Shrinking Projection Method of Common Fixed Point Problems for Discrete Asymptotically Strictly Pseudocontractive Semigroups and Mixed Equilibrium Problems in Hilbert Spaces

Abstract

This paper is concerned with a common element of the set of common fixed point for a discrete asymptotically strictly pseudocontractive semigroup and the set of solutions of the mixed equilibrium problems in Hilbert spaces. The strong convergence theorem for the above two sets is obtained by a general iterative scheme based on the shrinking projection method which extends and improves the corresponding ones due to Kim [Proceedings of the Asian Conference on Nonlinear Analysis and Optimization (Matsue, Japan, 2008), 139–162].

1. Introduction

Throughout this paper, we always assume that 𝐶 is a nonempty closed convex subset of a real Hilbert space 𝐻 with inner product and norm denoted by ⟨⋅,⋅⟩ and ‖⋅‖, respectively. The domain of the function 𝜑∶𝐶→ℝ∪{+∞} is the set dom𝜑={𝑥∈𝐶∶𝜑(𝑥)<+∞}.(1.1) Let 𝜑∶𝐶→ℝ∪{+∞} be a proper extended real-valued function, and let Φ be a bifunction from 𝐶×𝐶 into ℝ such that 𝐶∩dom𝜑≠∅, where ℝ is the set of real numbers. The so-called mixed equilibrium problem is to find 𝑥∈𝐶 such thatΦ(𝑥,𝑦)+𝜑(𝑦)−𝜑(𝑥)≥0,∀𝑦∈𝐶.(1.2) The set of solutions of problem (1.2) is denoted by 𝑀𝐸𝑃(Φ,𝜑); that is, 𝑀𝐸𝑃(Φ,𝜑)={𝑥∈𝐶∶Φ(𝑥,𝑦)+𝜑(𝑦)−𝜑(𝑥)≥,∀𝑦∈𝐶}.(1.3) It is obvious that if 𝑥 is a solution of problem (1.2) then 𝑥∈dom𝜑. As special cases of problem (1.2), we have the following. (i)If 𝜑=0, then problem (1.2) is reduced to find 𝑥∈𝐶 such that Φ(𝑥,𝑦)≥0,∀𝑦∈𝐶.(1.4) We denote by 𝐸𝑃(Φ) the set of solutions of equilibrium problem, problem (1.4) which was studied by Blum and Oettli [1]. (ii)If Φ(𝑥,𝑦)=⟨𝐵𝑥,𝑦−𝑥⟩ for all 𝑥,𝑦∈𝐶 where a mapping 𝐵∶𝐶→𝐻, then problem (1.4) is reduced to find 𝑥∈𝐶 such that ⟨𝐵𝑥,𝑦−𝑥⟩≥0,∀𝑦∈𝐶.(1.5) We denote by 𝑉𝐼(𝐶,𝐵) the set of solutions of variational inequality problem, problem (1.5) which was studied by Hartman and Stampacchia [2]. (iii)If Φ=0, then problem (1.2) is reduced to find 𝑥∈𝐶 such that 𝜑(𝑦)−𝜑(𝑥)≥0,∀𝑦∈𝐶.(1.6) We denote by Argmin(𝜑) the set of solutions of minimize problem.

Recall that 𝑃_𝐶 is the metric projection of 𝐻 onto 𝐶; that is, for each 𝑥∈𝐻, there exists the unique point in 𝑃_𝐶𝑥∈𝐶 such that ‖𝑥−𝑃_𝐶𝑥‖=min_𝑦∈𝐶‖𝑥−𝑦‖. A mapping 𝑇∶𝐶→𝐶 is called nonexpansive if ‖𝑇𝑥−𝑇𝑦‖≤‖𝑥−𝑦‖ for all 𝑥,𝑦∈𝐶, and a mapping 𝑓∶𝐶→𝐶 is called a contraction if there exists a constant 𝛼∈(0,1) such that ‖𝑓(𝑥)−𝑓(𝑦)‖≤𝛼‖𝑥−𝑦‖ for all 𝑥,𝑦∈𝐶. A point 𝑥∈𝐶 is a fixed point of 𝑇 provided 𝑇𝑥=𝑥. We denote by 𝐹(𝑇) the set of fixed points of 𝑇; that is, 𝐹(𝑇)={𝑥∈𝐶∶𝑇𝑥=𝑥}. If 𝐶 is a nonempty bounded closed convex subset of 𝐻 and 𝑇 is a nonexpansive mapping of 𝐶 into itself, then 𝐹(𝑇) is nonempty (see [3]).

Iterative methods are often used to solve the fixed point equation 𝑇𝑥=𝑥. The most well-known method is perhaps the Picard successive iteration method when 𝑇 is a contraction. Picard's method generates a sequence {𝑥_𝑛} successively as 𝑥_𝑛+1=𝑇𝑥_𝑛 for all 𝑛∈ℕ with 𝑥₁=𝑥 chosen arbitrarily, and this sequence converges in norm to the unique fixed point of 𝑇. However, if 𝑇 is not a contraction (for instance, if 𝑇 is a nonexpansive), then Picard's successive iteration fails, in general, to converge. Instead, Mann's iteration method for a nonexpansive mapping 𝑇 (see [4]) prevails and generates a sequence {𝑥_𝑛} recursively by𝑥_𝑛+1=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝑇𝑥_𝑛,∀𝑛∈ℕ,(1.7) where 𝑥₁=𝑥∈𝐶 chosen arbitrarily and the sequence {𝛼_𝑛} lies in the interval [0,1]. Recall that a mapping 𝑇∶𝐶→𝐶 is said to be as follows.(i)𝜅-strictly pseudocontractive (see [5]) if there exists a constant 𝜅∈[0,1) such that ‖𝑇𝑥−𝑇𝑦‖²≤‖𝑥−𝑦‖²+𝜅‖(𝐼−𝑇)𝑥−(𝐼−𝑇)𝑦‖²,∀𝑥,𝑦∈𝐶,(1.8) in brief, we use 𝜅-SPC to denote the 𝜅-strictly pseudocontractive, it is obvious that 𝑇 is a nonexpansive if and only if 𝑇 is a 0-SPC.(ii)Asymptotically 𝜅-SPC (see [6]) if there exists a constant 𝜅∈[0,1) and a sequence {𝛾_𝑛} of nonnegative real numbers with lim_𝑛→∞𝛾_𝑛=0 such that ‖𝑇^𝑛𝑥−𝑇^𝑛𝑦‖²≤1+𝛾_𝑛‖𝑥−𝑦‖²+𝜅‖(𝐼−𝑇^𝑛)𝑥−(𝐼−𝑇^𝑛)𝑦‖²,∀𝑥,𝑦∈𝐶,(1.9) for all 𝑛∈ℕ, if 𝜅=0 then 𝑇 is an asymptotically nonexpansive with 𝑘_𝑛=√1+𝛾_𝑛 for all 𝑛∈ℕ; that is, 𝑇 is an asymptotically nonexpansive (see [7]) if there exists a sequence {𝑘_𝑛}⊂[1,∞) with lim_𝑛→∞𝑘_𝑛=1 such that ‖𝑇^𝑛𝑥−𝑇^𝑛𝑦‖≤𝑘_𝑛‖𝑥−𝑦‖,∀𝑥,𝑦∈𝐶,(1.10) for all 𝑛∈ℕ, it is known that the class of 𝜅-SPC mappings, and the classes of asymptotically 𝜅-SPC mappings are independent (see [8]).

The Mann's algorithm for nonexpansive mappings has been extensively investigated (see [5, 9, 10] and the references therein). One of the well-known results is proven by Reich [10] for a nonexpansive mapping 𝑇 on 𝐶, which asserts the weak convergence of the sequence {𝑥_𝑛} generated by (1.7) in a uniformly convex Banach space with a Frechet differentiable norm under the control condition ∑^∞𝑛=1𝛼_𝑛(1−𝛼_𝑛)=∞. Recently, Marino and Xu [11] devoloped and extended Reich's result to SPC mapping in Hilbert space setting. More precisely, they proved the weak convergence of the Mann's iteration process (1.7) for a 𝜅-SPC mapping 𝑇 on 𝐶, and, subsequently, this result was improved and carried over the class of asymptotically 𝜅-SPC mappings by Kim and Xu [12].

It is known that the Mann's iteration (1.7) is in general not strongly convergent (see [13]). The strong convergence is guaranteed and has been proposed by Nakajo and Takahashi [14], they modified the Mann's iteration method (1.7) which is to find a fixed point of a nonexpansive mapping by a hybrid method, which called the shrinking projection method (or the CQ method) as the following theorem.

Theorem NT. Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let 𝑇 be a nonexpansive mapping of 𝐶 into itself such that 𝐹(𝑇)≠∅. Suppose that 𝑥₁=𝑥∈𝐶 chosen arbitrarily, and let {𝑥_𝑛} be the sequence defined by 𝐶_𝑛𝑦_𝑛=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝑇𝑥_𝑛,𝑄=𝑧∈𝐶∶‖‖𝑦_𝑛−𝑧‖‖≤‖‖𝑥_𝑛−𝑧_𝑛‖‖,=𝑥𝑧∈𝐶∶⟨𝑥_𝑛−𝑧,𝑥₁−𝑥_𝑛⟩≥0_𝑛+1,=𝑃_{𝐶𝑛∩𝑄𝑛}𝑥₁,∀𝑛∈ℕ,(1.11) where 0≤𝛼_𝑛≤𝛼<1. Then {𝑥_𝑛} converges strongly to 𝑃_𝐹(𝑇)(𝑥₁).
Subsequently, Marino and Xu [15] introduced an iterative scheme for finding a fixed point of a 𝜅-SPC mapping as the following theorem.

Theorem MX. Let 𝐶 be a closed convex subset of a Hilbert space 𝐻, and, 𝑇∶𝐶→𝐶 be a 𝜅-SPC mapping for some 0≤𝜅<1. Assume that 𝐹(𝑇)≠∅. Suppose that 𝑥₁=𝑥∈𝐶 chosen arbitrarily, and let {𝑥_𝑛} be the sequence defined by 𝐶_𝑛𝑦_𝑛=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝑇𝑥_𝑛,=𝑧∈𝐶∶‖‖𝑦𝑄^𝑛−_𝑛𝑧‖‖²≤‖‖𝑥_𝑛−𝑧‖‖²+1−𝛼_𝑛𝜅−𝛼_𝑛‖‖𝑥_𝑛−𝑇𝑥_𝑛‖‖²,=𝑥𝑧∈𝐶∶⟨𝑥_𝑛−𝑧,𝑥₁−𝑥_𝑛⟩≥0_𝑛+1,=𝑃_{𝐶𝑛∩𝑄𝑛}𝑥₁,∀𝑛∈ℕ,(1.12) where 0≤𝛼_𝑛<1. Then the sequence {𝑥_𝑛} converges strongly to 𝑃_𝐹(𝑇)(𝑥₁).
Quite recently, Kim and Xu [12] has improved and carried Theorem MX over the more wider class of asymptotically 𝜅-SPC mappings as the following theorem.

Theorem KX. Let 𝐶 be a closed convex subset of a Hilbert space 𝐻, and, 𝑇∶𝐶→𝐶 be an asymptotically 𝜅-SPC mapping for some 0≤𝜅<1 and a bounded sequence {𝛾_𝑛}⊂[0,∞) such that 𝑙𝑖𝑚_𝑛→∞𝛾_𝑛=0. Assume that 𝐹(𝑇) is a nonempty bounded subset of 𝐶. Suppose that 𝑥₁=𝑥∈𝐶 chosen arbitrarily, and let {𝑥_𝑛} be the sequence defined by 𝐶_𝑛𝑦_𝑛=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝑇^𝑛𝑥_𝑛,=𝑧∈𝐶∶‖‖𝑦_𝑛−𝑄𝑧_𝑛‖‖²≤‖‖𝑥_𝑛−𝑧‖‖²+𝜅−𝛼_𝑛1−𝛼_𝑛‖‖𝑥_𝑛−𝑇^𝑛𝑥_𝑛‖‖²+𝜃_𝑛,=𝑥𝑧∈𝐶∶⟨𝑥_𝑛−𝑧,𝑥₁−𝑥_𝑛⟩≥0_𝑛+1,=𝑃_{𝐶𝑛∩𝑄𝑛}𝑥₁,∀𝑛∈ℕ,(1.13) where 𝜃_𝑛=Δ^2𝑛(1−𝛼_𝑛)𝛾_𝑛→0 as 𝑛→∞, Δ_𝑛=sup{‖𝑥_𝑛−𝑧‖∶𝑧∈𝐹(𝑇)}<∞, and 0≤𝛼_𝑛<1 such that limsup_𝑛→∞𝛼_𝑛<1−𝜅. Then the sequence {𝑥_𝑛} converges strongly to 𝑃_𝐹(𝑇)(𝑥₁).

Recall that a discrete family 𝒮={𝑇_𝑛∶𝑛≥0} of self-mappings of 𝐶 is said to be a Lipschitzian semigroup on 𝐶 if the following conditions are satisfied. (1)𝑇₀=𝐼 where 𝐼 denotes the identity operator on 𝐶. (2)𝑇_𝑛+𝑚𝑥=𝑇_𝑛𝑇_𝑚𝑥,∀𝑛,𝑚≥0,∀𝑥∈𝐶. (3)There exists a sequence {𝐿_𝑛} of nonnegative real numbers such that ‖‖𝑇_𝑛𝑥−𝑇_𝑛𝑦‖‖≤𝐿_𝑛‖𝑥−𝑦‖,∀𝑥,𝑦∈𝐶,∀𝑛≥0.(1.14) A discrete Lipschitzian semigroup 𝒮 is called nonexpansive semigroup if 𝐿_𝑛=1 for all 𝑛≥0, contraction semigroup if 0<𝐿_𝑛<1 for all 𝑛≥0 and, asymptotically nonexpansive semigroup if limsup_𝑛→∞𝐿_𝑛≤1, respectively. We use 𝐹(𝒮) to denote the common fixed point set of the semigroup 𝒮; that is, 𝐹(𝒮)={𝑥∈𝐶∶𝑇_𝑛𝑥=𝑥,∀𝑛≥0}.

Very recently, Kim [16] introduced asymptotically 𝜅-SPC semigroup, a discrete family 𝒮={𝑇_𝑛∶𝑛≥0} of self-mappings of 𝐶 which is said to be asymptotically 𝜅-SPC semigroup on 𝐶 if, in addition to (1), (2) and the following condition (3^) are satisfied. ( 3′)There exists a constant 𝜅∈[0,1) and a bounded sequence {𝐿_𝑛} of nonnegative real numbers with limsup_𝑛→∞𝐿_𝑛≤1 such that‖‖𝑇_𝑛𝑥−𝑇_𝑛𝑦‖‖²≤𝐿_𝑛‖𝑥−𝑦‖²+𝜅‖‖𝐼−𝑇_𝑛𝑥−𝐼−𝑇_𝑛𝑦‖‖²,∀𝑥,𝑦∈𝐶,∀𝑛≥0.(1.15)

Note that for both discrete asymptotically nonexpansive semigroups and discrete asymptotically 𝜅-SPC semigroups, we can always assume that the Lipschitzian constants {𝐿_𝑛}_𝑛≥0 are such that 𝐿_𝑛≥1 for all 𝑛≥0 and lim_𝑛→∞𝐿_𝑛=1; otherwise, we replace 𝐿_𝑛 for all 𝑛≥0 with 𝐿^𝑛=max{sup_𝑚≥𝑛𝐿_𝑚,1}. Therefore, for a single asymptotically 𝜅-SPC mapping 𝑇∶𝐶→𝐶 note that (1.15) immediately reduces to (1.9) by taking 𝑇_𝑛≡𝑇^𝑛 and 𝛾_𝑛=𝐿_𝑛−1 such that 𝐿_𝑛≥1 for all 𝑛≥0 and lim_𝑛→∞𝐿_𝑛=1.

To be more precise, Kim also showed in the framework of Hilbert spaces for the asymptotically 𝜅-SPC semigroups that 𝑇_𝑛 is continuous on 𝐶 for all 𝑛≥0 and that 𝐹(𝒮) is closed and convex (see Lemma  3.2 in [16]), and the demiclosedness principle (see Theorem  3.3 in [16]) holds in the sense that if {𝑥_𝑛} is a sequence in 𝐶 such that 𝑥_𝑛⇀𝑧 and limsup_𝑚→∞limsup_𝑛→∞‖𝑥_𝑛−𝑇_𝑚𝑥_𝑛‖=0 then, z∈𝐹(𝒮)=⋂^∞𝑛=0𝐹(𝑇_𝑛), and he also introduced an iterative scheme to find a common fixed point of a discrete asymptotically 𝜅-SPC semigroup and a bounded sequence {𝐿_𝑛}⊂[1,∞) such that lim_𝑛→∞𝐿_𝑛=1 as follows:𝑥₀𝑦=𝑥∈𝐶chosenarbitrarily,_𝑛𝐶_𝑛=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝑇_𝑛𝑥_𝑛,=𝑧∈𝐶∶‖‖𝑦_𝑛−𝑧𝑄‖‖²≤‖‖𝑥_𝑛−𝑧‖‖²+1−𝛼_𝑛𝜃_𝑛+𝜅−𝛼_𝑛‖‖𝑥_𝑛−𝑇_𝑛𝑥_𝑛‖‖²_𝑛,𝑥=𝑧∈𝐶∶⟨𝑥_𝑛−𝑧,𝑥₀−𝑥_𝑛⟩≥0_𝑛+1,=𝑃_{𝐶𝑛∩𝑄𝑛}𝑥₀,∀𝑛∈ℕ∪{0},(1.16) where 𝜃_𝑛=(𝐿_𝑛−1)⋅sup{‖𝑥_𝑛−𝑧‖²∶𝑧∈𝐹(𝒮)}<∞. He proved that under the parameter 0≤𝛼_𝑛<1 for all 𝑛∈ℕ∪{0}, if 𝐹(𝒮) is a nonempty bounded subset of 𝐶, then the sequence {𝑥_𝑛} generated by (1.16) converges strongly to 𝑃_𝐹(𝒮)(𝑥₀).

Inspired and motivated by the works mentioned above, in this paper, we introduce a general iterative scheme (3.1) below to find a common element of the set of common fixed point for a discrete asymptotically 𝜅-SPC semigroup and the set of solutions of the mixed equilibrium problems in Hilbert spaces. The strong convergence theorem for the above two sets is obtained based on the shrinking projection method which extend and improve the corresponding ones due to Kim [16].

2. Preliminaries

Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. For a sequence {𝑥_𝑛} in 𝐻, we denote the strong convergence and the weak convergence of {𝑥_𝑛} to 𝑥∈𝐻 by 𝑥_𝑛→𝑥 and 𝑥_𝑛⇀𝑥, respectively, and the weak 𝜔-limit set of {𝑥_𝑛} by 𝜔_𝑤(𝑥_𝑛)={𝑥∶∃𝑥_𝑛𝑗⇀𝑥}.

For solving the mixed equilibrium problem, let us assume that the bifunction Φ∶𝐶×𝐶→ℝ, the function 𝜑∶𝐶→ℝ∪{+∞} and the set 𝐶 satisfy the following conditions. (A1)Φ(𝑥,𝑥)=0 for all 𝑥∈𝐶.(A2)Φ is monotone; that is, Φ(𝑥,𝑦)+Φ(𝑦,𝑥)≤0 for all 𝑥,𝑦∈𝐶.(A3)For each 𝑥,𝑦,𝑧∈𝐶, lim_𝑡↓0Φ(𝑡𝑧+(1−𝑡)𝑥,𝑦)≤Φ(𝑥,𝑦).(2.1)(A4)For each 𝑥∈𝐶,𝑦↦Φ(𝑥,𝑦) is convex and lower semicontinuous. (A5)For each 𝑦∈𝐶,𝑥↦Φ(𝑥,𝑦) is weakly upper semicontinuous. (B1)For each 𝑥∈𝐶 and 𝑟>0, there exists a bounded subset 𝐷_𝑥⊂𝐶 and 𝑦_𝑥∈𝐶 such that for any 𝑧∈𝐶⧵𝐷_𝑥, Φ𝑧,𝑦_𝑥+𝜑𝑦_𝑥−𝜑(𝑧)+1𝑟⟨𝑦^𝑥−𝑧,𝑧−𝑥⟩<0.(2.2)(B2)𝐶 is a bounded set.

Lemma 2.1 (see [17]). Let 𝐻 be a Hilbert space. For any 𝑥,𝑦∈𝐻 and 𝜆∈ℝ, one has ‖𝜆𝑥+(1−𝜆)𝑦‖²=𝜆‖𝑥‖²+(1−𝜆)‖𝑦‖²−𝜆(1−𝜆)‖𝑥−𝑦‖².(2.3)

Lemma 2.2 (see [3]). Let 𝐶 be a nonempty closed convex subset of a Hilbert space 𝐻. Then the following inequality holds: ⟨𝑥−𝑃_𝐶𝑥,𝑃_𝐶𝑥−𝑦⟩≥0,∀𝑥∈𝐻,𝑦∈𝐶.(2.4)

Lemma 2.3 (see [18]). Let 𝐶 be a nonempty closed convex subset of a Hilbert space 𝐻, Φ∶𝐶×𝐶→ℝ satisfying the conditions (𝐴1)-(𝐴5), and let 𝜑∶𝐶→ℝ∪{+∞} be a proper lower semicontinuous and convex function. Assume that either (𝐵1) or (𝐵2) holds. For 𝑟>0, define a mapping 𝑆_𝑟∶𝐶→𝐶 as follows: 𝑆_𝑟(𝑥)=𝑧∈𝐶∶Φ(𝑧,𝑦)+𝜑(𝑦)−𝜑(𝑧)+1𝑟⟨𝑦−𝑧,𝑧−𝑥⟩≥0,∀𝑦∈𝐶,(2.5) for all 𝑥∈𝐶. Then, the following statement hold. (1)For each 𝑥∈𝐶, 𝑆_𝑟(x)≠∅. (2)𝑆_𝑟 is single-valued.(3)𝑆_𝑟 is firmly nonexpansive; that is, for any 𝑥,𝑦∈𝐶, ‖‖𝑆_𝑟𝑥−𝑆_𝑟𝑦‖‖²≤⟨𝑆_𝑟𝑥−𝑆_𝑟𝑦,𝑥−𝑦⟩.(2.6)(4)𝐹(𝑆_𝑟)=𝑀𝐸𝑃(Φ,𝜑). (5)𝑀𝐸𝑃(Φ,𝜑) is closed and convex.

Lemma 2.4 (see [3]). Every Hilbert space 𝐻 has Radon-Riesz property or Kadec-Klee property; that is, for a sequence {𝑥_𝑛}⊂𝐻 with 𝑥_𝑛⇀𝑥 and ‖𝑥_𝑛‖→‖𝑥‖ then 𝑥_𝑛→𝑥.

Lemma 2.5 (see [16]). Let 𝐶 be a nonempty closed convex subset of a Hilbert space 𝐻, and let 𝒮={𝑇_𝑛∶𝑛≥0} be an asymptotically 𝜅-strictly pseudocontractive semigroup on 𝐶. Let {𝑥_𝑛} be a sequence in 𝐶 such that lim_𝑛→∞‖𝑥_𝑛−𝑥_𝑛+1‖=0 and lim_𝑛→∞‖𝑥_𝑛−𝑇_𝑛𝑥_𝑛‖=0. Then 𝜔_𝑤(𝑥_𝑛)⊂𝐹(𝒮).

3. Main Results

Theorem 3.1. Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻, let Φ be a bifunction from 𝐶×𝐶 into ℝ satisfying the conditions (A1)–(A5), and let 𝜑∶𝐶→ℝ∪{+∞} be a proper lower semicontinuous and convex function with either (B1) or (B2) holds. Let 𝒮={𝑇_𝑛∶𝑛≥0} be an asymptotically 𝜅-SPC semigroup on 𝐶 for some 𝜅∈[0,1) and a bounded sequence {𝐿_𝑛}⊂[1,∞) such that lim_𝑛→∞𝐿_𝑛=1. Assume that Ω∶=𝐹(𝒮)∩𝑀𝐸𝑃(Φ,𝜑) is a nonempty bounded subset of 𝐶. For 𝑥₀=𝑥∈𝐶 chosen arbitrarily, suppose that {𝑥_𝑛},{𝑦_𝑛}, and {𝑢_𝑛} are generated iteratively by 𝑢_𝑛∈𝐶suchthatΦ𝑢_𝑛,𝑦+𝜑(𝑦)−𝜑𝑢_𝑛+1𝑟_𝑛𝑦⟨𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛⟩≥0,∀𝑦∈𝐶,_𝑛𝐶_𝑛+1=𝛼_𝑛𝑢_𝑛+1−𝛼_𝑛𝑇_𝑛𝑢_𝑛,=𝑧∈𝐶_𝑛∩𝑄_𝑛𝑄∶‖‖𝑦_𝑛−𝑧‖‖²≤‖‖𝑥_𝑛−𝑧‖‖²+1−𝛼_𝑛𝜃_𝑛+𝜅−𝛼_𝑛‖‖𝑢_𝑛−𝑇_𝑛𝑢_𝑛‖‖²_𝑛+1,=𝑧∈𝐶_𝑛∩𝐶𝑄_𝑛∶⟨𝑥_𝑛−𝑧,𝑥₀−𝑥_𝑛⟩≥0₀,𝑥=𝑄₀=𝐶,_𝑛+1=𝑃_{𝐶𝑛+1∩𝑄𝑛+1}𝑥₀,∀𝑛∈ℕ∪{0},(3.1) where 𝜃_𝑛=(𝐿_𝑛−1)⋅sup{‖𝑥_𝑛−𝑧‖²∶𝑧∈Ω}<∞ satisfying the following conditions: (C1){𝛼_𝑛}⊂[𝑎,𝑏] such that 𝜅<𝑎<𝑏<1,(C2){𝑟_𝑛}⊂[𝑟,∞) for some 𝑟>0,(C3)∑^∞𝑛=0|𝑟_𝑛+1−𝑟_𝑛|<∞. Then the sequences {𝑥_𝑛},{𝑦_𝑛}, and {𝑢_𝑛} converge strongly to 𝑤=𝑃_Ω(𝑥₀).

Proof. Pick 𝑝∈Ω. Therefore, by (3.1) and the definition of 𝑆_𝑟𝑛 in Lemma 2.3, we have 𝑢_𝑛=𝑆_𝑟𝑛𝑥_𝑛∈dom𝜑,(3.2) and, by 𝐹(𝒮)=⋂^∞𝑛=0𝐹(𝑇_𝑛) and Lemma 2.3(4), we have 𝑇_𝑛𝑝=𝑝=𝑆_𝑟𝑛𝑝.(3.3) By (3.2), (3.3), and the nonexpansiveness of 𝑆_𝑟𝑛, we have ‖‖𝑢_𝑛−𝑝‖‖=‖‖𝑆_𝑟𝑛𝑥_𝑛−𝑆_𝑟𝑛𝑝‖‖≤‖‖𝑥_𝑛−𝑝‖‖.(3.4) By (3.3), (3.4), Lemma 2.1, and the asymptotically 𝜅-SPC semigroupness of 𝒮, we have ‖‖𝑦_𝑛−𝑝‖‖²=‖‖𝛼_𝑛𝑢_𝑛−𝑝+1−𝛼_𝑛𝑇_𝑛𝑢_𝑛−𝑝‖‖²=𝛼_𝑛‖‖𝑢_𝑛−𝑝‖‖²+1−𝛼_𝑛‖‖𝑇_𝑛𝑢_𝑛−𝑝‖‖²−𝛼_𝑛1−𝛼_𝑛‖‖𝑢_𝑛−𝑇_𝑛𝑢_𝑛‖‖²≤𝛼_𝑛‖‖𝑢_𝑛−𝑝‖‖²+1−𝛼_𝑛𝐿_𝑛‖‖𝑢_𝑛−𝑝‖‖²+𝜅‖‖𝑢_𝑛−𝑇_𝑛𝑢_𝑛‖‖²−𝛼^𝑛1−𝛼_𝑛‖‖𝑢_𝑛−𝑇_𝑛𝑢_𝑛‖‖²=1+1−𝛼_𝑛𝐿_𝑛−1‖‖𝑢_𝑛−𝑝‖‖²+1−𝛼_𝑛𝜅−𝛼_𝑛‖‖𝑢_𝑛−𝑇_𝑛𝑢_𝑛‖‖²≤‖‖𝑥_𝑛−𝑝‖‖²+1−𝛼_𝑛𝜃_𝑛+𝜅−𝛼_𝑛‖‖𝑢_𝑛−𝑇_𝑛𝑢_𝑛‖‖²,(3.5) where 𝜃_𝑛∶=(𝐿_𝑛−1)⋅sup{‖𝑥_𝑛−𝑧‖²∶𝑧∈Ω} for all 𝑛∈ℕ∪{0}.
Firstly, we prove that 𝐶_𝑛∩𝑄_𝑛 is closed and convex for all 𝑛∈ℕ∪{0}. It is obvious that 𝐶₀∩𝑄₀ is closed and by mathematical induction that 𝐶_𝑛∩𝑄_𝑛 is closed for all 𝑛≥1; that is 𝐶_𝑛∩𝑄_𝑛 is closed for all 𝑛∈ℕ∪{0}. Let 𝜖_𝑛=(1−𝛼_𝑛)(𝜃_𝑛+(𝑘−𝛼_𝑛)‖𝑢_𝑛−𝑇_𝑛𝑢_𝑛‖²) since, for any 𝑧∈𝐶, ‖𝑦_𝑛−𝑧‖²≤‖𝑥_𝑛−𝑧‖²+𝜖_𝑛 is equivalent to ‖‖𝑦_𝑛−𝑥_𝑛‖‖²+2⟨𝑦_𝑛−𝑥_𝑛,𝑥_𝑛−𝑧⟩−𝜖_𝑛≤0,(3.6) for all 𝑛∈ℕ∪{0}. Therefore, for any 𝑧₁,𝑧₂∈𝐶_𝑛+1∩𝑄_𝑛+1⊂𝐶_𝑛∩𝑄_𝑛 and 𝜖∈(0,1), we have ‖‖𝑦_𝑛−𝑥_𝑛‖‖²+2𝑦_𝑛−𝑥_𝑛,𝑥_𝑛−𝜖𝑧₁+(1−𝜖)𝑧₂−𝜖_𝑛=×𝜖‖‖‖‖𝑦𝑦^𝑛−𝑥_𝑛‖‖²+2⟨𝑦_𝑛−𝑥_𝑛,𝑥_𝑛−𝑧₁⟩−𝜖_𝑛+(1−𝜖)_𝑛≤0,−𝑥_𝑛‖‖²+2⟨𝑦_𝑛−𝑥_𝑛,𝑥_𝑛−𝑧₂⟩−𝜖_𝑛(3.7) for all 𝑛∈ℕ∪{0}, and we have 𝑥_𝑛−𝜖𝑧₁+(1−𝜖)𝑧₂,𝑥₀−𝑥_𝑛=≥0𝜖,⟨𝑥_𝑛−𝑧₁,𝑥₀−𝑥_𝑛⟩+(1−𝜖)⟨𝑥_𝑛−𝑧₂,𝑥₀−𝑥_𝑛⟩(3.8) for all 𝑛∈ℕ∪{0}. Since 𝐶₀∩𝑄₀ is convex and by putting 𝑛=0 in (3.6), (3.7), and (3.8), we have that 𝐶₁∩𝑄₁ is convex. Suppose that 𝑥_𝑘 is given and 𝐶_𝑘∩𝑄_𝑘 is convex for some 𝑘≥1. It follows by putting 𝑛=𝑘 in (3.6), (3.7), and (3.8) that 𝐶_𝑘+1∩𝑄_𝑘+1 is convex. Therefore, by mathematical induction, we have that 𝐶_𝑛∩𝑄_𝑛 is convex for all 𝑛≥1; that is, 𝐶_𝑛∩𝑄_𝑛 is convex for all 𝑛∈ℕ∪{0}. Hence, we obtain that 𝐶_𝑛∩𝑄_𝑛 is closed and convex for all 𝑛∈ℕ∪{0}.
Next, we prove that Ω⊂𝐶_𝑛∩𝑄_𝑛 for all 𝑛∈ℕ∪{0}. It is obvious that 𝑝∈Ω⊂𝐶=𝐶₀∩𝑄₀. Therefore, by (3.1) and (3.5), we have 𝑝∈𝐶₁ and note that 𝑝∈𝐶=𝑄₁, and so 𝑝∈𝐶₁∩𝑄₁. Hence, we have Ω⊂𝐶₁∩𝑄₁. Since 𝐶₁∩𝑄₁ is a nonempty closed convex subset of 𝐶, there exists a unique element 𝑥₁∈𝐶₁∩𝑄₁ such that 𝑥₁=𝑃_{𝐶1∩𝑄1}(𝑥₀). Suppose that 𝑥_𝑘∈𝐶_𝑘∩𝑄_𝑘 is given such that 𝑥_𝑘=𝑃_{𝐶𝑘∩𝑄𝑘}(𝑥₀) and 𝑝∈Ω⊂𝐶_𝑘∩𝑄_𝑘 for some 𝑘≥1. Therefore, by (3.1) and (3.5), we have 𝑝∈𝐶_𝑘+1. Since 𝑥_𝑘=𝑃_{𝐶𝑘∩Q𝑘}(𝑥₀); therefore, by Lemma 2.2, we have ⟨𝑥_𝑘−𝑧,𝑥₀−𝑥_𝑘⟩≥0,(3.9) for all 𝑧∈𝐶_𝑘∩𝑄_𝑘. Thus, by (3.1), we have 𝑝∈𝑄_𝑘+1, and so 𝑝∈𝐶_𝑘+1∩𝑄_𝑘+1. Hence, we have Ω⊂𝐶_𝑘+1∩𝑄_𝑘+1. Since 𝐶_𝑘+1∩𝑄_𝑘+1 is a nonempty closed convex subset of 𝐶, there exists a unique element 𝑥_𝑘+1∈𝐶_𝑘+1∩𝑄_𝑘+1 such that 𝑥_𝑘+1=𝑃_{𝐶𝑘+1∩𝑄𝑘+1}(𝑥₀). Therefore, by mathematical induction, we obtain Ω⊂𝐶_𝑛∩𝑄_𝑛 for all 𝑛≥1, and so Ω⊂𝐶_𝑛∩𝑄_𝑛 for all 𝑛∈ℕ∪{0}, and we can define 𝑥_𝑛+1=𝑃_{𝐶𝑛+1∩𝑄𝑛+1}(𝑥₀) for all 𝑛∈ℕ∪{0}. Hence, we obtain that the iteration (3.1) is well defined.
Next, we prove that {𝑥_𝑛} is bounded. Since 𝑥_𝑛=𝑃_{𝐶𝑛∩𝑄𝑛}(𝑥₀) for all 𝑛∈ℕ∪{0}, we have ‖‖𝑥_𝑛−𝑥₀‖‖≤‖‖𝑧−𝑥₀‖‖,(3.10) for all 𝑧∈𝐶_𝑛∩𝑄_𝑛. It follows by 𝑝∈Ω⊂𝐶_𝑛∩𝑄_𝑛 that ‖𝑥_𝑛−𝑥₀‖≤‖𝑝−𝑥₀‖ for all 𝑛∈ℕ∪{0}. This implies that {𝑥_𝑛} is bounded and so are {𝑦_𝑛} and {𝑢_𝑛}.
Next, we prove that ‖𝑥_𝑛−𝑥_𝑛+1‖→0 and ‖𝑢_𝑛−𝑢_𝑛+1‖→0 as 𝑛→∞. Since 𝑥_𝑛+1=𝑃_{𝐶𝑛+1∩𝑄𝑛+1}(𝑥₀)∈𝐶_𝑛+1∩𝑄_𝑛+1⊂𝐶_𝑛∩𝑄_𝑛; therefore, by (3.10), we have ‖𝑥_𝑛−𝑥₀‖≤‖𝑥_𝑛+1−𝑥₀‖ for all 𝑛∈ℕ∪{0}. This implies that {‖𝑥_𝑛−𝑥₀‖} is a bounded nondecreasing sequence; there exists the limit of ‖𝑥_𝑛−𝑥₀‖; that is, lim_𝑛→∞‖‖𝑥_𝑛−𝑥₀‖‖=𝑚,(3.11) for some 𝑚≥0. Since 𝑥_𝑛+1∈𝑄_𝑛+1; therefore, by (3.1), we have 𝑥_𝑛−𝑥_𝑛+1,𝑥₀−𝑥_𝑛≥0.(3.12) It follows by (3.12) that ‖‖𝑥_𝑛−𝑥_𝑛+1‖‖²=‖‖𝑥_𝑛−𝑥₀+𝑥₀−𝑥_𝑛+1‖‖²=+‖‖𝑥2_𝑛−𝑥_𝑛𝑥₀‖‖²+2⟨𝑥_𝑛−𝑥₀,𝑥₀−𝑥_𝑛⟩−𝑥₀,𝑥_𝑛−𝑥_𝑛+1+‖‖𝑥_𝑛+1−𝑥₀‖‖²≤‖‖𝑥_𝑛+1−𝑥₀‖‖²−‖‖𝑥_𝑛−𝑥₀‖‖².(3.13) Therefore, by (3.11), we obtain ‖‖𝑥_𝑛−𝑥_𝑛+1‖‖⟶0as𝑛⟶∞.(3.14) Indeed, from (3.1), we have Φ𝑢_𝑛,𝑦+𝜑(𝑦)−𝜑𝑢_𝑛+1𝑟_𝑛Φ𝑢⟨𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛⟩≥0,∀𝑦∈𝐶,(3.15)_𝑛+1,𝑦+𝜑(𝑦)−𝜑𝑢_𝑛+1+1𝑟_𝑛+1𝑦−𝑢_𝑛+1,𝑢_𝑛+1−𝑥_𝑛+1≥0,∀𝑦∈𝐶.(3.16) Substituting 𝑦=𝑢_𝑛+1 into (3.15) and 𝑦=𝑢_𝑛 into (3.16), we have Φ𝑢_𝑛,𝑢_𝑛+1+𝜑𝑢_𝑛+1−𝜑𝑢_𝑛+1𝑟_𝑛𝑢_𝑛+1Φ𝑢−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛≥0,_𝑛+1,𝑢_𝑛+𝜑𝑢_𝑛−𝜑𝑢_𝑛+1+1𝑟_𝑛+1𝑢_𝑛−𝑢_𝑛+1,𝑢_𝑛+1−𝑥_𝑛+1≥0.(3.17) Therefore, by the condition (A2), we get 0≤Φ𝑢_𝑛,𝑢_𝑛+1+Φ𝑢_𝑛+1,𝑢_𝑛+𝑢_𝑛+1−𝑢_𝑛,𝑢^𝑛−𝑥_𝑛𝑟_𝑛−𝑢^𝑛+1−𝑥_𝑛+1≤𝑢_𝑛+1𝑟_𝑛+1−𝑢_𝑛,𝑢^𝑛−𝑥_𝑛𝑟_𝑛−𝑢^𝑛+1−𝑥_𝑛+1𝑟_𝑛+1.(3.18) It follows that 0≤𝑢_𝑛+1−𝑢_𝑛,𝑢_𝑛−𝑢_𝑛+1+𝑢_𝑛+1−𝑥_𝑛−𝑟^𝑛𝑟_𝑛+1𝑢_𝑛+1=𝑢_𝑛+1−𝑥_𝑛+1−𝑢_𝑛,𝑢_𝑛−𝑢_𝑛+1+𝑢_𝑛+1−𝑢_𝑛,𝑢_𝑛+1−𝑥_𝑛+1+𝑥_𝑛+1−𝑥_𝑛−𝑟^𝑛𝑟_𝑛+1𝑢_𝑛+1−𝑥_𝑛+1.(3.19) Thus, we have ‖‖𝑢_𝑛+1−𝑢_𝑛‖‖²≤𝑢_𝑛+1−𝑢_𝑛,𝑥_𝑛+1−𝑥_𝑛+1−𝑟^𝑛𝑟_𝑛+1𝑢^𝑛+1≤‖‖𝑢_𝑛+1−𝑥_𝑛+1−𝑢_𝑛‖‖‖‖𝑥_𝑛+1−𝑥_𝑛‖‖+||||1−𝑟^𝑛𝑟_𝑛+1||||‖‖𝑢_𝑛+1−𝑥_𝑛+1‖‖.(3.20) It follows by the condition (C2) that ‖‖𝑢_𝑛+1−𝑢_𝑛‖‖≤‖‖𝑥_𝑛+1−𝑥_𝑛‖‖+||𝑟^𝑛+1−𝑟_𝑛||𝑟_𝑛+1≤‖‖𝑥_𝑛+1‖‖𝑢_𝑛+1−𝑥_𝑛+1‖‖−𝑥_𝑛‖‖+𝐿𝑟||𝑟^𝑛+1−𝑟_𝑛||,(3.21) where 𝐿=sup_𝑛≥0‖𝑢_𝑛−𝑥_𝑛‖<∞. Therefore, by the condition (C3) and (3.14), we obtain ‖‖𝑢_𝑛−𝑢_𝑛+1‖‖⟶0as𝑛⟶∞.(3.22)
Next, we prove that ‖𝑢_𝑛−𝑇_𝑛𝑢_𝑛‖→0, ‖𝑦_𝑛−𝑥_𝑛‖→0, and ‖𝑢_𝑛−𝑥_𝑛‖→0 as 𝑛→∞. Since 𝑥_𝑛+1∈𝐶_𝑛+1, by (3.1), we have ‖‖𝑦_𝑛−𝑥_𝑛+1‖‖²≤‖‖𝑥_𝑛−𝑥_𝑛+1‖‖²+1−𝛼_𝑛𝜃_𝑛+𝜅−𝛼_𝑛‖‖𝑢_𝑛−𝑇_𝑛𝑢_𝑛‖‖².(3.23) It follows that 1−𝛼_𝑛𝛼_𝑛−𝜅‖‖𝑢_𝑛−𝑇_𝑛𝑢_𝑛‖‖²≤‖‖𝑥_𝑛−𝑥_𝑛+1‖‖²+1−𝛼_𝑛𝜃_𝑛−‖‖𝑦_𝑛−𝑥_𝑛+1‖‖²≤‖‖𝑥_𝑛−𝑥_𝑛+1‖‖²+𝜃_𝑛.(3.24) Therefore, by the condition (C1), (3.14), and lim_𝑛→∞𝜃_𝑛=0, we obtain ‖‖𝑢_𝑛−𝑇_𝑛𝑢_𝑛‖‖⟶0as𝑛⟶∞.(3.25) From (3.23) and the condition (C1), we have ‖‖𝑦_𝑛−𝑥_𝑛+1‖‖²≤‖‖𝑥_𝑛−𝑥_𝑛+1‖‖²+𝜃_𝑛.(3.26) Therefore, ‖‖𝑦_𝑛−𝑥_𝑛‖‖²=‖‖𝑦_𝑛−𝑥_𝑛+1+𝑥_𝑛+1−𝑥_𝑛‖‖²=‖‖𝑦_𝑛−𝑥_𝑛+1‖‖²+2𝑦_𝑛−𝑥_𝑛+1,𝑥_𝑛+1−𝑥_𝑛+‖‖𝑥_𝑛+1−𝑥_𝑛‖‖²≤‖‖𝑦_𝑛−𝑥_𝑛+1‖‖²+2‖‖𝑦_𝑛−𝑥_𝑛+1‖‖‖‖𝑥_𝑛+1−𝑥_𝑛‖‖+‖‖𝑥_𝑛+1−𝑥_𝑛‖‖²≤2‖‖𝑥_𝑛+1−𝑥_𝑛‖‖‖‖𝑥_𝑛+1−𝑥_𝑛‖‖+‖‖𝑦_𝑛−𝑥_𝑛+1‖‖+𝜃_𝑛.(3.27) Hence, by (3.14) and lim_𝑛→∞𝜃_𝑛=0, we obtain ‖‖𝑦_𝑛−𝑥_𝑛‖‖⟶0as𝑛⟶∞.(3.28) By (3.2), (3.3), and the firmly nonexpansiveness of 𝑆_𝑟𝑛, we have ‖‖𝑢_𝑛−𝑝‖‖²≤=1𝑆_𝑟𝑛𝑥_𝑛−𝑆_𝑟𝑛𝑝,𝑥_𝑛−𝑝=⟨𝑢_𝑛−𝑝,𝑥_𝑛−𝑝⟩2‖‖𝑢^𝑛−𝑝‖‖²+‖‖𝑥_𝑛−𝑝‖‖²−‖‖𝑢_𝑛−𝑥_𝑛‖‖².(3.29) It follows that ‖‖𝑢_𝑛−𝑝‖‖²≤‖‖𝑥_𝑛−𝑝‖‖²−‖‖𝑢_𝑛−𝑥_𝑛‖‖².(3.30) Therefore, by the condition (C1) and (3.5), we have ‖‖𝑦_𝑛−𝑝‖‖²≤1+1−𝛼_𝑛𝐿_𝑛−1‖‖𝑢_𝑛−𝑝‖‖²+1−𝛼_𝑛𝜅−𝛼_𝑛‖‖𝑢_𝑛−𝑇_𝑛𝑢_𝑛‖‖²≤𝐿_𝑛−𝛼_𝑛𝐿_𝑛−1‖‖𝑢_𝑛−𝑝‖‖²≤𝐿_𝑛‖‖𝑢_𝑛−𝑝‖‖²≤𝐿_𝑛‖‖𝑥^𝑛−𝑝‖‖²−‖‖𝑢_𝑛−𝑥_𝑛‖‖².(3.31) It follows that 𝐿_𝑛‖‖𝑢_𝑛−𝑥_𝑛‖‖²≤𝐿_𝑛‖‖𝑥_𝑛−𝑝‖‖²−‖‖𝑦_𝑛−𝑝‖‖²=𝐿_𝑛−1‖‖𝑥_𝑛−𝑝‖‖²+‖‖𝑥_𝑛−𝑝‖‖²−‖‖𝑦_𝑛−𝑝‖‖²≤𝜃_𝑛+‖‖𝑥_𝑛−𝑦_𝑛‖‖‖‖𝑥_𝑛−𝑝‖‖+‖‖𝑦_𝑛−𝑝‖‖.(3.32) Hence, by (3.28) and lim_𝑛→∞𝜃_𝑛=0, we obtain ‖‖𝑢_𝑛−𝑥_𝑛‖‖⟶0as𝑛⟶∞.(3.33) Since {𝑢_𝑛} is bounded, there exists a subsequence {𝑢_𝑛𝑖} of {𝑢_𝑛} which converges weakly to 𝑤. Next, we prove that 𝑤∈Ω. From (3.22) and (3.25), we have ‖𝑢_𝑛𝑖−𝑢_𝑛𝑖+1‖→0, and ‖𝑢_𝑛𝑖−𝑇_𝑛𝑖𝑢_𝑛𝑖‖→0 as 𝑖→∞; therefore, by Lemma 2.5, we obtain 𝑤∈𝐹(𝒮). From (3.1), we have 0≤Φ𝑢_𝑛,𝑦+𝜑(𝑦)−𝜑𝑢_𝑛+1𝑟_𝑛⟨𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛⟩,∀𝑦∈𝐶.(3.34) It follows by the condition (A2) that Φ𝑦,𝑢_𝑛≤Φ𝑦,𝑢_𝑛+Φ𝑢_𝑛,𝑦+𝜑(𝑦)−𝜑𝑢_𝑛+1𝑟_𝑛≤𝜑(𝑦)−𝜑𝑢_𝑛⟨𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛⟩,∀𝑦∈𝐶+𝑟1_𝑛⟨𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛⟩,∀𝑦∈𝐶.(3.35) Hence, 𝜑(𝑦)−𝜑𝑢_𝑛𝑖+𝑦−𝑢_𝑛𝑖,𝑢^𝑛𝑖−𝑥_𝑛𝑖𝑟_𝑛𝑖≥Φ𝑦,𝑢^𝑛𝑖,∀𝑦∈𝐶.(3.36) Therefore, from (3.33) and by 𝑢_𝑛𝑖⇀𝑤 as 𝑖→∞, we obtain Φ𝑦,𝑤+𝜑𝑤−𝜑(𝑦)≤0,∀𝑦∈𝐶.(3.37) For a constant 𝑡 with 0<𝑡<1 and 𝑦∈𝐶, let 𝑦_𝑡=𝑡𝑦+(1−𝑡)𝑤. Since 𝑦,𝑤∈𝐶, thus, 𝑦_𝑡∈𝐶. So, from (3.37), we have Φ𝑦_𝑡,𝑤+𝜑𝑤−𝜑𝑦_𝑡≤0.(3.38) By (3.38), the conditions (A1) and (A4), and the convexity of 𝜑, we have 0=≤Φ𝑡Φ𝑦_𝑡,𝑦𝑦_𝑡𝑡+𝜑𝑦_𝑡−𝜑𝑦_𝑡=𝑡Φ𝑦_𝑡,𝑦+(1−𝑡)Φ𝑦_𝑡,𝑤+𝑡𝜑(𝑦)+(1−𝑡)𝜑𝑤−𝜑𝑦_𝑡≤𝑡Φ𝑦_𝑡,𝑦+𝜑(𝑦)−𝜑𝑦_𝑡+(1−𝑡)Φ𝑦_𝑡,𝑤+𝜑𝑤−𝜑𝑦_𝑡,𝑦+𝜑(𝑦)−𝜑𝑦_𝑡.(3.39) It follows that Φ𝑦_𝑡,𝑦+𝜑(𝑦)−𝜑𝑦_𝑡≥0.(3.40) Therefore, by the condition (A3) and the weakly lower semicontinuity of 𝜑, we have Φ(𝑤,𝑦)+𝜑(𝑦)−𝜑(𝑤)≥0 as 𝑡→0 for all 𝑦∈𝐶, and; hence, we obtain 𝑤∈𝑀𝐸𝑃(Φ,𝜑), and so 𝑤∈Ω.
Since Ω is a nonempty closed convex subset of 𝐶, there exists a unique 𝑤∈Ω such that 𝑤=𝑃_Ω(𝑥₀). Next, we prove that 𝑥_𝑛→𝑤 as 𝑛→∞. Since 𝑤=𝑃_Ω(𝑥₀), we have ‖𝑥₀−𝑤‖≤‖𝑥₀−𝑧‖ for all 𝑧∈Ω; it follows that ‖‖𝑥₀−𝑤‖‖≤‖‖𝑥₀−𝑤‖‖.(3.41) Since 𝑤∈Ω⊂𝐶_𝑛∩𝑄_𝑛; therefore, by (3.10), we have ‖‖𝑥₀−𝑥_𝑛‖‖≤‖‖𝑥₀−𝑤‖‖.(3.42) Since ‖𝑥_𝑛𝑖−𝑢_𝑛i‖→0 by (3.33) and 𝑢_𝑛𝑖⇀𝑤, we have 𝑥_𝑛𝑖⇀𝑤 as 𝑖→∞. Therefore, by (3.41), (3.42), and the weak lower semicontinuity of norm, we have ‖‖𝑥₀−𝑤‖‖≤‖‖𝑥₀−𝑤‖‖≤liminf_𝑖→∞‖‖𝑥₀−𝑥_𝑛𝑖‖‖≤limsup_𝑖→∞‖‖𝑥₀−𝑥_𝑛𝑖‖‖≤‖‖𝑥₀−𝑤‖‖.(3.43) It follows that ‖‖𝑥₀−𝑤‖‖=lim_𝑖→∞‖‖𝑥₀−𝑥_𝑛𝑖‖‖=‖‖𝑥₀−𝑤‖‖.(3.44) Since 𝑥_𝑛𝑖⇀𝑤 as 𝑖→∞; therefore, we have 𝑥₀−𝑥_𝑛𝑖⇀𝑥₀−𝑤as𝑖⟶∞.(3.45) Hence, from (3.44), (3.45), the Kadec-Klee property, and the uniqueness of 𝑤=𝑃_Ω(𝑥₀), we obtain 𝑥_𝑛𝑖⟶𝑤=𝑤as𝑖⟶∞.(3.46) It follows that {𝑥_𝑛} converges strongly to 𝑤 and so are {𝑦_𝑛} and {𝑢_𝑛}. This completes the proof.

Remark 3.2. The iteration (3.1) is the difference with the iterative scheme of Kim [16] as the following. (1)The sequence {𝑥_𝑛} is a projection sequence of 𝑥₀ onto 𝐶_𝑛∩𝑄_𝑛 for all 𝑛∈ℕ∪{0} such that𝐶₀∩𝑄₀⊃𝐶₁∩𝑄₁⊃⋯⊃𝐶_𝑛∩𝑄_𝑛⊃⋯⊃Ω.(3.47)(2)A solving of a common element of the set of common fixed point for a discrete asymptotically 𝜅-SPC semigroup and the set of solutions of the mixed equilibrium problems by iteration is obtained.

For solving the equilibrium problem, let us define the condition (𝐵3) as the condition (𝐵1) such that 𝜑=0. We have the following result.

Corollary 3.3. Let 𝐶 be a nonempty closed convex subset of a real Hilbert space H, and let Φ be a bifunction from 𝐶×𝐶 into ℝ satisfying the conditions (A1)–(A5) with either (𝐵2) or (𝐵3) holds. Let 𝒮={𝑇_𝑛∶𝑛≥0} be an asymptotically 𝜅-SPC semigroup on 𝐶 for some 𝜅∈[0,1) and a bounded sequence {𝐿_𝑛}⊂[1,∞) such that lim_𝑛→∞𝐿_𝑛=1. Assume that Ω∶=𝐹(𝒮)∩𝐸𝑃(Φ) is a nonempty bounded subset of 𝐶. For 𝑥₀=𝑥∈𝐶 chosen arbitrarily, suppose that {𝑥_𝑛},{𝑦_𝑛}, and {𝑢_𝑛} are generated iteratively by 𝑢_𝑛∈𝐶suchthatΦ𝑢_𝑛,𝑦+1𝑟_𝑛𝑦⟨𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛⟩≥0,∀𝑦∈𝐶,_𝑛𝐶_𝑛+1=𝛼_𝑛𝑢_𝑛+1−𝛼_𝑛𝑇_𝑛𝑢_𝑛,=𝑧∈𝐶_𝑛∩𝑄_𝑛𝑄∶‖‖𝑦_𝑛−𝑧‖‖²≤‖‖𝑥_𝑛−𝑧‖‖²+1−𝛼_𝑛𝜃_𝑛+𝜅−𝛼_𝑛‖‖𝑢_𝑛−𝑇_𝑛𝑢_𝑛‖‖²_𝑛+1,=𝑧∈𝐶_𝑛∩𝐶𝑄_𝑛∶⟨𝑥_𝑛−𝑧,𝑥₀−𝑥_𝑛⟩≥0₀,𝑥=𝑄₀=𝐶,_𝑛+1=𝑃_{𝐶𝑛+1∩𝑄𝑛+1}𝑥₀,∀𝑛∈ℕ∪{0},(3.48) where 𝜃_𝑛=(𝐿_𝑛−1)⋅sup{‖𝑥_𝑛−𝑧‖²∶𝑧∈Ω}<∞ satisfying the following conditions: (C1){𝛼_n}⊂[𝑎,𝑏] such that 𝜅<𝑎<𝑏<1,(C2){𝑟_𝑛}⊂[𝑟,∞) for some 𝑟>0,(C3)∑^∞𝑛=0|𝑟_𝑛+1−𝑟_𝑛|<∞.Then the sequences {𝑥_𝑛},{𝑦_𝑛}, and {𝑢_𝑛} converge strongly to 𝑤=𝑃_Ω(𝑥₀).

Proof. It is concluded from Theorem 3.1 immediately, by putting 𝜑=0.

Corollary 3.4. Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let 𝒮={𝑇_𝑛∶𝑛≥0} be an asymptotically 𝜅-SPC semigroup on 𝐶 for some 𝜅∈[0,1) and a bounded sequence {𝐿_𝑛}⊂[1,∞) such that lim_𝑛→∞𝐿_𝑛=1. Assume that 𝐹(𝒮) is a nonempty bounded subset of 𝐶. For 𝑥₀=𝑥∈𝐶 chosen arbitrarily, suppose that {𝑥_𝑛} and {𝑦_𝑛} are generated iteratively by 𝐶_𝑛+1𝑦_𝑛=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝑇_𝑛𝑥_𝑛,=𝑧∈𝐶_𝑛∩𝑄_𝑛∶𝑄‖‖𝑦_𝑛−𝑧‖‖²≤‖‖𝑥_𝑛−𝑧‖‖²+1−𝛼_𝑛𝜃_𝑛+𝜅−𝛼_𝑛‖‖𝑥_𝑛−𝑇_𝑛𝑥_𝑛‖‖²_𝑛+1,=𝑧∈𝐶_𝑛∩𝐶𝑄_𝑛∶⟨𝑥_𝑛−𝑧,𝑥₀−𝑥_𝑛⟩≥0₀,𝑥=𝑄₀=𝐶,_𝑛+1=𝑃_{𝐶𝑛+1∩𝑄𝑛+1}𝑥₀,∀𝑛∈ℕ∪{0},(3.49) where 𝜃_𝑛=(𝐿_𝑛−1)⋅sup{‖𝑥_𝑛−𝑧‖²∶𝑧∈𝐹(𝒮)}<∞ and {𝛼_𝑛}⊂[𝑎,𝑏] such that 𝜅<𝑎<𝑏<1. Then the sequences {𝑥_𝑛} and {𝑦_𝑛} converge strongly to 𝑤=𝑃_𝐹(𝒮)(𝑥₀).

Proof. It is concluded from Corollary 3.3 immediately, by putting Φ=0.

4. Applications

We introduce the equilibrium problem to the optimization problem:min_𝑥∈𝐶𝜁(𝑥),(4.1) where 𝐶 is a nonempty closed convex subset of a real Hilbert space 𝐻 and 𝜁∶𝐶→ℝ∪{+∞} is a proper convex and lower semicontinuous. We denote by Argmin(𝜁) the set of solutions of problem (4.1). We define the condition (𝐵4) as the condition (𝐵3) such that Φ∶𝐶×𝐶→ℝ is a bifunction defined by Φ(𝑥,𝑦)=𝜁(𝑦)−𝜁(𝑥) for all 𝑥,𝑦∈𝐶. Observe that 𝐸𝑃(Φ)=Argmin(𝜁). We give the interesting result as the following theorem.

Theorem 4.1. Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻, and let 𝜁∶𝐶→ℝ∪{+∞} be a proper lower semicontinuous and convex function with either (𝐵2) or (𝐵4) holds. Let 𝒮={𝑇_𝑛∶𝑛≥0} be an asymptotically 𝜅-SPC semigroup on 𝐶 for some 𝜅∈[0,1) and a bounded sequence {𝐿_𝑛}⊂[1,∞) such that lim_𝑛→∞𝐿_𝑛=1. Assume that Ω∶=𝐹(𝒮)∩𝐴𝑟𝑔𝑚𝑖𝑛(𝜁) is a nonempty bounded subset of 𝐶. For 𝑥₀=𝑥∈𝐶 chosen arbitrarily, suppose that {𝑥_𝑛},{𝑦_𝑛} and {𝑢_𝑛} are generated iteratively by 𝑢_𝑛∈𝐶suchthat𝜁(𝑦)−𝜁𝑢_𝑛+1𝑟_𝑛𝑦⟨𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛⟩≥0,∀𝑦∈𝐶,_𝑛𝐶_𝑛+1=𝛼_𝑛𝑢_𝑛+1−𝛼_𝑛𝑇_𝑛𝑢_𝑛,=𝑧∈𝐶_𝑛∩𝑄_𝑛𝑄∶‖‖𝑦_𝑛−𝑧‖‖²≤‖‖𝑥_𝑛−𝑧‖‖²+1−𝛼_𝑛𝜃_𝑛+𝜅−𝛼_𝑛‖‖𝑢_𝑛−𝑇_𝑛𝑢_𝑛‖‖²_𝑛+1,=𝑧∈𝐶_𝑛∩𝐶𝑄_𝑛∶⟨𝑥_𝑛−𝑧,𝑥₀−𝑥_𝑛⟩≥0₀,𝑥=𝑄₀=𝐶,_𝑛+1=𝑃_{𝐶𝑛+1∩𝑄𝑛+1}𝑥₀,∀𝑛∈ℕ∪{0},(4.2) where 𝜃_𝑛=(𝐿_𝑛−1)⋅sup{‖𝑥_𝑛−𝑧‖²∶𝑧∈Ω}<∞ satisfying the following conditions: (C1){𝛼_𝑛}⊂[𝑎,𝑏] such that 𝜅<𝑎<𝑏<1, (C2){𝑟_𝑛}⊂[𝑟,∞) for some 𝑟>0,(C3)∑^∞𝑛=0|𝑟_𝑛+1−𝑟_𝑛|<∞. Then the sequences {𝑥_𝑛},{𝑦_𝑛}, and {𝑢_𝑛} converge strongly to 𝑤=𝑃_Ω(𝑥₀).

Proof. It is concluded from Corollary 3.3 immediately, by defining Φ(𝑥,𝑦)=𝜁(𝑦)−𝜁(𝑥) for all 𝑥,𝑦∈𝐶.