Shrinking Projection Method of Fixed Point Problems for Asymptotically Pseudocontractive Mapping in the Intermediate Sense and Mixed Equilibrium Problems in Hilbert Spaces

Abstract

This paper is concerned with a common element of the set of fixed point for an asymptotically pseudocontractive mapping in the intermediate sense 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 that of Qin et al. (2010) and many others.

1. Introduction

Throughout this paper, we always assume that 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻 with inner product and norm denoted by ⟨⋅,⋅⟩ and ‖⋅‖, respectively. 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. 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 the mixed equilibrium problem is to find 𝑥∈𝐶 such thatΦ𝑥,𝑦+𝜑𝑦−𝜑(𝑥)≥0,∀𝑦∈𝐶. (1.2) The set of solution of problem (1.2) is denoted by MEP(Φ,𝜑), that is, MEPΦ,𝜑=𝑥∈𝐶∶Φ𝑥,𝑦+𝜑𝑦−𝜑(𝑥)≥0,∀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 Φ𝑥,𝑦≥0,∀𝑦∈𝐶. (1.4) We denote by EP(Φ) the set of solutions of equilibrium problem, which problem (1.4) 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, which problem (1.5) was studied by Hartman and Stampacchia [2]. (iii)If Φ=0, then problem (1.2) is reduced to find 𝑥∈𝐶 such that 𝜑𝑦−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 uniformly L-Lipschitzian if there exists a constant 𝐿>0 such that for each 𝑛∈ℕ, ‖𝑇^𝑛𝑥−𝑇^𝑛𝑦‖≤𝐿‖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, generates a sequence {𝑥_𝑛} recursively by𝑥_𝑛+1=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝑇𝑥_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 (i)asymptotically pseudocontractive [5, 6] if there exists a sequence 𝑘_𝑛⊂[1,∞) with lim_𝑛→∞𝑘_𝑛=1 such that 𝑇^𝑛𝑥−𝑇^𝑛𝑦,𝑥−𝑦≤𝑘_𝑛𝑥−𝑦²,∀𝑥,𝑦∈𝐶,∀𝑛∈ℕ; (1.8) it is easy to see that (1.8) is equivalent to 𝑇^𝑛𝑥−𝑇^𝑛𝑦²≤2𝑘_𝑛−1𝑥−𝑦²+𝐼−𝑇^𝑛𝑥−𝐼−𝑇^𝑛𝑦²,∀𝑥,𝑦∈𝐶, (1.9) for all 𝑛∈ℕ, (ii)asymptotically pseudocontractive in the intermediate sense [7] if there exists a sequence 𝑘_𝑛⊂[1,∞) with lim_𝑛→∞𝑘_𝑛=1 such that limsup_𝑛→∞sup_{𝑥,𝑦∈𝐶}𝑇^𝑛𝑥−𝑇^𝑛𝑦,𝑥−𝑦−𝑘_𝑛𝑥−𝑦²≤0; (1.10) if we define 𝜏_𝑛=max0,sup_{𝑥,𝑦∈𝐶}𝑇^𝑛𝑥−𝑇^𝑛𝑦,𝑥−𝑦−𝑘_𝑛𝑥−𝑦², (1.11) then lim_𝑛→∞𝜏_𝑛=0 and it follows that (1.10) is reduced to 𝑇^𝑛𝑥−𝑇^𝑛𝑦,𝑥−𝑦≤𝑘_𝑛𝑥−𝑦²+𝜏_𝑛,∀𝑥,𝑦∈𝐶,∀𝑛∈ℕ; (1.12) it is easy to see that (1.12) is equivalent to 𝑇^𝑛𝑥−𝑇^𝑛𝑦²≤2𝑘_𝑛−1𝑥−𝑦²+(𝐼−𝑇^𝑛)𝑥−(𝐼−𝑇^𝑛)𝑦²+2𝜏_𝑛,∀𝑥,𝑦∈𝐶, (1.13) for all 𝑛∈ℕ; it is obvious that if 𝜏_𝑛=0 for all 𝑛∈ℕ, then the class of asymptotically pseudocontractive mappings in the intermediate sense is reduced to the class of asymptotically pseudocontractive mappings.

The Mann’s algorithm for nonexpansive mappings has been extensively investigated (see [8–10] 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)=∞. It is known that the Mann's iteration (1.7) is in general not strongly convergent (see [11]). The strong convergence guaranteed has been proposed by Nakajo and Takahashi [12], 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 that 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 {𝑥_𝑛} the sequence defined by 𝑦_𝑛=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝑇𝑥_𝑛,𝐶_𝑛=𝑧∈𝐶∶𝑦_𝑛−𝑧≤𝑥_𝑛,𝑄−𝑧_𝑛=𝑧∈𝐶∶𝑥_𝑛−𝑧,𝑥₁−𝑥_𝑛,𝑥≥0_𝑛+1=𝑃_{𝐶𝑛∩𝑄𝑛}𝑥₁,∀𝑛∈ℕ, (1.14) where 0≤𝛼_𝑛≤𝛼<1. Then {𝑥_𝑛} converges strongly to 𝑃_𝐹(𝑇)(𝑥₁).

Subsequently, Schu [5] modified Ishikawa's iteration method (see [13]) for the class of asymptotically pseudocontractive mappings as the following theorem.

Theorem S. Let 𝐶 be a nonempty bounded closed convex subset of a real Hilbert space 𝐻. Let 𝑇∶𝐶→𝐶 be a completely continuous uniformly 𝐿-Lipschitzian such that 𝐿>0 and asymptotically pseudocontractive mapping defined as in (1.9) with the sequence {𝑘_𝑛}⊂[1,∞) such that lim_𝑛→∞𝑘_𝑛=1. Let 𝑞_𝑛=2𝑘_𝑛−1 for all 𝑛∈ℕ. Suppose that 𝑥₁=𝑥∈𝐶 chosen arbitrarily and {𝑥_𝑛} the sequence defined by 𝑦_𝑛=𝛽_𝑛𝑇^𝑛𝑥_𝑛+1−𝛽_𝑛𝑥_𝑛,𝑥_𝑛+1=𝛼_𝑛𝑇^𝑛𝑦_𝑛+1−𝛼_𝑛𝑥_𝑛,∀𝑛∈ℕ, (1.15) where {𝛼_𝑛},{𝛽_𝑛}⊂(0,1) such that 𝜖≤𝛼_𝑛≤𝛽_𝑛≤𝑏 for some 𝜖>0 and 𝑏∈(0,𝐿⁻²(√1+𝐿²−1)) and ∑^∞_𝑛=1(𝑞²_𝑛−1)<∞. Then {𝑥_𝑛} converges strongly to some fixed point of 𝑇.

Quite recently, Zhou [14] showed that every uniformly 𝐿-Lipschitzian and asymptotically pseudocontractions which are also uniformly asymptotically regular has a fixed point and the fixed point set is closed and convex, and he also introduced iterative scheme to find a fixed point of a uniformly L-Lipschitzian and asymptotically pseudocontractive mapping as the following theorem.

Theorem Z. Let 𝐶 be a nonempty bounded closed convex subset of a real Hilbert space 𝐻. Let 𝑇∶𝐶→𝐶 be a uniformly L-Lipschitzian such that 𝐿>0 and asymptotically pseudocontractive mapping with a fixed point defined as in (1.8) with the sequence {𝑘_𝑛}⊂[1,∞) such that lim_𝑛→∞𝑘_𝑛=1. Suppose that 𝑥₁=𝑥∈𝐶 chosen arbitrarily and {𝑥_𝑛} the sequence defined by 𝑦_𝑛=1−𝛼_𝑛𝑥_𝑛+𝛼_𝑛𝑇^𝑛𝑥_𝑛,𝐶_𝑛=𝑧∈𝐶∶𝛼_𝑛1−(1+𝐿)𝛼_𝑛𝑥_𝑛−𝑇^𝑛𝑥_𝑛²≤𝑥_𝑛−𝑧,𝑦_𝑛−𝑇^𝑛𝑦_𝑛+𝑘_𝑛−1(diam𝐶)²,𝑄_𝑛=𝑧∈𝐶∶𝑥_𝑛−𝑧,𝑥₁−𝑥_𝑛,𝑥≥0_𝑛+1=𝑃_{𝐶𝑛∩𝑄𝑛}𝑥₁,∀𝑛∈ℕ, (1.16) where {𝛼_𝑛}⊂[𝑎,𝑏] such that 0<𝑎<𝑏<1/(1+𝐿). Then {𝑥_𝑛} converges strongly to 𝑃_𝐹(𝑇)(𝑥₁).

To be more precisely, Qin et al. [7] showed in the framework of a real Hilbert spaces 𝐻 for the uniformly 𝐿-Lipschitzian and asymptotically pseudocontractive mapping in the intermediate sense that the fixed point set is closed and convex (see Lemma 1.4 in [7]) and the demiclosedness principle holds (see Lemma 1.5 in [7]), and they also introduced an iterative scheme to find a fixed point of a uniformly 𝐿-Lipschitzian such that 𝐿>0 and asymptotically pseudocontractive mapping in the intermediate sense on a nonempty bounded closed convex 𝐶⊂𝐻 defined as in (1.13) with the sequences {𝑘_𝑛}⊂[1,∞) and {𝜏_𝑛}⊂[0,∞) such that lim_𝑛→∞𝑘_𝑛=1 and lim_𝑛→∞𝜏_𝑛=0, and let 𝑞_𝑛=2𝑘_𝑛−1 for all 𝑛∈ℕ as follows:𝑥₁𝑧=𝑥∈𝐶chosenarbitrarily,_𝑛=1−𝛽_𝑛𝑥_𝑛+𝛽_𝑛𝑇^𝑛𝑥_𝑛,𝑦_𝑛=1−𝛼_𝑛𝑥_𝑛+𝛼_𝑛𝑇^𝑛𝑧_𝑛,𝐶_𝑛=𝑧∈𝐶∶𝑦_𝑛−𝑧²≤𝑥_𝑛−𝑧²+𝛼_𝑛𝜃_𝑛+𝛼_𝑛𝛽_n𝛽_𝑛𝑞_𝑛+𝛽²_𝑛𝐿²+𝛽_𝑛−1𝑇^𝑛𝑥_𝑛−𝑥_𝑛²,𝑄_𝑛=𝑧∈𝐶∶𝑥_𝑛−𝑧,𝑥₁−𝑥_𝑛,𝑥≥0_𝑛+1=𝑃_{𝐶𝑛∩𝑄𝑛}𝑥₁,∀𝑛∈ℕ, (1.17) where 𝜃_𝑛=(𝑞_𝑛[1+𝛽_𝑛(𝑞_𝑛−1)]−1)⋅(diam𝐶)²+2(𝛽_𝑛𝑞_𝑛+1)𝜏_𝑛<∞. They proved that under the sequences {𝛼_𝑛},{𝛽_𝑛}⊂(0,1) such that 𝑎≤𝛼_𝑛≤𝛽_𝑛≤𝑏 for some 𝑎>0 and 𝑏∈(0,𝐿⁻²(√1+𝐿²−1)), if 𝐹(𝑇) is nonempty, then the sequence {𝑥_𝑛} generated by (1.17) converges strongly to a fixed point of 𝑇.

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 fixed point for an asymptotically pseudocontractive mapping in the intermediate sense and the set of solutions of 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 Qin et al. [7] and many others.

2. Preliminaries

Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. 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↓0Φ𝑡𝑧+(1−𝑡)𝑥,𝑦≤Φ𝑥,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, Φ𝑧,𝑦_𝑥+𝜑𝑦_𝑥(𝑧)+1−𝜑𝑟𝑦_𝑥−𝑧,𝑧−𝑥<0; (2.2)(B2)𝐶 is a bounded set.

Lemma 2.1 (see [15]). Let 𝐻 be a Hilbert space. For any 𝑥,𝑦∈𝐻 and 𝜆∈ℝ, we have 𝜆𝑥+(1−𝜆)𝑦²=𝜆‖𝑥‖²+(1−𝜆)𝑦²−𝜆(1−𝜆)𝑥−𝑦². (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 [16]). 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)=𝑦(𝑧)+1𝑧∈𝐶∶Φ𝑧,𝑦+𝜑−𝜑𝑟𝑦−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_𝑟𝑦²≤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 [7]). Let 𝐶 be a nonempty closed convex of a real Hilbert space 𝐻, and let 𝑇∶𝐶→𝐶 be a uniformly 𝐿-Lipschitz and asymptotically pseudocontractive mapping in the intermediate sense such that 𝐹(𝑇) is nonempty. Then 𝐼−𝑇 is demiclosed at zero. That is, whenever {𝑥_𝑛} is a sequence in 𝐶 weakly converging to some 𝑥∈𝐶 and the sequence {(𝐼−𝑇)𝑥_𝑛} strongly converges to some 𝑦, it follows that (𝐼−𝑇)𝑥=𝑦.

3. Main Results

Theorem 3.1. Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻, Φ a bifunction from 𝐶×𝐶 into ℝ satisfying the conditions (A1)–(A5), and 𝜑∶𝐶→ℝ∪{+∞} a proper lower semicontinuous and convex function with either (𝐵1) or (𝐵2) holds. Let 𝑇∶𝐶→𝐶 be a uniformly L -Lipschitzian such that 𝐿>0 and asymptotically pseudocontractive mapping in the intermediate sense defined as in (1.13) with the sequences {𝑘_𝑛}⊂[1,∞) and {𝜏_𝑛}⊂[0,∞) such that lim_𝑛→∞𝑘_𝑛=1 and lim_𝑛→∞𝜏_𝑛=0. Let 𝑞_𝑛=2𝑘_𝑛−1 for all 𝑛∈ℕ. Assume that Ω∶=𝐹(𝑇)∩MEP(Φ,𝜑) be a nonempty bounded subset of 𝐶. For 𝑥₁=𝑥∈𝐶 chosen arbitrarily, suppose that {𝑥_𝑛},{𝑦_𝑛},{𝑧_𝑛}, and {𝑢_𝑛} are generated iteratively by 𝑢_𝑛Φ𝑢∈𝐶suchthat_𝑛,𝑦+𝜑𝑦−𝜑𝑢_𝑛+1𝑟_𝑛𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛𝑧≥0,∀𝑦∈𝐶,_𝑛=1−𝛽_𝑛𝑢_𝑛+𝛽_𝑛𝑇^𝑛𝑢_𝑛,𝑦_𝑛=1−𝛼_𝑛𝑢_𝑛+𝛼_𝑛𝑇^𝑛𝑧_𝑛,𝐶_𝑛+1=𝑧∈𝐶_𝑛∩𝑄_𝑛∶𝑦_𝑛−𝑧²≤𝑥_𝑛−𝑧²+𝛼_𝑛𝜃_𝑛+𝛼_𝑛𝛽_𝑛𝛽_𝑛𝑞_𝑛+𝛽²_𝑛𝐿²+𝛽_𝑛−1𝑇^𝑛𝑢_𝑛−𝑢_𝑛²,Q_𝑛+1=𝑧∈𝐶_𝑛∩𝑄_𝑛∶𝑥_𝑛−𝑧,𝑥₁−𝑥_𝑛,𝐶≥0₁=𝑄₁𝑥=𝐶,_𝑛+1=𝑃_{𝐶𝑛+1∩𝑄𝑛+1}𝑥₁,∀𝑛∈ℕ, (3.1) where 𝜃_𝑛=(𝑞_𝑛[1+𝛽_𝑛(𝑞_𝑛−1)]−1)⋅Δ²_𝑛+2(𝛽_𝑛𝑞_𝑛+1)𝜏_𝑛 and Δ_𝑛=sup{‖𝑥_𝑛−𝑧‖∶𝑧∈Ω}<∞ satisfying the following conditions: (C1){𝛼_𝑛},{𝛽_𝑛}⊂(0,1) such that 𝑎≤𝛼_𝑛≤𝛽_𝑛≤𝑏 for some 𝑎>0 and 𝑏∈(0,𝐿⁻²(√1+𝐿²−1)); (C2){𝑟_𝑛}⊂[𝑟,∞) for some 𝑟>0;(C3)∑^∞_𝑛=1|𝑟_𝑛+1−𝑟_𝑛|<∞. Then the sequences {𝑥_𝑛},{𝑦_𝑛},{𝑧_𝑛}, and {𝑢_𝑛} converge strongly to 𝑤=𝑃_Ω(𝑥₁).

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 𝑇𝑝=𝑝, 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), Lemma 2.1, the uniformly 𝐿-Lipschitzian of 𝑇, and the asymptotically pseudocontractive mapping in the intermediate sense of 𝑇, we have 𝑦_𝑛−𝑝²=(1−𝛼_𝑛)(𝑢_𝑛−𝑝)+𝛼_𝑛(𝑇^𝑛𝑧_𝑛−𝑝)²=1−𝛼_𝑛𝑢_𝑛−𝑝²+𝛼_𝑛𝑇^𝑛𝑧_𝑛−𝑝²−𝛼_𝑛1−𝛼_𝑛𝑇^𝑛𝑧_𝑛−𝑢_𝑛²≤1−𝛼_𝑛𝑢_𝑛−𝑝²+𝛼_𝑛𝑞_𝑛𝑧_𝑛−𝑝²+𝑧_𝑛−𝑇^𝑛𝑧_𝑛²+2𝜏_𝑛−𝛼_𝑛1−𝛼_𝑛𝑇^𝑛𝑧_𝑛−𝑢_𝑛²,𝑧_𝑛−𝑇^𝑛𝑧_𝑛²=1−𝛽_𝑛𝑢_𝑛−𝑇^𝑛𝑧_𝑛+𝛽_𝑛𝑇^𝑛𝑢_𝑛−𝑇^𝑛𝑧_𝑛²=1−𝛽_𝑛𝑢_𝑛−𝑇^𝑛𝑧_𝑛²+𝛽_𝑛𝑇^𝑛𝑢_𝑛−𝑇^𝑛𝑧_𝑛²−𝛽_𝑛1−𝛽_𝑛𝑇^𝑛𝑢_𝑛−𝑢_𝑛²≤1−𝛽_𝑛𝑢_𝑛−𝑇^𝑛𝑧_𝑛²+𝛽_𝑛𝐿²𝑢_𝑛−𝑧_𝑛²−𝛽_𝑛1−𝛽_𝑛𝑇^𝑛𝑢_𝑛−𝑢_𝑛²=1−𝛽_𝑛𝑢_𝑛−𝑇^𝑛𝑧_𝑛²+𝛽_𝑛𝛽²_𝑛𝐿²+𝛽_𝑛−1𝑇^𝑛𝑢_𝑛−𝑢_𝑛²,𝑧_𝑛−𝑝²=1−𝛽_𝑛𝑢_𝑛−𝑝+𝛽_𝑛𝑇^𝑛𝑢_𝑛−𝑝²=1−𝛽_𝑛𝑢_𝑛−𝑝²+𝛽_𝑛𝑇^𝑛𝑢_𝑛−𝑝²−𝛽_𝑛1−𝛽_𝑛𝑇^𝑛𝑢_𝑛−𝑢_𝑛²≤1−𝛽_𝑛𝑢_𝑛−𝑝²+𝛽_𝑛𝑞_𝑛𝑢_𝑛−𝑝²+𝑢_𝑛−𝑇^𝑛𝑢_𝑛²+2𝜏_𝑛−𝛽_𝑛1−𝛽_𝑛𝑇^𝑛𝑢_𝑛−𝑢_𝑛²=1+𝛽_𝑛𝑞_𝑛−1𝑢_𝑛−𝑝²+𝛽²_𝑛𝑇^𝑛𝑢_𝑛−𝑢_𝑛²+2𝛽_𝑛𝜏_𝑛. (3.5)(3.6)(3.7) Substituting (3.6) and (3.7) into (3.5), and by the condition (C1) and (3.4), we have 𝑦_𝑛−𝑝²≤𝑢_𝑛−𝑝²+𝛼_𝑛𝑞_𝑛1+𝛽_𝑛𝑞_𝑛−1−1𝑢_𝑛−𝑝²+2𝛼_𝑛𝛽_𝑛𝑞_𝑛+1𝜏_𝑛+𝛼_𝑛𝛽_𝑛𝛽_𝑛𝑞_𝑛+𝛽²_𝑛𝐿²+𝛽_𝑛−1𝑇^𝑛𝑢_𝑛−𝑢_𝑛²−𝛼_𝑛𝛽_𝑛−𝛼_𝑛𝑢_𝑛−𝑇^𝑛𝑧_𝑛²≤𝑢_𝑛−𝑝²+𝛼_𝑛𝑞_𝑛1+𝛽_𝑛𝑞_𝑛−1−1𝑢_𝑛−𝑝²+2𝛽_𝑛𝑞_𝑛+1𝜏_𝑛+𝛼_𝑛𝛽_𝑛𝛽_𝑛𝑞_𝑛+𝛽²_𝑛𝐿²+𝛽_𝑛−1𝑇^𝑛𝑢_𝑛−𝑢_𝑛²≤𝑢_𝑛−𝑝²+𝛼_𝑛𝜃_𝑛+𝛼_𝑛𝛽_𝑛𝛽_𝑛𝑞_𝑛+𝛽²_𝑛𝐿²+𝛽_𝑛−1𝑇^𝑛𝑢_𝑛−𝑢_𝑛²≤𝑥_𝑛−𝑝²+𝛼_𝑛𝜃_𝑛+𝛼_𝑛𝛽_𝑛𝛽_𝑛𝑞_𝑛+𝛽²_𝑛𝐿²+𝛽_𝑛−1𝑇^𝑛𝑢_𝑛−𝑢_𝑛², (3.8) where 𝜃_𝑛∶=(𝑞_𝑛[1+𝛽_𝑛(𝑞_𝑛−1)]−1)⋅Δ²_𝑛+2(𝛽_𝑛𝑞_𝑛+1)𝜏_𝑛 and Δ_𝑛∶=sup{‖𝑥_𝑛−𝑧‖∶𝑧∈Ω}.
Firstly, we prove that 𝐶_𝑛∩𝑄_𝑛 is closed and convex for all 𝑛∈ℕ. It is obvious that 𝐶₁∩𝑄₁ is closed, and by mathematical induction that 𝐶_𝑛∩𝑄_𝑛 is closed for all 𝑛≥2, that is 𝐶_𝑛∩𝑄_𝑛 is closed for all 𝑛∈ℕ. Let 𝜖_𝑛=𝛼_𝑛𝜃_𝑛+𝛼_𝑛𝛽_𝑛(𝛽_𝑛𝑞_𝑛+𝛽²_𝑛𝐿²+𝛽_𝑛−1)‖𝑇^𝑛𝑢_𝑛−𝑢_𝑛‖². Since for any 𝑧∈𝐶, ‖𝑦_𝑛−𝑧‖²≤‖𝑥_𝑛−𝑧‖²+𝜖_𝑛 is equivalent to 𝑦_𝑛−𝑥_𝑛²+2𝑦_𝑛−𝑥_𝑛,𝑥_𝑛−𝑧−𝜖_𝑛≤0, (3.9) for all 𝑛∈ℕ. 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+2𝑦_𝑛−𝑥_𝑛,𝑥_𝑛−𝜖𝑧₁+(1−𝜖)𝑧₂−𝜖_𝑛=𝜖𝑦_𝑛−𝑥_𝑛²+2𝑦_𝑛−𝑥_𝑛,𝑥_𝑛−𝑧₁−𝜖_𝑛+(1−𝜖)𝑦_𝑛−x_𝑛²+2𝑦_𝑛−𝑥_𝑛,𝑥_𝑛−𝑧₂−𝜖_𝑛≤0, (3.10) for all 𝑛∈ℕ, and we have 𝑥_𝑛−𝜖𝑧₁+(1−𝜖)𝑧₂,𝑥₁−𝑥_𝑛=𝜖𝑥_𝑛−𝑧₁,𝑥₁−𝑥_𝑛+(1−𝜖)𝑥_𝑛−𝑧₂,𝑥₁−𝑥_𝑛≥0, (3.11) for all 𝑛∈ℕ. Since 𝐶₁∩𝑄₁ is convex, and by putting 𝑛=1 in (3.9), (3.10), and (3.11), we have 𝐶₂∩𝑄₂ is convex. Suppose that 𝑥_𝑘 is given and 𝐶_𝑘∩𝑄_𝑘 is convex for some 𝑘≥2. It follows by putting 𝑛=𝑘 in (3.9), (3.10), and (3.11) that 𝐶_𝑘+1∩𝑄_𝑘+1 is convex. Therefore, by mathematical induction, we have 𝐶_𝑛∩𝑄_𝑛 is convex for all 𝑛≥2, that is, 𝐶_𝑛∩𝑄_𝑛 is convex for all 𝑛∈ℕ. Hence, we obtain that 𝐶_𝑛∩𝑄_𝑛 is closed and convex for all 𝑛∈ℕ.
Next, we prove that Ω⊂𝐶_𝑛∩𝑄_𝑛 for all 𝑛∈ℕ. It is obvious that 𝑝∈Ω⊂𝐶=𝐶₁∩𝑄₁. Therefore, by (3.1) and (3.8), we have 𝑝∈𝐶₂, and note that 𝑝∈𝐶=𝑄₂, and so 𝑝∈𝐶₂∩𝑄₂. Hence, we have Ω⊂𝐶₂∩𝑄₂. Since 𝐶₂∩𝑄₂ is a nonempty closed convex subset of 𝐶, there exists a unique element 𝑥₂∈𝐶₂∩𝑄₂ such that 𝑥₂=𝑃_{𝐶2∩𝑄2}(𝑥₁). Suppose that 𝑥_𝑘∈𝐶_𝑘∩𝑄_𝑘 is given such that 𝑥_𝑘=𝑃_{𝐶𝑘∩𝑄𝑘}(𝑥₁), and 𝑝∈Ω⊂𝐶_𝑘∩𝑄_𝑘 for some 𝑘≥2. Therefore, by (3.1) and (3.8), we have 𝑝∈𝐶_𝑘+1. Since 𝑥_𝑘=𝑃_{𝐶𝑘∩𝑄𝑘}(𝑥₁), therefore, by Lemma 2.2, we have 𝑥_𝑘−𝑧,𝑥₁−𝑥_𝑘≥0, (3.12) 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}(𝑥₁). Therefore, by mathematical induction, we obtain Ω⊂𝐶_𝑛∩𝑄_𝑛 for all 𝑛≥2, and so Ω⊂𝐶_𝑛∩𝑄_𝑛 for all 𝑛∈ℕ, and we can define 𝑥_𝑛+1=𝑃_{𝐶𝑛+1∩𝑄𝑛+1}(𝑥₁) for all 𝑛∈ℕ. 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}(𝑥₁) for all 𝑛∈ℕ, we have ‖𝑥_𝑛−𝑥₁‖≤‖𝑧−𝑥₁‖, (3.13) for all 𝑧∈𝐶_𝑛∩𝑄_𝑛. It follows by 𝑝∈Ω⊂𝐶_𝑛∩𝑄_𝑛 that ‖𝑥_𝑛−𝑥₁‖≤‖𝑝−𝑥₁‖ for all 𝑛∈ℕ. This implies that {𝑥_𝑛} is bounded, and so are {𝑦_𝑛},{𝑧_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}(𝑥₁)∈𝐶_𝑛+1∩𝑄_𝑛+1⊂𝐶_𝑛∩𝑄_𝑛, therefore, by (3.13), we have ‖𝑥_𝑛−𝑥₁‖≤‖𝑥_𝑛+1−𝑥₁‖ for all 𝑛∈ℕ. This implies that {‖𝑥_𝑛−𝑥₁‖} is a bounded nondecreasing sequence; there exists the limit of ‖𝑥_𝑛−𝑥₁‖, that is lim_𝑛→∞𝑥_𝑛−𝑥₁=𝑚, (3.14) for some 𝑚≥0. Since 𝑥_𝑛+1∈𝑄_𝑛+1, therefore, by (3.1), we have 𝑥_𝑛−𝑥_𝑛+1,𝑥₁−𝑥_𝑛≥0. (3.15) It follows by (3.15) that 𝑥_𝑛−𝑥_𝑛+1²=𝑥_𝑛−𝑥₁+𝑥₁−𝑥_𝑛+1²=𝑥_𝑛−𝑥₁²+2𝑥_𝑛−𝑥₁,𝑥₁−𝑥_𝑛+2𝑥_𝑛−𝑥₁,𝑥_𝑛−𝑥_𝑛+1+𝑥_𝑛+1−𝑥₁²≤𝑥_𝑛+1−𝑥₁²−𝑥_𝑛−𝑥₁². (3.16) Therefore, by (3.14), we obtain 𝑥_𝑛−𝑥_𝑛+1⟶0as𝑛⟶∞. (3.17) Indeed, from (3.1) we have Φ𝑢_𝑛,𝑦+𝜑𝑦−𝜑𝑢_𝑛+1𝑟_𝑛𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛Φ𝑢≥0,∀𝑦∈𝐶,_𝑛+1,𝑦+𝜑𝑦−𝜑𝑢_𝑛+1+1𝑟_𝑛+1𝑦−𝑢_𝑛+1,𝑢_𝑛+1−𝑥_𝑛+1≥0,∀𝑦∈𝐶; (3.18)(3.19) substituting 𝑦=𝑢_𝑛+1 into (3.18) and 𝑦=𝑢_𝑛 into (3.19), we have Φ𝑢_𝑛,𝑢_𝑛+1+𝜑𝑢_𝑛+1−𝜑𝑢_𝑛+1𝑟_𝑛𝑢_𝑛+1−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛Φ𝑢≥0,_𝑛+1,𝑢_𝑛+𝜑𝑢_𝑛−𝜑𝑢_𝑛+1+1𝑟_𝑛+1𝑢_𝑛−𝑢_𝑛+1,𝑢_𝑛+1−𝑥_𝑛+1≥0. (3.20) Therefore, by the condition (A2), we get 0≤Φ𝑢_𝑛,𝑢_𝑛+1+Φ𝑢_𝑛+1,𝑢_𝑛+𝑢_𝑛+1−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛𝑟_𝑛−𝑢_𝑛+1−𝑥_𝑛+1𝑟_𝑛+1≤𝑢_𝑛+1−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛𝑟_𝑛−𝑢_𝑛+1−𝑥_𝑛+1𝑟_𝑛+1. (3.21) It follows that 0≤𝑢_𝑛+1−𝑢_𝑛,𝑢_𝑛−𝑢_𝑛+1+𝑢_𝑛+1−𝑥_𝑛−𝑟_𝑛𝑟_𝑛+1𝑢_𝑛+1−𝑥_𝑛+1=𝑢_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+1𝑢_𝑛+1−𝑥_𝑛+1. (3.22) Thus, we have 𝑢_𝑛+1−𝑢_𝑛²≤𝑢_𝑛+1−𝑢_𝑛,𝑥_𝑛+1−𝑥_𝑛+𝑟1−^𝑛𝑟_𝑛+1𝑢_𝑛+1−𝑥_𝑛+1≤𝑢_𝑛+1−𝑢_𝑛𝑥_𝑛+1−𝑥_𝑛+𝑟1−^𝑛𝑟_𝑛+1𝑢_𝑛+1−𝑥_𝑛+1. (3.23) It follows by the condition (C2) that 𝑢_𝑛+1−𝑢_𝑛≤𝑥_𝑛+1−𝑥_𝑛+𝑟_𝑛+1−𝑟_𝑛𝑟_𝑛+1𝑢_𝑛+1−𝑥_𝑛+1≤𝑥_𝑛+1−𝑥_𝑛+𝑀𝑟𝑟_𝑛+1−𝑟_𝑛, (3.24) where 𝑀=sup_𝑛≥1‖𝑢_𝑛−𝑥_𝑛‖<∞. Therefore, by the condition (C3) and (3.17), we obtain 𝑢_𝑛−𝑢_𝑛+1⟶0as𝑛⟶∞. (3.25)
Next, we prove that ‖𝑇^𝑛𝑢_𝑛−𝑢_𝑛‖→0 and ‖𝑇𝑢_𝑛−𝑢_𝑛‖→0 as 𝑛→∞. Since 𝑥_𝑛+1∈𝐶_𝑛+1, by (3.1), we have 𝑦_𝑛−𝑥_𝑛+1²≤𝑥_𝑛−𝑥_𝑛+1²+𝛼_𝑛𝜃_𝑛+𝛼_𝑛𝛽_𝑛𝛽_𝑛𝑞_𝑛+𝛽²_𝑛𝐿²+𝛽_𝑛−1𝑇^𝑛𝑢_𝑛−𝑢_𝑛², (3.26) it follows by the condition (C1) that 𝑎²1−𝑏𝑞_𝑛−𝑏²𝐿²−𝑏‖𝑇^𝑛𝑢_𝑛−𝑢_𝑛‖²≤𝛼_𝑛𝛽_𝑛1−𝛽_𝑛𝑞_𝑛−𝛽²_𝑛𝐿²−𝛽_𝑛𝑇^𝑛𝑢_𝑛−𝑢_𝑛²≤𝑥_𝑛−𝑥_𝑛+1²+𝛼_𝑛𝜃_𝑛−𝑦_𝑛−𝑥_𝑛+1²≤𝑥_𝑛−𝑥_𝑛+1²+𝑏𝜃_𝑛. (3.27) Since lim_𝑛→∞𝑞_𝑛=1 and the condition (C1), we have lim_𝑛→∞(1−𝑏𝑞_𝑛−𝑏²𝐿²−𝑏)=1−2𝑏−𝑏²𝐿²>0. Therefore, from (3.27) by (3.17) and lim_𝑛→∞𝜃_𝑛=0, we obtain 𝑇^𝑛𝑢_𝑛−𝑢_𝑛⟶0as𝑛⟶∞. (3.28) By the uniformly 𝐿-Lipschitzian of 𝑇, we have 𝑢_𝑛−𝑇𝑢_𝑛≤𝑢_𝑛−𝑢_𝑛+1+𝑢_𝑛+1−𝑇^𝑛+1𝑢_𝑛+1+𝑇^𝑛+1𝑢_𝑛+1−𝑇^𝑛+1𝑢_𝑛+𝑇^𝑛+1𝑢_𝑛−𝑇𝑢_𝑛≤(1+𝐿)𝑢_𝑛−𝑢_𝑛+1+𝑢_𝑛+1−𝑇^𝑛+1𝑢_𝑛+1+𝐿𝑇^𝑛𝑢_𝑛−𝑢_𝑛. (3.29) Therefore, by (3.25) and (3.28), we obtain 𝑢_𝑛−𝑇𝑢_𝑛⟶0as𝑛⟶∞. (3.30)
Next, we prove that ‖𝑦_𝑛−𝑥_𝑛‖→0, ‖𝑢_𝑛−𝑧_𝑛‖→0 and ‖𝑢_𝑛−𝑥_𝑛‖→0 as 𝑛→∞. From (3.26), by the condition (C1), we have 𝑦_𝑛−𝑥_𝑛+1²≤𝑥_𝑛−𝑥_𝑛+1²+𝑏𝜃_𝑛+𝑏²𝑏𝑞_𝑛+𝑏²𝐿²+𝑏−1𝑇^𝑛𝑢_𝑛−𝑢_𝑛²; (3.31) it follows that 𝑦_𝑛−𝑥_𝑛²=𝑦_𝑛−𝑥_𝑛+1+𝑥_𝑛+1−𝑥_𝑛²=𝑦_𝑛−𝑥_𝑛+1²+2𝑦_𝑛−𝑥_𝑛+1,𝑥_𝑛+1−𝑥_𝑛+𝑥_𝑛+1−𝑥_𝑛²≤𝑦_𝑛−𝑥_𝑛+1²+2𝑦_𝑛−𝑥_𝑛+1𝑥_𝑛+1−𝑥_𝑛+𝑥_𝑛+1−𝑥_𝑛²≤2𝑥_𝑛+1−𝑥_𝑛𝑥_𝑛+1−𝑥_𝑛+𝑦_𝑛−𝑥_𝑛+1+𝑏𝜃_𝑛−𝑏²1−𝑏𝑞_𝑛−𝑏²𝐿²−𝑏𝑇^𝑛𝑢_𝑛−𝑢_𝑛². (3.32) Therefore, by (3.17), lim_𝑛→∞𝜃_𝑛=0, and lim_𝑛→∞(1−𝑏𝑞_𝑛−𝑏²𝐿²−𝑏)=1−2𝑏−𝑏²𝐿²>0, we obtain 𝑦_𝑛−𝑥_𝑛⟶0as𝑛⟶∞. (3.33) From (3.1), we have ‖𝑢_𝑛−𝑧_𝑛‖=𝛽_𝑛‖𝑢_𝑛−𝑇^𝑛𝑢_𝑛‖; therefore, by (3.28), we obtain 𝑢_𝑛−𝑧_𝑛⟶0as𝑛⟶∞. (3.34) By (3.2), (3.3), and the firmly nonexpansiveness of 𝑆_𝑟𝑛, we have 𝑢_𝑛−𝑝²≤𝑆_𝑟𝑛𝑥_𝑛−𝑆_𝑟𝑛𝑝,𝑥_𝑛−𝑝=𝑢_𝑛−𝑝,𝑥_𝑛=1−𝑝2𝑢_𝑛−𝑝²+𝑥_𝑛−𝑝²−𝑢_𝑛−𝑥_𝑛², (3.35) it follows that ‖𝑢_𝑛−𝑝‖²≤‖𝑥_𝑛−𝑝‖²−‖𝑢_𝑛−𝑥_𝑛‖². (3.36) Therefore, from (3.8), we have ‖𝑦_𝑛−𝑝‖²≤‖𝑢_𝑛−𝑝‖²+𝛼_𝑛𝜃_𝑛+𝛼_𝑛𝛽_𝑛𝛽_𝑛𝑞_𝑛+𝛽²_𝑛𝐿²+𝛽_𝑛−1‖𝑇^𝑛𝑢_𝑛−𝑢_𝑛‖²≤‖𝑥_𝑛−𝑝‖²−‖𝑢_𝑛−𝑥_𝑛‖²+𝛼_𝑛𝜃_𝑛+𝛼_𝑛𝛽_𝑛𝛽_𝑛𝑞_𝑛+𝛽²_𝑛𝐿²+𝛽_𝑛−1‖𝑇^𝑛𝑢_𝑛−𝑢_𝑛‖²; (3.37) it follows by the condition (C1) that 𝑢_𝑛−𝑥_𝑛²≤𝑥_𝑛−𝑝²−𝑦_𝑛−𝑝²+𝛼_𝑛𝜃_𝑛+𝛼_𝑛𝛽_𝑛𝛽_𝑛𝑞_𝑛+𝛽²_𝑛𝐿²+𝛽_𝑛−1𝑇^𝑛𝑢_𝑛−𝑢_𝑛²≤𝑥_𝑛−𝑦_𝑛𝑥_𝑛−𝑝+𝑦_𝑛−𝑝+𝑏𝜃_𝑛−𝑏²1−𝑏𝑞_𝑛−𝑏²𝐿²−𝑏𝑇^𝑛𝑢_𝑛−𝑢_𝑛². (3.38) Therefore, by (3.33), lim_𝑛→∞𝜃_𝑛=0, and lim_𝑛→∞(1−𝑏𝑞_𝑛−𝑏²𝐿²−𝑏)=1−2𝑏−𝑏²𝐿²>0, we obtain ‖𝑢_𝑛−𝑥_𝑛‖⟶0as𝑛⟶∞. (3.39) 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 𝑢_𝑛𝑖⇀𝑤 and ‖𝑢_𝑛𝑖−𝑇𝑢_𝑛𝑖‖→0 as 𝑖→∞ by (3.30), therefore, by Lemma 2.5, we obtain 𝑤∈𝐹(𝑇). From (3.1), we have 0≤Φ𝑢_𝑛,𝑦+𝜑𝑦−𝜑𝑢_𝑛+1𝑟_𝑛𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛,∀𝑦∈𝐶; (3.40) it follows by the condition (A2) that Φ𝑦,𝑢_𝑛≤Φ𝑦,𝑢_𝑛+Φ𝑢_𝑛,𝑦+𝜑𝑦−𝜑𝑢_𝑛+1𝑟_𝑛𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛𝑦𝑢,∀𝑦∈𝐶≤𝜑−𝜑_𝑛+1𝑟_𝑛𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛,∀𝑦∈𝐶. (3.41) Hence, 𝜑𝑦−𝜑𝑢_𝑛𝑖+𝑦−𝑢_𝑛𝑖,𝑢_𝑛𝑖−𝑥_𝑛𝑖𝑟_𝑛𝑖≥Φ𝑦,𝑢_𝑛𝑖,∀𝑦∈𝐶. (3.42) Therefore, from (3.39) and by 𝑢_𝑛𝑖⇀𝑤 as 𝑖→∞, we obtain Φ𝑦,𝑤+𝜑(𝑤)−𝜑𝑦≤0,∀𝑦∈𝐶. (3.43) 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 (3.43), we have Φ𝑦_𝑡,𝑤+𝜑(𝑤)−𝜑𝑦_𝑡≤0. (3.44) By (3.44), the conditions (A1) and (A4), and the convexity of 𝜑, we have 0=Φ𝑦_𝑡,𝑦_𝑡+𝜑𝑦_𝑡−𝜑𝑦_𝑡≤𝑡Φ𝑦_𝑡,𝑦+(1−𝑡)Φ𝑦_𝑡,𝑤+𝑡𝜑𝑦+(1−𝑡)𝜑(𝑤)−𝜑𝑦_𝑡=𝑡Φ𝑦_𝑡,𝑦+𝜑𝑦−𝜑𝑦_𝑡+(1−𝑡)Φ𝑦_𝑡,𝑤+𝜑(𝑤)−𝜑𝑦_𝑡≤𝑡Φ𝑦_𝑡,𝑦+𝜑𝑦−𝜑𝑦_𝑡; (3.45) it follows that Φ𝑦_𝑡,𝑦+𝜑𝑦−𝜑𝑦_𝑡≥0. (3.46) 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(𝑥₁). Next, we prove that 𝑥_𝑛→𝑤 as 𝑛→∞. Since 𝑤=𝑃_Ω(𝑥₁), we have ‖𝑥₁−𝑤‖≤‖𝑥₁−𝑧‖ for all 𝑧∈Ω; it follows that ‖𝑥₁−𝑤‖≤‖𝑥₁−𝑤‖. (3.47) Since 𝑤∈Ω⊂𝐶_𝑛∩𝑄_𝑛, therefore, by (3.13), we have 𝑥₁−𝑥_𝑛≤𝑥₁−𝑤. (3.48) Since ‖𝑥_𝑛𝑖−𝑢_𝑛𝑖‖→0 by (3.39) and 𝑢_𝑛𝑖⇀𝑤, we have 𝑥_𝑛𝑖⇀𝑤 as 𝑖→∞. Therefore, by (3.47), (3.48) and the weak lower semicontinuity of norm, we have 𝑥₁−𝑤≤𝑥₁−𝑤≤liminf_𝑖→∞𝑥₁−𝑥_𝑛𝑖≤limsup_𝑖→∞𝑥₁−𝑥_𝑛𝑖≤𝑥₁−𝑤. (3.49) It follows that 𝑥₁−𝑤=lim_𝑖→∞𝑥₁−𝑥_𝑛𝑖=𝑥₁−𝑤. (3.50) Since 𝑥_𝑛𝑖⇀𝑤 as 𝑖→∞, therefore, we have 𝑥₁−𝑥_𝑛𝑖⇀𝑥₁−𝑤as𝑖⟶∞. (3.51) Hence, from (3.50), (3.51), Kadec-Klee property, and the uniqueness of 𝑤=𝑃_Ω(𝑥₁), we obtain 𝑥_𝑛𝑖⟶𝑤=𝑤as𝑖⟶∞. (3.52) It follows that {𝑥_𝑛} converges strongly to 𝑤 and so are {𝑦_𝑛},{𝑧_𝑛}, and {𝑢_𝑛}. This completes the proof.

Remark 3.2. The iteration (3.1) is the difference with the iterative scheme of Qin et al. [7] as follows.(1)The sequence {𝑥_𝑛} is a projection sequence of 𝑥₁ onto 𝐶_𝑛∩𝑄_𝑛 for all 𝑛∈ℕ such that 𝐶₁∩𝑄₁⊃𝐶₂∩𝑄₂⊃⋯⊃𝐶_𝑛∩𝑄_𝑛⊃⋯⊃Ω. (3.53)(2) A solving of a common element of the set of fixed point for an asymptotically pseudocontractive mapping in the intermediate sense and the set of solutions of the mixed equilibrium problems by iteration is obtained.
We define the condition (B3) as the condition (B1) such that 𝜑=0. If 𝜑=0, then Theorem 3.1 is reduced immediately to the following result.

Corollary 3.3. Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻, and let Φ a bifunction from 𝐶×𝐶 into ℝ satisfying the conditions (A1)–(A5) with either (B2) or (B3) holds. Let 𝑇∶𝐶→𝐶 be a uniformly 𝐿-Lipschitzian such that 𝐿>0 and asymptotically pseudocontractive mapping in the intermediate sense defined as in (1.13) with the sequences {𝑘_𝑛}⊂[1,∞) and {𝜏_𝑛}⊂[0,∞) such that lim_𝑛→∞𝑘_𝑛=1 and lim_𝑛→∞𝜏_𝑛=0. Let 𝑞_𝑛=2𝑘_𝑛−1 for all 𝑛∈ℕ. Assume that Ω∶=𝐹(𝑇)∩EP(Φ) be a nonempty bounded subset of 𝐶. For 𝑥₁=𝑥∈𝐶 chosen arbitrarily, suppose that {𝑥_𝑛},{𝑦_𝑛},{𝑧_𝑛}, and {𝑢_𝑛} are generated iteratively by 𝑢_𝑛∈𝐶suchthatΦ𝑢_𝑛+1,𝑦𝑟_𝑛𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛𝑧≥0,∀𝑦∈𝐶,_𝑛=1−𝛽_𝑛𝑢_𝑛+𝛽_𝑛𝑇^𝑛𝑢_𝑛,𝑦_𝑛=1−𝛼_𝑛𝑢_𝑛+𝛼_𝑛𝑇^𝑛𝑧_𝑛,𝐶_𝑛+1=𝑧∈𝐶_𝑛∩𝑄_𝑛∶𝑦_𝑛−𝑧²≤𝑥_𝑛−𝑧²+𝛼_𝑛𝜃_𝑛+𝛼_𝑛𝛽_𝑛𝛽_𝑛𝑞_𝑛+𝛽²_𝑛𝐿²+𝛽_𝑛−1𝑇^𝑛𝑢_𝑛−𝑢_𝑛²,𝑄_𝑛+1=𝑧∈𝐶_𝑛∩𝑄_𝑛∶𝑥_𝑛−𝑧,𝑥₁−𝑥_𝑛,𝐶≥0₁=𝑄₁𝑥=𝐶,_𝑛+1=𝑃_{𝐶𝑛+1∩𝑄𝑛+1}𝑥₁,∀𝑛∈ℕ, (3.54) where 𝜃_𝑛=(𝑞_𝑛[1+𝛽_𝑛(𝑞_𝑛−1)]−1)⋅Δ²_𝑛+2(𝛽_𝑛𝑞_𝑛+1)𝜏_𝑛 and Δ_𝑛=sup{‖𝑥_𝑛−𝑧‖∶𝑧∈Ω}<∞ satisfying the following conditions: (C1){𝛼_𝑛},{𝛽_𝑛}⊂(0,1) such that 𝑎≤𝛼_𝑛≤𝛽_𝑛≤𝑏 for some 𝑎>0 and 𝑏∈(0,𝐿⁻²(√1+𝐿²−1)); (C2){𝑟_𝑛}⊂[𝑟,∞) for some 𝑟>0; (C3)∑^∞_𝑛=1|𝑟_𝑛+1−𝑟_𝑛|<∞. Then the sequences {𝑥_𝑛},{𝑦_𝑛},{𝑧_𝑛}, and {𝑢_𝑛} converge strongly to 𝑤=𝑃_Ω(𝑥₁).

If Φ=0, then Corollary 3.3 is reduced immediately to the following result.

Corollary 3.4. Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let 𝑇∶𝐶→𝐶 be a uniformly 𝐿-Lipschitzian such that 𝐿>0 and asymptotically pseudocontractive mapping in the intermediate sense defined as in (1.13) with the sequences {𝑘_𝑛}⊂[1,∞) and {𝜏_𝑛}⊂[0,∞) such that lim_𝑛→∞𝑘_𝑛=1 and lim_𝑛→∞𝜏_𝑛=0. Let 𝑞_𝑛=2𝑘_𝑛−1 for all 𝑛∈ℕ. Assume that 𝐹(𝑇) be a nonempty bounded subset of 𝐶. For 𝑥₁=𝑥∈𝐶 chosen arbitrarily, suppose that {𝑥_𝑛},{𝑦_𝑛}, and {𝑧_𝑛} are generated iteratively by 𝑧_𝑛=1−𝛽_𝑛𝑥_𝑛+𝛽_𝑛𝑇^𝑛𝑥_𝑛,𝑦_𝑛=1−𝛼_𝑛𝑥_𝑛+𝛼_𝑛𝑇^𝑛𝑧_𝑛,𝐶_𝑛+1=𝑧∈𝐶_𝑛∩𝑄_𝑛∶𝑦_𝑛−𝑧²≤𝑥_𝑛−𝑧²+𝛼_𝑛𝜃_𝑛+𝛼_𝑛𝛽_𝑛𝛽_𝑛𝑞_𝑛+𝛽²_𝑛𝐿²+𝛽_𝑛−1𝑇^𝑛𝑥_𝑛−𝑥_𝑛²,𝑄_𝑛+1=𝑧∈𝐶_𝑛∩𝑄_𝑛∶𝑥_𝑛−𝑧,𝑥₁−𝑥_𝑛,𝐶≥0₁=𝑄₁𝑥=𝐶,_𝑛+1=𝑃_{𝐶𝑛+1∩𝑄𝑛+1}𝑥₁,∀𝑛∈ℕ, (3.55) where 𝜃_𝑛=(𝑞_𝑛[1+𝛽_𝑛(𝑞_𝑛−1)]−1)⋅Δ²_𝑛+2(𝛽_𝑛𝑞_𝑛+1)𝜏_𝑛, Δ_𝑛=sup{‖𝑥_𝑛−𝑧‖∶𝑧∈F(𝑇)}<∞ and {𝛼_𝑛},{𝛽_𝑛}⊂(0,1) such that 𝑎≤𝛼_𝑛≤𝛽_𝑛≤𝑏 for some 𝑎>0 and 𝑏∈(0,𝐿⁻²(√1+𝐿²−1)). Then the sequences {𝑥_𝑛},{𝑦_𝑛} and {𝑧_𝑛} converge strongly to 𝑤=𝑃_𝐹(𝑇)(𝑥₁).

We introduce the equilibrium problem to the optimization problem: min_𝑥∈𝐶𝜁(𝑥), (3.56) where 𝐶 is a nonempty closed convex subset of a real Hilbert space 𝐻 and let 𝜁∶𝐶→ℝ∪{+∞} is a proper convex and lower semicontinuous. We denote by Argmin(𝜁) the set of solution of problem (3.56). 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 obtain that Corollary 3.3 is reduced immediately to the following result.

Corollary 3.5. Let 𝐶 is a nonempty closed convex subset of a real Hilbert space 𝐻, and 𝜁∶𝐶→ℝ∪{+∞} be a proper lower semicontinuous and convex function with either (B2) or (B4) holds. Let 𝑇∶𝐶→𝐶 be a uniformly 𝐿-Lipschitzian such that 𝐿>0 and asymptotically pseudocontractive mapping in the intermediate sense defined as in (1.13) with the sequences {𝑘_𝑛}⊂[1,∞) and {𝜏_𝑛}⊂[0,∞) such that lim_𝑛→∞𝑘_𝑛=1 and lim_𝑛→∞𝜏_𝑛=0. Let 𝑞_𝑛=2𝑘_𝑛−1 for all 𝑛∈ℕ. Assume that Ω∶=𝐹(𝑇)∩Argmin(𝜁) be a nonempty bounded subset of 𝐶. For 𝑥₁=𝑥∈𝐶 chosen arbitrarily, suppose that {𝑥_𝑛},{𝑦_𝑛},{𝑧_𝑛} and {𝑢_𝑛} are generated iteratively by 𝑢_𝑛∈𝐶suchthat𝜁𝑦−𝜁𝑢_𝑛+1𝑟_𝑛𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛𝑧≥0,∀𝑦∈𝐶,_𝑛=1−𝛽_𝑛𝑢_𝑛+𝛽_𝑛𝑇^𝑛𝑢_𝑛,𝑦_𝑛=1−𝛼_𝑛𝑢_𝑛+𝛼_𝑛𝑇^𝑛𝑧_𝑛,𝐶_𝑛+1=𝑧∈𝐶_𝑛∩𝑄_𝑛∶𝑦_𝑛−𝑧²≤𝑥_𝑛−𝑧²+𝛼_𝑛𝜃_𝑛+𝛼_𝑛𝛽_𝑛𝛽_𝑛𝑞_𝑛+𝛽²_𝑛𝐿²+𝛽_𝑛−1𝑇^𝑛𝑢_𝑛−𝑢_𝑛²,𝑄_𝑛+1=𝑧∈𝐶_𝑛∩𝑄_𝑛∶𝑥_𝑛−𝑧,𝑥₁−𝑥_𝑛,𝐶≥0₁=𝑄₁𝑥=𝐶,_𝑛+1=𝑃_{𝐶𝑛+1∩𝑄𝑛+1}𝑥₁,∀𝑛∈ℕ, (3.57) where 𝜃_𝑛=(𝑞_𝑛[1+𝛽_𝑛(𝑞_𝑛−1)]−1)⋅Δ²_𝑛+2(𝛽_𝑛𝑞_𝑛+1)𝜏_𝑛 and Δ_𝑛=sup{‖𝑥_𝑛−𝑧‖∶𝑧∈Ω}<∞ satisfying the following conditions: (C1){𝛼_𝑛},{𝛽_𝑛}⊂(0,1) such that 𝑎≤𝛼_𝑛≤𝛽_𝑛≤𝑏 for some 𝑎>0 and 𝑏∈(0,𝐿⁻²(√1+𝐿²−1));(C2){𝑟_𝑛}⊂[𝑟,∞) for some 𝑟>0; (C3)∑^∞_𝑛=1|𝑟_𝑛+1−𝑟_𝑛|<∞. Then the sequences {𝑥_𝑛},{𝑦_𝑛},{𝑧_𝑛}, and {𝑢_𝑛} converge strongly to 𝑤=𝑃_Ω(𝑥₁).