Existence and Modification of Halpern-Mann Iterations for Fixed Point and Generalized Mixed Equilibrium Problems with a Bifunction Defined on the Dual Space

Abstract

We study and establish the existence of a solution for a generalized mixed equilibrium problem with a bifunction defined on the dual space of a Banach space. Furthermore, we also modify Halpern-Mann iterations for finding a common solution of a generalized mixed equilibrium problem and a fixed point problem. Under suitable conditions of the purposed iterative sequences, the strong convergence theorems are established by using sunny generalized nonexpansive retraction in Banach spaces. Our results extend and improve various results existing in the current literature.

1. Introduction

In the past years, variational inequalities is among the most important and interesting mathematical problems, since they have wide applications in the optimization and control, economics and equilibrium, engineering science and physical sciences.

Equilibrium problem represents an important area of mathematical sciences such as optimization, operations research, game theory, financial mathematics, and mechanics. Equilibrium problems include variational inequalities, optimization problems, Nash equilibria problems, saddle point problems, and fixed point problems as special cases.

Throughout this paper, we denote the strong convergence and weak convergence {𝑥_𝑛} by 𝑥_𝑛→𝑥, 𝑥_𝑛⇀𝑥, respectively.

Let 𝐶 be a closed and convex subset of a real Banach space𝐸with the dual space 𝐸^∗. Let 𝐶^∗ be a closed and convex subset of 𝐸^∗. We recall the following definitions.(1)A mapping 𝐴∶𝐶→𝐸^∗ is said to be monotone if for each 𝑥,𝑦∈𝐶 such that 𝑥−𝑦,𝐴𝑥−𝐴𝑦≥0. (1)(2)A mapping 𝐴∶𝐶→𝐸^∗ is said to be 𝛿-𝑠𝑡𝑟𝑜𝑛𝑔𝑙𝑦𝑚𝑜𝑛𝑜𝑡𝑜𝑛𝑒, if there exists a constant 𝛿>0 such that 𝑥−𝑦,𝐴𝑥−𝐴𝑦≥𝛿𝑥−𝑦²,∀𝑥,𝑦∈𝐶. (2)(3)A mapping 𝐴∶𝐶→𝐸^∗ is said to be 𝛿-𝑖𝑛𝑣𝑒𝑟𝑠𝑒𝑠𝑡𝑟𝑜𝑛𝑔𝑙𝑦𝑚𝑜𝑛𝑜𝑡𝑜𝑛𝑒, if there exists a constant 𝛿>0 such that 𝑥−𝑦,𝐴𝑥−𝐴𝑦≥𝛿𝐴𝑥−𝐴𝑦²,∀𝑥,𝑦∈𝐶. (3)(4)A mapping 𝐴∶𝐶^∗→𝐸 is said to be skew monotone if for each 𝑥^∗,𝑦^∗∈𝐶^∗ such that 𝐴𝑥^∗−𝐴𝑦^∗,𝑥^∗−𝑦^∗≥0. (4)(5)A mapping 𝐴∶𝐷(𝐴)⊂𝐸^∗→𝐸 is said to be 𝛼-inverse strongly skew monotone if there exists a constant 𝛼>0 such that 𝐴𝑥^∗−𝐴𝑦^∗,𝑥^∗−𝑦^∗≥𝛼𝐴𝑥^∗−𝐴𝑦^∗2,∀𝑥^∗,𝑦^∗∈𝐷(𝐴). (5)

Definition 1. Let𝐸be a Banach space. Then,(1)𝐸 is said to be strictly convex if (‖𝑥+𝑦‖/2)<1 for all 𝑥,𝑦∈𝑈_𝐸={𝑧∈𝐸∶‖𝑧‖=1} with 𝑥≠𝑦;(2)𝐸 is said to be uniformly convex if for each 𝜖∈(0,2], there exists 𝛿>0 such that (‖𝑥+𝑦‖/2)≤1−𝛿, for all 𝑥,𝑦∈𝑈_𝐸 with ‖𝑥+𝑦‖>𝜖;(3)𝐸 is said to be smooth if the limit lim_𝑡→0𝑥+𝑡𝑦−‖𝑥‖𝑡 (6) exists, for each 𝑥,𝑦∈𝑈_𝐸;(4)𝐸 is said to be uniformly smooth if the limit (6) is attained uniformly, for all 𝑥,𝑦∈𝑈_𝐸;(5)𝐸 is said to have uniformly Gâteaux differentiable norm if for all 𝑦∈𝑈(𝐸), the the limit (6) converges uniformly, for 𝑥∈𝑈_𝐸.

Definition 2. 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. (7)(1)𝐸 is said to be smooth if 𝜌_𝐸(𝑡)>0,∀𝑡>0.(2)𝐸 is said to be uniformly smooth if and only if lim_𝑡→0⁺(𝜌_𝐸(𝑡)/𝑡)=0.

Definition 3. Let 𝐸 be a Banach space. Then, 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. (8)(1)𝐸 is said to be uniformly convex if and only if 𝛿_𝐸(𝜖)>0 for all 𝜖∈(0,2].(2)Let 𝑝 be a fixed real number 𝑝>1. Then, 𝐸 is said to be 𝑝-uniformly convex if there exists a constant 𝑐>0 such that 𝛿_𝐸(𝜖)≥𝑐𝜖^𝑝 for all 𝜖∈[0,2].
Observe that every 𝑝-uniformly convex is uniformly convex. One should note that no a Banach space is 𝑝-uniformly convex, for 1<𝑝<2. It is well known that a Hilbert space is 2-uniformly convex and uniformly smooth.

For any 𝑝>1, the generalized duality mapping 𝐽_𝑝∶𝐸→2^𝐸∗ is defined by 𝐽_𝑝𝑥=𝑓^∗∈𝐸^∗∶𝑥,𝑓^∗=‖𝑥‖^𝑝,𝑓^∗=‖𝑥‖^𝑝−1,∀𝑥∈𝐸. (9)

In particular, 𝐽=𝐽₂ is called the normalized duality mapping. If 𝐸 is a Hilbert space, then 𝐽=𝐼, where 𝐼 is the identity mapping. That is, 𝐽₂=𝑓𝑥=𝐽𝑥_∗∈𝐸^∗∶𝑥,𝑓^∗=‖𝑥‖²=𝑓^∗2,∀𝑥∈𝐸. (10)

Remark 4. The basic properties below hold (see [1–3]). (1)If 𝐸 is uniformly smooth real Banach space, then 𝐽 is uniformly continuous on each bounded subset of 𝐸.(2)If 𝐸 is uniformly smooth real Banach space, then 𝐽^∗∶𝐸^∗→2^𝐸 is a normalized duality mapping on 𝐸^∗, and then 𝐽⁻¹=𝐽^∗, (𝐽^∗)𝐽=𝐼_𝐸, and 𝐽(𝐽^∗)=𝐼_𝐸^∗, where on 𝐼_𝐸 and 𝐼_𝐸^∗ are the identity mappings on 𝐸 and 𝐸^∗, respectively.(3)Let 𝐸 be a smooth, strictly convex reflexive Banach space, and let 𝐽 be the duality mapping from 𝐸 into 𝐸^∗. Then, 𝐽⁻¹ is also single-valued, one-to-one, and onto, and it is also the duality mapping from 𝐸^∗ into 𝐸.(4)If 𝐸 is a reflexive, strictly convex Banach space, then 𝐽⁻¹ is hemicontinuous; that is, 𝐽⁻¹ is norm-to-weak*-continuous. (5)If 𝐸 is a reflexive, smooth, and strictly convex Banach space, then 𝐽 is single-valued, one-to-one, and onto. (6)A Banach space 𝐸 is uniformly smooth if and only if 𝐸^∗ is uniformly convex. (7)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.(8)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. (11)(9)Both uniformly smooth Banach space and uniformly convex Banach space are reflexive. (10)If 𝐸^∗ is uniformly convex, then 𝐽 is uniformly norm-to-norm continuous on each bounded subset of 𝐸.(11)If 𝐸^∗ is strictly convex Banach space, then 𝐽 is one-to-one; that is, 𝑥≠𝑦 implies 𝐽𝑥∩𝐽𝑦≠∅.

Let 𝐽 be the normalized duality mapping; then 𝐽 is said to be weakly sequentially continuous if the strong convergence of a sequence {𝑥_𝑛} to 𝑥∈𝐸 implies the weak* convergence of a sequence {𝐽𝑥_𝑛} to 𝐽𝑥 in 𝐸^∗.

Let 𝐸 be a smooth and strictly convex reflexive Banach space, and let 𝐶 be a nonempty, closed, and convex subset of 𝐸. We assume that the Lyapunov functional 𝜙∶𝐸×𝐸→ℝ⁺ is defined by [3, 4] 𝜙𝑥,𝑦=‖𝑥‖²−2𝑥,𝐽𝑦+𝑦²,∀𝑥,𝑦∈𝐸. (12)

Let 𝐶 is nonempty, closed, and convex subset of a Banach space 𝐸. The generalized projection [3] Π_𝐶∶𝐸→𝐶 is defined by for each 𝑥∈𝐸, Π_𝐶(𝑥)=argmin_𝑦∈𝐶𝜙𝑥,𝑦. (13)

Remark 5. From the definition of 𝜙. It is easy to see that (1)(‖𝑥‖−‖𝑦‖)²≤𝜙(𝑥,𝑦)≤(‖𝑥‖+‖𝑦‖)², for all 𝑥,𝑦∈𝐸;(2)𝜙(𝑥,𝑦)=𝜙(𝑥,𝑧)+𝜙(𝑧,𝑦)+2⟨𝑥−𝑧,𝐽𝑧−𝐽𝑦⟩, for all 𝑥,𝑦,𝑧∈𝐸;(3)𝜙(𝑥,𝑦)=⟨𝑥,𝐽𝑥−𝐽𝑦⟩+⟨𝑦−𝑥,𝐽𝑦⟩≤‖𝑥‖‖𝐽𝑥−𝐽𝑦‖+‖𝑦−𝑥‖‖𝑦‖, for all 𝑥,𝑦,𝑧∈𝐸;(4)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 6 (see [3, 4]). If 𝐶 is a nonempty, closed, and convex subset of a smooth and strictly convex reflexive real Banach space 𝐸, then(1)for 𝑥∈𝐸 and 𝑢∈𝐶, one has 𝑢=Π_𝐶(𝑥)⟺𝑢−𝑦,𝐽𝑥−𝐽𝑢≥0,∀𝑦∈𝐶; (14)(2)𝜙(𝑥,Π_𝐶(𝑦))+𝜙(Π_𝐶(𝑦),𝑦)≤𝜙(𝑥,𝑦),∀𝑥∈𝐶, and𝑦∈𝐸;(3)𝜙(𝑥,𝑦)=0 if and only if 𝑥=𝑦, ∀𝑥,𝑦∈𝐸.

Let 𝐶 be a nonempty, closed subset of a smooth, strictly convex, and reflexive Banach space 𝐸 such that 𝐽(𝐶) is closed and convex. For solving the equilibrium problem, let us assume that a bifunction 𝐹∶𝐽(𝐶)×𝐽(𝐶)→ℝ satisfies the following conditions:(DA1)𝐹(𝑥^∗,𝑥^∗)=0, for all 𝑥^∗∈𝐽(𝐶); (DA2)𝐹is monotone, that is, 𝐹(𝑥^∗,𝑦^∗)+𝐹(𝑦^∗,𝑥^∗)≤0, for all 𝑥^∗,𝑦^∗∈𝐽(𝐶)(DA3) for all 𝑥^∗,𝑦^∗,𝑧^∗∈𝐽(𝐶),limsup_𝑡↓0𝐹𝑡𝑧^∗+(1−𝑡)𝑥^∗,𝑦^∗≤𝐹𝑥^∗,𝑦^∗; (15)(DA4) for all 𝑥^∗∈𝐽(𝐶),  𝐹(𝑥^∗,⋅) is convex and lower semicontinuous.

The following result is in Blum and Oettli [5], and see proof in [6].

Let ℝ be the set of real numbers, let 𝐸 be a real Banach space with the norm ‖⋅‖, and let ⟨⋅,⋅⟩ that is the dual pair between 𝐸 and 𝐸^∗ by 𝐸^∗ be the dual space of 𝐸. Let 𝐶 be a nonempty, closed, and convex subset of 𝐸, let 𝐽 be the duality mapping from 𝐸 into 𝐸^∗ such that 𝐽(𝐶) is closed and convex of 𝐸^∗, let us assume that a bifunction 𝐹∶𝐽(𝐶)×𝐽(𝐶)→ℝ satisfies suitable conditions, let 𝐴∶𝐶^∗→𝐸 be a skew monotone operator from 𝐽(𝐶) into 𝐸, and let 𝜑∶𝐽(𝐶)→ℝ be a real-valued function.

The generalized mixed equilibrium problem is to find 𝑧∈𝐶 such that 𝐹𝐽𝑧,𝐽𝑦+𝐴𝐽𝑧,𝐽𝑦−𝐽𝑧+𝜑𝐽𝑦−𝜑(𝐽𝑧)≥0,∀𝑦∈𝐶. (16)

The set of solutions of (16) is denoted by GMEP(𝐹,𝐴,𝜑); that is, =𝐽+𝑧(𝐽𝑧).GMEP𝐹,𝐴,𝜑𝑧∈𝐶∶𝐹𝑧,𝐽𝑦𝐴𝐽𝑧,𝐽𝑦−𝐽+𝜑𝐽𝑦−𝜑≥0,∀𝑦∈𝐶 (17)

If 𝐴≡0, then the problem (16) reduces to the mixed equilibrium problem which is to find 𝑧∈𝐶 such that 𝐹𝐽𝑧,𝐽𝑦+𝜑𝐽𝑦−𝜑(𝐽𝑧)≥0,∀𝑦∈𝐶. (18)

The set of solution of problem (18) is denoted by MEP(𝐹,𝜑); that is, =𝐽(𝐽𝑧).MEP𝐹,𝜑𝑧∈𝐶∶𝐹𝑧,𝐽𝑦+𝜑𝐽𝑦−𝜑≥0,∀𝑦∈𝐶 (19)

If 𝐴≡0 and 𝜑≡0, then the problem (16) reduces to the equilibrium problem which is to find 𝑧∈𝐶 such that 𝐹𝐽𝑧,𝐽𝑦≥0,∀𝑦∈𝐶. (20)

The set of solution of problem (20) is denoted by EP(𝐹); that is, EP(𝐹)=𝑧∈𝐶∶𝐹𝐽𝑧,𝐽𝑦≥0,∀𝑦∈𝐶. (21)

The above formulation (20) was considered in Takahashi and Zembayashi [7], and they proved a strong convergence theorem for finding a solution of the equilibrium problem (20) in Banach spaces.

If 𝐹≡0 and 𝐴≡0, then the problem (16) reduces to variational inequality, which is to find 𝑧∈𝐶 such that 𝐴𝐽𝑧,𝐽𝑦−𝐽𝑧≥0,∀𝑦∈𝐶. (22)

The set of solution of problem (22) is denoted by VI(𝐽(𝐶),𝐴); that is, VI(𝐽(𝐶),𝐴)=𝑧∈𝐶∶𝐴𝐽𝑧,𝐽𝑦−𝐽𝑧≥0,∀𝑦∈𝐶. (23)

In the sequel, let 𝑇∶𝐶→𝐶 be a mapping, we denote by Fix(𝑇) the set of fixed points of 𝑇; that is, Fix(𝑇)={𝑥∈𝐶∶𝑇𝑥=𝑥}. (24)

In 1953, Mann [8] introduced an iterative algorithm which is defined by the initial point 𝑥₀ is taken in 𝐶 arbitrarily and 𝑥_𝑛+1=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝑇𝑥_𝑛,𝑛≥0 (25) where the sequence 𝛼_𝑛∈[0,1]. Mann’s iteration can yield only weak convergence.

In 1967, Halpern [9] introduced another iterative algorithm which is defined by the initial point 𝑥₀ is taken in 𝐶 arbitrarily and 𝑥₀𝑥=𝑢∈𝐶chosenarbitrary,_𝑛+1=𝛼_𝑛𝑢+1−𝛼_𝑛𝑇𝑥_𝑛,𝑛≥0, (26) which satisfied the conditions lim_𝑛→∞𝛼_𝑛=0 and ∑^∞_𝑛=1𝛼_𝑛=∞. Then, the sequence {𝑥_𝑛} is converges strongly to a fixed point of 𝑇.

In 2007, Takahashi and Zembayashi [7] introduced an iterative algorithm for finding a solution of an equilibrium problem with a bifunction defined on the dual space of a Banach space by using the shrinking projection method, and they established the strong convergence of the following results.

Theorem TZ. Let 𝐸 be a uniformly convex Banach space whose norm is uniformly Gâteaux differentiable, and let 𝐶 be a nonempty, closed and convex subset of 𝐸 such that 𝐽(𝐶) is closed, and convex of 𝐸^∗. Assume that a mapping 𝐹∶𝐽(𝐶)×𝐽(𝐶)→ℝ satisfied the conditions (DA1)–(DA4) such that EP(𝐹)≠∅. Let {𝑥_𝑛} be a sequence generated by the following algorithm: 𝑥₀∈𝐶,𝐶₀𝑢=𝐶𝑐ℎ𝑜𝑠𝑒𝑛𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑦,_𝑛∈𝐶𝑠𝑢𝑐ℎ𝑡ℎ𝑎𝑡𝐹𝐽𝑢_𝑛+1,𝐽𝑦𝑟_𝑛𝑢_𝑛−𝑥_𝑛,𝐽𝑦−𝐽𝑢_𝑛𝑦≥0,∀𝑦∈𝐶,_𝑛=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝑢_𝑛,𝐶_𝑛+1=𝑧∈𝐶_𝑛∶𝜙𝑦_𝑛,𝑧≤𝜙𝑥_𝑛,𝑥,𝑧_𝑛+1=𝑅_𝐶𝑛+1𝑥₀,∀𝑛∈ℕ∪{0}, (27) where 𝐽 is the duality mapping on 𝐸, the sequence {𝛼_𝑛}⊂[0,1] such thatlimsup_𝑛→∞𝛼_𝑛<1,𝑟_𝑛⊂[𝑎,∞) for some 𝑎>0, and 𝑅_𝐶𝑛+1 is the sunny generalized nonexpansive retraction from 𝐸 onto 𝐶_𝑛+1. Then, the sequence {𝑥_𝑛} converges strongly to some point 𝑝=𝑅_EF(𝐹)(𝑥₀), where 𝑅_EP(𝐹) is the sunny generalized nonexpansive retraction from 𝐸 to EP(𝐹).

In 2010, Plubtieng and Sriprad [10] proved the existence theorem of the variational inequality problem for skew monotone operator defined on the dual space of a smooth Banach space, and they established weak convergence theorem for finding a solution of the variational inequality problem using projection algorithm method with a new projection which was introduced by Ibaraki and Takahashi [11] and Iiduka and Takahashi [12] in Banach spaces.

Let 𝐸 be a smooth, strictly convex, and reflexive Banach space, let 𝐸^∗ be the dual space of 𝐸, and let 𝐶 be a nonempty, closed, and convex subset of 𝐸 such that 𝐽(𝐶) is closed and convex of 𝐸^∗, where 𝐽 is the duality mapping on 𝐸. Let 𝐴 be a skew monotone operator from 𝐽(𝐶) into 𝐸. Then, the variational inequality problem is to find 𝑧∈𝐶 such that 𝐴𝐽𝑧,𝐽𝑦−𝐽𝑧≥0,∀𝑦∈𝐶. (28)

The set of solution of problem (28) is denoted by VI(𝐽(𝐶),𝐴); that is, VI(𝐽(𝐶),𝐴)=𝑧∈𝐶∶𝐴𝐽𝑧,𝐽𝑦−𝐽𝑧≥0,∀𝑦∈𝐶. (29)

Theorem PS. Let 𝐸 be a uniformly convex and 2-uniformly smooth Banach space whose duality mapping 𝐽 is weakly sequentially continuous. Let 𝐶 be a nonempty, closed and convex subset of 𝐸 such that 𝐽(𝐶) is closed, and convex, and let 𝐴 be an 𝛼-inverse-strongly-skew-monotone operator of 𝐽𝐶 into 𝐸 such that VI(𝐽(𝐶),𝐴)≠∅, and ‖𝐴𝐽𝑧‖≤‖𝐴𝐽𝑦−𝐴𝐽𝑧‖, for all 𝑦∈𝐶 and 𝑧∈VI(𝐽(𝐶),𝐴). Let {𝑥_𝑛} be a sequence defined by 𝑥₁=𝑥∈𝐶 and 𝑥_𝑛+1=𝑅_𝐶𝑥_𝑛−𝜆_𝑛𝐴𝐽𝑥_𝑛, (30) for every 𝑛=1,2,3,…, where 𝑅_𝐶 is the sunny generalized nonexpansive retraction of 𝐸 into 𝐶,  {𝛼_𝑛}⊂[𝑎,𝑏] and for some 𝑎,𝑏 with 0<𝑎<𝑏<(𝛼/𝑐), where 𝑐>0 is a constant that satisfies ‖𝐽𝑥−𝐽𝑦‖≤𝑐‖𝑥−𝑦‖, for all 𝑥,𝑦∈𝐶. Then, the sequence {𝑥_𝑛} converges weakly to some element 𝑧∈VI(𝐽(𝐶),𝐴). Further 𝑧=lim_𝑛→∞𝑅_{VI(𝐽(𝐶),𝐴)}𝑥_𝑛.

In 2011, Chen et al. [13] introduced a new iterative method for finding a solution of equilibrium with a bifunction defined on the dual space of a Banach space. They established the strong convergence theorem by using the sunny generalized nonexpansive retraction in Banach spaces.

Theorem CCW. Let 𝐶 be a nonempty, closed and convex subset of a uniformly convex and uniformly smooth Banach space such that 𝐽(𝐶) is closed and convex. Assume that a bifunction 𝐹∶𝐽(𝐶)×𝐽(𝐶)→ℝ satisfies the conditions (DA1)–(DA4). Define a sequence {𝑥_𝑛} in C by the following algorithm: 𝑥₀𝑢∈𝐶𝑐ℎ𝑜𝑠𝑒𝑛𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑦,_𝑛∈𝐶𝑠𝑢𝑐ℎ𝑡ℎ𝑎𝑡𝐹𝐽𝑢_𝑛+1,𝐽𝑦𝑟_𝑛𝑢_𝑛−𝑥_𝑛,𝐽𝑦−𝐽𝑢_𝑛𝑥≥0,∀𝑦∈𝐶,_𝑛+1=𝛼_𝑛𝑥₀+1−𝛼_𝑛𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝑢_𝑛,∀𝑛∈ℕ, (31) where 𝐽 is the duality mapping on 𝐸, the sequences {𝛼_𝑛}, {𝛽_𝑛}⊂[0,1], and {𝑟_𝑛}⊂[𝑎,∞), for some 𝑎>0 such that ^∞_𝑛=0𝛼_𝑛<∞,liminf_𝑛→∞𝛽_𝑛1−𝛽_𝑛>0,liminf_𝑛→∞𝑟_𝑛>0. (32) Then, the sequence 𝑅_EP(𝐹){𝑥_𝑛} converges strongly to some point 𝑝∈EF(𝐹), where 𝑅_EP(𝐹) is the sunny generalized nonexpansive retraction from 𝐸 to EP(𝐹).

In 2012, Saewan et al. [14] introduced a new iterative scheme for finding a common element of the set of solutions of the mixed equilibrium problems and the set of fixed points for a -𝜙-nonexpansive mapping in Banach spaces by using sunny generalized nonexpansive retraction in Banach spaces.

Theorem SCK. Let 𝐸 be a uniformly smooth and uniformly convex Banach space, and let 𝐶 be a nonempty, closed, and convex subset of 𝐸 such that 𝐽(𝐶) is closed and convex of 𝐸^∗. Let 𝐹∶𝐽(𝐶)×𝐽(𝐶)→ℝ be a bifunction that satisfies the conditions (DA1)–(DA4), and let 𝑇∶𝐶→𝐶 be a closed and -𝜙-nonexpansive mapping. Assume that 𝔉∶=Fix(𝑇)∩MEP(𝐹,𝜑)≠∅. For an initial point 𝑥₀∈𝐶 and define a sequence {𝑥_𝑛} in𝐶by the following algorithm: 𝑥₀𝑢∈𝐶𝑐ℎ𝑜𝑠𝑒𝑛𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑦,_𝑛𝐹∈𝐶𝑠𝑢𝑐ℎ𝑡ℎ𝑎𝑡𝐽𝑢_𝑛,𝐽𝑦+𝜑𝐽𝑦−𝜑𝐽𝑢_𝑛+1𝑟_𝑛𝑢_𝑛−𝑥_𝑛,𝐽𝑦−𝐽𝑢_𝑛𝑥≥0,∀𝑦∈𝐶,_𝑛+1=𝛼_𝑛𝑥₀+1−𝛼_𝑛𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝑇𝑢_𝑛,∀𝑛∈ℕ, (33) where 𝐽 is the duality mapping on 𝐸, the sequence {𝛼_𝑛}, {𝛽_𝑛}⊂[𝑎,𝑏], and {𝑟_𝑛}⊂[𝑐,∞),for some𝑎,𝑏∈(0,1)and𝑐>0. If the following conditions are satisfied: ^∞_𝑛=0𝛼_𝑛<∞,^∞_𝑛=0𝛽_𝑛<∞,liminf_𝑛→∞𝑟_𝑛>0. (34) Then, the sequence 𝑅_𝔉{𝑥_𝑛} converges strongly to some point 𝑝∈𝔉, where 𝑅_𝔉 is the sunny generalized nonexpansive retraction from 𝐸 onto 𝔉.

In this paper, Motivated and inspired by the previously mentioned above results, we study and investigate the existence of theorem for a generalized mixed equilibrium problem with a bifunction defined on the dual space of a Banach space, and we construct an iterative procedure generated by the conditions for solving the common solution of a generalized mixed equilibrium problem and a fixed point problem by using the sunny generalized nonexpansive retraction. Under some suitable assumptions, the strong convergence theorem are established in Banach spaces. The results obtained in this paper extend and improve several recent results in this area.

2. Preliminaries

Definition 7. Let 𝐶 be a nonempty, closed subset of a smooth Banach space. (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 nonexpansive if 𝑇𝑥−𝑇𝑦≤𝑥−𝑦,∀𝑥,𝑦∈𝐶. (35)(3)A mapping 𝑇∶𝐶→𝐶 is said to be -𝜙-nonexpansive if Fix(𝑇)≠∅, and 𝜙𝑇𝑥,𝑇𝑦≤𝜙𝑥,𝑦,∀𝑥,𝑦∈𝐶. (36)(4)A mapping 𝑇∶𝐶→𝐶 is said to be generalized nonexpansive [15] if Fix(𝑇)≠∅ and 𝜙𝑇𝑥,𝑝≤𝜙𝑥,𝑝,∀𝑥∈𝐶,𝑝∈Fix(𝑇). (37)

Definition 8 (see [15]). Let 𝐶 be a nonempty, closed subset of a smooth Banach space 𝐸. A mapping 𝑅∶𝐸→𝐶 is called(1)a retraction if 𝑅²=𝑅;(2)a sunny if 𝑅(𝑅𝑥+𝑡(𝑥−𝑅𝑥))=𝑅𝑥, for all 𝑥∈𝐸 and 𝑡≥0.

We also know that if 𝐸 is a smooth, strictly convex, and reflexive Banach space and 𝐶 is nonempty, closed, and convex subset of 𝐸, then there exists a sunny generalized nonexpansive retraction 𝑅_𝐶 of 𝐸 onto 𝐶 if and only if 𝐽(𝐶) is closed and convex. In this case 𝑅_𝐶 is given by 𝑅_𝐶=𝐽⁻¹Π_𝐽(𝐶)𝐽. (38)

Definition 9 (see [15]). A nonempty, closed subset 𝐶 of a smooth Banach space 𝐸 is said to be a sunny generalized nonexpansive retraction of 𝐸 if there exists a sunny generalized nonexpansive 𝑅 from 𝐸 onto 𝐶.

Lemma 10 (see [11]). Let 𝐶 be a nonempty, closed, and subset of a smooth and strictly convex Banach space 𝐸, and let 𝑅 be a retraction from 𝐸 onto 𝐶. Then, the following are equivalent: (1)𝑅 is sunny generalized nonexpansive; (2)⟨𝑥−𝑅𝑥,𝐽𝑦−𝐽𝑅𝑥⟩≤0, for all 𝑥∈𝐸 and 𝑦∈𝐶.

Lemma 11 (see [11]). Let 𝐶 be a nonempty, closed, and sunny generalized nonexpansive retraction of a smooth and strictly convex Banach space 𝐸. Then, the sunny generalized nonexpansive retraction from 𝐸 onto 𝐶 is uniquely determined.

Lemma 12 (see [11]). Let 𝐶 be a nonempty, closed, and subset of a smooth and strictly convex Banach space 𝐸 such that there exists a sunny generalized nonexpansive retraction 𝑅 from 𝐸 onto 𝐶. Let 𝑥∈𝐸 and 𝑧∈𝐶. Then, the following hold: (1)𝑧=𝑅𝑥 if and only if ⟨𝑥−𝑧,𝐽𝑦−𝐽𝑧⟩≤0, for all 𝑦∈𝐶; (2)𝜙(𝑥,𝑅𝑥)+𝜙(𝑅𝑥,𝑧)≤𝜙(𝑥,𝑧).

Lemma 13 (see [16]). Let 𝐶 be a nonempty, closed, and subset of a smooth, strictly convex, and reflexive Banach space 𝐸. Then, the following are equivalent: (1)𝐶 is sunny generalized nonexpansive retraction of 𝐸; (2)𝐽(𝐶) is closed and convex.

Remark 14. Let 𝐸 be a Hilbert space. By the Lemmas 11 and 13, a sunny generalized nonexpansive retraction from 𝐸 onto 𝐶 reduces to a metric projection operator 𝑃 from 𝐸 onto 𝐶.

Lemma 15 (see [16]). Let 𝐸 be a smooth, strictly convex, and reflexive Banach space, let 𝐶 be a nonempty, closed, and sunny generalized nonexpansive retraction of 𝐸, and let 𝑅 be the sunny generalized nonexpansive retraction from 𝐸 onto 𝐶. Let 𝑥∈𝐸 and 𝑧∈𝐶. Then, the following are equivalent: (1)𝑧=𝑅𝑥; (2)𝜙(𝑥,𝑧)=min_𝑦∈𝐶𝜙(𝑥,𝑦).

Lemma 16 (see [4]). Let E be a uniformly smooth and strictly convex 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 17 (see [17]). Let E be a uniformly smooth and strictly convex real Banach space with the Kadec-Klee property, and let 𝐶 be a nonempty, closed, and convex subset of 𝐸. Let {𝑥_𝑛} and {𝑦_𝑛} be two sequences in 𝐶 and 𝑝∈𝐸. If 𝑥_𝑛→𝑝 and 𝜙(𝑥_𝑛,𝑦_𝑛)→0, then 𝑦_𝑛→𝑝.

Lemma 18 (see [18]). Let {𝑎_𝑛} and {𝑏_𝑛} be two sequences of nonnegative real numbers satisfying the inequality 𝑎_𝑛+1≤𝑎_𝑛+𝑏_𝑛,∀𝑛≥1. (39) If ∑^∞_𝑛=0𝑏_𝑛<∞, then lim_𝑛→∞𝑎_𝑛 exists.

Lemma 19 (see [19]). Let 𝐸 be a uniformly convex Banach space. Then, for any 𝑟>0, there exists a strictly increasing, continuous, and convex function ℎ∶[0,2𝑟]→ℝ such that ℎ(0)=0 and 𝑡𝑥+(1−𝑡)𝑦²≤𝑡‖𝑥‖²+(1−𝑡)𝑦²−𝑡(1−𝑡)ℎ𝑥−𝑦,∀𝑥,𝑦∈𝐵_𝑟(𝑧),𝑡∈[0,1], (40) where 𝐵_𝑟(𝑧)={𝑧∈𝐸∶‖𝑧‖≤𝑟}.

Lemma 20 (see [4]). Let 𝐸 be a smooth and uniformly convex Banach space. Then, for any 𝑟>0, there exists a strictly increasing, continuous, and convex function ℎ∶[0,2𝑟]→ℝ such that ℎ(0)=0 and ℎ𝑥−𝑦≤𝜙𝑥,𝑦,∀𝑥,𝑦∈𝐵_𝑟(𝑧),𝑡∈[0,1], (41) where 𝐵_𝑟(𝑧)={𝑧∈𝐸∶‖𝑧‖≤𝑟}.

Now, let us recall the following well-known concept and result.

Definition 21 (see [20]). Let 𝐵 be a subset of a topological vector space 𝑋. A mapping 𝐺∶𝐵→2^𝑋 is called a KKM mapping if conv{𝑥₁,𝑥₂,𝑥₃,…,𝑥_𝑚}⊂⋃^𝑚_𝑖=1𝐺(𝑥_𝑖) for 𝑥_𝑖∈𝐵 and 𝑖=1,2,3,…,𝑚, where conv𝐴 denotes the convex hull of the set 𝐴.

In [21], Fan gave the following famous infinite-dimensional generalization of Knaster, Kuratowski, and Mazurkiewicz’s classical finite-dimensional result.

Lemma 22 (see [21]). Let 𝐵 be a subset of a Hausdorff topological vector space 𝑋, and let 𝐺∶𝐵→2^𝑋 be a KKM mapping. If 𝐺(𝑥) is closed, for all 𝑥∈𝐵 and is compact for at least one 𝑥∈𝐵, then ⋂_𝑥∈𝐵𝐺(𝑥)≠∅.

Lemma 23 (see [14]). Let 𝐶 be a nonempty, closed, and convex subset of a smooth and strictly convex Banach space, and let 𝑇∶𝐶→𝐶 be a closed and -𝜙-nonexpansive mapping. Then, Fix(𝑇) is a closed and convex subset of 𝐶.

3. Existence Theorem

In this section, we prove the existence theorem of a solution for a generalized mixed equilibrium problem with a bifunction defined on the dual space of a Banach space.

Lemma 24. Let 𝐶 be a nonempty, compact, and convex subset of a uniformly smooth, strictly convex, and reflexive Banach space 𝐸, let 𝐽 be the duality mapping from 𝐸 into 𝐸^∗ such that 𝐽(𝐶) is closed and convex, let us assume that a bifunction 𝐹∶𝐽(𝐶)×𝐽(𝐶)→ℝ satisfies the following conditions (DA1)–(DA4), let 𝐶^∗ be a nonempty, closed, and convex subset of 𝐸^∗, let 𝐴∶𝐶^∗→𝐸 be an 𝛼-inverse strongly skew monotone and let 𝜑∶𝐽(𝐶)→ℝ be a convex and lower semicontinuous. Let𝑟>0 be given real number and 𝑥∈𝐸 be any point. Then, there exists 𝑧∈𝐶 such that 𝐹++1𝐽𝑧,𝐽𝑦𝐴𝐽𝑧,𝐽𝑦−𝐽𝑧𝑟()𝑧−𝑥,𝐽𝑦−𝑧+𝜑𝐽𝑦−𝜑𝐽𝑧≥0,∀𝑦∈𝐶. (42)

Proof. Let 𝑥₀ be any point in 𝐸. For each 𝑦∈𝐶, we define the mapping 𝐺∶𝐶→2^𝐸 as follows: 𝐺𝑦=++1𝑧∈𝐶∶𝐹𝐽𝑧,𝐽𝑦𝐴𝐽𝑧,𝐽𝑦−𝐽𝑧𝑟𝑧−𝑥₀().,𝐽𝑦−𝑧+𝜑𝐽𝑦−𝜑𝐽𝑧≥0,∀𝑦∈𝐶 (43) It is easy to see that 𝑦∈𝐺(𝑦), and hence 𝐺(𝑦)≠∅.(a) First, we will show that 𝐺 is a KKM mapping.
Suppose that 𝐺 is not a KKM mapping. Then, there exists a finite subset {𝑦₁,𝑦₂,𝑦₃,…,𝑦_𝑚} of 𝐶 and 𝛼_𝑖>0 with ∑^𝑚_𝑖=1𝛼_𝑖=1 such that 𝑥=∑^𝑚_𝑖=1𝛼_𝑖𝑦_𝑖∉⋃^𝑚_𝑖=1𝐺(𝑦_𝑖) for all 𝑖=1,2,3,…,𝑚.
It follows from the definition of a mapping 𝐺 that 𝐹𝐽𝑥,𝐽𝑦_𝑖+𝐴𝐽𝑥,𝐽𝑦_𝑖𝑥+1−𝐽𝑟𝑥−𝑥₀,𝐽𝑦_𝑖−𝑥+𝜑𝐽𝑦_𝑖−𝜑(𝐽𝑥)<0,∀𝑖=1,2,3,…,𝑚. (44) By the assumptions of (DA1) and (DA4), we get (𝐽𝑥)+⟨𝑥⟩+10=𝐹𝑥,𝐽𝐴𝐽𝑥,𝐽𝑥−𝐽𝑟𝑥−𝑥₀(𝑥(𝐽𝑥)(𝐽𝑥)≤,𝐽𝑥−)+𝜑−𝜑_𝑚𝑖=1𝛼_𝑖𝐹𝐽𝑥,𝐽𝑦_𝑖+𝐴𝐽𝑥,𝐽𝑦_𝑖𝑥+1−𝐽𝑟𝑥−𝑥₀,𝐽𝑦_𝑖−𝑥+𝜑𝐽𝑦_𝑖−𝜑(𝐽𝑥)<0, (45) which is a contradiction. Thus, 𝐺 is a KKM mapping on 𝐶.(b) Next, we will show that 𝐺(𝑦) is closed, for all 𝑦∈𝐶.Let {𝑧_𝑛} be a sequence in 𝐺(𝑦) such that 𝑧_𝑛→𝑧₀ as 𝑛→∞.
It then follows from 𝑧_𝑛∈𝐺(𝑦) that 𝐹𝐽𝑧_𝑛,𝐽𝑦+𝐴𝐽𝑧_𝑛,𝐽𝑦−𝐽𝑧_𝑛+1𝑟𝑧_𝑛−𝑥₀,𝐽𝑦−𝑧_𝑛+𝜑𝐽𝑦−𝜑𝐽𝑧_𝑛≥0. (46) By the assumption (DA3), the continuity of 𝐽, and the lower semicontinuity of 𝜑 and ‖⋅‖², it follows from (46) that 𝜑𝑧₀≤liminf_𝑛→∞𝜑𝑧_𝑛≤liminf_𝑛→∞𝐹𝐽𝑧_𝑛,𝐽𝑦+𝐴𝐽𝑧_𝑛,𝐽𝑦−𝐽𝑧_𝑛+1𝑟𝑧_𝑛−𝑥₀,𝐽𝑦−𝑧_𝑛+𝜑𝐽𝑦≤limsup_𝑛→∞𝐹𝐽𝑧_𝑛,𝐽𝑦+𝐴𝐽𝑧_𝑛,𝐽𝑦−𝐽𝑧_𝑛+1𝑟𝑧_𝑛−𝑥₀,𝐽𝑦−𝑧_𝑛+𝜑𝐽𝑦=limsup_𝑛→∞𝐹𝐽𝑧_𝑛,𝐽𝑦+𝐴𝐽𝑧_𝑛,𝐽𝑦−𝐽𝑧_𝑛+1𝑟𝑧_𝑛−𝑦+𝑦−𝑥₀,𝐽𝑦−𝑧_𝑛+𝜑𝐽𝑦=limsup_𝑛→∞𝐹𝐽𝑧_𝑛,𝐽𝑦+𝐴𝐽𝑧_𝑛,𝐽𝑦−𝐽𝑧_𝑛+1𝑟𝑧_𝑛−𝑦,𝐽𝑦−𝑧_𝑛+1𝑟𝑦−𝑥₀,𝐽𝑦−𝑧_𝑛+𝜑𝐽𝑦=limsup_𝑛→∞𝐹𝐽𝑧_𝑛,𝐽𝑦+𝐴𝐽𝑧_𝑛,𝐽𝑦−𝐽𝑧_𝑛+1𝑟𝑦−𝑥₀,𝐽𝑦−𝑧_𝑛−1𝑟𝑦−𝑧_𝑛,𝐽𝑦−𝑧_𝑛+𝜑𝐽𝑦=limsup_𝑛→∞𝐹𝐽𝑧_𝑛,𝐽𝑦+𝐴𝐽𝑧_𝑛,𝐽𝑦−𝐽𝑧_𝑛+1𝑟𝑦−𝑥₀,𝐽𝑦−𝑧_𝑛−1𝑟𝑦−𝑧_𝑛²+𝜑𝐽𝑦≤limsup_𝑛→∞𝐹𝐽𝑧_𝑛,𝐽𝑦+limsup_𝑛→∞𝐴𝐽𝑧_𝑛,𝐽𝑦−𝐽𝑧_𝑛+1𝑟limsup_𝑛→∞𝑦−𝑥₀,𝐽𝑦−𝑧_𝑛−liminf_𝑛→∞1𝑟𝑦−𝑧_𝑛²+𝜑𝐽𝑦≤𝐹𝐽𝑧₀,𝐽𝑦+𝐴𝐽𝑧₀,𝐽𝑦−𝐽𝑧₀+1𝑟𝑦−𝑥₀,𝐽𝑦−𝑧₀−1𝑟𝑦−𝑧₀²+𝜑𝐽𝑦=𝐹𝐽𝑧₀,𝐽𝑦+𝐴𝐽𝑧₀,𝐽𝑦−𝐽𝑧₀+1𝑟𝑦−𝑥₀,𝐽𝑦−𝑧₀−1𝑟𝑦−𝑧₀,𝐽𝑦−𝑧₀+𝜑𝐽𝑦=𝐹𝐽𝑧₀,𝐽𝑦+𝐴𝐽𝑧₀,𝐽𝑦−𝐽𝑧₀+1𝑟𝑦−𝑥₀,𝐽𝑦−𝑧₀+1𝑟𝑧₀−𝑦,𝐽𝑦−𝑧₀+𝜑𝐽𝑦=𝐹𝐽𝑧₀,𝐽𝑦+𝐴𝐽𝑧₀,𝐽𝑦−𝐽𝑧₀+1𝑟𝑦−𝑥₀+𝑧₀−𝑦,𝐽𝑦−𝑧₀+𝜑𝐽𝑦=𝐹𝐽𝑧₀,𝐽𝑦+𝐴𝐽𝑧₀,𝐽𝑦−𝐽𝑧₀+1𝑟𝑧₀−𝑥₀,𝐽𝑦−𝑧₀+𝜑𝐽𝑦. (47) Now, we get 𝐹𝐽𝑧₀,𝐽𝑦+𝐴𝐽𝑧₀,𝐽𝑦−𝐽𝑧₀+1𝑟𝑧₀−𝑥₀,𝐽𝑦−𝑧₀+𝜑𝐽𝑦−𝜑𝐽𝑧₀≥0. (48) Therefore, 𝑧₀∈𝐺(𝑦), and so 𝐺(𝑦) is closed, for all 𝑦∈𝐶.(c) We will show that 𝐺(𝑦) is weakly compact.
Now, we know that 𝐺(𝑦) is closed and subset of 𝐶.
Since 𝐶 is compact. Therefore, 𝐺(𝑦) is compact, and then 𝐺(𝑦) is weakly compact.
By using (a), (b), and (c) and Lemma 22, we can conclude that ⋂_𝑦∈𝐶𝐺(𝑦)≠∅.
Therefore, there exists 𝑧∈𝐶 such that 𝐹++1𝐽𝑧,𝐽𝑦𝐴𝐽𝑧,𝐽𝑦−𝐽𝑧𝑟()𝑧−𝑥,𝐽𝑦−𝑧+𝜑𝐽𝑦−𝜑𝐽𝑧≥0,∀𝑦∈𝐶. (49)

Theorem 25. Let 𝐶 be a nonempty, closed, and convex subset of a uniformly smooth and strictly convex real Banach space 𝐸 such that 𝐽(𝐶) is closed and convex, let us assume that a bifunction 𝐹∶𝐽(𝐶)×𝐽(𝐶)→ℝ satisfies the following conditions (DA1)–(DA4), let 𝐶^∗ be a nonempty, closed, and convex subset of 𝐸^∗, let 𝐴∶𝐶^∗→𝐸 be an 𝛼-inverse strongly skew monotone and let 𝜑∶𝐽(𝐶)→ℝ be a convex and lower semicontinuous. Let𝑟>0 be given real number and 𝑥∈𝐸 be any point. We define a mapping 𝑆_𝑟∶𝐸→𝐶 as follows: 𝑆_𝑟(𝑥)=++1𝑧∈𝐶∶𝐹𝐽𝑧,𝐽𝑦𝐴𝐽𝑧,𝐽𝑦−𝐽𝑧𝑟().𝑧−𝑥,𝐽𝑦−𝑧+𝜑𝐽𝑦−𝜑𝐽𝑧≥0,∀𝑦∈𝐶 (50) Then, the following conclusion hold:(1)𝑆_𝑟 is single-valued; (2)⟨𝑆_𝑟𝑥−𝑆_𝑟𝑦,𝐽(𝑆_𝑟𝑥−𝑆_𝑟𝑦)⟩≤⟨𝑥−𝑦,𝐽(𝑆_𝑟𝑥−𝑆_𝑟𝑦)⟩, ∀𝑥,𝑦∈𝐸; (3)Fix(𝑆_𝑟)=GMEP(𝐹,𝐴,𝜑); (4)𝐽(GMEP(𝐹,𝐴,𝜑)) is closed and convex;

Proof. We will complete this proof by the following four items.
(1) We will show that 𝑆_𝑟 is single-valued.
Indeed, for any 𝑥∈𝐸 and 𝑟>0, let 𝑧₁,𝑧₂∈𝑆_𝑟(𝑥). Then, 𝐹𝐽𝑧₁,𝐽𝑧₂+𝐴𝐽𝑧₁,𝐽𝑧₂−𝐽𝑧₁+1𝑟𝑧₁−𝑥,𝐽𝑧₂−𝑧₁+𝜑𝐽𝑧₂−𝜑𝐽𝑧₁𝐹≥0,𝐽𝑧₂,𝐽𝑧₁+𝐴𝐽𝑧₂,𝐽𝑧₁−𝐽𝑧₂+1𝑟𝑧₂−𝑥,𝐽𝑧₁−𝑧₂+𝜑𝐽𝑧₁−𝜑𝐽𝑧₂≥0. (51) Adding the two inequalities, we have 0≤𝐹𝐽𝑧₁,𝐽𝑧₂+𝐹𝐽𝑧₂,𝐽𝑧₁+𝐴𝐽𝑧₁,𝐽𝑧₂−𝐽𝑧₁+𝐴𝐽𝑧₂,𝐽𝑧₁−𝐽𝑧₂+1𝑟𝑧₁−𝑥,𝐽𝑧₂−𝑧₁+1𝑟𝑧₂−𝑥,𝐽𝑧₁−𝑧₂+𝜑𝐽𝑧₂−𝜑𝐽𝑧₁+𝜑𝐽𝑧₁−𝜑𝐽𝑧₂=𝐹𝐽𝑧₁,𝐽𝑧₂+𝐹𝐽𝑧₂,𝐽𝑧₁+𝐴𝐽𝑧₁,𝐽𝑧₂−𝐽𝑧₁−𝐴𝐽𝑧₂,𝐽𝑧₂−𝐽𝑧₁+1𝑟𝑧₁−𝑥,𝐽𝑧₂−𝑧₁−1𝑟𝑧₂−𝑥,𝐽𝑧₂−𝑧₁=𝐹𝐽𝑧₁,𝐽𝑧₂+𝐹𝐽𝑧₂,𝐽𝑧₁+𝐴𝐽𝑧₁−𝐴𝐽𝑧₂,𝐽𝑧₂−𝐽𝑧₁+1𝑟𝑧₁−𝑥−𝑧₂−𝑥,𝐽𝑧₂−𝑧₁=𝐹𝐽𝑧₁,𝐽𝑧₂+𝐹𝐽𝑧₂,𝐽𝑧₁+𝐴𝐽𝑧₁−𝐴𝐽𝑧₂,𝐽𝑧₂−𝐽𝑧₁+1𝑟𝑧₁−𝑧₂,𝐽𝑧₂−𝑧₁=𝐹𝐽𝑧₁,𝐽𝑧₂+𝐹𝐽𝑧₂,𝐽𝑧₁−𝐴𝐽𝑧₁−𝐴𝐽𝑧₂,𝐽𝑧₁−𝐽𝑧₂+1𝑟𝑧₁−𝑧₂,𝐽𝑧₂−𝑧₁. (52) Therefore, 𝐹𝐽𝑧₁,𝐽𝑧₂+𝐹𝐽𝑧₂,𝐽𝑧₁−𝐴𝐽𝑧₁−𝐴𝐽𝑧₂,𝐽𝑧₁−𝐽𝑧₂+1𝑟𝑧₁−𝑧₂,𝐽𝑧₂−𝑧₁≥0. (53) From the condition (DA2), 𝐴 is an 𝛼-inverse strongly skew monotone, and we have 0≤𝐹𝐽𝑧₁,𝐽𝑧₂+𝐹𝐽𝑧₂,𝐽𝑧₁−𝐴𝐽𝑧₁−𝐴𝐽𝑧₂,𝐽𝑧₁−𝐽𝑧₂+1𝑟𝑧₁−𝑧₂,𝐽𝑧₂−𝑧₁≤−𝛼𝐴𝐽𝑧₁−𝐴𝐽𝑧₂²+1𝑟𝑧₁−𝑧₂,𝐽𝑧₂−𝑧₁≤1𝑟𝑧₁−𝑧₂,𝐽𝑧₂−𝑧₁. (54) Since 𝑟>0, 𝐽 is monotone, 𝐸 is strictly convex, and we obtain 𝑧₁=𝑧₂. (55) This implies that 𝑆_𝑟 is single-valued.
(2) We will show that ⟨𝑆_𝑟𝑥−𝑆_𝑟𝑦,𝐽(𝑆_𝑟𝑥−𝑆_𝑟𝑦)⟩≤⟨𝑥−𝑦,𝐽(𝑆_𝑟𝑥−𝑆_𝑟𝑦)⟩, for all 𝑥,𝑦∈𝐸.
Indeed, for any 𝑥,𝑦∈𝐶 and 𝑟>0, we have 𝐹𝐽𝑆_𝑟𝑥,𝐽𝑆_𝑟𝑦+𝐴𝐽𝑆_𝑟𝑥,𝐽𝑆_𝑟𝑦−𝐽𝑆_𝑟𝑥+1𝑟𝑆_𝑟𝑥−𝑥,𝐽𝑆_𝑟𝑦−𝑆_𝑟𝑥+𝜑𝐽𝑆_𝑟𝑦−𝜑𝐽𝑆_𝑟𝑥𝐹≥0,𝐽𝑆_𝑟𝑦,𝐽𝑆_𝑟𝑥+𝐴𝐽𝑆_𝑟𝑦,𝐽𝑆_𝑟𝑥−𝐽𝑆_𝑟𝑦+1𝑟𝑆_𝑟𝑦−𝑦,𝐽𝑆_𝑟𝑥−𝑆_𝑟𝑦+𝜑𝐽𝑆_𝑟𝑥−𝜑𝐽𝑆_𝑟𝑦≥0. (56) Adding the two inequalities, we have 0≤𝐹𝐽𝑆_𝑟𝑥,𝐽𝑆_𝑟𝑦+𝐹𝐽𝑆_𝑟𝑦,𝐽𝑆_𝑟𝑥+𝐴𝐽𝑆_𝑟𝑥,𝐽𝑆_𝑟𝑦−𝐽𝑆_𝑟𝑥+𝐴𝐽𝑆_𝑟𝑦,𝐽𝑆_𝑟𝑥−𝐽𝑆_𝑟𝑦+1𝑟𝑆_𝑟𝑥−𝑥,𝐽𝑆_𝑟𝑦−𝑆_𝑟𝑥+1𝑟𝑆_𝑟𝑦−𝑦,𝐽𝑆_𝑟𝑥−𝑆_𝑟𝑦+𝜑𝐽𝑆_𝑟𝑦−𝜑𝐽𝑆_𝑟𝑥+𝜑𝐽𝑆_𝑟𝑥−𝜑𝐽𝑆_𝑟𝑦=𝐹𝐽𝑆_𝑟𝑥,𝐽𝑆_𝑟𝑦+𝐹𝐽𝑆_𝑟𝑦,𝐽𝑆_𝑟𝑥+𝐴𝐽𝑆_𝑟𝑥,𝐽𝑆_𝑟𝑦−𝐽𝑆_𝑟𝑥−𝐴𝐽𝑆_𝑟𝑦,𝐽𝑆_𝑟𝑦−𝐽𝑆_𝑟𝑥+1𝑟𝑆_𝑟𝑥−𝑥,𝐽𝑆_𝑟𝑦−𝑆_𝑟𝑥−1𝑟𝑆_𝑟𝑦−𝑦,𝐽𝑆_𝑟𝑦−𝑆_𝑟𝑥=𝐹𝐽𝑆_𝑟𝑥,𝐽𝑆_𝑟𝑦+𝐹𝐽𝑆_𝑟𝑦,𝐽𝑆_𝑟𝑥+𝐴𝐽𝑆_𝑟𝑥−𝐴𝐽𝑆_𝑟𝑦,𝐽𝑆_𝑟𝑦−𝐽𝑆_𝑟𝑥+1𝑟𝑆_𝑟𝑥−𝑥−𝑆_𝑟𝑦−𝑦,𝐽𝑆_𝑟𝑦−𝑆_𝑟𝑥=𝐹𝐽𝑆_𝑟𝑥,𝐽𝑆_𝑟𝑦+𝐹𝐽𝑆_𝑟𝑦,𝐽𝑆_𝑟𝑥+𝐴𝐽𝑆_𝑟𝑥−𝐴𝐽𝑆_𝑟𝑦,𝐽𝑆_𝑟𝑦−𝐽𝑆_𝑟𝑥+1𝑟𝑆_𝑟𝑥−𝑆_𝑟𝑦−𝑥−𝑦,𝐽𝑆_𝑟𝑦−𝑆_𝑟𝑥=𝐹𝐽𝑆_𝑟𝑥,𝐽𝑆_𝑟𝑦+𝐹𝐽𝑆_𝑟𝑦,𝐽𝑆_𝑟𝑥−𝐴𝐽𝑆_𝑟𝑥−𝐴𝐽𝑆_𝑟𝑦,𝐽𝑆_𝑟𝑥−𝐽𝑆_𝑟𝑦+1𝑟𝑆_𝑟𝑥−𝑆_𝑟𝑦−𝑥−𝑦,𝐽𝑆_𝑟𝑦−𝑆_𝑟𝑥. (57) Therefore, 𝐹𝐽𝑆_𝑟𝑥,𝐽𝑆_𝑟𝑦+𝐹𝐽𝑆_𝑟𝑦,𝐽𝑆_𝑟𝑥−𝐴𝐽𝑆_𝑟𝑥−𝐴𝐽𝑆_𝑟𝑦,𝐽𝑆_𝑟𝑥−𝐽𝑆_𝑟𝑦+1𝑟𝑆_𝑟𝑥−𝑆_𝑟𝑦−𝑥−𝑦,𝐽𝑆_𝑟𝑦−𝑆_𝑟𝑥≥0. (58) From the condition (DA2), 𝐴 is an 𝛼-inverse strongly skew monotone, and we have 0≤𝐹𝐽𝑆_𝑟𝑥,𝐽𝑆_𝑟𝑦+𝐹𝐽𝑆_𝑟𝑦,𝐽𝑆_𝑟𝑥−𝐴𝐽𝑆_𝑟𝑥−𝐴𝐽𝑆_𝑟𝑦,𝐽𝑆_𝑟𝑥−𝐽𝑆_𝑟𝑦+1𝑟𝑆_𝑟𝑥−𝑆_𝑟𝑦−𝑥−𝑦,𝐽𝑆_𝑟𝑦−𝑆_𝑟𝑥≤−𝛼𝐴𝐽𝑆_𝑟𝑥−𝐴𝐽𝑆_𝑟𝑦²+1𝑟𝑆_𝑟𝑥−𝑆_𝑟𝑦−𝑥−𝑦,𝐽𝑆_𝑟𝑦−𝑆_𝑟𝑥≤1𝑟𝑆_𝑟𝑥−𝑆_𝑟𝑦−𝑥−𝑦,𝐽𝑆_𝑟𝑦−𝑆_𝑟𝑥1≤−𝑟𝑆_𝑟𝑥−𝑆_𝑟𝑦−𝑥−𝑦,𝐽𝑆_𝑟𝑥−𝑆_𝑟𝑦. (59) Since 𝑟>0, we have 𝑆_𝑟𝑥−𝑆_𝑟𝑦−𝑥−𝑦,𝐽𝑆_𝑟𝑥−𝑆_𝑟𝑦≤0. (60) Therefore, 𝑆_𝑟𝑥−𝑆_𝑟𝑦,𝐽𝑆_𝑟𝑥−𝑆_𝑟𝑦−𝑥−𝑦,𝐽𝑆_𝑟𝑥−𝑆_𝑟𝑦≤0. (61) This implies that 𝑆_𝑟𝑥−𝑆_𝑟𝑦,𝐽𝑆_𝑟𝑥−𝑆_𝑟𝑦≤𝑥−𝑦,𝐽𝑆_𝑟𝑥−𝑆_𝑟𝑦. (62)
(3) We will show that Fix(𝑆_𝑟)=GMEP(𝐹,𝐴,𝜑).
It is easy to see that 𝑧∈Fix𝑆_𝑟⟺𝑧=𝑆_𝑟𝑧++1⟺𝐹𝐽𝑧,𝐽𝑦𝐴𝐽𝑧,𝐽𝑦−𝐽𝑧𝑟()+().𝑧−𝑧,𝐽𝑦−𝑧+𝜑𝐽𝑦−𝜑𝐽𝑧≥0⟺𝐹𝐽𝑧,𝐽𝑦𝐴𝐽𝑧,𝐽𝑦−𝐽𝑧+𝜑𝐽𝑦−𝜑𝐽𝑧≥0⟺𝑧∈GMEP𝐹,𝐴,𝜑 (63) This implies that Fix(𝑆_𝑟)=GMEP(𝐹,𝐴,𝜑).
(4) We will show that 𝐽(GMEP(𝐹,𝐴,𝜑)) is closed and convex.
For each 𝑦∈𝐶, we define the mapping𝐻∶𝐶→2^𝐸as follows: 𝐻𝑦=+().𝑧∈𝐶∶𝐹𝐽𝑧,𝐽𝑦𝐴𝐽𝑧,𝐽𝑦−𝐽𝑧+𝜑𝐽𝑦−𝜑𝐽𝑧≥0 (64) It is easy to see that 𝑦∈𝐻(𝑦), so that 𝐻(𝑦)≠∅.
Next, we will show that 𝐻 is a KKM mapping.
Suppose that there exists a finite subset {𝑧₁,𝑧₂,…,𝑧_𝑚} of 𝐶 and 𝛽_𝑖>0 with ∑^𝑚_𝑖=1𝛽_𝑖=1 such that 𝑧=∑^𝑚_𝑖=1𝛽_𝑖𝑧_𝑖∉𝐻(𝑧_𝑖), for all 𝑖=1,2,3,…,𝑚. Then, 𝐹𝐽𝑧,𝐽𝑧_𝑖+𝐴𝐽𝑧,𝐽𝑧_𝑖−𝐽𝑧+𝜑𝐽𝑧_𝑖−𝜑(𝐽𝑧)<0,𝑖=1,2,3,…,𝑚. (65) It follows from (DA1) and (DA4) that (𝐽𝑧)+⟨𝑧⟩(𝐽𝑧)(𝐽𝑧)≤0=𝐹𝑧,𝐽𝐴𝐽𝑧,𝐽𝑧−𝐽+𝜑−𝜑_𝑚𝑖=1𝛽_𝑖𝐹𝐽𝑧,𝐽𝑧_𝑖+𝐴𝐽𝑧,𝐽𝑧_𝑖−𝐽𝑧+𝜑𝐽𝑧_𝑖−𝜑(𝐽𝑧)<0. (66) which is the contradiction. Hence, 𝐻 is a KKM mapping on 𝐶.
(4.1) Next, we will show that 𝐻(𝑦) is closed, for each 𝑦∈𝐶.
For any 𝑦∈𝐶, let {𝑧_𝑛} be any sequence in 𝐻(𝑦) such that 𝑧_𝑛→𝑧₀ as 𝑛→∞.
Hence, 𝑧_𝑛−𝑥₀→𝑧−𝑥₀ as 𝑛→∞. Next, we will show that 𝑧₀∈𝐻(𝑦). Then, for each 𝑦∈𝐶, we have 𝐹𝐽𝑧_𝑛,𝐽𝑦+𝐴𝐽𝑧_𝑛,𝐽𝑦−𝐽𝑧_𝑛+𝜑𝐽𝑦−𝜑𝐽𝑧_𝑛≥0. (67)
It follows from the assumption (DA3) that 𝐹𝐽𝑧₀,𝐽𝑦+𝐴𝐽𝑧₀,𝐽𝑦−𝐽𝑧₀+𝜑𝐽𝑦−𝜑𝐽𝑧₀≥limsup_𝑛→∞𝐹𝐽𝑧_𝑛,𝐽𝑦+lim_𝑛→∞𝐴𝐽𝑧_𝑛,𝐽𝑦−𝐽𝑧_𝑛+𝜑𝐽𝑦−𝜑𝐽𝑧_𝑛≥0. (68)
This implies that 𝑧₀∈𝐻(𝑦), and hence 𝐻(𝑦) is closed, for each 𝑦∈𝐶.
Therefore, ⋂_𝑦∈𝐶𝐻(𝑦)=𝐽(GMEP(𝐹,𝐴,𝜑)) is closed.
(4.2) Next, we will show that 𝐽(GMEP(𝐹,𝐴,𝜑)) is convex.
Let 𝑧^∗₁,𝑧^∗₂∈𝐽(GMEP(𝐹,𝐴,𝜑)); then, we have 𝑧^∗₁=𝐽𝑧₁∈𝐽(𝐶) and 𝑧^∗₂=𝐽𝑧₂∈𝐽(𝐶), where 𝑧₁,𝑧₂∈𝐶.
For 𝑘,𝑡∈(0,1), let 𝑧^∗=𝑘𝑧^∗₁+(1−𝑘)𝑧^∗₂, and for any 𝑦∈𝐶, we set 𝑥^∗_𝑡=𝑡𝐽𝑦+(1−𝑡)𝑧^∗.
It follows from (DA1) and (DA4) that 0=𝐹𝑥^∗_𝑡,𝑥^∗_𝑡+𝜑𝑥^∗_𝑡−𝜑𝑥^∗_𝑡≤𝐹𝑥^∗_𝑡,𝑡𝐽𝑦+(1−𝑡)𝑧^∗+𝜑𝑡𝐽𝑦+(1−𝑡)𝑧^∗−𝜑𝑥^∗_𝑡≤𝑡𝐹𝑥^∗_𝑡,𝐽𝑦+(1−𝑡)𝐹𝑥^∗_𝑡,𝑧^∗+𝑡𝜑𝑥^∗_𝑡+(1−𝑡)𝜑𝐽𝑦−𝜑𝑥^∗_𝑡≤𝑡𝐹𝑥^∗_𝑡,𝐽𝑦+(1−𝑡)𝜑𝐽𝑦−(1−𝑡)𝜑𝑥^∗_𝑡≤𝐹𝑥^∗_𝑡,𝐽𝑦+𝜑𝐽𝑦−𝜑𝑥^∗_𝑡,0=𝐴𝑥^∗_𝑡,𝑥^∗_𝑡−𝑥^∗_𝑡=𝐴𝑥^∗_𝑡,𝑥^∗_𝑡−𝐽𝑦+𝐽𝑦−𝑥^∗_𝑡=𝐴𝑥^∗_𝑡,𝑥^∗_𝑡−𝐽𝑦+𝐴𝑥^∗_𝑡,𝐽𝑦−𝑥^∗_𝑡=𝐴𝑥^∗_𝑡,𝑥^∗_𝑡−𝐽𝑦−𝐴𝑥^∗_𝑡,𝑥^∗_𝑡≤−𝐽𝑦𝐴𝑥^∗_𝑡,𝑥^∗_𝑡=−𝐽𝑦𝐴𝑥^∗_𝑡,𝑡𝐽𝑦+(1−𝑡)𝑧^∗=−𝐽𝑦𝐴𝑥^∗_𝑡,𝑡𝐽𝑦+(1−𝑡)𝑧^∗−((=𝑡+1−𝑡))𝐽𝑦𝐴𝑥^∗_𝑡,(1−𝑡)𝑧^∗=(−𝐽𝑦1−𝑡)𝐴𝑥^∗_𝑡,𝑧^∗=(−𝐽𝑦𝑡−1)𝐴𝑥^∗_𝑡,𝐽𝑦−𝑧^∗≤𝐴𝑥^∗_𝑡,𝐽𝑦−𝑧^∗. (69)(70) Adding two inequalities (69) and (70), we get 0≤𝐹𝑥^∗_𝑡,𝐽𝑦+𝐴𝑥^∗_𝑡,𝐽𝑦−𝑧^∗+𝜑𝐽𝑦−𝜑𝑥^∗_𝑡. (71) Letting 𝑡↓0. It follows from (DA3) and the hemicontinuous of 𝐴 that 𝐹𝑧^∗,𝐽𝑦+𝐴𝑧^∗,𝐽𝑦−𝑧^∗+𝜑𝐽𝑦−𝜑𝑧^∗≥0,∀𝑦∈𝐶. (72) Hence, 𝑧^∗∈𝐽(GMEP(𝐹,𝐴,𝜑)), and thus 𝐽(GMEP(𝐹,𝐴,𝜑)) is convex.
This completes the proof.

4. Iterative Algorithm for Strong Convergence Theorem

In this section, we modify Halpern-Mann iteration to find the common solution of a generalized mixed equilibrium problem and a fixed point problem in Banach spaces.

Theorem 26. Let 𝐸 be a uniformly smooth and strictly convex real Banach space, let 𝐶 be a nonempty, closed and convex subset of 𝐸 such that 𝐽(𝐶) is closed, and convex of 𝐸^∗, let us assume that a bifunction 𝐹∶𝐽(𝐶)×𝐽(𝐶)→ℝ satisfies the following conditions (DA1)–(DA4), let 𝐶^∗ be a nonempty, closed and convex subset of 𝐸^∗, let 𝐴∶𝐶^∗→𝐸 be an 𝛼-inverse strongly skew monotone, let 𝜑∶𝐽(𝐶)→ℝ be a convex, and lower semicontinuous, and let 𝑇∶𝐶→𝐶 be a closed and -𝜙-nonexpansive mapping. Assume that Ω∶=GMEP(𝐹,𝐴,𝜑)∩Fix(𝑇) is nonempty. Let {𝑥_𝑛} be a sequence generated by 𝑥₀𝑢∈𝐶𝑐ℎ𝑜𝑠𝑒𝑛𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑦,_𝑛𝐹∈𝐶𝑠𝑢𝑐ℎ𝑡ℎ𝑎𝑡𝐽𝑢_𝑛,𝐽𝑦+𝐴𝐽𝑢_𝑛,𝐽𝑦−𝐽𝑢_𝑛+1𝑟_𝑛𝑢_𝑛−𝑥_𝑛,𝐽𝑦−𝑢_𝑛+𝜑𝐽𝑦−𝜑𝐽𝑢_𝑛𝑥≥0,∀𝑦∈𝐶,_𝑛+1=𝛼_𝑛𝑥₀+1−𝛼_𝑛𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝑇𝑢_𝑛,∀𝑛∈ℕ, (73) where 𝐽 is the duality mapping on 𝐸 and {𝛼_𝑛} is a sequence in [0,1], {𝛽_𝑛}⊂[𝑎,𝑏], for some 0<𝑎<𝑏<1 and {𝑟_𝑛}⊂[𝑐,∞), for some 𝑐>0 such that ^∞_𝑛=0𝛼_𝑛<∞,^∞_𝑛=0𝛽_𝑛<∞,liminf_𝑛→∞𝑟_𝑛>0. (74) Then, the sequence {𝑅_Ω(𝑥_𝑛)} converges strongly to some point in Ω, where 𝑅_Ω is the sunny generalized nonexpansive retraction from 𝐸 onto Ω.

Proof. We will complete this proof by the following three steps.
Step 1. We will show that the sequences{𝑥_𝑛}and{𝑢_𝑛}are bounded.
Let 𝑢_𝑛=𝑆_𝑟𝑛𝑥_𝑛 and 𝑦_𝑛=𝛽_𝑛𝑥_𝑛+(1−𝛽_𝑛)𝑇𝑢_𝑛, for any 𝑛≥1. Then, 𝑥_𝑛+1=𝛼_𝑛𝑥₀+1−𝛼_𝑛𝑦_𝑛. (75) From Theorem 25 and Lemma 23, we know that GMEP(𝐹,𝐴,𝜑) and Fix(𝑇) are closed, and convex subset of 𝐸. Therefore, Ω is nonempty, closed and convex subset of 𝐸.
For any 𝑝∈Ω, 𝑇 is a closed and -𝜙-nonexpansive mapping, we compute 𝜙𝑦_𝑛,𝑝≤𝜙𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝑇𝑢_𝑛=𝛽,𝑝_𝑛𝑥_𝑛+1−𝛽_𝑛𝑇𝑢_𝑛²−2𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝑇𝑢_𝑛,𝐽𝑝+𝑝²=𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝑇𝑢_𝑛²−2𝛽_𝑛𝑥_𝑛,𝐽𝑝−21−𝛽_𝑛𝑇𝑢_𝑛+𝛽,𝐽𝑝_𝑛+1−𝛽_𝑛𝑝²≤𝛽_𝑛𝑥_𝑛²+1−𝛽_𝑛𝑇𝑢_𝑛²−2𝛽_𝑛𝑥_𝑛,𝐽𝑝−21−𝛽_𝑛𝑇𝑢_𝑛+𝛽,𝐽𝑝_𝑛+1−𝛽_𝑛𝑝²=𝛽_𝑛𝑥_𝑛²−2𝑥_𝑛,𝐽𝑝+𝑝²+1−𝛽_𝑛𝑇𝑢_𝑛²−2𝑇𝑢_𝑛,𝐽𝑝+𝑝²=𝛽_𝑛𝜙𝑥_𝑛,𝑝+1−𝛽_𝑛𝜙𝑇𝑢_𝑛,𝑝≤𝛽_𝑛𝜙𝑥_𝑛,𝑝+1−𝛽_𝑛𝜙𝑢_𝑛,𝑝=𝛽_𝑛𝜙𝑥_𝑛,𝑝+1−𝛽_𝑛𝜙𝑆_𝑟𝑛𝑥_𝑛,𝑝≤𝛽_𝑛𝜙𝑥_𝑛,𝑝+1−𝛽_𝑛𝜙𝑥_𝑛𝑥,𝑝=𝜙_𝑛,𝑝, (76) and we have 𝜙𝑥_𝑛+1,𝑝≤𝜙𝛼_𝑛𝑥₀+1−𝛼_𝑛𝑦_𝑛=𝛼,𝑝_𝑛𝑥₀+1−𝛼_𝑛𝑦_𝑛²−2𝛼_𝑛𝑥₀+1−𝛼_𝑛𝑦_𝑛,𝐽𝑝+𝑝²=𝛼_𝑛𝑥₀+1−𝛼_𝑛𝑦_𝑛²−2𝛼_𝑛𝑥₀,𝐽𝑝−21−𝛼_𝑛𝑦_𝑛+𝛼,𝐽𝑝_𝑛+1−𝛼_𝑛𝑝²≤𝛼_𝑛𝑥₀²+(1−𝛼)𝑦_𝑛²−2𝛼_𝑛𝑥₀,𝐽𝑝−21−𝛼_𝑛𝑦_𝑛+𝛼,𝐽𝑝_𝑛+1−𝛼_𝑛𝑝²=𝛼_𝑛𝑥₀²−2𝑥₀,𝐽𝑝+𝑝²+1−𝛼_𝑛𝑦_𝑛²−2𝑦_𝑛,𝐽𝑝+𝑝²=𝛼_𝑛𝜙𝑥₀,𝑝+1−𝛼_𝑛𝜙𝑦_𝑛,𝑝≤𝛼_𝑛𝜙𝑥₀,𝑝+1−𝛼_𝑛𝜙𝑥_𝑛,𝑝≤𝛼_𝑛𝜙𝑥₀,𝑝+𝜙𝑥_𝑛,𝑝. (77) By virtue of ∑^∞_𝑛=0𝛼_𝑛<∞, it follows from Lemma 16 that lim_𝑛→∞𝜙(𝑥_𝑛,𝑝) exists.
Therefore, {𝜙(𝑥_𝑛,𝑝)} is bounded, and so {𝑥_𝑛} is bounded. Hence, {𝑢_𝑛} and {𝑦_𝑛} are also bounded.
Step 2. We will show that𝑅_Ω(𝑥_𝑛)is bounded.
Let 𝑧_𝑛=𝑅_Ω(𝑥_𝑛)and𝑝∈Ω. Then, 𝑧_𝑛∈Ω.
It follows from Lemma 12(2) that 𝜙𝑥_𝑛,𝑧_𝑛=𝜙𝑥_𝑛,𝑅_Ω𝑥_𝑛≤𝜙𝑥_𝑛,𝑝−𝜙𝑅_Ω𝑥_𝑛𝑥,𝑝≤𝜙_𝑛,𝑝. (78) Since {𝑥_𝑛} is bounded. Therefore, {𝑧_𝑛} is bounded.
Hence, 𝑅_Ω(𝑥_𝑛) is bounded.
Step 3. We will show that{𝑅_Ω(𝑥_𝑛)}converges strongly to some point inΩ.
From (78), we have𝜙(𝑥_𝑛,𝑧_𝑛)≤𝜙(𝑥_𝑛,𝑝). Replacing 𝑥_𝑛 by 𝑥₀, we get 𝜙(𝑥₀,𝑧_𝑛)≤𝜙(𝑥₀,𝑝).
Therefore, {𝜙(𝑥₀,𝑧_𝑛)} is bounded.
Now, we know that𝜙(𝑥_𝑛+1,𝑧_𝑛)≤𝛼_𝑛𝜙(𝑥₀,𝑧_𝑛)+𝜙(𝑥_𝑛,𝑧_𝑛). By Lemma 12(2), we get 𝜙𝑥_𝑛+1,𝑧_𝑛+1=𝜙𝑥_𝑛+1,𝑅_Ω𝑥_𝑛+1≤𝜙𝑥_𝑛+1,𝑧_𝑛−𝜙𝑅_Ω𝑥_𝑛+1,𝑧_𝑛≤𝜙𝑥_𝑛+1,𝑧_𝑛≤𝛼_𝑛𝜙𝑥₀,𝑧_𝑛+𝜙𝑥_𝑛,𝑧_𝑛. (79)
Since {𝜙(𝑥₀,𝑧_𝑛)} is bounded. There exists 𝑀>0 such that |𝜙(𝑥₀,𝑧_𝑛)|≤𝑀.
By the assumption ∑^∞_𝑛=0𝛼_𝑛<∞, we have ^∞_𝑛=0𝜙𝑥₀,𝑧_𝑛≤𝑀^∞_𝑛=0𝛼_𝑛<∞. (80) It follows from Lemma 16 that lim_𝑛→∞𝜙(𝑥_𝑛,𝑧_𝑛) exists.
For any 𝑚∈ℕ, we get 𝜙𝑥_𝑛+𝑚,𝑝≤𝜙𝑥_𝑛,𝑝+^𝑚−1_𝑗=0𝛼_𝑛+𝑗𝜙𝑥₀,𝜙𝑥,𝑝_𝑛+𝑚,𝑧_𝑛≤𝜙𝑥_𝑛,𝑧_𝑛+^𝑚−1_𝑗=0𝛼_𝑛+𝑗𝜙𝑥₀,𝑧_𝑛. (81) Since 𝑧_𝑛+𝑚=𝑅_Ω(𝑥_𝑛+𝑚) and from Lemma 12(2), we have 𝜙𝑥_𝑛+𝑚,𝑧_𝑛+𝑚+𝜙𝑧_𝑛+𝑚,𝑧_𝑛≤𝜙𝑥_𝑛+𝑚,𝑧_𝑛≤𝜙𝑥_𝑛,𝑧_𝑛+^𝑚−1_𝑗=0𝛼_𝑛+𝑗𝜙𝑥₀,𝑝. (82) Hence, 𝜙𝑧_𝑛+𝑚,𝑧_𝑛≤𝜙𝑥_𝑛,𝑧_𝑛−𝜙𝑥_𝑛+𝑚,𝑧_𝑛+𝑚+^𝑚−1_𝑗=0𝛼_𝑛+𝑗𝜙𝑥₀,𝑝. (83) We set 𝑟=sup{‖𝑧_𝑛‖∶𝑛∈ℕ}. From Lemma 20, it follows that there exists a strictly increasing, continuous, and convex function ℎ∶[0,2𝑟]→ℝ such that ℎ(0)=0 and ℎ𝑧_𝑛−𝑧_𝑛+𝑚≤𝜙𝑧_𝑛+𝑚,𝑧_𝑛≤𝜙𝑥_𝑛,𝑧_𝑛−𝜙𝑥_𝑛+𝑚,𝑧_𝑛+𝑚+^𝑚−1_𝑗=0𝛼_𝑛+𝑗𝜙𝑥₀,𝑧_𝑛. (84) Since {𝜙(𝑥_𝑛,𝑧_𝑛)} is convergent sequence, {𝜙(𝑥₀,𝑧_𝑛)} is bounded, and ∑^∞_𝑛=0𝛼_𝑛<∞, then, we obtain lim_𝑛→∞𝑧_𝑛−𝑧_𝑛+𝑚=0,∀𝑚∈ℕ. (85) This implies that {𝑧_𝑛} is a Cauchy sequence.
Note that Ω is closed.
Thus, there exists 𝑝∈Ω such that 𝑧_𝑛→𝑝.
Therefore, the sequence 𝑅_Ω(𝑥_𝑛) converges strongly to some point 𝑝∈Ω.
This completes the proof.

If we set 𝐴≡0 in Theorem 26, then Theorem 26 reduces to the following corollary which extends and improves the following result of Saewan et al. [14].

Corollary 27. Let 𝐸 be a uniformly smooth and strictly convex real Banach space, let 𝐶 be a nonempty, closed, and convex subset of 𝐸 such that 𝐽(𝐶) is closed and convex of 𝐸^∗, let us assume that a bifunction 𝐹∶𝐽(𝐶)×𝐽(𝐶)→ℝ satisfies the following conditions (DA1)–(DA4), let 𝐶^∗ be a nonempty, closed, and convex subset of 𝐸^∗, let 𝜑∶𝐽(𝐶)→ℝ be a convex and lower semicontinuous, and let 𝑇∶𝐶→𝐶 be a closed and -𝜙-nonexpansive mapping. Assume that Ω∶=MEP(𝐹,𝜑)∩Fix(𝑇) is nonempty. Let {𝑥_𝑛} be a sequence generated by 𝑥₀𝑢∈𝐶𝑐ℎ𝑜𝑠𝑒𝑛𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑦,_𝑛𝐹∈𝐶𝑠𝑢𝑐ℎ𝑡ℎ𝑎𝑡𝐽𝑢_𝑛+1,𝐽𝑦𝑟_𝑛𝑢_𝑛−𝑥_𝑛,𝐽𝑦−𝑢_𝑛+𝜑𝐽𝑦−𝜑𝐽𝑢_𝑛𝑥≥0,∀𝑦∈𝐶,_𝑛+1=𝛼_𝑛𝑥₀+1−𝛼_𝑛𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝑇𝑢_𝑛,∀𝑛∈ℕ, (86) where 𝐽 is the duality mapping on 𝐸 and {𝛼_𝑛} is a sequence in [0,1], {𝛽_𝑛}⊂[𝑎,𝑏], for some 0<𝑎<𝑏<1 and {𝑟_𝑛}⊂[𝑐,∞), for some 𝑐>0 such that ^∞_𝑛=0𝛼_𝑛<∞,^∞_𝑛=0𝛽_𝑛<∞,liminf_𝑛→∞𝑟_𝑛>0. (87) Then, the sequence {𝑅_Ω(𝑥_𝑛)} converges strongly to some point in Ω, where 𝑅_Ω is the sunny generalized nonexpansive retraction from 𝐸 onto Ω.

If we set 𝐴≡0 and 𝜑≡0 in Theorem 26, then Theorem 26 reduces to the following corollary which extends and improves the following result of Chen et al. [13].

Corollary 28. Let 𝐸 be a uniformly smooth and strictly convex real Banach space, let 𝐶 be a nonempty, closed, and convex subset of 𝐸 such that 𝐽(𝐶) is closed and convex of 𝐸^∗, let us assume that a bifunction 𝐹∶𝐽(𝐶)×𝐽(𝐶)→ℝ satisfies the following conditions (DA1)–(DA4), let 𝐶^∗ be a nonempty, closed, and convex subset of 𝐸^∗, and let 𝑇∶𝐶→𝐶 be a closed and -𝜙-nonexpansive mapping. Assume that Ω∶=EP(𝐹)∩Fix(𝑇) is nonempty. Let {𝑥_𝑛} be a sequence generated by 𝑥₀𝑢∈𝐶𝑐ℎ𝑜𝑠𝑒𝑛𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑦,_𝑛𝐹∈𝐶𝑠𝑢𝑐ℎ𝑡ℎ𝑎𝑡𝐽𝑢_𝑛+1,𝐽𝑦𝑟_𝑛𝑢_𝑛−𝑥_𝑛,𝐽𝑦−𝑢_𝑛𝑥≥0,∀𝑦∈𝐶,_𝑛+1=𝛼_𝑛𝑥₀+1−𝛼_𝑛𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝑇𝑢_𝑛,∀𝑛∈ℕ, (88) where 𝐽 is the duality mapping on 𝐸 and {𝛼_𝑛} is a sequence in [0,1], {𝛽_𝑛}⊂[𝑎,𝑏], for some 0<𝑎<𝑏<1 and {𝑟_𝑛}⊂[𝑐,∞), for some 𝑐>0 such that ^∞_𝑛=0𝛼_𝑛<∞,^∞_𝑛=0𝛽_𝑛<∞,liminf_𝑛→∞𝑟_𝑛>0. (89) Then, the sequence {𝑅_Ω(𝑥_𝑛)} converges strongly to some point in Ω, where 𝑅_Ω is the sunny generalized nonexpansive retraction from 𝐸 onto Ω.

If we set 𝐴≡0, 𝜑≡0, and 𝑇≡𝐼 (identity mapping) in Theorem 26, then Theorem 26 reduces to the following corollary which extends and improves the following result of Chen et al. [13].

Corollary 29. Let 𝐸 be a uniformly smooth and strictly convex real Banach space, let 𝐶 be a nonempty, closed, and convex subset of 𝐸 such that 𝐽(𝐶) is closed and convex of 𝐸^∗, let us assume that a bifunction 𝐹∶𝐽(𝐶)×𝐽(𝐶)→ℝ satisfies the following conditions (DA1)–(DA4), and let 𝐶^∗ be a nonempty, closed, and convex subset of 𝐸^∗. Assume that 𝐸𝑃(𝐹) is nonempty. Let {𝑥_𝑛} be a sequence generated by 𝑥₀𝑢∈𝐶𝑐ℎ𝑜𝑠𝑒𝑛𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑟𝑦,_𝑛𝐹∈𝐶𝑠𝑢𝑐ℎ𝑡ℎ𝑎𝑡𝐽𝑢_𝑛+1,𝐽𝑦𝑟_𝑛𝑢_𝑛−𝑥_𝑛,𝐽𝑦−𝑢_𝑛𝑥≥0,∀𝑦∈𝐶,_𝑛+1=𝛼_𝑛𝑥₀+1−𝛼_𝑛𝛽_𝑛𝑥_𝑛+1−𝛽_𝑛𝑢_𝑛,∀𝑛∈ℕ, (90) where 𝐽 is the duality mapping on 𝐸 and {𝛼_𝑛} is a sequence in [0,1], {𝛽_𝑛}⊂[𝑎,𝑏], for some 0<𝑎<𝑏<1 and {𝑟_𝑛}⊂[𝑐,∞), for some 𝑐>0 such that ^∞_𝑛=0𝛼_𝑛<∞,^∞_𝑛=0𝛽_𝑛<∞,liminf_𝑛→∞𝑟_𝑛>0. (91) Then, the sequence {𝑅_EP(𝐹)(𝑥_𝑛)} converges strongly to some point in Ω, where 𝑅_EP(𝐹) is the sunny generalized nonexpansive retraction from 𝐸 onto EP(𝐹).

Acknowledgments

The first author would like to thank the Bansomdejchaopraya Rajabhat University for financial support. The authors would like to thank the Higher Education Research Promotion and National Research University Project of Thailand’s Office of the Higher Education Commission for financial support (Under NRU-CSEC Project no. 56000508).