A New Hybrid Projection Algorithm for System of Equilibrium Problems and Variational Inequality Problems and Two Finite Families of Quasi-𝜙-Nonexpansive Mappings

Abstract

We introduce a modified Mann’s iterative procedure by using the hybrid projection method for solving the common solution of the system of equilibrium problems for a finite family of bifunctions satisfying certain condition, the common solution of fixed point problems for two finite families of quasi-𝜙-nonexpansive mappings, and the common solution of variational inequality problems for a finite family of continuous monotone mappings in a uniformly smooth and strictly convex real Banach space. Then, we prove a strong convergence theorem of the iterative procedure generated by some mild conditions. Our result presented in this paper improves and generalizes some well-known results in the literature.

1. Introduction

Throughout this paper, we denote by ℝ and ℕ the set of all real numbers and the set of all positive numbers, respectively. We also assume that 𝐸 is a real Banach space and 𝐸^∗ is the dual space of 𝐸. Let 𝐶 be a nonempty, closed and convex subset of a real Banach space 𝐸 with the dual 𝐸^∗. We recall the following definitions.

A mapping 𝐴∶𝐶→𝐶 is said to be nonexpansive if 𝐴𝑥−𝐴𝑦≤𝑥−𝑦,∀𝑥,𝑦∈𝐶. (1)

A mapping 𝐴∶𝐶→𝐸^∗ is said to be monotone if for each 𝑥,𝑦∈𝐶, such that 𝑥−𝑦,𝐴𝑥−𝐴𝑦≥0. (2)

A mapping 𝐴∶𝐶→𝐸^∗ is said to be 𝛿-strongly monotone if there exists a constant 𝛿>0 such that 𝑥−𝑦,𝐴𝑥−𝐴𝑦≥𝛿𝑥−𝑦²,∀𝑥,𝑦∈𝐶. (3)

A mapping 𝐴∶𝐶→𝐸^∗ is said to be 𝛿-inverse strongly monotone if there exists a constant 𝛿>0 such that 𝑥−𝑦,𝐴𝑥−𝐴𝑦≥𝛿𝐴𝑥−𝐴𝑦²,∀𝑥,𝑦∈𝐶. (4)

If 𝐴 is 𝛿-inverse strongly monotone, then it is Lipschitz continuous with constant 1/𝛿, that is, for all 𝑥,𝑦∈𝐶, ‖𝐴𝑥−𝐴𝑦‖≤(1/𝛿)‖𝑥−𝑦‖.

Clearly, the class of monotone mappings include the class of 𝛿-inverse strongly monotone mappings.

Let 𝐴∶𝐶→𝐸^∗ be a monotone mapping. The variational inequality problem is to find a point 𝑧∈𝐶 such that 𝑦−𝑧,𝐴𝑧≥0,∀𝑦∈𝐶. (5)

The set of the solution of the variational inequality problem is denoted by VI(𝐶,𝐴).

Let 𝐶 be a nonempty, closed and convex subset of a smooth, strictly convex and reflexive Banach space 𝐸, let 𝑇∶𝐶→𝐶 be a mapping, and 𝐹(𝑇) be the set of fixed points of 𝑇.

A point 𝑥∈𝐶 is said to be a fixed point of 𝑇 if 𝑇𝑥=𝑥. The set of the solution of the fixed point of 𝑇 is denoted by 𝐹(𝑇)∶={𝑥∈𝐶∶𝑇𝑥=𝑥}.

A point 𝑝∈𝐶 is said to be an asymptotic fixed point of 𝑇 if there exists a sequence {𝑥_𝑛}⊂𝐶 such that 𝑥_𝑛⇀𝑝 and ‖𝑥_𝑛−𝑇𝑥_𝑛‖→0. We denoted the set of all asymptotic fixed points of 𝑇 by 𝐹(𝑇).

A point 𝑝∈𝐶 is said to be a strong asymptotic fixed point of 𝑇 if there exists a sequence {𝑥_𝑛}⊂𝐶 such that 𝑥_𝑛→𝑝 and ‖𝑥_𝑛−𝑇𝑥_𝑛‖→0. We denoted the set of all strong asymptotic fixed points of 𝑇 by 𝐹(𝑇).

Let 𝐹∶𝐶×𝐶→ℝ be a bifunction. The equilibrium problem is to find a point 𝑧∈𝐶 such that 𝐹𝑧,𝑦≥0,∀𝑦∈𝐶. (6) The set of the solution of equilibrium problem is denoted by EP(𝐹). Numerous problems in sciences, mathematics, optimizations, and economics reduced to find a solution of equilibrium problems. The equilibrium problems include variational inequality problems and fixed point problem, and optimization problems as special cases (see, e.g., [1–3]). Recently, many authors have considered the problem for finding the common solution of fixed point problems, the common solution of equilibrium problems, and the common solution of variational inequality problems.

In 1953, Mann [4] introduced the iterative sequence {𝑥_𝑛}_𝑛∈ℕ which is defined by 𝑥_𝑛+1=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝑇𝑥_𝑛, (7) where the initial element 𝑥₀∈𝐶 is arbitrary, 𝑇 is a nonexpansive mapping, and {𝛼_𝑛} is the sequence in [0,1] such that lim_𝑛→∞𝛼_𝑛=0 and ∑^∞_𝑛=1𝛼_𝑛=∞. The sequence of (7) is generally referred to as the Mann iteration.

In 2009, Takahashi and Zembayashi [5] introduced the following iterative scheme by the shrinking projection method, and they proved that {𝑥_𝑛}_𝑛∈ℕ converges strongly to 𝑞=Π_{EP(𝐹)∩𝐹(𝑆)}, under appropriate conditions.

Theorem TZ.  Let 𝐸 be a uniformly smooth and uniformly convex real Banach space, and let 𝐶 be a nonempty, closed and convex subset of 𝐸. Let 𝐹 be a bifunction from 𝐶×𝐶 to ℝ satisfying (A1)–(A4) and let 𝑇 be a relatively nonexpansive mapping from 𝐶 into itself such that 𝐹(𝑆)∩EP(𝐹)≠∅. Let {𝑥_𝑛} be a sequence generated by𝑥₀∈𝐶₀𝑦=𝐶𝑐ℎ𝑜𝑠𝑒𝑛𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑦,_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, (8)for every 𝑛∈ℕ∪{0}, where 𝐽 is the duality mapping on 𝐸, the sequence 𝛼_𝑛⊂[0,1] satisfies liminf_𝑛→∞𝛼_𝑛(1−𝛼_𝑛)>0, and {𝑟_𝑛}_𝑛∈ℕ⊂(0,∞) satisfies liminf_𝑛→∞𝑟_𝑛>0. Then, the sequence {𝑥_𝑛} converges strongly to Π_{𝐹(𝑆)∩EP(𝑓)}𝑥₀, where Π_{𝐹(𝑆)∩EP(𝐹)}𝑥₀ is the generalized projection of 𝐸 onto 𝐹(𝑆)∩EP(𝐹).

In 2009, Qin et al. [6] extended the iterative process (8) from a single relatively nonexpansive mapping to two relatively quasi-nonexpansive mappings. In 2011, Zegeye and Shahzad [7] introduced an iterative process for finding an element in the common fixed point set of finite family of closed relatively quasi-nonexpansive mappings, common solutions of finite family of equilibrium problems, and common solutions of the finite family of variational inequality problems for monotone mappings in Banach spaces.

Theorem ZS.  Let 𝐶 be a nonempty, closed and convex subset of a uniformly smooth and strictly convex real Banach space 𝐸 with the dual 𝐸^∗. Let 𝑓_𝑘∶𝐶×𝐶→ℝ, 𝑘=1,2,3,…,𝐿, be a finite family of bifunctions. Let 𝑆_𝑗∶𝐶→𝐶, 𝑗=1,2,3,…,𝐷 be a finite family relatively quasi-nonexpansive mappings and 𝐴_𝑖∶𝐶→𝐸^∗, 𝑖=1,2,3,…,𝑁 be a finite family of continuous monotone mappings.

For 𝑥∈𝐸, they define mappings 𝐹_𝑟𝑛,𝑇_𝑟𝑛∶𝐸→𝐶 by𝐹_𝑟𝑛𝑥∶=𝑧∈𝐶∶𝑦−𝑧,𝐴_𝑛𝑧+1𝑟_𝑛,𝑇𝑦−𝑧,𝐽𝑧−𝐽𝑥≥0,∀𝑦∈𝐶_𝑟𝑛𝑥∶=𝑧∈𝐶∶𝑓_𝑛+1𝑧,𝑦𝑟_𝑛𝑦−𝑧,𝐽𝑧−𝐽𝑥≥0,∀𝑦∈𝐶, (9)where 𝑆_𝑛≡𝑆_𝑛(mod D), 𝐴_𝑛≡𝐴_𝑛(mod N), 𝑓_𝑛(⋅,⋅)≡𝑓_𝑛(mod L)(⋅,⋅), and {𝑟_𝑛}_𝑛∈ℕ⊂[𝑐₁,∞) for some 𝑐₁>0. Assume that ℱ∶=(⋂^𝐷_𝑗=1𝐹(𝑆_𝑗))⋂(⋂^𝑁_𝑖=1𝑉𝐼(𝐶,𝐴_𝑖))⋂(⋂^𝐿_𝑘=1EP(𝑓_𝑘))≠∅. Let {𝑥_𝑛} be a sequence generated by𝑥₀∈𝐶₀𝑧=𝐶𝑐ℎ𝑜𝑠𝑒𝑛𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑦,_𝑛=𝐹_𝑟𝑛𝑥_𝑛,𝑢_𝑛=𝑇_𝑟𝑛𝑥_𝑛,𝑦_𝑛=𝐽⁻¹𝛼₀𝐽𝑥_𝑛+𝛼₁𝐽𝑧_𝑛+𝛼₂𝐽𝑆_𝑛𝑢_𝑛,𝐶_𝑛+1=𝑧∈𝐶_𝑛∶𝜙𝑧,𝑦_𝑛≤𝜙𝑧,𝑥_𝑛,𝑥_𝑛+1=Π_𝐶𝑛+1𝑥₀,∀𝑛≥0, (10) where the real numbers 𝛼₀,𝛼₁,𝛼₂∈(0,1) such that 𝛼₀+𝛼₁+𝛼₂=1. Then, {𝑥_𝑛}_𝑛∈ℕ converges strongly to an element of ℱ.

In this paper, motivated and inspired by the previously mentioned above results, we introduce a new iterative procedure for solving the common solution of system of equilibrium problems for a finite family of bifunctions satisfying certain conditions and the common solution of fixed point problems for two countable families of quasi-𝜙-nonexpansive mappings and the common solution of variational inequality problems for a finite family of monotone mappings in a uniformly smooth and strictly convex real Banach space. Then, we prove a strong convergence theorem of the iterative procedure generated by the conditions. The results obtained in this paper extend and improve several recent results in this area.

2. Preliminaries

A Banach space 𝐸 is said to be strictly convex if ‖𝑥+𝑦‖/2<1 for all 𝑥,𝑦∈𝐸 with ‖𝑥‖=‖𝑦‖=1 and 𝑥≠𝑦. It is said to be uniformly convex if lim_𝑛→∞‖𝑥_𝑛−𝑦_𝑛‖=0 for any two sequences {𝑥_𝑛} and {𝑦_𝑛} in 𝐸 such that ‖𝑥_𝑛‖≤1, ‖𝑦_𝑛‖≤1 and lim_𝑛→∞𝑥_𝑛+𝑦_𝑛2=1. (11)

Let 𝑈_𝐸 = {𝑥∈𝐸∶‖𝑥‖=1} be the unit sphere of 𝐸. Then the Banach space 𝐸 is said to be smooth if lim_𝑡→0𝑥+𝑡𝑦−‖𝑥‖𝑡 (12) exists for each 𝑥,𝑦∈𝑈_𝐸. It is said to be uniformly smooth if the limit (12) is attained uniformly for all 𝑥,𝑦∈𝑈_𝐸.

Let 𝐸 be a Banach space. Then a function 𝜌_𝐸∶ℝ⁺→ℝ⁺ is said to be the modulus of smoothness of 𝐸 if 𝜌_𝐸(𝑡)+=sup𝑥+𝑦𝑥−𝑦2−1∶‖𝑥‖=1,𝑦=𝑡. (13)

The space 𝐸 is said to be smooth if 𝜌_𝐸(𝑡)>0, ∀𝑡>0 and is said to be 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‖≤1,𝑦≤1;𝑥−𝑦≥𝜖. (14)

A Banach space 𝐸 is said to be uniformly convex if and only if 𝛿_𝐸(𝜖)>0 for all 𝜖∈(0,2].

We recall the following definitions.

Definition 1. Let 𝐶 be a nonempty set.(1)A mapping 𝑇∶𝐶→𝐶 is said to be closed if for each {𝑥_𝑛}⊂𝐶,𝑥_𝑛→𝑥 and 𝑇𝑥_𝑛→𝑦 imply 𝑇𝑥=𝑦.(2)A mapping 𝑇∶𝐶→𝐶 is said to be quasi-𝜙-nonexpansive (relatively quasi-nonexpansive) if 𝐹(𝑇)≠∅, and 𝜙𝑝,𝑇𝑥≤𝜙𝑝,𝑥,∀𝑥∈𝐶,𝑝∈𝐹(𝑇). (15)(3)A mapping 𝑇∶𝐶→𝐶 is said to be relatively nonexpansive [8, 9] if 𝐹(𝑇)≠∅,  𝐹(𝑇)=𝐹(𝑇) and 𝜙𝑝,𝑇𝑥≤𝜙𝑝,𝑥,∀𝑥∈𝐶,𝑝∈𝐹(𝑇). (16)(4)A mapping 𝑇∶𝐶→𝐶 is said to be weak relatively nonexpansive [10] if 𝐹(𝑇)≠∅,  𝐹(𝑇)=𝐹(𝑇) and 𝜙𝑝,𝑇𝑥≤𝜙𝑝,𝑥,∀𝑥∈𝐶,𝑝∈𝐹(𝑇). (17)

Remark 2. We here the following basic properties.(1)Each relatively nonexpansive mapping is closed.(2)The class of quasi-𝜙-nonexpansive mappings contains properly the class of weak relatively nonexpansive mappings as a subclass, but the converse may be not true.(3)The class of weak relatively nonexpansive mappings contains properly the class of relatively nonexpansive mappings as a subclass, but the converse may be not true.(4)The class of quasi-𝜙-nonexpansive mappings contains properly the class of relatively nonexpansive mappings as a subclass, but the converse may be not true.(5)If 𝐸 is a real uniformly smooth Banach space, then 𝐽 is uniformly continuous on each bounded subset of 𝐸.(6)If 𝐸 is a strictly convex reflexive Banach space, then 𝐽⁻¹ is hemicontinuous, that is, 𝐽⁻¹ is norm-to-weak^∗ continuous.(7)If 𝐸 is a smooth and strictly convex reflexive Banach space, then 𝐽 is single-valued, one-to-one, and onto.(8)A Banach space 𝐸 is uniformly smooth if and only if 𝐸^∗ is uniformly convex.(9)Each uniformly convex Banach space 𝐸 has the Kadec-Klee property, that is, for any sequence {𝑥_𝑛}⊂𝐸, if {𝑥_𝑛}⇀𝑥∈𝐸 and ‖𝑥_𝑛‖→‖𝑥‖, then 𝑥_𝑛→𝑥.(10)A Banach space 𝐸 is strictly convex if and only if 𝐽 is strictly monotone, that is, 𝑥−𝑦,𝑥^∗−𝑦^∗>0,whenever𝑥,𝑦∈𝐸,𝑥≠𝑦,𝑥^∗∈𝐽𝑥,𝑦^∗∈𝐽𝑦. (18)(11)Both uniformly smooth Banach spaces and uniformly convex Banach spaces are reflexive.(12)𝐸^∗ is uniformly convex, then 𝐽 is uniformly norm-to-norm continuous on each bounded subset of 𝐸.
Let 𝐸 be a real Banach space and {𝑥_𝑛} be a sequence in 𝐸. We denote by 𝑥_𝑛→𝑥 and 𝑥_𝑛⇀𝑥 the strong convergence and weak convergence of {𝑥_𝑛}, respectively. The normalized duality mapping 𝐽 from 𝐸 to 2^𝐸∗ is defined by 𝐽𝑥=𝑓^∗∈𝐸^∗∶𝑥,𝑓^∗=‖𝑥‖²=𝑓^∗2,∀𝑥∈𝐸, (19) where ⟨⋅,⋅⟩ denotes the duality pairing. It is well known that if 𝐸 is smooth, then 𝐽 is single-valued and demicontinuous, and if 𝐸 is uniformly smooth, then 𝐽 is uniformly continuous on bounded subset of 𝐸. Moreover, if 𝐸 is reflexive and strictly convex Banach space with a strictly convex dual, then 𝐽⁻¹ is single-valued, one-to-one, surjective, and it is the duality mapping from 𝐸^∗ to 𝐸 and so 𝐽𝐽⁻¹=𝐼_𝐸^∗ and 𝐽⁻¹𝐽=𝐼_𝐸 (see [11, 12]). We note that in a Hilbert space 𝐻, the mapping 𝐽 is the identity operator.
Now, let 𝐸 be a smooth and strictly convex reflexive Banach space. As Alber (see [13]) and Kamimura and Takahashi (see [14]) did, the Lyapunov functional 𝜙∶𝐸×𝐸→ℝ⁺ is defined by 𝜙𝑥,𝑦=‖𝑥‖²−2𝑥,𝐽𝑦+𝑦²,∀𝑥,𝑦∈𝐸. (20)
It follows from Kohsaka and Takahashi (see [15]) that 𝜙(𝑥,𝑦)=0 if and only if 𝑥=𝑦 and that ‖𝑥‖−𝑦²≤𝜙𝑥,𝑦≤‖𝑥‖+𝑦². (21)
Further suppose that 𝐶 is nonempty, closed and convex subset of 𝐸. The generalized, projection (Alber see [13]) Π_𝐶∶𝐸→𝐶 is defined by for each 𝑥∈𝐸, Π_𝐶(𝑥)=argmin_𝑦∈𝐶𝜙𝑥,𝑦. (22)

Remark 3. If 𝐸 is a real Hilbert space 𝐻, then 𝜙(𝑥,𝑦)=‖𝑥−𝑦‖² and Π_𝐶=𝑃_𝐶 (the metric projection of 𝐻 onto 𝐶).

Lemma 4 (Alber [13]). Let 𝐶 be a nonempty, closed and convex subset of a smooth and strictly convex reflexive Banach space 𝐸, and let 𝑥∈𝐸. Then 𝜙𝑥,Π_𝐶𝑦+𝜙Π_𝐶𝑦,𝑦≤𝜙𝑥,𝑦,∀𝑥∈𝐶,∀𝑦∈𝐸. (23)

Lemma 5 (Kamimura and Takahashi [14]). Let 𝐶 be a nonempty, closed and convex subset of a smooth and strictly convex reflexive Banach space 𝐸, and let 𝑥∈𝐸 and 𝑝∈𝐶. Then, 𝑝=Π_𝐶(𝑥)⟺𝑝−𝑦,𝐽𝑥−𝐽𝑝≥0,∀𝑦∈𝐶. (24)

Lemma 6 (Qin et al. [6] and Kohsaka and Takahashi [16]). Let 𝐸 be a smooth, strictly convex and reflexive Banach space, and 𝐴⊂𝐸×𝐸^∗ is a continuous monotone mapping with 𝐴⁻¹(0)≠𝜙. Then, it is proved in [16] that the resolvent 𝐽_𝑟∶=(𝐽+𝑟𝐴)⁻¹𝐽, for 𝑟>0 is quasi-𝜙-nonexpansive. Moreover, if 𝑇∶𝐸→𝐸 is quasi-𝜙-nonexpansive, then using the definition of 𝜙 one can show that 𝐹(𝑇) is closed and convex (see [6]).

Lemma 7 (Kamimura and Takahashi [14]). Let 𝐸 be a uniformly convex and smooth real Banach space and let {𝑥_𝑛} and {𝑦_𝑛} be two sequences of 𝐸. If 𝜙(𝑥_𝑛,𝑦_𝑛)→0 and either {𝑥_𝑛} or {𝑦_𝑛} is bounded, then ‖𝑥_𝑛−𝑦_𝑛‖→0.

Lemma 8. Let 𝐸 be a uniformly smooth and strictly convex Banach space with the Kadec-Klee property, let {𝑥_𝑛} and {𝑦_𝑛} be two sequences of 𝐸, and 𝑝∈𝐸. If 𝑥_𝑛→𝑝 and 𝜙(𝑥_𝑛,𝑦_𝑛)→0, then 𝑦_𝑛→𝑝.

For solving the equilibrium problem, let us assume that the bifunction 𝐹∶𝐶×𝐶→ℝ 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,∀𝑥,𝑦∈𝐶;(A3)lim_𝑡→0𝐹(𝑡𝑧+(1−𝑡)𝑥,𝑦)≤𝐹(𝑥,𝑦),∀𝑥,𝑦,𝑧∈𝐶; (A4) for any 𝑥∈𝐶, 𝑦↦𝐹(𝑥,𝑦) is convex and lower semicontinuous.

Lemma 9 (Blum and Oettli [1]). Let 𝐶 be a nonempty, closed and convex subset of a smooth and strictly convex reflexive Banach space 𝐸 and let 𝐹∶𝐶×𝐶→ℝ be a bifunction satisfying the following conditions (A1)–(A4). Let 𝑟>0 be any given number and 𝑥∈𝐸 be any point. Then, there exists a point 𝑧∈𝐶 such that 𝐹+1𝑧,𝑦𝑟𝑦−𝑧,𝐽𝑧−𝐽𝑥≥0,∀𝑦∈𝐶. (25)

Lemma 10 (Takahashi and Zembayashi [5]). Let 𝐶 be a nonempty, closed and convex subset of a uniformly smooth and strictly convex real Banach space 𝐸, and 𝐹∶𝐶×𝐶→ℝ be a bifunction satisfying the following conditions (A1)–(A4). Let 𝑟>0 be any given number and 𝑥∈𝐸 be any point defined a mapping 𝑇_𝑟∶𝐸→𝐶 as follows. Then, there exists a point 𝑧∈𝐶 such that 𝑇_𝑟(𝑥)=+1𝑧∈𝐶∶𝐹𝑧,𝑦𝑟𝑦−𝑧,𝐽𝑧−𝐽𝑥≥0,∀𝑦∈𝐶. (26) Then, the following conclusions hold:(1)𝑇_𝑟 is single-valued; (2)𝑇_𝑟 is a firmly nonexpansive type mapping, that is, 𝑇_𝑟𝑥−𝑇_𝑟𝑦,𝐽𝑇_𝑟𝑥−𝐽𝑇_𝑟𝑦≤𝑇_𝑟𝑥−𝑇_𝑟𝑦,𝐽𝑥−𝐽𝑦,∀𝑥,𝑦∈𝐸; (27)(3)𝐹(𝑇_𝑟)=EP(𝐹);(4)EP(𝐹) is a closed and convex subset of 𝐶; (5)𝜙(𝑝,𝑇_𝑟𝑥)+𝜙(𝑇_𝑟𝑥,𝑥)≤𝜙(𝑝,𝑥),∀𝑝∈𝐹(𝑇_𝑟).

Lemma 11 (Zegeye and Shahzad [7]). Let 𝐶 be a nonempty, closed and convex subset of a uniformly smooth and strictly convex real Banach space 𝐸, and 𝐴∶𝐶×𝐸^∗→ℝ be a continuous monotone mapping. Let 𝑟>0 be any given number and 𝑥∈𝐸 be any point defined a mapping 𝐾_𝑟∶𝐸→𝐶 as follows: 𝐾_𝑟(𝑥)=+1𝑧∈𝐶∶𝑦−𝑧,𝐴𝑧𝑟𝑦−𝑧,𝐽𝑧−𝐽𝑥≥0,∀𝑦∈𝐶. (28) Then, the following conclusions hold:(1)𝐾_𝑟 is single valued; (2)𝐹(𝐾_𝑟) is a firmly nonexpansive type mapping, that is, 𝐾_𝑟𝑥−𝐾_𝑟𝑦,𝐽𝐾_𝑟𝑥−𝐽𝐾_𝑟𝑦≤𝐾_𝑟𝑥−𝐾_𝑟𝑦,𝐽𝑥−𝐽𝑦,∀𝑥,𝑦∈𝐸; (29)(3)𝐹(𝐾_𝑟)=VI(𝐶,𝐴);(4)VI(𝐶,𝐴) is a closed and convex subset of 𝐶; (5)𝜙(𝑝,𝐾_𝑟𝑥)+𝜙(𝐾_𝑟𝑥,𝑥)≤𝜙(𝑝,𝑥),∀𝑝∈𝐹(𝐾_𝑟).

Lemma 12 (Xu [17]). Let 𝐸 be a uniformly convex Banach space, let 𝐵_𝑟(0) be a closed ball of 𝐸, where 𝑟>0. Then, there exists a continuous, strictly increasing, and convex function 𝑔∶[0,∞)→[0,∞) with 𝑔(0)=0 such that 𝛼𝑥+𝛽𝑦²≤𝛼‖𝑥‖²+𝛽𝑦²−𝛼𝛽𝑔𝑥−𝑦, (30) for all 𝑥,𝑦∈𝐵_𝑟(0) and all 𝛼,𝛽∈[0,1] with 𝛼+𝛽=1.

3. Main Results

In this section, we prove a strong convergence theorem which solves the problem for finding a common solution of the system of equilibrium problems and variational inequality problems and fixed point problems in Banach spaces.

Theorem 13. Let 𝐶 be a nonempty, closed and convex subset of a uniformly smooth and strictly convex real Banach space 𝐸 which has the Kadec-Klee property. Suppose that(1){𝐹_𝑘}^𝑀_𝑘=1∶𝐶×𝐶→ℝ is a finite family of bifunctions satisfying conditions (A1)–(A4), where 𝑘=1,2,3,…,𝑀;(2){𝑇_𝑖}^𝐷_𝑖=1 and {𝑆_𝑖}^𝐷_𝑖=1 are two finite families of quasi-𝜙-nonexpansive mappings from 𝐶 into 𝐸, where 𝑖=1,2,3,…,𝐷;(3){𝐴_𝑛}^𝑁_𝑛=1∶𝐶→𝐸^∗ is a finite family of continuous monotone mappings, where 𝑛=1,2,3,…,𝑁;(4)For 𝑥∈𝐸, one defines the mappings 𝐾_𝑟𝑛,𝑇_𝑟𝑛∶𝐸→𝐶 by 𝐾_𝑟𝑛(𝑥)∶=𝑧∈𝐶∶𝑦−𝑧,𝐴_𝑛𝑧+1𝑟_𝑛,𝑇𝑦−𝑧,𝐽𝑧−𝐽𝑥≥0,∀𝑦∈𝐶_𝑟𝑛(𝑥)∶=𝑧∈𝐶∶𝐹_𝑛+1𝑧,𝑦𝑟_𝑛𝑦−𝑧,𝐽𝑧−𝐽𝑥≥0,∀𝑦∈𝐶. (31)
Assume that Ω∶=(⋂^𝐷_𝑖=1𝐹(𝑇_𝑖))⋂(⋂^𝐷_𝑖=1𝐹(𝑆_𝑖))⋂(⋂^𝑀_𝑘=1SEP(𝐹_𝑘))⋂(⋂^𝑁_𝑛=1VI(𝐶,𝐴_𝑛)) is a nonempty and bounded in 𝐶. Let {𝑥_𝑛} be a sequence generated by 𝑥₀∈𝐶₀𝑧=𝐶,𝑐ℎ𝑜𝑠𝑒𝑛𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑦,_𝑛=𝐾_𝑟𝑛𝑥_𝑛,𝑢_𝑛=𝑇_𝑟𝑛𝑥_𝑛,𝑦_𝑛=𝐽⁻¹𝛼𝐽𝑥_𝑛+𝛽𝐽𝑇_𝑛𝑧_𝑛+𝛾𝐽𝑆_𝑛𝑢_𝑛,𝐶_𝑛+1=𝑧∈𝐶_𝑛∶𝜙𝑧,𝑦_𝑛≤𝜙𝑧,𝑥_𝑛,𝑥_𝑛+1=Π_𝐶𝑛+1𝑥₀,∀𝑛≥0, (32) where 𝐴_𝑛≡𝐴_𝑛(mod𝑁), 𝐹_𝑛≡𝐹_𝑛(mod𝑀), 𝑇_𝑛≡𝑇_𝑛(mod𝐷), 𝑆_𝑛≡𝑆_𝑛(mod𝐷), {𝑟_𝑛}⊂[𝑎,∞) for some 𝑎>0 and 𝑛∈ℕ and 𝛼,𝛽,𝛾 are real numbers in (0,1) such that 𝛼+𝛽+𝛾=1.
Then, the sequence {𝑥_𝑛}^∞_𝑛=0 converges strongly to 𝑝=Π_Ω(𝑥₀).

Proof. We will complete this proof by seven steps below.
Step  1. We will show that 𝐶_𝑛 is closed and convex for each 𝑛≥0.
From the definition of 𝐶_𝑛, it is obvious that 𝐶_𝑛 is closed. Moreover, since 𝜙𝑧,𝑦_𝑛≤𝜙𝑧,𝑥_𝑛isequivalentto2𝑧,𝐽𝑥_𝑛−𝐽𝑦_𝑛−𝑥_𝑛²+𝑦_𝑛²≤0. (33) It follows that 𝐶_𝑛 is convex for each 𝑛≥0. Therefore, 𝐶_𝑛 is closed and convex for each 𝑛≥0.
Step  2. We will show that Ω⊂𝐶_𝑛 for each 𝑛≥0.
From the assumption, we see that 𝐹⊂𝐶₀=𝐶. Suppose that 𝐹⊂𝐶_𝑘 for some 𝑘≥1. Now, for 𝑝∈Ω, Since 𝑆_𝑘 and 𝑇_𝑘 are quasi-𝜙-nonexpansive and by Lemmas 10 and 11, we compute 𝜙𝑝,𝑦_𝑘=𝜙𝑝,𝐽⁻¹𝛼𝐽𝑥_𝑘+𝛽𝐽𝑇_𝑘𝑧_𝑘+𝛾𝐽𝑆_𝑘𝑢_𝑘=𝑝²−2𝑝,𝛼𝐽𝑥_𝑘+𝛽𝐽𝑇_𝑘𝑧_𝑘+𝛾𝐽𝑆_𝑘𝑢_𝑘+𝛼𝐽𝑥_𝑘+𝛽𝐽𝑇_𝑘𝑧_𝑘+𝛾𝐽𝑆_𝑘𝑢_𝑘²≤𝑝²−2𝛼𝑝,𝐽𝑥_𝑘−2𝛽𝑝,𝐽𝑇_𝑘𝑧_𝑘−2𝛾𝑝,𝐽𝑆_𝑘𝑢_𝑘+𝛼𝐽𝑥_𝑘²+𝛽𝐽𝑇_𝑘𝑧_𝑘²+𝛾𝐽𝑆_𝑘𝑢_𝑘²=𝛼𝜙𝑝,𝑥_𝑘+𝛽𝜙𝑝,𝑇_𝑘𝑧_𝑘+𝛾𝜙𝑝,𝑆_𝑘𝑢_𝑘≤𝛼𝜙𝑝,𝑥_𝑘+𝛽𝜙𝑝,𝑧_𝑘+𝛾𝜙𝑝,𝑢_𝑘=𝛼𝜙𝑝,𝑥_𝑘+𝛽𝜙𝑝,𝐾_𝑟𝑘𝑥_𝑘+𝛾𝜙𝑝,𝑇_𝑟𝑘𝑥_𝑘≤𝛼𝜙𝑝,𝑥_𝑘+𝛽𝜙𝑝,𝑥_𝑘+𝛾𝜙𝑝,𝑥_𝑘=𝛼+𝛽+𝛾𝜙𝑝,𝑥_𝑘=𝜙𝑝,𝑥_𝑘. (34) Therefore, 𝑝∈𝐶_𝑘+1. By the induction, this implies that Ω⊂𝐶_𝑛 for each 𝑛≥0.
Hence, the sequence {𝑥_𝑛} is well defined.
Step  3. We will show that the sequence{𝑥_𝑛} is bounded.
Let 𝐺∶=⋂^∞_𝑛=0𝐶_𝑛. From Ω⊂𝐶_𝑛 for each 𝑛≥0 and {𝑥_𝑛} is well defined. From the assumption of 𝐺, we see that 𝐺 is closed and convex subset of 𝐶.
Let 𝑝=Π_𝐺(𝑥₀), where 𝑝 is the unique element that satisfies inf_𝑥∈𝐺𝜙(𝑥,𝑥₀)=𝜙(𝑝,𝑥₀). Now, we will show that ‖𝑥_𝑛−𝑝‖→0 as 𝑛→∞.
From the assumption of 𝐶_𝑛, we know that 𝐶⊃𝐶₁⊃𝐶₂⊃𝐶₃⊃⋯ and since 𝑥_𝑛=Π_𝐶𝑛(𝑥₀) and 𝑥_𝑛+1=Π_𝐶𝑛+1(𝑥₀)∈𝐶_𝑛+1⊂𝐶_𝑛. Using Lemma 4, we get 𝜙𝑥_𝑛,𝑥₀≤𝜙𝑥_𝑛+1,𝑥₀≤⋯≤𝜙𝑝,𝑥₀, (35) for all 𝑝∈Ω⊂𝐶_𝑛, where 𝑛≥1. Then, the sequence {𝜙(𝑥_𝑛,𝑥₀)} is bounded. Hence, the sequence {𝑥_𝑛} is also bounded.
Step  4. We will show that there exists 𝑝∈𝐶 such that 𝑥_𝑛→𝑝, as 𝑛→∞.
Since 𝑥_𝑛=Π_𝐶𝑛(𝑥₀) and 𝑥_𝑛+1=Π_𝐶𝑛+1𝑥₀∈𝐶_𝑛+1⊂𝐶_𝑛, we have 𝜙𝑥_𝑛,𝑥₀≤𝜙𝑥_𝑛+1,𝑥₀,∀𝑛≥0. (36) Therefore, the sequence {𝜙(𝑥_𝑛,𝑥₀)} is nondecreasing. Hence lim_𝑛→∞𝜙(𝑥_𝑛,𝑥₀) exists. By the definition of 𝐶_𝑛, one has that 𝐶_𝑚⊂𝐶_𝑛 and 𝑥_𝑚=Π_𝐶𝑚𝑥₀∈𝐶_𝑛 for any positive integer 𝑚≥𝑛.
It follows that 𝜙𝑥_𝑚,𝑥_𝑛=𝜙𝑥_𝑚,Π_𝐶𝑛𝑥₀≤𝜙𝑥_𝑚,𝑥₀−𝜙Π_𝐶𝑛𝑥₀,𝑥₀=𝜙𝑥_𝑚,𝑥₀−𝜙𝑥_𝑛,𝑥₀. (37) Since lim_𝑛→∞𝜙(𝑥_𝑛,𝑥₀) exists, by taking 𝑚,𝑛→∞ in (37), we have 𝜙(𝑥_𝑚,𝑥_𝑛)→0.
From Lemma 7, it follows that ‖𝑥_𝑚−𝑥_𝑛‖→0 as 𝑚,𝑛→∞. Thus {𝑥_𝑛} is a Cauchy sequence.
Without a loss of generalization, we can assume that 𝑥_𝑛⇀𝑝₀∈𝐸. Since {𝑥_𝑛} is bounded and 𝐸 is reflexive. Since 𝐺∶=⋂^∞_𝑛=0𝐶_𝑛 is closed and convex, it follows that 𝑝₀∈𝐺, ∀𝑛≥0. Moreover, by using the weak lower semicontinuous of the norm on 𝐸 and (35), we obtain 𝜙𝑝,𝑥₀≤𝜙𝑝₀,𝑥₀≤liminf_𝑛→∞𝜙𝑥_𝑛,𝑥₀≤limsup_𝑛→∞𝜙𝑥_𝑛,𝑥₀≤inf_𝑥∈𝐺𝜙𝑥,𝑥₀=𝜙𝑝,𝑥₀. (38) This implies that lim_𝑛→∞𝜙𝑥_𝑛,𝑥₀=𝜙𝑝,𝑥₀=𝜙𝑝₀,𝑥₀=inf_𝑥∈𝐺𝜙𝑥,𝑥₀. (39) By using Lemma 5, we have ⟨𝑝−𝑝₀,𝐽𝑝−𝐽𝑝₀⟩=0 and hence, 𝑝=𝑝₀. By the definition of 𝜙, we get liminf_𝑛→∞𝜙𝑥_𝑛,𝑥₀=liminf_𝑛→∞𝑥_𝑛²−2𝑥_𝑛,𝐽𝑥₀+𝑥₀²≥𝑝²−2𝑝,𝐽𝑥₀+𝑥₀²=𝜙𝑝,𝑥₀. (40) Therefore, ‖𝑥_𝑛‖→‖𝑝‖. Since 𝑥_𝑛⇀𝑝, by the Kadec-Klee property of 𝐸, we obtain lim_𝑛→∞𝑥_𝑛=𝑝. (41) From 𝐽 is uniformly continuous, we also have lim_𝑛→∞𝐽𝑥_𝑛=𝐽𝑝. (42)Step  5. We will show that 𝑦_𝑛→𝑝, 𝑧_𝑛→𝑝 and 𝑢_𝑛→𝑝 as 𝑛→∞.
Since 𝑥_𝑛+1∈𝐶_𝑛+1, we have 𝜙(𝑥_𝑛+1,𝑦_𝑛)≤𝜙(𝑥_𝑛+1,𝑥_𝑛)→0, as 𝑛→∞. Thus, from (21), we obtain 𝑦_𝑛→𝑝,as𝑛→∞, (43) and so 𝐽𝑦_𝑛→𝐽𝑝,as𝑛→∞. (44) This implies that {𝐽𝑦_𝑛} is bounded. Note that reflexivity of 𝐸 implies reflexivity of 𝐸^∗. Thus, we assume that 𝐽𝑦_𝑛⇀𝑦∈𝐸^∗. Furthermore, the reflexivity of 𝐸 implies there exists 𝑥∈𝐸 such that 𝑦=𝐽𝑥. Then, it follows that 𝜙𝑥_𝑛+1,𝑦_𝑛=𝑥_𝑛+1²−2𝑥_𝑛+1,𝐽𝑦_𝑛+𝑦_𝑛²=𝑥_𝑛+1²−2𝑥_𝑛+1,𝐽𝑦_𝑛+𝐽𝑦_𝑛². (45) Taking the liminf_𝑛→∞ on both sides of (45) and using weak lower semicontinuous of norm to get that 0≥𝑝²−2𝑝,𝑦+𝑦²=𝑝²−2𝑝,𝐽𝑥+‖𝐽𝑥‖²=𝑝²−2𝑝,𝐽𝑥+‖𝑥‖²=𝜙𝑝,𝑥. (46) Thus 𝑝=𝑥, and so 𝑦=𝐽𝑥=𝐽𝑝. It follows that 𝐽𝑦_𝑛⇀𝐽𝑝. Now, from (44) and the Kadec-Klee property of 𝐸^∗, we obtain 𝐽𝑦_𝑛→𝐽𝑝,as𝑛→∞. (47) Thus the demicontinuity of 𝐽⁻¹ implies that 𝑦_𝑛⇀𝑝. Now, from (43) and the fact that 𝐸 has the Kadec-Klee property, we obtain lim_𝑛→∞𝑦_𝑛=𝑝. (48) In the fact that 𝑥_𝑛→𝑝 and 𝑦_𝑛→𝑝 as 𝑛→∞, we get lim_𝑛→∞𝜙𝑝,𝑧_𝑛=𝜙𝑝,𝑝. (49) Since 𝑧_𝑛=𝐾_𝑟𝑛𝑥_𝑛, it follows from Lemma 11 that 𝜙𝑧_𝑛,𝑥_𝑛=𝜙𝐾_𝑟𝑛𝑥_𝑛,𝑥_𝑛≤𝜙𝑝,𝑥_𝑛−𝜙𝑝,𝐾_𝑟𝑛𝑥_𝑛≤𝜙𝑝,𝑥_𝑛−𝜙𝑝,𝑧_𝑛→0,as𝑛→∞. (50) From (21), we obtain 𝑧_𝑛→𝑝, (51) and so {𝑧_𝑛} is bounded. Since 𝐸 is reflexive, we assume that 𝑧_𝑛⇀𝑧∈𝐸. It follows that 𝜙𝑧_𝑛,𝑥_𝑛=𝑧_𝑛²−2𝑧_𝑛,𝐽𝑥_𝑛+𝐽𝑥_𝑛². (52) Taking the liminf_𝑛→∞ on both sides of (52) and using the continuity of 𝐽, we get that 0≥‖𝑧‖²−2𝑧,𝐽𝑝+𝐽𝑝²=𝜙𝑧,𝑝. (53) This implies that 𝑝=𝑧 and hence 𝑧_𝑛⇀𝑝.
Now, from (51) and the Kadec-Klee property of 𝐸, we obtain lim_𝑛→∞𝑧_𝑛=𝑝. (54) In the fact that 𝑥_𝑛→𝑝, 𝑦_𝑛→𝑝 and 𝑧_𝑛→𝑝 as 𝑛→∞, we get lim_𝑛→∞𝜙𝑝,𝑢_𝑛=𝜙𝑝,𝑝. (55) Since 𝑢_𝑛=𝑇_𝑟𝑛𝑥_𝑛, it follows from Lemma 10 that 𝜙𝑢_𝑛,𝑥_𝑛=𝜙𝑇_𝑟𝑛𝑥_𝑛,𝑥_𝑛≤𝜙𝑝,𝑥_𝑛−𝜙𝑝,𝑇_𝑟𝑛𝑥_𝑛≤𝜙𝑝,𝑥_𝑛−𝜙𝑝,𝑢_𝑛→0,as𝑛→∞. (56) From (21), we obtain 𝑢_𝑛→𝑝, (57) and so {𝑢_𝑛} is bounded. Since 𝐸 is reflexive, we assume that 𝑢_𝑛⇀𝑢∈𝐸. It follows that 𝜙𝑢_𝑛,𝑥_𝑛=𝑢_𝑛²−2𝑢_𝑛,𝐽𝑥_𝑛+𝐽𝑥_𝑛². (58) Taking the liminf_𝑛→∞ on both sides of (58) and using the continuity of 𝐽, we get that 0≥‖𝑢‖²−2𝑢,𝐽𝑝+𝐽𝑝²=𝜙𝑢,𝑝. (59) This implies that 𝑝=𝑢 and hence 𝑢_𝑛⇀𝑝.
Now, from (57) and the Kadec-Klee property of 𝐸, we obtain lim_𝑛→∞𝑢_𝑛=𝑝. (60)Step  6. We will show that 𝑝∈Ω.
Substep 1. We will show that 𝑝∈⋂^𝑁_𝑛=1VI(𝐶,𝐴_𝑛).
From the definition of 𝐾_𝑟𝑛 of algorithm (32), we have 𝑦−𝑧_𝑛,𝐴_𝑛𝑧_𝑛+1𝑟_𝑛𝑦−𝑧_𝑛,𝐽𝑧_𝑛−𝐽𝑥_𝑛≥0,∀𝑦∈𝐶. (61) Let {𝑛_𝑘}_𝑘∈ℕ⊂ℕ be such that 𝐴_𝑛𝑘=𝐴₁, for all 𝑘∈ℕ. Then from (61), we obtain 𝑦−𝑧_𝑛𝑘,𝐴₁𝑧_𝑛𝑘+1𝑟_𝑛𝑘𝑦−𝑧_𝑛𝑘,𝐽𝑧_𝑛𝑘−𝐽𝑥_𝑛𝑘≥0,∀𝑦∈𝐶, (62) and that is 𝑦−𝑧_𝑛𝑘,𝐴₁𝑧_𝑛𝑘+𝑦−𝑧_𝑛𝑘,𝐽𝑧_𝑛𝑘−𝐽𝑥_𝑛𝑘𝑟_𝑛𝑘≥0,∀𝑦∈𝐶. (63) Now, we set 𝑣_𝑡=𝑡𝑣+(1−𝑡)𝑝, for all 𝑡∈(0,1] and 𝑣∈𝐶.
Therefore, we get 𝑣_𝑡∈𝐶. From (63), it follows that 𝑣_𝑡−𝑧_𝑛𝑘,𝐴₁𝑣_𝑡≥𝑣_𝑡−𝑧_𝑛𝑘,𝐴₁𝑣_𝑡−𝑣_𝑡−𝑧_𝑛𝑘,𝐴₁𝑧_𝑛𝑘−𝑣_𝑡−𝑧_𝑛𝑘,𝐽𝑧_𝑛𝑘−𝐽𝑥_𝑛𝑘𝑟_𝑛𝑘=𝑣_𝑡−𝑧_𝑛𝑘,𝐴₁𝑣_𝑡−𝐴₁𝑧_𝑛𝑘−𝑣_𝑡−𝑧_𝑛𝑘,𝐽𝑧_𝑛𝑘−𝐽𝑥_𝑛𝑘𝑟_𝑛𝑘. (64) From the continuity of 𝐽 and (41) and (54), we have 𝑥_𝑛𝑘→𝑝, 𝑧_𝑛𝑘→𝑝, as 𝑘→∞, we obtain 𝐽𝑧_𝑛𝑘−𝐽𝑥_𝑛𝑘𝑟_𝑛𝑘→0,as𝑘→∞. (65) Since 𝐴₁ is a monotone mapping, we also have ⟨𝑣_𝑡−𝑧_𝑛𝑘,𝐴₁𝑣_𝑡−𝐴₁𝑧_𝑛𝑘⟩≥0.
Thus, it follows that 0≤lim_𝑛→∞𝑣_𝑡−𝑧_𝑛𝑘,𝐴₁𝑣_𝑡=𝑣_𝑡−𝑝,𝐴₁𝑣_𝑡, (66) and hence 𝑦−𝑝,𝐴₁𝑣_𝑡≥0,∀𝑦∈𝐶. (67) If 𝑡→0, we obtain 𝑦−𝑝,𝐴₁𝑝≥0,∀𝑦∈𝐶. (68) This implies that 𝑝∈VI(𝐶,𝐴₁).
Similarly, let {𝑛_𝑘}_𝑘∈ℕ⊂ℕ be such that 𝐴_𝑛𝑘=𝐴₂, for all 𝑘∈ℕ.
Then, we have again that 𝑝∈VI(𝐶,𝐴₂).
Continuing in the same way, we obtain that 𝑝∈VI(𝐶,𝐴_𝑛), where 𝑛=3,4,5,…,𝑁.
Hence, 𝑝∈⋂^𝑁_𝑛=1VI(𝐶,𝐴_𝑛).
Substep 2. We will show that 𝑝∈⋂^𝑀_𝑘=1SEP(𝐹_𝑘).
From the definition of 𝑇_𝑟𝑛 of algorithm (32) and (A2), we have 1𝑟_𝑛𝑦−𝑢_𝑛,𝐽𝑢_𝑛−𝐽𝑥_𝑛≥−𝐹_𝑛𝑢_𝑛,𝑦≥𝐹𝑦,𝑢_𝑛,∀𝑦∈𝐶. (69) Let {𝑛_𝑘}_𝑘∈ℕ⊂ℕ be such that 𝐹_𝑛𝑘=𝐹₁, for all 𝑘∈ℕ. Then from (69), we obtain 1𝑟_𝑛𝑘𝑦−𝑢_𝑛𝑘,𝐽𝑢_𝑛𝑘−𝐽𝑥_𝑛𝑘=𝑦−𝑢_𝑛𝑘,𝐽𝑢_𝑛𝑘−𝐽𝑥_𝑛𝑘𝑟_𝑛𝑘≥𝐹₁𝑦,𝑢_𝑛𝑘,∀𝑦∈𝐶. (70) From the continuity of 𝐽 and (41) and (60), we have 𝑥_𝑛𝑘→𝑝, 𝑢_𝑛𝑘→𝑝, as 𝑘→∞ and we obtain 𝐽𝑢_𝑛𝑘−𝐽𝑥_𝑛𝑘𝑟_𝑛𝑘→0,as𝑘→∞. (71) Therefore, 𝐹₁(𝑦,𝑢_𝑛𝑘)≤0,∀𝑦∈𝐶.
Now, we set 𝑤_𝑡=𝑡𝑤+(1−𝑡)𝑝, for all 𝑡∈(0,1] and 𝑤∈𝐶.
Consequently, we get 𝑤_𝑡∈𝐶. And so 𝐹₁(𝑤_𝑡,𝑝)≤0.
Therefore, from (A1), we obtain 0=𝐹₁𝑤_𝑡,𝑤_𝑡≤𝑡𝐹₁𝑤_𝑡,𝑦+(1−𝑡)𝐹₁𝑤_𝑡,𝑝≤𝑡𝐹₁𝑤_𝑡,𝑦. (72) Thus, 𝐹₁(𝑤_𝑡,𝑦)≥0,∀𝑦∈𝐶. From (A3), if 𝑡→0, then we get 𝐹₁(𝑝,𝑦)≥0,∀𝑦∈𝐶. This implies that 𝑝∈EP(𝐹₁).
Similarly, let {𝑛_𝑘}_𝑘∈ℕ⊂ℕ be such that 𝐹_𝑛𝑘=𝐹₂, for all 𝑘∈ℕ. Then, we have again that 𝑝∈EP(𝐹₂).
Continuing in the same way, we obtain that 𝑝∈EP(𝐹_𝑘), where 𝑘=3,4,5,…,𝑀. Hence, 𝑝∈⋂^𝑀_𝑘=1SEP(𝐹_𝑘).
Substep 3. We will show that 𝑝∈⋂^𝐷_𝑖=1𝐹(𝑆_𝑖).
From algorithm (32) and Lemma 12, we compute 𝜙𝑝,𝑦_𝑛=𝜙𝑝,𝐽⁻¹𝛼𝐽𝑥_𝑛+𝛽𝐽𝑇_𝑛𝑧_𝑛+𝛾𝐽𝑆_𝑛𝑢_𝑛=𝑝²−2𝑝,𝛼𝐽𝑥_𝑛+𝛽𝐽𝑇_𝑛𝑧_𝑛+𝛾𝐽𝑆_𝑛𝑢_𝑛+𝛼𝐽𝑥_𝑛+𝛽𝐽𝑇_𝑛𝑧_𝑛+𝛾𝐽𝑆_𝑛𝑢_𝑛²≤𝑝²−2𝛼𝑝,𝐽𝑥_𝑛−2𝛽𝑝,𝐽𝑇_𝑛𝑧_𝑛−2𝛾𝑝,𝐽𝑆_𝑛𝑢_𝑛+𝛼𝑥_𝑛²+𝛽𝑇_𝑛𝑧_𝑛²+𝛾𝑆_𝑛𝑢_𝑛²−𝛼𝛽𝑔𝐽𝑥_𝑛−𝐽𝑇_𝑛𝑧_𝑛−𝛼𝛾𝑔𝐽𝑥_𝑛−𝐽𝑆_𝑛𝑢_𝑛=𝛼𝜙𝑝,𝑥_𝑛+𝛽𝜙𝑝,𝑇_𝑛𝑧_𝑛+𝛾𝜙𝑝,𝑆_𝑛𝑢_𝑛−𝛼𝛽𝑔𝐽𝑥_𝑛−𝐽𝑇_𝑛𝑧_𝑛−𝛼𝛾𝑔𝐽𝑥_𝑛−𝐽𝑆_𝑛𝑢_𝑛≤𝛼𝜙𝑝,𝑥_𝑛+𝛽𝜙𝑝,𝑧_𝑛+𝛾𝜙𝑝,𝑢_𝑛−𝛼𝛽𝑔𝐽𝑥_𝑛−𝐽𝑇_𝑛𝑧_𝑛−𝛼𝛾𝑔𝐽𝑥_𝑛−𝐽𝑆_𝑛𝑢_𝑛=𝛼𝜙𝑝,𝑥_𝑛+𝛽𝜙𝑝,𝐾_𝑟𝑛𝑥_𝑛+𝛾𝜙𝑝,𝑇_𝑟𝑛𝑥_𝑛−𝛼𝛽𝑔𝐽𝑥_𝑛−𝐽𝑇_𝑛𝑧_𝑛−𝛼𝛾𝑔𝐽𝑥_𝑛−𝐽𝑆_𝑛𝑢_𝑛≤𝛼𝜙𝑝,𝑥_𝑛+𝛽𝜙𝑝,𝑥_𝑛+𝛾𝜙𝑝,𝑥_𝑛−𝛼𝛽𝑔𝐽𝑥_𝑛−𝐽𝑇_𝑛𝑧_𝑛−𝛼𝛾𝑔𝐽𝑥_𝑛−𝐽𝑆_𝑛𝑢_𝑛=𝛼+𝛽+𝛾𝜙𝑝,𝑥_𝑛−𝛼𝛽𝑔𝐽𝑥_𝑛−𝐽𝑇_𝑛𝑧_𝑛−𝛼𝛾𝑔𝐽𝑥_𝑛−𝐽𝑆_𝑛𝑢_𝑛=𝜙𝑝,𝑥_𝑛−𝛼𝛽𝑔𝐽𝑥_𝑛−𝐽𝑇_𝑛𝑧_𝑛−𝛼𝛾𝑔𝐽𝑥_𝑛−𝐽𝑆_𝑛𝑢_𝑛≤𝜙𝑝,𝑥_𝑛. (73) From (73), we have 𝜙𝑝,𝑦_𝑛≤𝜙𝑝,𝑥_𝑛−𝛼𝛽𝑔𝐽𝑥_𝑛−𝐽𝑇_𝑛𝑧_𝑛−𝛼𝛾𝑔𝐽𝑥_𝑛−𝐽𝑆_𝑛𝑢_𝑛≤𝜙𝑝,𝑥_𝑛−𝛼𝛾𝑔𝐽𝑥_𝑛−𝐽𝑆_𝑛𝑢_𝑛. (74) From 𝑥_𝑛→𝑝, 𝑦_𝑛→𝑝, and 𝛼,𝛾>0, we obtain 𝛼𝛾𝑔𝐽𝑥_𝑛−𝐽𝑆_𝑛𝑢_𝑛≤𝜙𝑝,𝑥_𝑛−𝜙𝑝,𝑦_𝑛→0,as𝑛→∞. (75) Therefore, 𝑔𝐽𝑥_𝑛−𝐽𝑆_𝑛𝑢_𝑛→0,as𝑛→∞. (76) It follows from the property of 𝑔 that 𝐽𝑥_𝑛−𝐽𝑆_𝑛𝑢_𝑛→0,as𝑛→∞. (77) From (42), we have 𝐽𝑥_𝑛→𝐽𝑝, as 𝑛→∞. Then, 𝐽𝑆_𝑛𝑢_𝑛→𝐽𝑝,as𝑛→∞, (78) and so 𝑆_𝑛𝑢_𝑛→𝑝,as𝑛→∞. (79) Moreover, the demicontinuity of 𝐽⁻¹ implies that 𝑆_𝑛𝑢_𝑛⇀𝑝 as 𝑛→∞. Thus, the Kadec-Klee property of 𝐸, we obtain 𝑆_𝑛𝑢_𝑛→𝑝,as𝑛→∞. (80) Let {𝑛_𝑘}_𝑘∈ℕ⊂ℕ be such that 𝑆_𝑛𝑘=𝑆₁, for all 𝑘∈ℕ.
Then, from (60), we have 𝑢_𝑛𝑘→𝑝, as 𝑛→∞. It follows from (80) and the closedness of 𝑆₁ that 𝑝=lim_𝑘→∞𝑆_𝑛𝑘𝑢_𝑛𝑘=lim_𝑘→∞𝑆₁𝑢_𝑛𝑘=𝑆₁𝑝. (81) This implies that 𝑝∈𝐹(𝑆₁).
Similarly, let {𝑛_𝑘}_𝑘∈ℕ⊂ℕ be such that 𝑆_𝑛𝑘=𝑆₂, for all 𝑘∈ℕ. Then, we have again that 𝑝∈𝐹(𝑆₂).
Continuing in the same way, we obtain that 𝑝∈𝐹(𝑆_𝑖), where 𝑖=3,4,5,…,𝐷.
Hence, 𝑝∈⋂^𝐷_𝑖=1𝐹(𝑆_𝑖).
Substep 4. We will show that 𝑝∈⋂^𝐷_𝑖=1𝐹(𝑇_𝑖).
From (73), we have 𝜙𝑝,𝑦_𝑛≤𝜙𝑝,𝑥_𝑛−𝛼𝛽𝑔𝐽𝑥_𝑛−𝐽𝑇_𝑛𝑧_𝑛−𝛼𝛾𝑔𝐽𝑥_𝑛−𝐽𝑆_𝑛𝑢_𝑛≤𝜙𝑝,𝑥_𝑛−𝛼𝛽𝑔𝐽𝑥_𝑛−𝐽𝑇_𝑛𝑧_𝑛. (82) From 𝑥_𝑛→𝑝, 𝑦_𝑛→𝑝, and 𝛼,𝛽>0, we obtain 𝛼𝛽𝑔𝐽𝑥_𝑛−𝐽𝑇_𝑛𝑧_𝑛≤𝜙𝑝,𝑥_𝑛−𝜙𝑝,𝑦_𝑛→0,as𝑛→∞. (83) Therefore, 𝑔𝐽𝑥_𝑛−𝐽𝑇_𝑛𝑧_𝑛→0,as𝑛→∞. (84) It follows from the property of 𝑔 that 𝐽𝑥_𝑛−𝐽𝑇_𝑛𝑧_𝑛→0,as𝑛→∞. (85) From (42), we have 𝐽𝑥_𝑛→𝐽𝑝, as 𝑛→∞. Then, 𝐽𝑇_𝑛𝑧_𝑛→𝐽𝑝,as𝑛→∞, (86) and so 𝑇_𝑛𝑧_𝑛→𝑝,as𝑛→∞. (87) Moreover, the demicontinuity of 𝐽⁻¹ implies that 𝑇_𝑛𝑧_𝑛⇀𝑝 as 𝑛→∞. Thus, the Kadec-Klee property of 𝐸, we obtain 𝑇_𝑛𝑧_𝑛→𝑝,as𝑛→∞. (88) Let {𝑛_𝑘}_𝑘∈ℕ⊂ℕ be such that 𝑇_𝑛𝑘=𝑇₁, for all 𝑘∈ℕ.
Then, from (54), we have 𝑧_𝑛𝑘→𝑝, as 𝑛→∞. It follows from (88) and the closedness of 𝑇₁ that 𝑝=lim_𝑘→∞𝑇_𝑛𝑘𝑧_𝑛𝑘=lim_𝑘→∞𝑇₁𝑧_𝑛𝑘=𝑇₁𝑝. (89) This implies that 𝑝∈𝐹(𝑇₁).
Similarly, let {𝑛_𝑘}_𝑘∈ℕ⊂ℕ be such that 𝑇_𝑛𝑘=𝑇₂, for all 𝑘∈ℕ. Then, we have again that 𝑝∈𝐹(𝑇₂).
Continuing in the same way, we obtain that 𝑝∈𝐹(𝑇_𝑖), where 𝑖=3,4,5,…,𝐷.
Hence, 𝑝∈⋂^𝐷_𝑖=1𝐹(𝑇_𝑖).
From Substeps (6.1)–(6.4), we can conclude that 𝑝∈Ω∶=^𝐷_𝑖=1𝐹𝑇_𝑖^𝐷_𝑖=1𝐹𝑆_𝑖^𝑀_𝑘=1SEP𝐹_{𝑘𝑁𝑛=1}VI𝐶,𝐴_𝑛. (90)
Step  7. Finally, we will show that 𝑝=Π_Ω(𝑥₀).
From 𝑥_𝑛=Π_𝐶𝑛(𝑥₀), we have 𝐽𝑥₀−𝐽𝑥_𝑛,𝑥_𝑛−𝑝≥0,∀𝑝∈Ω. (91) Taking 𝑛→∞ in (91), one has 𝐽𝑥₀−𝐽𝑝,𝑝−𝑝≥0,∀𝑝∈Ω. (92) Now, we have 𝑝∈Ω and by Lemma 5, we get 𝑝=Π_Ω𝑥₀. (93) This completes the proof of Theorem 13.

If we set 𝑀=𝑁=𝐷=1 in Theorem 13, then we obtain the following result.

Corollary 14. Let 𝐶 be a nonempty, closed and convex subset of a uniformly smooth and strictly convex real Banach space 𝐸 which has the Kadec-Klee property. Suppose that(1)𝐹∶𝐶×𝐶→ℝ is a bifunction satisfying conditions (A1)–(A4);(2)𝑇 and 𝑆 are two quasi-𝜙-nonexpansive mappings from 𝐶 into 𝐸;(3)𝐴∶𝐶→𝐸^∗ is a continuous monotone mapping.(4)For 𝑥∈𝐸, one defines the mappings 𝐾_𝑟𝑛,𝑇_𝑟𝑛∶𝐸→𝐶 as follows: 𝐾_𝑟𝑛(𝑥)+1∶=𝑧∈𝐶∶𝑦−𝑧,𝐴𝑧𝑟_𝑛;𝑇𝑦−𝑧,𝐽𝑧−𝐽𝑥≥0,∀𝑦∈𝐶_𝑟𝑛(𝑥)+1∶=𝑧∈𝐶∶𝐹𝑧,𝑦𝑟_𝑛⟨𝑦−𝑧,𝐽𝑧−𝐽𝑥⟩≥0,∀𝑦∈𝐶. (94)
Assume that Ω∶=𝐹(𝑇)∩𝐹(𝑆)∩EP(𝐹)∩VI(𝐶,𝐴) is a nonempty and bounded in 𝐶 and let {𝑥_𝑛} be a sequence generated by 𝑥₀∈𝐶₀𝑧=𝐶,𝑐ℎ𝑜𝑠𝑒𝑛𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑦,_𝑛=𝐾_𝑟𝑛𝑥_𝑛,𝑢_𝑛=𝑇_𝑟𝑛𝑥_𝑛,𝑦_𝑛=𝐽⁻¹𝛼𝐽𝑥_𝑛+𝛽𝐽𝑇𝑧_𝑛+𝛾𝐽𝑆𝑢_𝑛,𝐶_𝑛+1=𝑧∈𝐶_𝑛∶𝜙𝑧,𝑦_𝑛≤𝜙𝑧,𝑥_𝑛,𝑥_𝑛+1=Π_𝐶𝑛+1𝑥₀,∀𝑛≥0, (95) where {𝑟_𝑛}⊂[𝑎,∞) for some 𝑎>0 and 𝑛∈ℕ, and 𝛼, 𝛽, 𝛾 are real numbers in (0,1) such that 𝛼+𝛽+𝛾=1.
Then, the sequence {𝑥_𝑛}^∞_𝑛=0 converges strongly to 𝑝=Π_Ω(𝑥₀).

If we set 𝑇_𝑖=𝐼, for any 𝑖=1,2,3,…,𝐷 in Theorem 13, then we obtain the following result which extends and improves the result’s Zegeye and Shahzad [7].

Corollary 15 (Zegeye and Shahzad [7]). Let 𝐶 be a nonempty, closed and convex subset of a uniformly smooth and strictly convex real Banach space 𝐸 which has Kadec-Klee property. Suppose that(1){𝐹_𝑘}^𝑀_𝑘=1∶𝐶×𝐶→ℝ is a finite family of bifunctions satisfying conditions (A1)–(A4), where 𝑘=1,2,3,…,𝑀;(2){𝑆_𝑖}^𝐷_𝑖=1∶𝐶→𝐸 is a finite family of quasi-𝜙-nonexpansive mappings, where 𝑖=1,2,3,…,𝐷;(3){𝐴_𝑛}^𝑁_𝑛=1∶𝐶→𝐸^∗ is a finite family of continuous monotone mappings, where 𝑛=1,2,3,…,𝑁;(4)For 𝑥∈𝐸, one defines the mappings 𝐾_𝑟𝑛,𝑇_𝑟𝑛∶𝐸→𝐶 by 𝐾_𝑟𝑛(𝑥)∶=𝑧∈𝐶∶𝑦−𝑧,𝐴_𝑛𝑧+1𝑟_𝑛;𝑇𝑦−𝑧,𝐽𝑧−𝐽𝑥≥0,∀𝑦∈𝐶_𝑟𝑛(𝑥)∶=𝑧∈𝐶∶𝐹_𝑛+1𝑧,𝑦𝑟_𝑛𝑦−.