Convergence Theorems for a Common Point of Solutions of Equilibrium and Fixed Point of Relatively Nonexpansive Multivalued Mapping Problems

Abstract

We introduce an iterative process which converges strongly to a common point of set of solutions of equilibrium problem and set of fixed points of finite family of relatively nonexpansive multi-valued mappings in Banach spaces.

1. Introduction

Let 𝐸 be a real Banach space with dual 𝐸^∗. The function 𝜙∶𝐸×𝐸→ℝ⁺, defined by 𝜙𝑥,𝑦=‖𝑥‖²−2𝑥,𝐽𝑦+𝑦²,for𝑥,𝑦∈𝐸, (1.1) is studied by Alber [1] and Reich [2], where 𝐽 is the normalized duality mapping from 𝐸 to 2^𝐸∗ defined by 𝐽𝑥∶={𝑓^∗∈𝐸^∗∶⟨𝑥,𝑓^∗⟩=||𝑥||²=||𝑓^∗||²}, where ⟨⋅,⋅⟩ denotes the generalized duality pairing. It is well known that 𝐸 is smooth if and only if 𝐽 is single valued and if 𝐸 is uniformly smooth then 𝐽 is uniformly continuous on bounded subsets of 𝐸. We note that in a Hilbert space 𝐻,  𝐽 is the identity operator.

Let 𝐶 be a nonempty closed convex subset of a Hilbert space 𝐻. It is well known that the metric projection of 𝐻 onto 𝐶,  𝑃_𝐶∶𝐻→𝐶, is nonexpansive. This fact actually characterizes Hilbert spaces and consequently, it is not available in more general Banach spaces. In this direction, Alber [1] introduced a generalized projection operator Π_𝐶 in a Banach space 𝐸 which is an analogue of metric projection in Hilbert spaces.

Let 𝐶 be a nonempty closed and convex subset of a reflexive, strictly convex and smooth Banach space 𝐸. The generalized projection mapping, introduced by Alber [1], is a mapping Π_𝐶∶𝐸→𝐶, that assigns to an arbitrary point 𝑥∈𝐸 the minimum point of the functional 𝜙(𝑦,𝑥), that is, Π_𝐶𝑥=𝑥, where 𝑥 is the solution to the minimization problem 𝜙(𝑥,𝑥)=min𝜙𝑦,𝑥,𝑦∈𝐶. (1.2) Let 𝐶 be a nonempty closed convex subset of a Banach space 𝐸. Let 𝑇∶𝐶→𝐶 be a single-valued mapping. An element 𝑝∈𝐶 is called a fixed point of 𝑇 if 𝑇(𝑝)=𝑝. The set of fixed points of 𝑇 is denoted by 𝐹(𝑇). A point 𝑝 in 𝐶 is said to be an asymptotic fixed point of 𝑇 (see [2]) if 𝐶 contains a sequence {𝑥_𝑛} which converges weakly to 𝑝 such that lim_𝑛→∞||𝑥_𝑛−𝑇𝑥_𝑛||=0. The set of asymptotic fixed points of 𝑇 will be denoted by 𝐹(𝑇). 𝑇 is said to be nonexpansive if ||𝑇𝑥−𝑇𝑦||≤||𝑥−𝑦|| for each 𝑥,𝑦∈𝐶 and is called relatively nonexpansive if (A1) 𝐹(𝑇)≠∅; (A2) 𝜙(𝑝,𝑇𝑥)≤𝜙(𝑝,𝑥) for 𝑥∈𝐶 and 𝑝∈𝐹(𝑇) and (A3) 𝐹(𝑇)=𝐹(𝑇).

Let 𝐶 be a nonempty closed convex subset of a Banach space 𝐸 and let 𝑁(𝐶) and 𝐶𝐵(𝐶) denote the family of nonempty subsets and nonempty closed bounded subsets of 𝐶, respectively. Let 𝐻 be the Hausdorff metric on 𝐶𝐵(𝐶) defined by 𝐻(𝐴,𝐵)=maxsup_𝑎∈𝐴𝑑(𝑎,𝐵),sup_𝑏∈𝐵𝑑(𝑏,𝐴), (1.3) for all 𝐴,𝐵∈𝐶𝐵(𝐶), where 𝑑(𝑎,𝐵)=inf{||𝑎−𝑏||∶𝑏∈𝐵} is the distance from the point 𝑎 to the subset 𝐵.

Let 𝑇∶𝐶→𝐶𝐵(𝐶) be a multivalued mapping. 𝑇 is said to be a nonexpansive if 𝐻(𝑇𝑥,𝑇𝑦)≤||𝑥−𝑦||, for 𝑥,𝑦∈𝐶. An element 𝑝∈𝐶 is called a fixed point of 𝑇, if 𝑝∈𝐹(𝑇), where 𝐹(𝑇)∶={𝑝∈𝐶∶𝑝∈𝑇(𝑝)}. A point 𝑝∈𝐶 called an asymptotic fixed point of 𝑇, if there exists a sequence {𝑥_𝑛} in 𝐶 which converges weakly to 𝑝 such that lim_𝑛→∞𝑑(𝑥_𝑛,𝑇𝑥_𝑛)=0. 𝑇 is said to be relatively nonexpansive if (B1) 𝐹(𝑇)≠∅; (B2) 𝜙(𝑝,𝑢)≤𝜙(𝑝,𝑥) for 𝑥∈𝐶, 𝑢∈𝑇𝑥, 𝑝∈𝐹(𝑇) and (B3) 𝐹(𝑇)=𝐹(𝑇), where 𝐹(𝑇) is the set of asymptotic fixed points of 𝑇.

We remark that the class of relatively nonexpansive single-valued mappings is contained in a class of relatively nonexpansive multi-valued mappings. An example of relatively nonexpansive multi-valued mapping by Homaeipour and Razani [3] is given below.

Example 1.1. Let 𝐼=[0,1], 𝑋=𝐿^𝑝(𝐼), 1<𝑝<∞ and 𝐶={𝑓∈𝑋∶𝑓(𝑥)≥0,forall𝑥∈𝐼}. Let 𝑇∶𝐶→𝐶𝐵(𝐶) be defined by 𝑇𝑓=(𝑥)−3𝑔∈𝐶∶𝑓4(𝑥)(𝑥)−1≤𝑔≤𝑓4(𝑥){0},∀𝑥∈𝐼,if𝑓>1,𝑥∈𝐼,,otherwise. (1.4) It is shown in [3] that 𝑇 is relatively nonexpansive multi-valued mapping which is not nonexpansive.

The study of fixed points for multi-valued nonexpansive mappings in relation to Hausdorff metric was introduced by Markin [4] (see also [5]). Since then a lot of activity in this area and fixed point theory for multi-valued nonexpansive mappings has been developed which has some nontrivial applications in pure and applied sciences including control theory, convex optimization, differential inclusion, and economics (see, e.g., [6] and references therein). Later, Lim [7] established the existence of fixed points for multi-valued nonexpansive mappings in uniformly convex Banach spaces.

It is well known that the normal Mann’s iterative [8] algorithm has only weak convergence in an infinite-dimensional Hilbert space even for nonexpansive single-valued mappings. Consequently, in order to obtain strong convergence, one has to modify the normal Mann’s iteration algorithm, the so called hybrid projection iteration method is such a modification. The hybrid projection iteration algorithm (HPIA) was introduced initially by Haugazeau [9] in 1968. For 40 years, (HPIA) has received rapid developments. For details, the readers are referred to papers [10–12] and the references therein.

In 2003, Nakajo and Takahashi [12] proposed the following modification of the Mann iteration method for a nonexpansive single-valued mapping 𝑇 in a Hilbert space 𝐻: 𝑥₀𝑦∈𝐶,chosenarbitrary,_𝑛=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝑇𝑥_𝑛,𝐶_𝑛=𝑧∈𝐶∶𝑦_𝑛−𝑧≤𝑥_𝑛,𝑄−𝑧_𝑛=𝑧∈𝐶∶𝑥_𝑛−𝑧,𝑥₀−𝑥_𝑛,𝑥≥0_𝑛+1=𝑃_{𝐶𝑛∩𝑄𝑛}𝑥₀,𝑛≥0, (1.5) where 𝐶 is a closed convex subset of 𝐻, 𝑃_𝐶 denotes the metric projection from 𝐻 onto 𝐶. They proved that if the sequence {𝛼_𝑛} is bounded above from one then the sequence {𝑥_𝑛} generated by (1.5) converges strongly to 𝑃_𝐹(𝑇)(𝑥₀).

In spaces more general than Hilbert spaces, Matsushita and Takahashi [11] proposed the following hybrid iteration method with generalized projection for relatively nonexpansive single-valued mapping 𝑇 in a Banach space 𝐸: 𝑥₀𝑦∈𝐶,chosenarbitrary,_𝑛=𝐽⁻¹𝛼_𝑛𝐽𝑥_𝑛+1−𝛼_𝑛𝐽𝑇𝑥_𝑛,𝐶_𝑛=𝑧∈𝐶∶𝜙𝑧,𝑦_𝑛≤𝜙𝑧,𝑥_𝑛,𝑄_𝑛=𝑧∈𝐶∶𝑥_𝑛−𝑧,𝐽𝑥₀−𝐽𝑥_𝑛,𝑥≥0_𝑛+1=Π_{𝐶𝑛∩𝑄𝑛}𝑥₀,𝑛≥0. (1.6) They proved the following convergence theorem.

Theorem MT. Let 𝐸 be a uniformly convex and uniformly smooth Banach space, let 𝐶 be a nonempty closed convex subset of 𝐸, let 𝑇 be a relatively nonexpansive single-valued mapping from 𝐶 into itself, and let {𝛼_𝑛} be a sequence of real numbers such that 0≤𝛼_𝑛<1 and limsup_𝑛→∞𝛼_𝑛<1. Suppose that {𝑥_𝑛} is given by (1.6), where 𝐽 is the duality mapping on 𝐸. If 𝐹(𝑇) is nonempty, then {𝑥_𝑛} converges strongly to Π_𝐹(𝑇)𝑥₀, where Π_𝐹(𝑇)(⋅) is the generalized projection from 𝐸 onto 𝐹(𝑇).

Let 𝑓∶𝐶×𝐶→ℝ be a bifunction, where ℝ is the set of real numbers. The equilibrium problem for 𝑓 is finding𝑥^∗∈𝐶suchthat𝑓𝑥^∗,𝑦≥0,∀𝑦∈𝐶. (1.7) The solution set of (1.7) is denoted by 𝐸𝑃(𝑓).

If 𝑓(𝑥,𝑦)=⟨𝐴𝑥,𝑦−𝑥⟩, where 𝐴∶𝐶→𝐶 is a monotone mapping, then the problem (1.7) reduces to the system of variational inequality problem findanelement𝑥^∗∈𝐶suchthat𝐴𝑥^∗,𝑦−𝑥^∗≥0,∀𝑦∈𝐶. (1.8) That is, the problem (1.8) is a special case of (1.7). The set of solutions of inequality (1.8) is denoted by 𝑉𝐼(𝐶,𝐴).

For solving the equilibrium problem for a bifunction 𝑓∶𝐶×𝐶→𝑅, we assume that 𝑓 satisfies the following conditions: (A1) 𝑓(𝑥,𝑥)=0, for all 𝑥∈𝐶, (A2) 𝑓 is monotone, that is,  𝑓(𝑥,𝑦)+𝑓(𝑦,𝑥)≤0, for all 𝑥,𝑦∈𝐶, (A3) for each 𝑥,𝑦,𝑧∈𝐶, lim_𝑡→0⁺𝑓(𝑡𝑧+(1−𝑡)𝑥,𝑦)≤𝑓(𝑥,𝑦), (A4) for each 𝑥∈𝐶,  𝑦→𝑓(𝑥,𝑦) is convex and lower semicontinuous.

Recently, many authors studied the problem of finding a common element of the set of fixed points of nonexpansive or relatively nonexpansive single-valued mapping and the set of solutions of an equilibrium problems in the frame work of Hilbert spaces and Banach spaces, respectively: see, for instance, [2, 13–21] and the references therein.

In [22], Kumam introduced the following iterative scheme in a Hilbert space: 𝑥₀𝑢∈𝐻,_𝑛∈𝐶suchthat𝑓𝑢_𝑛+1,𝑦𝑟_𝑛𝑦−𝑢_𝑛,𝑢_𝑛−𝑦_𝑛𝑤≥0,∀𝑦∈𝐶,_𝑛=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝑇𝑢_𝑛,𝐶_𝑛=𝑧∈𝐻∶𝑤_𝑛−𝑧≤𝑥_𝑛,𝑄−𝑧_𝑛=𝑧∈𝐶∶𝑥_𝑛−𝑧,𝑥₀−𝑥_𝑛,𝑥≥0_𝑛+1=𝑃_{𝐶𝑛+1∩𝑄𝑛}𝑥₀,𝑛≥0, (1.9) for finding a common element of the set of fixed point of nonexpansive single-valued mapping 𝑇 and set of solution of equilibrium problems.

In the case that 𝐸 is a Banach space, Takahashi and Zembayashi [16] introduced the following iterative scheme which is called the shrinking projection method: 𝑥₀𝑦∈𝐶,chosenarbitrary,_𝑛=𝐽⁻¹𝛼_𝑛𝐽𝑥_𝑛+1−𝛼_𝑛𝐽𝑇𝑥_𝑛,𝑢_𝑛∈𝐶suchthat𝑓𝑢_𝑛+1,𝑦𝑟_𝑛𝑦−𝑢_𝑛,𝐽𝑢_𝑛−𝐽𝑦_𝑛𝐶≥0,∀𝑦∈𝐶,_𝑛+1=𝑧∈𝐶_𝑛∶𝜙𝑧,𝑢_𝑛≤𝜙𝑧,𝑥_𝑛,𝑥_𝑛+1=Π_𝐶𝑛+1𝑥₀,𝑛≥0, (1.10) where 𝐽 is the duality mapping on 𝐸, Π_𝐶 is the generalized projection from 𝐸 onto 𝐶 and 𝑇 is relatively nonexpansive single-valued mapping. They proved that the sequence {𝑥_𝑛} converges strongly to a common element of the set of fixed point of relatively nonexpansive single-valued mapping and set of solution of equilibrium problem under appropriate conditions.

We remark that the computation of 𝑥_𝑛+1 in (1.9) and (1.10) is not simple because of the involvement of computation of 𝐶_𝑛+1 from 𝐶_𝑛 for each 𝑛≥0.

More recently, Homaeipour and Razani [3] studied the following iterative scheme for a fixed point of relatively nonexpansive multi-valued mapping in uniformly convex and uniformly smooth Banach space 𝐸: 𝑥₀𝑥∈𝐶,chosenarbitrary,_𝑛+1=Π_𝐶𝐽⁻¹𝛼_𝑛𝐽𝑥_𝑛+1−𝛼_𝑛𝐽𝑧_𝑛,𝑧_𝑛∈𝑇𝑥_𝑛,𝑛≥0, (1.11) where {𝛼_𝑛}⊂(0,1) for all 𝑛≥0 and liminf_𝑛→∞𝛼_𝑛(1−𝛼_𝑛)>0. They proved that if 𝐽 is weakly sequentially continuous then the sequence {𝑥_𝑛} converges weakly to a fixed point of 𝑇. Furthermore, it is shown that the scheme converges strongly to a fixed point of 𝑇 if interior of 𝐹(𝑇) is nonempty.

But it is worth mentioning that the convergence of the scheme is either weak or it requires that the interior of  𝐹(𝑇) is nonempty.

In this paper, motivated by Kumam [22], Takahashi and Zembayashi [16], and Homaeipour and Razani [3], we construct an iterative scheme which converges strongly to a common point of set of solutions of equilibrium problem and set of fixed points of finite family of relatively nonexpansive multi-valued mappings in Banach spaces. Our scheme does not involve computation of 𝐶_𝑛 and 𝑄_𝑛, for each 𝑛≥0, and the requirement that the interior of 𝐹 is nonempty is dispensed with. Our theorems improve and unify most of the results that have been proved for this important class of nonlinear operators.

2. Preliminaries

Let 𝐸 be a normed linear space with  dim𝐸≥2. The modulus of smoothness of 𝐸 is the function 𝜌_𝐸∶[0,∞)→[0,∞) defined by 𝜌_𝐸(𝜏)+∶=sup𝑥+𝑦𝑥−𝑦2−1∶‖𝑥‖=1;𝑦=𝜏. (2.1) The space 𝐸 is said to be smooth if  𝜌_𝐸(𝜏)>0,  for all  𝜏>0 and 𝐸 is called uniformly smooth if and only if lim_𝑡→0⁺(𝜌_𝐸(𝑡)/𝑡)=0.

The modulus of convexity of 𝐸 is the function 𝛿_𝐸∶(0,2]→[0,1] defined by 𝛿_𝐸(𝜖)∶=inf1−𝑥+𝑦2∶‖𝑥‖=𝑦=1;𝜖=𝑥−𝑦. (2.2)𝐸 is called uniformly convex if and only if 𝛿_𝐸(𝜖)>0, for every 𝜖∈(0,2].

In the sequel, we will need the following results.

Lemma 2.1   (see [1]). Let 𝐾 be a nonempty closed and convex subset of a real reflexive, strictly convex, and smooth Banach space 𝐸 and let 𝑥∈𝐸. Then for all 𝑦∈𝐾, 𝜙𝑦,Π_𝐾𝑥+𝜙Π_𝐾𝑥,𝑥≤𝜙𝑦,𝑥. (2.3)

We make use of the function 𝑉∶𝐸×𝐸^∗→ℝ, defined by 𝑉𝑥,𝑥^∗=‖𝑥‖²−2𝑥,𝑥^∗+𝑥^∗2,∀𝑥∈𝐸,𝑥^∗∈𝐸^∗, (2.4) studied by Alber [1]. That is, 𝑉(𝑥,𝑥^∗)=𝜙(𝑥,𝐽⁻¹𝑥^∗) for all 𝑥∈𝐸 and 𝑥^∗∈𝐸^∗. We know the following lemma.

Lemma 2.2 (see [1]). Let 𝐸 be reflexive strictly convex and smooth Banach space with 𝐸^∗ as its dual. Then 𝑉𝑥,𝑥^∗+2𝐽⁻¹𝑥^∗−𝑥,𝑦^∗≤𝑉𝑥,𝑥^∗+𝑦^∗, (2.5) for all 𝑥∈𝐸 and 𝑥^∗,𝑦^∗∈𝐸^∗.

Lemma 2.3 (see [1]). Let 𝐶 be a convex subset of a real smooth Banach space 𝐸. Let 𝑥∈𝐸. Then 𝑥₀=Π_𝐶𝑥 if and only if 𝑧−𝑥₀,𝐽𝑥−𝐽𝑥₀≤0,∀𝑧∈𝐶. (2.6)

Lemma 2.4 (see [23]). Let 𝐸 be a uniformly convex Banach space and 𝐵_𝑅(0) be a closed ball of 𝐸. Then, there exists a continuous strictly increasing convex function 𝑔∶[0,∞)→[0,∞) with 𝑔(0)=0 such that 𝛼₁𝑥₁+𝛼₂𝑥₂+⋯+𝛼_𝑁𝑥_𝑁²≤^𝑁_𝑖=1𝛼_𝑖𝑥_𝑖²−𝛼_𝑖𝛼_𝑗𝑔𝑥_𝑖−𝑥_𝑗, (2.7) for 𝑖,𝑗∈{1,…,𝑁}, 𝛼_𝑖∈(0,1) such that ∑^𝑁_𝑖=1𝛼_𝑖=1, and 𝑥_𝑖∈𝐵_𝑅(0)∶={𝑥∈𝐸∶||𝑥||≤𝑅}, for 𝑖=1,2,…,𝑁.

Lemma 2.5 (see [24]). Let 𝐸 be a real smooth and uniformly convex Banach space and let {𝑥_𝑛} and {𝑦_𝑛} be two sequences of 𝐸. If either {𝑥_𝑛} or {𝑦_𝑛} is bounded and 𝜙(𝑥_𝑛,𝑦_𝑛)→0 as 𝑛→∞, then 𝑥_𝑛−𝑦_𝑛→0, as 𝑛→∞.

Proposition 2.6 (see [3]). Let 𝐸 be a strictly convex and smooth Banach space and 𝐶 be a nonempty closed convex subset of 𝐸. Let 𝑇∶𝐶→𝑁(𝐶) be a relatively nonexpansive multi-valued mapping. Then 𝐹(𝑇) is closed and convex.

Lemma 2.7 (see [16]). Let 𝐶 be a nonempty, closed and convex subset of a uniformly smooth, strictly convex and reflexive real Banach space 𝐸. Let 𝑓 be a bifunction from 𝐶×𝐶 to ℝ which satisfies conditions (A1)–(A4). For 𝑟>0 and 𝑥∈𝐸, define the mapping 𝐹_𝑟∶𝐸→𝐶 as follows: 𝐹_𝑟+1𝑥∶=𝑧∈𝐶∶𝑓𝑧,𝑦𝑟𝑦−𝑧,𝐽𝑧−𝐽𝑥≥0,∀𝑦∈𝐶. (2.8) Then the following statements hold:(1)𝐹_𝑟 is single-valued,(2)𝐹(𝐹_𝑟)=𝐸𝑃(𝑓),(3)𝜙(𝑞,𝐹_𝑟𝑥)+𝜙(𝐹_𝑟𝑥,𝑥)≤𝜙(𝑞,𝑥), for 𝑞∈𝐹(𝐹_𝑟),(4)𝐸𝑃(𝑓) is closed and convex.

Lemma 2.8 (see [25]). Let {𝑎_𝑛} be sequences of real numbers such that there exists a subsequence {𝑛_𝑖} of {𝑛} such that 𝑎_𝑛𝑖<𝑎_𝑛𝑖+1 for all 𝑖∈ℕ. Then there exists a nondecreasing sequence {𝑚_𝑘}⊂ℕ such that 𝑚_𝑘→∞ and the following properties are satisfied by all (sufficiently large) numbers 𝑘∈ℕ: 𝑎_𝑚𝑘≤𝑎_𝑚𝑘+1,𝑎_𝑘≤𝑎_𝑚𝑘+1. (2.9) In fact, 𝑚_𝑘=max{𝑗≤𝑘∶𝑎_𝑗<𝑎_𝑗+1}.

Lemma 2.9 (see [26]). Let {𝑎_𝑛} be a sequence of nonnegative real numbers satisfying the following relation: 𝑎_𝑛+1≤1−𝛽_𝑛𝑎_𝑛+𝛽_𝑛𝛿_𝑛,𝑛≥𝑛₀,𝑓𝑜𝑟𝑠𝑜𝑚𝑒𝑛₀∈ℕ, (2.10) where {𝛽_𝑛}⊂(0,1) and {𝛿_𝑛}⊂𝑅 satisfying the following conditions: lim_𝑛→∞𝛽_𝑛=0,∑^∞_𝑛=1𝛽_𝑛=∞, and limsup_𝑛→∞𝛿_𝑛≤0. Then, lim_𝑛→∞𝑎_𝑛=0.

3. Main Result

Let 𝐶 be a nonempty, closed and convex subset of a smooth, strictly convex and reflexive real Banach space 𝐸 with dual 𝐸^∗. Let 𝑓∶𝐶×𝐶→ℝ be a bifunction. For the rest of this paper, 𝐹_𝑟𝑛𝑥 is a mapping defined as follows. For 𝑥∈𝐸, let 𝐹_𝑟𝑛:𝐸→𝐶 be given by 𝐹_𝑟𝑛+1𝑥∶=𝑧∈𝐶∶𝑓𝑧,𝑦𝑟_𝑛𝑦−𝑧,𝐽𝑧−𝐽𝑥≥0,∀𝑦∈𝐶, (3.1) where {𝑟_𝑛}_𝑛∈ℕ⊂[𝑐₁,∞), for some 𝑐₁>0.

Theorem 3.1. Let 𝐶 be a nonempty, closed and convex subset of a uniformly smooth and uniformly convex real Banach space 𝐸. Let 𝑓∶𝐶×𝐶→ℝ, be a bifunction which satisfies conditions (A1)–(A4). Let 𝑇_𝑖∶𝐶→𝐶𝐵(𝐶), for 𝑖=1,2,…,𝑁, be a finite family of relatively nonexpansive multi-valued mappings. Assume that 𝐹∶=∩^𝑁_𝑖=1𝐹(𝑇_𝑖)∩𝐸𝑃(𝑓) is nonempty. Let {𝑥_𝑛} be a sequence generated by 𝑥₀𝑤=𝑤∈𝐶,𝑐ℎ𝑜𝑠𝑒𝑛𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑖𝑙𝑦,_𝑛=𝐹_𝑟𝑛𝑥_𝑛,𝑦_𝑛=Π_𝐶𝐽⁻¹𝛼_𝑛𝐽𝑤+1−𝛼_𝑛𝐽𝑤_𝑛,𝑥_𝑛+1=𝐽⁻¹𝛽_𝑛,0𝐽𝑤_𝑛+^𝑁_𝑖=1𝛽_𝑛,𝑖𝐽𝑢_𝑛,𝑖,𝑢_𝑛,𝑖∈𝑇_𝑖𝑦_𝑛,𝑛≥0, (3.2) where 𝛼_𝑛∈(0,1) such that lim_𝑛→∞𝛼_𝑛=0, ∑^∞_𝑛=1𝛼_𝑛=∞, {𝛽_𝑛,𝑖}⊂[𝑎,𝑏]⊂(0,1),  for 𝑖=1,2,…,𝑁, satisfying 𝛽_𝑛,0+𝛽_𝑛,1+⋯+𝛽_𝑛,𝑁=1, for each 𝑛≥0. Then {𝑥_𝑛} converges strongly to an element of 𝐹.

Proof. Since 𝐹 is nonempty closed and convex, put 𝑥^∗∶=Π_𝐹𝑤. Now from (3.2), Lemma 2.7(3) and property of 𝜙, we get that 𝜙𝑥^∗,𝑦_𝑛=𝜙𝑥^∗,Π_𝐶𝐽⁻¹𝛼_𝑛𝐽𝑤+1−𝛼_𝑛𝐽𝑤_𝑛≤𝜙𝑥^∗,𝐽⁻¹𝛼_𝑛𝐽𝑤+1−𝛼_𝑛𝐽𝑤_𝑛=𝑥^∗2−2𝑥^∗,𝛼_𝑛𝐽𝑤+1−𝛼_𝑛𝐽𝑤_𝑛+𝛼_𝑛𝐽𝑤+1−𝛼_𝑛𝐽𝑤_𝑛²≤𝑥^∗2−2𝛼_𝑛𝑥^∗,𝐽𝑤−21−𝛼_𝑛𝑥^∗,𝐽𝑤_𝑛+𝛼_𝑛‖𝑤‖²+1−𝛼_𝑛𝑤_𝑛²≤𝛼_𝑛𝜙𝑥^∗,𝑤+1−𝛼_𝑛𝜙𝑥^∗,𝑤_𝑛=𝛼_𝑛𝜙𝑥^∗,𝑤+1−𝛼_𝑛𝜙𝑥^∗,𝐹_𝑟𝑛𝑥_𝑛≤𝛼_𝑛𝜙𝑥^∗,𝑤+1−𝛼_𝑛𝜙𝑥^∗,𝑥_𝑛. (3.3) Now, from (3.2), Lemma 2.7(3), relatively nonexpansiveness of 𝑇_𝑖, property of 𝜙 and (3.3), we have that 𝜙𝑥^∗,𝑥_𝑛+1=𝜙𝑥^∗,𝐽⁻¹𝛽_𝑛,0𝐽𝑤_𝑛+^𝑁_𝑖=1𝛽_𝑛,𝑖𝐽𝑢_𝑛,𝑖≤𝛽_𝑛,0𝜙𝑥^∗,𝑤_𝑛+^𝑁_𝑖=1𝛽_𝑛,𝑖𝜙𝑥^∗,𝑢_𝑛,𝑖=𝛽_𝑛,0𝜙𝑥^∗,𝐹_𝑟𝑛𝑥_𝑛+^𝑁_𝑖=1𝛽_𝑛,𝑖𝜙𝑥^∗,𝑢_𝑛,𝑖≤𝛽_𝑛,0𝜙𝑥^∗,𝑥_𝑛+1−𝛽_𝑛,0𝜙𝑥^∗,𝑦_𝑛≤𝛽_𝑛,0𝜙𝑥^∗,𝑥_𝑛+1−𝛽_𝑛,0𝛼_𝑛𝜙𝑥^∗,𝑤+1−𝛼_𝑛𝜙𝑥^∗,𝑥_𝑛≤𝛿_𝑛𝜙𝑥^∗,𝑤+1−𝛿_𝑛𝜙𝑥^∗,𝑥_𝑛, (3.4) where 𝛿_𝑛=(1−𝛽_𝑛,0)𝛼_𝑛. Thus, by induction, 𝜙𝑥^∗,𝑥_𝑛+1≤max𝜙𝑥^∗,𝑥₀,𝜙𝑥^∗,𝑤,∀𝑛≥0, (3.5) which implies that {𝑥_𝑛} is bounded and hence {𝑦_𝑛} and {𝑤_𝑛} are bounded. Now let 𝑧_𝑛=𝐽⁻¹(𝛼_𝑛𝐽𝑤+(1−𝛼_𝑛)𝐽𝑤_𝑛). Then we have that 𝑦_𝑛=Π_𝐶𝑧_𝑛. Using Lemma 2.2 and property of 𝜙, we obtain that 𝜙𝑥^∗,𝑦_𝑛≤𝜙𝑥^∗,𝑧_𝑛=𝑉𝑥^∗,𝐽𝑧_𝑛≤𝑉𝑥^∗,𝐽𝑧_𝑛−𝛼_𝑛𝐽𝑤−𝐽𝑥^∗−2𝑧_𝑛−𝑥^∗,−𝛼_𝑛𝐽𝑤−𝐽𝑥^∗=𝜙𝑥^∗,𝐽⁻¹𝛼_𝑛𝐽𝑥^∗+1−𝛼_𝑛𝐽𝑤_𝑛+2𝛼_𝑛𝑧_𝑛−𝑥^∗,𝐽𝑤−𝐽𝑥^∗≤𝛼_𝑛𝜙𝑥^∗,𝑥^∗+1−𝛼_𝑛𝜙𝑥^∗,𝑤_𝑛+2𝛼_𝑛𝑧_𝑛−𝑥^∗,𝐽𝑤−𝐽𝑥^∗=1−𝛼_𝑛𝜙𝑥^∗,𝑤_𝑛+2𝛼_𝑛𝑧_𝑛−𝑥^∗,𝐽𝑤−𝐽𝑥^∗≤1−𝛼_𝑛𝜙𝑥^∗,𝑥_𝑛+2𝛼_𝑛𝑧_𝑛−𝑥^∗,𝐽𝑤−𝐽𝑥^∗. (3.6)
Furthermore, from (3.2), Lemma 2.4, relatively nonexpansiveness of 𝑇_𝑖, for each 𝑖=1,2,…,𝑁, Lemma 2.7(3), and (3.6) we have that 𝜙𝑥^∗,𝑥_𝑛+1=𝜙𝑥^∗,𝐽⁻¹𝛽_𝑛,0𝐽𝑤_𝑛+^𝑁_𝑖=1𝛽_𝑛,𝑖𝐽𝑢_𝑛,𝑖≤𝛽_𝑛,0𝜙𝑥^∗,𝑤_𝑛+^𝑁_𝑖=1𝛽_𝑛,𝑖𝜙𝑥^∗,𝑢_𝑛,𝑖−𝛽_𝑛,0𝛽_𝑛,𝑖𝑔𝐽𝑤_𝑛−𝐽𝑢_𝑛,𝑖=𝛽_𝑛,0𝜙𝑥^∗,𝐹_𝑟𝑛𝑥_𝑛+^𝑁_𝑖=1𝛽_𝑛,𝑖𝜙𝑥^∗,𝑢_𝑛,𝑖−𝛽_𝑛,0𝛽_𝑛,𝑖𝑔𝐽𝑤_𝑛−𝐽𝑢_𝑛,𝑖≤𝛽_𝑛,0𝜙𝑥^∗,𝑥_𝑛−𝜙𝑥_𝑛,𝑤_𝑛+1−𝛽_𝑛,0𝜙𝑥^∗,𝑦_𝑛−𝛽_𝑛,0𝛽_𝑛,𝑖𝑔𝐽𝑤_𝑛−𝐽𝑢_𝑛,𝑖≤𝛽_𝑛,0𝜙𝑥^∗,𝑥_𝑛−𝛽_𝑛,0𝜙𝑥_𝑛,𝑤_𝑛+1−𝛽_𝑛,0×1−𝛼_𝑛𝜙𝑥^∗,𝑥_𝑛+2𝛼_𝑛𝑧_𝑛−𝑥^∗,𝐽𝑤−𝐽𝑥^∗−𝛽_𝑛,0𝛽_𝑛,𝑖𝑔𝐽𝑤_𝑛−𝐽𝑢_𝑛,𝑖=1−𝛿_𝑛𝜙𝑥^∗,𝑥_𝑛+2𝛿_𝑛𝑧_𝑛−𝑥^∗,𝐽𝑤−𝐽𝑥^∗−𝛽_𝑛,0𝜙𝑥_𝑛,𝑤_𝑛−𝛽_𝑛,0𝛽_𝑛,𝑖𝑔𝐽𝑤_𝑛−𝐽𝑢_𝑛,𝑖, (3.7) and hence 𝜙𝑥^∗,𝑥_𝑛+1≤1−𝛿_𝑛𝜙𝑥^∗,𝑥_𝑛+2𝛿_𝑛𝑧_𝑛−𝑥^∗,𝐽𝑤−𝐽𝑥^∗, (3.8) where 𝛿_𝑛∶=𝛼_𝑛(1−𝛽_𝑛,0), for all 𝑛∈ℕ. Note that 𝛿_𝑛 satisfies lim_𝑛𝛿_𝑛=0 and ∑^∞_𝑛=1𝛿_𝑛=∞.

Now, we consider two cases.

Case 1. Suppose that there exists 𝑛₀∈ℕ such that {𝜙(𝑥^∗,𝑥_𝑛)} is nonincreasing for all 𝑛≥𝑛₀. In this situation, {𝜙(𝑥^∗,𝑥_𝑛)} is then convergent. Then from (3.7), we have that 𝜙(𝑥_𝑛,𝑤_𝑛)→0 and hence Lemma 2.5 implies that 𝑥_𝑛−𝑤_𝑛→0,as𝑛→∞. (3.9) Moreover, from (3.7), we have that 𝛽_𝑛,0𝛽_𝑛,𝑖𝑔(||𝐽𝑤_𝑛−𝐽𝑢_𝑛,𝑖||)→0, as 𝑛→∞, which implies by the property of 𝑔 that 𝐽𝑤_𝑛−𝐽𝑢_𝑛,𝑖→0, as 𝑛→∞, for each 𝑖∈{1,2,…,𝑁}, and hence, since 𝐽⁻¹ uniformly continuous on bounded sets, we obtain that 𝑤_𝑛−𝑢_𝑛,𝑖→0,as𝑛→∞. (3.10) Furthermore, by Lemma 2.1, property of 𝜙 and the fact that 𝛼_𝑛→0, as 𝑛→∞, imply that 𝜙𝑤_𝑛,𝑦_𝑛=𝜙𝑤_𝑛,Π_𝐶𝑧_𝑛≤𝜙𝑤_𝑛,𝑧_𝑛=𝜙𝑤_𝑛,𝐽⁻¹𝛼_𝑛𝐽𝑤+1−𝛼_𝑛𝐽𝑤_𝑛≤𝛼_𝑛𝜙𝑤_𝑛,𝑤+1−𝛼_𝑛𝜙𝑤_𝑛,𝑤_𝑛→0,as𝑛→∞, (3.11) and hence 𝑤_𝑛−𝑦_𝑛→0,𝑤_𝑛−𝑧_𝑛→0,as𝑛→∞. (3.12) Therefore, from (3.9), (3.10), and (3.12), we obtain that 𝑥_𝑛−𝑧_𝑛→0,𝑦_𝑛−𝑥_𝑛𝑑𝑦→0,as𝑛→∞,_𝑛,𝑇_𝑖𝑦_𝑛≤𝑦_𝑛−𝑢_𝑛,𝑖≤𝑦_𝑛−𝑤_𝑛+𝑤_𝑛−𝑢_𝑛,𝑖→0, (3.13)(3.14) as 𝑛→∞, for each 𝑖∈{1,2,…,𝑁}.

Let {𝑧_𝑛𝑖} be a subsequence of {𝑧_𝑛} such that 𝑧_𝑛𝑖⇀𝑧  and limsup_𝑛→∞⟨𝑧_𝑛−𝑥^∗,𝐽𝑤−𝐽𝑥^∗⟩=lim_𝑖→∞⟨𝑧_𝑛𝑖−𝑥^∗,𝐽𝑤−𝐽𝑥^∗⟩. Then, from (3.12), (3.13), and the uniform continuity of 𝐽, we get that 𝑥_𝑛𝑖,𝑤_𝑛𝑖,𝑦_𝑛𝑖⇀𝑧,𝐽𝑥_𝑛−𝐽𝑤_𝑛→0,as𝑛→∞. (3.15) Now, we show that 𝑧∈𝐸𝑃(𝑓). But, from the definition of 𝑤_𝑛 and (A2) we note that 1𝑟_𝑛𝑖𝑦−𝑤_𝑛𝑖,𝐽𝑤_𝑛𝑖−𝐽𝑥_𝑛𝑖≥−𝑓𝑤_𝑛𝑖,𝑦≥𝑓𝑦,𝑤_𝑛𝑖,∀𝑦∈𝐶. (3.16) Letting 𝑖→∞, we have from (3.15) and (A4) that 𝑓(𝑦,𝑧)≤0, for all 𝑦∈𝐶. Now, for 0<𝑡≤1 and 𝑦∈𝐶, let 𝑦_𝑡=𝑡𝑦+(1−𝑡)𝑧. Since 𝑦∈𝐶 and 𝑧∈𝐶, we have 𝑦_𝑡∈𝐶 and hence 𝑓(𝑦_𝑡,𝑧)≤0. So, from the convexity of the equilibrium bifunction 𝑓(𝑥,𝑦) on the second variable 𝑦, we have 0=𝑓𝑦_𝑡,𝑦_𝑡≤𝑡𝑓𝑦_𝑡,𝑦+(1−𝑡)𝑓𝑦_𝑡,𝑧≤𝑡𝑓𝑦_𝑡,𝑦, (3.17) and hence 𝑓(𝑦_𝑡,𝑦)≥0. Now, letting 𝑡→0 and condition (A3), we obtain that 𝑓(𝑧,𝑦)≥0, for all 𝑦∈𝐶, and hence 𝑧∈𝐸𝑃(𝑓).

Next, we show that 𝑧∈∩^𝑁_𝑖=1𝐹(𝑇_𝑖). But, since each 𝑇_𝑖 satisfies condition (B3) we obtain from (3.13) and (3.15) that 𝑧∈𝐹(𝑇_𝑖), for each 𝑖=1,2,…,𝑁, and hence 𝑧∈∩^𝑁_𝑖=1𝐹(𝑇_𝑖). Thus, from the above discussions we obtain that 𝑧∈𝐹∶=∩^𝑁_𝑖=1𝐹(𝑇_𝑖)∩𝐸𝑃(𝑓). Therefore, by Lemma 2.3, we immediately obtain that limsup_𝑛→∞⟨𝑧_𝑛−𝑥^∗,𝐽𝑤−𝐽𝑥^∗⟩=lim_𝑖→∞⟨𝑧_𝑛𝑖−𝑥^∗,𝐽𝑤−𝐽𝑥^∗⟩=⟨𝑧−𝑥^∗,𝐽𝑤−𝐽𝑥^∗⟩≤0. It follows from (3.8) and Lemma 2.9 that 𝜙(𝑥^∗,𝑥_𝑛)→0, as 𝑛→∞. Consequently, 𝑥_𝑛→𝑥^∗ by Lemma 2.5.

Case 2. Suppose that there exists a subsequence {𝑛_𝑖} of {𝑛} such that 𝜙𝑥^∗,𝑥_𝑛𝑖<𝜙𝑥^∗,𝑥_𝑛𝑖+1 (3.18) for all 𝑖∈ℕ. Then, by Lemma 2.8, there exist a nondecreasing sequence {𝑚_𝑘}⊂ℕ such that 𝑚_𝑘→∞, 𝜙(𝑥^∗,𝑥_𝑚𝑘)≤𝜙(𝑥^∗,𝑥_𝑚𝑘+1) and 𝜙(𝑥^∗,𝑥_𝑘)≤𝜙(𝑥^∗,𝑥_𝑚𝑘+1), for all 𝑘∈ℕ. Now, from (3.7) and the fact that 𝛿_𝑛→0, we have 𝛽_𝑚𝑘,0𝜙𝑥_𝑚𝑘,𝑤_𝑚𝑘+𝛽_𝑚𝑘,0𝛽_{𝑚𝑘,𝑖}𝑔𝐽𝑤_𝑚𝑘−𝐽𝑢_{𝑚𝑘,𝑖}≤𝜙𝑥^∗,𝑥_𝑚𝑘−𝜙𝑥^∗,𝑥_𝑚𝑘+1−𝛿_𝑚𝑘𝜙𝑥^∗,𝑥_𝑚𝑘+2𝛿_𝑚𝑘𝑧_𝑚𝑘−𝑥^∗,𝐽𝑤−𝐽𝑥^∗, (3.19) as 𝑘→∞. Thus, using the same proof of Case 1, we obtain that 𝑥_𝑚𝑘−𝑤_𝑚𝑘→0 and 𝑤_𝑚𝑘−𝑢_{𝑚𝑘,𝑖}→0, as 𝑘→∞, for each 𝑖=1,2,…,𝑁 and hence limsup_𝑛→∞𝑧_𝑚𝑘−𝑥^∗,𝐽𝑤−𝐽𝑥^∗≤0. (3.20) Then from (3.8), we have that 𝜙𝑥^∗,𝑥_𝑚𝑘+1≤1−𝛿_𝑚𝑘𝜙𝑥^∗,𝑥_𝑚𝑘+2𝛿_𝑚𝑘𝑧_𝑚𝑘−𝑥^∗,𝐽𝑤−𝐽𝑥^∗. (3.21) Since 𝜙(𝑥^∗,𝑥_𝑚𝑘)≤𝜙(𝑥^∗,𝑥_𝑚𝑘+1), (3.21) implies that 𝛿_𝑚𝑘𝜙𝑥^∗,𝑥_𝑚𝑘≤𝜙𝑥^∗,𝑥_𝑚𝑘−𝜙𝑥^∗,𝑥_𝑚𝑘+1+2𝛿_𝑚𝑘𝑧_𝑚𝑘−𝑥^∗,𝐽𝑤−𝐽𝑥^∗≤2𝛿_𝑚𝑘𝑧_𝑚𝑘−𝑥^∗,𝐽𝑤−𝐽𝑥^∗. (3.22) In particular, since 𝛿_𝑚𝑘>0, we get 𝜙𝑥^∗,𝑥_𝑚𝑘≤2𝑧_𝑚𝑘−𝑥^∗,𝐽𝑤−𝐽𝑥^∗. (3.23) Then, from (3.20), we obtain that 𝜙(𝑥^∗,𝑥_𝑚𝑘)→0, as 𝑘→∞. This together with (3.21) gives 𝜙(𝑥^∗,𝑥_𝑚𝑘+1)→0, as 𝑘→∞. But 𝜙(𝑥^∗,𝑥_𝑘)≤𝜙(𝑥^∗,𝑥_𝑚𝑘+1) for all 𝑘∈ℕ, thus we obtain that 𝑥_𝑘→𝑥^∗. Therefore, from the above two cases, we can conclude that {𝑥_𝑛} converges strongly to 𝑥^∗ and the proof is complete.

If in Theorem 3.1, we assume that 𝑓(𝑥,𝑦)=⟨𝐴𝑥,𝑦−𝑥⟩, for 𝐴 continuous monotone mapping, then we obtain the following corollary.

Corollary 3.2. Let 𝐶 be a nonempty, closed and convex subset of a uniformly smooth and uniformly convex real Banach space 𝐸. Let 𝐴∶𝐶→𝐸^∗ be a continuous monotone mapping. Let 𝑇_𝑖∶𝐶→𝐶𝐵(𝐶), for 𝑖=1,2,…,𝑁, be a finite family of relatively nonexpansive multi-valued mappings. Assume that 𝐹∶=∩^𝑁_𝑖=1𝐹(𝑇_𝑖)∩𝑉𝐼(𝐶,𝐴) is nonempty. Let {𝑥_𝑛} be a sequence generated by 𝑥₀𝑤=𝑤∈𝐶,𝑐ℎ𝑜𝑠𝑒𝑛𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑖𝑙𝑦,_𝑛∈𝐶𝑠𝑢𝑐ℎ𝑡ℎ𝑎𝑡𝐴𝑤_𝑛,𝑦−𝑤_𝑛+1𝑟_𝑛𝑦−𝑤_𝑛,𝐽𝑤_𝑛−𝐽𝑥_𝑛𝑦≥0,∀𝑦∈𝐶,_𝑛=Π_𝐶𝐽⁻¹𝛼_𝑛𝐽𝑤+1−𝛼_𝑛𝐽𝑤_𝑛,𝑥_𝑛+1=𝐽⁻¹𝛽_𝑛,0𝐽𝑤_𝑛+^𝑁_𝑖=1𝛽_𝑛,𝑖𝐽𝑢_𝑛,𝑖,𝑢_𝑛,𝑖∈𝑇_𝑖𝑦_𝑛,𝑛≥0, (3.24) where 𝛼_𝑛∈(0,1) such that lim_𝑛→∞𝛼_𝑛=0, ∑^∞_𝑛=1𝛼_𝑛=∞, {𝛽_𝑛,𝑖}⊂[𝑎,𝑏]⊂(0,1), for 𝑖=1,2,…,𝑁, satisfying 𝛽_𝑛,0+𝛽_𝑛,1+⋯+𝛽_𝑛,𝑁=1, for each 𝑛≥0. Then {𝑥_𝑛} converges strongly to an element of 𝐹.

Proof. Let 𝑓(𝑥,𝑦)=⟨𝐴𝑥,𝑦−𝑥⟩. Since 𝐴 is monotone and continuous, we get that a bifunction 𝑓 satisfies conditions (A1)–(A4). Thus, the conclusion follows from Theorem 3.1.

If in Theorem 3.1, we assume that 𝑁=1, then we get the following theorem.

Corollary 3.3. Let 𝐶 be a nonempty, closed, and convex subset of a uniformly smooth and uniformly convex real Banach space 𝐸. Let 𝑓∶𝐶×𝐶→ℝ, be a bifunction which satisfies conditions (A1)–(A4). Let 𝑇∶𝐶→𝐶𝐵(𝐶) be a relatively nonexpansive multi-valued mapping. Assume that 𝐹∶=𝐹(𝑇)∩𝐸𝑃(𝑓) is nonempty. Let {𝑥_𝑛} be a sequence generated by 𝑥₀𝑤=𝑤∈𝐶,𝑐ℎ𝑜𝑠𝑒𝑛𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑖𝑙𝑦,_𝑛=𝐹_𝑟𝑛𝑥_𝑛,𝑦_𝑛=Π_𝐶𝐽⁻¹𝛼_𝑛𝐽𝑤+1−𝛼_𝑛𝐽𝑤_𝑛,𝑥_𝑛+1=𝐽⁻¹𝛽_𝑛𝐽𝑤_𝑛+1−𝛽_𝑛𝐽𝑢_𝑛,𝑢_𝑛∈𝑇𝑦_𝑛,𝑛≥0, (3.25) where 𝛼_𝑛∈(0,1) such that lim_𝑛→∞𝛼_𝑛=0, ∑^∞_𝑛=1𝛼_𝑛=∞, {𝛽_𝑛}⊂[𝑎,𝑏]⊂(0,1), for each 𝑛≥0. Then {𝑥_𝑛} converges strongly to an element of 𝐹.

Proof. The proof follows from Theorem 3.1 with 𝑁=1.

If in Theorem 3.1, we assume that 𝑓≡0, we get the following corollary.

Corollary 3.4. Let 𝐶 be a nonempty, closed and convex subset of a uniformly smooth and uniformly convex real Banach space 𝐸. Let 𝑇_𝑖∶𝐶→𝐶𝐵(𝐶), for 𝑖=1,2,…,𝑁, be a finite family of relatively nonexpansive multi-valued mappings. Assume that 𝐹∶=∩^𝑁_𝑖=1𝐹(𝑇_𝑖) is nonempty. Let {𝑥_𝑛} be a sequence generated by 𝑥₀𝑦=𝑤∈𝐶,𝑐ℎ𝑜𝑠𝑒𝑛𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑖𝑙𝑦,_𝑛=Π_𝐶𝐽⁻¹𝛼_𝑛𝐽𝑤+1−𝛼_𝑛𝐽𝑥_𝑛,𝑥_𝑛+1=𝐽⁻¹𝛽_𝑛,0𝐽𝑥_𝑛+^𝑁_𝑖=1𝛽_𝑛,𝑖𝐽𝑢_𝑛,𝑖,𝑢_𝑛,𝑖∈𝑇_𝑖𝑦_𝑛,𝑛≥0, (3.26) where 𝛼_𝑛∈(0,1) such that lim_𝑛→∞𝛼_𝑛=0, ∑^∞_𝑛=1𝛼_𝑛=∞, {𝛽_𝑛,𝑖}⊂[𝑎,𝑏]⊂(0,1), for 𝑖=1,2,…,𝑁 satisfying 𝛽_𝑛,0+𝛽_𝑛,1+⋯+𝛽_𝑛,𝑁=1, for each 𝑛≥0. Then {𝑥_𝑛} converges strongly to an element of 𝐹.

If in Theorem 3.1, we assume that each 𝑇_𝑖,  𝑖=1,2,…,𝑁 is single valued, we get the following corollary.

Corollary 3.5. Let 𝐶 be a nonempty, closed and convex subset of a uniformly smooth and uniformly convex real Banach space 𝐸. Let 𝑓∶𝐶×𝐶→ℝ, be a bifunction which satisfies conditions (A1)–(A4). Let 𝑇_𝑖∶𝐶→𝐶, for 𝑖=1,2,…,𝑁, be a finite family of relatively nonexpansive single-valued mappings. Assume that 𝐹∶=∩^𝑁_𝑖=1𝐹(𝑇_𝑖)∩𝐸𝑃(𝑓) is nonempty. Let {𝑥_𝑛} be a sequence generated by 𝑥₀𝑤=𝑤∈𝐶,𝑐ℎ𝑜𝑠𝑒𝑛𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑖𝑙𝑦,_𝑛=𝐹_𝑟𝑛𝑥_𝑛,𝑦_𝑛=Π_𝐶𝐽⁻¹𝛼_𝑛𝐽𝑤+1−𝛼_𝑛𝐽𝑤_𝑛,𝑥_𝑛+1=𝐽⁻¹𝛽_𝑛,0𝐽𝑤_𝑛+^𝑁_𝑖=1𝛽_𝑛,𝑖𝐽𝑇_𝑖𝑦_𝑛,𝑛≥0, (3.27) where 𝛼_𝑛∈(0,1) such that lim_𝑛→∞𝛼_𝑛=0, ∑^∞_𝑛=1𝛼_𝑛=∞, {𝛽_𝑛,𝑖}⊂[𝑎,𝑏]⊂(0,1), for 𝑖=1,2,…,𝑁, satisfying 𝛽_𝑛,0+𝛽_𝑛,1+⋯+𝛽_𝑛,𝑁=1, for each 𝑛≥0. Then {𝑥_𝑛} converges strongly to an element of 𝐹.

If 𝐸=𝐻, a real Hilbert space, then 𝐸 is uniformly convex and uniformly smooth real Banach space. In this case, 𝐽=𝐼, identity map on 𝐻 and Π_𝐶=𝑃_𝐶, projection mapping from 𝐻 onto 𝐶. Thus, the following corollary holds.

Corollary 3.6. Let 𝐶 be a nonempty, closed, and convex subset of a Hilbert space 𝐻. Let 𝑓∶𝐶×𝐶→ℝ, be a bifunction which satisfies conditions (A1)–(A4). Let 𝑇_𝑖∶𝐶→𝐶𝐵(𝐶), for 𝑖=1,2,…,𝑁, be a finite family of relatively nonexpansive multi-valued mappings. Assume that 𝐹∶=∩^𝑁_𝑖=1𝐹(𝑇_𝑖)∩𝐸𝑃(𝑓) is nonempty. Let {𝑥_𝑛} be a sequence generated by 𝑥₀𝑤=𝑤∈𝐶,𝑐ℎ𝑜𝑠𝑒𝑛𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑖𝑙𝑦,_𝑛=𝐹_𝑟𝑛𝑥_𝑛,𝑦_𝑛=𝑃_𝐶𝛼_𝑛𝑤+1−𝛼_𝑛𝑤_𝑛,𝑥_𝑛+1=𝛽_𝑛,0𝑤_𝑛+^𝑁_𝑖=1𝛽_𝑛,𝑖𝑢_𝑛,𝑖,𝑢_𝑛,𝑖∈𝑇_𝑖𝑦_𝑛,𝑛≥0, (3.28) where 𝛼_𝑛∈(0,1) such that lim_𝑛→∞𝛼_𝑛=0, ∑^∞_𝑛=1𝛼_𝑛=∞, {𝛽_𝑛,𝑖}⊂[𝑎,𝑏]⊂(0,1), for 𝑖=1,2,…,𝑁, satisfying 𝛽_𝑛,0+𝛽_𝑛,1+⋯+𝛽_𝑛,𝑁=1, for each 𝑛≥0. Then {𝑥_𝑛} converges strongly to an element of 𝐹.

Remark 3.7. (1) Theorem 3.1 improves and extends the corresponding results of Kumanm [22] and Takahashi and Zembayashi [16] in the sense that either our scheme does not require computation of 𝐶_𝑛+1, for each 𝑛≥1, or the space considered is more general.
(2) Theorem 3.1 improves the corresponding results of Homaeipour and Razani [3] in the sense that our convergence is strong and the requirement that the interior of 𝐹 is nonempty is dispensed with.

Acknowledgment

The authors thank the reviewers for their useful comments.