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Generalized mixed equilibrium problems and quasi- ϕ-asymptotically nonexpansive multivalued mappings in Banach spaces

  • Bashir AliEmail author
  • Lawal Umar
  • M. H. Harbau
Open Access
Original Research
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Abstract

In this paper, we introduce two iterative algorithms for finding a common element of the set of fixed points of a quasi- ϕ-asymptotically nonexpansive multivalued mapping and the sets of solutions of generalized mixed equilibrium problem in Banach space. Then, we prove strong and weak convergence of the sequences to element in the mentioned set. Our results generalize and improve recent results announced by many authors.

Keywords

Fixed point Generalized mixed equilibrium problem Quasi- Φ -asymptotically nonexpansive multivalued mappings Banach space 

Introduction

Let E be a real Banach space with norm ∥.∥,E∗ be the dual space of E, and C be a nonempty closed convex subset of E. Let f be a bifunction from C×C to \(\mathbb {R}\), where \(\mathbb {R}\) is the set of real numbers. The equilibrium problem is to find \(\hat {x} \in C\) such that
$$\begin{array}{@{}rcl@{}} f(\hat{x},y) \geq 0,\forall y \in C. \end{array} $$
(1)

This problem was first studied by Blum and Oettli [1]. The set of solutions of equilibrium problem (1.1) is denoted by EP(f) that is EP(f) = \(\lbrace \hat {x}\in C : f(\hat {x},y)\geq 0, \forall y \in C \rbrace \). Let A:C→E∗ be a nonlinear mapping. The variational inequality problem with respect to A and C is to find u∈C such that 〈Au,v−u〉≥0 for all v∈C. The set of solutions of variational inequality problem with respect to C and A is denoted by VI(C,A). Setting f(x,y)=〈Ax,y−x〉 for all x,y∈C, then \(\hat {x}\in EP(f)\) if and only if \(\langle A\hat {x}, y-\hat {x} \rangle \geq 0,\) for all y∈C, i.e., \(\hat {x}\) is a solution of the variational inequality with respect to A and C. Let \(\varphi : C \longrightarrow \mathbb {R}\cup \lbrace \infty \rbrace \) be proper, convex, and lower semi-continuous, then the minimization problem of φ is to find x∈C such that φ(x)≤φ(y) ∀y∈C.

The generalized equilibrium problem is to find \(\hat {x} \in C\) such that
$$\begin{array}{@{}rcl@{}} f(\hat{x},y) + \langle A \hat{x}, y-\hat{x} \rangle \geq 0, ~\forall y\in C. \end{array} $$
(2)
The set of solutions of (2) is denoted by
$$\begin{array}{@{}rcl@{}} GEP(f,A) = \left\{ \hat{x}\in C : f(\hat{x},y) + \left\langle A \hat{x},y-\hat{x} \right\rangle \geq 0, ~\forall y\in C\right\}. \end{array} $$
In this paper, we are interested in solving equilibrium problem with respect to f given by
$$\begin{array}{@{}rcl@{}} f(x,y) = \sum_{i=1}^{k} f_{i}(x,y), \forall x,y\in C, \end{array} $$
(3)

where \(f_{i} : C \times C \longrightarrow \mathbb {R}\) are bifunctions for i = 1,2,3,...k, satisfying the following conditions (A1)–(A4) below;(A1) fi(x,x)=0, for all x∈C, for i=1,2,3...k(A2) fi is monotone, i.e., fi(x,y)+fi(y,x)≤0, for each i∈{1,2,3,...,k} and x,y∈C(A3) for all x,y,z∈C, we have \(\underset {t\to \infty }{\limsup } f_{i}(tz + (1-t)x,y)\leq f_{i} (x, y)\)(A4) for all x∈C,fi(x,.) is convex and lower semi-continuous ∀i∈{1,2,3,...,k}.

The mixed equilibrium problem is to find \(\widehat {x}\in C\) such that
$$\begin{array}{@{}rcl@{}} \sum_{i=1}^{k}f_{i} \left(\hat{x},y\right) + \varphi (y)- \varphi \left(\hat{x} \right) \geq 0, \forall y\in C. \end{array} $$
(4)
The set of solution of (4) is denoted by
$$\begin{array}{@{}rcl@{}} GMEP \left(f_{i},\varphi\right) = \left\{ \hat{x} \in C : \sum_{i=1}^{k} f_{i} \left(\hat{x}, y\right) + \varphi (y)- \varphi \left(\hat{x} \right) \geq 0, \forall y \in C\right\}. \end{array} $$
The generalized mixed equilibrium problem is to find \(\widehat {x}\in C\) such that
$$\begin{array}{@{}rcl@{}} \sum_{i=1}^{k}f_{i} (\hat{x},y) + \langle A \hat{x}, y-\hat{x}\rangle + \varphi (y)- \varphi (\hat{x}) \geq 0, \forall y\in C. \end{array} $$
(5)
The set of solution of (5) is denoted by
$$\begin{array}{@{}rcl@{}} \begin{aligned} GMEP (f_{i},A,\varphi)& = \left\lbrace \hat{x} \in C : \sum_{i=1}^{k} f_{i} (\hat{x}, y) + \langle A \hat{x}, y-\hat{x} \rangle + \varphi (y)- \varphi (\hat{x}) \geq 0, \forall y \in C\right\rbrace. \end{aligned} \end{array} $$

The generalized mixed equilibrium problems are problems that arises in various applications such as in economics, mathematical physics, engineering, and other fields. Moreover, equilibrium problems are closely related with other general problems in nonlinear analysis such as fixed point, game theory, variational inequality, and optimization problems. Some methods have been proposed to solve the equilibrium problem in Hilbert spaces, see for example [2, 3, 4] and references contained therein.

In 2007, Tada and Takahashi [5, 6] and Takahashi and Takahashi [7] proved weak and strong convergence theorems for finding a common element of the set of an equilibrium problem and the set of fixed points of a nonexpansive mapping in a Hilbert space. In 2009, Takahashi and Zembayashi [8] introduced two iterative sequences for finding a common element of the set of fixed points of a relatively nonexpansive mapping and the set of solutions of an equilibrium problem in Banach space as follows :
$$\begin{array}{@{}rcl@{}} \left\{\begin{array}{ll} x_{0} = x\in C \\ y_{n} = J^{-1} \left(\alpha_{n}Jx_{n} + (1-\alpha_{n})\right.\!\!JSx_{n},\\ u_{n} \in C \; \text{such that} \; f (u_{n}, y) + \frac{1}{r_{n}} \left\langle y - u_{n}, Ju_{n} - Jy_{n} \right\rangle \geq 0, \forall y\in C\\ H_{n} = \lbrace z\in C : \phi (z, u_{n}) \leq \phi (z, x_{n})\rbrace \\ W_{n} = \lbrace z\in C : \left\langle x_{n} - z, Jx - Jx_{n} \right\rangle \geq 0\rbrace \\ x_{n+1} = \Pi_{H_{n} \cap W_{n}}x, n\geq 0, \end{array}\right. \end{array} $$
(6)
$$\begin{array}{@{}rcl@{}} \left\{\begin{array}{ll} x_{n} \in C \; \text{such that}\; f (x_{n}, y) + \frac{1}{r_{n}} \left\langle y - x_{n}, Jx_{n} - Ju_{n} \right\rangle \geq 0, \forall y\in C\\ u_{n+1} = J^{-1} \left(\alpha_{n}Jx_{n} + (1-\alpha_{n})JSx_{n},~~\forall n\geq 0,\right. \end{array}\right. \end{array} $$
(7)

where J is the normalized duality mapping on E, {αn}⊂[0,1] satisfies \(\underset {n\to \infty }{\liminf }\alpha _{n}(1-\alpha _{n}) > 0\) and {rn}⊂[a,∞] for some a>0. They proved strong convergence of the scheme (6) to a common element of the set of fixed points of relatively nonexpansive mapping and the set of solution of an equilibrium problem in a Banach space. Moreover, they proved weak convergence using scheme (7). In 2012, Chang et al. [9] considered the class of uniformly quasi- ϕ-asymptotically nonexpansive nonself mappings and studied in a uniformly convex and uniformly smooth real Banach space. In 2014, Deng et al. [10] proved strong convergence theorems of the hybrid algorithm for common fixed point problem of finite family of asymptotically nonexpansive mappings and the set of solution of mixed equilibrium problem in uniformly smooth and uniformly convex Banach spaces. In 2016, Ezeora [11] proved strong convergence theorems for a common element of the set of solution of generalized mixed equilibrium problem and the set of common fixed points of a finite family of multivalued strictly pseudocontractive mappings in real Hilbert spaces.

In this paper, motivated and inspired by the results mentioned above, we prove strong and weak convergence theorems for finding a common element of the set of fixed point of a quasi- ϕ-asymptotically nonexpansive multivalued mapping and the sets of solutions of generalized mixed equilibrium problem in Banach space. Our results generalized and improve recent results announced by many authors.

Preliminaries

Throughout this paper, we denoted by \(\mathbb {N}\) and \(\mathbb {R}\) the sets of positive integer and real numbers, respectively. Let E be a Banach space and E∗ be the dual of E; we denote the strong convergence and the weak convergence of a sequence {xn} to x in E by xn→x and \(x_{n}\rightharpoonup x\) respectively. We also denote the weak ∗ convergence of a sequence \(\lbrace x_{n}^{*}\rbrace \) to x∗ in E∗ by \(x_{n}^{*} \rightharpoonup x^{*}\); for all x∈E and x∗∈E∗, we denote the value of x∗ at x by 〈x,x∗〉, which is called duality pairing. The normalized duality mapping J on E is defined by
$$J (x) = \left\lbrace x^{*} \in E^{*}: \left\langle x, x^{*}\right\rangle =\left\Vert x \right\Vert^{2} =\left\Vert x^{*}\right\Vert^{2} \right\rbrace.$$
for every x∈E. A Banach space E is said to be strictly convex if \(\frac {\parallel x + y \parallel }{2} < 1\) for all x,y∈E with ∥x∥=∥y∥=1 and x≠y. The space E is also said to be uniformly convex if for each ε∈(0,2], there exists δ>0 such that \(\frac {\parallel x + y \parallel }{2} \leq 1 - \delta \) for all x,y∈E with ∥x∥=∥y∥=1 and ∥x−y∥≥ε. A Banach space is said to have Kadec-Klee property, if for \(x_{n} \rightharpoonup x\) and ∥xn∥→∥x∥ imply xn→x. Every Hilbert space and uniformly convex Banach space has Kadec-Klee property. The space E is said to be smooth if the \(\underset {n\to 0}{\lim }\frac {\parallel x + ty \parallel - \parallel x \parallel }{t}\) exists for all x,y∈S(E)={z∈E:∥z∥=1}. It is also said to be uniformly smooth if the limit exists uniformly in x,y∈S(E). It is known that if E is smooth, strictly convex, and reflexive, then the duality mapping J is single-valued, one-to-one, and onto. The duality mapping J is said to be weakly sequentially continuous if for any sequence {xn} in E, \(x_{n} \rightharpoonup x\) implies \(Jx_{n} \rightharpoonup Jx\), see [12].

Let E be a smooth, strictly convex, and reflexive Banach space and C be a nonempty closed convex subset of E. Throughout this paper, we denote by ϕ the function defined by ϕ(y,x)=∥y∥2−2〈y,Jx〉+∥x∥2,∀x,y∈E. It is clear from the definition of the function ϕ that for all x,y,z∈E, we have (∥y∥−∥x∥)2≤ϕ(x,y)≤(∥y∥+∥x∥)2 and ϕ(x,y)=ϕ(x,z)+ϕ(z,y)+〈x−z,Jz−Jy〉. Following Albert [13], the generalized projection ΠC from E onto C is defined by ΠC(x) = argmin ϕ(y,x),∀x∈E and y∈C. If E is a Hilbert space H, then ϕ(y,x)=∥y−x∥2 and ΠC become the metric projection of H onto C.

The following lemmas for generalized projections are well known.

Lemma 1

see [13] Let C be a nonempty closed convex subset of a smooth, strictly convex, and reflexive Banach space E. Then, ϕ(x,ΠCy)+ϕ(ΠCy,y)≤ϕ(x,y),∀x∈C and y∈E.

Lemma 2

see [13, 14] Let C be a nonempty closed convex subset of a smooth, strictly convex, and reflexive Banach space. Let x∈E and z∈C. Then, z=ΠCx⇔〈y−z,Jx−Jy〉≤0,∀y∈C.

A mapping T:C→C is called nonexpansive if ∥Tx−Ty∥≤∥x−y∥,∀x,y∈C. We denote by F(T) the set of fixed points of T see [15]. A point p∈C is said to be an asymptotic fixed point of T if there exists a sequence {xn} in C which converges weakly to p and lim n→∞∥xn−Txn∥=0. We denote the set of all asymptotic fixed points of T by \(\hat {F}(T).\) Following Matsushita and Takahashi [16, 17, 18], a mapping T of C into itself is said to be relatively nonexpansive if the following conditions are satisfied:(i) F(T)≠∅(ii) ϕ(p,Tx)≤ϕ(p,x),∀x∈C,p∈F(T) and (iii) \(F(T) = \hat {F}(T)\).Let C be a nonempty closed convex subset of a Banach space E. Let \(\hat {C}B(C)\) be the families of nonempty, closed, and bounded subsets of C

Definition 1

A multivalued mapping \(T : C \longrightarrow \hat {C}B(C)\) is said to be relatively nonexpansive if(i) F(T)≠∅(ii) ϕ(p,ω)≤ϕ(p,x),∀x∈C,ω∈Tx,p∈F(T) and (iii) \(F(T) = \hat {F}(T)\).

A multivalued mapping \(T : C\longrightarrow \hat {C}B(C)\) is said to be closed if for any sequence {xn}⊂C with xn→x and ωn∈T(xn) with ωn→y then y∈Tx.

Definition 2

A multivalued mapping \(T : C \longrightarrow \hat {C}B(C)\) is said to be quasi- ϕ- nonexpansive if(i) F(T)≠∅ and (ii) ϕ(p,ω)≤ϕ(p,x),∀x∈C,ω∈Tx,p∈F(T).

A multivalued mapping \(T : C\longrightarrow \hat {C}B(C)\) is a said be quasi- ϕ-asymptotically nonexpansive if(i) F(T)≠∅(ii) There exists a real sequence {kn}⊂[1,∞) with kn→1 such that ϕ(p,ωn)≤knϕ(p,x),∀n≥1,x∈C,ωn∈Tnx,p∈F(T).

Lemma 3

see [14] Let E be a smooth and uniformly convex Banach space and let {xn} and {yn} be sequences in E such that either {xn} or {yn} is bounded. If \(\underset {n\to \infty }{\lim } \phi (x_{n}, y_{n}) = 0,\) then \(\underset {n\to \infty }{\lim } \parallel x_{n} - y_{n} \parallel = 0\).

Lemma 4

see [19] Let E be a uniformly convex Banach space. For arbitrary r>0, let Br(0):={∥x∈E:∥x∥≤r}. Then, for any given sequence \(\left \lbrace x_{n} \right \rbrace ^{\infty }_{n=1} \subset B_{r} (0)\)and for any given sequence \(\lbrace \lambda \rbrace ^{\infty }_{n=1}\) of positive numbers such that \(\sum _{n=1}^{\infty } \lambda _{n} = 1\), there exists a continuous strictly increasing convex function
$$g : [0, 2r]\longrightarrow \mathbb{R}, g(0) = 0 $$
such that for any positive integers i,j with i<j, the following inequality holds:
$$\left\Vert \sum_{n=1}^{\infty} \lambda_{n}x_{n}\right\Vert^{2} \leq \sum_{n=1}^{\infty} \lambda_{n} \left\Vert x_{n}\right\Vert^{2} - \lambda_{i}\lambda_{j}g \left(\left\Vert x_{i} - x_{j}\right\Vert\right). $$

Lemma 5

see [20] Let E be a smooth and uniformly convex Banach space and let r>0. Then, there exists a strictly increasing continuous and convex function \(g : [ 0, 2r ]\longrightarrow \mathbb {R}\) such that g(0)=0 and g1(∥x−y∥)≤ϕ(x,y), for all x,y∈Br.

Lemma 6

see [21] Let {an},{bn}, and {cn} be sequences of nonnegative real numbers satisfying an+1≤(1+cn)an+bn, for all \(n\in \mathbb {N},\) where \(\sum _{n=1}^{\infty } b_{n} < \infty \) and \(\sum _{n=1}^{\infty } c_{n} <\infty \). Then,(i) \(\underset {n\to \infty }{\lim } a_{n}\) exists.(ii)if \(\underset {n\to \infty }{\liminf } a_{n} = 0,\) then \(\underset {n\to \infty }{\lim } a_{n} = 0.\)

Lemma 7

see [22] Let E be a smooth, strictly convex, and reflexive Banach space and C be a nonempty closed convex subset of E. Let B:C→E∗ be a continuous and monotone mapping, \(\zeta : C \longrightarrow \mathbb {R}\) be a lower semi-continuous and convex function, and \(h : C \times C \longrightarrow \mathbb {R}\) be a bifunction satisfying the conditions (A1)−(A4). Let r>0 be any given number and u∈E be any given point. Then, the following hold:(1) There exists z∈C such that
$$h(z, v) + \zeta (v) - \zeta (z) + \langle v - z, Bz \rangle + \frac{1}{r}\left\langle v - z, Jz - Ju \right\rangle \geq 0, \forall v\in C. $$
(2) If we define a mapping Ar:E→C by
$${}A_{r}(u) \,=\, \left\lbrace\! z\in C : h(z, v) \,+\, \zeta (v) - \zeta (z) \,+\, \left\langle v - z, Bz \right\rangle + \frac{1}{r}\left\langle v - z, Jz - Ju \right\rangle \!\geq\! 0, \forall v\in C\! \right\rbrace\!, u\in E, $$
the mapping Ar has the following properties:(a) Ar is single-valued;(b) \(F(A_{r}) = GMEP(h, A, \zeta) = \hat {F}(A_{r})\)(c) GMEP(h,A,ζ) is a closed convex subset of C;(d) ϕ(q,Aru)+ϕ(Aru,u)≤ϕ(q,u),∀q∈F(Ar),u∈E.where \(\hat {F}(A_{r})\) denotes the set of asymptotic fixed points of Ar, i.e.,
$$\hat{F}(A_{r}) := \left\lbrace x\in C : \exists \left\{ x_{n}\right\} \subset C,~~ s.t ~~ x_{n}\rightharpoonup x, \left\Vert x_{n} - A_{r}x_{n} \right\Vert \longrightarrow 0 (n\longrightarrow \infty)\right\rbrace $$

Strong convergence theorem

In this section, we prove a strong convergence theorem for finding a common element of the set of solutions of generalized mixed equilibrium problems and the set of fixed point of quasi- ϕ-asymptotically nonexpansive multivalued mapping in Banach space.

Theorem 1

Let E be a uniformly smooth and uniformly convex Banach space, and Let C be a nonempty closed convex subset of E and \(\hat {C}B(C)\) be the family of nonempty, closed, and bounded subsets of C. Let \(f_{i} : C \times C \longrightarrow \mathbb {R}, i = 1,2,3,...k\) be bi functions which satisfy the conditions (A1)−(A4),A:C→E∗ be a nonlinear mapping, and \(\varphi : C \longrightarrow \mathbb {R} \cup \lbrace \infty \rbrace \) be a proper, convex, and lower semi-continuous function. Let Ti,i=1,2,3,...N be a quasi- ϕ-asymptotically nonexpansive multivalued mapping from C into \(\hat {C}B(C)\) such that F(T)∩GMEP(f,A,φ)≠∅. Let {xn} be a sequence generated by
$$\begin{array}{@{}rcl@{}} \left\{\begin{array}{ll} x_{0} = x\in C \\ y_{n} = J^{-1} \left(\alpha_{0,n}Jx_{n} + \sum_{i=1}^{N} \alpha_{i,n}Jw_{i,n}\right), \\ u_{n} \in C \; such\; that\; \sum_{i=1}^{K} f_{i}(u_{n}, y)+ \varphi(y) -\varphi (u_{n})+ \left\langle y - u_{n}, Au_{n} \right\rangle\\ + \frac{1}{r_{n}}\left\langle y - u_{n}, Ju_{n} - Jy_{n}\right\rangle \geq 0,\forall y\in C\\ M_{n} = \left\lbrace z\in C : \phi(z, u_{n})\leq k_{n}^{2} \phi(z, x_{n})\right\rbrace\\ W_{n}=\left\lbrace z\in C : \left\langle x_{n}- z, Jx - Jx_{n}\right\rangle \geq 0\right\rbrace\\ x_{n+1} = \Pi_{M_{n}\cap W_{n}}x, ~\forall n\geq 0, \end{array}\right. \end{array} $$
(8)

where J is the normalized duality mapping of E, {αi,n}⊂[0,1] satisfies \(\underset {n\to \infty }{\liminf }\alpha _{0,n}\alpha _{i,n} > 0, \sum _{i=0}^{N} \alpha _{i,n} = 1\) and \(w_{i,n} \in T_{i}^{n}x_{n}, \forall _{i} = 1,2,3,...N. \lbrace r_{n} \rbrace \subset [ a, \infty ]\), some a>0. Then, {xn} converges strongly to ΠF(T)∩GMEP(f,A,φ)x, where ΠF(T)∩GMEP(f,A,φ) is the generalized projection of E onto F(T)∩GMEP(f,A,φ),

Proof

Let two functions \(\tau : C\times C\longrightarrow \mathbb {R}\) and Tr:E→C be defined by
$$\begin{array}{@{}rcl@{}} \tau (x,y)= \sum_{i=1}^{k} f_{i}(x, y) + \langle Ax, y - x \rangle + \varphi (y) - \varphi(x), ~\forall x,y\in C \end{array} $$
and
$$\begin{array}{@{}rcl@{}} T_{r}(x) = \left\lbrace u\in C : \tau \left(u, y\right) + \frac{1}{r}\left\langle y - u, Ju - Jx\right\rangle \geq 0, \forall y\in C, \right\rbrace ~\forall x \in E \end{array} $$
respectively. Now, the function τ satisfies conditions (A1)−(A4) and Tr has the properties (a)−(d). Therefore, iterative sequence (8) can be rewritten as
$$\begin{array}{@{}rcl@{}} \left\{\begin{array}{ll} x_{0} = x\in C \\ y_{n} = J^{-1}\left(\alpha_{0,n}Jx_{n} + \sum_{i=1}^{N} \alpha_{i,n}Jw_{i,n}\right), w_{i,n}\in T_{i}^{n}x_{n}, n\geq 1\\ u_{n} \in C \; \text{such that}\; \tau\left(u_{n}, y\right) + \frac{1}{r_{n}}\left\langle y - u_{n}, Ju_{n} - Jy_{n} \right\rangle \geq 0,\forall y\in C\\ M_{n} = \left\lbrace z\in C : \phi (z, u_{n})\leq k_{n}^{2}\phi (z, x_{n})\right\rbrace\\ W_{n} =\left\lbrace z\in C : \left\langle x_{n} - z, Jx - Jx_{n} \right\rangle\geq 0\right\rbrace \\ x_{n+1} = \Pi_{M_{n} \cap W_{n}x},~ N \in \mathbb{N}.\end{array}\right. \end{array} $$
(9)

□

We first show that Mn∩Wn is closed and convex, and it is obvious that Mn is closed and convex since \(\phi (z,u_{n})\leq k_{n}^{2}\phi (z,x_{n})\Longleftrightarrow \left (1-k_{n}^{2} \right)\left \Vert z \right \Vert ^{2} -2\left (1- k_{n}^{2}\right)\left \langle z, Ju_{n}\right \rangle + 2k_{n}^{2} \left \langle z,Jx_{n}-Ju_{n}\right \rangle \leq k_{n}^{2}\left \Vert x_{n} \Vert ^{2}-\Vert u_{n}\right \Vert ^{2}. \)

Thus, Mn∩Wn is a closed and convex subset of E for all \(n\in \mathbb {N}\cup \lbrace 0\rbrace \), so that {xn} is well defined.

Let u∈F(T)∩GMEP(f,A,φ), putting \(\phantom {\dot {i}\!}u_{n} = \omega _{n}\in T_{r_{n}}y_{n}\) for all \(n\in \mathbb {N}\cup \lbrace 0\rbrace \), and since \(T_{r_{n}}\) are quasi- ϕ-asymptotically nonexpansive multivalued, we have
$$\begin{array}{@{}rcl@{}} \phi(u, u_{n}) &=&\phi(u, \omega_{n})\\ &\leq& k_{n} \phi (u, y_{n}) \\ &=& k_{n} \left[ \phi\left(u,J^{-1} \left(\alpha_{0,n} Jx_{n} + \sum_{i=1}^{N} \alpha_{i,n} Jw_{i,n} \right) \right)\right], \\ \\ &=& k_{n} \left[ \Vert u \Vert^{2} - 2 \left\langle u, \alpha_{i,0} Jx_{n} + \sum_{i=1}^{N} \alpha_{i,n}Jw_{i,n} \right\rangle + \left\Vert \alpha_{0,n}Jx_{n} + \sum_{i=1}^{N} \alpha_{i,n}Jw_{i,n} \right\Vert^{2}\right.\\ &\leq& k_{n} \left[ \Vert u \Vert^{2}- 2\left(\!\left\langle u, \alpha_{0,n}Jx_{n} \right\rangle \!+ \! \left\langle u, \sum_{i=1}^{N} \alpha_{i,n}Jw_{i,n} \right\rangle \!\right) \!+ \! \alpha_{0,n} \left\| Jx_{n} \right\|^{2} \!+ \!\sum_{i=1}^{N}\alpha_{i,n} \Vert Jw_{i,n} \Vert^{2} \right]\\ &=&k_{n} \left[ \alpha_{0,n}\left(\parallel u \parallel^{2} - 2 \left\langle u, Jx_{n} \right\rangle + \Vert x_{n} \Vert^{2}\right) \!+ \! \sum_{i=1}^{N} \alpha_{i,n} \left(\Vert u \Vert^{2} - 2 \langle u, Jw_{i,n} \rangle + \Vert w_{i,n} \Vert^{2} \right)\right]\\ &=& k_{n}\left[ \alpha_{0,n} \phi \left(u, x_{n} \right) + \sum_{i=1}^{N} \alpha_{i,n} \phi (u,w_{i,n}) \right]\\ &\leq& k_{n} \left[ \alpha_{0,n} \phi (u, x_{n}) +k_{n} \sum_{i=1}^{N}\alpha_{i,n} \phi(u, x_{n})\right]\\ &\leq& k_{n} \left[ k_{n} \alpha_{0,n} \phi (u, x_{n}) + k_{n}\sum_{i=1}^{N} \alpha_{i,n} \phi (u, x_{n})\right] \\ &=& k_{n}\left[\left(\alpha_{0,n} + \sum_{i=1}^{N}\alpha_{i,n}\right)k_{n}\phi\left(u, x_{n}\right)\right]\\ &=&k_{n}^{2}\phi(u, x_{n}). \end{array} $$

Hence, we have u∈Mn. This implies that \(F(T) \cap GMEP(f, A,\varphi) \subset M_{n}, \forall n \in \mathbb {N}\cup \lbrace 0 \rbrace \).

Next, we show by induction that \(F(T) \cap GMEP (f,A,\varphi) \subset M_{n} \cap W_{n}, \forall n \in \mathbb {N}\cup \lbrace 0\rbrace \).

From W0=C, we have
$$\begin{array}{@{}rcl@{}} F(T) \cap GMEP(f,A,\varphi) \subset M_{0}\cap W_{0},. \end{array} $$

Suppose that F(T)∩GMEP(f,A,φ)⊂Mk∩Wk, for some \(k \in \mathbb {N}\cup \lbrace 0\rbrace.\) Then, there exists xk+1∈Mk∩Wk such that \(x_{k+1} =\Pi _{M_{k} \cap W_{k}}x \)

From the definition of xk+1, we have for all z∈Mk∩Wk,
$$\begin{array}{@{}rcl@{}} \left\langle x_{k+1} - z, Jx - Jx_{k+1} \right\rangle \geq 0. \end{array} $$
Since F(T)∩GMEP(f,A,φ)⊂Mk∩Wk, we have
$$\begin{array}{@{}rcl@{}} \left\langle x_{k+1} - z, Jx - Jx_{k+1} \right\rangle \geq 0. \end{array} $$
∀z∈F(T)∩GMEP(f,A,φ) and so z∈Wk+1. Thus, F(T)∩GMEP(f,A,φ)⊂Wk+1. Therefore, we have F(T)∩GMEP(f,A,φ)⊂Mk+1∩Wk+1. Therefore, we obtain
$$\begin{array}{@{}rcl@{}} F(T)\cap GMEP\left(f, A,\varphi\right)\subset M_{n} \cap W_{n}, ~\forall n \in \mathbb{N} \cup \lbrace 0 \rbrace. \end{array} $$
From the definition of Wn, we have \(\phantom {\dot {i}\!}x_{n} = \Pi _{W_{n}}x\); using this and Lemma 1, we have
$$\begin{array}{@{}rcl@{}} \phi (x_{n}, x) &=& \phi \left(\Pi_{W_{n}}x, x \right) \leq \phi (u, x) - \phi \left(u, \Pi_{W_{n}}x, \right)\\ &\leq& \phi(u, x) \end{array} $$
(10)

for all u∈F(T)∩GMEP(f,A,φ)⊂Wn. Therefore, ϕ(xn,x) is bounded, and consequently {xn} and \(\left \lbrace T_{i}^{n}x_{n} \right \rbrace \) are bounded.

Since \(x_{n+1} = \Pi _{M_{n} \cap W_{n}}x \in M_{n} \cap W_{n}\subset W_{n}\) and \(\phantom {\dot {i}\!}x_{n} = \Pi _{W_{n}}x,\) we have
$$\begin{array}{@{}rcl@{}} \phi (x_{n}, x) \leq \phi (x_{n+1}, x),~ \forall n \in \mathbb{N}\cup \lbrace 0 \rbrace. \end{array} $$
(11)

Thus, {ϕ(xn,x)} is nondecreasing. Using (10) and (11), we have the limit of {ϕ(xn,x)} exists.

From \(\phantom {\dot {i}\!}x_{n} = \Pi _{W_{n}}x\) and Lemma 1, we also have
$$\begin{array}{@{}rcl@{}} \phi (x_{n+1}, x_{n}) = \phi (x_{n+1}, \Pi_{W_{n}}x,) \leq \phi (x_{n+1}, x) - \phi (\Pi_{W_{n}}x,, x) = \phi (x_{n+1}, x) - \phi (x_{n}, x) \end{array} $$

for all \(n \in \mathbb {N} \cup \lbrace 0 \rbrace.\) This means that \(\underset {n\to \infty }{lim} \phi (x_{n+1}, x_{n}) = 0\).

From \(x_{n+1} = \Pi _{M_{n} \cap W_{n}}x \in M_{n}\) and the definition of Mn, we have
$$\begin{array}{@{}rcl@{}} \phi (x_{n+1}, u_{n})\leq k_{n}^{2} \phi(x_{n+1}, x_{n}),\forall n \in \mathbb{N} \cup \lbrace 0 \rbrace. \end{array} $$
Therefore, we have\(\phantom {\dot {i}\!} \underset {n\to \infty }{\lim } \phi (x_{n+1}, u_{n})= 0\). As E is uniformly convex and smooth, we have from Lemma 3 that
$$\begin{array}{@{}rcl@{}} \underset{n\to \infty}{\lim} \Vert x_{n+1}- x_{n} \Vert =\underset{n\to \infty}{\lim} \Vert x_{n+1} - u_{n} \Vert =0. \end{array} $$
From which, we have
$$\begin{array}{@{}rcl@{}} \underset{n\to \infty}{\lim} \Vert x_{n} - u_{n} \Vert = 0. \end{array} $$
Since J is uniformly norm-to-norm continuous on bounded sets, we have
$$\begin{array}{@{}rcl@{}} \underset{n\to \infty}{\lim} \Vert Jx_{n} - Ju_{n}\Vert = 0. \end{array} $$
Let r = sup\(_{n \in \mathbb {N}} \left \lbrace \Vert x_{n} \Vert, \Vert T_{i}^{n} x_{n}\Vert \right \rbrace.\) Since E is a uniformly smooth Banach space, we know that E∗ is a uniformly convex Banach space. So, for u∈F(T)∩GMEP(f,A,φ), putting \(\phantom {\dot {i}\!}u_{n} = \omega _{n} = T_{r_{n}}y_{n}\) and using Lemma 4, we have :
$$\begin{array}{@{}rcl@{}} \phi (u, u_{n}) &=& \phi (u, \omega_{n})\\ &\leq& k_{n} \phi (u, y_{n})\\ &=& k_{n} \left[ \phi\left(u,J^{-1} \left(\alpha_{0,n} Jx_{n} + \sum_{i=1}^{N} \alpha_{i,n} Jw_{i,n} \right) \right) \right],\\ &=& k_{n} \left[ \Vert u \Vert_{2} - 2 \left\langle u, \alpha_{i,0} Jx_{n} + \sum_{i=1}^{N} \alpha_{i,n}Jw_{i,n} \right\rangle + \left\Vert \alpha_{0,n}Jx_{n} + \sum_{i=1}^{N} \alpha_{i,n}Jw_{i,n} \right\Vert^{2}\right] \\ &\leq& k_{n} \left[ \Vert u \Vert^{2} - 2 (\langle u, \alpha_{0,n}Jx_{n} \rangle + \langle u, \sum_{i=1}^{N} \alpha_{i,n}Jw_{i,n} \rangle) + \alpha_{0,n} \Vert Jx_{n} \Vert^{2} + \sum_{i=1}^{N} \alpha_{i,n} \Vert Jw_{i,n} \Vert^{2}\right.\\ & & {} - \left.\alpha_{0,n}\alpha_{i,n} g (\Vert Jx_{n} - Jw_{i,n} \Vert){\vphantom{ \sum_{i=1}^{N}}}\right] \\ &=& k_{n} \left[ \alpha_{0,n} \left[ \Vert u \Vert^{2} - 2 \langle u, Jx_{n} \rangle + \Vert x_{n} \Vert^{2} \right] + \sum_{i=1}^{N} \alpha_{i,n} \left[ \Vert u \Vert^{2} - 2 \langle u, Jw_{i,n} \rangle + \Vert w_{i,n} \Vert^{2}\right]\right.\\ & & {} -\left.{\vphantom{\sum_{i=1}^{N}}} \alpha_{0,n}\alpha_{i,n} g \left(\Vert Jx_{n} - Jw_{i,n} \Vert \right)\right]\\ &=& k_{n} \left[ \alpha_{0,n} \phi (u, x_{n}) + \sum_{i=1}^{N} \alpha_{i,n} \phi (u, w_{i,n}) - \alpha_{0,n}\alpha_{i,n} g (\Vert Jx_{n} - Jw_{i,n} \Vert)\right]\\ &\leq& k_{n} \left[ \alpha_{0,n} \phi (u, x_{n}) + \sum_{i=1}^{N} k_{n} \alpha_{i,n} \phi (u, x_{n}) - \alpha_{0,n}\alpha_{i,n} g (\Vert Jx_{n} - Jw_{i,n} \Vert)\right]\\ &\leq& k_{n} \left[ k_{n} \alpha_{0,n} \phi (u, x_{n}) + k_{n} \sum_{i=1}^{N} \alpha_{i,n} \phi (u, x_{n}) - \alpha_{0,n}\alpha_{i,n} g (\Vert Jx_{n} - Jw_{i,n} \Vert)\right]\\ &=& k_{n} \left[ k_{n} \phi(u, x_{n}) - \alpha_{0,n}\alpha_{i,n} g (\Vert Jx_{n} - Jw_{i,n} \Vert)\right] \end{array} $$
$$\begin{array}{@{}rcl@{}} &=& k_{n}^{2} \phi (u, x_{n}) -k_{n} \alpha_{0,n}\alpha_{i,n} g (\Vert Jx_{n} - Jw_{i,n} \Vert). \end{array} $$
(12)

Therefore, from (12), we have\(k_{n}\alpha _{0,n}\alpha _{i,n} g (\Vert Jx_{n} - Jw_{i,n} \Vert)\leq k_{n}^{2} \phi (u, x_{n}) - \phi (u, u_{n}),\forall n\in \mathbb {N} \cup \lbrace 0 \rbrace.\)

But
$$\begin{array}{@{}rcl@{}} \lefteqn{k_{n}^{2} \phi (u, x_{n})- \phi(u, u_{n}) = k_{n}^{2} \left[ \Vert u \Vert^{2} - 2 \langle u, Jx_{n} \rangle + \left\Vert x_{n} \right\Vert^{2} \right] - \left[ \Vert u \parallel^{2} - 2 \langle u, Ju_{n} \rangle + \parallel u_{n} \Vert^{2}\right]} \\ &=&\left(k_{n}^{2} -1 \right) \parallel u \Vert^{2}-2\left(k_{n}^{2}-1\right) \left\langle u, Ju_{n}\right\rangle - 2k_{n}^{2} \langle u, Jx_{n} - Ju_{n}\rangle + k_{n}^{2} \parallel x_{n} \Vert^{2}- \parallel u_{n} \Vert^{2}\\ &=& \left(k_{n}^{2} -1 \right) \parallel u \Vert^{2}-2\left(k_{n}^{2}-1\right) \langle u, Ju_{n}\rangle - 2k_{n}^{2} \langle u, Jx_{n} - Ju_{n}\rangle + \left(k_{n}^{2}-1\right) \parallel x_{n} \Vert^{2}\\ & &+\parallel x_{n} \Vert^{2}- \parallel u_{n} \Vert^{2}\\ &\leq& |\left(k_{n}^{2} -1 \right) \parallel u \Vert^{2}|+|2(k_{n}^{2}-1) \langle u, Ju_{n}\rangle| + |2k_{n}^{2} \langle u, Jx_{n} - Ju_{n}\rangle| + |\left(k_{n}^{2}-1\right) \parallel x_{n} \Vert^{2}|\\ & &+|\parallel x_{n} \Vert^{2}- \parallel u_{n} \Vert^{2}|\\ &\leq& \left(k_{n}^{2} -1 \right) \parallel u \Vert^{2}+2\left(k_{n}^{2}-1\right) \Vert u\Vert \Vert Ju_{n}\Vert + 2k_{n}^{2} \Vert u \Vert \Vert Jx_{n} - Ju_{n}\Vert + \left(k_{n}^{2}-1\right) \parallel x_{n} \Vert^{2} \\ & &+(\parallel x_{n} - u_{n} \Vert) (\parallel x_{n}\Vert + \parallel u_{n} \Vert)\\ \end{array} $$
Hence,
$$\begin{array}{@{}rcl@{}} \underset{n\to \infty}{\lim} \left(k_{n}^{2} \phi(u,x_{n}) -\phi(u, u_{n})\right) = 0.\end{array} $$
(13)
Since \(\underset {n\to \infty }{\liminf } \alpha _{0,n} \alpha _{i,n} > 0,\) we have
$$\begin{array}{@{}rcl@{}} \underset{n\to \infty}{\lim}g(\Vert Jx_{n} - Jw_{i,n}\Vert) = 0. \end{array} $$

From the property g, we have \(\underset {n\to \infty }{\lim } \Vert Jx_{n} - Jw_{i,n}\Vert = 0\).

Since J−1 is uniformly norm-to-norm continuous on bounded sets, we have
$$\begin{array}{@{}rcl@{}} \underset{n\to \infty}{\lim} \Vert x_{n} - w_{i,n} \Vert = 0. \end{array} $$

Since {xn} is bounded, there exists a subsequence \(\left \lbrace x_{n_{k}}\right \rbrace \) of {xn} such that \(x_{n_{k}} \rightharpoonup \hat {x},\) for some \(\hat {x}\in E\). Since T is quasi- ϕ-asymptotically nonexpansive multivalued mapping and E is a reflexive space, then we have \(\hat {x}\in F(T_{i})\).

Next, we show that \(\hat {x}\in GMEP\left (f, A,\varphi \right).\) From \(\phantom {\dot {i}\!}u_{n} = T_{r_{n}}y_{n}\) Lemma 7 (d) and (13), we have
$$\begin{array}{@{}rcl@{}} \phi (u_{n}, y_{n})&=& \phi(T_{r_{n} }y_{n}, y_{n})\\ &\leq& \phi(u, y_{n}) - \phi(u, T_{r_{n}}y_{n})\\ &\leq& k_{n}^{2}(u, x_{n}) -\phi(u,T_{r_{n}} y_{n})\\ &=& k_{n}^{2}\phi (u, x_{n}) - \phi(u, u_{n}). \end{array} $$

Thus, \(\underset {n\to \infty }{\lim } \phi (u_{n}, y_{n}) = 0\).

Since E is uniformly convex and smooth, we have from Lemma 3 that
$$\begin{array}{@{}rcl@{}} \underset{n\to \infty}{\lim}\parallel u_{n} - y_{n}\parallel = 0. \end{array} $$
(14)

From \(x_{n_{k}} \rightharpoonup \hat {x}, ~ \Vert x_{n} - u_{n} \Vert \longrightarrow 0\) and (14), we have \(y_{n_{k}} \rightharpoonup \hat {x}\) and \(u_{n_{k}} \rightharpoonup \hat {x}.\)

As J is uniformly norm-to-norm continuous on bounded sets and (14), we have\(\underset {n\to \infty }{\lim } \left \Vert Ju_{n} - Jy_{n} \right \Vert = 0\). From rn≥a, we have
$$\begin{array}{@{}rcl@{}} \underset{n\to \infty}{\lim}\left\Vert \frac{Ju_{n} - Jy_{n}}{r_{n}}\right\Vert = 0. \end{array} $$
(15)
By \(\phantom {\dot {i}\!}u_{n} = T_{r_{n}}y_{n}\), we have
$$\begin{array}{@{}rcl@{}} \tau\left(u_{n}, y\right) + \frac{1}{r_{n}}\langle y-u_{n}, Ju_{n} - Jy_{n} \rangle \geq 0, ~ \forall y\in C. \end{array} $$
Replacing n by nk, we have from (A2) that
$$\begin{array}{@{}rcl@{}} \frac{1}{r_{n_{k}}}\left\langle y - u_{n_{n}}, Ju_{n_{k}} - Jy_{n_{k}} \right\rangle \geq - \tau \left(u_{n_{k}}, y\right) \geq \tau\left(y, u_{n_{k}}\right), \forall y\in C \end{array} $$
(16)
Letting k→∞, in (16) and using (A4), we obtain
$$\begin{array}{@{}rcl@{}} \tau \left(y, \hat{x}\right)\leq 0, \forall y\in C. \end{array} $$
For t with 0<t≤1 and y∈C, let \(y_{t}= ty +(1-t)\hat {x}\). Since y∈C and \(\hat {x}\in C\), we have yt∈C and \(\tau (y_{t}, \hat {x})\leq 0, \forall y\in C\). Now, using (A1) and (A3), we have
$$\begin{array}{@{}rcl@{}} 0 &=& \tau(y_{t}, y_{t})\\ &\leq& t \tau \left(y_{t}, y\right) + (1-t) \tau\left(y_{t}, \hat{x}\right)\\ &\leq& t \tau \left(y_{t}, y\right)\!. \end{array} $$
Dividing by, t we have
$$\begin{array}{@{}rcl@{}} \tau \left(y_{t}, y \right)\geq 0, \forall y \in C. \end{array} $$
Letting t→0, and using (A3), we have
$$\begin{array}{@{}rcl@{}} \tau \left(\hat{x}, y \right)\geq 0, \forall y \in C. \end{array} $$

This shows that \(\hat {x}\in GMEP\left (f, A, \varphi \right)\).

Let ω=ΠF(T)∩GMEP(f,A,φ)x, From \(x_{n+1} = \Pi _{M_{n}\cap W_{n}}x\) and ω∈F(T)∩GMEP(f,A,φ)⊂Mn∩Wn, we have
$$\begin{array}{@{}rcl@{}} \phi(x_{n+1},x) \leq \phi(\omega,x). \end{array} $$
Since the norm is weakly lower semi-continuous and \(x_{n_{k}}\rightharpoonup \hat {x}\), we have
$$\begin{array}{@{}rcl@{}} \phi(\hat{x},x)&=& \left\Vert \hat{x} \right\Vert^{2}- 2 \left\langle \hat{x}, Jx \right\rangle + \left\Vert x \right\Vert^{2} \\&\leq& \underset{k\to \infty}{\liminf}\left(\Vert x_{n_{k}} \parallel^{2}-2\left\langle x_{n_{k}}, Jx \right\rangle + \Vert x\parallel^{2}\right)\\ &=& \underset{k\to \infty}{\liminf} \phi \left(x_{n_{k}}, x \right)\\ &\leq& \underset{k\to \infty}{\limsup}\phi \left(x_{n_{k}}, x \right)\\ &\leq& \phi(\omega, x). \end{array} $$
From the definition of ΠF(T)∩GMEP(f,A,φ), we have \(\hat {x} = \omega.\) Hence, \(\underset {k \rightarrow \infty }{lim}\phi \left (x_{n_{k}}, x \right) = \phi (\omega, x),\) Therefore,
$$\begin{array}{@{}rcl@{}} 0 &=& \underset{k\to\infty}{\lim}\left(\phi \left(x_{n_{k}}, x\right) - \phi (\omega, x)\right)\\&=& \underset{k \to \infty}{\lim}\left(\Vert x_{n_{k}} \Vert^{2} - \parallel \omega\Vert^{2} - 2 \left\langle x_{n_{k}} -\omega, Jx \right\rangle \right)\\ &=& \underset{k\to \infty}{\lim} \left(\parallel x_{n_{k}} \Vert^{2} - \parallel \omega \Vert^{2} \right). \end{array} $$

Since E has the Kadec-Klee property, we have that \(x_{n_{k}} \longrightarrow \omega = \Pi _{F(T)\cap GMEP\left (f, A, \varphi \right)}x.\)

Therefore, {xn} converges strongly to ΠF(T)∩GMEP(f,A,φ)x.

Weak convergence theorem

In this section, we prove a weak convergence theorem for finding a common element of the set of solutions of generalized mixed equilibrium problem and the set of fixed point of quasi- ϕ-asymptotically nonexpansive multivalued mapping in Banach space. Before proving the Theorem, we need the following proposition.

Proposition 1

Let E be a uniformly smooth and uniformly convex Banach space and let C be a nonempty closed convex subset of E and \(\hat {C}B(C)\) be the family of nonempty, closed, and bounded subsets of C. Let \(f_{i} : C \times C \longrightarrow \mathbb {R},(i = 1,2,3,...K \in \mathbb {N})\) be bifunctions satisfying (A1)−(A4),A:C→E∗ be a nonlinear mapping and let \(\varphi : C \longrightarrow \mathbb {R} \cup \lbrace \infty \rbrace \) be a proper, convex, and lower semi-continuous function. Let T be a quasi- ϕ-asymptotically nonexpansive multivalued mapping from C into \(\hat {C}B(C)\) such that F(T)∩GMEP(f,A,φ)≠∅. Let {xn} be a sequence generated by u1∈E
$$\begin{array}{@{}rcl@{}} \left\{\begin{array}{ll} x_{n} \in C \;\text{such that}\; \tau(x_{n}, y) + \frac{1}{r_{n}} \left\langle y - x_{n}, Jx_{n} - Ju_{n} \right\rangle \geq 0, \forall y\in C\\ u_{n+1} = J^{-1} \left(\alpha_{0,n}Jx_{n} + \sum_{i=1}^{N} Jw_{i,n} \right),~~~ w_{i,n} \in T_{i}^{n}x_{n}, K,N,\in \mathbb{N} \end{array}\right. \end{array} $$

for every \(n\in \mathbb {N}\), where J is the normalized duality mapping on E, {αi,n}⊂[0,∞) satisfying \(\underset {n\to \infty }{\liminf }\alpha _{0,n}\alpha _{i,n} > 0, \sum _{i=0}^{N}\alpha _{i,n} = 1\) and \(w_{i,n} \in T_{i}^{n}x_{n}, \forall i = 1,2,3,...N \in \mathbb {N}.\)

Let {rn}⊂(0,∞). Then, {ΠF(T)∩GMEP(f,A,φ)xn} converges strongly to z∈F(T)∩GMEP(f,A,φ), where ΠF(T)∩GMEP(f,A,φ) is generalized projection of E onto F(T)∩GMEP(f,A,φ).

Proof

Let u∈F(T)∩GMEP(f,A,φ). Putting \(\phantom {\dot {i}\!}x_{n} = \omega _{n}\in T_{r_{n}}u_{n}\) for all \(n\in \mathbb {N}\), we know that \(T_{r_{n}}\) are quasi- ϕ-asymptotically nonexpansive multivalued, and we have
$$\begin{array}{@{}rcl@{}} {}&&{}\phi\left(u, x_{n+1}\right)\\ {}&=&\phi\left(u, \omega_{n+1} \right)\\ {}&\leq& k_{n} \phi \left(u, u_{n+1}\right) \\ {}&=& k_{n} \left[ \phi(u,J^{-1} (\alpha_{0,n} Jx_{n} + \sum_{i=1}^{N} \alpha_{i,n} Jw_{i,n}))\right], ~~~w_{i,n}\in T_{i}^{n} x_{n}, n \geq 1\\ {}&=& k_{n} \left[ \Vert u \Vert^{2} - 2 \left\langle u, \alpha_{i,0} Jx_{n} + \sum_{i=1}^{N} \alpha_{i,n}Jw_{i,n} \right\rangle + \left\Vert \alpha_{0,n}Jx_{n} + \sum_{i=1}^{N} \alpha_{i,n}Jw_{i,n} \right\Vert^{2}\right] \end{array} $$
$$\begin{array}{@{}rcl@{}} &\leq& k_{n} \left[ \Vert u \Vert^{2}- 2\left(\left\langle u, \alpha_{0,n}Jx_{n} \right\rangle + \left\langle u, \sum_{i=1}^{N} \alpha_{i,n}Jw_{i,n} \right\rangle \right) + \alpha_{0,n} {\Vert Jx_{n} \Vert}^{2} + \sum_{i=1}^{N}\alpha_{i,n} {\Vert Jw_{i,n} \Vert}^{2}\right]\\ &=&k_{n} \left[ \alpha_{0,n}\left(\parallel u \parallel^{2} - 2 \langle u, Jx_{n} \rangle + \Vert x_{n} \Vert^{2}\right) + \sum_{i=1}^{N} \alpha_{i,n} \left(\Vert u \Vert^{2} - 2 \langle u, Jw_{i,n} \rangle + \Vert w_{i,n} \Vert^{2} \right)\right] \\ &\leq& k_{n} \left[\alpha_{0,n} \phi (u, x_{n}) +k_{n} \sum_{i=1}^{N}\alpha_{i,n} \phi(u, x_{n})\right]\\ &\leq& k_{n} \left[ k_{n} \alpha_{0,n} \phi (u, x_{n}) + k_{n}\sum_{i=1}^{N} \alpha_{i,n} \phi (u, x_{n}) \right]\\ &=& k_{n}^{2}\phi(u, x_{n}). \end{array} $$
Thus,
$$\begin{array}{@{}rcl@{}} \phi(u, x_{n+1}) &\leq&k_{n}^{2}\phi(u, x_{n}) \end{array} $$
(17)
Hence, we have
$$\begin{array}{@{}rcl@{}} \phi(u, x_{n+1}) \leq k_{n}^{2}\phi(u, x_{n}) = \left(1 + \left(k_{n}^{2} - 1\right)\right)\phi(u, x_{n}) \end{array} $$

By Lemma 6, \( \sum _{i=1}^{\infty }\left (k_{n}^{2} - 1\right) <\infty \), we obtain \(\underset {n\to \infty }{\lim }\phi (u, x_{n})\) exists. It follows that {xn} and {wi,n} are bounded.

Define yn=ΠF(T)∩GMEP(f,A,φ)xn for all \(n\in \mathbb {N}\). Then, yn∈F(T)∩GMEP(f,A,φ); therefore, from (17), we have
$$\begin{array}{@{}rcl@{}} \phi\left(y_{n}, x_{n+1}\right)\leq k_{n}^{2}\phi(y_{n}, x_{n}). \end{array} $$
(18)
Thus,
$$\begin{array}{@{}rcl@{}} \phi(y_{n+1}, x_{n+1}) &=& \phi \left(\Pi_{F(T)\cap GMEP\left(f, A,\varphi\right)}x_{n+1}, x_{n+1} \right)\\ &\leq& \phi (y_{n}, x_{n+1}) - \phi \left(y_{n}, \Pi_{F(T)\cap GMEP\left(f, A,\varphi\right)}x_{n+1}\right)\\ &=& \phi(y_{n}, x_{n+1}) - \phi(y_{n},y_{n+1})\\ &\leq& \phi(y_{n}, x_{n+1}). \end{array} $$

Hence, from (18), we have \(\phi (y_{n+1}, x_{n+1}) \leq k_{n}^{2} \phi (y_{n}, x_{n}) = \left (1 + \left (k_{n}^{2} - 1\right)\right)\phi (y_{n}, x_{n})\).

By the assumption \( \sum _{i=1}^{\infty }\left (k_{n}^{2} - 1\right) <\infty \) and using Lemma 6, we have \( \underset {n\to \infty }{lim}\phi (y_{n}, x_{n})\). For \(m\in \mathbb {N}\) such that m>n, we also have from (18) that
$$\begin{array}{@{}rcl@{}} \phi \left(y_{n}, x_{n+m}\right) \leq \left(k_{n}^{2}\right)^{m} \phi(y_{n}, x_{n}). \end{array} $$
From yn+m=ΠF(T)∩GMEP(f,A,φ)xn+m and Lemma 1, we have
$$\begin{array}{@{}rcl@{}} \phi\left(y_{n}, y_{n+m}\right) + \phi (y_{n+m}, x_{n+m}) &\leq& \phi(y_{n}, x_{n+m})\\ &\leq& \left(k_{n}^{2}\right)^{m}\phi (y_{n}, x_{n}) \end{array} $$
Hence,\(\phi (y_{n}, y_{n+m})\leq \left (k_{n}^{2}\right)^{m} \phi (y_{n}, x_{n}) - \phi (y_{n+m}, x_{n+m}).\) Let \(r ={\sup }_{n\in \mathbb {N}} \Vert y_{n} \Vert \);from Lemma 5, we have
$$\begin{array}{@{}rcl@{}} g (\Vert y_{n} - y_{n+m} \Vert)&\leq& \phi(y_{n}, y_{n+m})\leq \left(k_{n}^{2}\right)^{m} \phi(y_{n}, x_{n}) - \phi(y_{n+m}, x_{n+m}) \end{array} $$

Since \( \underset {n\to \infty }{lim}\phi (y_{n}, x_{n})\) exists, from the property of g, we have that {yn} is Cauchy. Since F(T)∩GMEP(f,A,φ) is closed, {yn} converges strongly to z∈F(T)∩GMEP(f,A,φ). □

Now, we prove the following theorem.

Theorem 2

Let E be a real uniformly smooth and uniformly convex Banach space, let C be a nonempty, closed, and convex subset of E, and let \(\hat {C}B(C)\) be family of nonempty, closed, and bounded subsets of C. Let \(f_{i} : C \times C \longrightarrow \mathbb {R},(i = 1,2,3,...K)\) be bifunctions satisfying (A1)−(A4),A:C→E∗ be a nonlinear mapping, and \(\varphi : C \longrightarrow \mathbb {R} \cup \lbrace \infty \rbrace \) be a proper, convex, and lower semi-continuous function. Let T be a quasi- ϕ-asymptotically nonexpansive multivalued mapping from C into \(\hat {C}B(C)\) such that F(T)∩GMEP(f,A,φ)≠∅. Let {xn} be a sequence generated by u1∈E
$$\begin{array}{@{}rcl@{}} \left\{\begin{array}{ll} x_{n} \in C \; \text{such that}\; \tau (x_{n}, y) + \frac{1}{r_{n}} \left\langle y - x_{n}, Jx_{n} - Ju_{n} \right\rangle \geq 0, \forall y\in C.\\ u_{n+1} = J^{-1} \left(\alpha_{0,n}Jx_{n} + \sum_{i=1}^{N} Jw_{i,n} \right), ~~~w_{i,n} \in T_{i}^{n}x_{n}, K,N,\in \mathbb{N} \end{array}\right. \end{array} $$

for every \(n\in \mathbb {N}\), where J is the normalized duality mapping of E, {αi,n}⊂[0,∞] satisfies \(\underset {n\to \infty }{\liminf }\alpha _{0,n}\alpha _{i,n} > 0, \sum _{i=0}^{N}\alpha _{i,n} = 1\) and \(w_{i,n} \in T_{i}^{n}x_{n}, \forall i = 1,2,3,...N \in \mathbb {N}.\)

Let {rn}⊂[a,∞)for some a>0. If J is weakly sequentially continuous, then {xn} converges weakly to z∈F(T)∩GMEP(f,A,φ), where \(z =\underset {n\to \infty }{lim}\Pi _{F(T)\cap GMEP(f, A,\varphi)}x_{n}.\)

Proof

As in the proof of proposition 17, we have that {xn} and {wi,n} and \(\lbrace T^{n}_{i}x_{n}\rbrace \) are bounded sequences. Let \(r ={\sup }_{n\in \mathbb {N}}\lbrace \Vert x_{n} \Vert, \Vert w_{i,n}\Vert \rbrace.\) For u∈F(T)∩GMEP(f,A,φ), we have
$$\begin{array}{@{}rcl@{}} \phi(u, x_{n+1}) &=&\phi(u, \omega_{n+1})\\ &\leq& k_{n} \phi \left(u, u_{n+1}\right) \\ &=& k_{n} \left[ \phi(u,J^{-1} \left(\alpha_{0,n} Jx_{n} + \sum_{i=1}^{N} \alpha_{i,n} Jw_{i,n} \right))\right]\\ &=& k_{n} \left[ \Vert u \Vert^{2} - 2 \left\langle u, \alpha_{i,0} Jx_{n} + \sum_{i=1}^{N}\alpha_{i,n}Jw_{i,n} \right\rangle + \left\Vert \alpha_{0,n}Jx_{n} + \sum_{i=1}^{N} \alpha_{i,n}Jw_{i,n} \right\Vert^{2}\right.\\ \end{array} $$
$$\begin{array}{@{}rcl@{}} &\leq& k_{n} \left[ \Vert u \Vert^{2}- 2(\langle u, \alpha_{0,n}Jx_{n} \rangle + \left\langle u, \sum_{i=1}^{N} \alpha_{i,n}Jw_{i,n} \right\rangle \right)\\& & {} \left.+ \alpha_{0,n} \Vert x_{n} \Vert^{2} + \sum_{i=1}^{N}\alpha_{i,n} \Vert w_{i,n} \Vert^{2} - \alpha_{0,n}\alpha_{i,n} g(\Vert Jx_{n} - Jw_{i,n} \Vert)\right] \\&=&k_{n} \left[ \alpha_{0,n}(\parallel u \parallel^{2} - 2 \langle u, Jx_{n} \rangle + \Vert x_{n} \Vert^{2}) + \sum_{i=1}^{N} \alpha_{i,n} (\Vert u \Vert^{2} - 2 \langle u, Jw_{i,n} \rangle + \Vert w_{i,n} \Vert^{2})\right.\\& & \left. {}{\vphantom{\sum_{i=1}^{N}}} - \alpha_{0,n}\alpha_{i,n} g(\Vert Jx_{n} -Jw_{i,n} \Vert)\right] \end{array} $$
$$\begin{array}{@{}rcl@{}} &=& k_{n} \left[ \alpha_{0,n} \phi (u, x_{n}) + \sum_{i=1}^{N} \alpha_{i,n} \phi (u,w_{i,n}) - \alpha_{0,n}\alpha_{i,n} g(\Vert Jx_{n} - Jw_{i,n} \Vert)\right]\\ &\leq & k_{n} \left[ k_{n} \alpha_{0,n} \phi (u, x_{n}) + k_{n}\sum_{i=1}^{N} \alpha_{i,n} \phi (u, x_{n})- \alpha_{0,n}\alpha_{i,n} g(\Vert Jx_{n} - Jw_{i,n} \Vert)\right]\\ &=& k_{n} \left[ k_{n} \phi (u,x_{n})- \alpha_{0,n}\alpha_{i,n} g(\Vert Jx_{n} - Jw_{i,n} \Vert) \right]\\ &=&k_{n}^{2}\phi(u, x_{n})- k_{n} \alpha_{0,n}\alpha_{i,n} g(\Vert Jx_{n} - Jw_{i,n} \Vert). \end{array} $$
Thus,
$$\begin{array}{@{}rcl@{}} \phi(u, x_{n+1}) &\leq&k_{n}^{2}\phi(u, x_{n}) - k_{n} \alpha_{0,n}\alpha_{i,n} g(\Vert Jx_{n} - Jw_{i,n} \Vert). \end{array} $$
(19)
Hence, we have
$$\begin{array}{@{}rcl@{}} k_{n} \alpha_{0,n}\alpha_{i,n} g(\Vert Jx_{n} - Jw_{i,n} \Vert) \leq k_{n}^{2}\phi(u, x_{n})- \phi(u, x_{n+1}). \end{array} $$
Since {ϕ(u,xn)} is convergent, we have
$$\begin{array}{@{}rcl@{}} \underset{n\to \infty}{\lim} \left(k_{n}^{2} \phi (u, x_{n})-\phi(u, x_{n+1})\right) = 0. \end{array} $$
As, we get \(\underset {n\to \infty }{\lim } \alpha _{0,n}\alpha _{i,n} > 0\)
$$\begin{array}{@{}rcl@{}} \underset{n\to \infty}{\lim}g(\Vert Jx_{n} - Jw_{i,n} \Vert) = 0. \end{array} $$
From the property of g, we have
$$\begin{array}{@{}rcl@{}} \underset{n\to \infty}{\lim} \Vert Jx_{n} - Jw_{i,n} \Vert = 0. \end{array} $$
Since J−1 is uniformly norm-to-norm continuous on bounded sets, we have
$$\begin{array}{@{}rcl@{}} \underset{n\to \infty}{\lim} \Vert x_{n} - w_{i,n} \Vert = 0. \end{array} $$
(20)

Since {xn} is bounded, there exists a subsequence \(\left \lbrace x_{n_{k}}\right \rbrace \) of {xn} such that \(\left \{x_{n_{k}}\right \}\) converges weakly to \(\hat {x}\in C\). From (20) and F(T), we have \(\hat {x}\in F(T).\)

Next, we show that \(\hat {x}\in GMEP(f, A, \varphi).\) Let \(T = {\sup }_{n\in \mathbb {N}}\lbrace \Vert x_{n} \Vert, \Vert u_{n} \Vert \rbrace \) from Lemma 5, and putting \(\phantom {\dot {i}\!}x_{n} = T_{r_{n}}u_{n}\), we have from Lemma 7 (d) and(19) that for u∈F(T)∩GMEP(f,A,φ).
$$\begin{array}{@{}rcl@{}} g_{1} (\Vert x_{n} - u_{n} \Vert)&\leq& \phi(x_{n}, u_{n})\\&\leq& \phi(u, u_{n}) - \phi(u, x_{n})\\ &\leq& k_{n}\phi(u, u_{n}) - \phi(u, x_{n}) \\ &\leq& k_{n}^{2} \phi(u, x_{n-1}) - \phi(u, x_{n}). \end{array} $$
Since {ϕ(u,xn)} converges, we have
$$\begin{array}{@{}rcl@{}} \underset{n\to \infty}{\lim}g_{1}(\parallel x_{n} - u_{n}\parallel) = 0. \end{array} $$
From the property of g1, we have
$$\begin{array}{@{}rcl@{}} \underset{n\to \infty}{\lim} \parallel x_{n} - u_{n}\parallel = 0. \end{array} $$
Since J is uniformly norm-to-norm continuous on bounded sets, we have
$$\begin{array}{@{}rcl@{}} \underset{n\to \infty}{\lim}\left\| Jx_{n} - Ju_{n}\right\| = 0. \end{array} $$
From rn≥a, we have
$$\begin{array}{@{}rcl@{}} \underset{n\to \infty}{\lim} \left\| \frac{Jx_{n} - Ju_{n}}{r_{n}}\right\|= 0. \end{array} $$
By \(\phantom {\dot {i}\!}x_{n} = T_{r_{n}}u_{n}\), we have
$$\begin{array}{@{}rcl@{}} \tau(x_{n}, y)+ \frac{1}{r_{n}}\left\langle y-x_{n}, Jx_{n}-Ju_{n} \right\rangle \geq 0, \forall y\in C. \end{array} $$
As in the proof of Theorem 1, we have \(\hat {x}\in GMEP\left (f, A, \varphi \right).\) Therefore, \(\hat {x}\in F(T)\cap GMEP\left (f, A, \varphi \right).\) Let yn=ΠF(T)∩GMEP(f,A,φ)xn. From Lemma 2 and \(\hat {x}\in F(T)\cap GMEP\left (f, A, \varphi \right),\) we have
$$\begin{array}{@{}rcl@{}} \left\langle y_{n_{k}} - \hat{x}, Jx_{n_{k}} - Jy_{n_{k}} \right\rangle \geq 0. \end{array} $$
From proposition 17, we also have that {yn} converges strongly to z∈F(T)∩GMEP(f,A,φ), since J is weakly sequentially continuous, as k→∞ we have
$$\begin{array}{@{}rcl@{}} \left\langle z - \hat{x}, J\hat{x} - Jz \right\rangle \geq 0. \end{array} $$
(21)
On the other hand, since J is monotone, we have
$$\begin{array}{@{}rcl@{}} \left\langle z - \hat{x}, J\hat{x} - Jz \right\rangle \leq 0. \end{array} $$
(22)
Hence by (21) and (22), we have
$$\begin{array}{@{}rcl@{}} \left\langle z - \hat{x}, J\hat{x} - Jz \right\rangle = 0. \end{array} $$
From the strict convexity of E, we have
$$\begin{array}{@{}rcl@{}} z = \hat{x}. \end{array} $$
Therefore, {xn} converges weakly to \(\hat {x}\in F(T)\cap GMEP\left (f, A, \varphi \right)\), and we have
$$\begin{array}{@{}rcl@{}} \hat{x} = \underset{n\to \infty}{\lim}\Pi_{F(T)\cap GMEP\left(f, A, \varphi\right)}x_{n}. \end{array} $$

□

Notes

Acknowledgements

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Authors and Affiliations

  1. 1.Department of Mathematical Sciences, Bayero UniversityKanoNigeria
  2. 2.Department of Mathematics, Federal College of EducationKadunaNigeria
  3. 3.Department of Science and Technology Education, Bayero UniversityKanoNigeria
  4. 4.Department of Mathematics, Ahmadu Bello UniversityKadunaNigeria

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