Strong Convergence Results for Equilibrium Problems and Fixed Point Problems for Multivalued Mappings

Abstract

Using viscosity approximation method, we study strong convergence to a common element of the set of solutions of an equilibrium problem and the set of common fixed points of a finite family of multivalued mappings satisfying the condition (𝐸) in the setting of Hilbert space. Our results improve and extend some recent results in the literature.

1. Introduction

Let 𝐻 be a real Hilbert space with inner product ⟨⋅,⋅⟩ and norm ‖⋅‖. Let 𝐶 be a nonempty closed convex subset 𝐻. A subset 𝐶⊂𝐻 is called proximal if, for each 𝑥∈𝐻, there exists an element 𝑦∈𝐶 such that 𝑥−𝑦=dist(𝑥,𝐶)=inf{‖𝑥−𝑧‖∶𝑧∈𝐶}. (1)

A single-valued mapping 𝑇∶𝐶→𝐶 is said to be nonexpansive, if 𝑇𝑥−𝑇𝑦≤𝑥−𝑦,∀𝑥,𝑦∈𝐶. (2)

Let 𝑃_𝐶 be a nearest point projection of 𝐻 into 𝐶; that is, for 𝑥∈𝐻, 𝑃_𝐶𝑥 is a unique nearest point in 𝐶 with the property 𝑥−𝑃_𝐶𝑥∶=inf𝑥−𝑦∶𝑦∈𝐶. (3)

We denote by 𝐶𝐵(𝐶), 𝐾(𝐶), and 𝑃(𝐶) the collection of all nonempty closed bounded subsets, nonempty compact subsets, and nonempty proximal bounded subsets of 𝐶 respectively. The Hausdorff metric 𝐻 on 𝐶𝐵(𝐻) is defined by 𝐻(𝐴,𝐵)∶=maxsup_𝑥∈𝐴dist(𝑥,𝐵),sup_𝑦∈𝐵dist𝑦,𝐴, (4) for all 𝐴,𝐵∈𝐶𝐵(𝐻).

Let 𝑇∶𝐻→2^𝐻 be a multivalued mapping. An element 𝑥∈𝐻 is said to be a fixed point of 𝑇, if 𝑥∈𝑇𝑥 and the set of fixed points of 𝑇 is denoted by 𝐹(𝑇).

A multivalued mapping 𝑇∶𝐻→𝐶𝐵(𝐻) is called (i)nonexpansive if 𝐻𝑇𝑥,𝑇𝑦≤𝑥−𝑦,𝑥,𝑦∈𝐻; (5)(ii)quasi-nonexpansive if 𝐹(𝑇)≠∅ and 𝐻(𝑇𝑥,𝑇𝑝)≤‖𝑥−𝑝‖ for all 𝑥∈𝐻 and all 𝑝∈𝐹(𝑇).

Recently, García-Falset et al. [1] introduced a new condition on single-valued mappings, called condition (𝐸), which is weaker than nonexpansiveness.

Definition 1. A mapping 𝑇∶𝐻→𝐻 is said to satisfy condition (𝐸_𝜇) provided that 𝑥−𝑇𝑦≤𝜇‖𝑥−𝑇𝑥‖+𝑥−𝑦,𝑥,𝑦∈𝐻. (6)
We say that 𝑇 satisfies condition (𝐸) whenever 𝑇 satisfies (𝐸_𝜇) for some 𝜇≥1.

Recently, Abkar and Eslamian [2, 3] generalized this condition for multivalued mappings as follows.

Definition 2. A multivalued mapping 𝑇∶𝐻→𝐶𝐵(𝐻) is said to satisfy condition (𝐸) provided that 𝐻𝑇𝑥,𝑇𝑦≤𝜇dist(𝑥,𝑇𝑥)+𝑥−𝑦,𝑥,𝑦∈𝐻, (7) for some 𝜇≥1.

It is obvious that every nonexpansive multivalued mapping 𝑇∶𝐻→𝐶𝐵(𝐻) satisfies the condition (𝐸), and every mapping 𝑇∶𝐻→𝐶𝐵(𝐻) which satisfies the condition (𝐸) with nonempty fixed point set 𝐹(𝑇) is quasi-nonexpansive.

Example 3. Let us define a mapping 𝑇 on [0,3] by 𝑇(𝑥)=𝑥0,3[],𝑥≠31,2𝑥=3. (8) It is easy to see that 𝑇 satisfies the condition (𝐸) but is not nonexpansive. Indeed, for 𝑥,𝑦∈[0,3), 𝐻(𝑇𝑥,𝑇𝑦)=|(𝑥−𝑦)/3|≤|𝑥−𝑦|. Let 𝑥=0 and 𝑦=3. Then 𝐻(𝑇𝑥,𝑇𝑦)=2≤3=|𝑥−𝑦|. If 𝑥∈(0,3) and 𝑦=3, then, we have dist(𝑥,𝑇𝑥)=2𝑥/3 and dist(𝑦,𝑇𝑦)=1; hence 𝐻𝑥𝑇𝑥,𝑇𝑦=2−3≤3−𝑥+4𝑥3=𝑥−𝑦+2dist(𝑥,𝑇𝑥). (9) Thus, 𝑇 satisfies the condition (𝐸). However, 𝑇 is not nonexpansive; indeed for 𝑥=3 and 𝑦=7/3, 𝐻(𝑇𝑥,𝑇𝑦)=11/9>2/3=|𝑥−𝑦|.
Let Ψ∶𝐶×𝐶→ℝ be a bifunction. The equilibrium problem associated with the bifunction Ψ and the set 𝐶 is: find𝑥∈𝐶suchthatΨ𝑥,𝑦≥0,∀𝑦∈𝐶. (10) Such a point 𝑥∈𝐶 is called the solution of the equilibrium problem. The set of solutions is denoted by 𝐸𝑃(Ψ).
A broad class of problems in optimization theory, such as variational inequality, convex minimization, and fixed point problems, can be formulated as an equilibrium problem; see [4, 5]. In the literature, many techniques and algorithms have been proposed to analyze the existence and approximation of a solution to equilibrium problem; see [6]. Many researchers have studied various iteration processes for finding a common element of the set of solutions of the equilibrium problems and the set of fixed points of a class of nonlinear mappings. For example, see [7–22].
Fixed points and fixed point iteration process for nonexpansive mappings have been studied extensively by many authors to solve nonlinear operator equations, as well as variational inequalities; see, for example, [23–28]. In the recent years, fixed point theory for multivalued mappings has been studied by many authors; see [29–40] and the references therein.
In this paper, using viscosity approximation method, we study strong convergence to a common element of the set of solutions of an equilibrium problem and the set of common fixed points of a finite family of multivalued mappings satisfying the condition (𝐸) in the setting of Hilbert space. Our results improve and extend some recent results in the literature.

2. Preliminaries

For solving the equilibrium problem, we assume that the bifunction Ψ satisfies the following conditions: (A1)Ψ(𝑥,𝑥)=0 for any 𝑥∈𝐶; (A2)Ψ is monotone; that is, Ψ(𝑥,𝑦)+Ψ(𝑦,𝑥)≤0 for any 𝑥,𝑦∈𝐶;(A3)Ψ is upper-hemicontinuous; that is, for each 𝑥,𝑦,𝑧∈𝐶, limsup_𝑡→0+Ψ𝑡𝑧+(1−𝑡)𝑥,𝑦≤Ψ𝑥,𝑦; (11)(A4)Ψ(𝑥,.) is convex and lower semicontinuous for each 𝑥∈𝐶.

Lemma 4 (see [4]). Let 𝐶 be a nonempty closed convex subset of 𝐻 and let Ψ be a bifunction of 𝐶×𝐶 into ℝ satisfying (𝐴1)--(𝐴4). Let 𝑟>0 and 𝑥∈𝐻. Then, there exists 𝑧∈𝐶 such that Ψ+1𝑧,𝑦𝑟𝑦−𝑧,𝑧−𝑥≥0∀𝑦∈𝐶. (12)

Lemma 5 (see [6]). Assume that Ψ∶𝐶×𝐶→ℝ satisfies (𝐴1)--(𝐴4). For 𝑟>0 and 𝑥∈𝐻, define a mapping 𝑆_𝑟∶𝐻→𝐶 as follows:𝑆_𝑟+1𝑥=𝑧∈𝐶∶Ψ𝑧,𝑦𝑟𝑦−𝑧,𝑧−𝑥≥0,∀𝑦∈𝐶. (13) Then, the following hold: (i)𝑆_𝑟 is single valued; (ii)𝑆_𝑟 is firmly nonexpansive; that is, for any 𝑥,𝑦∈𝐻, 𝑆_𝑟𝑥−𝑆_𝑟𝑦²≤𝑆_𝑟𝑥−𝑆_𝑟𝑦,𝑥−𝑦; (14)(iii)𝐹(𝑆_𝑟)=𝐸𝑃(Ψ); (iv)𝐸𝑃(Ψ) is closed and convex.

Lemma 6 (see [41]). Let 𝐻 be a real Hilbert space. Then, for all 𝑥,𝑦,𝑧∈𝐻 and 𝛼,𝛽,𝛾∈[0,1] with 𝛼+𝛽+𝛾=1 one has 𝛼𝑥+𝛽𝑦+𝛾𝑧²=𝛼‖𝑥‖²+𝛽𝑦²+𝛾‖𝑧‖²−𝛼𝛽𝑥−𝑦²−𝛼𝛾‖𝑥−𝑧‖²−𝛽𝛾𝑧−𝑦². (15)

Lemma 7. For every 𝑥 and 𝑦 in a Hilbert space 𝐻, the following inequality holds: 𝑥+𝑦²≤‖𝑥‖²+2⟨𝑦,𝑥+𝑦⟩. (16)

Lemma 8 (see [42]). Let {𝑎_𝑛} be a sequence of nonnegative real numbers, {𝛼_𝑛} a sequence in (0,1) with ∑^∞_𝑛=1𝛼_𝑛=∞, {𝛾_𝑛} a sequence of nonnegative real numbers with ∑^∞_𝑛=1𝛾_𝑛<∞, and {𝛽_𝑛} a sequence of real numbers with limsup_𝑛→∞𝛽_𝑛≤0. Suppose that the following inequality holds: 𝑎_𝑛+1≤1−𝛼_𝑛𝑎_𝑛+𝛼_𝑛𝛽_𝑛+𝛾_𝑛,𝑛≥0. (17) Then, lim_𝑛→∞𝑎_𝑛=0.

Lemma 9 (see [43]). Let {𝑢_𝑛} be a sequence of real numbers that does not decrease at infinity, in the sense that there exists a subsequence {𝑢_𝑛𝑖} of {𝑢_𝑛} such that 𝑢_𝑛𝑖<𝑢_𝑛𝑖+1 for all 𝑖≥0. For every sufficiently large number 𝑛≥𝑛₀, define an integer sequence {𝜏(𝑛)} as 𝜏(𝑛)=max𝑘≤𝑛∶𝑢_𝑘<𝑢_𝑘+1. (18) Then, 𝜏(𝑛)→∞ as 𝑛→∞ and for all 𝑛≥𝑛₀, max𝑢_𝜏(𝑛),𝑢_𝑛≤𝑢_𝜏(𝑛)+1. (19)

Lemma 10 (see [20]). Let 𝐶 be a closed convex subset of a real Hilbert space 𝐻. Let 𝑇∶𝐶→𝐶𝐵(𝐶) be a quasi-nonexpansive multivalued mapping. If 𝐹(𝑇)≠∅ and 𝑇(𝑝)={𝑝} for all 𝑝∈𝐹(𝑇). Then 𝐹(𝑇) is closed and convex.

Lemma 11 (see [20]). Let 𝐶 be a closed convex subset of a real Hilbert space 𝐻. Let 𝑇∶𝐶→𝑃(𝐶) be a multivalued mapping such that 𝑃_𝑇 is quasi-nonexpansive and 𝐹(𝑇)≠∅, where 𝑃_𝑇(𝑥)={𝑦∈𝑇𝑥∶‖𝑥−𝑦‖=dist(𝑥,𝑇𝑥)}. Then, 𝐹(𝑇) is closed and convex.

Lemma 12 (see [16, 20]). Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let 𝑇∶𝐶→𝐾(𝐶) be a multivalued mapping satisfying the condition (𝐸). If 𝑥_𝑛 converges weakly to 𝑣 and lim_𝑛→∞dist(𝑥_𝑛,𝑇𝑥_𝑛)=0, then 𝑣∈𝑇𝑣.

3. A Strong Convergence Theorem

Theorem 13. Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻 and Ψ a bifunction of 𝐶×𝐶 into ℝ satisfying (𝐴1)--(𝐴4). Let 𝑇_𝑖∶𝐶→𝐶𝐵(𝐶) (𝑖=1,2,…,𝑚) be a finite family of multivalued mappings, each satisfying condition (𝐸). Assume further that ℱ=⋂^𝑚_𝑖=1𝐹(𝑇_𝑖)⋂𝐸𝑃(Ψ)≠∅ and 𝑇_𝑖(𝑝)={𝑝},(𝑖=1,2,…,𝑚) for each 𝑝∈ℱ. Let 𝑓 be a 𝑘-contraction of 𝐶 into itself. Let {𝑥_𝑛} and {𝑢_𝑛} be sequences generated the following algorithm: 𝑥₀𝑢∈𝐶,_𝑛∈𝐶𝑠𝑢𝑐ℎ𝑡ℎ𝑎𝑡Ψ𝑢_𝑛+1,𝑦𝑟_𝑛𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛𝑦≥0,∀𝑦∈𝐶_𝑛,1=𝑎_𝑛,1𝑢_𝑛+𝑏_𝑛,1𝑥_𝑛+𝑐_𝑛,1𝑧_𝑛,1,𝑦_𝑛,2=𝑎_𝑛,2𝑢_𝑛+𝑏_𝑛,2𝑧_𝑛,1+𝑐_𝑛,2𝑧_𝑛,2,𝑦_𝑛,3=𝑎_𝑛,3𝑢_𝑛+𝑏_𝑛,3𝑧_𝑛,2+𝑐_𝑛,3𝑧_𝑛,3⋮𝑦_𝑛,𝑚=𝑎_𝑛,𝑚𝑢_𝑛+𝑏_𝑛,𝑚𝑧_{𝑛,𝑚−1}+𝑐_𝑛,𝑚𝑧_𝑛,𝑚,𝑥_𝑛+1=𝜗_𝑛𝑓𝑥_𝑛+1−𝜗_𝑛𝑦_𝑛,𝑚,∀𝑛≥0, (20) where 𝑧_𝑛,1∈𝑇₁(𝑢_𝑛), 𝑧_𝑛,𝑘∈𝑇_𝑘(𝑦_{𝑛,𝑘−1}) for 𝑘=2,…,𝑚, and {𝑎_𝑛,𝑖}, {𝑏_𝑛,𝑖}, {𝑐_𝑛,𝑖}, {𝜗_𝑛}, and {𝑟_𝑛} satisfy the following conditions: (i){𝑎_𝑛,𝑖},{𝑏_𝑛,𝑖},{𝑐_𝑛,𝑖}⊂[𝑎,𝑏]⊂(0,1),𝑎_𝑛,𝑖+𝑏_𝑛,𝑖+𝑐_𝑛,𝑖=1,(𝑖=1,2,…,𝑚), (ii){𝜗_𝑛}⊂(0,1), lim_𝑛→∞𝜗_𝑛=0, ∑^∞_𝑛=1𝜗_𝑛=∞, (iii){𝑟_𝑛}⊂(0,∞), and liminf_𝑛→∞𝑟_𝑛>0. Then, the sequences {𝑥_𝑛} and {𝑢_𝑛} converge strongly to 𝑞∈ℱ, where 𝑞=𝑃_ℱ𝑓(𝑞).

Proof. Let 𝑄=𝑃_ℱ. It is easy to see that 𝑄𝑓 is a contraction. By Banach contraction principle, there exists a 𝑞∈ℱ such that 𝑞=𝑃_ℱ𝑓(𝑞). From Lemma 5 for all 𝑛≥0, we have 𝑢_𝑛−𝑞=𝑆_𝑟𝑛𝑥_𝑛−𝑆_𝑟𝑛𝑞≤𝑥_𝑛−𝑞. (21) We show that {𝑥_𝑛} is bounded. Since, for each 𝑖=1,2,…,𝑚, 𝑇_𝑖 satisfies the condition (𝐸) and we have 𝑦_𝑛,1=𝑎−𝑞_𝑛,1𝑢_𝑛+𝑏_𝑛,1𝑥_𝑛+𝑐_𝑛,1𝑧_𝑛,1−𝑞≤𝑎_𝑛,1𝑢_𝑛−𝑞+𝑏_𝑛,1𝑥_𝑛−𝑞+𝑐_𝑛,1𝑧_𝑛,1−𝑞=𝑎_𝑛,1𝑢_𝑛−𝑞+𝑏_𝑛,1𝑥_𝑛−𝑞+𝑐_𝑛,1dist𝑧_𝑛,1,𝑇₁𝑞≤𝑎_𝑛,1𝑢_𝑛−𝑞+𝑏_𝑛,1𝑥_𝑛−𝑞+𝑐_𝑛,1𝐻𝑇₁𝑢_𝑛,𝑇₁𝑞≤𝑎_𝑛,1𝑢_𝑛−𝑞+𝑏_𝑛,1𝑥_𝑛−𝑞+𝑐_𝑛,1𝑢_𝑛≤𝑥−𝑞_𝑛,𝑦−𝑞_𝑛,2=𝑎−𝑞_𝑛,2𝑢_𝑛+𝑏_𝑛,2𝑧_𝑛,1+𝑐_𝑛,2𝑧_𝑛,2−𝑞≤𝑎_𝑛,2𝑢_𝑛−𝑞+𝑏_𝑛,2𝑧_𝑛,1−𝑞+𝑐_𝑛,2𝑧_𝑛,2−𝑞=𝑎_𝑛,2𝑢_𝑛−𝑞+𝑏_𝑛,2dist𝑧_𝑛,1,𝑇₁𝑞+𝑐_𝑛,2dist𝑧_𝑛,2,𝑇₂𝑞≤𝑎_𝑛,2𝑢_𝑛−𝑞+𝑏_𝑛,2𝐻𝑇₁𝑢_𝑛,𝑇₁𝑞+𝑐_𝑛,2𝐻𝑇₂𝑦_𝑛,1,𝑇₂𝑞≤𝑎_𝑛,2𝑢_𝑛−𝑞+𝑏_𝑛,2𝑢_𝑛−𝑞+𝑐_𝑛,2𝑦_𝑛,1−𝑞≤‖𝑥_𝑛−𝑞‖. (22)(23)
By continuing this process, we obtain 𝑦_𝑛,𝑚−𝑞≤𝑥_𝑛−𝑞. (24)
This implies that 𝑥_𝑛+1=𝜗−𝑞_𝑛𝑓𝑥_𝑛+1−𝜗_𝑛𝑦_𝑛−𝑞≤𝜗_𝑛𝑓𝑥_𝑛−𝑞+1−𝜗_𝑛𝑦_𝑛−𝑞≤𝜗_𝑛𝑓𝑥_𝑛−𝑓𝑞+𝑓𝑞−𝑞+1−𝜗_𝑛𝑥_𝑛−𝑞≤𝜗_𝑛𝑘𝑥_𝑛−𝑞+𝜗_𝑛𝑓𝑞−𝑞+1−𝜗_𝑛𝑥_𝑛=−𝑞1−𝜗_𝑛(1−𝑘)𝑥_𝑛−𝑞+𝜗_𝑛𝑥𝑓𝑞−𝑞≤max_𝑛−𝑞,𝑓𝑞−𝑞.1−𝑘 (25) By induction, we get 𝑥_𝑛−𝑞≤max𝑥₀−𝑞,𝑓𝑞−𝑞,1−𝑘 (26) for all 𝑛∈ℕ. This implies that {𝑥_𝑛} is bounded and we also obtain that {𝑢_𝑛},{𝑦_𝑛},{𝑓𝑥_𝑛}, and {𝑧_𝑛,𝑖} are bounded. Next, we show that lim_𝑛→∞dist(𝑢_𝑛,𝑇_𝑖𝑢_𝑛)=0 for each 𝑖∈ℕ. By Lemma 6, we have 𝑦_𝑛,1−𝑞²=𝑎_𝑛,1𝑢_𝑛+𝑏_𝑛,1𝑥_𝑛+𝑐_𝑛,1𝑧_𝑛,1−𝑞²≤𝑎_𝑛,1𝑢_𝑛−𝑞²+𝑏_𝑛,1𝑥_𝑛−𝑞²+𝑐_𝑛,1𝑧_𝑛,1−𝑞²−𝑎_𝑛,1𝑏_𝑛,1𝑥_𝑛−𝑢_𝑛²−𝑎_𝑛,1𝑐_𝑛,1𝑢_𝑛−𝑧_𝑛,1²=𝑎_𝑛,1𝑢_𝑛−𝑞²+𝑏_𝑛,1𝑥_𝑛−q²+𝑐_𝑛,1dist𝑧_𝑛,1,𝑇₁𝑞²−𝑎_𝑛,1𝑏_𝑛,1𝑥_𝑛−𝑢_𝑛²−𝑎_𝑛,1𝑐_𝑛,1𝑢_𝑛−𝑧_𝑛,1²≤𝑎_𝑛,1𝑢_𝑛−𝑞²+𝑏_𝑛,1𝑥_𝑛−𝑞²+𝑐_𝑛,1𝐻𝑇₁𝑢_𝑛,𝑇₁𝑞²−𝑎_𝑛,1𝑏_𝑛,1𝑥_𝑛−𝑢_𝑛²−𝑎_𝑛,1𝑐_𝑛,1𝑢_𝑛−𝑧_𝑛,1²≤𝑎_𝑛,1𝑢_𝑛−𝑞²+𝑏_𝑛,1𝑥_𝑛−𝑞²+𝑐_𝑛,1𝑢_𝑛−𝑞²−𝑎_𝑛,1𝑏_𝑛,1𝑥_𝑛−𝑢_𝑛²−𝑎_𝑛,1𝑐_𝑛,1𝑢_𝑛−𝑧_𝑛,1²≤𝑥_𝑛−𝑞²−𝑎_𝑛,1𝑏_𝑛,1𝑥_𝑛−𝑢_𝑛²−𝑎_𝑛,1𝑐_𝑛,1𝑢_𝑛−𝑧_𝑛,1². (27) Applying Lemma 6 once more, we have 𝑦_𝑛,2−𝑞²=𝑎_𝑛,2𝑢_𝑛+𝑏_𝑛,2𝑧_𝑛,1+𝑐_𝑛,2𝑧_𝑛,2−𝑞²≤𝑎_𝑛,2𝑢_𝑛−𝑞²+𝑏_𝑛,2𝑧_𝑛,1−𝑞²+𝑐_𝑛,2𝑧_𝑛,2−𝑞²−𝑎_𝑛,2𝑐_𝑛,2𝑢_𝑛−𝑧_𝑛,2²=𝑎_𝑛,2𝑢_𝑛−𝑞²+𝑏_𝑛,2dist𝑧_𝑛,1,𝑇₁𝑞²+𝑐_𝑛,2dist𝑧_𝑛,2,𝑇₂𝑞²−𝑎_𝑛,2𝑐_𝑛,2𝑢_𝑛−𝑧_𝑛,2²≤𝑎_𝑛,2𝑢_𝑛−𝑞²+𝑏_𝑛,2𝐻𝑇₁𝑢_𝑛,𝑇₁𝑞²+𝑐_𝑛,2𝐻𝑇₁𝑦_𝑛,1,𝑇₂𝑞²−𝑎_𝑛,2𝑐_𝑛,2𝑢_𝑛−𝑧_𝑛,2²≤𝑎_𝑛,2𝑢_𝑛−𝑞²+𝑏_𝑛,2𝑢_𝑛−𝑞²+𝑐_𝑛,2𝑦_𝑛,1−𝑞²−𝑎_𝑛,2𝑐_𝑛,2𝑢_𝑛−𝑧_𝑛,2²≤𝑥_𝑛−𝑞²−𝑎_𝑛,2𝑐_𝑛,2𝑢_𝑛−𝑧_𝑛,2²−𝑎_𝑛,1𝑐_𝑛,1𝑐_𝑛,2𝑢_𝑛−𝑧_𝑛,1²−𝑎_𝑛,1𝑏_𝑛,1𝑐_𝑛,2𝑥_𝑛−𝑢_𝑛². (28) By continuing this process we have 𝑦_𝑛,𝑚−𝑞²=𝑎_𝑛,𝑚𝑢_𝑛+𝑏_𝑛,𝑚𝑧_{𝑛,𝑚−1}+𝑐_𝑛,𝑚𝑧_𝑛,𝑚−𝑞²≤𝑎_𝑛,𝑚𝑢_𝑛−𝑞²+𝑏_𝑛,𝑚𝑧_{𝑛,𝑚−1}−𝑞²+𝑐_𝑛,𝑚𝑧_𝑛,𝑚−𝑞²−𝑎_𝑛,𝑚𝑐_𝑛,𝑚𝑢_𝑛−𝑧_𝑛,𝑚²=𝑎_𝑛,𝑚𝑢_𝑛−𝑞²+𝑏_𝑛,𝑚dist𝑧_{𝑛,𝑚−1},𝑇_𝑚−1𝑞²+𝑐_𝑛,𝑚dist𝑧_𝑛,𝑚,𝑇_𝑚𝑞²−𝑎_𝑛,𝑚𝑐_𝑛,𝑚𝑢_𝑛−𝑧_𝑛,𝑚²≤𝑎_𝑛,𝑚𝑢_𝑛−𝑞²+𝑏_𝑛,𝑚𝐻𝑇_𝑚−1𝑦_{𝑛,𝑚−2},𝑇_𝑚−1𝑞²+𝑐_𝑛,𝑚𝐻(𝑇_𝑚𝑦_{𝑛,𝑚−1},𝑇_𝑚𝑞)²−𝑎_𝑛,𝑚𝑐_𝑛,𝑚𝑢_𝑛−𝑧_𝑛,𝑚²≤𝑎_𝑛,𝑚𝑢_𝑛−𝑞²+𝑏_𝑛,𝑚𝑦_{𝑛,𝑚−2}−𝑞²+𝑐_𝑛,𝑚𝑦_{𝑛,𝑚−1}−𝑞²−𝑎_𝑛,𝑚𝑐_𝑛,𝑚𝑢_𝑛−𝑧_𝑛,𝑚²≤𝑢_𝑛−𝑞²−𝑎_𝑛,𝑚𝑐_𝑛,𝑚𝑢_𝑛−𝑧_𝑛,𝑚²−𝑎_{𝑛,𝑚−1}𝑐_{𝑛,𝑚−1}𝑐_𝑛,𝑚𝑢_𝑛−𝑧_{𝑛,𝑚−1}²−⋯−𝑎_𝑛,1𝑐_𝑛,1𝑐_𝑛,2…𝑐_𝑛,𝑚𝑢_𝑛−𝑧_𝑛,1²−𝑎_𝑛,1𝑏_𝑛,1𝑐_𝑛,2…𝑐_𝑛,𝑚𝑢_𝑛−𝑥_𝑛², (29) which implies that 𝑥_𝑛+1−𝑞²=𝜗_𝑛𝑓𝑥_𝑛+1−𝜗_𝑛𝑦_𝑛,𝑚−𝑞²≤𝜗_𝑛𝑓𝑥_𝑛−𝑞²+1−𝜗_𝑛𝑦_𝑛,𝑚−𝑞²≤𝜗_𝑛𝑓𝑥_𝑛−𝑞²+1−𝜗_𝑛𝑢_𝑛−𝑞²−1−𝜗_𝑛𝑎_𝑛,𝑚𝑐_𝑛,𝑚𝑢_𝑛−𝑧_𝑛,𝑚²−1−𝜗_𝑛𝑎_{𝑛,𝑚−1}𝑐_{𝑛,𝑚−1}𝑐_𝑛,𝑚𝑢_𝑛−𝑧_{𝑛,𝑚−1}²−⋯−1−𝜗_𝑛𝑎_𝑛,1𝑐_𝑛,1𝑐_𝑛,2…𝑐_𝑛,𝑚𝑢_𝑛−𝑧_𝑛,1²−1−𝜗_𝑛𝑎_𝑛,1𝑏_𝑛,1𝑐_𝑛,2…𝑐_𝑛,𝑚𝑢_𝑛−𝑥_𝑛². (30) Therefore, we have that 1−𝜗_𝑛𝑎_𝑛,1𝑏_𝑛,1𝑐_𝑛,2…𝑐_𝑛,𝑚𝑢_𝑛−𝑥_𝑛²≤𝑥_𝑛−𝑞²−𝑥_𝑛+1−𝑞²+𝜗_𝑛𝛾𝑓𝑥_𝑛−𝑞. (31) In order to prove that 𝑥_𝑛→𝑞 as 𝑛→∞, we consider the following two cases.
Case 1. Suppose that there exists 𝑛₀ such that {‖𝑥_𝑛−𝑞‖} is nonincreasing, for all 𝑛≥𝑛₀. Boundedness of {‖𝑥_𝑛−𝑞‖} implies that ‖𝑥_𝑛−𝑞‖ is convergent. From (31) and conditions (i), (ii) we have that lim_𝑛→∞𝑢_𝑛−𝑥_𝑛=0. (32) By a similar argument, for 𝑘=1,2,…,𝑚, we obtain that lim_𝑛→∞𝑢_𝑛−𝑧_𝑛,𝑘=0. (33) Hence, lim_𝑛→∞dist𝑢_𝑛,𝑇₁𝑢_𝑛≤lim_𝑛→∞𝑢_𝑛−𝑧_𝑛,1=0,lim_𝑛→∞dist𝑢_𝑛,𝑇_𝑘𝑦_{𝑛,𝑘−1}≤lim_𝑛→∞𝑢_𝑛−𝑧_𝑛,𝑘().=0,𝑘=2,…,𝑚 (34) Therefore, we have lim_𝑛→∞𝑢_𝑛−𝑦_𝑛,1≤lim_𝑛→∞𝑏_𝑛,1𝑢_𝑛−𝑥_𝑛+lim_𝑛→∞𝑐_𝑛,1𝑢_𝑛−𝑧_𝑛,1=0. (35) For 𝑘=2,…,𝑚, we have lim_𝑛→∞𝑢_𝑛−𝑦_𝑛,𝑘≤lim_𝑛→∞𝑏_𝑛,𝑘𝑢_𝑛−𝑧_{𝑛,𝑘−1}+lim_𝑛→∞𝑐_𝑛,𝑘𝑢_𝑛−𝑧_𝑛,𝑘=0. (36) Using the previous inequality for 𝑘=2,…,𝑚, we have dist𝑢_𝑛,𝑇_𝑘𝑢_𝑛≤dist𝑢_𝑛,𝑇_𝑘𝑦_{𝑛,𝑘−1}+𝐻𝑇_𝑘𝑦_{𝑛,𝑘−1},𝑇_𝑘𝑢_𝑛≤dist𝑢_𝑛,𝑇_𝑘𝑦_{𝑛,𝑘−1}+𝜇dist𝑦_{𝑛,𝑘−1},𝑇_𝑘𝑦_{𝑛,𝑘−1}+𝑦_{𝑛,𝑘−1}−𝑢_𝑛≤𝜇+1dist𝑢_𝑛,𝑇_𝑘𝑦_{𝑛,𝑘−1}+𝜇+1𝑦_{𝑛,𝑘−1}−𝑢_𝑛≤𝜇+1𝑢_𝑛−𝑧_𝑛,𝑘+𝜇+1𝑦_{𝑛,𝑘−1}−𝑢_𝑛⟶0,𝑛⟶∞. (37) Next, we show that limsup_𝑛→∞𝑞−𝑓𝑞,𝑞−𝑥_𝑛≤0, (38) where 𝑞=𝑃_ℱ𝑓(𝑞). To show this inequality, we choose a subsequence {𝑥_𝑛𝑖} of {𝑥_𝑛} such that lim_𝑖→∞𝑞−𝑓𝑞,𝑞−𝑥_𝑛𝑖=limsup_𝑛→∞𝑞−𝑓𝑞,𝑞−𝑥_𝑛. (39) Since {𝑥_𝑛𝑖} is bounded, there exists a subsequence {𝑥_𝑛𝑖𝑗} of {𝑥_𝑛𝑖} which converges weakly to 𝑣. Without loss of generality, we can assume that 𝑥_𝑛𝑖 converges weakly to 𝑣. Since lim_𝑛→∞‖𝑥_𝑛−𝑢_𝑛‖=0, we have 𝑢_𝑛𝑖 converges weakly to 𝑣. We show that 𝑣∈ℱ. Let us show 𝑣∈𝐸𝑃(Ψ). Since 𝑢_𝑛=𝑆_𝑟𝑛𝑥_𝑛, we have Ψ𝑢_𝑛+1,𝑦𝑟_𝑛𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛≥0∀𝑦∈𝐶. (40) From (A2), we have 1𝑟_𝑛𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛≥Ψ𝑦,𝑢_𝑛. (41) Replacing 𝑛 with 𝑛_𝑖, we have 𝑦−𝑢_𝑛𝑖,𝑢_𝑛𝑖−𝑥_𝑛𝑖𝑟_𝑛𝑖≥Ψ𝑦,𝑢_𝑛𝑖. (42) From (A4), we have 0≥Ψ𝑦,𝑣,∀𝑦∈𝐶. (43) For 𝑡∈(0,1] and 𝑦∈𝐶, let 𝑦_𝑡=𝑡𝑦+(1−𝑡)𝑣. Since 𝑦,𝑣∈𝐶, and 𝐶 is convex, we have 𝑦_𝑡∈𝐶 and hence Ψ(𝑦_𝑡,𝑣)≤0. So, from (A1) and (A4) we have 0=Ψ𝑦_𝑡,𝑦_𝑡≤𝑡Ψ𝑦_𝑡,𝑦+(1−𝑡)Ψ𝑦_𝑡,𝑣≤𝑡Ψ𝑦_𝑡,𝑦, (44) which gives 0≤Ψ(𝑦_𝑡,𝑦). Letting 𝑡→0, we have, for each 𝑦∈𝐶, 0≤Ψ(𝑣,𝑦) Also, since 𝑢_𝑛𝑖⇀𝑣 and lim_𝑛→∞dist(𝑢_𝑛,𝑇_𝑖𝑢_𝑛)=0, by Lemma 12 we have 𝑣∈⋂^𝑚_𝑖=1𝐹(𝑇_𝑖). Hence, 𝑣∈ℱ. Since 𝑞=𝑃_ℱ𝑓(𝑞) and 𝑣∈ℱ, it follows that limsup_𝑛→∞𝑞−𝑓𝑞,𝑞−𝑥_𝑛=lim_𝑖→∞𝑞−𝑓𝑞,𝑞−𝑥_𝑛𝑖=⟨𝑞−𝑓𝑞,𝑞−𝑣⟩≤0. (45) By using Lemma 7 and inequality (31) we have 𝑥_𝑛+1−𝑞²≤1−𝜗_𝑛𝑦_𝑛,𝑚−𝑞²+2𝜗_𝑛𝑓𝑥_𝑛−𝑞,𝑥_𝑛+1≤−𝑞1−𝜗_𝑛²𝑦_𝑛,𝑚−𝑞²+2𝜗_𝑛𝑓𝑥_𝑛−𝑓𝑞,𝑥_𝑛+1−𝑞+2𝜗_𝑛𝑓𝑞−𝑞,𝑥_𝑛+1≤−𝑞1−𝜗_𝑛²𝑥_𝑛−𝑞²+2𝜗_𝑛𝑘𝑥_𝑛−𝑞𝑥_𝑛+1−𝑞+2𝜗_𝑛𝑓𝑞−𝑞,𝑥_𝑛+1≤−𝑞1−𝜗_𝑛²𝑥_𝑛−𝑞²+𝜗_𝑛𝑘𝑥_𝑛−𝑞²+𝑥_𝑛+1−𝑞²+2𝜗_𝑛𝑓𝑞−𝑞,𝑥_𝑛+1≤−𝑞1−𝜗_𝑛²+𝜗_𝑛𝑘𝑥_𝑛−𝑞²+𝜗_𝑛𝑘𝑥_𝑛+1−𝑞²+2𝜗_𝑛𝑓𝑞−𝑞,𝑥_𝑛+1−𝑞. (46)
This implies that 𝑥_𝑛+1−𝑞²≤2()𝜗1−1−𝑘_𝑛1−𝜗_𝑛𝑘𝑥_𝑛−𝑞²+𝜗²_𝑛1−𝜗_𝑛𝑘𝑥_𝑛−𝑞²+2𝜗_𝑛1−𝜗_𝑛𝑘𝑓𝑞−𝑞,𝑥_𝑛+1−𝑞. (47) From Lemma 8, we conclude that the sequence {𝑥_𝑛} converges strongly to 𝑞.
Case 2. Assume that there exists a subsequence {𝑥_𝑛𝑗} of {𝑥_𝑛} such that 𝑥_𝑛𝑗−𝑞<𝑥_𝑛𝑗+1−𝑞, (48) for all 𝑗∈ℕ. In this case, from Lemma 9, there exists a nondecreasing sequence {𝜏(𝑛)} of ℕ for all 𝑛≥𝑛₀ (for some 𝑛₀ large enough) such that 𝜏(𝑛)→∞ as 𝑛→∞ and the following inequalities hold for all 𝑛≥𝑛₀∶𝑥_𝜏(𝑛)−𝑞≤𝑥_𝜏(𝑛)+1−𝑞,𝑥_𝑛−𝑞≤𝑥_𝜏(𝑛)+1−𝑞. (49) From (31) we obtain lim_𝑛→∞‖𝑢_𝜏(𝑛)−𝑇_𝑖𝑢_𝜏(𝑛)‖=0, and lim_𝑛→∞‖𝑢_𝜏(𝑛)−𝑥_𝜏(𝑛)‖=0. Following an argument similar to that in Case 1, we have lim_𝑛→∞𝑥_𝜏(𝑛)−𝑞=0,lim_𝑛→∞𝑥_𝜏(𝑛)+1−𝑞=0. (50) Thus, by Lemma 9 we have 0≤𝑥_𝑛−𝑞≤max𝑥_𝜏(𝑛)−𝑞,𝑥_𝑛−𝑞≤𝑥_𝜏(𝑛)+1−𝑞. (51) Therefore, {𝑥_𝑛} converges strongly to 𝑞=𝑃_ℱ𝑓(𝑞). This completes the proof.

Now, we remove the condition that 𝑇(𝑝)={𝑝} for all 𝑝∈ℱ, and state the following theorem.

Theorem 14. Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻 and Ψ a bifunction of 𝐶×𝐶 into ℝ satisfying (𝐴1)--(𝐴4). Let, for each 1≤𝑖≤𝑚, 𝑇_𝑖∶𝐶→𝑃(𝐶) be multivalued mappings such that 𝑃_𝑇𝑖 satisfies the condition (𝐸). Assume that ℱ=⋂^𝑚_𝑖=1𝐹(𝑇_𝑖)⋂𝐸𝑃(Ψ)≠∅. Let 𝑓 be a 𝑘-contraction of 𝐶 into itself. Let {𝑥_𝑛} and {𝑢_𝑛} be sequences generated the following algorithm: 𝑥₀𝑢∈𝐶,_𝑛∈𝐶𝑠𝑢𝑐ℎ𝑡ℎ𝑎𝑡Ψ𝑢_𝑛+1,𝑦r_𝑛𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛𝑦≥0,∀𝑦∈𝐶_𝑛,1=𝑎_𝑛,1𝑢_𝑛+𝑏_𝑛,1𝑥_𝑛+𝑐_𝑛,1𝑧_𝑛,1,𝑦_𝑛,2=𝑎_𝑛,2𝑢_𝑛+𝑏_𝑛,2𝑧_𝑛,1+𝑐_𝑛,2𝑧_𝑛,2,𝑦_𝑛,3=𝑎_𝑛,3𝑢_𝑛+𝑏_𝑛,3𝑧_𝑛,2+𝑐_𝑛,3𝑧_𝑛,3⋮𝑦_𝑛,𝑚=𝑎_𝑛,𝑚𝑢_𝑛+𝑏_𝑛,𝑚𝑧_{𝑛,𝑚−1}+𝑐_𝑛,𝑚𝑧_𝑛,𝑚,𝑥_𝑛+1=𝜗_𝑛𝑓𝑥_𝑛+1−𝜗_𝑛𝑦_𝑛,𝑚,∀𝑛≥0, (52) where 𝑧_𝑛,1∈𝑃_𝑇1(𝑢_𝑛), 𝑧_𝑛,𝑘∈𝑃_𝑇𝑘(𝑦_{𝑛,𝑘−1}) for 𝑘=2,…,𝑚, and {𝑎_𝑛,𝑖}, {𝑏_𝑛,𝑖}, {𝑐_𝑛,𝑖}, {𝜗_𝑛} and, {𝑟_𝑛} satisfy the following conditions: (i){𝑎_𝑛,𝑖},{𝑏_𝑛,𝑖},{𝑐_𝑛,𝑖}⊂[𝑎,𝑏]⊂(0,1),𝑎_𝑛,𝑖+𝑏_𝑛,𝑖+𝑐_𝑛,𝑖=1,(𝑖=1,2,…,𝑚), (ii){𝜗_𝑛}⊂(0,1), lim_𝑛→∞𝜗_𝑛=0, ∑^∞_𝑛=1𝜗_𝑛=∞, (iii){𝑟_𝑛}⊂(0,∞), and liminf_𝑛→∞𝑟_𝑛>0. Then, the sequences {𝑥_𝑛} and {𝑢_𝑛} converge strongly to 𝑞∈ℱ, where 𝑞=𝑃_ℱ𝑓(𝑞).

Proof. Let 𝑝∈ℱ; then 𝑃_𝑇𝑖(𝑝)={𝑝},(𝑖=1,2,…,𝑚). Now by substituting 𝑃_𝑇𝑖 instead of 𝑇_𝑖, and using a similar argument as in the proof of Theorem 13, the desired result follows.

As a corollary for single-valued mappings, we obtain the following result.

Corollary 15. Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻 and Ψ a bifunction of 𝐶×𝐶 into ℝ satisfying (𝐴1)--(𝐴4). Let, for each 1≤𝑖≤𝑚, 𝑇_𝑖∶𝐶→𝐶 be a finite family of mappings satisfying condition (𝐸). Assume that ℱ=⋂^𝑚_𝑖=1𝐹(𝑇_𝑖)⋂𝐸𝑃(Ψ)≠∅. Let 𝑓 be a 𝑘-contraction of 𝐶 into itself. Let {𝑥_𝑛} and {𝑢_𝑛} be sequences generated the following algorithm: 𝑥₀𝑢∈𝐶,_𝑛∈𝐶𝑠𝑢𝑐ℎ𝑡ℎ𝑎𝑡Ψ𝑢_𝑛+1,𝑦𝑟_𝑛𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛𝑦≥0,∀𝑦∈𝐶_𝑛,1=𝑎_𝑛,1𝑢_𝑛+𝑏_𝑛,1𝑥_𝑛+𝑐_𝑛,1𝑇₁𝑢_𝑛,𝑦_𝑛,2=𝑎_𝑛,2𝑢_𝑛+𝑏_𝑛,2𝑇₁𝑢_𝑛+𝑐_𝑛,2𝑇₂𝑦_𝑛,1⋮𝑦_𝑛,𝑚=𝑎_𝑛,𝑚𝑢_𝑛+𝑏_𝑛,𝑚𝑇_𝑚−1𝑦_{𝑛,𝑚−2}+𝑇_𝑚𝑦_{𝑛,𝑚−1},𝑥_𝑛+1=𝜗_𝑛𝑓𝑥_𝑛+1−𝜗_𝑛𝑦_𝑛,𝑚,∀𝑛≥0, (53) where {𝑎_𝑛,𝑖}, {𝑏_𝑛,𝑖},{𝑐_𝑛,𝑖}, {𝜗_𝑛}, and {𝑟_𝑛} satisfy the following conditions: (i){𝑎_𝑛,𝑖},{𝑏_𝑛,𝑖},{𝑐_𝑛,𝑖}⊂[𝑎,𝑏]⊂(0,1),𝑎_𝑛,𝑖+𝑏_𝑛,𝑖+𝑐_𝑛,𝑖=1, (𝑖=1,2,…,𝑚),(ii){𝜗_𝑛}⊂(0,1), lim_𝑛→∞𝜗_𝑛=0, ∑^∞_𝑛=1𝜗_𝑛=∞(iii){𝑟_𝑛}⊂(0,∞), and liminf_𝑛→∞𝑟_𝑛>0. Then, the sequences {𝑥_𝑛} and {𝑢_𝑛} converge strongly to 𝑞∈ℱ, where 𝑞=𝑃_ℱ𝑓(𝑞).

Remark 16. Our results generalize the corresponding results of S. Takahashi and W. Takahashi [9] from a single valued nonexpansive mapping to a finite family of multivalued mappings satisfying the condition (𝐸). Our results also improve the recent results of Eslamian [16].

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This paper was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. The authors therefore, acknowledge with thanks DSR for technical and financial support. The authors are also thankful to the referees for their valuable suggestion/comments.