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

Shih-Sen Chang; Lin Wang; Yong-Kun Tang; Yun-He Zhao; Zao-Li Ma

doi:10.1186/1687-1812-2012-69

Research
Open Access

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

Shih-Sen Chang¹Email author,
Lin Wang¹,
Yong-Kun Tang¹,
Yun-He Zhao¹ and
Zao-Li Ma²

Fixed Point Theory and Applications20122012:69

https://doi.org/10.1186/1687-1812-2012-69

Received: 16 January 2012
Accepted: 30 April 2012
Published: 30 April 2012

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.

Keywords

multi-valued total quasi-ϕ-asymptotically nonexpansive mappings
quasi-ϕ-asymptotically nonexpansive multi-valued mappings
quasi-ϕ-nonexpansive multi-valued mappings
relatively nonexpansive multi-valued mappings
generalized projection

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 (x) = {f^{*} \in X^{*} : 〈 x, f^{*} 〉 = ǁ x ǁ^{2} = ǁ f^{*} ǁ^{2}}, x \in E .

In the sequel, we use F(T ) to denote the set of fixed points of a mapping T , and use and $R^{+}$ 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

Θ : C \times C \to R

be a bifunction,

ψ : C \to 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

Θ (u, y) + ⟨ A u, y - u ⟩ + ψ (y) - ψ (u) \geq 0, \forall y \in C .

(1.1)

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

Ω = {u \in C : Θ (u, y) + ⟨ A u, y - u ⟩ + ψ (y) - ψ (u) \geq 0, \forall y \in C} .

(1.2)

Special examples:

(I)

If A ≡ 0, the problem (1.1) is equivalent to finding u ∈ C such that
$Θ (u, y) + ψ (y) - ψ (u) \geq 0, \forall y \in C .$

(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
$⟨ A u, y - u ⟩ + ψ (y) - ψ (u) \geq 0, \forall y \in C .$

(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 - δ

for all x, y ∈ U with ||x - y|| ≥ ϵ. X is said to be smooth if the limit

lim_{t \to 0} \frac{| | x + t y | | - | | x | |}{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

ϕ : X \times X \to R^{+}

to denote the Lyapunov functional which is defined by

ϕ (x, y) = ǁ x ǁ^{2} - 2 ⟨ x, J y ⟩ + ǁ y ǁ^{2}, \forall x, y \in X .

(1.5)

It is obvious from the definition of ϕ that

{(| | x | | - | | y | |)}^{2} \leq ϕ (x, y) \leq {(| | x | | + | | y | |)}^{2}, \forall x, y \in X .

(1.6)

and

ϕ (x, J^{- 1} (λ J y + (1 - λ) J z)) \leq λ ϕ (x, y) + (1 - λ) ϕ (x, z),

(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

Π_{C} (x) = arg inf_{y \in C} ϕ (y, x), \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, ∏ _Cy) + ϕ (∏ _Cy, y) ≤ ϕ (x, y) for all x ∈ C and y ∈ X;
(b)

If x ∈ X and z ∈ C, then
$z = Π_{C} x \Leftrightarrow ⟨ z - y, J x - J z ⟩ \geq 0, \forall y \in C;$
(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
$lim_{n \to \infty} d (x_{n}, T (x_{n})) : = lim_{n \to \infty} inf_{y \in T (x_{n})} | | x_{n} - y | | = 0 .$

In the sequel we use

\hat{F} (T)

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)

$\hat{F} (T) = F (T)$ .

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

ϕ (p, w) \leq ϕ (p, x), \forall x \in C, w \in T x, p \in F (T) .

(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
$ϕ (p, w_{n}) \leq k_{n} ϕ (p, x), \forall n \geq 1, x \in C, w_{n} \in T^{n} x, p \in F (T) .$

(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 $ζ : R^{+} \to R^{+}$ with ζ (0) = 0 such that for all x ∈ C, p ∈ F(T )
$ϕ (p, w_{n}) \leq ϕ (p, x) + ν_{n} ζ (ϕ (p, x)) + μ_{n}, \forall n \geq 1, w_{n} \in T^{n} x .$

(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
$| | w_{n} - s_{n} | | \leq L | | x - y | |, \forall x, y \in C, w_{n} \in T^{n} x, s_{n} \in T^{n} y, n \geq 1 .$
(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 ${T_{i}}_{i = 1}^{\infty} : C \to 2^{C}$ is said to be uniformly ({ν_n }, {μ_n }, ζ)-total quasi-ϕ-asymptotically nonexpansive, if $F : = ⋂_{i = 1}^{\infty} F (T_{i}) \neq \emptyset$ and there exist nonnegative real sequences ({ν_n }, {μ_n } with ν_n → 0, μ_n → 0 and a strictly increasing continuous function $ζ : R^{+} \to R^{+}$ with ζ(0) = 0 such that for all x ∈ C, $p \in F$
$ϕ (p, w_{n, i}) \leq ϕ (p, x) + ν_{n} ζ (ϕ (p, x)) + μ_{n}, \forall n \geq 1, w_{n, i} \in T_{i}^{n} x, i = 1, 2, \dots .$

(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
$ϕ (p, w_{n}) \leq ϕ (p, x) + ν_{n} ζ (ϕ (p, x)) + μ_{n}, \forall n \geq 1, x \in C, w_{n} \in T^{n} x, p \in F (T),$

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 : (x_{1}, x_{2}, \dots,) \to (0, x_{1}^{2}, a_{2} x_{2}, a_{3} x_{3}, \dots), (x_{1}, x_{2}, \dots,) \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 (f) = \{\begin{matrix} {g \in D : a \leq f (x) - g (x) \leq b, \forall x \in I}, i f f (x) > 1, \forall x \in I; \\ {0}, o t h e r w i s e . \end{matrix}

(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
$ϕ (p, g) = ϕ (0, g) = ǁ g ǁ_{C}^{2} \leq ǁ f ǁ_{C}^{2} = ϕ (0, f) = ϕ (p, f) .$

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

ϕ (p, g) = ϕ (0, 0) = 0 \leq ǁ f ǁ_{C}^{2} = ϕ (0, f) = ϕ (p, f) .

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

ϕ (p, g) \leq ϕ (p, f), \forall p \in F (T), g \in T (f),

(1.13)

(II)

For any $g \in T^{2} (f) = T (T (f)) = ⋃_{g_{1} \in T (f)} T (g_{1})$ , there exists some $g_{1}^{*} \in T (f)$ such that $g \in T (g_{1}^{*})$ .
(1)

If $g_{1}^{*} > 1, \forall x \in I$ , then we have $a \leq g_{1}^{*} - g < b$ . By (1.13), for any p ∈ F(T) = {0}, we have
$ϕ (p, g) = ϕ (0, g) = ǁ g ǁ_{C}^{2} \leq ǁ g_{1}^{*} ǁ_{C}^{2} = ϕ (0, g_{1}^{*}) = ϕ (p, g_{1}^{*}) \leq ϕ (p, f) .$
(2)

If there exists x ₁ ∈ I such that $0 \leq g_{1}^{*} (x_{1}) \leq 1$ , then by the definition of T , we have $T g_{1}^{*} = {0}$ . Since $g \in T g_{1}^{*} = {0}$ , and so g = 0. Hence for any p ∈ F(T), by (1.13) we have
$ϕ (p, g) = ϕ (0, 0) = 0 \leq ǁ g_{1}^{*} ǁ^{2} = ϕ (0, g_{1}^{*}) = ϕ (p, g_{1}^{*}) \leq ϕ (p, f) .$

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

ϕ (p, g) \leq ϕ (p, f), \forall p \in F (T), g \in T^{2} (f),

(1.14)

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

ϕ (p, g) \leq ϕ (p, f) .

(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

ϕ (p, g) \leq ϕ (p, f) + ν_{n} ζ (ϕ (p, f)) + μ_{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
$a \leq f_{n} (x) - g_{n} (x) \leq b, \forall n \geq 1 a n d x \in I .$

(1.16)

Letting n → ∞ in (1.16), we have

a \leq f (x) - g (x) \leq b, \forall n \geq 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

ϕ (x_{n}, u) \leq ϕ (x_{n}, p) + ν_{1} ζ (ϕ (x_{n}, p)), \forall u \in T p,

and

\begin{aligned} ϕ (p, u) & = lim_{n \to \infty} ϕ (x_{n}, u) \\ \leq lim_{n \to \infty} (ϕ (x_{n}, p) + ν_{1} ζ (ϕ (x_{n}, p))) = 0, \forall u \in T p . \end{aligned}

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{gathered} u_{1} \in T q, 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{gathered}

(2.1)

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

\begin{aligned} ϕ (q, u_{n}) & = | | q | |^{2} - 2 ⟨ q, J u_{n} ⟩ + | | u_{n} | |^{2} \\ = | | q | |^{2} - 2 t ⟨ x, J u_{n} ⟩ - 2 (1 - t) ⟨ y, J u_{n} ⟩ + | | u_{n} | |^{2} \\ = | | q | |^{2} + t ϕ (x, u_{n}) + (1 - t) ϕ (y, u_{n}) - t | | x | |^{2} - (1 - t) | | y | |^{2} \end{aligned}

(2.2)

Since

\begin{aligned} t ϕ (x, u_{n}) + (1 - t) ϕ (y, u_{n}) \\ \leq t (ϕ (x, q) + ν_{n} ζ (ϕ (x, q)) + μ_{n}) + (1 - t) (ϕ (y, q) + ν_{n} ζ (ϕ (y, q)) + μ_{n}) \\ = t (ǁ x ǁ^{2} - 2 ⟨ x, J q ⟩ + ǁ q ǁ^{2} + ν_{n} ζ (ϕ (x, q)) + μ_{n}) \\ + (1 - t) (ǁ y ǁ^{2} - 2 ⟨ y, J q ⟩ + ǁ q ǁ^{2} + ν_{n} ζ (ϕ (y, q)) + μ_{n}) \\ = t ǁ x ǁ^{2} + (1 - t) ǁ y ǁ^{2} - ǁ q ǁ^{2} + t ν_{n} ζ (ϕ (x, q)) + (1 - t) ν_{n} ζ (ϕ (y, q)) + μ_{n} \end{aligned}

(2.3)

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

ϕ (q, u_{n}) \leq t ν_{n} ζ (ϕ (x, q)) + (1 - t) ν_{n} ζ (ϕ (y, q)) + μ_{n} \to 0 (n \to \infty) .

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

{x_{i}}_{i = 1}^{ω} \subset B_{r} (0)

(where ω is any positive integer or +∞) and for any sequence

{λ_{i}}_{i = 1}^{ω}

of positive numbers with

\sum_{n = 1}^{ω} λ_{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:

| | \sum_{n = 1}^{ω} λ_{n} x_{n} | |^{2} \leq \sum_{n = 1}^{ω} λ_{n} | | x_{n} | |^{2} - λ_{1} λ_{i} g (| | x_{1} - x_{i} | |),

(2.4)

and for all x ∈ X

ϕ (x, J^{- 1} (\sum_{i = 1}^{ω} λ_{i} J x_{i}) \leq \sum_{i = 1}^{ω} λ_{i} ϕ (x, x_{i}) - λ_{1} λ_{i} g (| | J x_{1} - J x_{i} | |) .

(2.5)

For solving the generalized MEP, let us assume that the function $ψ : C \to R$ is convex and lower semi-continuous, the nonlinear mapping A : C → X* is continuous and monotone, and the bifunction $Θ : C \times C \to R$ 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

Θ : C \times C \to 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
$Θ (z, y) + \frac{1}{r} ⟨ y - z, J z - J x ⟩ \geq 0, \forall y \in C .$
(ii)

[13] Define a mapping T_r : X → C by
$T_{r} x = \{z \in C : Θ (z, y) + \frac{1}{r} ⟨ y - z, J z - J x ⟩ \geq 0, \forall y \in C\}, x \in X .$

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,
$⟨T_{r} (z) - T_{r} (y), J T_{r} (z) - J T_{r} (y)⟩ \leq ⟨T_{r} (z) - T_{r} (y), J z - J y⟩;$
(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,

ψ : C \to R

be a lower semi-continuous and convex function, and

Θ : C \times C \to 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
$Θ (u, y) + ⟨ A u, y - u ⟩ + ψ (y) - ψ (u) + \frac{1}{r} ⟨ y - u, J u - J x ⟩ \geq 0 .$

(2.6)
(ii)

If we define a mapping K_r : C → C by
$\begin{gathered} K_{r} (x) = \{u \in C : Θ (u, y) + ⟨ A u, y - u ⟩ + ψ (y) - ψ (u) \\ + \frac{1}{r} ⟨ y - u, J u - J x ⟩ \geq 0, \forall y \in C\}, x \in C, \end{gathered}$

(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
$⟨K_{r} (z) - K_{r} (y), J K_{r} (z) - J K_{r} (y)⟩ \leq ⟨K_{r} (z) - K_{r} (y), J z - J y⟩;$
(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) $Θ : C \times C \to R$ is a bifunction satisfying the conditions (A₁)-(A₄), A : C → X* is a continuous and monotone mapping, and $ψ : C \to R$ is a lower semi-continuous and convex function.

(H3) ${\{T_{i}\}}_{i = 1}^{\infty} : C \to 2^{C}$ 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,

{\{T_{i}\}}_{i = 1}^{\infty}

satisfy the above conditions (H1)-(H3). Let {x_n } be the sequence generated by

\{\begin{gathered} x_{0} \in C chosen arbitrary, C_{0} = C, \\ y_{n} = J^{- 1} (α_{n} J x_{n} + (1 - α_{n}) J z_{n}), \forall n \geq 1, \\ z_{n} = J^{- 1} (β_{n, 0} J x_{n} + \sum_{i = 1}^{\infty} β_{n, i} J w_{n, i}), (w_{n, i} \in T_{i}^{n} x_{n}, i \geq 1), \forall n \geq 1, \\ u_{n} \in C s u c h t h a t \forall y \in C, \forall n \geq 1, \\ Θ (u_{n}, y) + ⟨ A u_{n}, y - u_{n} ⟩ + ψ (y) - ψ (u_{n}) + \frac{1}{r_{n}} ⟨ y - u_{n}, J u_{n} - J y_{n} ⟩ \geq 0, \\ C_{n + 1} = {ν \in C_{n} : ϕ (ν, u_{n}) \leq ϕ (ν, x_{n}) + ξ_{n}}, \forall n \geq 0, \\ {x_{n}}_{+ 1} = Π_{C_{n + 1}} x_{0}, \forall n \geq 0, \end{gathered}

(3.1)

where

\prod_{C_{n + 1}}

is the generalized projection of X onto C_n+1,

F : = \cap_{i = 1}^{\infty} F (T_{i})

,

ξ_{n} = ν_{n} {sup}_{p \in F} ζ (ϕ (p, x_{n})) + μ_{n}, {α_{n}}

and {β_n,0,β_n,i} are sequences in 0[1] satisfying the following conditions:

(i)

for each n ≥ 0, $\sum_{i = 0}^{\infty} β_{n, i} = 1$ ;
(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 $G : = F \cap Ω = \cap_{i = 1}^{\infty} F (T_{i}) \cap Ω$ is nonempty and is a bounded subset of C, then the sequence {x_n } converges strongly to ∏ _Gx₀.

Proof. First, we define two functions

H : C \times C \to R

and K_r : C → C by

\begin{gathered} H (x, y) = Θ (x, y) + ⟨ A x, y - x ⟩ + ψ (y) - ψ (x), \forall x, y \in C, \\ K_{r} (x) = ⟨ u \in C : H (u, y) + \frac{1}{r} ⟨ y - u, J u - J x ⟩ \geq 0, \forall y \in C}, x \in C . \end{gathered}

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

\{\begin{gathered} x_{0} \in C chosen arbitrary, C_{0} = C, \\ y_{n} = J^{- 1} (α_{n} J x_{n} + (1 - α_{n}) J z_{n}), \forall n \geq 1, \\ z_{n} = J^{- 1} (β_{n, 0} J x_{n} + \sum_{i = 1}^{\infty} β_{n, i} J w_{n, i}) (w_{n, i} \in T_{i}^{n} x_{n}, i \geq 1), \forall n \geq 1, \\ u_{n} \in C such that \\ H (u_{n}, y) + \frac{1}{r_{n}} ⟨ y - u_{n}, J u_{n} - J y_{n} ⟩ \geq 0, \forall y \in C, \forall n \geq 1, \\ C_{n + 1} = {ν \in C_{n} : ϕ (ν, u_{n}) \leq ϕ (ν, x_{n}) + ξ_{n}}, \forall n \geq 0, \\ {x_{n}}_{+ 1} = Π_{C_{n + 1}} x_{0} \forall n \geq 0 . \end{gathered}

(3.2)

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

(I) and C_nare 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

2 ⟨ ν, J x_{n} - J y_{n} ⟩ \leq {∥x_{n}∥}^{2} - {∥y_{n}∥}^{2} + ξ_{n}, n = 1,2, \dots,

hence the set

{C_{n}}_{+ 1} = {ν \in C_{n} : 2 ⟨ ν, J x_{n} - J y_{n} ⟩ \leq {∥x_{n}∥}^{2} - {∥y_{n}∥}^{2} + ξ_{n}}

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 )

ϕ (x_{n}, x_{0}) = ϕ (Π_{C_{n}} x_{0}, x_{0}) \leq ϕ (u, x_{0}) - ϕ (u, Π_{C_{n}} x_{0}) \leq ϕ (u, x_{0}) .

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

ϕ (x_{n}, x_{0}) \leq ϕ ({x_{n}}_{+ 1}, x_{0}), \forall n \geq 0 .

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

(III) $G : = F \cap Ω \subset C_{n}$ for all n ≥ 0.

Indeed, it is obvious that

G \subset C_{0} = C

. Suppose that

G \subset C_{n}

for some

n \in 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 G \subset C_{n}

and n ≥ 1, it follows from (1.7) that

\begin{aligned} ϕ (u, u_{n}) & = ϕ (u, K_{r_{n}} y_{n}) \leq ϕ (u, y_{n}) \\ = ϕ (u, J^{- 1} (α_{n} J x_{n} + (1 - α_{n}) J z_{n})) \\ \leq α_{n} ϕ (u, x_{n}) + (1 - α_{n}) ϕ (u, z_{n}) . \end{aligned}

(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{aligned} ϕ (u, z_{n}) & = ϕ (u, J^{- 1} (β_{n, 0} J x_{n} + \sum_{i = 1}^{\infty} β_{n, i} J w_{n, i})) \\ \leq β_{n, 0} ϕ (u, x_{n}) + \sum_{i = 1}^{\infty} β_{n, i} ϕ (u, w_{n, i}) - β_{n, 0} β_{n, l} g (| | J x_{n} - J w_{n, l} | |) \\ \leq β_{n, 0} ϕ (u, x_{n}) + \sum_{i = 1}^{\infty} β_{n, i} (ϕ (u, w_{n, i}) + ν_{n} ζ (ϕ (u, w_{n, i})) + μ_{n}) \\ - β_{n, 0} β_{n, l} g (| | J x_{n} - J w_{n, l} | |) \\ \leq ϕ (u, x_{n}) + ν_{n} sup_{p \in F} ζ (ϕ (p, x_{n})) + μ_{n} - β_{n, 0} β_{n, l} g (∥J x_{n} - J w_{n, l}∥) \\ = ϕ (u, x_{n}) + ξ_{n} - β_{n, 0} β_{n, l} g (∥J x_{n} - J w_{n, l}∥), \end{aligned}

(3.4)

where

ξ_{n} = ν_{n} {sup}_{p \in F} ζ (ϕ (p, x_{n}))

. Substituting (3.4) into (3.3) and simplifying,

\forall u \in G

we have

\begin{aligned} ϕ (u, u_{n}) & \leq ϕ (u, y_{n}) \\ \leq ϕ (u, x_{n}) + (1 - α_{n}) ξ_{n} - (1 - α_{n}) β_{n, 0} β_{n, l} g (| | J x_{n} - J w_{n, l} | |) \\ \leq ϕ (u, x_{n}) + ξ_{n} - (1 - α_{n}) β_{n, 0} β_{n, l} g (| | J x_{n} - J w_{n, l} | |) \\ \leq ϕ (u, x_{n}) + ξ_{n}, \end{aligned}

(3.5)

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

By the way, in view of the assumption on {ν_n }, {μ_n } we have

ξ_{n} = ν_{n} sup_{p \in F} ζ (ϕ (p, x_{n})) + μ_{n} \to 0 (n \to \infty) .

(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

{x_{n_{i}}} \subset {x_{n}}

such that

x_{n_{i}} ⇀ 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}} = Π_{C_{n_{i}}} x_{0}

, we have

ϕ (x_{n_{i}}, x_{0}) \leq ϕ (p^{*}, x_{0}), \forall n_{i} \geq 0 .

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

\begin{aligned} \underset{n_{i} \to \infty}{lim inf} ϕ (x_{n_{i}}, x_{0}) & = \underset{n_{i} \to \infty}{lim inf} (| | x_{n_{i}} | |^{2} - 2 ⟨ x_{n i}, J x_{0} ⟩ + | | x_{0} | |^{2}) \\ \geq | | p^{*} | |^{2} - 2 ⟨ p^{*}, J x_{0} ⟩ + | | x_{0} | |^{2} = ϕ (p^{*}, x_{0}), \end{aligned}

and so

ϕ (p^{*}, x_{0}) \leq \underset{n_{i} \to \infty}{lim inf} ϕ (x_{n_{i}}, x_{0}) \leq \underset{n_{i} \to \infty}{lim sup} ϕ (x_{n_{i}}, x_{0}) \leq ϕ (p^{*}, x_{0}) .

This implies that

lim_{n_{i} \to \infty} ϕ (x_{n_{i}}, x_{0}) = ϕ (p^{*}, x_{0})

, and so

| | x_{n_{i}} | | \to | | p^{*} | |

. Since

x_{n_{i}} ⇀ p^{*}

, by virtue of Kadec-Klee property of X, we obtain that

lim_{n_{i} \to \infty} x_{n_{i}} = p^{*} .

Since {ϕ(x_n , x₀)} is convergent, this together with

{lim}_{n_{i} \to \infty} ϕ (x_{n_{i}}, x_{0}) = ϕ (p^{*}, x_{0})

, which shows that lim_n→∞ϕ(x_n , x₀) = ϕ(p*, x₀). If there exists some sequence

{x_{n_{i}}} \subset {x_{n}}

such that

x_{n_{j}} \to q

, then from Lemma 1.2(a) we have that

\begin{aligned} ϕ (p^{*}, q) & = lim_{n_{i}, n_{j} \to \infty} ϕ (x_{n_{i}}, x_{n_{j}}) = lim_{n_{i}, n_{j} \to \infty} ϕ (x_{n_{i}}, Π_{C_{n_{j}}} x_{0}) \\ \leq lim_{n_{i}, n_{j} \to \infty} (ϕ (x_{n_{i}}, x_{0}) - ϕ (Π_{C_{n_{j}}} x_{0}, x_{0})) \\ = lim_{n_{i}, n_{j} \to \infty} (ϕ (x_{n_{i}}, x_{0}) - ϕ (x_{n_{j}}, x_{0})) \\ = ϕ (p^{*}, x_{0}) - ϕ (p^{*}, x_{0}) = 0 . \end{aligned}

This implies that p* = q and

lim_{n \to \infty} x_{n} = p^{*} .

(3.6)

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

First, we prove that

p^{*} \in F

. In fact, since x_n+1∈ C_n+1⊂ C_n , it follows from (3.1) and (3.6) that

ϕ ({x_{n}}_{+ 1}, y_{n}) \leq ϕ ({x_{n}}_{+ 1}, x_{n}) + ξ_{n} \to 0 (n \to \infty) .

By the virtue of Lemma 2.1, we have

lim_{n \to \infty} y_{n} = p^{*} .

(3.7)

From (3.5), for any

u \in F

and

w_{n, i} \in T_{i}^{n} x_{n}

, we have

ϕ (u, y_{n}) \leq ϕ (u, x_{n}) + ξ_{n} - (1 - α_{n}) β_{n, 0} β_{n, l} g (| | J x_{n} - J w_{n, l} | |),

i.e.,

(1 - α_{n}) β_{n, 0} β_{n, l} g (| | J x_{n} - J w_{n, l} | |) \leq ϕ (u, x_{n}) + ξ_{n} - ϕ (u, y_{n}) \to 0 (n \to \infty) .

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

| | J x_{n} - J w_{n, l} | | \to 0 (n \to \infty), \forall l \geq 1 .

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

w_{n, l} ⇀ p^{*} (n \to \infty), \forall l \geq 1 .

(3.8)

Again since

| | | w_{n, l} | | - | | p^{*} | | | = | | | J w_{n, l} | | - | | J p^{*} | | | \leq | | J w_{n, l} - J p^{*} | | \to 0 (n \to \infty),

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

lim_{n \to \infty} w_{n, l} = p^{*}, \forall l \geq 1 .

(3.9)

Let {s_n,l} be a sequence generated by

\begin{gathered} 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 \geq 1 \end{gathered}

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{aligned} | | s_{n + 1, l} - w_{n, l} | | & \leq | | s_{n + 1, l} - w_{n + 1, l} | | + | | w_{n + 1, l} - x_{n + 1} | | + | | x_{n + 1} - x_{n} | | + | | x_{n} - w_{n, l} | | \\ \leq (L_{l} + 1) | | x_{n + 1} - x_{n} | | + | | w_{n + 1, l} - x_{n + 1} | | + | | x_{n} - w_{n, l} | | . \end{aligned}

(3.10)

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

\begin{gathered} lim_{n \to \infty} | | s_{n + 1, l} - w_{n, l} | | = 0, a n d \\ lim_{n \to \infty} s_{n + 1, l} = p^{*} . \end{gathered}

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 F = \cap_{i = 1}^{\infty} F (T_{i}) .

Next, we prove that p* ∈ Ω. Since

x_{n + 1} = Π_{C_{n + 1}} x_{0} \in C_{n}

, it follows from (3.1) and (3.6) that

ϕ (x_{n + 1}, u_{n}) \leq ϕ (x_{n + 1}, x_{n}) + ξ_{n} \to 0 (n \to \infty) .

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

lim_{n \to \infty} 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

lim_{n \to \infty} \frac{| | J u_{n} - J y_{n} | |}{r_{n}} = 0 .

(3.12)

Since

H (u_{n}, y) + \frac{1}{r_{n}} ⟨ y - u_{n}, J u_{n} - J y_{n} ⟩ \geq 0, \forall y \in C

, by condition (A₁), we have

\frac{1}{r_{n}} ⟨ y - u_{n}, J u_{n} - J y_{n} ⟩ \geq - H (u_{n}, y) \geq H (y, u_{n}), \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 (y_{t}, y_{t}) \leq t H (y_{t}, y) + (1 - t) H (y_{t}, p^{*}) \leq t H (y_{t}, y) .

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 $p^{*} \in G = F ⋂ Ω$ .

(VI) we prove that $x_{n} \to p^{*} = Π_{G} x_{0}$ .

Let q = ∏ _Gx₀. Since

q \in G \subset C_{n}

and x_n = ∏ _Cn x₀, we have

ϕ (x_{n}, x_{0}) \leq ϕ (q, x_{0}), \forall n \geq 0 .

This implies that

ϕ (p^{*}, x_{0}) = lim_{n \to \infty} ϕ (x_{n}, x_{0}) \leq ϕ (q, x_{0}) .

(3.14)

In view of the definition of $Π_{G} x_{0}$ , from (3.14) we have p* = q. Therefore, $x_{n} \to p^{*} = Π_{G} x_{0}$ . This completes the proof of Theorem 3.1.

Definition 3.2 A finite family of multi-valued mappings

{T_{i}}_{i = 1}^{\infty} : C \to 2^{C}

is said to be uniformly quasi-ϕ-asymptotically nonexpansive, if

F = ⋂_{i = 1}^{\infty} F (T_{i}) \neq \emptyset

and there exists a real sequence {k_n } ⊂ [1, ∞), k_n → 1 such that for each i = 1, 2, . . . , N

ϕ (p, w_{n, i}) \leq k_{n} ϕ (p, x), \forall x \in C, p \in ⋂_{i = 1}^{\infty} F (T_{i}), 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

{\{T_{i}\}}_{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

\{\begin{gathered} x_{0} \in C chosen arbitrary, C_{0} = C, \\ y_{n} = J^{- 1} (α_{n} J x_{n} + (1 - α_{n}) J z_{n}), \forall n \geq 1, \\ z_{n} = J^{- 1} (β_{n, 0} J x_{n} + \sum_{i = 1}^{\infty} β_{n, i} J w_{n, i}), (w_{n, i} \in T_{i}^{n} x_{n}, i \geq 1), \forall n \geq 1, \\ u_{n} \in C s u c h t h a t \forall y \in C \\ Θ (u_{n}, y) + ⟨ A u_{n}, y - u_{n} ⟩ + ψ (y) - ψ (u_{n}) + \frac{1}{r_{n}} ⟨ y - u_{n}, J u_{n} - J y_{n} ⟩ \geq 0, \\ C_{n + 1} = {ν \in C_{n} : ϕ (ν, u_{n}) \leq ϕ (ν, x_{n}) + ξ_{n}}, \forall n \geq 0, \\ x_{n + 1} = Π_{C_{n + 1}} x_{0}, \forall n \geq 0, \end{gathered}

(3.16)

where $F : = ⋂_{i = 1}^{\infty} F (T_{i}), ξ_{n} = (k_{n} - 1) {sup}_{p \in F} ζ (ϕ (p, x_{n})), {β_{n, 0}, β_{n, i}}_{i = 1}^{\infty},$ and {α_n } are sequences in 0[1] satisfying the conditions (i), (ii), (iii) in Theorem 3.1. If $F : = ⋃_{i = 1}^{\infty} F (T_{i})$ is a bounded subset of C, then {x_n } converges strongly to $Π_{G} x_{0}$ .

Proof. Since ${\{T_{i}\}}_{i = 1}^{\infty}$ 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 $ξ_{n} = (k_{n} - 1) {sup}_{p \in F} ϕ (p, x_{n}) \to 0$ (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

{\{T_{i}\}}_{i = 1}^{\infty}

be a countable family of closed and quasi-ϕ-nonexpansive multi-valued mappings. Let {x_n } be the sequence generated by

\{\begin{gathered} x_{0} \in C chosen arbitrary, C_{0} = C, \\ y_{n} = J^{- 1} (α_{n} J x_{n} + (1 - α_{n}) J z_{n}), \forall n \geq 1, \\ z_{n} = J^{- 1} (β_{n, 0} J x_{n} + \sum_{i = 1}^{\infty} β_{n, i} J w_{n, i}), (w_{n, i} \in T_{i}^{n} x_{n}, i \geq 1), \forall n \geq 1, \\ u_{n} \in C s u c h t h a t \forall y \in C, \forall n \geq 1, \\ Θ (u_{n}, y) + ⟨ A u_{n}, y - u_{n} ⟩ + ψ (y) - ψ (u_{n}) + \frac{1}{r_{n}} ⟨ y - u_{n}, J u_{n} - J y_{n} ⟩ \geq 0, \\ C_{n + 1} = {ν \in C_{n} : ϕ (ν, u_{n}) \leq ϕ (ν, x_{n})}, \forall n \geq 0, \\ x_{n + 1} = Π_{C_{n + 1}} x_{0}, \forall n \geq 0, \end{gathered}

(3.17)

where ${β_{n, 0}, β_{n, i}}_{i = 1}^{\infty}$ and {α_n } are sequences in 0[1] satisfying the conditions (i), (ii), (iii) in Theorem 3.1. If $F : = ⋂_{i = 1}^{\infty} F (T_{i}) \neq \emptyset$ , then {x_n } converges strongly to $Π_{G} x_{0}$ .

Proof. Since ${\{T_{i}\}}_{i = 1}^{\infty}$ 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 $ξ_{n} = (k_{n} - 1) {sup}_{u \in F} ϕ (u, x_{n}) = 0$ 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 ${T_{i}}_{i = 1}^{ω}$ (where ω is a finite positive integer or +∞) is to finding a point in the nonempty intersection $⋂_{i = 1}^{ω} C_{i}$ , 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

\{\begin{gathered} x_{0} \in C chosen arbitrary, C_{0} = C, \\ y_{n} = J^{- 1} (α_{n} J x_{n} + (1 - α_{n}) J z_{n}), \forall n \geq 1, \\ z_{n} = J^{- 1} (β_{n, 0} J x_{n} + \sum_{i = 1}^{\infty} β_{n, i} J w_{n, i}), (w_{n, i} \in T_{i}^{n} x_{n}, i \geq 1), \forall n \geq 1, \\ C_{n + 1} = {ν \in C_{n} : ϕ (ν, u_{n}) \leq ϕ (ν, x_{n})}, u_{n} = Π_{C} y_{n}, \forall n \geq 1, \\ x_{n + 1} = Π_{C_{n + 1}} x_{0}, \forall n \geq 0, \end{gathered}

(4.1)

converges strongly to a point $p^{*} = \prod_{F} x_{0}$ , which is a solution of the convex feasibility problem for a countable family of closed and quasi-ϕ-nonexpansive multi-valued mappings ${\{T_{i}\}}_{i = 1}^{\infty}$ where $F = ⋂_{i = 1}^{\infty} F (T_{i})$ .

(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 \geq 1, F = C, G = Ω

. By Theorem 3.4 the sequence {x_n } defined by

\{\begin{gathered} x_{0} \in C chosen arbitrary, C_{0} = C, \\ u_{n} \in C s u c h t h a t \forall y \in C, \forall n \geq 1, \\ Θ (u_{n}, y) + ⟨ A u_{n}, y - u_{n} ⟩ + ψ (y) - ψ (u_{n}) + \frac{1}{r_{n}} ⟨ y - u_{n}, J u_{n} - J x_{n} ⟩ \geq 0, \\ C_{n + 1} = {ν \in C_{n} : ϕ (ν, u_{n}) \leq ϕ (ν, x_{n})}, \forall n \geq 0, \\ x_{n + 1} = Π_{C_{n + 1}} x_{0}, \forall n \geq 0 . \end{gathered}

(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

\{\begin{gathered} x_{0} \in C chosen arbitrary, C_{0} = C, \\ u_{n} \in C s u c h t h a t \forall y \in C \\ ψ (y) - ψ (u_{n}) + \frac{1}{r_{n}} ⟨ y - u_{n}, J u_{n} - J x_{n} ⟩ \geq 0, \\ C_{n + 1} = {ν \in C_{n} : ϕ (ν, u_{n}) \leq ϕ (ν, x_{n})}, \forall n \geq 0, \\ x_{n + 1} = Π_{C_{n + 1}} x_{0} . \end{gathered}

(4.3)

converges strongly to a point p* = ∏ _Kx₀ 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

\{\begin{gathered} x_{0} \in C chosen arbitrary, C_{0} = C, \\ u_{n} \in C s u c h t h a t \forall y \in C \\ ⟨ A u_{n}, y - u_{n} ⟩ + ψ (y) - ψ (u_{n}) + \frac{1}{r_{n}} ⟨ y - u_{n}, J u_{n} - J x_{n} ⟩ \geq 0, \\ C_{n + 1} = {ν \in C_{n} : ϕ (ν, u_{n}) \leq ϕ (ν, x_{n})}, \forall n \geq 0, \\ x_{n + 1} = Π_{C_{n + 1}} x_{0} . \end{gathered}

(4.4)

converges strongly to a point p* = ∏ _Qx₀ 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).

Declarations

Acknowledgements

This study was supported by the Natural Science Foundation of Yunnan Province (Grant No. 2011FB074).

Authors’ Affiliations

(1)

College of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming, Yunnan, 650221, China

(2)

School of Information Engineering, the College of Arts and Sciences, Yunnan Normal University, Kunming, Yunnan, 650222, China

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