Strong convergence theorems of nonlinear operator equations for countable family of multi-valued total quasi-ϕ-asymptotically nonexpansive mappings with applications

Abstract

The purpose of this article is first to introduce the concept of total quasi-ϕ-asymptotically nonexpansive multi-valued mapping which contains many kinds of mappings as its special cases, and then by using the hybrid shrinking technique to propose an iterative algorithm for finding a common element of the set of solutions for a generalized mixed equilibrium problem, the set of solutions for variational inequality problems, and the set of common fixed points for a countable family of multi-valued total quasi-ϕ-asymptotically nonexpansive mappings in a real uniformly smooth and strictly convex Banach space with Kadec-Klee property. The results presented in the article not only generalize some recent results from single-valued mappings to multi-valued mappings, but also improve and extend the main results of Homaeipour and Razani.

2000 AMS Subject Classification: 47J06; 47J25.

1. Introduction

Throughout this article, we always assume that X is a real Banach space with the dual X*, C is a nonempty closed convex subset of X, and J : X → 2 ^X is the normalized duality mapping defined by

$$J\left(x\right)=\left\{f^{*}\in X^{*}:〈x,f^{*}〉=ǁx{ǁ}^2=ǁf^{*}{ǁ}^2\right\},x\in E.$$

In the sequel, we use F(T ) to denote the set of fixed points of a mapping T , and use and ${\mathcal{R}}^{+}$ to denote the set of all real numbers and the set of all nonnegative real numbers, respectively. We denote by x _n → x and x _n ⇀ x the strong convergence and weak convergence of a sequence {x _n }, respectively.

Let $\mathrm{\Theta }:C\times C\to \mathcal{R}$ be a bifunction, $\psi :C\to \mathcal{R}$ be a real valued function, and A : C → X* be a nonlinear mapping. The so-called generalized mixed equilibrium problem is to find u ∈ C such that

$$\mathrm{\Theta }\left(u,y\right)+⟨Au,y-u⟩+\psi \left(y\right)-\psi \left(u\right)\ge 0,\forall y\in C.$$

(1.1)

The set of solutions to (1.1) is denoted by Ω, i.e.,

$$\mathrm{\Omega }=\left\{u\in C:\mathrm{\Theta }\left(u,y\right)+⟨Au,y-u⟩+\psi \left(y\right)-\psi \left(u\right)\ge 0,\forall y\in C\right\}.$$

(1.2)

Special examples:

• (I) If A ≡ 0, the problem (1.1) is equivalent to finding u ∈ C such that

  $$\mathrm{\Theta }\left(u,y\right)+\psi \left(y\right)-\psi \left(u\right)\ge 0,\forall y\in C.$$

  (1.3)

which is called the mixed equilibrium problem (MEP) [1].

• (II) If Θ ≡ 0, the problem (1.1) is equivalent to finding u ∈ C such that

  $$⟨Au,y-u⟩+\psi \left(y\right)-\psi \left(u\right)\ge 0,\forall y\in C.$$

  (1.4)

which is called the mixed variational inequality of Browder type (VI)[2].

A Banach space X is said to be strictly convex, if $\frac{∥x+y∥}{2}<1$ for all x, y ∈ U = {z ∈ X : ||z|| = 1} with x ≠ y. X is said to be uniformly convex if, for each ϵ ∈ (0, 2], there exists δ > 0 such that $\frac{∥x+y∥}{2}<1-\delta$ for all x, y ∈ U with ||x - y|| ≥ ϵ. X is said to be smooth if the limit

$$\underset{t\to 0}{lim}\frac{\left|\right|x+ty\left|\right|-\left|\right|x\left|\right|}{t}$$

exists for all x, y ∈ U. X is said to be uniformly smooth if the above limit is attained uniformly in x, y ∈ U.

Remark 1.1 The following basic properties of a Banach space X can be found in Cioranescu [1].

• (i) If X is uniformly smooth, then X is reflexive and the normalized duality mapping J is uniformly continuous on each bounded subset of X;

• (ii) If X is a reflexive and strictly convex Banach space, then J ^-1 is norm-weak-continuous;

• (iii) If X is a smooth, strictly convex, and reflexive Banach space, then J is single-valued, one-to-one and onto;

• (iv) A Banach space X is uniformly smooth if and only if X* is uniformly convex;

• (v) Each uniformly convex Banach space X has the Kadec-Klee property, i.e., for any sequence {x _n } ⊂ X, if x _n ⇀ x ∈ X and ||x _n || → ||x||, then x _n → x.

Let X be a smooth Banach space. In the sequel, we use $\varphi :X\times X\to {\mathcal{R}}^{+}$ to denote the Lyapunov functional which is defined by

$$\varphi \left(x,y\right)=ǁx{ǁ}^2-2⟨x,Jy⟩+ǁy{ǁ}^2,\forall x,y\in X.$$

(1.5)

It is obvious from the definition of ϕ that

$${\left(\left|\right|x\left|\right|-\left|\right|y\left|\right|\right)}^2\le \varphi \left(x,y\right)\le {\left(\left|\right|x\left|\right|+\left|\right|y\left|\right|\right)}^2,\forall x,y\in X.$$

(1.6)

and

$$\varphi \left(x,J^{-1}\left(\lambda Jy+\left(1-\lambda \right)Jz\right)\right)\le \lambda \varphi \left(x,y\right)+\left(1-\lambda \right)\varphi \left(x,z\right),$$

(1.7)

for all λ ∈ [0, 1] and x, y, z ∈ X. If X is a smooth, strictly convex, and reflexive Banach space, following Alber [2], the generalized projection ∏ _C : X → C is defined by

$${\mathrm{\Pi }}_C\left(x\right)=arg\underset{y\in C}{inf}\varphi \left(y,x\right),\forall x\in X.$$

Lemma 1.2[2] Let X be a smooth, strictly convex, and reflexive Banach space and C be a nonempty closed convex subset of X. Then the following conclusions hold:

• (a) ϕ (x, ∏ _C y) + ϕ (∏ _C y, y) ≤ ϕ (x, y) for all x ∈ C and y ∈ X;

• (b) If x ∈ X and z ∈ C, then

  $$z={\mathrm{\Pi }}_{C}x\iff ⟨z-y,Jx-Jz⟩\ge 0,\forall y\in C;$$

• (c) For x, y ∈ X, ϕ(x, y) = 0 if and only if x = y.

In the sequel, we denote by 2 ^C the family of all nonempty subsets of C.

Definition 1.3 Let T : C → 2 ^C be a multi-valued mapping.

• (1) A point p ∈ C is said to be an asymptotic fixed point of T, if there exists a sequence {x _n } in C such that {x _n } converges weakly to p and

  $$\underset{n\to \infty }{lim}d\left(x_n,T\left(x_n\right)\right):=\underset{n\to \infty }{lim}\underset{y\in T\left(x_n\right)}{inf}\left|\right|x_n-y\left|\right|=0.$$

In the sequel we use $\widehat{F}\left(T\right)$ to denote the set of all asymptotic fixed points of T;

• (2) A multi-valued mapping T : C → 2 ^C is said to be relatively nonexpansive [3], if

• (a) F(T ) ≠ Ø;

• (b) ϕ (p, w) ≤ ϕ (p, x), ∀x ∈ C, w ∈ Tx, p ∈ F(T)

• (c) $\widehat{F}\left(T\right)=F\left(T\right)$.

Definition 1.4 (1) A multi-valued mapping T : C → 2 ^C is said to be quasi-ϕ-nonexpansive, if F (T ) ≠ Ø and

$$\varphi \left(p,w\right)\le \varphi \left(p,x\right),\forall x\in C,w\in Tx,p\in F\left(T\right).$$

• (2) A multi-valued mapping T : C → 2 ^C is said to be quasi-ϕ-asymptotically nonexpansive if F(T ) ≠ Ø and there exists a real sequence {k _n } ⊂ [1, ∞) with k _n → 1 such that

  $$\varphi \left(p,w_n\right)\le k_n\varphi \left(p,x\right),\forall n\ge 1,x\in C,w_n\in T^{n}x,p\in F\left(T\right).$$

  (1.8)

• (3) A multi-valued mapping T : C → 2 ^C is said to be ({ν _n }, {μ _n },ζ)-total quasi-ϕ-asymptotically nonexpansive, if F(T) ≠ Ø and there exist nonnegative real sequences {ν _n }, {μ _n } with ν _n → 0, μ _n → 0 (as n → ∞) and a strictly increasing continuous function $\zeta :{\mathcal{R}}^{+}\to {\mathcal{R}}^{+}$ with ζ (0) = 0 such that for all x ∈ C, p ∈ F(T )

  $$\varphi \left(p,w_n\right)\le \varphi \left(p,x\right)+{\nu }_n\zeta \left(\varphi \left(p,x\right)\right)+{\mu }_n,\forall n\ge 1,w_n\in T^{n}x.$$

  (1.9)

• (4) A total quasi-ϕ-asymptotically nonexpansive multi-valued mapping T : C → 2 ^C is said to be uniformly L-Lipschitz continuous if there exists a constant L > 0 such that

  $$\left|\right|w_n-s_n\left|\right|\le L\left|\right|x-y\left|\right|,\forall x,y\in C,w_n\in T^{n}x,s_n\in T^{n}y,n\ge 1.$$

• (5) A multi-valued mapping T : C → 2 ^C is said to be closed if, for any sequences {x _n } and {w _n } in C with w _n ∈ T (x _n ), if x _n → x and w _n → y, then y ∈ Tx.

• (6) A countable family of multi-valued mappings ${\left\{T_i\right\}}_{i=1}^{\infty }:C\to 2^C$ is said to be uniformly ({ν _n }, {μ _n }, ζ)-total quasi-ϕ-asymptotically nonexpansive, if $\mathcal{F}:={\bigcap }_{i=1}^{\infty }F\left(T_i\right)\ne \varnothing$ and there exist nonnegative real sequences ({ν _n }, {μ _n } with ν _n → 0, μ _n → 0 and a strictly increasing continuous function $\zeta :{\mathcal{R}}^{+}\to {\mathcal{R}}^{+}$ with ζ(0) = 0 such that for all x ∈ C, $p\in \mathcal{F}$

  $$\varphi \left(p,w_{n,i}\right)\le \varphi \left(p,x\right)+{\nu }_n\zeta \left(\varphi \left(p,x\right)\right)+{\mu }_n,\forall n\ge 1,w_{n,i}\in T_i^{n}x,i=1,2,\dots .$$

  (1.10)

Remark 1.5 From the definitions, it is easy to know that

• (1) Every quasi-ϕ-asymptotically nonexpansive multi-valued mapping must be a total quasi-ϕ-asymptotically nonexpansive multi-valued mapping. In fact, taking ζ(t) = t, t ≥ 0, k _n = ν _n + 1 and μ _n = 0, then (1.6) can be rewritten as

  $$\varphi \left(p,w_n\right)\le \varphi \left(p,x\right)+{\nu }_n\zeta \left(\varphi \left(p,x\right)\right)+{\mu }_n,\forall n\ge 1,x\in C,w_n\in T^{n}x,p\in F\left(T\right),$$

where ν _n → 0 (as n → ∞).

• (2) The class of quasi-ϕ-asymptotically nonexpansive multi-valued mappings contains properly the class of quasi-ϕ-nonexpansive multi-valued mappings as a subclass, but the converse is not true.

• (3) The class of quasi-ϕ-nonexpansive multi-valued mappings contains properly the class of relatively nonexpansive multi-valued mappings as a subclass, but the converse is not true.

Example 1.6 Now we give some examples of single-valued and multi-valued total quasi-ϕ-asymptotically nonexpansive mappings.

(1) Single-valued total quasi- ϕ -asymptotically nonexpansive mapping.

Let C be a unit ball in a real Hilbert space l² and let T : C → C be a mapping defined by

$$T:\left(x_1,x_2,\dots ,\right)\to \left(0,x_1^2,a_2{x}_2,a_3{x}_3,\dots \right),\left(x_1,x_2,\dots ,\right)\in l^2,$$

(1.11)

where {a _i } is a sequence in (0, 1) such that ${\prod }_{i=2}^{\infty }{a}_i=\frac{1}{2}$. It is proved in [4] that T is total quasi-ϕ-asymptotically nonexpansive.

(2) Multi-valued total quasi- ϕ -asymptotically nonexpansive mappings.

Let I = 0[1], X = C(I) (the Banach space of continuous functions defined on I with the uniform convergence norm || f || _C = sup _t∈I |f(t)|), D = {f ∈ X : f (x) ≥ 0, ∀x ∈ I} and a, b be two constants in (0, 1) with a < b. Let T : D → 2 ^D be a multi-valued mapping defined by

$$T\left(f\right)=\left\{\begin{array}{c}\left\{g\in D:a\le f\left(x\right)-g\left(x\right)\le b,\forall x\in I\right\},iff\left(x\right)>1,\forall x\in I;\hfill \\ \left\{0\right\},otherwise.\hfill \end{array}\right.$$

(1.12)

It is easy to see that F (T ) = {0}, therefore F(T) is nonempty.

Next, we prove that T : D → 2 ^D is a closed total quasi-ϕ-asymptotically nonexpansive multi-valued mapping. In fact, for any given f ∈ D:

• (I) if f(x) > 1, ∀x ∈ I, then for any g ∈ T(f), we have a ≤ f(x) - g(x) ≤ b. Hence for any p ∈ F(T ) = {0} we have

  $$\varphi \left(p,g\right)=\varphi \left(0,g\right)=ǁg{ǁ}_C^2\le ǁf{ǁ}_C^2=\varphi \left(0,f\right)=\varphi \left(p,f\right).$$

If there exists some point x₀ ∈ I such that 0 ≤ f (x₀) ≤ 1, then from the definition of mapping T, we have T(f) = {0}. Hence for any p ∈ F(T) and g ∈ T(f) = {0}, we have

$$\varphi \left(p,g\right)=\varphi \left(0,0\right)=0\le ǁf{ǁ}_C^2=\varphi \left(0,f\right)=\varphi \left(p,f\right).$$

Summing up the above arguments we have that for any given f ∈ D

$$\varphi \left(p,g\right)\le \varphi \left(p,f\right),\forall p\in F\left(T\right),g\in T\left(f\right),$$

(1.13)

• (II) For any $g\in T^2\left(f\right)=T\left(T\left(f\right)\right)={\bigcup }_{g_1\in T\left(f\right)}T\left(g_1\right)$, there exists some $g_1^{*}\in T\left(f\right)$ such that $g\in T\left(g_1^{*}\right)$.

• (1) If $g_1^{*}>1,\forall x\in I$, then we have $a\le g_1^{*}-g<b$. By (1.13), for any p ∈ F(T) = {0}, we have

  $$\varphi \left(p,g\right)=\varphi \left(0,g\right)=ǁg{ǁ}_C^2\le ǁg_1^{*}{ǁ}_C^2=\varphi \left(0,g_1^{*}\right)=\varphi \left(p,g_1^{*}\right)\le \varphi \left(p,f\right).$$

• (2) If there exists x ₁ ∈ I such that $0\le g_1^{*}\left(x_1\right)\le 1$, then by the definition of T , we have $T{g}_1^{*}=\left\{0\right\}$. Since $g\in T{g}_1^{*}=\left\{0\right\}$, and so g = 0. Hence for any p ∈ F(T), by (1.13) we have

  $$\varphi \left(p,g\right)=\varphi \left(0,0\right)=0\le ǁg_1^{*}{ǁ}^2=\varphi \left(0,g_1^{*}\right)=\varphi \left(p,g_1^{*}\right)\le \varphi \left(p,f\right).$$

From (1) and (2) we have that for any given f ∈ D

$$\varphi \left(p,g\right)\le \varphi \left(p,f\right),\forall p\in F\left(T\right),g\in T^2\left(f\right),$$

(1.14)

By induction, we can prove that for any given f ∈ D, g ∈ Tⁿ (f), n ≥ 1, p ∈ F(T),

$$\varphi \left(p,g\right)\le \varphi \left(p,f\right).$$

(1.15)

Letting {μ _n } and {ν _n } be two any nonnegative sequences with μ _n → 0 and ν _n → 0 and ζ(t) = t, t ≥ 0, then (1.15) can be rewritten as

$$\varphi \left(p,g\right)\le \varphi \left(p,f\right)+{\nu }_n\zeta \left(\varphi \left(p,f\right)\right)+{\mu }_n$$

for any f ∈ D, g ∈ Tⁿ (f), n ≥ 1, p ∈ F(T). This shows that T : C → 2 ^C is a total quasi-ϕ-asymptotically nonexpansive multi-valued mapping.

Next, we prove that T is a closed mapping. In fact, let {f _n } and {g _n } be two sequences in D with g _n ∈ T(f _n ) such that || f _n - f || _C → 0, ||g _n - g|| _C → 0 as n → ∞.

• (1) If f(x) > 1, ∀x ∈ I, since {f _n } converges uniformly to f, then there exists n ₀ ≥ 1 such that f _n (x) > 1, ∀x ∈ I, ∀n ≥ n ₀. By the definition of T, we have

  $$a\le f_n\left(x\right)-g_n\left(x\right)\le b,\forall n\ge 1andx\in I.$$

  (1.16)

Letting n → ∞ in (1.16), we have

$$a\le f\left(x\right)-g\left(x\right)\le b,\forall n\ge 1.$$

This implies that g ∈ T (f).

• (2) If there exists some point x ₂ ∈ I such that 0 ≤ f (x ₂) ≤ 1, then T(f) = {0}. Since {f _n } converges uniformly to f, then there exists a positive integer n ₂ such that 0 ≤ f _n (x ₂) ≤ 1, ∀n ≥ n ₂. By the definition of T, this implies that T(f _n ) = 0, ∀n ≥ n ₂. Since g _n ∈ T(f _n ), this implies that g _n = 0, ∀n ≥ n ₂. Since g _n → g, g = 0. Therefore g ∈ T(f).

These show that T is a closed mapping.

Concerning the weak and strong convergence of iterative sequences to approximate a common element of the set of solutions for a generalized MEP, the set of solutions for variational inequality problems, and the set of common fixed points for single-valued relatively non-expansive mappings, single-valued quasi-ϕ-nonexpansive mappings, single-valued quasi-ϕ-asymptotically nonexpansive mappings and single-valued total quasi-ϕ-asymptotically non-expansive mappings have been studied by many authors in the setting of Hilbert or Banach spaces (see, for example, [4–21] and the references therein). Very recently, in 2011, Homaeipour and Razani [3] introduced the concept of multi-valued relatively nonexpansive mappings and proved some weak and strong convergence theorems to approximation a fixed point for a single relatively nonexpansive multi-valued mapping in a uniformly convex and uniformly smooth Banach space X which improve and extend the corresponding results of Matsushita and Takahashi [5].

Motivated and inspired by the researches going on in this direction, the purpose of this article is first to introduce the concept of total quasi-ϕ-asymptotically nonexpansive multi-valued mapping which contains multi-valued relatively nonexpansive mappings and many other kinds of mappings as its special cases, and then by using the hybrid shirking iterative algorithm for finding a common element of the set of solutions for a generalized MEP, the set of solutions for variational inequality problems, and the set of common fixed points for a countable family of multi-valued total quasi-ϕ-asymptotically nonexpansive mappings in a real uniformly smooth and strictly convex Banach space with Kadec-Klee property. The results presented in the article not only generalize the corresponding results of [4–21] from single-valued mappings to multi-valued mappings, but also improve and extend the main results of Homaeipour and Razani [3]. The method given in this article is quite different from that one adopted in [3].

2. Preliminaries

In order to prove our main results, the following conclusions and notations will be needed.

Lemma 2.1[8] Let X be a real uniformly smooth and strictly convex Banach space with Kadec-Klee property, and C be a nonempty closed convex set of X. Let {x _n } and {y _n } be two sequences in C such that x _n → p and ϕ(x _n , y _n ) → 0, where ϕ is the function defined by (1.1), then y _n → p.

Lemma 2.2 Let X and C be as in Lemma 2.1. Let T : C → 2 ^C be a closed and ({ν _n }, {μ _n }, ζ)-total quasi-ϕ-asymptotically nonexpansive multi-valued mapping. If μ₁ = 0, then the fixed point set F (T) of T is a closed and convex subset of C.

Proof Let {x _n } be a sequence in F(T) with x _n → p(as n → ∞), we prove that p ∈ F(T). In fact, by the assumption that T is a ({ν _n }, {μ _n }, ζ)-total quasi-ϕ-asymptotically nonexpansive multi-valued mapping with μ₁ = 0, hence we have

$$\varphi \left(x_n,u\right)\le \varphi \left(x_n,p\right)+{\nu }_1\zeta \left(\varphi \left(x_n,p\right)\right),\forall u\in Tp,$$

and

$$\begin{array}{cc}\hfill \varphi \left(p,u\right)& =\underset{n\to \infty }{lim}\varphi \left(x_n,u\right)\hfill \\ \le \underset{n\to \infty }{lim}\left(\varphi \left(x_n,p\right)+{\nu }_1\zeta \left(\varphi \left(x_n,p\right)\right)\right)=0,\forall u\in Tp.\hfill \end{array}$$

By Lemma 1.2(c), p = u. Hence, p ∈ Tp. This implies that F (T ) is a closed set in C.

Next, we prove that F (T) is convex. For any x, y ∈ F(T), t ∈ (0, 1), putting q = tx + (1 - t)y, we prove that q ∈ F (T ). Indeed, let {u _n } be a sequence generated by

$$\begin{array}{c}{u}_1\in Tq,u_2\in T{u}_1\subset T^{2}q,u_3\in T{u}_2\subset T^{3}q,\dots \\ u_n\in T{u}_{n-1}\subset T^{n}q,\dots \end{array}$$

(2.1)

Therefore for each u _n ∈ Tu_{n- 1}⊂ Tⁿq, we have

$$\begin{array}{cc}\hfill \varphi \left(q,u_n\right)& =\left|\right|q|{|}^2-2⟨q,J{u}_n⟩+|\left|u_n\right|{|}^2\hfill \\ =\left|\right|q|{|}^2-2t⟨x,J{u}_n⟩-2\left(1-t\right)⟨y,J{u}_n⟩+|\left|u_n\right|{|}^2\hfill \\ =\left|\right|q|{|}^2+t\varphi \left(x,u_n\right)+\left(1-t\right)\varphi \left(y,u_n\right)-t|\left|x\right|{|}^2-\left(1-t\right)\left|\right|y|{|}^2\hfill \end{array}$$

(2.2)

Since

$$\begin{array}{c}t\varphi \left(x,u_n\right)+\left(1-t\right)\varphi \left(y,u_n\right)\hfill \\ \le t\left(\varphi \left(x,q\right)+{\nu }_n\zeta \left(\varphi \left(x,q\right)\right)+{\mu }_n\right)+\left(1-t\right)\left(\varphi \left(y,q\right)+{\nu }_n\zeta \left(\varphi \left(y,q\right)\right)+{\mu }_n\right)\hfill \\ =t\left(ǁx{ǁ}^2-2⟨x,Jq⟩+ǁq{ǁ}^2+{\nu }_n\zeta \left(\varphi \left(x,q\right)\right)+{\mu }_n\right)\hfill \\ +\left(1-t\right)\left(ǁy{ǁ}^2-2⟨y,Jq⟩+ǁq{ǁ}^2+{\nu }_n\zeta \left(\varphi \left(y,q\right)\right)+{\mu }_n\right)\hfill \\ =tǁx{ǁ}^2+\left(1-t\right)ǁy{ǁ}^2-ǁq{ǁ}^2+t{\nu }_n\zeta \left(\varphi \left(x,q\right)\right)+\left(1-t\right){\nu }_n\zeta \left(\varphi \left(y,q\right)\right)+{\mu }_n\hfill \end{array}$$

(2.3)

Substituting (2.3) into (2.2) and simplifying we have

$$\varphi \left(q,u_n\right)\le t{\nu }_n\zeta \left(\varphi \left(x,q\right)\right)+\left(1-t\right){\nu }_n\zeta \left(\varphi \left(y,q\right)\right)+{\mu }_n\to 0\left(n\to \infty \right).$$

By Lemma 2.1, we have u _n → q (as n → ∞). This implies that u_n+1→ q (as n → ∞). Since u_n+1∈ Tu _n and T is closed, we have q ∈ Tq, i.e., q ∈ F(T).

This completes the proof of Lemma 2.2.

Lemma 2.3[8] Let X be a uniformly convex Banach space, r > 0 be a positive number and B _r (0) be a closed ball of X. Then for any sequence ${\left\{x_i\right\}}_{i=1}^{\omega }\subset B_r\left(0\right)$ (where ω is any positive integer or +∞) and for any sequence ${\left\{{\lambda }_i\right\}}_{i=1}^{\omega }$ of positive numbers with ${\sum }_{n=1}^{\omega }{\lambda }_n=1$, there exists a continuous, strictly increasing, and convex function g : [0, 2r) → [0, ∞), g(0) = 0 such that for any positive integer i ≠ 1, the following hold:

$$\left|\right|\sum _{n=1}^{\omega }{\lambda }_n{x}_n|{|}^2\le \sum _{n=1}^{\omega }{\lambda }_n|\left|x_n\right|{|}^2-{\lambda }_1{\lambda }_{i}g\left(\left|\right|x_1-x_i\left|\right|\right),$$

(2.4)

and for all x ∈ X

$$\varphi (x,J^{-1}\left(\sum _{i=1}^{\omega }{\lambda }_{i}J{x}_i\right)\le \sum _{i=1}^{\omega }{\lambda }_i\varphi \left(x,x_i\right)-{\lambda }_1{\lambda }_{i}g\left(\left|\right|J{x}_1-J{x}_i\left|\right|\right).$$

(2.5)

For solving the generalized MEP, let us assume that the function $\psi :C\to \mathcal{R}$ is convex and lower semi-continuous, the nonlinear mapping A : C → X* is continuous and monotone, and the bifunction $\mathrm{\Theta }:C\times C\to \mathcal{R}$ satisfies the following conditions:

(A₁) Θ(x, x) = 0, ∀x ∈ C.

(A₂) Θ is monotone, i.e., Θ(x, y) + Θ(y, x) ≤ 0, ∀x, y ∈ C.

(A₃) lim sup _{t ↓0} Θ(x + t(z - x), y) ≤ Θ(x, y), ∀x, y, z ∈ C.

(A₄) The function y ↦ Θ (x, y) is convex and lower semicontinuous.

Lemma 2.4 Let X be a smooth, strictly convex, and reflexive Banach space, and C be a nonempty closed convex subset of X. Let $\mathrm{\Theta }:C\times C\to \mathcal{R}$ be a bifunction satisfying the conditions (A₁)-(A₄). Let r > 0 and x ∈ X. Then, the following hold:

• (i) [12] There exists z ∈ C such that

  $$\mathrm{\Theta }\left(z,y\right)+\frac{1}{r}⟨y-z,Jz-Jx⟩\ge 0,\forall y\in C.$$

• (ii) [13] Define a mapping T _r : X → C by

  $$T_{r}x=\left\{z\in C:\mathrm{\Theta }\left(z,y\right)+\frac{1}{r}⟨y-z,Jz-Jx⟩\ge 0,\forall y\in C\right\},x\in X.$$

Then, the following conclusions hold:

• (a) T _r is single-valued;

• (b) T _r is a firmly nonexpansive-type mapping, i.e., ∀z, y ∈ X,

  $$⟨T_r\left(z\right)-T_r\left(y\right),J{T}_r\left(z\right)-J{T}_r\left(y\right)⟩\le ⟨T_r\left(z\right)-T_r\left(y\right),Jz-Jy⟩;$$

• (c) F(T _r ) = EP(Θ) = F(T _r );

• (d) EP(Θ) is closed and convex;

• (e) ϕ(q, T _r (x)) + ϕ(T _r (x), x) ≤ ϕ(q, x), ∀q ∈ F(T _r ).

Lemma 2.5[18] Let X be a smooth, strictly convex, and reflexive Banach space, and C be a nonempty closed convex subset of X. Let A : C → X* be a continuous and monotone mapping, $\psi :C\to \mathcal{R}$ be a lower semi-continuous and convex function, and $\mathrm{\Theta }:C\times C\to \mathcal{R}$ be a bifunction satisfying the conditions (A₁)-(A₄). Let r > 0 be any given number and x ∈ X be any given point. Then, the following conclusions hold:

• (i) There exists u ∈ C such that ∀y ∈ C

  $$\mathrm{\Theta }\left(u,y\right)+⟨Au,y-u⟩+\psi \left(y\right)-\psi \left(u\right)+\frac{1}{r}⟨y-u,Ju-Jx⟩\ge 0.$$

  (2.6)

• (ii) If we define a mapping K _r : C → C by

  $$\begin{array}{c}{K}_r\left(x\right)=\left\{u\in C:\mathrm{\Theta }\left(u,y\right)+⟨Au,y-u⟩+\psi \left(y\right)-\psi \left(u\right)\right.\\ \left.+\frac{1}{r}⟨y-u,Ju-Jx⟩\ge 0,\forall y\in C\right\},x\in C,\end{array}$$

  (2.7)

then, the mapping K _r has the following properties:

• (a) K _r is single-valued;

• (b) K _r is a firmly nonexpansive-type mapping, i.e., ∀z, y ∈ X

  $$⟨K_r\left(z\right)-K_r\left(y\right),J{K}_r\left(z\right)-J{K}_r\left(y\right)⟩\le ⟨K_r\left(z\right)-K_r\left(y\right),Jz-Jy⟩;$$

• (c) F(K _r ) = Ω = F(K _r );

• (d) Ω is a closed convex set of C;

• (e) ϕ (p, K _r (z)) + ϕ (K _r (z), z) ≤ ϕ (p, z), ∀p ∈ F(K _r ), z ∈ X.

Remark 2.6 It follows from Lemma 2.4 that the mapping K _r : C → C defined by (2.6) is a relatively nonexpansive mapping. Thus, it is quasi-ϕ-nonexpansive.

3. Main results

In this section, we shall use the hybrid iterative algorithm to find a common element of the set of solutions of a generalized MEP, the set of solutions for variational inequality problems, and the set of fixed points of a infinite family of total quasi-ϕ-asymptotically nonexpansive multi-valued mappings. For the purpose we give the following hypotheses:

(H1) X is a uniformly smooth and strictly convex Banach space with Kadec-Klee property and C is a nonempty closed convex subset of X;

(H2) $\mathrm{\Theta }:C\times C\to \mathcal{R}$ is a bifunction satisfying the conditions (A₁)-(A₄), A : C → X* is a continuous and monotone mapping, and $\psi :C\to \mathcal{R}$ is a lower semi-continuous and convex function.

(H3) ${\left\{T_i\right\}}_{i=1}^{\infty }:C\to 2^C$ is a countable family of closed and uniformly ({ν _n }, {μ _n }, ζ)-total quasi-ϕ-asymptotically nonexpansive multi-valued mappings and for each i = 1, 2, . . . , T _i is uniformly L _i -Lipschitzian with μ₁ = 0.

We have the following

Theorem 3.1. Let X, C, Θ, A, ψ, ${\left\{T_i\right\}}_{i=1}^{\infty }$ satisfy the above conditions (H1)-(H3). Let {x _n } be the sequence generated by

$$\left\{\begin{array}{c}{x}_0\in C\mathsf{\text{ chosen arbitrary,}}{C}_0=C,\hfill \\ y_n=J^{-1}\left({\alpha }_{n}J{x}_n+\left(\mathsf{\text{1}}-{\alpha }_n\right)J{z}_n\right),\forall n\ge \mathsf{\text{1,}}\hfill \\ z_n=J^{-1}\left({\beta }_{n,0}J{x}_n+\sum _{i=1}^{\infty }{\beta }_{n,i}J{w}_{n,i}\right),\left(w_{n,i}\in T_i^n{x}_n,i\ge \mathsf{\text{1}}\right),\forall n\ge \mathsf{\text{1,}}\hfill \\ u_n\in Csuchthat\forall y\in C,\forall n\ge \mathsf{\text{1,}}\hfill \\ \mathrm{\Theta }\left(u_n,y\right)+⟨A{u}_n,y-u_n⟩+\psi \left(y\right)-\psi \left(u_n\right)+\frac{1}{r_n}⟨y-u_n,J{u}_n-J{y}_n⟩\ge 0,\hfill \\ C_{n+1}=\left\{\nu \in C_n:\varphi \left(\nu ,u_n\right)\le \varphi \left(\nu ,x_n\right)+{\xi }_n\right\},\forall n\ge 0,\hfill \\ {x_n}_{+\mathsf{\text{1}}}={\mathrm{\Pi }}_{C_{n+1}}{x}_0,\forall n\ge 0,\hfill \end{array}\right.$$

(3.1)

where ${\prod }_{C_{n+1}}$ is the generalized projection of X onto C_n+1, $\mathcal{F}:={\cap }_{i=1}^{\infty }F\left(T_i\right)$, ${\xi }_n={\nu }_n{sup}_{p\in \mathcal{F}}\zeta \left(\varphi \left(p,x_n\right)\right)+{\mu }_n,\left\{{\alpha }_n\right\}$ and {β_n,0,β_n,i} are sequences in 0[1] satisfying the following conditions:

• (i) for each n ≥ 0, ${\sum }_{i=0}^{\infty }{\beta }_{n,i}=1$;

• (ii) lim inf_n→∞ β _{n ,0,} β _ni > 0 for any i ≥ 1;

• (iii) 0 ≤ α _n ≤ α < 1 for some α ∈ (0, 1).

If $\mathcal{G}:=\mathcal{F}\cap \mathrm{\Omega }={\cap }_{i=1}^{\infty }F\left(T_i\right)\cap \mathrm{\Omega }$ is nonempty and is a bounded subset of C, then the sequence {x _n } converges strongly to ∏ _G x₀.

Proof. First, we define two functions $H:C\times C\to \mathcal{R}$ and K _r : C → C by

$$\begin{array}{c}H\left(x,y\right)=\mathrm{\Theta }\left(x,y\right)+⟨Ax,y-x⟩+\psi \left(y\right)-\psi \left(x\right),\forall x,y\in C,\\ K_r\left(x\right)=⟨u\in C:H\left(u,y\right)+\frac{1}{r}⟨y-u,Ju-Jx⟩\ge 0,\forall y\in C\},x\in C.\end{array}$$

By Lemma 2.5, we know that the function H satisfies the conditions (A₁)-(A₄) and K _r has the property (a)-(e). Therefore, (3.1) can be rewritten as

$$\left\{\begin{array}{c}{x}_0\in C\mathsf{\text{ chosen arbitrary,}}{C}_0=C,\hfill \\ y_n=J^{-1}\left({\alpha }_{n}J{x}_n+\left(\mathsf{\text{1}}-{\alpha }_n\right)J{z}_n\right),\forall n\ge \mathsf{\text{1,}}\hfill \\ z_n=J^{-1}\left({\beta }_{n,0}J{x}_n+\sum _{i=1}^{\infty }{\beta }_{n,i}J{w}_{n,i}\right)\left(w_{n,i}\in T_i^n{x}_n,i\ge \mathsf{\text{1}}\right),\forall n\ge \mathsf{\text{1,}}\hfill \\ u_n\in C\mathsf{\text{ such that}}\hfill \\ H\left(u_n,y\right)+\frac{1}{r_n}⟨y-u_n,J{u}_n-J{y}_n⟩\ge 0,\forall y\in C,\forall n\ge 1,\hfill \\ C_{n+1}=\left\{\nu \in C_n:\varphi \left(\nu ,u_n\right)\le \varphi \left(\nu ,x_n\right)+{\xi }_n\right\},\forall n\ge 0,\hfill \\ {x_n}_{+\mathsf{\text{1}}}={\mathrm{\Pi }}_{C_{n+1}}{x}_0\forall n\ge 0.\hfill \end{array}\right.$$

(3.2)

Now we divide the proof of Theorem 3.1 into six steps.

(I)and C _n are closed and convex for each n ≥ 0.

In fact, it follows from Lemma 2.2 that F(T _i ), i ≥ 1 is closed and convex subsets of C. Therefore is a closed and convex subsets in C.

Again by the assumption, C₀ = C is closed and convex. Suppose that C _n is closed and convex for some n ≥ 1. Since the condition ϕ(ν, y _n ) ≤ ϕ (ν, x _n ) + ξ _n is equivalent to

$$\mathsf{\text{2}}⟨\nu ,J{x}_n-J{y}_n⟩\le {∥x_n∥}^{\mathsf{\text{2}}}-{∥y_n∥}^{\mathsf{\text{2}}}\mathsf{\text{ + }}{\xi }_n,n=\mathsf{\text{1,2,}}\dots \mathsf{\text{,}}$$

hence the set

$${C_n}_{+\mathsf{\text{1}}}=\left\{\nu \in C_n:2⟨\nu ,J{x}_n-J{y}_n⟩\le {∥x_n∥}^2-{∥y_n∥}^2+{\xi }_n\right\}$$

is closed and convex. Therefore C _n is closed and convex for each n ≥ 0.

(II) {x _n } is bounded and {ϕ (x _n , x₀)} is a convergent sequence.

Indeed, it follows from (3.1) and Lemma 1.2(a) that for all n ≥ 0, u ∈ F(T )

$$\varphi \left(x_n,x_0\right)=\varphi \left({\mathrm{\Pi }}_{C_n}{x}_0,x_0\right)\le \varphi \left(u,x_0\right)-\varphi \left(u,{\mathrm{\Pi }}_{C_n}{x}_0\right)\le \varphi \left(u,x_0\right).$$

This implies that {ϕ (x _n , x₀)} is bounded. By virtue of (1.6), we know that {x _n } is bounded.

In view of structure of {C _n }, we have $C_{n+1}\subset C_n,x_n={\prod }_{C_n}{x}_0$ and $x_{n+1}={\prod }_{C_{n+1}}{x}_0$. This implies that x_n+1∈ C _n and

$$\varphi \left(x_n,x_0\right)\le \varphi \left({x_n}_{+\mathsf{\text{1}}},x_0\right),\forall n\ge 0.$$

Therefore {ϕ(x _n , x₀)} is a convergent sequence.

(III)$\mathcal{G}:=\mathcal{F}\cap \mathrm{\Omega }\subset C_n$for all n ≥ 0.

Indeed, it is obvious that $\mathcal{G}\subset C_0=C$. Suppose that $\mathcal{G}\subset C_n$ for some $n\in \mathcal{N}$. Since $u_n=K_{r_n}{y}_n$, by Lemma 2.5 and Remark 2.6, $K_{r_n}$ is quasi-ϕ-nonexpansive. Hence, for any given $u\in \mathcal{G}\subset C_n$ and n ≥ 1, it follows from (1.7) that

$$\begin{array}{cc}\hfill \varphi \left(u,u_n\right)& =\varphi \left(u,K_{r_n}{y}_n\right)\le \varphi \left(u,y_n\right)\hfill \\ =\varphi \left(u,J^{-1}\left({\alpha }_{n}J{x}_n+\left(\mathsf{\text{1}}-{\alpha }_n\right)J{z}_n\right)\right)\hfill \\ \le {\alpha }_n\varphi \left(u,x_n\right)+\left(\mathsf{\text{1}}-{\alpha }_n\right)\varphi \left(u,z_n\right).\hfill \end{array}$$

(3.3)

Furthermore, it follows from Lemma 2.3 that for any u ∈ G ⊂ C _n , $w_{n,i}\in T_i^n{x}_n$ and i ≥ 1 we have

$$\begin{array}{cc}\hfill \varphi \left(u,z_n\right)& =\varphi \left(u,J^{-1}\left({\beta }_{n,0}J{x}_n+\sum _{i=1}^{\infty }{\beta }_{n,i}J{w}_{n,i}\right)\right)\hfill \\ \le {\beta }_{n,0}\varphi \left(u,x_n\right)+\sum _{i=1}^{\infty }{\beta }_{n,i}\varphi \left(u,w_{n,i}\right)-{\beta }_{n,0}{\beta }_{n,l}g\left(\left|\right|J{x}_n-J{w}_{n,l}\left|\right|\right)\hfill \\ \le {\beta }_{n,0}\varphi \left(u,x_n\right)+\sum _{i=1}^{\infty }{\beta }_{n,i}\left(\varphi \left(u,w_{n,i}\right)+{\nu }_n\zeta \left(\varphi \left(u,w_{n,i}\right)\right)+{\mu }_n\right)\hfill \\ -{\beta }_{n,0}{\beta }_{n,l}g\left(\left|\right|J{x}_n-J{w}_{n,l}\left|\right|\right)\hfill \\ \le \varphi \left(u,x_n\right)+{\nu }_n\underset{p\in \mathcal{F}}{sup}\zeta \left(\varphi \left(p,x_n\right)\right)+{\mu }_n-{\beta }_{n,0}{\beta }_{n,l}g\left(∥J{x}_n-J{w}_{n,l}∥\right)\hfill \\ =\varphi \left(u,x_n\right)+{\xi }_n-{\beta }_{n,0}{\beta }_{n,l}g\left(∥J{x}_n-J{w}_{n,l}∥\right),\hfill \end{array}$$

(3.4)

where ${\xi }_n={\nu }_n{sup}_{p\in \mathcal{F}}\zeta \left(\varphi \left(p,x_n\right)\right)$. Substituting (3.4) into (3.3) and simplifying, $\forall u\in \mathcal{G}$ we have

$$\begin{array}{cc}\hfill \varphi \left(u,u_n\right)& \le \varphi \left(u,y_n\right)\hfill \\ \le \varphi \left(u,x_n\right)+\left(1-{\alpha }_n\right){\xi }_n-\left(\mathsf{\text{1}}-{\alpha }_n\right){\beta }_{n,0}{\beta }_{n,l}g\left(\left|\right|J{x}_n-J{w}_{n,l}\left|\right|\right)\hfill \\ \le \varphi \left(u,x_n\right)+{\xi }_n-\left(\mathsf{\text{1}}-{\alpha }_n\right){\beta }_{n,0}{\beta }_{n,l}g\left(\left|\right|J{x}_n-J{w}_{n,l}\left|\right|\right)\hfill \\ \le \varphi \left(u,x_n\right)+{\xi }_n,\hfill \end{array}$$

(3.5)

i.e., u ∈ C_n+1and so $\mathcal{G}\subset C_{n+1}$ for all n ≥ 0.

By the way, in view of the assumption on {ν _n }, {μ _n } we have

$${\xi }_n={\nu }_n\underset{p\in \mathcal{F}}{sup}\zeta \left(\varphi \left(p,x_n\right)\right)+{\mu }_n\to 0\left(n\to \infty \right).$$

(IV) {x _n } converges strongly to some point p* ∈ C.

In fact, since {x _n } is bounded and X is reflexive, there exists a subsequence $\left\{x_{n_i}\right\}\subset \left\{x_n\right\}$ such that $x_{n_i}\rightharpoonup p^{*}$ (some point in C). Since C _n is closed and convex and C_n+1⊂ C _n , this implies that C _n is weakly closed and p* ∈ C _n for each n ≥ 0. In view of $x_{n_i}={\mathrm{\Pi }}_{C_{n_i}}{x}_0$, we have

$$\varphi \left(x_{n_i},x_0\right)\le \varphi \left(p^{*},x_0\right),\forall n_i\ge 0.$$

Since the norm || · || is weakly lower semi-continuous, we have

$$\begin{array}{cc}\hfill \underset{n_i\to \infty }{liminf}\varphi \left(x_{n_i},x_0\right)& =\underset{n_i\to \infty }{liminf}\left(\left|\right|x_{n_i}|{|}^2-\mathsf{\text{2}}⟨x_{ni},J{x}_0⟩+|\left|x_0\right|{|}^2\right)\hfill \\ \ge \left|\right|p^{*}|{|}^{\mathsf{\text{2}}}-\mathsf{\text{2}}⟨p^{*},J{x}_0⟩+|\left|x_0\right|{|}^{\mathsf{\text{2}}}=\varphi \left(p^{*},x_0\right),\hfill \end{array}$$

and so

$$\varphi \left(p^{*},x_0\right)\le \underset{n_i\to \infty }{liminf}\varphi \left(x_{n_i},x_0\right)\le \underset{n_i\to \infty }{limsup}\varphi \left(x_{n_i},x_0\right)\le \varphi \left(p^{*},x_0\right).$$

This implies that $\underset{n_i\to \infty }{lim}\varphi \left(x_{n_i},x_0\right)=\varphi \left(p^{*},x_0\right)$, and so $\left|\right|x_{n_i}\left|\right|\to \left|\right|p^{*}\left|\right|$. Since $x_{n_i}\rightharpoonup p^{*}$, by virtue of Kadec-Klee property of X, we obtain that

$$\underset{n_i\to \infty }{lim}{x}_{n_i}=p^{*}.$$

Since {ϕ(x _n , x₀)} is convergent, this together with ${lim}_{n_i\to \infty }\varphi \left(x_{n_i},x_0\right)=\varphi \left(p^{*},x_0\right)$, which shows that lim_n→∞ϕ(x _n , x₀) = ϕ(p*, x₀). If there exists some sequence $\left\{x_{n_i}\right\}\subset \left\{x_n\right\}$ such that $x_{n_j}\to q$, then from Lemma 1.2(a) we have that

$$\begin{array}{cc}\hfill \varphi \left(p^{*},q\right)& =\underset{n_i,n_j\to \infty }{lim}\varphi \left(x_{n_i},x_{n_j}\right)=\underset{n_i,n_j\to \infty }{lim}\varphi \left(x_{n_i},{\mathrm{\Pi }}_{C_{n_j}}{x}_0\right)\hfill \\ \le \underset{n_i,n_j\to \infty }{lim}\left(\varphi \left(x_{n_i},x_0\right)-\varphi \left({\mathrm{\Pi }}_{C_{n_j}}{x}_0,x_0\right)\right)\hfill \\ =\underset{n_i,n_j\to \infty }{lim}\left(\varphi \left(x_{n_i},x_0\right)-\varphi \left(x_{n_j},x_0\right)\right)\hfill \\ =\varphi \left(p^{*},x_0\right)-\varphi \left(p^{*},x_0\right)=0.\hfill \end{array}$$

This implies that p* = q and

$$\underset{n\to \infty }{lim}{x}_n=p^{*}.$$

(3.6)

(V) Now we prove that$p^{*}\in G=\mathcal{F}\cap \mathrm{\Omega }$.

First, we prove that $p^{*}\in \mathcal{F}$. In fact, since x_n+1∈ C_n+1⊂ C _n , it follows from (3.1) and (3.6) that

$$\varphi \left({x_n}_{+\mathsf{\text{1}}},y_n\right)\le \varphi \left({x_n}_{+\mathsf{\text{1}}},x_n\right)+{\xi }_n\to 0\left(n\to \infty \right).$$

By the virtue of Lemma 2.1, we have

$$\underset{n\to \infty }{lim}{y}_n=p^{*}.$$

(3.7)

From (3.5), for any $u\in \mathcal{F}$ and $w_{n,i}\in T_i^n{x}_n$, we have

$$\varphi \left(u,y_n\right)\le \varphi \left(u,x_n\right)+{\xi }_n-\left(\mathsf{\text{1}}-{\alpha }_n\right){\beta }_{n,0}{\beta }_{n,l}g\left(\left|\right|J{x}_n-J{w}_{n,l}\left|\right|\right),$$

i.e.,

$$\left(\mathsf{\text{1}}-{\alpha }_n\right){\beta }_{n,0}{\beta }_{n,l}g\left(\left|\right|J{x}_n-J{w}_{n,l}\left|\right|\right)\le \varphi \left(u,x_n\right)\mathsf{\text{ + }}{\xi }_n-\varphi \left(u,y_n\right)\to 0\left(n\to \infty \right).$$

By conditions (ii) and (iii) it shows that lim_n→∞g(||Jx _n - Jw_n,l||) = 0. In view of property of g, we have

$$\left|\right|J{x}_n-J{w}_{n,l}\left|\right|\to 0\left(n\to \infty \right),\forall l\ge 1.$$

Since Jx _n → Jp*, this implies that Jw_n,l→ Jp*. From Remark 1.1 (ii) it yields

$$w_{n,l}\rightharpoonup p^{*}\left(n\to \infty \right),\forall l\ge 1.$$

(3.8)

Again since

$$\left|\right|\left|w_{n,l}\right||-|\left|p^{*}\right|\left|\right|=\left|\right|\left|J{w}_{n,l}\right||-|\left|J{p}^{*}\right|\left|\right|\le \left|\right|J{w}_{n,l}-J{p}^{*}\left|\right|\to 0\left(n\to \infty \right),$$

this together with (3.8) and the Kadec-Klee property of X shows that

$$\underset{n\to \infty }{lim}{w}_{n,l}=p^{*},\forall l\ge 1.$$

(3.9)

Let {s_n,l} be a sequence generated by

$$\begin{array}{c}{s}_{2,l}\in T_l{w}_{1,l}\subset T_l^2{x}_1,s_{3,l}\in T_l{w}_{2,l}\subset T_l^3{x}_2,\dots ,\\ s_{n+1,l}\in T_l{w}_{n,l}\subset T_l^{n+1}{x}_n,\dots ,l\ge 1\end{array}$$

By the assumption that each T _i is uniformly L _i -Lipschitz continuous, hence for any $w_{n,l}\in T_l^n{x}_n$ and $s_{n+1,l}\in T_l{w}_n\subset T_l^{n+1}{x}_n$ we have

$$\begin{array}{cc}\hfill \left|\right|s_{n+1,l}-w_{n,l}\left|\right|& \le \left|\right|s_{n+1,l}-w_{n+1,l}\left|\right|+\left|\right|w_{n+1,l}-x_{n+1}\left|\right|+\left|\right|x_{n+1}-x_n\left|\right|+\left|\right|x_n-w_{n,l}\left|\right|\hfill \\ \le \left(L_l+1\right)\left|\right|x_{n+1}-x_n\left|\right|+\left|\right|w_{n+1,l}-x_{n+1}\left|\right|+\left|\right|x_n-w_{n,l}\left|\right|.\hfill \end{array}$$

(3.10)

This together with (3.6) and (3.10) shows that

$$\begin{array}{c}\underset{n\to \infty }{lim}\left|\right|s_{n+1,l}-w_{n,l}\left|\right|=0,and\hfill \\ \underset{n\to \infty }{lim}{s}_{n+1,l}=p^{*}.\hfill \end{array}$$

In view of the closeness of T _l , it yields that p* ∈ Tp*, i.e., p* ∈ F (T _l ). By the arbitrariness of l ≥ 1, we have

$$p^{*}\in \mathcal{F}={\cap }_{i=1}^{\infty }F\left(T_i\right).$$

Next, we prove that p* ∈ Ω. Since $x_{n+1}={\mathrm{\Pi }}_{C_{n+1}}{x}_0\in C_n$, it follows from (3.1) and (3.6) that

$$\varphi \left(x_{n+1},u_n\right)\le \varphi \left(x_{n+1},x_n\right)+{\xi }_n\to 0\left(n\to \infty \right).$$

Since x _n → p*, by virtue of Lemma 2.1 we have

$$\underset{n\to \infty }{lim}{u}_n=p^{*}.$$

(3.11)

This together with (3.7) shows that ||u _n - y _n || → 0 and lim _{n →∞} ||Ju _n - Jy _n || → 0. By the assumption that r _n ≥ a, ∀n ≥ 0, we have

$$\underset{n\to \infty }{lim}\frac{\left|\right|J{u}_n-J{y}_n\left|\right|}{r_n}=0.$$

(3.12)

Since $H\left(u_n,y\right)+\frac{1}{r_n}⟨y-u_n,J{u}_n-J{y}_n⟩\ge 0,\forall y\in C$, by condition (A₁), we have

$$\frac{1}{r_n}⟨y-u_n,J{u}_n-J{y}_n⟩\ge -H\left(u_n,y\right)\ge H\left(y,u_n\right),\forall y\in C.$$

(3.13)

By the assumption that y ↦ H(x, y) is convex and lower semi-continuous, letting n → ∞ in (3.13), from (3.11) and (3.12), we have H(y, p*) ≤ 0, ∀y ∈ C.

For t ∈ (0, 1] and y ∈ C, letting y _t = ty + (1 - t)p*, therefore y _t ∈ C and H(y _t , p*) ≤ 0. By condition (A₁) and (A₄), we have

$$0=H\left(y_t,y_t\right)\le tH\left(y_t,y\right)+\left(1-t\right)H\left(y_t,p^{*}\right)\le tH\left(y_t,y\right).$$

Dividing both sides of the above equation by t, we have H(y _t , y) ≤ 0, ∀y ∈ C. Letting t ↓ 0, from condition (A₃), we have H(p*, y) ≤ 0, ∀y ∈ C, i.e., p* ∈ Ω, and $p^{*}\in \mathcal{G}=\mathcal{F}\bigcap \mathrm{\Omega }$.

(VI) we prove that$x_n\to p^{*}={\mathrm{\Pi }}_{\mathcal{G}}{x}_0$.

Let q = ∏ _G x₀. Since $q\in \mathcal{G}\subset C_n$ and x _n = ∏ _Cn x₀, we have

$$\varphi \left(x_n,x_0\right)\le \varphi \left(q,x_0\right),\forall n\ge 0.$$

This implies that

$$\varphi \left(p^{*},x_0\right)=\underset{n\to \infty }{lim}\varphi \left(x_n,x_0\right)\le \varphi \left(q,x_0\right).$$

(3.14)

In view of the definition of ${\mathrm{\Pi }}_{\mathcal{G}}{x}_0$, from (3.14) we have p* = q. Therefore, $x_n\to p^{*}={\mathrm{\Pi }}_{\mathcal{G}}{x}_0$. This completes the proof of Theorem 3.1.

Definition 3.2 A finite family of multi-valued mappings ${\left\{T_i\right\}}_{i=1}^{\infty }:C\to 2^C$ is said to be uniformly quasi-ϕ-asymptotically nonexpansive, if $\mathcal{F}={\bigcap }_{i=1}^{\infty }F\left(T_i\right)\ne \varnothing$ and there exists a real sequence {k _n } ⊂ [1, ∞), k _n → 1 such that for each i = 1, 2, . . . , N

$$\varphi \left(p,w_{n,i}\right)\le k_n\varphi \left(p,x\right),\forall x\in C,p\in \bigcap _{i=1}^{\infty }F\left(T_i\right),w_{n,i}\in T_i^{n}x$$

(3.15)

The following theorems can be obtained from Theorem 3.1 immediately.

Theorem 3.3 Let X, C, Θ, A, ψ be as in Theorem 3.1. Let ${\left\{T_i\right\}}_{i=1}^{\infty }$ be a countable family of closed and uniformly quasi-ϕ-asymptotically nonexpansive multi-valued mappings with a real sequence {k _n } ⊂ [1, ∞), k _n → 1 and for each i = 1, 2, . . . , T _i be uniformly L _i -Lipschitzian. Let {x _n } be the sequence generated by

$$\left\{\begin{array}{c}{x}_0\in C\mathsf{\text{chosen arbitrary}},C_0=C,\hfill \\ y_n=J^{-1}\left({\alpha }_{n}J{x}_n+\left(1-{\alpha }_n\right)J{z}_n\right),\forall n\ge 1,\hfill \\ z_n=J^{-1}\left({\beta }_{n,0}J{x}_n+\sum _{i=1}^{\infty }{\beta }_{n,i}J{w}_{n,i}\right),\left(w_{n,i}\in T_i^n{x}_n,i\ge 1\right),\forall n\ge 1,\hfill \\ u_n\in Csuchthat\forall y\in C\hfill \\ \mathrm{\Theta }\left(u_n,y\right)+⟨A{u}_n,y-u_n⟩+\psi \left(y\right)-\psi \left(u_n\right)+\frac{1}{r_n}⟨y-u_n,J{u}_n-J{y}_n⟩\ge 0,\hfill \\ C_{n+1}=\left\{\nu \in C_n:\varphi \left(\nu ,u_n\right)\le \varphi \left(\nu ,x_n\right)+{\xi }_n\right\},\forall n\ge 0,\hfill \\ x_{n+1}={\mathrm{\Pi }}_{C_{n+1}}{x}_0,\forall n\ge 0,\hfill \end{array}\right.$$

(3.16)

where $\mathcal{F}:={\bigcap }_{i=1}^{\infty }F\left(T_i\right),{\xi }_n=\left(k_n-1\right){sup}_{p\in \mathcal{F}}\zeta \left(\varphi \left(p,x_n\right)\right),{\left\{{\beta }_{n,0},{\beta }_{n,i}\right\}}_{i=1}^{\infty },$and {α _n } are sequences in 0[1] satisfying the conditions (i), (ii), (iii) in Theorem 3.1. If $\mathcal{F}:={\bigcup }_{i=1}^{\infty }F\left(T_i\right)$ is a bounded subset of C, then {x _n } converges strongly to ${\mathrm{\Pi }}_{\mathcal{G}}{x}_0$.

Proof. Since ${\left\{T_i\right\}}_{i=1}^{\infty }$ is a countable family of closed and uniformly quasi-ϕ-asymptotically nonexpansive multi-valued mappings, by Remark 1.5(2), it is a countable family of closed and uniformly total quasi-ϕ-asymptotically nonexpansive multi-valued mappings with non-negative sequences {ν _n = (k _n - 1)}, {μ _n = 0} and a strictly increasing and continuous function ζ(t) = t, t ≥ 0. Hence ${\xi }_n=\left(k_n-1\right){sup}_{p\in \mathcal{F}}\varphi \left(p,x_n\right)\to 0$ (as n → ∞). Therefore all conditions in Theorem 3.1 are satisfied. The conclusion of Theorem 3.3 can be obtained from Theorem 3.1 immediately.

Theorem 3.4 Let X, C, Θ, A, ψ be as in Theorem 3.1. Let ${\left\{T_i\right\}}_{i=1}^{\infty }$ be a countable family of closed and quasi-ϕ-nonexpansive multi-valued mappings. Let {x _n } be the sequence generated by

$$\left\{\begin{array}{c}{x}_0\in C\mathsf{\text{chosen arbitrary}},C_0=C,\hfill \\ y_n=J^{-1}\left({\alpha }_{n}J{x}_n+\left(1-{\alpha }_n\right)J{z}_n\right),\forall n\ge 1,\hfill \\ z_n=J^{-1}\left({\beta }_{n,0}J{x}_n+\sum _{i=1}^{\infty }{\beta }_{n,i}J{w}_{n,i}\right),\left(w_{n,i}\in T_i^n{x}_n,i\ge 1\right),\forall n\ge 1,\hfill \\ u_n\in Csuchthat\forall y\in C,\forall n\ge 1,\hfill \\ \mathrm{\Theta }\left(u_n,y\right)+⟨A{u}_n,y-u_n⟩+\psi \left(y\right)-\psi \left(u_n\right)+\frac{1}{r_n}⟨y-u_n,J{u}_n-J{y}_n⟩\ge 0,\hfill \\ C_{n+1}=\left\{\nu \in C_n:\varphi \left(\nu ,u_n\right)\le \varphi \left(\nu ,x_n\right)\right\},\forall n\ge 0,\hfill \\ x_{n+1}={\mathrm{\Pi }}_{C_{n+1}}{x}_0,\forall n\ge 0,\hfill \end{array}\right.$$

(3.17)

where ${\left\{{\beta }_{n,0},{\beta }_{n,i}\right\}}_{i=1}^{\infty }$ and {α _n } are sequences in 0[1] satisfying the conditions (i), (ii), (iii) in Theorem 3.1. If $\mathcal{F}:={\bigcap }_{i=1}^{\infty }F\left(T_i\right)\ne \varnothing$, then {x _n } converges strongly to ${\mathrm{\Pi }}_{\mathcal{G}}{x}_0$.

Proof. Since ${\left\{T_i\right\}}_{i=1}^{\infty }$ is a countable family of closed quasi-ϕ-nonexpansive multi-valued mappings, by Remark 1.5(3), it is a countable of closed and uniformly quasi-ϕ- asymptotically nonexpansive multi-valued mappings with sequence {k _n = 1}. Hence ${\xi }_n=\left(k_n-1\right){sup}_{u\in \mathcal{F}}\varphi \left(u,x_n\right)=0$ Therefore, the conditions appearing in Theorem 3.3: " is a bounded subset in C" and "for each i ≥ 1, T _i is uniformly L _i -Lipschitz" is no use here. Therefore all conditions in Theorem 3.3 are satisfied. The conclusion of Theorem 3.4 can be obtained from Theorem 3.3 immediately.

Remark 3.5 Theorems 3.1, 3.3, and 3.4 not only generalize the corresponding results of Matsushita and Takahashi [5], Plubtieng and Ungchittrakool [6], Ceng et al. [9], Su et al. [10], Ofoedu and Malonza [11], Wang et al. [12], Chang et al. [4, 7, 8, 13, 17, 19, 20], Yao et al. [14], Zegeye et al. [15] and Nilsrakoo and Saejung [16] from single-valued mappings to multi-valued mappings, but also improve and extend the main results of Homaeipour and Razani [3] and the method adopted in this article is also different from that one adopted in [3].

4. Applications

In this section, we shall utilize the results presented in Section 3 to study some problems.

(I) Application to convex feasibility problem.

The "so called" convex feasibility problem for a family of mappings ${\left\{T_i\right\}}_{i=1}^{\omega }$ (where ω is a finite positive integer or +∞) is to finding a point in the nonempty intersection ${\bigcap }_{i=1}^{\omega }{C}_i$, where C _i is a fixed point set of T _i , i = 1, 2, . . . , ω.

In Theorem 3.4 if Θ = 0, A = 0, ψ = 0, then by Lemma 1.2(c), the condition "u _n ∈ C such that ∀y ∈ C, 〈y - u _n , Ju _n - Jy _n 〉 ≥ 0" is equivalent to u _n = Π _C (y _n ). Hence from Theorem 3.4, the iterative sequence {x _n } defined by

$$\left\{\begin{array}{c}{x}_0\in C\mathsf{\text{chosen arbitrary}},C_0=C,\hfill \\ y_n=J^{-1}\left({\alpha }_{n}J{x}_n+\left(1-{\alpha }_n\right)J{z}_n\right),\forall n\ge 1,\hfill \\ z_n=J^{-1}\left({\beta }_{n,0}J{x}_n+\sum _{i=1}^{\infty }{\beta }_{n,i}J{w}_{n,i}\right),\left(w_{n,i}\in T_i^n{x}_n,i\ge 1\right),\forall n\ge 1,\hfill \\ C_{n+1}=\left\{\nu \in C_n:\varphi \left(\nu ,u_n\right)\le \varphi \left(\nu ,x_n\right)\right\},u_n={\mathrm{\Pi }}_C{y}_n,\forall n\ge 1,\hfill \\ x_{n+1}={\mathrm{\Pi }}_{C_{n+1}}{x}_0,\forall n\ge 0,\hfill \end{array}\right.$$

(4.1)

converges strongly to a point $p^{*}={\prod }_{\mathcal{F}}{x}_0$, which is a solution of the convex feasibility problem for a countable family of closed and quasi-ϕ-nonexpansive multi-valued mappings ${\left\{T_i\right\}}_{i=1}^{\infty }$ where $\mathcal{F}={\bigcap }_{i=1}^{\infty }F\left(T_i\right)$.

(II) Application to generalized MEP

In Theorem 3.4 taking T _i = I, ∀i ≥ 1, (the identity mapping on C), then $z_n=y_n=x_n,\forall n\ge 1,\mathcal{F}=C,\mathcal{G}=\mathrm{\Omega }$. By Theorem 3.4 the sequence {x _n } defined by

$$\left\{\begin{array}{c}{x}_0\in C\mathsf{\text{chosen arbitrary}},C_0=C,\hfill \\ u_n\in Csuchthat\forall y\in C,\forall n\ge 1,\hfill \\ \mathrm{\Theta }\left(u_n,y\right)+⟨A{u}_n,y-u_n⟩+\psi \left(y\right)-\psi \left(u_n\right)+\frac{1}{r_n}⟨y-u_n,J{u}_n-J{x}_n⟩\ge 0,\hfill \\ C_{n+1}=\left\{\nu \in C_n:\varphi \left(\nu ,u_n\right)\le \varphi \left(\nu ,x_n\right)\right\},\forall n\ge 0,\hfill \\ x_{n+1}={\mathrm{\Pi }}_{C_{n+1}}{x}_0,\forall n\ge 0.\hfill \end{array}\right.$$

(4.2)

converges strongly to a point p* = ∏_Ωx₀, which is a solution of the generalized MEP (1.1).

(III) Application to optimization problem

In (4.2), if Θ = 0, A = 0, then from Theorem 3.4 the sequence {x _n } defined by

$$\left\{\begin{array}{c}{x}_0\in C\mathsf{\text{chosen arbitrary}},C_0=C,\hfill \\ u_n\in Csuchthat\forall y\in C\hfill \\ \psi \left(y\right)-\psi \left(u_n\right)+\frac{1}{r_n}⟨y-u_n,J{u}_n-J{x}_n⟩\ge 0,\hfill \\ C_{n+1}=\left\{\nu \in C_n:\varphi \left(\nu ,u_n\right)\le \varphi \left(\nu ,x_n\right)\right\},\forall n\ge 0,\hfill \\ x_{n+1}={\mathrm{\Pi }}_{C_{n+1}}{x}_0.\hfill \end{array}\right.$$

(4.3)

converges strongly to a point p* = ∏ _K x₀ which is a solution of the optimization problem min _x∈C ψ(x), where K ⊂ C is the set of solutions to this optimization problem.

(IV) Application to the mixed variational inequality problem of Browder type

In (4.2), if Θ = 0, then the iterative sequence {x _n } defined by

$$\left\{\begin{array}{c}{x}_0\in C\mathsf{\text{chosen arbitrary}},C_0=C,\hfill \\ u_n\in Csuchthat\forall y\in C\hfill \\ ⟨A{u}_n,y-u_n⟩+\psi \left(y\right)-\psi \left(u_n\right)+\frac{1}{r_n}⟨y-u_n,J{u}_n-J{x}_n⟩\ge 0,\hfill \\ C_{n+1}=\left\{\nu \in C_n:\varphi \left(\nu ,u_n\right)\le \varphi \left(\nu ,x_n\right)\right\},\forall n\ge 0,\hfill \\ x_{n+1}={\mathrm{\Pi }}_{C_{n+1}}{x}_0.\hfill \end{array}\right.$$

(4.4)

converges strongly to a point p* = ∏ _Q x₀ which is a solution of the mixed variational inequality of Browder type (1.4), where Q is the set of solutions to equation (1.4).