Strong Convergence Theorems for Modifying Halpern Iterations for Quasi-𝝓-Asymptotically Nonexpansive Multivalued Mapping in Banach Spaces with Applications

Abstract

An iterative sequence for quasi-𝜙-asymptotically nonexpansive multivalued mapping for modifying Halpern's iterations is introduced. Under suitable conditions, some strong convergence theorems are proved. The results presented in the paper improve and extend the corresponding results in the work by Chang et al. 2011.

1. Introduction

Throughout this paper, we denote by 𝑁 and 𝑅 the sets of positive integers and real numbers, respectively. Let 𝐷 be a nonempty closed subset of a real Banach space 𝑋. A mapping 𝑇∶𝐷→𝐷 is said to be nonexpansive, if ‖𝑇𝑥−𝑇𝑦‖≤‖𝑥−𝑦‖, for all 𝑥,𝑦∈𝐷. Let 𝑁(𝐷) and CB(𝐷) denote the family of nonempty subsets and nonempty closed bounded subsets of 𝐷, respectively. The Hausdorff metric on CB(𝐷) is defined by 𝐻𝐴₁,𝐴₂=maxsup_𝑥∈𝐴1𝑑𝑥,𝐴₂,sup_𝑦∈𝐴2𝑑𝑦,𝐴₁ (1.1) for 𝐴₁, 𝐴₂∈CB(𝐷), where 𝑑(𝑥,𝐴₁)=inf{‖𝑥−𝑦‖,𝑦∈𝐴₁}. The multivalued mapping 𝑇∶𝐷→CB(𝐷) is called nonexpansive, if 𝐻(𝑇𝑥,𝑇𝑦)≤‖𝑥−𝑦‖, for all 𝑥,𝑦∈𝐷. An element 𝑝∈𝐷 is called a fixed point of 𝑇∶𝐷→𝑁(𝐷), if 𝑝∈𝑇(𝑝). The set of fixed points of 𝑇 is represented by 𝐹(𝑇).

Let 𝑋 be a real Banach space with dual 𝑋^∗. We denote by 𝐽 the normalized duality mapping from 𝑋 to 2^𝑋∗ which is defined by 𝐽(𝑥)=𝑥^∗∈𝑋^∗∶𝑥,𝑥^∗=‖𝑥‖²=𝑥^∗2,𝑥∈𝑋, (1.2) where ⟨⋅,⋅⟩ denotes the generalized duality pairing.

A Banach space 𝑋 is said to be strictly convex, if ‖(𝑥+𝑦)/2‖≤1 for all 𝑥,𝑦∈𝑋 with ‖𝑥‖=‖𝑦‖=1 and 𝑥≠𝑦. A Banach space is said to be uniformly convex, if lim_𝑛→∞‖𝑥_𝑛−𝑦_𝑛‖=0 for any two sequences {𝑥_𝑛},{𝑦_𝑛}⊂𝑋 with ‖𝑥_𝑛‖=‖𝑦_𝑛‖=1 and lim_𝑛→∞‖(𝑥_𝑛+𝑦_𝑛)/2‖=0.

The norm of Banach space 𝑋 is said to be Gâteaux differentiable, if for each 𝑥,𝑦∈𝑆(𝑥), the limit lim_𝑡→0𝑥+𝑡𝑦−‖𝑥‖𝑡 (1.3) exists, where 𝑆(𝑥)={𝑥∶‖𝑥‖=1,𝑥∈𝑋}. In this case, 𝑋 is said to be smooth. The norm of Banach space 𝑋 is said to be Fréchet differentiable, if for each 𝑥∈𝑆(𝑥), the limit (1.3) is attained uniformly, for 𝑦∈𝑆(𝑥), and the norm is uniformly Fréchet differentiable if the limit (1.3) is attained uniformly for 𝑥,𝑦∈𝑆(𝑥). In this case, 𝑋 is said to be uniformly smooth.

Remark 1.1. The following basic properties for Banach space X and for the normalized duality mapping 𝐽 can be found in Cioranescu [1]. (1)𝑋 (𝑋^∗, resp.) is uniformly convex if and only if 𝑋^∗ (𝑋, resp.) is uniformly smooth.(2)If 𝑋 is smooth, then 𝐽 is single-valued and norm-to-weak* continuous.(3)If 𝑋 is reflexive, then 𝐽 is onto.(4)If 𝑋 is strictly convex, then 𝐽𝑥⋂𝐽𝑦≠∅, for all 𝑥,𝑦∈𝑋.(5)If 𝑋 has a Fréchet differentiable norm, then 𝐽 is norm-to-norm continuous.(6)If 𝑋 is uniformly smooth, then 𝐽 is uniformly norm-to-norm continuous on each bounded subset of 𝑋.(7)Each uniformly convex Banach space 𝑋 has the Kadec-Klee property, that is, for any sequence {𝑥_𝑛}⊂𝑋, if 𝑥_𝑛⇀𝑥∈𝑋 and ‖𝑥_𝑛‖→‖𝑥‖, then 𝑥_𝑛→𝑥∈𝑋.(8)If 𝑋 is a reflexive and strictly convex Banach space with a strictly convex dual 𝑋^∗ and 𝐽^∗∶𝑋^∗→𝑋 is the normalized duality mapping in 𝑋^∗, then 𝐽⁻¹=𝐽^∗, 𝐽𝐽^∗=𝐼_𝑋^∗ and 𝐽^∗𝐽=𝐼_𝑋.

Next, we assume that 𝑋 is a smooth, strictly convex, and reflexive Banach space and 𝐷 is a nonempty, closed and convex subset of 𝑋. In the sequel, we always use 𝜙∶𝑋×𝑋→𝑅⁺ to denote the Lyapunov functional defined by 𝜙𝑥,𝑦=‖𝑥‖²−2⟨𝑥,𝐽𝑦⟩+𝑦²,𝑥,𝑦∈𝑋. (1.4) It is obvious from the definition of the function 𝜙 that ‖𝑥‖−𝑦²≤𝜙𝑥,𝑦≤‖𝑥‖+𝑦²,𝜙𝑦,𝑥=𝜙𝑦,𝑧+𝜙(𝑧,𝑥)+2⟨𝑧−𝑦,𝐽𝑥−𝐽𝑧⟩,𝑥,𝑦,𝑧∈𝑋,𝜙𝑥,𝐽⁻¹𝜆𝐽𝑦+(1−𝜆)𝐽𝑧≤𝜆𝜙𝑥,𝑦+(1−𝜆)𝜙(𝑥,𝑧), (1.5)(1.6) for all 𝜆∈[0,1] and 𝑥,𝑦,𝑧∈𝑋.

Following Alber [2], the generalized projection Π_𝐷∶𝑋→𝐷 is defined by Π_𝐷(𝑥)=arginf_𝑦∈𝐷𝜙𝑦,𝑥,∀𝑥∈𝑋. (1.7) Many problems in nonlinear analysis can be reformulated as a problem of finding a fixed point of a nonexpansive mapping.

Example 1.2 (see [3]). Let Π_𝐷 be the generalized projection from a smooth, reflexive and strictly convex Banach space 𝑋 onto a nonempty closed convex subset 𝐷 of 𝑋, then Π_𝐷 is a closed and quasi-𝜙-nonexpansive from 𝑋 onto 𝐷.

In 1953, Mann [4] introduced the following iterative sequence {𝑥_𝑛}: 𝑥_𝑛+1=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝑇𝑥_𝑛, (1.8) where the initial guess 𝑥₁∈𝐷 is arbitrary, and {𝛼_𝑛} is a real sequence in [0,1]. It is known that under appropriate settings the sequence {𝑥_𝑛} converges weakly to a fixed point of 𝑇. However, even in a Hilbert space, Mann iteration may fail to converge strongly [5]. Some attempts to construct iteration method guaranteeing the strong convergence have been made. For example, Halpern [6] proposed the following so-called Halpern iteration: 𝑥_𝑛+1=𝛼_𝑛𝑢+1−𝛼_𝑛𝑇𝑥_𝑛, (1.9) where 𝑢, 𝑥₁∈𝐷 are arbitrary given and {𝛼_𝑛} is a real sequence in [0,1]. Another approach was proposed by Nakajo and Takahashi [7]. They generated a sequence as follows: 𝑥₁∈𝑋isarbitrary,𝑦_𝑛=𝛼_𝑛𝑢+1−𝛼_𝑛𝑇𝑥_𝑛,𝐶_𝑛=𝑧∈𝐷∶𝑦_𝑛−𝑧≤𝑥_𝑛,𝑄−𝑧_𝑛=𝑧∈𝐷∶𝑥_𝑛−𝑧,𝑥₁−𝑥_𝑛,𝑥≥0_𝑛+1=𝑃_{𝐶𝑛⋂𝑄𝑛}𝑥₁(𝑛=1,2,…), (1.10) where {𝛼_𝑛} is a real sequence in [0,1] and 𝑃_𝐾 denotes the metric projection from a Hilbert space 𝐻 onto a closed and convex subset 𝐾 of 𝐻. It should be noted here that the iteration previous works only in Hilbert space setting. To extend this iteration to a Banach space, the concept of relatively nonexpansive mappings are introduced (see [8–12]).

Inspired by Matsushita and Takahashi, in this paper, we introduce modifying Halpern-Mann iterations sequence for finding a fixed point of multivalued mapping 𝑇∶𝐷→CB(𝐷).

2. Preliminaries

In the sequel, we denote the strong convergence and weak convergence of the sequence {𝑥_𝑛} by 𝑥_𝑛→𝑥 and 𝑥_𝑛⇀𝑥, respectively.

Lemma 2.1 (see [2]). Let 𝑋 be a smooth, strictly convex, and reflexive Banach space, and let 𝐷 be a nonempty closed and convex subset of 𝑋. Then the following conclusions hold (a)𝜙(𝑥,y)=0 if and only if 𝑥=𝑦, for all 𝑥,𝑦∈𝑋; (b)𝜙(𝑥,Π_𝐷𝑦)+𝜙(Π_𝐷𝑦,𝑦)≤𝜙(𝑥,𝑦), for all 𝑥∈𝐷, for all 𝑦∈𝑋; (c)if 𝑥∈𝑋 and 𝑧∈𝐷, then 𝑧=Π_𝐷𝑥⇔⟨𝑧−𝑦,𝐽𝑥−𝐽𝑧⟩≥0, for all 𝑦∈𝐷.

Remark 2.2. If 𝐻 is a real Hilbert space, then 𝜙(𝑥,𝑦)=‖𝑥−𝑦‖² and Π_𝐷 is the metric projection 𝑃_𝐷 of 𝐻 onto 𝐷.

Definition 2.3. A point 𝑝∈𝐷 is said to be an asymptotic fixed point of 𝑇∶𝐷→CB(𝐷), if there exists a sequence {𝑥_𝑛}⊂𝐷 such that 𝑥_𝑛⇀𝑥∈𝑋 and 𝑑(𝑥_𝑛,𝑇(𝑥_𝑛))→0. Denote the set of all asymptotic fixed points of 𝑇 by 𝐹(𝑇).

Definition 2.4. (1) A multivalued mapping 𝑇∶𝐷→CB(𝐷) is said to be relatively nonexpansive, if 𝐹(𝑇)≠∅, 𝐹(𝑇)=𝐹(𝑇), and 𝜙(𝑝,𝑧)≤𝜙(𝑝,𝑥), for all 𝑥∈𝐷, 𝑝∈𝐹(𝑇), 𝑧∈𝑇(𝑥).
(2) A multivalued mapping 𝑇∶𝐷→CB(𝐷) is said to be closed, if for any sequence {𝑥_𝑛}⊂𝐷 with 𝑥_𝑛→𝑥∈𝐷 and 𝑑(𝑦,𝑇(𝑥_𝑛))→0, then 𝑑(𝑦,𝑇(𝑥))=0.

Next, we present an example of relatively nonexpansive multivalued mapping.

Example 2.5 (see [13]). Let 𝑋 be a smooth, strictly convex, and reflexive Banach space, let 𝐷 be a nonempty closed and convex subset of 𝑋, and let 𝑓∶𝐷×𝐷→𝑅 be a bifunction satisfying the conditions: (A1) 𝑓(𝑥,𝑥)=0, for all 𝑥∈𝐷; (A2) 𝑓(𝑥,𝑦)+𝑓(𝑦,𝑥)≤0, for all 𝑥,𝑦∈𝐷; (A3) lim_𝑡→0𝑓(𝑡𝑧+(1−𝑡)𝑥,𝑦)≤𝑓(𝑥,𝑦), for each 𝑥,𝑦,𝑧∈𝐷; (A4) the function 𝑦↦𝑓(𝑥,𝑦) is convex and lower semicontinuous, for each 𝑥∈𝐷. The “so-called” equilibrium problem for 𝑓 is to find a 𝑥^∗∈𝐷 such that 𝑓(𝑥^∗,𝑦)≥0, for all 𝑦∈𝐷. The set of its solutions is denoted by EP(𝑓).
Let 𝑟>0, 𝑥∈𝑋 and define mapping 𝑇_𝑟∶𝑋→𝐷 as follows: 𝑇_𝑟(𝑥)=+1𝑥∈𝐷,𝑓𝑧,𝑦𝑟𝑦−𝑧,𝐽𝑧−𝐽𝑥≥0,∀𝑦∈𝐷,∀𝑥∈𝑋, (2.1) then (1) 𝑇_𝑟 is single-valued, and so {𝑧}=𝑇_𝑟(𝑥); (2) 𝑇_𝑟 is a relatively nonexpansive mapping, therefore it is a closed and quasi-𝜙-nonexpansive mapping; (3) 𝐹(𝑇_𝑟)=EP(𝑓).

Definition 2.6. (1) A multivalued mapping 𝑇∶𝐷→CB(𝐷) is said to be quasi-𝜙-nonexpansive, if 𝐹(𝑇)≠∅, and 𝜙(𝑝,𝑧)≤𝜙(𝑝,𝑥), for all 𝑥∈𝐷, 𝑝∈𝐹(𝑇), 𝑧∈𝑇(𝑥).
(2) A multivalued mapping 𝑇∶𝐷→CB(𝐷) is said to be quasi-𝜙-asymptotically nonexpansive, if 𝐹(𝑇)≠∅, and there exists a real sequence 𝑘_𝑛⊂[1,+∞), 𝑘_𝑛→1 such that 𝜙𝑝,𝑧_𝑛≤𝑘_𝑛𝜙𝑝,𝑥,∀𝑥∈𝐷,𝑝∈𝐹(𝑇),𝑧_𝑛∈𝑇^𝑛(𝑥). (2.2)
(3) A multivalued mapping 𝑇∶𝐷→CB(𝐷) is said to be totally quasi-𝜙-asymptotically nonexpansive, if 𝐹(𝑇)≠∅, and there exist nonnegative real sequences {𝑣_𝑛}, {𝜇_𝑛} with 𝑣_𝑛, 𝜇_𝑛→0 (as 𝑛→∞) and a strictly increasing continuous function 𝜁∶𝑅⁺→𝑅⁺ with 𝜁(0)=0 such that 𝜙𝑝,𝑧_𝑛≤𝜙𝑝,𝑥+𝑣_𝑛𝜁𝜙𝑝,𝑥+𝜇_𝑛,∀𝑥∈𝐷,∀𝑛≥1,𝑝∈𝐹(𝑇),𝑧_𝑛∈𝑇^𝑛(𝑥). (2.3)

Remark 2.7. From the definitions, it is obvious that a relatively nonexpansive multivalued mapping is a quasi-𝜙-nonexpansive multivalued mapping, and a quasi-𝜙-nonexpansive multivalued mapping is a quasi-𝜙-asymptotically nonexpansive multivalued mapping, but the converse is not true.

Lemma 2.8. Let 𝑋 be a real uniformly smooth and strictly convex Banach space with Kadec-Klee property, and let 𝐷 be a nonempty closed and convex subset of 𝑋. Let {𝑥_𝑛} and {𝑦_𝑛} be two sequences in 𝐷 such that 𝑥_𝑛→𝑝 and 𝜙(𝑥_𝑛,𝑦_𝑛)→0, where 𝜙 is the function defined by (1.4), then 𝑦_𝑛→𝑝.

Proof. For 𝜙(𝑥_𝑛,𝑦_𝑛)→0, we have (‖𝑥_𝑛‖−‖𝑦_𝑛‖)²→0. This implies that ‖𝑦_𝑛‖→‖𝑝‖ and so ‖𝐽𝑦_𝑛‖→‖𝐽𝑝‖. Since 𝐷 is uniformly smooth, 𝑋^∗ is reflexive and 𝐽𝑋=𝑋^∗, therefore, there exist a subsequence {𝐽𝑦_𝑛𝑖}⊂{𝐽𝑦_𝑛} and a point 𝑥∈𝑋 such that 𝐽𝑦_𝑛𝑖⇀𝐽𝑥. Because the norm ‖⋅‖ is weakly lower semi continuous, we have 0=lim_{𝑛𝑖→∞}𝜙𝑥_𝑛𝑖,𝑦_𝑛𝑖=lim_{𝑛𝑖→∞}𝑥_𝑛𝑖²−2𝑥_𝑛𝑖,𝐽𝑦_𝑛𝑖+𝐽𝑦_𝑛𝑖²≥𝑝²−2⟨𝑝,𝐽𝑥⟩+‖𝐽𝑥‖²=𝜙𝑝,𝑥. (2.4) By Lemma 2.1(a), we have 𝑝=𝑥. Hence we have 𝐽𝑦_𝑛𝑖⇀𝐽𝑝. Since ‖𝐽𝑦_𝑛𝑖‖→‖𝐽𝑝‖ and 𝑋^∗ has the Kadec-Klee property, we have 𝐽𝑦_𝑛𝑖→𝐽𝑝. By Remark 1.1, it follows that 𝑦_𝑛𝑖⇀𝑝. Since ‖𝐽𝑦_𝑛𝑖‖→‖𝐽𝑝‖, by using the Kadec-Klee property of 𝑋, we get 𝑦_𝑛𝑖→𝑝. If there exists another subsequence {𝐽𝑦_𝑛𝑗}⊂{𝐽𝑦_𝑛} such that 𝑦_𝑛𝑗→𝑞, then we have 0=lim_{𝑛𝑗→∞}𝜙𝑥_𝑛𝑗,𝑦_𝑛𝑗=lim_{𝑛𝑗→∞}𝑥_𝑛𝑗²−2𝑥_𝑛𝑗,𝐽𝑦_𝑛𝑗+𝐽𝑦_𝑛𝑗²=𝑝²−2⟨𝑝,𝐽𝑞⟩+𝑞²=𝜙𝑝,𝑞. (2.5) This implies that 𝑝=𝑞. So 𝑦_𝑛→𝑝. The conclusion of Lemma 2.8 is proved.

Lemma 2.9. Let 𝑋 and 𝐷 be as in Lemma 2.8. Let 𝑇∶𝐷→CB(𝐷) be a closed and quasi-𝜙-asymptotically nonexpansive multivalued mapping with nonnegative real sequences {𝑘_𝑛}⊂[1,+∞), if 𝑘_𝑛→1, then the fixed point set 𝐹(𝑇) of 𝑇 is a closed and convex subset of 𝐷.

Proof. Let {𝑥_𝑛} be a sequence in 𝐹(𝑇), such that 𝑥_𝑛→𝑥^∗. Since 𝑇 is quasi-𝜙-asymptotically nonexpansive multivalued mapping, we have 𝜙𝑥_𝑛,𝑧≤𝑘₁𝜙𝑥_𝑛,𝑥^∗ (2.6) for all 𝑧∈𝑇𝑥^∗ and for all 𝑛∈𝑁. Therefore, 𝜙𝑥^∗,𝑧=lim_𝑛→∞𝜙𝑥_𝑛,𝑧≤lim_𝑛→∞𝑘₁𝜙𝑥_𝑛,𝑥^∗=𝑘₁𝜙𝑥^∗,𝑥^∗=0. (2.7) By Lemma 2.1, we obtain 𝑧=𝑥^∗, Hence, 𝑇𝑥^∗={𝑥^∗}. So, we have 𝑥^∗∈𝐹(𝑇). This implies that 𝐹(𝑇) is closed.
Let 𝑝, 𝑞∈𝐹(𝑇) and 𝑡∈(0,1), and put 𝑤=𝑡𝑝+(1−𝑡)𝑞. we prove that 𝑤∈𝐹(𝑇). Indeed, in view of the definition of 𝜙, let 𝑧_𝑛∈𝑇^𝑛𝑤, we have 𝜙𝑤,𝑧_𝑛=‖𝑤‖²−2⟨𝑤,𝐽𝑧_𝑛⟩+𝑧_𝑛²=‖𝑤‖²−2⟨𝑡𝑝+(1−𝑡)𝑞,𝐽𝑧_𝑛⟩+𝑧_𝑛²=‖𝑤‖²+𝑡𝜙𝑝,𝑧_𝑛+(1−𝑡)𝜙𝑞,𝑧_𝑛−𝑡𝑝²−(1−𝑡)𝑞². (2.8) Since 𝑡𝜙𝑝,𝑧_𝑛+(1−𝑡)𝜙𝑞,𝑧_𝑛≤𝑡𝑘_𝑛𝜙𝑝,𝑤+(1−𝑡)𝑘_𝑛𝜙𝑝𝑞,𝑤=𝑡²−2⟨𝑝,𝐽𝑤⟩+‖𝑤‖²+𝑘_𝑛𝜙+(𝑞−1𝑝,𝑤1−𝑡)²−2𝑞,𝐽𝑤+‖𝑤‖²+𝑘_𝑛𝜙𝑝−1𝑞,𝑤=𝑡²+(1−𝑡)𝑞²−‖𝑤‖²+𝑡𝑘_𝑛−1𝜙𝑝,𝑤+(1−𝑡)𝑘_𝑛−1𝜙𝑞,𝑤. (2.9) Substituting (2.8) into (2.9) and simplifying it, we have 𝜙𝑤,𝑧_𝑛≤𝑡𝑘_𝑛−1𝜙𝑝,𝑤+(1−𝑡)𝑘_𝑛−1𝜙𝑞,𝑤→0,(as𝑛→∞). (2.10) Hence, we have 𝑧_𝑛→𝑤. This implies that 𝑧_𝑛+1(∈𝑇𝑇^𝑛𝑤)→𝑤. Since 𝑇 is closed, we have 𝑇𝑤={𝑤}, that is, 𝑤∈𝐹(𝑇). This completes the proof of Lemma 2.9.

Definition 2.10. A mapping 𝑇∶𝐷→CB(𝐷) is said to be uniformly 𝐿-Lipschitz continuous, if there exists a constant 𝐿>0 such that ‖𝑥_𝑛−𝑦_𝑛‖≤𝐿‖𝑥−𝑦‖, where 𝑥,𝑦∈𝐷, 𝑥_𝑛∈𝑇^𝑛𝑥, 𝑦_𝑛∈𝑇^𝑛𝑦.

3. Main Results

Theorem 3.1. Let 𝑋 be a real uniformly smooth and strictly convex Banach space with Kadec-Klee property, let 𝐷 be a nonempty, closed and convex subset of 𝑋, and let 𝑇∶𝐷→CB(𝐷) be a closed and uniformly 𝐿-Lipschitz continuous quasi-𝜙-asymptotically nonexpansive multivalued mapping with nonnegative real sequences {𝑘_𝑛}⊂[1,+∞) and 𝑘_𝑛→1 satisfying condition (2.2). Let {𝛼_𝑛} be a sequence in (0,1). If {𝑥_𝑛} is the sequence generated by 𝑥₁∈𝑋isarbitrary;𝐷₁𝑦=𝐷,_𝑛=𝐽⁻¹𝛼_𝑛𝐽𝑥₁+1−𝛼_𝑛𝐽𝑧_𝑛,𝑧_𝑛∈𝑇^𝑛𝑥_𝑛,𝐷_𝑛+1=𝑧∈𝐷_𝑛∶𝜙𝑧,𝑦_𝑛≤𝛼_𝑛𝜙𝑧,𝑥₁+1−𝛼_𝑛𝜙𝑧,𝑥_𝑛+𝜉_𝑛,𝑥_𝑛+1=Π_𝐷𝑛+1𝑥₁(𝑛=1,2,…), (3.1) where 𝜉_𝑛=(𝑘_𝑛−1)sup_{𝑝∈𝐹(𝑇)}𝜙(𝑝,𝑥_𝑛), 𝐹(𝑇) is the fixed point set of 𝑇, and Π_𝐷𝑛+1 is the generalized projection of 𝑋 onto 𝐷_𝑛+1. If 𝐹(𝑇) is nonempty, then {𝑥_𝑛} converges strongly to Π_𝐹(𝑇)𝑥₁.

Proof. (I) First, we prove that 𝐷_𝑛 are closed and convex subsets in 𝐷. By the assumption that 𝐷₁=𝐷 is closed and convex. Suppose that 𝐷_𝑛 is closed and convex for some 𝑛≥1. In view of the definition of 𝜙, we have 𝐷_𝑛+1=𝑧∈𝐷_𝑛∶𝜙𝑧,𝑦_𝑛≤𝛼_𝑛𝜙𝑧,𝑥₁+1−𝛼_𝑛𝜙𝑧,𝑥_𝑛+𝜉_𝑛=𝑧∈𝐷∶𝜙𝑧,𝑦_𝑛≤𝛼_𝑛𝜙𝑧,𝑥₁+1−𝛼_𝑛𝜙𝑧,𝑥_𝑛+𝜉_𝑛∩𝐷_𝑛=𝑧∈𝐷∶2𝛼_𝑛⟨𝑧,𝐽𝑥₁⟩+21−𝛼_𝑛⟨𝑧,𝐽𝑥_𝑛⟩−2⟨𝑧,𝐽𝑧_𝑛⟩≤𝛼_𝑛𝑥₁²+1−𝛼_𝑛𝑥_𝑛²−𝑧_𝑛²∩𝐷_𝑛. (3.2) This shows that 𝐷_𝑛+1 is closed and convex. The conclusions are proved.
(II) Next, we prove that 𝐹(𝑇)⊂𝐷_𝑛, for all 𝑛≥1. In fact, it is obvious that 𝐹(𝑇)⊂𝐷₁. Suppose 𝐹(𝑇)⊂𝐷_𝑛, for some 𝑛≥1. Hence, for any 𝑢∈𝐹(𝑇)⊂𝐷_𝑛, by (1.6), we have 𝜙𝑢,𝑦_𝑛=𝜙𝑢,𝐽⁻¹𝛼_𝑛𝐽𝑥₁+1−𝛼_𝑛𝐽𝑧_𝑛≤𝛼_𝑛𝜙𝑢,𝑥₁+1−𝛼_𝑛𝜙𝑢,𝑧_𝑛≤𝛼_𝑛𝜙𝑢,𝑥₁+1−𝛼_𝑛𝑘_𝑛𝜙𝑢,𝑥_𝑛=𝛼_𝑛𝜙𝑢,𝑥₁+1−𝛼_𝑛𝜙𝑢,𝑥_𝑛+𝑘_𝑛−1𝜙𝑢,𝑥_𝑛≤𝛼_𝑛𝜙𝑢,𝑥₁+1−𝛼_𝑛𝜙𝑢,𝑥_𝑛+𝑘_𝑛−1sup_{𝑢∈𝐹(𝑇)}𝜙𝑢,𝑥_𝑛=𝛼_𝑛𝜙𝑢,𝑥₁+1−𝛼_𝑛𝜙𝑢,𝑥_𝑛+𝜉_𝑛. (3.3) This shows that 𝑢∈𝐹(𝑇)⊂𝐷_𝑛+1 and so 𝐹(𝑇)⊂𝐷_𝑛.
(III) Now, we prove that {𝑥_𝑛} converges strongly to some point 𝑝^∗. In fact, since 𝑥_𝑛=Π_𝐷𝑛𝑥₁, from Lemma 2.1(c), we have ⟨𝑥_𝑛−𝑦,𝐽𝑥₁−𝐽𝑥_𝑛⟩≥0,∀𝑦∈𝐷_𝑛. (3.4) Again since 𝐹(𝑇)⊂𝐷_𝑛, we have ⟨𝑥_𝑛−𝑢,𝐽𝑥₁−𝐽𝑥_𝑛⟩≥0,∀𝑢∈𝐹(𝑇). (3.5) It follows from Lemma 2.1(b) that for each 𝑢∈𝐹(𝑇) and for each 𝑛≥1, 𝜙𝑥_𝑛,𝑥₁=𝜙Π_𝐷𝑛𝑥₁,𝑥₁≤𝜙𝑢,𝑥₁−𝜙𝑢,𝑥_𝑛≤𝜙𝑢,𝑥₁. (3.6) Therefore, {𝜙(𝑥_𝑛,𝑥₁)} is bounded, and so is {𝑥_𝑛}. Since 𝑥_𝑛=Π_𝐷𝑛𝑥₁ and 𝑥_𝑛+1=Π_𝐷𝑛+1𝑥₁∈𝐷_𝑛+1⊂𝐷_𝑛, we have 𝜙(𝑥_𝑛,𝑥₁)≤𝜙(𝑥_𝑛+1,𝑥₁). This implies that {𝜙(𝑥_𝑛,𝑥₁)} is nondecreasing. Hence lim_𝑛→∞𝜙(𝑥_𝑛,𝑥₁) exists. Since 𝑋 is reflexive, there exists a subsequence {𝑥_𝑛𝑖}⊂{𝑥_𝑛} such that 𝑥_𝑛𝑖⇀𝑝^∗ (some point in 𝐷=𝐷₁). Since 𝐷_𝑛 is closed and convex and 𝐷_𝑛+1⊂𝐷_𝑛. This implies that 𝐷_𝑛 is weakly closed and 𝑝^∗∈𝐷_𝑛 for each 𝑛≥1. In view of 𝑥_𝑛𝑖=Π_𝐷𝑛𝑖𝑥₁, we have 𝜙𝑥_𝑛𝑖,𝑥₁≤𝜙𝑝^∗,𝑥₁,∀𝑛_𝑖≥1. (3.7) Since the norm ‖⋅‖ is weakly lower semicontinuous, we have lim_{𝑛𝑖→∞}inf𝜙𝑥_𝑛,𝑥₁=lim_{𝑛𝑖→∞}inf𝑥_𝑛𝑖²−2𝑥_𝑛𝑖,𝐽𝑥₁+𝑥₁²≥𝑝^∗2−2𝑝^∗,𝐽𝑥₁+𝑥₁²=𝜙𝑝^∗,𝑥₁, (3.8) and so 𝜙𝑝^∗,𝑥₁≤lim_{𝑛𝑖→∞}inf𝜙𝑥_𝑛,𝑥₁≤lim_{𝑛𝑖→∞}sup𝜙𝑥_𝑛,𝑥₁=𝜙𝑝^∗,𝑥₁. (3.9) This shows that lim_{𝑛𝑖→∞}𝜙(𝑥_𝑛𝑖,𝑥₁)=𝜙(𝑝^∗,𝑥₁), and we have ‖𝑥_𝑛𝑖‖→‖𝑝^∗‖. Since 𝑥_𝑛𝑖⇀𝑝^∗, by the virtue of Kadec-Klee property of 𝑋, we obtain that 𝑥_𝑛𝑖→𝑝^∗. Since {𝜙(𝑥_𝑛,𝑥₁)} is convergent, this together with lim_{𝑛𝑖→∞}𝜙(𝑥_𝑛𝑖,𝑥₁)=𝜙(𝑝^∗,𝑥₁) shows that lim_{𝑛𝑖→∞}𝜙(𝑥_𝑛,𝑥₁)=𝜙(𝑝^∗,𝑥₁). If there exists some subsequence {𝑥_𝑛𝑗}⊂{𝑥_𝑛} such that 𝑥_𝑛→𝑞, then from Lemma 2.1, we have 𝜙𝑝^∗,𝑞=lim_{𝑛𝑖,𝑛𝑗→∞}𝜙𝑥_𝑛𝑖,𝑥_𝑛𝑗=lim_{𝑛𝑖,𝑛𝑗→∞}𝜙𝑥_𝑛𝑖,Π_𝐷𝑛𝑗𝑥₁≤lim_{𝑛𝑖,𝑛𝑗→∞}𝜙𝑥_𝑛𝑖,𝑥₁−𝜙Π_𝐷𝑛𝑗𝑥₁,𝑥₁=lim_{𝑛𝑖,𝑛𝑗→∞}𝜙𝑥_𝑛𝑖,𝑥₁−𝜙𝑥_𝑛𝑗,𝑥₁=𝜙𝑝^∗,𝑥₁−𝜙𝑝^∗,𝑥₁=0, (3.10) that is, 𝑝^∗=𝑞 and hence 𝑥_𝑛→𝑝^∗. (3.11) By the way, from (3.11), it is easy to see that 𝜉_𝑛=𝑘_𝑛−1sup_{𝑝∈𝐹(𝑇)}𝜙𝑝,𝑥_𝑛→0. (3.12)
(IV) Now, we prove that 𝑝^∗∈𝐹(𝑇). In fact, since 𝑥_𝑛+1∈𝐷_𝑛+1, from (3.1), (3.11), and (3.12), we have 𝜙𝑥_𝑛+1,𝑦_𝑛≤𝛼_𝑛𝜙𝑥_𝑛+1,𝑥₁+1−𝛼_𝑛𝜙𝑥_𝑛+1,𝑥_𝑛+𝜉_𝑛→0. (3.13) Since 𝑥_𝑛→𝑝^∗, it follows from (3.13) and Lemma 2.8 that 𝑦_𝑛→𝑝^∗. (3.14) Since {𝑥_𝑛} is bounded and 𝑇 is quasi-𝜙-asymptotically nonexpansive multivalued mapping, 𝑇^𝑛𝑥_𝑛 is bounded. In view of 𝛼_𝑛→0. Hence from (3.1), we have that lim_𝑛→∞𝐽𝑦_𝑛−𝐽𝑧_𝑛=lim_𝑛→∞𝐽𝑥₁−𝐽𝑧_𝑛=0. (3.15) Since 𝐽𝑦_𝑛→𝐽𝑝^∗, this implies 𝐽𝑧_𝑛→𝐽𝑝^∗. From Remark 1.1, it yields that 𝑧_𝑛⇀𝑝^∗. (3.16) Again since 𝑧_𝑛−𝑝^∗=𝐽𝑧_𝑛−𝐽𝑝^∗≤𝐽𝑧_𝑛−𝐽𝑝^∗→0, (3.17) this together with (3.16) and the Kadec-Klee-property of 𝑋 shows that 𝑧_𝑛→𝑝^∗. (3.18) On the other hand, by the assumptions that 𝑇 is 𝐿-Lipschitz continuous, thus we have 𝑑𝑇𝑧_𝑛,𝑧_𝑛≤𝑑𝑇𝑧_𝑛,𝑧_𝑛+1+𝑧_𝑛+1−𝑥_𝑛+1+𝑥_𝑛+1−𝑥_𝑛+𝑥_𝑛−𝑧_𝑛≤(𝐿+1)𝑥_𝑛+1−𝑥_𝑛+𝑧_𝑛+1−𝑥_𝑛+1+𝑥_𝑛−𝑧_𝑛. (3.19) From (3.18) and 𝑥_𝑛→𝑝^∗, we have that 𝑑(𝑇𝑧_𝑛,𝑧_𝑛)→0. In view of the closeness of 𝑇, it yields that 𝑇(𝑝^∗)={𝑝^∗}, this implies that 𝑝^∗∈𝐹(𝑇).
(V) Finally, we prove that 𝑝^∗=Π_𝐹(𝑇)𝑥₁  and so 𝑥_𝑛→Π_𝐹(𝑇)𝑥₁. Let 𝑤=Π_𝐹(𝑇)𝑥₁. Since 𝑤∈𝐹(𝑇)⊂𝐷_𝑛, we have 𝜙(𝑝^∗,𝑥₁)≤𝜙(𝑤,𝑥₁). This implies that 𝜙𝑝^∗,𝑥₁=lim_𝑛→∞𝜙𝑥_𝑛,𝑥₁≤𝜙𝑤,𝑥₁, (3.20) which yields that 𝑝^∗=𝑤=Π_𝐹(𝑇)𝑥₁. Therefore, 𝑥_𝑛→Π_𝐹(𝑇)𝑥₁. This completes the proof of Theorem 3.1.

By Remark 2.7, the following corollaries are obtained.

Corollary 3.2. Let 𝑋 and 𝐷 be as in Theorem 3.1, and let 𝑇∶𝐷→CB(𝐷) be a closed and uniformly 𝐿-Lipschitz continuous a relatively nonexpansive multivalued mapping. Let {𝛼_𝑛} in (0,1) with lim_𝑛→∞𝛼_𝑛=0. Let {𝑥_𝑛} be the sequence generated by 𝑥₁∈𝑋isarbitrary;𝐷₁𝑦=𝐷,_𝑛=𝐽⁻¹𝛼_𝑛𝐽𝑥₁+1−𝛼_𝑛𝐽𝑧_𝑛,𝑧_𝑛∈𝑇𝑥_𝑛,𝐷_𝑛+1=𝑧∈𝐷_𝑛∶𝜙𝑧,𝑦_𝑛≤𝛼_𝑛𝜙𝑧,𝑥₁+1−𝛼_𝑛𝜙𝑧,𝑥_𝑛,𝑥_𝑛+1=Π_𝐷𝑛+1𝑥₁(𝑛=1,2,…), (3.21) where 𝐹(𝑇) is the set of fixed points of 𝑇, and Π_𝐷𝑛+1 is the generalized projection of 𝑋 onto 𝐷_𝑛+1, then {𝑥_𝑛} converges strongly to Π_𝐹(𝑇)𝑥₁.

Corollary 3.3. Let 𝑋 and 𝐷 be as in Theorem 3.1, and let 𝑇∶𝐷→CB(𝐷) be a closed and uniformly 𝐿-Lipschitz continuous quasi-𝜙-nonexpansive multivalued mapping. Let {𝛼_𝑛} be a sequence of real numbers such that 𝛼_𝑛∈(0,1) for all 𝑛∈𝑁, and satisfying: lim_𝑛→∞𝛼_𝑛=0. Let {𝑥_𝑛} be the sequence generated by (3.21). Then, {𝑥_𝑛} converges strongly to Π_𝐹(𝑇)𝑥₁.

4. Application

We utilize Corollary 3.3 to study a modified Halpern iterative algorithm for a system of equilibrium problems.

Theorem 4.1. Let 𝐷, 𝑋, and {𝛼_𝑛} be the same as in Theorem 3.1. Let 𝑓∶𝐷×𝐷→𝑅 be a bifunction satisfying conditions (A1)–(A4) as given in Example 2.5. Let 𝑇_𝑟∶𝑋→𝐷 be a mapping defined by (2.1), that is, 𝑇_𝑟(𝑥)=+1𝑥∈𝐷,𝑓𝑧,𝑦𝑟⟨𝑦−𝑧,𝐽𝑧−𝐽𝑥⟩≥0,∀𝑦∈𝐷,∀𝑥∈𝑋. (4.1) Let {𝑥_𝑛} be the sequence generated by 𝑥₁∈𝑋isarbitrary;𝐷₁𝑓𝑢=𝐷,_𝑛+1,𝑦𝑟𝑦−𝑢_𝑛,𝐽𝑢_𝑛−𝐽𝑥_𝑛≥0,∀𝑦∈𝐷,𝑟>0,𝑢_𝑛∈𝑇_𝑟𝑥_𝑛,𝑦_𝑛=𝐽⁻¹𝛼_𝑛𝐽𝑥₁+1−𝛼_𝑛𝐽𝑢_𝑛,𝐷_𝑛+1=𝑧∈𝐷_𝑛∶𝜙𝑧,𝑦_𝑛≤𝛼_𝑛𝜙𝑧,𝑥₁+1−𝛼_𝑛𝜙𝑧,𝑥_𝑛,𝑥_𝑛+1=Π_𝐷𝑛+1𝑥₁(𝑛=1,2,…). (4.2) If 𝐹(𝑇_𝑟)≠∅, then {𝑥_𝑛} converges strongly to Π_𝐹(𝑇)𝑥₁ which is a common solution of the system of equilibrium problems for 𝑓.

Proof. In Example 2.5, we have pointed out that 𝑢_𝑛=𝑇_𝑟(𝑥_𝑛), 𝐹(𝑇_𝑟)=EP(𝑓), and 𝑇_𝑟 is a closed and quasi-𝜙-nonexpansive mapping. Hence (4.2) can be rewritten as follows: 𝑥₁∈𝑋isarbitrary;𝐷₁𝑦=𝐷,_𝑛=𝐽⁻¹𝛼_𝑛𝐽𝑥₁+1−𝛼_𝑛𝐽𝑢_𝑛,𝑢_𝑛∈𝑇_𝑟𝑥_𝑛,𝐷_𝑛+1=𝑧∈𝐷_𝑛∶𝜙𝑧,𝑦_𝑛≤𝛼_𝑛𝜙𝑧,𝑥₁+1−𝛼_𝑛𝜙𝑧,𝑥_𝑛,𝑥_𝑛+1=Π_𝐷𝑛+1𝑥₁(𝑛=1,2,…). (4.3) Therefore the conclusion of Theorem 4.1 can be obtained from Corollary 3.3.