Journal of Applied Mathematics

Journal of Applied Mathematics / 2012 / Article
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Numerical and Analytical Methods for Variational Inequalities and Related Problems with Applications

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Research Article | Open Access

Volume 2012 |Article ID 174318 | 20 pages | https://doi.org/10.1155/2012/174318

General Iterative Algorithms for Hierarchical Fixed Points Approach to Variational Inequalities

Academic Editor: Zhenyu Huang
Received24 Mar 2012
Accepted16 May 2012
Published12 Jul 2012

Abstract

This paper deals with new methods for approximating a solution to the fixed point problem; find Μƒπ‘₯∈𝐹(𝑇), where 𝐻 is a Hilbert space, 𝐢 is a closed convex subset of 𝐻, 𝑓 is a 𝜌-contraction from 𝐢 into 𝐻, 0<𝜌<1, 𝐴 is a strongly positive linear-bounded operator with coefficient 𝛾>0, 0<𝛾<𝛾/𝜌, 𝑇 is a nonexpansive mapping on 𝐢, and 𝑃𝐹(𝑇) denotes the metric projection on the set of fixed point of 𝑇. Under a suitable different parameter, we obtain strong convergence theorems by using the projection method which solves the variational inequality ⟨(π΄βˆ’π›Ύπ‘“)Μƒπ‘₯+𝜏(πΌβˆ’π‘†)Μƒπ‘₯,π‘₯βˆ’Μƒπ‘₯⟩β‰₯0 for π‘₯∈𝐹(𝑇), where 𝜏∈[0,∞). Our results generalize and improve the corresponding results of Yao et al. (2010) and some authors. Furthermore, we give an example which supports our main theorem in the last part.

1. Introduction

Throughout this paper, we assume that 𝐻 is a real Hilbert space where inner product and norm are denoted by βŸ¨β‹…,β‹…βŸ© and β€–β‹…β€–, respectively, and let 𝐢 be a nonempty closed convex subset of 𝐻. A mapping π‘‡βˆΆπΆβ†’πΆ is called nonexpansive if ‖𝑇π‘₯βˆ’π‘‡π‘¦β€–β‰€β€–π‘₯βˆ’π‘¦β€–,βˆ€π‘₯,π‘¦βˆˆπΆ.(1.1)

We use 𝐹(𝑇) to denote the set of fixed points of 𝑇, that is, 𝐹(𝑇)={π‘₯βˆˆπΆβˆΆπ‘‡π‘₯=π‘₯}. It is assumed throughout the paper that 𝑇 is a nonexpansive mapping such that 𝐹(𝑇)β‰ βˆ….

Recall that a mapping π‘“βˆΆπΆβ†’π» is a contraction on 𝐢 if there exists a constant 𝜌∈(0,1) such that ‖𝑓(π‘₯)βˆ’π‘“(𝑦)β€–β‰€πœŒβ€–π‘₯βˆ’π‘¦β€–,βˆ€π‘₯,π‘¦βˆˆπΆ.(1.2)

A mapping π΄βˆΆπ»β†’π» is called a strongly positive linear bounded operator on 𝐻 if there exists a constant 𝛾>0 with property ⟨𝐴π‘₯,π‘₯⟩β‰₯𝛾‖π‘₯β€–2,βˆ€π‘₯∈𝐻.(1.3)

A mapping π‘€βˆΆπ»β†’π» is called a strongly monotone operator with 𝛼 if ⟨π‘₯βˆ’π‘¦,𝑀π‘₯βˆ’π‘€π‘¦βŸ©β‰₯𝛼‖π‘₯βˆ’π‘¦β€–2,βˆ€π‘₯,π‘¦βˆˆπ»,(1.4) and 𝑀 is called a monotone operator if ⟨π‘₯βˆ’π‘¦,𝑀π‘₯βˆ’π‘€π‘¦βŸ©β‰₯0,βˆ€π‘₯,π‘¦βˆˆπ».(1.5) We easily prove that the mapping (πΌβˆ’π‘‡) is monotone operator, if 𝑇 is nonexpansive mapping.

The metric (or nearest point) projection from 𝐻 onto 𝐢 is mapping 𝑃𝐢[β‹…]βˆΆπ»β†’πΆ which assigns to each point π‘₯∈𝐢 the unique point 𝑃𝐢[π‘₯]∈𝐢 satisfying the property β€–β€–π‘₯βˆ’π‘ƒπΆ[π‘₯]β€–β€–=infπ‘¦βˆˆπΆβ€–π‘₯βˆ’π‘¦β€–=βˆΆπ‘‘(π‘₯,𝐢).(1.6)

The variational inequality for a monotone operator, π‘€βˆΆπ»β†’π» over 𝐢, is to find a point in VI(𝐢,𝑀)∢={Μƒπ‘₯∈𝐢∢⟨π‘₯βˆ’Μƒπ‘₯,𝑀̃π‘₯⟩β‰₯0,βˆ€π‘₯∈𝐢}.(1.7)

A hierarchical fixed point problem is equivalent to the variational inequality for a monotone operator over the fixed point set. Moreover, to find a hierarchically fixed point of a nonexpansive mapping 𝑇 with respect to another nonexpansive mapping 𝑆, namely, we find Μƒπ‘₯∈𝐹(𝑇) such that ⟨π‘₯βˆ’Μƒπ‘₯,(πΌβˆ’π‘†)Μƒπ‘₯⟩β‰₯0,βˆ€π‘₯∈𝐹(𝑇).(1.8)

Iterative methods for nonexpansive mappings have recently been applied to solve a convex minimization problem; see, for example, [1–5] and the references therein. A typical problem is to minimize a quadratic function over the set of the fixed points of a nonexpansive mapping on a real Hilbert space 𝐻: minπ‘₯∈𝐹(𝑇)12⟨𝐴π‘₯,π‘₯βŸ©βˆ’βŸ¨π‘₯,π‘βŸ©,(1.9) where 𝑏 is a given point in 𝐻. In [5], it is proved that the sequence {π‘₯𝑛} defined by the iterative method below, with the initial guess π‘₯0∈𝐻 chosen arbitrarily, π‘₯𝑛+1=ξ€·πΌβˆ’π›Όπ‘›π΄ξ€Έπ‘‡π‘₯𝑛+𝛼𝑛𝑏,𝑛β‰₯0,(1.10) converges strongly to the unique solution of the minimization problem (1.9) provided the sequence {𝛼𝑛} of parameters satisfies certain appropriate conditions.

On the other hand, Moudafi [6] introduced the viscosity approximation method for nonexpansive mappings (see [7] for further developments in both Hilbert and Banach spaces). Starting with an arbitrary initial π‘₯0∈𝐻, define a sequence {π‘₯𝑛} recursively by π‘₯𝑛+1=πœŽπ‘›π‘“ξ€·π‘₯𝑛+ξ€·1βˆ’πœŽπ‘›ξ€Έπ‘‡π‘₯𝑛,𝑛β‰₯0,(1.11) where {πœŽπ‘›} is a sequence in (0,1). It is proved in [6, 7] that under certain appropriate conditions imposed on {πœŽπ‘›}, the sequence {π‘₯𝑛} generated by (1.11) strongly converges to the unique solution π‘₯βˆ— in 𝐢 of the variational inequality ⟨(πΌβˆ’π‘“)π‘₯βˆ—,π‘₯βˆ’π‘₯βˆ—βŸ©β‰₯0,π‘₯∈𝐢.(1.12)

In 2006, Marino and Xu [8] introduced a general iterative method for nonexpansive mapping. Starting with an arbitrary initial π‘₯0∈𝐻, define a sequence {π‘₯𝑛} recursively by π‘₯𝑛+1=πœ–π‘›ξ€·π‘₯𝛾𝑓𝑛+ξ€·πΌβˆ’πœ–π‘›π΄ξ€Έπ‘‡π‘₯𝑛,𝑛β‰₯0.(1.13) They proved that if the sequence {πœ–π‘›} of parameters satisfies appropriate conditions, then the sequence {π‘₯𝑛} generated by (1.13) strongly converges to the unique solution Μƒπ‘₯=𝑃𝐹(𝑇)(πΌβˆ’π΄+𝛾𝑓)Μƒπ‘₯ of the variational inequality ⟨(π΄βˆ’π›Ύπ‘“)Μƒπ‘₯,π‘₯βˆ’Μƒπ‘₯⟩β‰₯0,βˆ€π‘₯∈𝐹(𝑇),(1.14) which is the optimality condition for the minimization problem minπ‘₯∈𝐹(𝑇)12⟨𝐴π‘₯,π‘₯βŸ©βˆ’β„Ž(π‘₯),(1.15) where β„Ž is a potential function for 𝛾𝑓 (i.e., β„Žβ€²(π‘₯)=𝛾𝑓(π‘₯) for π‘₯∈𝐻).

In 2010, Yao et al. [9] introduced an iterative algorithm for solving some hierarchical fixed point problem, starting with an arbitrary initial guess π‘₯0∈𝐢, define a sequence {π‘₯𝑛} iteratively by 𝑦𝑛=𝛽𝑛𝑆π‘₯𝑛+ξ€·1βˆ’π›½π‘›ξ€Έπ‘₯𝑛,π‘₯𝑛+1=𝑃𝐢𝛼𝑛𝑓π‘₯𝑛+ξ€·1βˆ’π›Όπ‘›ξ€Έπ‘‡π‘¦π‘›ξ€»,βˆ€π‘›β‰₯1.(1.16) They proved that if the sequences {𝛼𝑛} and {𝛽𝑛} of parameters satisfies appropriate conditions, then the sequence {π‘₯𝑛} generated by (1.16) strongly converges to the unique solution 𝑧 in 𝐻 of the variational inequality π‘§βˆˆπΉ(𝑇),⟨(πΌβˆ’π‘“)𝑧,π‘₯βˆ’π‘§βŸ©β‰₯0,βˆ€π‘₯∈𝐹(𝑇).(1.17)

In this paper we will combine the general iterative method (1.13) with the iterative algorithm (1.16) and consider the following iterative algorithm: 𝑦𝑛=𝛽𝑛𝑆π‘₯𝑛+ξ€·1βˆ’π›½π‘›ξ€Έπ‘₯𝑛,π‘₯𝑛+1=𝑃𝐢𝛼𝑛π‘₯𝛾𝑓𝑛+ξ€·πΌβˆ’π›Όπ‘›π΄ξ€Έπ‘‡π‘¦π‘›ξ€»,βˆ€π‘›β‰₯1.(1.18) We will prove in Section 3 that if the sequences {𝛼𝑛} and {𝛽𝑛} of parameters satisfy appropriate conditions and limπ‘›β†’βˆž(𝛽𝑛/𝛼𝑛)=𝜏∈(0,∞) then the sequence {π‘₯𝑛} generated by (1.18) converges strongly to the unique solution Μƒπ‘₯ in 𝐻 of the following variational inequality 1Μƒπ‘₯∈𝐹(𝑇),πœξ‚­(π΄βˆ’π›Ύπ‘“)Μƒπ‘₯+(πΌβˆ’π‘†)Μƒπ‘₯,π‘₯βˆ’Μƒπ‘₯β‰₯0,βˆ€π‘₯∈𝐹(𝑇).(1.19) In particular, if we take 𝜏=0, under suitable difference assumptions on parameter, then the sequence {π‘₯𝑛} generated by (1.18) converges strongly to the unique solution Μƒπ‘₯ in 𝐻 of the following variational inequality Μƒπ‘₯∈𝐹(𝑇),⟨(π΄βˆ’π›Ύπ‘“)Μƒπ‘₯,π‘₯βˆ’Μƒπ‘₯⟩β‰₯0,βˆ€π‘₯∈𝐹(𝑇).(1.20) Our results improve and extend the recent results of Yao et al. [9] and some authors. Furthermore, we give an example which supports our main theorem in the last part.

2. Preliminaries

This section collects some lemma which can be used in the proofs for the main results in the next section. Some of them are known, others are not hard to derive.

Lemma 2.1 (Browder [10]). Let 𝐻 be a Hilbert space, 𝐢 be a closed convex subset of 𝐻, and π‘‡βˆΆπΆβ†’πΆ be a nonexpansive mapping with 𝐹(𝑇)β‰ βˆ…. If {π‘₯𝑛} is a sequence in 𝐢 weakly converging to π‘₯ and if {(πΌβˆ’π‘‡)π‘₯𝑛} converges strongly to 𝑦, then (πΌβˆ’π‘‡)π‘₯=𝑦; in particular, if 𝑦=0 then π‘₯∈𝐹(𝑇).

Lemma 2.2. Let π‘₯∈𝐻 and π‘§βˆˆπΆ be any points. Then one has the following:
(1) That 𝑧=𝑃𝐢[π‘₯] if and only if there holds the relation: ⟨π‘₯βˆ’π‘§,π‘¦βˆ’π‘§βŸ©β‰€0,βˆ€π‘¦βˆˆπΆ.(2.1)(2) That 𝑧=𝑃𝐢[π‘₯] if and only if there holds the relation: β€–π‘₯βˆ’π‘§β€–2≀‖π‘₯βˆ’π‘¦β€–2βˆ’β€–π‘¦βˆ’π‘§β€–2,βˆ€π‘¦βˆˆπΆ.(2.2)(3) There holds the relation: βŸ¨π‘ƒπΆ[π‘₯]βˆ’π‘ƒπΆ[𝑦]‖‖𝑃,π‘₯βˆ’π‘¦βŸ©β‰₯𝐢[π‘₯]βˆ’π‘ƒπΆ[𝑦]β€–β€–2,βˆ€π‘₯,π‘¦βˆˆπ».(2.3) Consequently, 𝑃𝐢 is nonexpansive and monotone.

Lemma 2.3 (Marino and Xu [8]). Let 𝐻 be a Hilbert space, 𝐢 be a closed convex subset of 𝐻, π‘“βˆΆπΆβ†’π» be a contraction with coefficient 0<𝜌<1, and π‘‡βˆΆπΆβ†’πΆ be nonexpansive mapping. Let 𝐴 be a strongly positive linear bounded operator on a Hilbert space 𝐻 with coefficient βˆ’π›Ύ>0. Then, for 0<𝛾<βˆ’π›Ύ/𝜌, for π‘₯,π‘¦βˆˆπΆ, (1)the mapping (πΌβˆ’π‘“) is strongly monotone with coefficient (1βˆ’πœŒ), that is, ⟨π‘₯βˆ’π‘¦,(πΌβˆ’π‘“)π‘₯βˆ’(πΌβˆ’π‘“)π‘¦βŸ©β‰₯(1βˆ’πœŒ)β€–π‘₯βˆ’π‘¦β€–2,(2.4)(2)the mapping (π΄βˆ’π›Ύπ‘“) is strongly monotone with coefficient βˆ’π›Ύβˆ’π›ΎπœŒ that is ξ‚€βŸ¨π‘₯βˆ’π‘¦,(π΄βˆ’π›Ύπ‘“)π‘₯βˆ’(π΄βˆ’π›Ύπ‘“)π‘¦βŸ©β‰₯βˆ’ξ‚β€–π›Ύβˆ’π›ΎπœŒπ‘₯βˆ’π‘¦β€–2.(2.5)

Lemma 2.4 (Xu [4]). Assume that {π‘Žπ‘›} is a sequence of nonnegative numbers such that π‘Žπ‘›+1≀1βˆ’π›Ύπ‘›ξ€Έπ‘Žπ‘›+𝛿𝑛,βˆ€π‘›β‰₯0,(2.6) where {𝛾𝑛} is a sequence in (0,1) and {𝛿𝑛} is a sequence in ℝ such that (1)βˆ‘βˆžπ‘›=1𝛾𝑛=∞, (2)limsupπ‘›β†’βˆž(𝛿𝑛/𝛾𝑛)≀0 or βˆ‘βˆžπ‘›=1|𝛿𝑛|<∞. Then limπ‘›β†’βˆžπ‘Žπ‘›=0.

Lemma 2.5 (Marino and Xu [8]). Assume 𝐴 is a strongly positive linear bounded operator on a Hilbert space 𝐻 with coefficient βˆ’π›Ύ>0 and 0<π›Όβ‰€β€–π΄β€–βˆ’1. Then β€–πΌβˆ’π›Όπ΄β€–β‰€1βˆ’π›Όβˆ’π›Ύ.

Lemma 2.6 (Acedo and Xu [11]). Let 𝐢 be a closed convex subset of 𝐻. Let {π‘₯𝑛} be a bounded sequence in 𝐻. Assume that (1)The weak πœ”-limit set πœ”π‘€(π‘₯𝑛)βŠ‚πΆ,(2)For each π‘§βˆˆπΆ, limπ‘›β†’βˆžβ€–π‘₯π‘›βˆ’π‘§β€– exists. Then {π‘₯𝑛} is weakly convergent to a point in 𝐢.

NotationWe use β†’ for strong convergence and ⇀ for weak convergence.

3. Main Results

Theorem 3.1. Let 𝐢 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let π‘“βˆΆπΆβ†’π» be a 𝜌-contraction with 𝜌∈(0,1). Let 𝑆,π‘‡βˆΆπΆβ†’πΆ be two nonexpansive mappings with 𝐹(𝑇)β‰ βˆ…. Let 𝐴 be a strongly positive linear bounded operator on 𝐻 with coefficient βˆ’π›Ύ>0. {𝛼𝑛} and {𝛽𝑛} are two sequences in (0,1) and 0<𝛾<βˆ’π›Ύ/𝜌. Starting with an arbitrary initial guess π‘₯0∈𝐢 and {π‘₯𝑛} is a sequence generated by 𝑦𝑛=𝛽𝑛𝑆π‘₯𝑛+ξ€·1βˆ’π›½π‘›ξ€Έπ‘₯𝑛,π‘₯𝑛+1=𝑃𝐢𝛼𝑛π‘₯𝛾𝑓𝑛+ξ€·πΌβˆ’π›Όπ‘›π΄ξ€Έπ‘‡π‘¦π‘›ξ€»,βˆ€π‘›β‰₯1.(3.1)
Suppose that the following conditions are satisfied: (C1)  limπ‘›β†’βˆžπ›Όπ‘›=0 and βˆ‘βˆžπ‘›=1𝛼𝑛=∞, (C2)  limπ‘›β†’βˆž(𝛽𝑛/𝛼𝑛)=𝜏=0, (C3)  limπ‘›β†’βˆž(|π›Όπ‘›βˆ’π›Όπ‘›βˆ’1|/𝛼𝑛)=0 and limπ‘›β†’βˆž(|π›½π‘›βˆ’π›½π‘›βˆ’1|/𝛽𝑛)=0, or (C4)β€‰β€‰βˆ‘βˆžπ‘›=1|π›Όπ‘›βˆ’π›Όπ‘›βˆ’1|<∞ and βˆ‘βˆžπ‘›=1|π›½π‘›βˆ’π›½π‘›βˆ’1|<∞. Then the sequence {π‘₯𝑛} converges strongly to a point Μƒπ‘₯∈𝐻, which is the unique solution of the variational inequality: Μƒπ‘₯∈𝐹(𝑇),⟨(π΄βˆ’π›Ύπ‘“)Μƒπ‘₯,π‘₯βˆ’Μƒπ‘₯⟩β‰₯0,βˆ€π‘₯∈𝐹(𝑇).(3.2) Equivalently, one has 𝑃𝐹(𝑇)(πΌβˆ’π΄+𝛾𝑓)Μƒπ‘₯=Μƒπ‘₯.

Proof. We first show the uniqueness of a solution of the variational inequality (3.2), which is indeed a consequence of the strong monotonicity of π΄βˆ’π›Ύπ‘“. Suppose βˆ’π‘₯∈𝐹(𝑇) and Μƒπ‘₯∈𝐹(𝑇) both are solutions to (3.2), then ⟨(π΄βˆ’π›Ύπ‘“)βˆ’π‘₯,βˆ’π‘₯βˆ’Μƒπ‘₯βŸ©β‰€0 and ⟨(π΄βˆ’π›Ύπ‘“)Μƒπ‘₯,Μƒπ‘₯βˆ’βˆ’π‘₯βŸ©β‰€0. It follows that (π΄βˆ’π›Ύπ‘“)βˆ’π‘₯,βˆ’π‘₯ξ‚­+ξ‚¬βˆ’Μƒπ‘₯(π΄βˆ’π›Ύπ‘“)Μƒπ‘₯,Μƒπ‘₯βˆ’βˆ’π‘₯ξ‚­=(π΄βˆ’π›Ύπ‘“)βˆ’π‘₯,βˆ’π‘₯ξ‚­βˆ’ξ‚¬βˆ’Μƒπ‘₯(π΄βˆ’π›Ύπ‘“)Μƒπ‘₯,βˆ’π‘₯ξ‚­βˆ’Μƒπ‘₯=⟨(π΄βˆ’π›Ύπ‘“)βˆ’π‘₯βˆ’(π΄βˆ’π›Ύπ‘“)Μƒπ‘₯,βˆ’π‘₯βˆ’Μƒπ‘₯βŸ©β‰€0.(3.3) The strong monotonicity of π΄βˆ’π›Ύπ‘“ (Lemma 2.3) implies that βˆ’π‘₯=Μƒπ‘₯ and the uniqueness is proved.
Next, we prove that the sequence {π‘₯𝑛} is bounded. Since 𝛼𝑛→0 and limπ‘›β†’βˆž(𝛽𝑛/𝛼𝑛)=0 by condition (C1) and (C2), respectively, we can assume, without loss of generality, that 𝛼𝑛<β€–π΄β€–βˆ’1 and 𝛽𝑛<𝛼𝑛 for all 𝑛β‰₯1. Take π‘’βˆˆπΉ(𝑇) and from (3.1), we have β€–β€–π‘₯𝑛+1β€–β€–=β€–β€–π‘ƒβˆ’π‘’πΆξ€Ίπ›Όπ‘›ξ€·π‘₯𝛾𝑓𝑛+ξ€·πΌβˆ’π›Όπ‘›π΄ξ€Έπ‘‡π‘¦π‘›ξ€»βˆ’π‘ƒπΆ[𝑒]‖‖≀‖‖𝛼𝑛π‘₯𝛾𝑓𝑛+ξ€·πΌβˆ’π›Όπ‘›π΄ξ€Έπ‘‡π‘¦π‘›β€–β€–βˆ’π‘’β‰€π›Όπ‘›π›Ύβ€–β€–π‘“ξ€·π‘₯π‘›ξ€Έβ€–β€–βˆ’π‘“(𝑒)+𝛼𝑛‖‖‖𝛾𝑓(𝑒)βˆ’π΄π‘’β€–+πΌβˆ’π›Όπ‘›π΄ξ€Έξ€·π‘‡π‘¦π‘›ξ€Έβ€–β€–.βˆ’π‘’(3.4) Since β€–πΌβˆ’π›Όπ‘›π΄β€–β‰€1βˆ’π›Όπ‘›βˆ’π›Ύ and by Lemma 2.5, we note that β€–β€–π‘₯𝑛+1β€–β€–βˆ’π‘’β‰€π›Όπ‘›π›Ύβ€–β€–π‘“ξ€·π‘₯π‘›ξ€Έβ€–β€–βˆ’π‘“(𝑒)+𝛼𝑛‖𝛾𝑓(𝑒)βˆ’π΄π‘’β€–+1βˆ’π›Όπ‘›βˆ’π›Ύξ‚β€–β€–π‘‡π‘¦π‘›β€–β€–βˆ’π‘’β‰€π›Όπ‘›β€–β€–π‘₯π›ΎπœŒπ‘›β€–β€–βˆ’π‘’+𝛼𝑛‖𝛾𝑓(𝑒)βˆ’π΄π‘’β€–+1βˆ’π›Όπ‘›βˆ’π›Ύξ‚β€–β€–π‘‡π‘¦π‘›β€–β€–βˆ’π‘‡π‘’β‰€π›Όπ‘›β€–β€–π‘₯π›ΎπœŒπ‘›β€–β€–βˆ’π‘’+𝛼𝑛‖𝛾𝑓(𝑒)βˆ’π΄π‘’β€–+1βˆ’π›Όπ‘›βˆ’π›Ύξ‚β€–β€–π‘¦π‘›β€–β€–βˆ’π‘’β‰€π›Όπ‘›β€–β€–π‘₯π›ΎπœŒπ‘›β€–β€–βˆ’π‘’+𝛼𝑛+‖𝛾𝑓(𝑒)βˆ’π΄π‘’β€–1βˆ’π›Όπ‘›βˆ’π›Ύξ‚ξ€Ίπ›½π‘›β€–β€–π‘†π‘₯π‘›β€–β€–βˆ’π‘†π‘’+π›½π‘›ξ€·β€–π‘†π‘’βˆ’π‘’β€–+1βˆ’π›½π‘›ξ€Έβ€–β€–π‘₯π‘›β€–β€–ξ€»βˆ’π‘’β‰€π›Όπ‘›β€–β€–π‘₯π›ΎπœŒπ‘›β€–β€–βˆ’π‘’+𝛼𝑛‖+‖𝛾𝑓(𝑒)βˆ’π΄π‘’1βˆ’π›Όπ‘›βˆ’π›Ύξ‚ξ€Ίπ›½π‘›β€–β€–π‘₯π‘›β€–β€–βˆ’π‘’+π›½π‘›ξ€·β€–π‘†π‘’βˆ’π‘’β€–+1βˆ’π›½π‘›ξ€Έβ€–β€–π‘₯𝑛‖‖=ξ‚€βˆ’π‘’1βˆ’π›Όπ‘›ξ‚€βˆ’β€–β€–π‘₯π›Ύβˆ’π›ΎπœŒξ‚ξ‚π‘›β€–β€–βˆ’π‘’+𝛼𝑛‖𝛾𝑓(𝑒)βˆ’π΄π‘’β€–+1βˆ’π›Όπ‘›βˆ’π›Ύξ‚π›½π‘›β‰€ξ‚€β€–π‘†π‘’βˆ’π‘’β€–1βˆ’π›Όπ‘›ξ‚€βˆ’β€–β€–π‘₯π›Ύβˆ’π›ΎπœŒξ‚ξ‚π‘›β€–β€–βˆ’π‘’+𝛼𝑛‖𝛾𝑓(𝑒)βˆ’π΄π‘’β€–+π›½π‘›β‰€ξ‚€β€–π‘†π‘’βˆ’π‘’β€–1βˆ’π›Όπ‘›ξ‚€βˆ’β€–β€–π‘₯π›Ύβˆ’π›ΎπœŒξ‚ξ‚π‘›β€–β€–βˆ’π‘’+𝛼𝑛‖𝛾𝑓(𝑒)βˆ’π΄π‘’β€–+𝛼𝑛=ξ‚€β€–π‘†π‘’βˆ’π‘’β€–1βˆ’π›Όπ‘›ξ‚€βˆ’β€–β€–π‘₯π›Ύβˆ’π›ΎπœŒξ‚ξ‚π‘›β€–β€–βˆ’π‘’+𝛼𝑛[(]=‖𝛾𝑓𝑒)βˆ’π΄π‘’β€–+β€–π‘†π‘’βˆ’π‘’β€–1βˆ’π›Όπ‘›ξ‚€βˆ’β€–β€–π‘₯π›Ύβˆ’π›ΎπœŒξ‚ξ‚π‘›β€–β€–βˆ’π‘’+π›Όπ‘›ξ‚€βˆ’ξ‚π›Ύβˆ’π›ΎπœŒβ€–π›Ύπ‘“(𝑒)βˆ’π΄π‘’β€–+β€–π‘†π‘’βˆ’π‘’β€–ξ‚€βˆ’ξ‚.π›Ύβˆ’π›ΎπœŒ(3.5) By induction, we can obtain β€–β€–π‘₯𝑛+1β€–β€–βŽ§βŽͺ⎨βŽͺβŽ©β€–β€–π‘₯βˆ’π‘’β‰€max0β€–β€–,βˆ’π‘’β€–π›Ύπ‘“(𝑒)βˆ’π΄π‘’β€–+β€–π‘†π‘’βˆ’π‘’β€–ξ‚€βˆ’ξ‚βŽ«βŽͺ⎬βŽͺ⎭,π›Ύβˆ’π›ΎπœŒ(3.6) which implies that the sequence {π‘₯𝑛} is bounded and so are the sequences {𝑓(π‘₯𝑛)}, {𝑆π‘₯𝑛}, and {𝐴𝑇𝑦𝑛}.
Set π‘€π‘›βˆΆ=𝛼𝑛𝛾𝑓(π‘₯𝑛)+(πΌβˆ’π›Όπ‘›π΄)𝑇𝑦𝑛,𝑛β‰₯1. We get β€–β€–π‘₯𝑛+1βˆ’π‘₯𝑛‖‖=‖‖𝑃𝐢𝑀𝑛+1ξ€»βˆ’π‘ƒπΆξ€Ίπ‘€π‘›ξ€»β€–β€–β‰€β€–β€–π‘€π‘›+1βˆ’π‘€π‘›β€–β€–.(3.7) It follows that β€–β€–π‘₯𝑛+1βˆ’π‘₯𝑛‖‖≀‖‖𝛼𝑛π‘₯𝛾𝑓𝑛+ξ€·πΌβˆ’π›Όπ‘›π΄ξ€Έπ‘‡π‘¦π‘›ξ€Έβˆ’ξ€·π›Όπ‘›βˆ’1ξ€·π‘₯π›Ύπ‘“π‘›βˆ’1ξ€Έ+ξ€·πΌβˆ’π›Όπ‘›βˆ’1π΄ξ€Έπ‘‡π‘¦π‘›βˆ’1‖‖≀𝛼𝑛𝛾‖‖𝑓π‘₯𝑛π‘₯βˆ’π‘“π‘›βˆ’1ξ€Έβ€–β€–+||π›Όπ‘›βˆ’π›Όπ‘›βˆ’1||β€–β€–ξ€·π‘₯π›Ύπ‘“π‘›βˆ’1ξ€Έβˆ’π΄π‘‡π‘¦π‘›βˆ’1β€–β€–+ξ‚€1βˆ’π›Όπ‘›βˆ’π›Ύξ‚β€–β€–π‘‡π‘¦π‘›βˆ’π‘‡π‘¦π‘›βˆ’1‖‖≀𝛼𝑛‖‖π‘₯π›ΎπœŒπ‘›βˆ’π‘₯π‘›βˆ’1β€–β€–+||π›Όπ‘›βˆ’π›Όπ‘›βˆ’1||β€–β€–ξ€·π‘₯π›Ύπ‘“π‘›βˆ’1ξ€Έβˆ’π΄π‘‡π‘¦π‘›βˆ’1β€–β€–+ξ‚€1βˆ’π›Όπ‘›βˆ’π›Ύξ‚β€–β€–π‘¦π‘›βˆ’π‘¦π‘›βˆ’1β€–β€–.(3.8) By (3.7) and (3.8), we get β€–β€–π‘₯𝑛+1βˆ’π‘₯π‘›β€–β€–β‰€π›Όπ‘›β€–β€–π‘€π›ΎπœŒπ‘›βˆ’π‘€π‘›βˆ’1β€–β€–+||π›Όπ‘›βˆ’π›Όπ‘›βˆ’1||β€–β€–ξ€·π‘₯π›Ύπ‘“π‘›βˆ’1ξ€Έβˆ’π΄π‘‡π‘¦π‘›βˆ’1β€–β€–+ξ‚€1βˆ’π›Όπ‘›βˆ’π›Ύξ‚β€–β€–π‘¦π‘›βˆ’π‘¦π‘›βˆ’1β€–β€–.(3.9) From (3.1), we obtain β€–β€–π‘¦π‘›βˆ’π‘¦π‘›βˆ’1β€–β€–=‖‖𝛽𝑛𝑆π‘₯𝑛+ξ€·1βˆ’π›½π‘›ξ€Έπ‘₯π‘›ξ€Έβˆ’ξ€·π›½π‘›βˆ’1𝑆π‘₯π‘›βˆ’1+ξ€·1βˆ’π›½π‘›βˆ’1ξ€Έπ‘₯π‘›βˆ’1ξ€Έβ€–β€–=‖‖𝛽𝑛𝑆π‘₯π‘›βˆ’π‘†π‘₯π‘›βˆ’1ξ€Έ+ξ€·π›½π‘›βˆ’π›½π‘›βˆ’1𝑆π‘₯π‘›βˆ’1βˆ’π‘₯π‘›βˆ’1ξ€Έ+ξ€·1βˆ’π›½π‘›π‘₯ξ€Έξ€·π‘›βˆ’π‘₯π‘›βˆ’1‖‖≀‖‖π‘₯π‘›βˆ’π‘₯π‘›βˆ’1β€–β€–+||π›½π‘›βˆ’π›½π‘›βˆ’1||‖‖𝑆π‘₯π‘›βˆ’1βˆ’π‘₯π‘›βˆ’1‖‖≀‖‖π‘₯π‘›βˆ’π‘₯π‘›βˆ’1β€–β€–+||π›½π‘›βˆ’π›½π‘›βˆ’1||𝑀,(3.10) where 𝑀 is a constant such that supπ‘›βˆˆβ„•ξ€½β€–β€–ξ€·π‘₯π›Ύπ‘“π‘›βˆ’1ξ€Έβˆ’π΄π‘‡π‘¦π‘›βˆ’1β€–β€–+‖‖𝑆π‘₯π‘›βˆ’1βˆ’π‘₯π‘›βˆ’1‖‖≀𝑀.(3.11) Substituting (3.10) into (3.8) to obtain β€–β€–π‘₯𝑛+1βˆ’π‘₯𝑛‖‖≀𝛼𝑛‖‖π‘₯π›ΎπœŒπ‘›βˆ’π‘₯π‘›βˆ’1β€–β€–+||π›Όπ‘›βˆ’π›Όπ‘›βˆ’1||β€–β€–ξ€·π‘₯π›Ύπ‘“π‘›βˆ’1ξ€Έβˆ’π΄π‘‡π‘¦π‘›βˆ’1β€–β€–+ξ‚€1βˆ’π›Όπ‘›βˆ’π›Ύξ‚ξ€Ίβ€–β€–π‘₯π‘›βˆ’π‘₯π‘›βˆ’1β€–β€–+||π›½π‘›βˆ’π›½π‘›βˆ’1||𝑀≀𝛼𝑛‖‖π‘₯π›ΎπœŒπ‘›βˆ’π‘₯π‘›βˆ’1β€–β€–+||π›Όπ‘›βˆ’π›Όπ‘›βˆ’1||𝑀+ξ‚€1βˆ’π›Όπ‘›βˆ’π›Ύξ‚ξ€Ίβ€–β€–π‘₯π‘›βˆ’π‘₯π‘›βˆ’1β€–β€–+||π›½π‘›βˆ’π›½π‘›βˆ’1||𝑀=ξ‚€1βˆ’π›Όπ‘›ξ‚€βˆ’β€–β€–π‘₯π›Ύβˆ’π›ΎπœŒξ‚ξ‚π‘›βˆ’π‘₯π‘›βˆ’1β€–β€–ξ€Ί||𝛼+π‘€π‘›βˆ’π›Όπ‘›βˆ’1||+||π›½π‘›βˆ’π›½π‘›βˆ’1||≀1βˆ’π›Όπ‘›ξ‚€βˆ’β€–β€–π‘€π›Ύβˆ’π›ΎπœŒξ‚ξ‚π‘›βˆ’π‘€π‘›βˆ’1β€–β€–ξ€Ί||𝛼+π‘€π‘›βˆ’π›Όπ‘›βˆ’1||+||π›½π‘›βˆ’π›½π‘›βˆ’1||ξ€».(3.12) At the same time, we can write (3.12) as β€–β€–π‘₯𝑛+1βˆ’π‘₯𝑛‖‖≀1βˆ’π›Όπ‘›ξ‚€βˆ’β€–β€–π‘€π›Ύβˆ’π›ΎπœŒξ‚ξ‚π‘›βˆ’π‘€π‘›βˆ’1β€–β€–+𝑀𝛼𝑛||π›Όπ‘›βˆ’π›Όπ‘›βˆ’1||𝛼𝑛+||π›½π‘›βˆ’π›½π‘›βˆ’1||𝛼𝑛≀1βˆ’π›Όπ‘›ξ‚€βˆ’β€–β€–π‘€π›Ύβˆ’π›ΎπœŒξ‚ξ‚π‘›βˆ’π‘€π‘›βˆ’1β€–β€–+𝑀𝛼𝑛||π›Όπ‘›βˆ’π›Όπ‘›βˆ’1||𝛽𝑛+||π›½π‘›βˆ’π›½π‘›βˆ’1||𝛽𝑛.(3.13) From (3.12), (C4), and Lemma 2.5 or from (3.13), (C3), and Lemma 2.5, we can deduce that β€–π‘₯𝑛+1βˆ’π‘₯𝑛‖→0, respectively.
From (3.1), we have β€–β€–π‘₯π‘›βˆ’π‘‡π‘₯𝑛‖‖≀‖‖π‘₯π‘›βˆ’π‘₯𝑛+1β€–β€–+β€–β€–π‘₯𝑛+1βˆ’π‘‡π‘₯𝑛‖‖=β€–β€–π‘₯π‘›βˆ’π‘₯𝑛+1β€–β€–+β€–β€–π‘ƒπΆξ€Ίπ‘€π‘›ξ€»βˆ’π‘ƒπΆξ€Ίπ‘‡π‘₯𝑛‖‖≀‖‖π‘₯π‘›βˆ’π‘₯𝑛+1β€–β€–+β€–β€–π‘€π‘›βˆ’π‘‡π‘₯𝑛‖‖=β€–β€–π‘₯π‘›βˆ’π‘₯𝑛+1β€–β€–+‖‖𝛼𝑛π‘₯𝛾𝑓𝑛+ξ€·πΌβˆ’π›Όπ‘›π΄ξ€Έπ‘‡π‘¦π‘›βˆ’π‘‡π‘₯𝑛‖‖≀‖‖π‘₯π‘›βˆ’π‘₯𝑛+1β€–β€–+𝛼𝑛‖‖π‘₯π›Ύπ‘“π‘›ξ€Έβˆ’π΄π‘‡π‘₯𝑛‖‖+ξ‚€1βˆ’π›Όπ‘›βˆ’π›Ύξ‚β€–β€–π‘‡π‘¦π‘›βˆ’π‘‡π‘₯𝑛‖‖≀‖‖π‘₯π‘›βˆ’π‘₯𝑛+1β€–β€–+𝛼𝑛‖‖π‘₯π›Ύπ‘“π‘›ξ€Έβˆ’π΄π‘‡π‘₯𝑛‖‖+ξ‚€1βˆ’π›Όπ‘›βˆ’π›Ύξ‚β€–β€–π‘¦π‘›βˆ’π‘₯𝑛‖‖=β€–β€–π‘₯π‘›βˆ’π‘₯𝑛+1β€–β€–+𝛼𝑛‖‖π‘₯π›Ύπ‘“π‘›ξ€Έβˆ’π΄π‘‡π‘₯𝑛‖‖+ξ‚€1βˆ’π›Όπ‘›βˆ’π›Ύξ‚π›½π‘›β€–β€–π‘†π‘₯π‘›βˆ’π‘₯𝑛‖‖.(3.14) Notice that 𝛼𝑛→0, 𝛽𝑛→0, and β€–π‘₯𝑛+1βˆ’π‘₯𝑛‖→0, so we obtain β€–β€–π‘₯π‘›βˆ’π‘‡π‘₯π‘›β€–β€–βŸΆ0.(3.15) Next, we prove limsupπ‘›β†’βˆžβŸ¨π›Ύπ‘“(𝑧)βˆ’π΄π‘§,π‘₯π‘›βˆ’π‘§βŸ©β‰€0,(3.16) where 𝑧=𝑃𝐹(𝑇)(πΌβˆ’π΄+𝛾𝑓)𝑧. Since the sequence {π‘₯𝑛} is bounded we can take a subsequence {π‘₯π‘›π‘˜} of {π‘₯𝑛} such that limsupπ‘›β†’βˆžβŸ¨π›Ύπ‘“(𝑧)βˆ’π΄π‘§,π‘₯π‘›βˆ’π‘§βŸ©=limπ‘˜β†’βˆžξ«π›Ύπ‘“(𝑧)βˆ’π΄π‘§,π‘₯π‘›π‘˜ξ¬,βˆ’π‘§(3.17) and π‘₯π‘›π‘˜β‡€0π‘₯00086Μƒπ‘₯. From (3.15) and by Lemma 2.1, it follows that Μƒπ‘₯∈𝐹(𝑇). Hence, by Lemma 2.2(1) that limsupπ‘›β†’βˆžβŸ¨π›Ύπ‘“(𝑧)βˆ’π΄π‘§,π‘₯π‘›βˆ’π‘§βŸ©=limπ‘˜β†’βˆžξ«π›Ύπ‘“(𝑧)βˆ’π΄π‘§,π‘₯π‘›π‘˜ξ¬βˆ’π‘§=βŸ¨π›Ύπ‘“(𝑧)βˆ’π΄π‘§,Μƒπ‘₯βˆ’π‘§βŸ©=⟨(πΌβˆ’π΄+𝛾𝑓)π‘§βˆ’π‘§,Μƒπ‘₯βˆ’π‘§βŸ©β‰€0.(3.18) Now, by Lemma 2.2(1), we observe that ξ«π‘ƒπΆξ€Ίπ‘€π‘›ξ€»βˆ’π‘€π‘›,π‘ƒπΆξ€Ίπ‘€π‘›ξ€»ξ¬βˆ’π‘§β‰€0,(3.19) and so β€–β€–π‘₯𝑛+1β€–β€–βˆ’π‘§2=ξ«π‘ƒπΆξ€Ίπ‘€π‘›ξ€»βˆ’π‘§,𝑃𝐢𝑀𝑛=ξ«π‘ƒβˆ’π‘§πΆξ€Ίπ‘€π‘›ξ€»βˆ’π‘€π‘›,𝑃𝐢𝑀𝑛+ξ«π‘€βˆ’π‘§π‘›βˆ’π‘§,π‘ƒπΆξ€Ίπ‘€π‘›ξ€»ξ¬β‰€ξ«π‘€βˆ’π‘§π‘›βˆ’π‘§,𝑃𝐢𝑀𝑛=ξ«π›Όβˆ’π‘§π‘›ξ€·π‘₯𝛾𝑓𝑛+ξ€·πΌβˆ’π›Όπ‘›π΄ξ€Έπ‘‡π‘¦π‘›βˆ’π‘§,π‘₯𝑛+1ξ¬βˆ’π‘§β‰€π›Όπ‘›π›Ύβ€–β€–π‘“ξ€·π‘₯𝑛‖‖‖‖π‘₯βˆ’π‘“(𝑧)𝑛+1β€–β€–βˆ’π‘§+𝛼𝑛𝛾𝑓(𝑧)βˆ’π΄π‘§,π‘₯𝑛+1+ξ‚€βˆ’π‘§1βˆ’π›Όπ‘›βˆ’π›Ύξ‚β€–β€–π‘‡π‘¦π‘›β€–β€–β€–β€–π‘₯βˆ’π‘§π‘›+1β€–β€–βˆ’π‘§β‰€π›Όπ‘›β€–β€–π‘₯π›ΎπœŒπ‘›β€–β€–β€–β€–π‘₯βˆ’π‘§π‘›+1β€–β€–βˆ’π‘§+𝛼𝑛𝛾𝑓(𝑧)βˆ’π΄π‘§,π‘₯𝑛+1+ξ‚€βˆ’π‘§1βˆ’π›Όπ‘›βˆ’π›Ύξ‚β€–β€–π‘¦π‘›β€–β€–β€–β€–π‘₯βˆ’π‘§π‘›+1β€–β€–βˆ’π‘§=𝛼𝑛‖‖π‘₯π›ΎπœŒπ‘›β€–β€–β€–β€–π‘₯βˆ’π‘§π‘›+1β€–β€–βˆ’π‘§+𝛼𝑛𝛾𝑓(𝑧)βˆ’π΄π‘§,π‘₯𝑛+1+ξ‚€βˆ’π‘§1βˆ’π›Όπ‘›βˆ’π›Ύξ‚β€–β€–π›½π‘›π‘†π‘₯𝑛+ξ€·1βˆ’π›½π‘›ξ€Έπ‘₯𝑛‖‖‖‖π‘₯βˆ’π‘§π‘›+1β€–β€–βˆ’π‘§β‰€π›Όπ‘›β€–β€–π‘₯π›ΎπœŒπ‘›β€–β€–β€–β€–π‘₯βˆ’π‘§π‘›+1β€–β€–βˆ’π‘§+𝛼𝑛𝛾𝑓(𝑧)βˆ’π΄π‘§,π‘₯𝑛+1+ξ‚€βˆ’π‘§1βˆ’π›Όπ‘›βˆ’π›Ύξ‚ξ€Ίπ›½π‘›β€–β€–π‘†π‘₯π‘›β€–β€–βˆ’π‘†π‘§+π›½π‘›β€–ξ€·π‘†π‘§βˆ’π‘§β€–+1βˆ’π›½π‘›ξ€Έβ€–β€–π‘₯π‘›β€–β€–ξ€»βˆ’π‘§β€–π‘₯𝑛+1βˆ’π‘§β€–β‰€π›Όπ‘›β€–β€–π‘₯π›ΎπœŒπ‘›β€–β€–β€–β€–π‘₯βˆ’π‘§π‘›+1β€–β€–βˆ’π‘§+𝛼𝑛𝛾𝑓(𝑧)βˆ’π΄π‘§,π‘₯𝑛+1+ξ‚€βˆ’π‘§1βˆ’π›Όπ‘›βˆ’π›Ύξ‚ξ€Ίπ›½π‘›β€–β€–π‘₯π‘›β€–β€–βˆ’π‘§+π›½π‘›β€–ξ€·π‘†π‘§βˆ’π‘§β€–+1βˆ’π›½π‘›ξ€Έβ€–β€–π‘₯𝑛‖‖‖‖π‘₯βˆ’π‘§π‘›+1β€–β€–=ξ‚€βˆ’π‘§1βˆ’π›Όπ‘›ξ‚€βˆ’β€–β€–π‘₯π›Ύβˆ’π›ΎπœŒξ‚ξ‚π‘›β€–β€–β€–β€–π‘₯βˆ’π‘§π‘›+1β€–β€–βˆ’π‘§+𝛼𝑛𝛾𝑓(𝑧)βˆ’π΄π‘§,π‘₯𝑛+1+ξ‚€βˆ’π‘§1βˆ’π›Όπ‘›βˆ’π›Ύξ‚π›½π‘›β€–β€–π‘₯β€–π‘†π‘§βˆ’π‘§β€–π‘›+1β€–β€–β‰€ξ‚ƒβˆ’π‘§1βˆ’π›Όπ‘›ξ‚€βˆ’π›Ύβˆ’π›ΎπœŒξ‚ξ‚„2‖‖π‘₯π‘›β€–β€–βˆ’π‘§2+β€–β€–π‘₯𝑛+1β€–β€–βˆ’π‘§2ξ‚„+𝛼𝑛𝛾𝑓(𝑧)βˆ’π΄π‘§,π‘₯𝑛+1+ξ‚€βˆ’π‘§1βˆ’π›Όπ‘›βˆ’π›Ύξ‚π›½π‘›β€–β€–π‘₯β€–π‘†π‘§βˆ’π‘§β€–π‘›+1β€–β€–.βˆ’π‘§(3.20) Hence, it follows that β€–β€–π‘₯𝑛+1β€–β€–βˆ’π‘§2≀1βˆ’π›Όπ‘›ξ‚€βˆ’ξ‚π›Ύβˆ’π›ΎπœŒ1+π›Όπ‘›ξ‚€βˆ’ξ‚β€–β€–π‘₯π›Ύβˆ’π›ΎπœŒπ‘›β€–β€–βˆ’π‘§2+2𝛼𝑛1+π›Όπ‘›ξ‚€βˆ’ξ‚ξ«π›Ύβˆ’π›ΎπœŒπ›Ύπ‘“(𝑧)βˆ’π΄π‘§,π‘₯𝑛+1+2ξ‚€βˆ’π‘§1βˆ’π›Όπ‘›βˆ’π›Ύξ‚π›½π‘›1+π›Όπ‘›ξ‚€βˆ’ξ‚β€–β€–π‘₯π›Ύβˆ’π›ΎπœŒβ€–π‘†π‘§βˆ’π‘§β€–π‘›+1β€–β€–=βŽ‘βŽ’βŽ’βŽ£βˆ’π‘§2π›Όπ‘›ξ‚€βˆ’ξ‚π›Ύβˆ’π›ΎπœŒ1+π›Όπ‘›ξ‚€βˆ’ξ‚βŽ€βŽ₯βŽ₯⎦⎑⎒⎒⎣1π›Ύβˆ’π›ΎπœŒπ›Όπ‘›ξ‚€βˆ’ξ‚ξ«π›Ύβˆ’π›ΎπœŒπ›Ύπ‘“(𝑧)βˆ’π΄π‘§,π‘₯𝑛+1+π›½βˆ’π‘§π‘›ξ‚€1βˆ’π›Όπ‘›βˆ’π›Ύξ‚π›Όπ‘›ξ‚€βˆ’ξ‚β€–β€–β€–π‘₯π›Ύβˆ’π›ΎπœŒπ‘†π‘§βˆ’π‘§β€–π‘›+1β€–β€–βŽ€βŽ₯βŽ₯βŽ¦Γ—βŽ‘βŽ’βŽ’βŽ£βˆ’π‘§1βˆ’2π›Όπ‘›ξ‚€βˆ’ξ‚π›Ύβˆ’π›ΎπœŒ1+π›Όπ‘›ξ‚€βˆ’ξ‚βŽ€βŽ₯βŽ₯βŽ¦β€–β€–π‘₯π›Ύβˆ’π›ΎπœŒπ‘›β€–β€–βˆ’π‘§2.(3.21) We observe that limsupπ‘›β†’βˆžβŽ‘βŽ’βŽ’βŽ£1π›Όπ‘›ξ‚€βˆ’ξ‚ξ«π›Ύβˆ’π›ΎπœŒπ›Ύπ‘“(𝑧)βˆ’π΄π‘§,π‘₯𝑛+1+π›½βˆ’π‘§π‘›ξ‚€1βˆ’π›Όπ‘›βˆ’π›Ύξ‚π›Όπ‘›ξ‚€βˆ’ξ‚β€–β€–π‘₯π›Ύβˆ’π›ΎπœŒβ€–π‘†π‘§βˆ’π‘§β€–π‘›+1β€–β€–βŽ€βŽ₯βŽ₯βŽ¦βˆ’π‘§β‰€0.(3.22) Thus, by Lemma 2.4, π‘₯𝑛→𝑧 as π‘›β†’βˆž. This is completes.

From Theorem 3.1, we can deduce the following interesting corollary.

Corollary 3.2 (Yao et al. [9]). Let 𝐢 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let π‘“βˆΆπΆβ†’π» be a 𝜌-contraction (possibly nonself) with 𝜌∈(0,1). Let 𝑆,π‘‡βˆΆπΆβ†’πΆ be two nonexpansive mappings with 𝐹(𝑇)β‰ βˆ….  {𝛼𝑛} and {𝛽𝑛} are two sequences in (0,1). Starting with an arbitrary initial guess π‘₯0∈𝐢 and {π‘₯𝑛} is a sequence generated by 𝑦𝑛=𝛽𝑛𝑆π‘₯𝑛+ξ€·1βˆ’π›½π‘›ξ€Έπ‘₯𝑛,π‘₯𝑛+1=𝑃𝐢𝛼𝑛𝑓π‘₯𝑛+ξ€·1βˆ’π›Όπ‘›ξ€Έπ‘‡π‘¦π‘›ξ€»,βˆ€π‘›β‰₯1.(3.23) Suppose that the following conditions are satisfied: (C1)  limπ‘›β†’βˆžπ›Όπ‘›=0 and βˆ‘βˆžπ‘›=1𝛼𝑛=∞, (C2)  limπ‘›β†’βˆž(𝛽𝑛/𝛼𝑛)=0, (C3)  limπ‘›β†’βˆž(|π›Όπ‘›βˆ’π›Όπ‘›βˆ’1|/𝛼𝑛)=0 and limπ‘›β†’βˆž(|π›½π‘›βˆ’π›½π‘›βˆ’1|/𝛽𝑛)=0, or (C4)β€‰β€‰βˆ‘βˆžπ‘›=1|π›Όπ‘›βˆ’π›Όπ‘›βˆ’1|<∞ and βˆ‘βˆžπ‘›=1|π›½π‘›βˆ’π›½π‘›βˆ’1|<∞. Then the sequence {π‘₯𝑛} converges strongly to a point Μƒπ‘₯∈𝐻, which is the unique solution of the variational inequality: Μƒπ‘₯∈𝐹(𝑇),⟨(πΌβˆ’π‘“)Μƒπ‘₯,π‘₯βˆ’Μƒπ‘₯⟩β‰₯0,βˆ€π‘₯∈𝐹(𝑇).(3.24) Equivalently, one has 𝑃𝐹(𝑇)(𝑓)Μƒπ‘₯=Μƒπ‘₯. In particular, if one takes 𝑓=0, then the sequence {π‘₯𝑛} converges in norm to the Minimum norm fixed point Μƒπ‘₯ of 𝑇, namely, the point Μƒπ‘₯ is the unique solution to the quadratic minimization problem: 𝑧=argminπ‘₯∈𝐹(𝑇)β€–π‘₯β€–2.(3.25)

Proof. As a matter of fact, if we take 𝐴=𝐼 and 𝛾=1 in Theorem 3.1. This completes the proof.

Under different conditions on data we obtain the following result.

Theorem 3.3. Let 𝐢 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let π‘“βˆΆπΆβ†’π» be a 𝜌-contraction (possibly nonself) with 𝜌∈(0,1). Let 𝑆,π‘‡βˆΆπΆβ†’πΆ be two nonexpansive mappings with 𝐹(𝑇)β‰ βˆ…. Let 𝐴 be a strongly positive linear bounded operator on a Hilbert space 𝐻 with coefficient βˆ’π›Ύ>0 and 0<𝛾<βˆ’π›Ύ/𝜌. {𝛼𝑛} and {𝛽𝑛} are two sequences in (0,1). Starting with an arbitrary initial guess π‘₯0∈𝐢 and {π‘₯𝑛} is a sequence generated by 𝑦𝑛=𝛽𝑛𝑆π‘₯𝑛+ξ€·1βˆ’π›½π‘›ξ€Έπ‘₯𝑛,π‘₯𝑛+1=𝑃𝐢𝛼𝑛π‘₯𝛾𝑓𝑛+ξ€·πΌβˆ’π›Όπ‘›π΄ξ€Έπ‘‡π‘¦π‘›ξ€»,βˆ€π‘›β‰₯1.(3.26) Suppose that the following conditions are satisfied: (C1)  limπ‘›β†’βˆžπ›Όπ‘›=0 and βˆ‘βˆžπ‘›=1𝛼𝑛=∞, (C2)  limπ‘›β†’βˆž(𝛽𝑛/𝛼𝑛)=𝜏∈(0,∞), (C5)  limπ‘›β†’βˆž((|π›Όπ‘›βˆ’π›Όπ‘›βˆ’1|+|π›½π‘›βˆ’π›½π‘›βˆ’1|)/𝛼𝑛𝛽𝑛)=0, (C6)  there exists a constant 𝐾>0 such that (1/𝛼𝑛)|1/π›½π‘›βˆ’1/π›½π‘›βˆ’1|≀𝐾. Then the sequence {π‘₯𝑛} converges strongly to a point Μƒπ‘₯∈𝐻, which is the unique solution of the variational inequality: 1Μƒπ‘₯∈𝐹(𝑇),πœξ‚­(π΄βˆ’π›Ύπ‘“)Μƒπ‘₯+(πΌβˆ’π‘†)Μƒπ‘₯,π‘₯βˆ’Μƒπ‘₯β‰₯0,βˆ€π‘₯∈𝐹(𝑇).(3.27)

Proof. First of all, we show that (3.27) has the unique solution. Indeed, let βˆ’π‘₯ and Μƒπ‘₯ be two solutions. Then (π΄βˆ’π›Ύπ‘“)Μƒπ‘₯,Μƒπ‘₯βˆ’βˆ’π‘₯ξ‚­ξ‚¬β‰€πœ(πΌβˆ’π‘†)Μƒπ‘₯,βˆ’π‘₯ξ‚­.βˆ’Μƒπ‘₯(3.28) Analogously, we have (π΄βˆ’π›Ύπ‘“)βˆ’π‘₯,βˆ’π‘₯ξ‚­ξ‚¬βˆ’Μƒπ‘₯β‰€πœ(πΌβˆ’π‘†)βˆ’π‘₯,Μƒπ‘₯βˆ’βˆ’π‘₯ξ‚­.(3.29) Adding (3.28) and (3.29), by Lemma 2.3, we obtain ξ‚€βˆ’ξ‚β€–β€–π›Ύβˆ’π›ΎπœŒΜƒπ‘₯βˆ’βˆ’π‘₯β€–β€–2≀(π΄βˆ’π›Ύπ‘“)Μƒπ‘₯βˆ’(π΄βˆ’π›Ύπ‘“)βˆ’π‘₯,Μƒπ‘₯βˆ’βˆ’π‘₯ξ‚­ξ‚¬β‰€βˆ’πœ(πΌβˆ’π‘†)Μƒπ‘₯βˆ’(πΌβˆ’π‘†)βˆ’π‘₯,Μƒπ‘₯βˆ’βˆ’π‘₯≀0,(3.30) and so Μƒπ‘₯=βˆ’π‘₯. From (C2), we can assume, without loss of generality, that 𝛽𝑛≀(𝜏+1)𝛼𝑛 for all 𝑛β‰₯1. By a similar argument in Theorem 3.1, we have β€–β€–π‘₯𝑛+1β€–β€–βˆ’π‘’β‰€π›Όπ‘›β€–β€–π‘₯π›ΎπœŒπ‘›β€–β€–βˆ’π‘’+𝛼𝑛‖+‖𝛾𝑓(𝑒)βˆ’π΄π‘’1βˆ’π›Όπ‘›βˆ’π›Ύξ‚ξ€Ίβ€–β€–π‘₯π‘›β€–β€–βˆ’π‘’+π›½π‘›ξ€·β€–π‘†π‘’βˆ’π‘’β€–+1βˆ’π›½π‘›ξ€Έβ€–β€–π‘₯𝑛‖‖=ξ‚€βˆ’π‘’1βˆ’π›Όπ‘›ξ‚€βˆ’β€–β€–π‘₯π›Ύβˆ’π›ΎπœŒξ‚ξ‚π‘›β€–β€–βˆ’π‘’+𝛼𝑛‖𝛾𝑓(𝑒)βˆ’π΄π‘’β€–+1βˆ’π›Όπ‘›βˆ’π›Ύξ‚π›½π‘›β‰€ξ‚€β€–π‘†π‘’βˆ’π‘’β€–1βˆ’π›Όπ‘›ξ‚€βˆ’β€–β€–π‘₯π›Ύβˆ’π›ΎπœŒξ‚ξ‚π‘›β€–β€–βˆ’π‘’+𝛼𝑛(‖𝛾𝑓𝑒)βˆ’π΄π‘’β€–+π›½π‘›β‰€ξ‚€β€–π‘†π‘’βˆ’π‘’β€–1βˆ’π›Όπ‘›ξ‚€βˆ’β€–β€–π‘₯π›Ύβˆ’π›ΎπœŒξ‚ξ‚π‘›β€–β€–βˆ’π‘’+𝛼𝑛‖𝛾𝑓(𝑒)βˆ’π΄π‘’β€–+(𝜏+1)𝛼𝑛=ξ‚€β€–π‘†π‘’βˆ’π‘’β€–1βˆ’π›Όπ‘›ξ‚€βˆ’β€–β€–π‘₯π›Ύβˆ’π›ΎπœŒξ‚ξ‚π‘›β€–β€–βˆ’π‘’+𝛼𝑛[(]=‖𝛾𝑓𝑒)βˆ’π΄π‘’β€–+(𝜏+1)β€–π‘†π‘’βˆ’π‘’β€–1βˆ’π›Όπ‘›ξ‚€βˆ’β€–β€–π‘₯π›Ύβˆ’π›ΎπœŒξ‚ξ‚π‘›β€–β€–βˆ’π‘’+π›Όπ‘›ξ‚€βˆ’ξ‚π›Ύβˆ’π›ΎπœŒβ€–π›Ύπ‘“(𝑒)βˆ’π΄π‘’β€–+(𝜏+1)β€–π‘†π‘’βˆ’π‘’β€–ξ‚€βˆ’ξ‚.π›Ύβˆ’π›ΎπœŒ(3.31) By induction, we obtain β€–β€–π‘₯𝑛‖‖‖‖π‘₯βˆ’π‘’β‰€max0β€–β€–,1βˆ’π‘’βˆ’[β€–]ξƒ°,π›Ύβˆ’π›ΎπœŒπ›Ύπ‘“(𝑒)βˆ’π΄π‘’β€–+(𝜏+1)β€–π‘†π‘’βˆ’π‘’β€–(3.32) which implies that the sequence {π‘₯𝑛} is bounded. Since (C5) implies (C4) then, from Theorem 3.1, we can deduce β€–π‘₯𝑛+1βˆ’π‘₯𝑛‖→0.
From (3.1), we note that π‘₯𝑛+1=π‘ƒπΆξ€Ίπ‘€π‘›ξ€»βˆ’π‘€π‘›+𝑀𝑛+π‘¦π‘›βˆ’π‘¦π‘›=π‘ƒπΆξ€Ίπ‘€π‘›ξ€»βˆ’π‘€π‘›+𝛼𝑛π‘₯𝛾𝑓𝑛+ξ€·π‘‡π‘¦π‘›βˆ’π‘¦π‘›ξ€Έ+ξ€·π‘¦π‘›βˆ’π›Όπ‘›π΄π‘‡π‘¦π‘›ξ€Έ.(3.33) Hence, it follows that π‘₯π‘›βˆ’π‘₯𝑛+1=ξ€·π‘€π‘›βˆ’π‘ƒπΆξ€Ίπ‘€π‘›ξ€»ξ€Έ+𝛼𝑛𝐴π‘₯𝑛π‘₯βˆ’π›Ύπ‘“π‘›+ξ€·π‘¦ξ€Έξ€Έπ‘›βˆ’π‘‡π‘¦π‘›ξ€Έ+ξ€·π‘₯π‘›βˆ’π‘¦π‘›ξ€Έ+π›Όπ‘›ξ€·π΄π‘‡π‘¦π‘›βˆ’π΄π‘₯𝑛=ξ€·π‘€π‘›βˆ’π‘ƒπΆξ€Ίπ‘€π‘›ξ€»ξ€Έ+𝛼𝑛(π΄βˆ’π›Ύπ‘“)π‘₯𝑛+(πΌβˆ’π‘‡)𝑦𝑛+𝛽𝑛(πΌβˆ’π‘†)π‘₯𝑛+π›Όπ‘›π΄ξ€·π‘‡π‘¦π‘›βˆ’π‘₯𝑛,(3.34) and so π‘₯π‘›βˆ’π‘₯𝑛+1ξ€·1βˆ’π›Όπ‘›ξ€Έπ›½π‘›=1ξ€·1βˆ’π›Όπ‘›ξ€Έπ›½π‘›ξ€·π‘€π‘›βˆ’π‘ƒπΆξ€Ίπ‘€π‘›+𝛼𝑛1βˆ’π›Όπ‘›ξ€Έπ›½π‘›(π΄βˆ’π›Ύπ‘“)π‘₯𝑛+1ξ€·1βˆ’π›Όπ‘›ξ€Έπ›½π‘›(πΌβˆ’π‘‡)𝑦𝑛+1ξ€·1βˆ’π›Όπ‘›ξ€Έ(πΌβˆ’π‘†)π‘₯𝑛+𝛼𝑛1βˆ’π›Όπ‘›ξ€Έπ›½π‘›π΄ξ€·π‘‡π‘¦π‘›βˆ’π‘₯𝑛.(3.35) Set π‘£π‘›βˆΆ=(π‘₯π‘›βˆ’π‘₯𝑛+1)/(1βˆ’π›Όπ‘›)𝛽𝑛. Then, we have 𝑣𝑛=1ξ€·1βˆ’π›Όπ‘›ξ€Έπ›½π‘›ξ€·π‘€π‘›βˆ’π‘ƒπΆξ€Ίπ‘€π‘›+𝛼𝑛1βˆ’π›Όπ‘›ξ€Έπ›½π‘›(π΄βˆ’π›Ύπ‘“)π‘₯𝑛+1ξ€·1βˆ’π›Όπ‘›ξ€Έπ›½π‘›(πΌβˆ’π‘‡)𝑦𝑛+1ξ€·1βˆ’π›Όπ‘›ξ€Έ(πΌβˆ’π‘†)π‘₯𝑛+𝛼𝑛1βˆ’π›Όπ‘›ξ€Έπ›½π‘›π΄ξ€·π‘‡π‘¦π‘›βˆ’π‘₯𝑛.(3.36) From (3.12) in Theorem 3.1 and (C6), we obtain β€–β€–π‘₯𝑛+1βˆ’π‘₯𝑛‖‖𝛽𝑛≀1βˆ’π›Όπ‘›ξ‚€βˆ’β€–β€–π‘₯π›Ύβˆ’π›ΎπœŒξ‚ξ‚π‘›βˆ’π‘₯π‘›βˆ’1‖‖𝛽𝑛||𝛼+π‘€π‘›βˆ’π›Όπ‘›βˆ’1||𝛽𝑛+||π›½π‘›βˆ’π›½π‘›βˆ’1||𝛽𝑛=ξ‚€1βˆ’π›Όπ‘›ξ‚€βˆ’β€–β€–π‘₯π›Ύβˆ’π›ΎπœŒξ‚ξ‚π‘›βˆ’π‘₯π‘›βˆ’1‖‖𝛽𝑛+ξ‚€1βˆ’π›Όπ‘›ξ‚€βˆ’β€–β€–π‘₯π›Ύβˆ’π›ΎπœŒξ‚ξ‚π‘›βˆ’π‘₯π‘›βˆ’1β€–β€–π›½π‘›βˆ’1βˆ’ξ‚€1βˆ’π›Όπ‘›ξ‚€βˆ’β€–β€–π‘₯π›Ύβˆ’π›ΎπœŒξ‚ξ‚π‘›βˆ’π‘₯π‘›βˆ’1β€–β€–π›½π‘›βˆ’1ξ‚Έ||𝛼+π‘€π‘›βˆ’π›Όπ‘›βˆ’1||𝛽𝑛+||π›½π‘›βˆ’π›½π‘›βˆ’1||𝛽𝑛=ξ‚€1βˆ’π›Όπ‘›ξ‚€βˆ’β€–β€–π‘₯π›Ύβˆ’π›ΎπœŒξ‚ξ‚π‘›βˆ’π‘₯π‘›βˆ’1β€–β€–π›½π‘›βˆ’1+ξ‚€1βˆ’π›Όπ‘›ξ‚€βˆ’β€–β€–π‘₯π›Ύβˆ’π›ΎπœŒξ‚ξ‚π‘›βˆ’π‘₯π‘›βˆ’1β€–β€–ξ‚Έ1π›½π‘›βˆ’1π›½π‘›βˆ’1ξ‚Ήξ‚Έ||𝛼+π‘€π‘›βˆ’π›Όπ‘›βˆ’1||𝛽𝑛+||π›½π‘›βˆ’π›½π‘›βˆ’1||𝛽𝑛≀1βˆ’π›Όπ‘›ξ‚€βˆ’β€–β€–π‘₯π›Ύβˆ’π›ΎπœŒξ‚ξ‚π‘›βˆ’π‘₯π‘›βˆ’1β€–β€–π›½π‘›βˆ’1+β€–β€–π‘₯π‘›βˆ’π‘₯π‘›βˆ’1β€–β€–||||1π›½π‘›βˆ’1π›½π‘›βˆ’1||||ξ‚Έ||𝛼+π‘€π‘›βˆ’π›Όπ‘›βˆ’1||𝛽𝑛+||π›½π‘›βˆ’π›½π‘›βˆ’1||𝛽𝑛≀1βˆ’π›Όπ‘›ξ‚€βˆ’β€–β€–π‘₯π›Ύβˆ’π›ΎπœŒξ‚ξ‚π‘›βˆ’π‘₯π‘›βˆ’1β€–β€–π›½π‘›βˆ’1+𝛼𝑛𝐾‖‖π‘₯π‘›βˆ’π‘₯π‘›βˆ’1β€–β€–ξ‚Έ||𝛼+π‘€π‘›βˆ’π›Όπ‘›βˆ’1||𝛽𝑛+||π›½π‘›βˆ’π›½π‘›βˆ’1||𝛽𝑛≀1βˆ’π›Όπ‘›ξ‚€βˆ’β€–β€–π‘€π›Ύβˆ’π›ΎπœŒξ‚ξ‚π‘›βˆ’π‘€π‘›βˆ’1β€–β€–π›½π‘›βˆ’1+𝛼𝑛𝐾‖‖π‘₯π‘›βˆ’π‘₯π‘›βˆ’1β€–β€–ξ‚Έ||𝛼+π‘€π‘›βˆ’π›Όπ‘›βˆ’1||𝛽𝑛+||π›½π‘›βˆ’π›½π‘›βˆ’1||𝛽𝑛.(3.37) This together with Lemma 2.4 and (C2) imply that limπ‘›β†’βˆžβ€–β€–π‘₯𝑛+1βˆ’π‘₯𝑛‖‖𝛽𝑛=limπ‘›β†’βˆžβ€–β€–π‘€π‘›+1βˆ’π‘€π‘›β€–β€–π›½π‘›=limπ‘›β†’βˆžβ€–β€–π‘€π‘›+1βˆ’π‘€π‘›β€–β€–π›Όπ‘›=0.(3.38) From (3.36), for π‘§βˆˆπΉ(𝑇), we have βŸ¨π‘£π‘›,π‘₯𝑛1βˆ’π‘§βŸ©=ξ€·1βˆ’π›Όπ‘›ξ€Έπ›½π‘›ξ«π‘€π‘›βˆ’π‘ƒπΆξ€Ίπ‘€π‘›ξ€»,π‘ƒπΆξ€Ίπ‘€π‘›βˆ’1+π›Όβˆ’π‘§π‘›ξ€·1βˆ’π›Όπ‘›ξ€Έπ›½π‘›βŸ¨(π΄βˆ’π›Ύπ‘“)π‘₯𝑛,π‘₯𝑛+1βˆ’π‘§βŸ©ξ€·1βˆ’π›Όπ‘›ξ€Έπ›½π‘›βŸ¨(πΌβˆ’π‘‡)𝑦𝑛,π‘₯𝑛1βˆ’π‘§βŸ©+ξ€·1βˆ’π›Όπ‘›ξ€ΈβŸ¨(πΌβˆ’π‘†)π‘₯𝑛,π‘₯𝑛+π›Όβˆ’π‘§βŸ©π‘›ξ€·1βˆ’π›Όπ‘›ξ€Έπ›½π‘›ξ«π΄ξ€·π‘‡π‘¦π‘›βˆ’π‘₯𝑛,π‘₯𝑛=1βˆ’π‘§ξ€·1βˆ’π›Όπ‘›ξ€Έπ›½π‘›ξ«π‘€π‘›βˆ’π‘ƒπΆξ€Ίπ‘€π‘›ξ€»,𝑃𝐢𝑀𝑛+1βˆ’π‘§ξ€·1βˆ’π›Όπ‘›ξ€Έπ›½π‘›ξ«π‘€π‘›βˆ’π‘ƒπΆξ€Ίπ‘€π‘›ξ€»,π‘ƒπΆξ€Ίπ‘€π‘›βˆ’1ξ€»βˆ’π‘ƒπΆξ€Ίπ‘€π‘›+𝛼𝑛1βˆ’π›Όπ‘›ξ€Έπ›½π‘›βŸ¨(π΄βˆ’π›Ύπ‘“)π‘₯π‘›βˆ’(π΄βˆ’π›Ύπ‘“)𝑧,π‘₯π‘›π›Όβˆ’π‘§βŸ©+𝑛1βˆ’π›Όπ‘›ξ€Έπ›½π‘›βŸ¨(π΄βˆ’π›Ύπ‘“)𝑧,π‘₯𝑛+1βˆ’π‘§βŸ©ξ€·1βˆ’π›Όπ‘›ξ€ΈβŸ¨(πΌβˆ’π‘†)π‘₯π‘›βˆ’(πΌβˆ’π‘†)𝑧,π‘₯𝑛1βˆ’π‘§βŸ©+ξ€·1βˆ’π›Όπ‘›ξ€ΈβŸ¨(πΌβˆ’π‘†)𝑧,π‘₯𝑛+1βˆ’π‘§βŸ©ξ€·1βˆ’π›Όπ‘›ξ€Έπ›½π‘›βŸ¨(πΌβˆ’π‘‡)𝑦𝑛,π‘₯π‘›π›Όβˆ’π‘§βŸ©+𝑛1βˆ’π›Όπ‘›ξ€Έπ›½π‘›ξ«π΄ξ€·π‘‡π‘¦π‘›βˆ’π‘₯𝑛,π‘₯𝑛.βˆ’π‘§(3.39) By Lemmas 2.2 and 2.3, we obtain βŸ¨π‘£π‘›,π‘₯𝑛1βˆ’π‘§βŸ©β‰₯ξ€·1βˆ’π›Όπ‘›ξ€Έπ›½π‘›ξ«π‘€π‘›βˆ’π‘ƒπΆξ€Ίπ‘€π‘›ξ€»,π‘ƒπΆξ€Ίπ‘€π‘›βˆ’1ξ€»βˆ’π‘ƒπΆξ€Ίπ‘€π‘›+ξ‚€ξ€»ξ¬βˆ’ξ‚π›Όπ›Ύβˆ’π›ΎπœŒπ‘›ξ€·1βˆ’π›Όπ‘›ξ€Έπ›½π‘›β€–β€–π‘₯π‘›β€–β€–βˆ’π‘§2+𝛼𝑛1βˆ’π›Όπ‘›ξ€Έπ›½π‘›βŸ¨(π΄βˆ’π›Ύπ‘“)𝑧,π‘₯𝑛1βˆ’π‘§βŸ©+ξ€·1βˆ’π›Όπ‘›ξ€ΈβŸ¨(πΌβˆ’π‘†)𝑧,π‘₯𝑛+1βˆ’π‘§βŸ©ξ€·1βˆ’π›Όπ‘›ξ€Έπ›½π‘›βŸ¨(πΌβˆ’π‘‡)𝑦𝑛,π‘₯π‘›π›Όβˆ’π‘§βŸ©+𝑛1βˆ’π›Όπ‘›ξ€Έπ›½π‘›ξ«π΄ξ€·π‘‡π‘¦π‘›βˆ’π‘₯𝑛,π‘₯𝑛.βˆ’π‘§(3.40) Now, we observe that β€–β€–π‘₯π‘›β€–β€–βˆ’π‘§2≀1βˆ’π›Όπ‘›ξ€Έπ›½π‘›ξ‚€βˆ’ξ‚π›Όπ›Ύβˆ’π›ΎπœŒπ‘›βŸ¨π‘£,π‘₯π‘›π›½βˆ’π‘§βŸ©βˆ’π‘›ξ‚€βˆ’ξ‚π›Όπ›Ύβˆ’π›ΎπœŒπ‘›βŸ¨(πΌβˆ’π‘†)𝑧,π‘₯π‘›βˆ’1βˆ’π‘§βŸ©ξ‚€βˆ’ξ‚π›Ύβˆ’π›ΎπœŒβŸ¨(π΄βˆ’π›Ύπ‘“)𝑧,π‘₯𝑛1βˆ’π‘§βŸ©βˆ’ξ‚€βˆ’ξ‚π›Όπ›Ύβˆ’π›ΎπœŒπ‘›βŸ¨(πΌβˆ’π‘‡)𝑦𝑛,π‘₯π‘›βˆ’1βˆ’π‘§βŸ©ξ‚€βˆ’ξ‚ξ«π΄ξ€·π›Ύβˆ’π›ΎπœŒπ‘‡π‘¦π‘›βˆ’π‘₯𝑛,π‘₯π‘›ξ¬βˆ’1βˆ’π‘§ξ‚€βˆ’ξ‚π›Όπ›Ύβˆ’π›ΎπœŒπ‘›ξ«π‘€π‘›βˆ’π‘ƒπΆξ€Ίπ‘€π‘›ξ€»,π‘ƒπΆξ€Ίπ‘€π‘›βˆ’1ξ€»βˆ’π‘ƒπΆξ€Ίπ‘€π‘›β‰€ξ€·ξ€»ξ¬1βˆ’π›Όπ‘›ξ€Έπ›½π‘›ξ‚€βˆ’ξ‚π›Όπ›Ύβˆ’π›ΎπœŒπ‘›βŸ¨π‘£,π‘₯π‘›π›½βˆ’π‘§βŸ©βˆ’π‘›ξ‚€βˆ’ξ‚π›Όπ›Ύβˆ’π›ΎπœŒπ‘›βŸ¨(πΌβˆ’π‘†)𝑧,π‘₯π‘›βˆ’1βˆ’π‘§βŸ©ξ‚€βˆ’ξ‚π›Ύβˆ’π›ΎπœŒβŸ¨(π΄βˆ’π›Ύπ‘“)𝑧,π‘₯𝑛1βˆ’π‘§βŸ©βˆ’ξ‚€βˆ’ξ‚π›Όπ›Ύβˆ’π›ΎπœŒπ‘›βŸ¨(πΌβˆ’π‘‡)𝑦𝑛,π‘₯π‘›βˆ’1βˆ’π‘§βŸ©ξ‚€βˆ’ξ‚ξ«π΄ξ€·π›Ύβˆ’π›ΎπœŒπ‘‡π‘¦π‘›βˆ’π‘₯𝑛,π‘₯𝑛+β€–β€–π‘€βˆ’π‘§π‘›βˆ’π‘€π‘›βˆ’1β€–β€–ξ‚€βˆ’ξ‚β€–β€–π‘€π›Ύβˆ’π›ΎπœŒπ‘›βˆ’π‘ƒπΆξ€Ίπ‘€π‘›ξ€»β€–β€–.(3.41) From (C1) and (C2), we have 𝛽𝑛→0. Hence, from (3.1), we deduce β€–π‘¦π‘›βˆ’π‘₯𝑛‖→0 and β€–π‘₯𝑛+1βˆ’π‘‡π‘¦π‘›β€–β†’0. Therefore, β€–β€–π‘¦π‘›βˆ’π‘‡π‘¦π‘›β€–β€–β‰€β€–β€–π‘¦π‘›βˆ’π‘₯𝑛‖‖+β€–β€–π‘₯π‘›βˆ’π‘₯𝑛+1β€–β€–+β€–β€–π‘₯𝑛+1βˆ’π‘‡π‘¦π‘›β€–β€–β†’0.(3.42)
Since 𝑣𝑛→0, (πΌβˆ’π‘‡)𝑦𝑛→0, 𝐴(π‘‡π‘¦π‘›βˆ’π‘₯𝑛)β†’0, and β€–π‘€π‘›βˆ’π‘€π‘›βˆ’1β€–/(βˆ’π›Ύβˆ’π›ΎπœŒ)β†’0, every weak cluster point of {π‘₯𝑛} is also a strong cluster point. Note that the sequence {π‘₯𝑛} is bounded, thus there exists a subsequence {π‘₯π‘›π‘˜} converging to a point Μƒπ‘₯∈𝐻. For all π‘§βˆˆπΉ(𝑇), it follows from (3.39) that (π΄βˆ’π›Ύπ‘“)π‘₯π‘›π‘˜,π‘₯π‘›π‘˜ξ¬=ξ€·βˆ’π‘§1βˆ’π›Όπ‘›π‘˜ξ€Έπ›½π‘›π‘˜π›Όπ‘›π‘˜ξ«π‘£π‘›π‘˜,π‘₯π‘›π‘˜ξ¬βˆ’1βˆ’π‘§π›Όπ‘›π‘˜ξ«(πΌβˆ’π‘‡)π‘¦π‘›π‘˜,π‘₯π‘›π‘˜ξ¬βˆ’π›½βˆ’π‘§π‘›π‘˜π›Όπ‘›π‘˜ξ«(πΌβˆ’π‘†)π‘₯π‘›π‘˜,π‘₯π‘›π‘˜ξ¬βˆ’ξ«π΄ξ€·βˆ’π‘§π‘‡π‘¦π‘›π‘˜βˆ’π‘₯π‘›π‘˜ξ€Έ,π‘₯π‘›π‘˜ξ¬βˆ’1βˆ’π‘§π›Όπ‘›π‘˜ξ«π‘€π‘›π‘˜βˆ’π‘ƒπΆξ€Ίπ‘€π‘›π‘˜ξ€»,π‘ƒπΆξ€Ίπ‘€π‘›π‘˜βˆ’1ξ€»ξ¬β‰€ξ€·βˆ’π‘§1βˆ’π›Όπ‘›π‘˜ξ€Έπ›½π‘›π‘˜π›Όπ‘›π‘˜ξ«π‘£π‘›π‘˜,π‘₯π‘›π‘˜ξ¬βˆ’1βˆ’π‘§π›Όπ‘›π‘˜ξ«(πΌβˆ’π‘‡)π‘¦π‘›π‘˜,π‘₯π‘›π‘˜ξ¬βˆ’π›½βˆ’π‘§π‘›π‘˜π›Όπ‘›π‘˜ξ«(πΌβˆ’π‘†)π‘₯π‘›π‘˜,π‘₯π‘›π‘˜ξ¬βˆ’ξ«π΄ξ€·βˆ’π‘§π‘‡π‘¦π‘›π‘˜βˆ’π‘₯π‘›π‘˜ξ€Έ,π‘₯π‘›π‘˜ξ¬βˆ’1βˆ’π‘§π›Όπ‘›π‘˜ξ«π‘€π‘›π‘˜βˆ’π‘ƒπΆξ€Ίπ‘€π‘›π‘˜ξ€»,π‘ƒπΆξ€Ίπ‘€π‘›π‘˜βˆ’1ξ€»βˆ’π‘ƒπΆξ€Ίπ‘€π‘›π‘˜βˆ’ξ«π΄ξ€·ξ€»ξ¬π‘‡π‘¦π‘›π‘˜βˆ’π‘₯π‘›π‘˜ξ€Έ,π‘₯π‘›π‘˜ξ¬βˆ’1βˆ’π‘§π›Όπ‘›π‘˜ξ«π‘€π‘›π‘˜βˆ’π‘ƒπΆξ€Ίπ‘€π‘›π‘˜ξ€»,π‘ƒπΆξ€Ίπ‘€π‘›π‘˜βˆ’1ξ€»ξ¬β‰€ξ€·βˆ’π‘§1βˆ’π›Όπ‘›π‘˜ξ€Έπ›½π‘›π‘˜π›Όπ‘›π‘˜ξ«π‘£π‘›π‘˜,π‘₯π‘›π‘˜ξ¬βˆ’1βˆ’π‘§π›Όπ‘›π‘˜ξ«(πΌβˆ’π‘‡)π‘¦π‘›π‘˜,π‘₯π‘›π‘˜ξ¬βˆ’π›½βˆ’π‘§π‘›π‘˜π›Όπ‘›π‘˜ξ«(πΌβˆ’