.2. Letting 𝑀=𝑋, EP(𝑔)=𝑋 and 𝑢_𝑛=𝑦_𝑛 in (3.1), then 𝐻_𝑛+1={𝑧∈𝐻_𝑛∶𝐺(𝑧,𝐽𝑦_𝑛)≤𝐺(𝑧,𝐽𝑥_𝑛)} and so Theorem 3.1 reduces to Theorem 4.1 of Li et al. [10].

Acknowledgments

This work was supported by the Key Program of NSFC (Grant no. 70831005) and the National Natural Science Foundation of China (11171237, 11101069). The work is also supported by HK CityU, CTTFS (9360142).