Abstract

This paper deals with new methods for approximating a solution to the fixed point problem; find ฬƒ๐‘ฅโˆˆ๐น(๐‘‡), where ๐ป is a Hilbert space, ๐ถ is a closed convex subset of ๐ป, ๐‘“ is a ๐œŒ-contraction from ๐ถ into ๐ป, 0<๐œŒ<1, ๐ด is a strongly positive linear-bounded operator with coefficient ๐›พ>0, 0<๐›พ<๐›พ/๐œŒ, ๐‘‡ is a nonexpansive mapping on ๐ถ, and ๐‘ƒ๐น(๐‘‡) denotes the metric projection on the set of fixed point of ๐‘‡. Under a suitable different parameter, we obtain strong convergence theorems by using the projection method which solves the variational inequality โŸจ(๐ดโˆ’๐›พ๐‘“)ฬƒ๐‘ฅ+๐œ(๐ผโˆ’๐‘†)ฬƒ๐‘ฅ,๐‘ฅโˆ’ฬƒ๐‘ฅโŸฉโ‰ฅ0 for ๐‘ฅโˆˆ๐น(๐‘‡), where ๐œโˆˆ[0,โˆž). 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 ๐ป is a real Hilbert space where inner product and norm are denoted by โŸจโ‹…,โ‹…โŸฉ and โ€–โ‹…โ€–, respectively, and let ๐ถ be a nonempty closed convex subset of ๐ป. A mapping ๐‘‡โˆถ๐ถโ†’๐ถ is called nonexpansive if โ€–๐‘‡๐‘ฅโˆ’๐‘‡๐‘ฆโ€–โ‰คโ€–๐‘ฅโˆ’๐‘ฆโ€–,โˆ€๐‘ฅ,๐‘ฆโˆˆ๐ถ.(1.1)

We use ๐น(๐‘‡) to denote the set of fixed points of ๐‘‡, that is, ๐น(๐‘‡)={๐‘ฅโˆˆ๐ถโˆถ๐‘‡๐‘ฅ=๐‘ฅ}. It is assumed throughout the paper that ๐‘‡ is a nonexpansive mapping such that ๐น(๐‘‡)โ‰ โˆ….

Recall that a mapping ๐‘“โˆถ๐ถโ†’๐ป is a contraction on ๐ถ if there exists a constant ๐œŒโˆˆ(0,1) such that โ€–๐‘“(๐‘ฅ)โˆ’๐‘“(๐‘ฆ)โ€–โ‰ค๐œŒโ€–๐‘ฅโˆ’๐‘ฆโ€–,โˆ€๐‘ฅ,๐‘ฆโˆˆ๐ถ.(1.2)

A mapping ๐ดโˆถ๐ปโ†’๐ป is called a strongly positive linear bounded operator on ๐ป if there exists a constant ๐›พ>0 with property โŸจ๐ด๐‘ฅ,๐‘ฅโŸฉโ‰ฅ๐›พโ€–๐‘ฅโ€–2,โˆ€๐‘ฅโˆˆ๐ป.(1.3)

A mapping ๐‘€โˆถ๐ปโ†’๐ป is called a strongly monotone operator with ๐›ผ if โŸจ๐‘ฅโˆ’๐‘ฆ,๐‘€๐‘ฅโˆ’๐‘€๐‘ฆโŸฉโ‰ฅ๐›ผโ€–๐‘ฅโˆ’๐‘ฆโ€–2,โˆ€๐‘ฅ,๐‘ฆโˆˆ๐ป,(1.4) and ๐‘€ is called a monotone operator if โŸจ๐‘ฅโˆ’๐‘ฆ,๐‘€๐‘ฅโˆ’๐‘€๐‘ฆโŸฉโ‰ฅ0,โˆ€๐‘ฅ,๐‘ฆโˆˆ๐ป.(1.5) We easily prove that the mapping (๐ผโˆ’๐‘‡) is monotone operator, if ๐‘‡ is nonexpansive mapping.

The metric (or nearest point) projection from ๐ป onto ๐ถ is mapping ๐‘ƒ๐ถ[โ‹…]โˆถ๐ปโ†’๐ถ which assigns to each point ๐‘ฅโˆˆ๐ถ the unique point ๐‘ƒ๐ถ[๐‘ฅ]โˆˆ๐ถ satisfying the property โ€–โ€–๐‘ฅโˆ’๐‘ƒ๐ถ[๐‘ฅ]โ€–โ€–=inf๐‘ฆโˆˆ๐ถโ€–๐‘ฅโˆ’๐‘ฆโ€–=โˆถ๐‘‘(๐‘ฅ,๐ถ).(1.6)

The variational inequality for a monotone operator, ๐‘€โˆถ๐ปโ†’๐ป over ๐ถ, is to find a point in VI(๐ถ,๐‘€)โˆถ={ฬƒ๐‘ฅโˆˆ๐ถโˆถโŸจ๐‘ฅโˆ’ฬƒ๐‘ฅ,๐‘€ฬƒ๐‘ฅโŸฉโ‰ฅ0,โˆ€๐‘ฅโˆˆ๐ถ}.(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 ๐‘‡ with respect to another nonexpansive mapping ๐‘†, namely, we find ฬƒ๐‘ฅโˆˆ๐น(๐‘‡) such that โŸจ๐‘ฅโˆ’ฬƒ๐‘ฅ,(๐ผโˆ’๐‘†)ฬƒ๐‘ฅโŸฉโ‰ฅ0,โˆ€๐‘ฅโˆˆ๐น(๐‘‡).(1.8)

Iterative methods for nonexpansive mappings have recently been applied to solve a convex minimization problem; see, for example, [1โ€“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 ๐ป: min๐‘ฅโˆˆ๐น(๐‘‡)12โŸจ๐ด๐‘ฅ,๐‘ฅโŸฉโˆ’โŸจ๐‘ฅ,๐‘โŸฉ,(1.9) where ๐‘ is a given point in ๐ป. In [5], it is proved that the sequence {๐‘ฅ๐‘›} defined by the iterative method below, with the initial guess ๐‘ฅ0โˆˆ๐ป chosen arbitrarily, ๐‘ฅ๐‘›+1=๎€ท๐ผโˆ’๐›ผ๐‘›๐ด๎€ธ๐‘‡๐‘ฅ๐‘›+๐›ผ๐‘›๐‘,๐‘›โ‰ฅ0,(1.10) converges strongly to the unique solution of the minimization problem (1.9) provided the sequence {๐›ผ๐‘›} 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 ๐‘ฅ0โˆˆ๐ป, define a sequence {๐‘ฅ๐‘›} recursively by ๐‘ฅ๐‘›+1=๐œŽ๐‘›๐‘“๎€ท๐‘ฅ๐‘›๎€ธ+๎€ท1โˆ’๐œŽ๐‘›๎€ธ๐‘‡๐‘ฅ๐‘›,๐‘›โ‰ฅ0,(1.11) where {๐œŽ๐‘›} is a sequence in (0,1). It is proved in [6, 7] that under certain appropriate conditions imposed on {๐œŽ๐‘›}, the sequence {๐‘ฅ๐‘›} generated by (1.11) strongly converges to the unique solution ๐‘ฅโˆ— in ๐ถ of the variational inequality โŸจ(๐ผโˆ’๐‘“)๐‘ฅโˆ—,๐‘ฅโˆ’๐‘ฅโˆ—โŸฉโ‰ฅ0,๐‘ฅโˆˆ๐ถ.(1.12)

In 2006, Marino and Xu [8] introduced a general iterative method for nonexpansive mapping. Starting with an arbitrary initial ๐‘ฅ0โˆˆ๐ป, define a sequence {๐‘ฅ๐‘›} recursively by ๐‘ฅ๐‘›+1=๐œ–๐‘›๎€ท๐‘ฅ๐›พ๐‘“๐‘›๎€ธ+๎€ท๐ผโˆ’๐œ–๐‘›๐ด๎€ธ๐‘‡๐‘ฅ๐‘›,๐‘›โ‰ฅ0.(1.13) They proved that if the sequence {๐œ–๐‘›} of parameters satisfies appropriate conditions, then the sequence {๐‘ฅ๐‘›} generated by (1.13) strongly converges to the unique solution ฬƒ๐‘ฅ=๐‘ƒ๐น(๐‘‡)(๐ผโˆ’๐ด+๐›พ๐‘“)ฬƒ๐‘ฅ of the variational inequality โŸจ(๐ดโˆ’๐›พ๐‘“)ฬƒ๐‘ฅ,๐‘ฅโˆ’ฬƒ๐‘ฅโŸฉโ‰ฅ0,โˆ€๐‘ฅโˆˆ๐น(๐‘‡),(1.14) which is the optimality condition for the minimization problem min๐‘ฅโˆˆ๐น(๐‘‡)12โŸจ๐ด๐‘ฅ,๐‘ฅโŸฉโˆ’โ„Ž(๐‘ฅ),(1.15) where โ„Ž is a potential function for ๐›พ๐‘“ (i.e., โ„Žโ€ฒ(๐‘ฅ)=๐›พ๐‘“(๐‘ฅ) for ๐‘ฅโˆˆ๐ป).

In 2010, Yao et al. [9] introduced an iterative algorithm for solving some hierarchical fixed point problem, starting with an arbitrary initial guess ๐‘ฅ0โˆˆ๐ถ, define a sequence {๐‘ฅ๐‘›} iteratively by ๐‘ฆ๐‘›=๐›ฝ๐‘›๐‘†๐‘ฅ๐‘›+๎€ท1โˆ’๐›ฝ๐‘›๎€ธ๐‘ฅ๐‘›,๐‘ฅ๐‘›+1=๐‘ƒ๐ถ๎€บ๐›ผ๐‘›๐‘“๎€ท๐‘ฅ๐‘›๎€ธ+๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐‘‡๐‘ฆ๐‘›๎€ป,โˆ€๐‘›โ‰ฅ1.(1.16) They proved that if the sequences {๐›ผ๐‘›} and {๐›ฝ๐‘›} of parameters satisfies appropriate conditions, then the sequence {๐‘ฅ๐‘›} generated by (1.16) strongly converges to the unique solution ๐‘ง in ๐ป of the variational inequality ๐‘งโˆˆ๐น(๐‘‡),โŸจ(๐ผโˆ’๐‘“)๐‘ง,๐‘ฅโˆ’๐‘งโŸฉโ‰ฅ0,โˆ€๐‘ฅโˆˆ๐น(๐‘‡).(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: ๐‘ฆ๐‘›=๐›ฝ๐‘›๐‘†๐‘ฅ๐‘›+๎€ท1โˆ’๐›ฝ๐‘›๎€ธ๐‘ฅ๐‘›,๐‘ฅ๐‘›+1=๐‘ƒ๐ถ๎€บ๐›ผ๐‘›๎€ท๐‘ฅ๐›พ๐‘“๐‘›๎€ธ+๎€ท๐ผโˆ’๐›ผ๐‘›๐ด๎€ธ๐‘‡๐‘ฆ๐‘›๎€ป,โˆ€๐‘›โ‰ฅ1.(1.18) We will prove in Section 3 that if the sequences {๐›ผ๐‘›} and {๐›ฝ๐‘›} of parameters satisfy appropriate conditions and lim๐‘›โ†’โˆž(๐›ฝ๐‘›/๐›ผ๐‘›)=๐œโˆˆ(0,โˆž) then the sequence {๐‘ฅ๐‘›} generated by (1.18) converges strongly to the unique solution ฬƒ๐‘ฅ in ๐ป of the following variational inequality ๎‚ฌ1ฬƒ๐‘ฅโˆˆ๐น(๐‘‡),๐œ๎‚ญ(๐ดโˆ’๐›พ๐‘“)ฬƒ๐‘ฅ+(๐ผโˆ’๐‘†)ฬƒ๐‘ฅ,๐‘ฅโˆ’ฬƒ๐‘ฅโ‰ฅ0,โˆ€๐‘ฅโˆˆ๐น(๐‘‡).(1.19) In particular, if we take ๐œ=0, under suitable difference assumptions on parameter, then the sequence {๐‘ฅ๐‘›} generated by (1.18) converges strongly to the unique solution ฬƒ๐‘ฅ in ๐ป of the following variational inequality ฬƒ๐‘ฅโˆˆ๐น(๐‘‡),โŸจ(๐ดโˆ’๐›พ๐‘“)ฬƒ๐‘ฅ,๐‘ฅโˆ’ฬƒ๐‘ฅโŸฉโ‰ฅ0,โˆ€๐‘ฅโˆˆ๐น(๐‘‡).(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 ๐ป be a Hilbert space, ๐ถ be a closed convex subset of ๐ป, and ๐‘‡โˆถ๐ถโ†’๐ถ be a nonexpansive mapping with ๐น(๐‘‡)โ‰ โˆ…. If {๐‘ฅ๐‘›} is a sequence in ๐ถ weakly converging to ๐‘ฅ and if {(๐ผโˆ’๐‘‡)๐‘ฅ๐‘›} converges strongly to ๐‘ฆ, then (๐ผโˆ’๐‘‡)๐‘ฅ=๐‘ฆ; in particular, if ๐‘ฆ=0 then ๐‘ฅโˆˆ๐น(๐‘‡).

Lemma 2.2. Let ๐‘ฅโˆˆ๐ป and ๐‘งโˆˆ๐ถ be any points. Then one has the following:
(1) That ๐‘ง=๐‘ƒ๐ถ[๐‘ฅ] if and only if there holds the relation: โŸจ๐‘ฅโˆ’๐‘ง,๐‘ฆโˆ’๐‘งโŸฉโ‰ค0,โˆ€๐‘ฆโˆˆ๐ถ.(2.1)(2) That ๐‘ง=๐‘ƒ๐ถ[๐‘ฅ] if and only if there holds the relation: โ€–๐‘ฅโˆ’๐‘งโ€–2โ‰คโ€–๐‘ฅโˆ’๐‘ฆโ€–2โˆ’โ€–๐‘ฆโˆ’๐‘งโ€–2,โˆ€๐‘ฆโˆˆ๐ถ.(2.2)(3) There holds the relation: โŸจ๐‘ƒ๐ถ[๐‘ฅ]โˆ’๐‘ƒ๐ถ[๐‘ฆ]โ€–โ€–๐‘ƒ,๐‘ฅโˆ’๐‘ฆโŸฉโ‰ฅ๐ถ[๐‘ฅ]โˆ’๐‘ƒ๐ถ[๐‘ฆ]โ€–โ€–2,โˆ€๐‘ฅ,๐‘ฆโˆˆ๐ป.(2.3) Consequently, ๐‘ƒ๐ถ is nonexpansive and monotone.

Lemma 2.3 (Marino and Xu [8]). Let ๐ป be a Hilbert space, ๐ถ be a closed convex subset of ๐ป, ๐‘“โˆถ๐ถโ†’๐ป be a contraction with coefficient 0<๐œŒ<1, and ๐‘‡โˆถ๐ถโ†’๐ถ be nonexpansive mapping. Let ๐ด be a strongly positive linear bounded operator on a Hilbert space ๐ป with coefficient โˆ’๐›พ>0. Then, for 0<๐›พ<โˆ’๐›พ/๐œŒ, for ๐‘ฅ,๐‘ฆโˆˆ๐ถ, (1)the mapping (๐ผโˆ’๐‘“) is strongly monotone with coefficient (1โˆ’๐œŒ), that is, โŸจ๐‘ฅโˆ’๐‘ฆ,(๐ผโˆ’๐‘“)๐‘ฅโˆ’(๐ผโˆ’๐‘“)๐‘ฆโŸฉโ‰ฅ(1โˆ’๐œŒ)โ€–๐‘ฅโˆ’๐‘ฆโ€–2,(2.4)(2)the mapping (๐ดโˆ’๐›พ๐‘“) is strongly monotone with coefficient โˆ’๐›พโˆ’๐›พ๐œŒ that is ๎‚€โŸจ๐‘ฅโˆ’๐‘ฆ,(๐ดโˆ’๐›พ๐‘“)๐‘ฅโˆ’(๐ดโˆ’๐›พ๐‘“)๐‘ฆโŸฉโ‰ฅโˆ’๎‚โ€–๐›พโˆ’๐›พ๐œŒ๐‘ฅโˆ’๐‘ฆโ€–2.(2.5)

Lemma 2.4 (Xu [4]). Assume that {๐‘Ž๐‘›} is a sequence of nonnegative numbers such that ๐‘Ž๐‘›+1โ‰ค๎€ท1โˆ’๐›พ๐‘›๎€ธ๐‘Ž๐‘›+๐›ฟ๐‘›,โˆ€๐‘›โ‰ฅ0,(2.6) where {๐›พ๐‘›} is a sequence in (0,1) and {๐›ฟ๐‘›} is a sequence in โ„ such that (1)โˆ‘โˆž๐‘›=1๐›พ๐‘›=โˆž, (2)limsup๐‘›โ†’โˆž(๐›ฟ๐‘›/๐›พ๐‘›)โ‰ค0 or โˆ‘โˆž๐‘›=1|๐›ฟ๐‘›|<โˆž. Then lim๐‘›โ†’โˆž๐‘Ž๐‘›=0.

Lemma 2.5 (Marino and Xu [8]). Assume ๐ด is a strongly positive linear bounded operator on a Hilbert space ๐ป with coefficient โˆ’๐›พ>0 and 0<๐›ผโ‰คโ€–๐ดโ€–โˆ’1. Then โ€–๐ผโˆ’๐›ผ๐ดโ€–โ‰ค1โˆ’๐›ผโˆ’๐›พ.

Lemma 2.6 (Acedo and Xu [11]). Let ๐ถ be a closed convex subset of ๐ป. Let {๐‘ฅ๐‘›} be a bounded sequence in ๐ป. Assume that (1)The weak ๐œ”-limit set ๐œ”๐‘ค(๐‘ฅ๐‘›)โŠ‚๐ถ,(2)For each ๐‘งโˆˆ๐ถ, lim๐‘›โ†’โˆžโ€–๐‘ฅ๐‘›โˆ’๐‘งโ€– exists. Then {๐‘ฅ๐‘›} is weakly convergent to a point in ๐ถ.

NotationWe use โ†’ for strong convergence and โ‡€ for weak convergence.

3. Main Results

Theorem 3.1. Let ๐ถ be a nonempty closed convex subset of a real Hilbert space ๐ป. Let ๐‘“โˆถ๐ถโ†’๐ป be a ๐œŒ-contraction with ๐œŒโˆˆ(0,1). Let ๐‘†,๐‘‡โˆถ๐ถโ†’๐ถ be two nonexpansive mappings with ๐น(๐‘‡)โ‰ โˆ…. Let ๐ด be a strongly positive linear bounded operator on ๐ป with coefficient โˆ’๐›พ>0. {๐›ผ๐‘›} and {๐›ฝ๐‘›} are two sequences in (0,1) and 0<๐›พ<โˆ’๐›พ/๐œŒ. Starting with an arbitrary initial guess ๐‘ฅ0โˆˆ๐ถ and {๐‘ฅ๐‘›} is a sequence generated by ๐‘ฆ๐‘›=๐›ฝ๐‘›๐‘†๐‘ฅ๐‘›+๎€ท1โˆ’๐›ฝ๐‘›๎€ธ๐‘ฅ๐‘›,๐‘ฅ๐‘›+1=๐‘ƒ๐ถ๎€บ๐›ผ๐‘›๎€ท๐‘ฅ๐›พ๐‘“๐‘›๎€ธ+๎€ท๐ผโˆ’๐›ผ๐‘›๐ด๎€ธ๐‘‡๐‘ฆ๐‘›๎€ป,โˆ€๐‘›โ‰ฅ1.(3.1)
Suppose that the following conditions are satisfied: (C1)โ€‰โ€‰lim๐‘›โ†’โˆž๐›ผ๐‘›=0 and โˆ‘โˆž๐‘›=1๐›ผ๐‘›=โˆž, (C2)โ€‰โ€‰lim๐‘›โ†’โˆž(๐›ฝ๐‘›/๐›ผ๐‘›)=๐œ=0, (C3)โ€‰โ€‰lim๐‘›โ†’โˆž(|๐›ผ๐‘›โˆ’๐›ผ๐‘›โˆ’1|/๐›ผ๐‘›)=0 and lim๐‘›โ†’โˆž(|๐›ฝ๐‘›โˆ’๐›ฝ๐‘›โˆ’1|/๐›ฝ๐‘›)=0, or (C4)โ€‰โ€‰โˆ‘โˆž๐‘›=1|๐›ผ๐‘›โˆ’๐›ผ๐‘›โˆ’1|<โˆž and โˆ‘โˆž๐‘›=1|๐›ฝ๐‘›โˆ’๐›ฝ๐‘›โˆ’1|<โˆž. Then the sequence {๐‘ฅ๐‘›} converges strongly to a point ฬƒ๐‘ฅโˆˆ๐ป, which is the unique solution of the variational inequality: ฬƒ๐‘ฅโˆˆ๐น(๐‘‡),โŸจ(๐ดโˆ’๐›พ๐‘“)ฬƒ๐‘ฅ,๐‘ฅโˆ’ฬƒ๐‘ฅโŸฉโ‰ฅ0,โˆ€๐‘ฅโˆˆ๐น(๐‘‡).(3.2) Equivalently, one has ๐‘ƒ๐น(๐‘‡)(๐ผโˆ’๐ด+๐›พ๐‘“)ฬƒ๐‘ฅ=ฬƒ๐‘ฅ.

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 ๐ดโˆ’๐›พ๐‘“. Suppose โˆ’๐‘ฅโˆˆ๐น(๐‘‡) and ฬƒ๐‘ฅโˆˆ๐น(๐‘‡) both are solutions to (3.2), then โŸจ(๐ดโˆ’๐›พ๐‘“)โˆ’๐‘ฅ,โˆ’๐‘ฅโˆ’ฬƒ๐‘ฅโŸฉโ‰ค0 and โŸจ(๐ดโˆ’๐›พ๐‘“)ฬƒ๐‘ฅ,ฬƒ๐‘ฅโˆ’โˆ’๐‘ฅโŸฉโ‰ค0. It follows that ๎‚ฌ(๐ดโˆ’๐›พ๐‘“)โˆ’๐‘ฅ,โˆ’๐‘ฅ๎‚ญ+๎‚ฌโˆ’ฬƒ๐‘ฅ(๐ดโˆ’๐›พ๐‘“)ฬƒ๐‘ฅ,ฬƒ๐‘ฅโˆ’โˆ’๐‘ฅ๎‚ญ=๎‚ฌ(๐ดโˆ’๐›พ๐‘“)โˆ’๐‘ฅ,โˆ’๐‘ฅ๎‚ญโˆ’๎‚ฌโˆ’ฬƒ๐‘ฅ(๐ดโˆ’๐›พ๐‘“)ฬƒ๐‘ฅ,โˆ’๐‘ฅ๎‚ญโˆ’ฬƒ๐‘ฅ=โŸจ(๐ดโˆ’๐›พ๐‘“)โˆ’๐‘ฅโˆ’(๐ดโˆ’๐›พ๐‘“)ฬƒ๐‘ฅ,โˆ’๐‘ฅโˆ’ฬƒ๐‘ฅโŸฉโ‰ค0.(3.3) The strong monotonicity of ๐ดโˆ’๐›พ๐‘“ (Lemma 2.3) implies that โˆ’๐‘ฅ=ฬƒ๐‘ฅ and the uniqueness is proved.
Next, we prove that the sequence {๐‘ฅ๐‘›} is bounded. Since ๐›ผ๐‘›โ†’0 and lim๐‘›โ†’โˆž(๐›ฝ๐‘›/๐›ผ๐‘›)=0 by condition (C1) and (C2), respectively, we can assume, without loss of generality, that ๐›ผ๐‘›<โ€–๐ดโ€–โˆ’1 and ๐›ฝ๐‘›<๐›ผ๐‘› for all ๐‘›โ‰ฅ1. Take ๐‘ขโˆˆ๐น(๐‘‡) and from (3.1), we have โ€–โ€–๐‘ฅ๐‘›+1โ€–โ€–=โ€–โ€–๐‘ƒโˆ’๐‘ข๐ถ๎€บ๐›ผ๐‘›๎€ท๐‘ฅ๐›พ๐‘“๐‘›๎€ธ+๎€ท๐ผโˆ’๐›ผ๐‘›๐ด๎€ธ๐‘‡๐‘ฆ๐‘›๎€ปโˆ’๐‘ƒ๐ถ[๐‘ข]โ€–โ€–โ‰คโ€–โ€–๐›ผ๐‘›๎€ท๐‘ฅ๐›พ๐‘“๐‘›๎€ธ+๎€ท๐ผโˆ’๐›ผ๐‘›๐ด๎€ธ๐‘‡๐‘ฆ๐‘›โ€–โ€–โˆ’๐‘ขโ‰ค๐›ผ๐‘›๐›พโ€–โ€–๐‘“๎€ท๐‘ฅ๐‘›๎€ธโ€–โ€–โˆ’๐‘“(๐‘ข)+๐›ผ๐‘›โ€–โ€–๎€ทโ€–๐›พ๐‘“(๐‘ข)โˆ’๐ด๐‘ขโ€–+๐ผโˆ’๐›ผ๐‘›๐ด๎€ธ๎€ท๐‘‡๐‘ฆ๐‘›๎€ธโ€–โ€–.โˆ’๐‘ข(3.4) Since โ€–๐ผโˆ’๐›ผ๐‘›๐ดโ€–โ‰ค1โˆ’๐›ผ๐‘›โˆ’๐›พ and by Lemma 2.5, we note that โ€–โ€–๐‘ฅ๐‘›+1โ€–โ€–โˆ’๐‘ขโ‰ค๐›ผ๐‘›๐›พโ€–โ€–๐‘“๎€ท๐‘ฅ๐‘›๎€ธโ€–โ€–โˆ’๐‘“(๐‘ข)+๐›ผ๐‘›โ€–๎‚€๐›พ๐‘“(๐‘ข)โˆ’๐ด๐‘ขโ€–+1โˆ’๐›ผ๐‘›โˆ’๐›พ๎‚โ€–โ€–๐‘‡๐‘ฆ๐‘›โ€–โ€–โˆ’๐‘ขโ‰ค๐›ผ๐‘›โ€–โ€–๐‘ฅ๐›พ๐œŒ๐‘›โ€–โ€–โˆ’๐‘ข+๐›ผ๐‘›๎‚€โ€–๐›พ๐‘“(๐‘ข)โˆ’๐ด๐‘ขโ€–+1โˆ’๐›ผ๐‘›โˆ’๐›พ๎‚โ€–โ€–๐‘‡๐‘ฆ๐‘›โ€–โ€–โˆ’๐‘‡๐‘ขโ‰ค๐›ผ๐‘›โ€–โ€–๐‘ฅ๐›พ๐œŒ๐‘›โ€–โ€–โˆ’๐‘ข+๐›ผ๐‘›โ€–๎‚€๐›พ๐‘“(๐‘ข)โˆ’๐ด๐‘ขโ€–+1โˆ’๐›ผ๐‘›โˆ’๐›พ๎‚โ€–โ€–๐‘ฆ๐‘›โ€–โ€–โˆ’๐‘ขโ‰ค๐›ผ๐‘›โ€–โ€–๐‘ฅ๐›พ๐œŒ๐‘›โ€–โ€–โˆ’๐‘ข+๐›ผ๐‘›+๎‚€โ€–๐›พ๐‘“(๐‘ข)โˆ’๐ด๐‘ขโ€–1โˆ’๐›ผ๐‘›โˆ’๐›พ๎‚๎€บ๐›ฝ๐‘›โ€–โ€–๐‘†๐‘ฅ๐‘›โ€–โ€–โˆ’๐‘†๐‘ข+๐›ฝ๐‘›๎€ทโ€–๐‘†๐‘ขโˆ’๐‘ขโ€–+1โˆ’๐›ฝ๐‘›๎€ธโ€–โ€–๐‘ฅ๐‘›โ€–โ€–๎€ปโˆ’๐‘ขโ‰ค๐›ผ๐‘›โ€–โ€–๐‘ฅ๐›พ๐œŒ๐‘›โ€–โ€–โˆ’๐‘ข+๐›ผ๐‘›โ€–+๎‚€โ€–๐›พ๐‘“(๐‘ข)โˆ’๐ด๐‘ข1โˆ’๐›ผ๐‘›โˆ’๐›พ๎‚๎€บ๐›ฝ๐‘›โ€–โ€–๐‘ฅ๐‘›โ€–โ€–โˆ’๐‘ข+๐›ฝ๐‘›๎€ทโ€–๐‘†๐‘ขโˆ’๐‘ขโ€–+1โˆ’๐›ฝ๐‘›๎€ธโ€–โ€–๐‘ฅ๐‘›โ€–โ€–๎€ป=๎‚€โˆ’๐‘ข1โˆ’๐›ผ๐‘›๎‚€โˆ’โ€–โ€–๐‘ฅ๐›พโˆ’๐›พ๐œŒ๎‚๎‚๐‘›โ€–โ€–โˆ’๐‘ข+๐›ผ๐‘›๎‚€โ€–๐›พ๐‘“(๐‘ข)โˆ’๐ด๐‘ขโ€–+1โˆ’๐›ผ๐‘›โˆ’๐›พ๎‚๐›ฝ๐‘›โ‰ค๎‚€โ€–๐‘†๐‘ขโˆ’๐‘ขโ€–1โˆ’๐›ผ๐‘›๎‚€โˆ’โ€–โ€–๐‘ฅ๐›พโˆ’๐›พ๐œŒ๎‚๎‚๐‘›โ€–โ€–โˆ’๐‘ข+๐›ผ๐‘›โ€–๐›พ๐‘“(๐‘ข)โˆ’๐ด๐‘ขโ€–+๐›ฝ๐‘›โ‰ค๎‚€โ€–๐‘†๐‘ขโˆ’๐‘ขโ€–1โˆ’๐›ผ๐‘›๎‚€โˆ’โ€–โ€–๐‘ฅ๐›พโˆ’๐›พ๐œŒ๎‚๎‚๐‘›โ€–โ€–โˆ’๐‘ข+๐›ผ๐‘›โ€–๐›พ๐‘“(๐‘ข)โˆ’๐ด๐‘ขโ€–+๐›ผ๐‘›=๎‚€โ€–๐‘†๐‘ขโˆ’๐‘ขโ€–1โˆ’๐›ผ๐‘›๎‚€โˆ’โ€–โ€–๐‘ฅ๐›พโˆ’๐›พ๐œŒ๎‚๎‚๐‘›โ€–โ€–โˆ’๐‘ข+๐›ผ๐‘›[(]=๎‚€โ€–๐›พ๐‘“๐‘ข)โˆ’๐ด๐‘ขโ€–+โ€–๐‘†๐‘ขโˆ’๐‘ขโ€–1โˆ’๐›ผ๐‘›๎‚€โˆ’โ€–โ€–๐‘ฅ๐›พโˆ’๐›พ๐œŒ๎‚๎‚๐‘›โ€–โ€–โˆ’๐‘ข+๐›ผ๐‘›๎‚€โˆ’๎‚๐›พโˆ’๐›พ๐œŒโ€–๐›พ๐‘“(๐‘ข)โˆ’๐ด๐‘ขโ€–+โ€–๐‘†๐‘ขโˆ’๐‘ขโ€–๎‚€โˆ’๎‚.๐›พโˆ’๐›พ๐œŒ(3.5) By induction, we can obtain โ€–โ€–๐‘ฅ๐‘›+1โ€–โ€–โŽงโŽชโŽจโŽชโŽฉโ€–โ€–๐‘ฅโˆ’๐‘ขโ‰คmax0โ€–โ€–,โˆ’๐‘ขโ€–๐›พ๐‘“(๐‘ข)โˆ’๐ด๐‘ขโ€–+โ€–๐‘†๐‘ขโˆ’๐‘ขโ€–๎‚€โˆ’๎‚โŽซโŽชโŽฌโŽชโŽญ,๐›พโˆ’๐›พ๐œŒ(3.6) which implies that the sequence {๐‘ฅ๐‘›} is bounded and so are the sequences {๐‘“(๐‘ฅ๐‘›)}, {๐‘†๐‘ฅ๐‘›}, and {๐ด๐‘‡๐‘ฆ๐‘›}.
Set ๐‘ค๐‘›โˆถ=๐›ผ๐‘›๐›พ๐‘“(๐‘ฅ๐‘›)+(๐ผโˆ’๐›ผ๐‘›๐ด)๐‘‡๐‘ฆ๐‘›,๐‘›โ‰ฅ1. We get โ€–โ€–๐‘ฅ๐‘›+1โˆ’๐‘ฅ๐‘›โ€–โ€–=โ€–โ€–๐‘ƒ๐ถ๎€บ๐‘ค๐‘›+1๎€ปโˆ’๐‘ƒ๐ถ๎€บ๐‘ค๐‘›๎€ปโ€–โ€–โ‰คโ€–โ€–๐‘ค๐‘›+1โˆ’๐‘ค๐‘›โ€–โ€–.(3.7) It follows that โ€–โ€–๐‘ฅ๐‘›+1โˆ’๐‘ฅ๐‘›โ€–โ€–โ‰คโ€–โ€–๎€ท๐›ผ๐‘›๎€ท๐‘ฅ๐›พ๐‘“๐‘›๎€ธ+๎€ท๐ผโˆ’๐›ผ๐‘›๐ด๎€ธ๐‘‡๐‘ฆ๐‘›๎€ธโˆ’๎€ท๐›ผ๐‘›โˆ’1๎€ท๐‘ฅ๐›พ๐‘“๐‘›โˆ’1๎€ธ+๎€ท๐ผโˆ’๐›ผ๐‘›โˆ’1๐ด๎€ธ๐‘‡๐‘ฆ๐‘›โˆ’1๎€ธโ€–โ€–โ‰ค๐›ผ๐‘›๐›พโ€–โ€–๐‘“๎€ท๐‘ฅ๐‘›๎€ธ๎€ท๐‘ฅโˆ’๐‘“๐‘›โˆ’1๎€ธโ€–โ€–+||๐›ผ๐‘›โˆ’๐›ผ๐‘›โˆ’1||โ€–โ€–๎€ท๐‘ฅ๐›พ๐‘“๐‘›โˆ’1๎€ธโˆ’๐ด๐‘‡๐‘ฆ๐‘›โˆ’1โ€–โ€–+๎‚€1โˆ’๐›ผ๐‘›โˆ’๐›พ๎‚โ€–โ€–๐‘‡๐‘ฆ๐‘›โˆ’๐‘‡๐‘ฆ๐‘›โˆ’1โ€–โ€–โ‰ค๐›ผ๐‘›โ€–โ€–๐‘ฅ๐›พ๐œŒ๐‘›โˆ’๐‘ฅ๐‘›โˆ’1โ€–โ€–+||๐›ผ๐‘›โˆ’๐›ผ๐‘›โˆ’1||โ€–โ€–๎€ท๐‘ฅ๐›พ๐‘“๐‘›โˆ’1๎€ธโˆ’๐ด๐‘‡๐‘ฆ๐‘›โˆ’1โ€–โ€–+๎‚€1โˆ’๐›ผ๐‘›โˆ’๐›พ๎‚โ€–โ€–๐‘ฆ๐‘›โˆ’๐‘ฆ๐‘›โˆ’1โ€–โ€–.(3.8) By (3.7) and (3.8), we get โ€–โ€–๐‘ฅ๐‘›+1โˆ’๐‘ฅ๐‘›โ€–โ€–โ‰ค๐›ผ๐‘›โ€–โ€–๐‘ค๐›พ๐œŒ๐‘›โˆ’๐‘ค๐‘›โˆ’1โ€–โ€–+||๐›ผ๐‘›โˆ’๐›ผ๐‘›โˆ’1||โ€–โ€–๎€ท๐‘ฅ๐›พ๐‘“๐‘›โˆ’1๎€ธโˆ’๐ด๐‘‡๐‘ฆ๐‘›โˆ’1โ€–โ€–+๎‚€1โˆ’๐›ผ๐‘›โˆ’๐›พ๎‚โ€–โ€–๐‘ฆ๐‘›โˆ’๐‘ฆ๐‘›โˆ’1โ€–โ€–.(3.9) From (3.1), we obtain โ€–โ€–๐‘ฆ๐‘›โˆ’๐‘ฆ๐‘›โˆ’1โ€–โ€–=โ€–โ€–๎€ท๐›ฝ๐‘›๐‘†๐‘ฅ๐‘›+๎€ท1โˆ’๐›ฝ๐‘›๎€ธ๐‘ฅ๐‘›๎€ธโˆ’๎€ท๐›ฝ๐‘›โˆ’1๐‘†๐‘ฅ๐‘›โˆ’1+๎€ท1โˆ’๐›ฝ๐‘›โˆ’1๎€ธ๐‘ฅ๐‘›โˆ’1๎€ธโ€–โ€–=โ€–โ€–๐›ฝ๐‘›๎€ท๐‘†๐‘ฅ๐‘›โˆ’๐‘†๐‘ฅ๐‘›โˆ’1๎€ธ+๎€ท๐›ฝ๐‘›โˆ’๐›ฝ๐‘›โˆ’1๎€ธ๎€ท๐‘†๐‘ฅ๐‘›โˆ’1โˆ’๐‘ฅ๐‘›โˆ’1๎€ธ+๎€ท1โˆ’๐›ฝ๐‘›๐‘ฅ๎€ธ๎€ท๐‘›โˆ’๐‘ฅ๐‘›โˆ’1๎€ธโ€–โ€–โ‰คโ€–โ€–๐‘ฅ๐‘›โˆ’๐‘ฅ๐‘›โˆ’1โ€–โ€–+||๐›ฝ๐‘›โˆ’๐›ฝ๐‘›โˆ’1||โ€–โ€–๐‘†๐‘ฅ๐‘›โˆ’1โˆ’๐‘ฅ๐‘›โˆ’1โ€–โ€–โ‰คโ€–โ€–๐‘ฅ๐‘›โˆ’๐‘ฅ๐‘›โˆ’1โ€–โ€–+||๐›ฝ๐‘›โˆ’๐›ฝ๐‘›โˆ’1||๐‘€,(3.10) where ๐‘€ is a constant such that sup๐‘›โˆˆโ„•๎€ฝโ€–โ€–๎€ท๐‘ฅ๐›พ๐‘“๐‘›โˆ’1๎€ธโˆ’๐ด๐‘‡๐‘ฆ๐‘›โˆ’1โ€–โ€–+โ€–โ€–๐‘†๐‘ฅ๐‘›โˆ’1โˆ’๐‘ฅ๐‘›โˆ’1โ€–โ€–๎€พโ‰ค๐‘€.(3.11) Substituting (3.10) into (3.8) to obtain โ€–โ€–๐‘ฅ๐‘›+1โˆ’๐‘ฅ๐‘›โ€–โ€–โ‰ค๐›ผ๐‘›โ€–โ€–๐‘ฅ๐›พ๐œŒ๐‘›โˆ’๐‘ฅ๐‘›โˆ’1โ€–โ€–+||๐›ผ๐‘›โˆ’๐›ผ๐‘›โˆ’1||โ€–โ€–๎€ท๐‘ฅ๐›พ๐‘“๐‘›โˆ’1๎€ธโˆ’๐ด๐‘‡๐‘ฆ๐‘›โˆ’1โ€–โ€–+๎‚€1โˆ’๐›ผ๐‘›โˆ’๐›พ๎‚๎€บโ€–โ€–๐‘ฅ๐‘›โˆ’๐‘ฅ๐‘›โˆ’1โ€–โ€–+||๐›ฝ๐‘›โˆ’๐›ฝ๐‘›โˆ’1||๐‘€๎€ปโ‰ค๐›ผ๐‘›โ€–โ€–๐‘ฅ๐›พ๐œŒ๐‘›โˆ’๐‘ฅ๐‘›โˆ’1โ€–โ€–+||๐›ผ๐‘›โˆ’๐›ผ๐‘›โˆ’1||๐‘€+๎‚€1โˆ’๐›ผ๐‘›โˆ’๐›พ๎‚๎€บโ€–โ€–๐‘ฅ๐‘›โˆ’๐‘ฅ๐‘›โˆ’1โ€–โ€–+||๐›ฝ๐‘›โˆ’๐›ฝ๐‘›โˆ’1||๐‘€๎€ป=๎‚€1โˆ’๐›ผ๐‘›๎‚€โˆ’โ€–โ€–๐‘ฅ๐›พโˆ’๐›พ๐œŒ๎‚๎‚๐‘›โˆ’๐‘ฅ๐‘›โˆ’1โ€–โ€–๎€บ||๐›ผ+๐‘€๐‘›โˆ’๐›ผ๐‘›โˆ’1||+||๐›ฝ๐‘›โˆ’๐›ฝ๐‘›โˆ’1||๎€ปโ‰ค๎‚€1โˆ’๐›ผ๐‘›๎‚€โˆ’โ€–โ€–๐‘ค๐›พโˆ’๐›พ๐œŒ๎‚๎‚๐‘›โˆ’๐‘ค๐‘›โˆ’1โ€–โ€–๎€บ||๐›ผ+๐‘€๐‘›โˆ’๐›ผ๐‘›โˆ’1||+||๐›ฝ๐‘›โˆ’๐›ฝ๐‘›โˆ’1||๎€ป.(3.12) At the same time, we can write (3.12) as โ€–โ€–๐‘ฅ๐‘›+1โˆ’๐‘ฅ๐‘›โ€–โ€–โ‰ค๎‚€1โˆ’๐›ผ๐‘›๎‚€โˆ’โ€–โ€–๐‘ค๐›พโˆ’๐›พ๐œŒ๎‚๎‚๐‘›โˆ’๐‘ค๐‘›โˆ’1โ€–โ€–+๐‘€๐›ผ๐‘›๎‚ธ||๐›ผ๐‘›โˆ’๐›ผ๐‘›โˆ’1||๐›ผ๐‘›+||๐›ฝ๐‘›โˆ’๐›ฝ๐‘›โˆ’1||๐›ผ๐‘›๎‚นโ‰ค๎‚€1โˆ’๐›ผ๐‘›๎‚€โˆ’โ€–โ€–๐‘ค๐›พโˆ’๐›พ๐œŒ๎‚๎‚๐‘›โˆ’๐‘ค๐‘›โˆ’1โ€–โ€–+๐‘€๐›ผ๐‘›๎‚ธ||๐›ผ๐‘›โˆ’๐›ผ๐‘›โˆ’1||๐›ฝ๐‘›+||๐›ฝ๐‘›โˆ’๐›ฝ๐‘›โˆ’1||๐›ฝ๐‘›๎‚น.(3.13) From (3.12), (C4), and Lemma 2.5 or from (3.13), (C3), and Lemma 2.5, we can deduce that โ€–๐‘ฅ๐‘›+1โˆ’๐‘ฅ๐‘›โ€–โ†’0, respectively.
From (3.1), we have โ€–โ€–๐‘ฅ๐‘›โˆ’๐‘‡๐‘ฅ๐‘›โ€–โ€–โ‰คโ€–โ€–๐‘ฅ๐‘›โˆ’๐‘ฅ๐‘›+1โ€–โ€–+โ€–โ€–๐‘ฅ๐‘›+1โˆ’๐‘‡๐‘ฅ๐‘›โ€–โ€–=โ€–โ€–๐‘ฅ๐‘›โˆ’๐‘ฅ๐‘›+1โ€–โ€–+โ€–โ€–๐‘ƒ๐ถ๎€บ๐‘ค๐‘›๎€ปโˆ’๐‘ƒ๐ถ๎€บ๐‘‡๐‘ฅ๐‘›๎€ปโ€–โ€–โ‰คโ€–โ€–๐‘ฅ๐‘›โˆ’๐‘ฅ๐‘›+1โ€–โ€–+โ€–โ€–๐‘ค๐‘›โˆ’๐‘‡๐‘ฅ๐‘›โ€–โ€–=โ€–โ€–๐‘ฅ๐‘›โˆ’๐‘ฅ๐‘›+1โ€–โ€–+โ€–โ€–๐›ผ๐‘›๎€ท๐‘ฅ๐›พ๐‘“๐‘›๎€ธ+๎€ท๐ผโˆ’๐›ผ๐‘›๐ด๎€ธ๐‘‡๐‘ฆ๐‘›โˆ’๐‘‡๐‘ฅ๐‘›โ€–โ€–โ‰คโ€–โ€–๐‘ฅ๐‘›โˆ’๐‘ฅ๐‘›+1โ€–โ€–+๐›ผ๐‘›โ€–โ€–๎€ท๐‘ฅ๐›พ๐‘“๐‘›๎€ธโˆ’๐ด๐‘‡๐‘ฅ๐‘›โ€–โ€–+๎‚€1โˆ’๐›ผ๐‘›โˆ’๐›พ๎‚โ€–โ€–๐‘‡๐‘ฆ๐‘›โˆ’๐‘‡๐‘ฅ๐‘›โ€–โ€–โ‰คโ€–โ€–๐‘ฅ๐‘›โˆ’๐‘ฅ๐‘›+1โ€–โ€–+๐›ผ๐‘›โ€–โ€–๎€ท๐‘ฅ๐›พ๐‘“๐‘›๎€ธโˆ’๐ด๐‘‡๐‘ฅ๐‘›โ€–โ€–+๎‚€1โˆ’๐›ผ๐‘›โˆ’๐›พ๎‚โ€–โ€–๐‘ฆ๐‘›โˆ’๐‘ฅ๐‘›โ€–โ€–=โ€–โ€–๐‘ฅ๐‘›โˆ’๐‘ฅ๐‘›+1โ€–โ€–+๐›ผ๐‘›โ€–โ€–๎€ท๐‘ฅ๐›พ๐‘“๐‘›๎€ธโˆ’๐ด๐‘‡๐‘ฅ๐‘›โ€–โ€–+๎‚€1โˆ’๐›ผ๐‘›โˆ’๐›พ๎‚๐›ฝ๐‘›โ€–โ€–๐‘†๐‘ฅ๐‘›โˆ’๐‘ฅ๐‘›โ€–โ€–.(3.14) Notice that ๐›ผ๐‘›โ†’0, ๐›ฝ๐‘›โ†’0, and โ€–๐‘ฅ๐‘›+1โˆ’๐‘ฅ๐‘›โ€–โ†’0, so we obtain โ€–โ€–๐‘ฅ๐‘›โˆ’๐‘‡๐‘ฅ๐‘›โ€–โ€–โŸถ0.(3.15) Next, we prove limsup๐‘›โ†’โˆžโŸจ๐›พ๐‘“(๐‘ง)โˆ’๐ด๐‘ง,๐‘ฅ๐‘›โˆ’๐‘งโŸฉโ‰ค0,(3.16) where ๐‘ง=๐‘ƒ๐น(๐‘‡)(๐ผโˆ’๐ด+๐›พ๐‘“)๐‘ง. Since the sequence {๐‘ฅ๐‘›} is bounded we can take a subsequence {๐‘ฅ๐‘›๐‘˜} of {๐‘ฅ๐‘›} such that limsup๐‘›โ†’โˆžโŸจ๐›พ๐‘“(๐‘ง)โˆ’๐ด๐‘ง,๐‘ฅ๐‘›โˆ’๐‘งโŸฉ=lim๐‘˜โ†’โˆž๎ซ๐›พ๐‘“(๐‘ง)โˆ’๐ด๐‘ง,๐‘ฅ๐‘›๐‘˜๎ฌ,โˆ’๐‘ง(3.17) and ๐‘ฅ๐‘›๐‘˜โ‡€0๐‘ฅ00086ฬƒ๐‘ฅ. From (3.15) and by Lemma 2.1, it follows that ฬƒ๐‘ฅโˆˆ๐น(๐‘‡). Hence, by Lemma 2.2(1) that limsup๐‘›โ†’โˆžโŸจ๐›พ๐‘“(๐‘ง)โˆ’๐ด๐‘ง,๐‘ฅ๐‘›โˆ’๐‘งโŸฉ=lim๐‘˜โ†’โˆž๎ซ๐›พ๐‘“(๐‘ง)โˆ’๐ด๐‘ง,๐‘ฅ๐‘›๐‘˜๎ฌโˆ’๐‘ง=โŸจ๐›พ๐‘“(๐‘ง)โˆ’๐ด๐‘ง,ฬƒ๐‘ฅโˆ’๐‘งโŸฉ=โŸจ(๐ผโˆ’๐ด+๐›พ๐‘“)๐‘งโˆ’๐‘ง,ฬƒ๐‘ฅโˆ’๐‘งโŸฉโ‰ค0.(3.18) Now, by Lemma 2.2(1), we observe that ๎ซ๐‘ƒ๐ถ๎€บ๐‘ค๐‘›๎€ปโˆ’๐‘ค๐‘›,๐‘ƒ๐ถ๎€บ๐‘ค๐‘›๎€ป๎ฌโˆ’๐‘งโ‰ค0,(3.19) and so โ€–โ€–๐‘ฅ๐‘›+1โ€–โ€–โˆ’๐‘ง2=๎ซ๐‘ƒ๐ถ๎€บ๐‘ค๐‘›๎€ปโˆ’๐‘ง,๐‘ƒ๐ถ๎€บ๐‘ค๐‘›๎€ป๎ฌ=๎ซ๐‘ƒโˆ’๐‘ง๐ถ๎€บ๐‘ค๐‘›๎€ปโˆ’๐‘ค๐‘›,๐‘ƒ๐ถ๎€บ๐‘ค๐‘›๎€ป๎ฌ+๎ซ๐‘คโˆ’๐‘ง๐‘›โˆ’๐‘ง,๐‘ƒ๐ถ๎€บ๐‘ค๐‘›๎€ป๎ฌโ‰ค๎ซ๐‘คโˆ’๐‘ง๐‘›โˆ’๐‘ง,๐‘ƒ๐ถ๎€บ๐‘ค๐‘›๎€ป๎ฌ=๎ซ๐›ผโˆ’๐‘ง๐‘›๎€ท๐‘ฅ๐›พ๐‘“๐‘›๎€ธ+๎€ท๐ผโˆ’๐›ผ๐‘›๐ด๎€ธ๐‘‡๐‘ฆ๐‘›โˆ’๐‘ง,๐‘ฅ๐‘›+1๎ฌโˆ’๐‘งโ‰ค๐›ผ๐‘›๐›พโ€–โ€–๐‘“๎€ท๐‘ฅ๐‘›๎€ธโ€–โ€–โ€–โ€–๐‘ฅโˆ’๐‘“(๐‘ง)๐‘›+1โ€–โ€–โˆ’๐‘ง+๐›ผ๐‘›๎ซ๐›พ๐‘“(๐‘ง)โˆ’๐ด๐‘ง,๐‘ฅ๐‘›+1๎ฌ+๎‚€โˆ’๐‘ง1โˆ’๐›ผ๐‘›โˆ’๐›พ๎‚โ€–โ€–๐‘‡๐‘ฆ๐‘›โ€–โ€–โ€–โ€–๐‘ฅโˆ’๐‘ง๐‘›+1โ€–โ€–โˆ’๐‘งโ‰ค๐›ผ๐‘›โ€–โ€–๐‘ฅ๐›พ๐œŒ๐‘›โ€–โ€–โ€–โ€–๐‘ฅโˆ’๐‘ง๐‘›+1โ€–โ€–โˆ’๐‘ง+๐›ผ๐‘›๎ซ๐›พ๐‘“(๐‘ง)โˆ’๐ด๐‘ง,๐‘ฅ๐‘›+1๎ฌ+๎‚€โˆ’๐‘ง1โˆ’๐›ผ๐‘›โˆ’๐›พ๎‚โ€–โ€–๐‘ฆ๐‘›โ€–โ€–โ€–โ€–๐‘ฅโˆ’๐‘ง๐‘›+1โ€–โ€–โˆ’๐‘ง=๐›ผ๐‘›โ€–โ€–๐‘ฅ๐›พ๐œŒ๐‘›โ€–โ€–โ€–โ€–๐‘ฅโˆ’๐‘ง๐‘›+1โ€–โ€–โˆ’๐‘ง+๐›ผ๐‘›๎ซ๐›พ๐‘“(๐‘ง)โˆ’๐ด๐‘ง,๐‘ฅ๐‘›+1๎ฌ+๎‚€โˆ’๐‘ง1โˆ’๐›ผ๐‘›โˆ’๐›พ๎‚โ€–โ€–๐›ฝ๐‘›๐‘†๐‘ฅ๐‘›+๎€ท1โˆ’๐›ฝ๐‘›๎€ธ๐‘ฅ๐‘›โ€–โ€–โ€–โ€–๐‘ฅโˆ’๐‘ง๐‘›+1โ€–โ€–โˆ’๐‘งโ‰ค๐›ผ๐‘›โ€–โ€–๐‘ฅ๐›พ๐œŒ๐‘›โ€–โ€–โ€–โ€–๐‘ฅโˆ’๐‘ง๐‘›+1โ€–โ€–โˆ’๐‘ง+๐›ผ๐‘›๎ซ๐›พ๐‘“(๐‘ง)โˆ’๐ด๐‘ง,๐‘ฅ๐‘›+1๎ฌ+๎‚€โˆ’๐‘ง1โˆ’๐›ผ๐‘›โˆ’๐›พ๎‚๎€บ๐›ฝ๐‘›โ€–โ€–๐‘†๐‘ฅ๐‘›โ€–โ€–โˆ’๐‘†๐‘ง+๐›ฝ๐‘›โ€–๎€ท๐‘†๐‘งโˆ’๐‘งโ€–+1โˆ’๐›ฝ๐‘›๎€ธโ€–โ€–๐‘ฅ๐‘›โ€–โ€–๎€ปโˆ’๐‘งโ€–๐‘ฅ๐‘›+1โˆ’๐‘งโ€–โ‰ค๐›ผ๐‘›โ€–โ€–๐‘ฅ๐›พ๐œŒ๐‘›โ€–โ€–โ€–โ€–๐‘ฅโˆ’๐‘ง๐‘›+1โ€–โ€–โˆ’๐‘ง+๐›ผ๐‘›๎ซ๐›พ๐‘“(๐‘ง)โˆ’๐ด๐‘ง,๐‘ฅ๐‘›+1๎ฌ+๎‚€โˆ’๐‘ง1โˆ’๐›ผ๐‘›โˆ’๐›พ๎‚๎€บ๐›ฝ๐‘›โ€–โ€–๐‘ฅ๐‘›โ€–โ€–โˆ’๐‘ง+๐›ฝ๐‘›โ€–๎€ท๐‘†๐‘งโˆ’๐‘งโ€–+1โˆ’๐›ฝ๐‘›๎€ธโ€–โ€–๐‘ฅ๐‘›โ€–โ€–๎€ปโ€–โ€–๐‘ฅโˆ’๐‘ง๐‘›+1โ€–โ€–=๎‚€โˆ’๐‘ง1โˆ’๐›ผ๐‘›๎‚€โˆ’โ€–โ€–๐‘ฅ๐›พโˆ’๐›พ๐œŒ๎‚๎‚๐‘›โ€–โ€–โ€–โ€–๐‘ฅโˆ’๐‘ง๐‘›+1โ€–โ€–โˆ’๐‘ง+๐›ผ๐‘›๎ซ๐›พ๐‘“(๐‘ง)โˆ’๐ด๐‘ง,๐‘ฅ๐‘›+1๎ฌ+๎‚€โˆ’๐‘ง1โˆ’๐›ผ๐‘›โˆ’๐›พ๎‚๐›ฝ๐‘›โ€–โ€–๐‘ฅโ€–๐‘†๐‘งโˆ’๐‘งโ€–๐‘›+1โ€–โ€–โ‰ค๎‚ƒโˆ’๐‘ง1โˆ’๐›ผ๐‘›๎‚€โˆ’๐›พโˆ’๐›พ๐œŒ๎‚๎‚„2๎‚ƒโ€–โ€–๐‘ฅ๐‘›โ€–โ€–โˆ’๐‘ง2+โ€–โ€–๐‘ฅ๐‘›+1โ€–โ€–โˆ’๐‘ง2๎‚„+๐›ผ๐‘›๎ซ๐›พ๐‘“(๐‘ง)โˆ’๐ด๐‘ง,๐‘ฅ๐‘›+1๎ฌ+๎‚€โˆ’๐‘ง1โˆ’๐›ผ๐‘›โˆ’๐›พ๎‚๐›ฝ๐‘›โ€–โ€–๐‘ฅโ€–๐‘†๐‘งโˆ’๐‘งโ€–๐‘›+1โ€–โ€–.โˆ’๐‘ง(3.20) Hence, it follows that โ€–โ€–๐‘ฅ๐‘›+1โ€–โ€–โˆ’๐‘ง2โ‰ค1โˆ’๐›ผ๐‘›๎‚€โˆ’๎‚๐›พโˆ’๐›พ๐œŒ1+๐›ผ๐‘›๎‚€โˆ’๎‚โ€–โ€–๐‘ฅ๐›พโˆ’๐›พ๐œŒ๐‘›โ€–โ€–โˆ’๐‘ง2+2๐›ผ๐‘›1+๐›ผ๐‘›๎‚€โˆ’๎‚๎ซ๐›พโˆ’๐›พ๐œŒ๐›พ๐‘“(๐‘ง)โˆ’๐ด๐‘ง,๐‘ฅ๐‘›+1๎ฌ+2๎‚€โˆ’๐‘ง1โˆ’๐›ผ๐‘›โˆ’๐›พ๎‚๐›ฝ๐‘›1+๐›ผ๐‘›๎‚€โˆ’๎‚โ€–โ€–๐‘ฅ๐›พโˆ’๐›พ๐œŒโ€–๐‘†๐‘งโˆ’๐‘งโ€–๐‘›+1โ€–โ€–=โŽกโŽขโŽขโŽฃโˆ’๐‘ง2๐›ผ๐‘›๎‚€โˆ’๎‚๐›พโˆ’๐›พ๐œŒ1+๐›ผ๐‘›๎‚€โˆ’๎‚โŽคโŽฅโŽฅโŽฆโŽกโŽขโŽขโŽฃ1๐›พโˆ’๐›พ๐œŒ๐›ผ๐‘›๎‚€โˆ’๎‚๎ซ๐›พโˆ’๐›พ๐œŒ๐›พ๐‘“(๐‘ง)โˆ’๐ด๐‘ง,๐‘ฅ๐‘›+1๎ฌ+๐›ฝโˆ’๐‘ง๐‘›๎‚€1โˆ’๐›ผ๐‘›โˆ’๐›พ๎‚๐›ผ๐‘›๎‚€โˆ’๎‚โ€–โ€–โ€–๐‘ฅ๐›พโˆ’๐›พ๐œŒ๐‘†๐‘งโˆ’๐‘งโ€–๐‘›+1โ€–โ€–โŽคโŽฅโŽฅโŽฆร—โŽกโŽขโŽขโŽฃโˆ’๐‘ง1โˆ’2๐›ผ๐‘›๎‚€โˆ’๎‚๐›พโˆ’๐›พ๐œŒ1+๐›ผ๐‘›๎‚€โˆ’๎‚โŽคโŽฅโŽฅโŽฆโ€–โ€–๐‘ฅ๐›พโˆ’๐›พ๐œŒ๐‘›โ€–โ€–โˆ’๐‘ง2.(3.21) We observe that limsup๐‘›โ†’โˆžโŽกโŽขโŽขโŽฃ1๐›ผ๐‘›๎‚€โˆ’๎‚๎ซ๐›พโˆ’๐›พ๐œŒ๐›พ๐‘“(๐‘ง)โˆ’๐ด๐‘ง,๐‘ฅ๐‘›+1๎ฌ+๐›ฝโˆ’๐‘ง๐‘›๎‚€1โˆ’๐›ผ๐‘›โˆ’๐›พ๎‚๐›ผ๐‘›๎‚€โˆ’๎‚โ€–โ€–๐‘ฅ๐›พโˆ’๐›พ๐œŒโ€–๐‘†๐‘งโˆ’๐‘งโ€–๐‘›+1โ€–โ€–โŽคโŽฅโŽฅโŽฆโˆ’๐‘งโ‰ค0.(3.22) Thus, by Lemma 2.4, ๐‘ฅ๐‘›โ†’๐‘ง as ๐‘›โ†’โˆž. This is completes.

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

Corollary 3.2 (Yao et al. [9]). Let ๐ถ be a nonempty closed convex subset of a real Hilbert space ๐ป. Let ๐‘“โˆถ๐ถโ†’๐ป be a ๐œŒ-contraction (possibly nonself) with ๐œŒโˆˆ(0,1). Let ๐‘†,๐‘‡โˆถ๐ถโ†’๐ถ be two nonexpansive mappings with ๐น(๐‘‡)โ‰ โˆ…. โ€‰{๐›ผ๐‘›} and {๐›ฝ๐‘›} are two sequences in (0,1). Starting with an arbitrary initial guess ๐‘ฅ0โˆˆ๐ถ and {๐‘ฅ๐‘›} is a sequence generated by ๐‘ฆ๐‘›=๐›ฝ๐‘›๐‘†๐‘ฅ๐‘›+๎€ท1โˆ’๐›ฝ๐‘›๎€ธ๐‘ฅ๐‘›,๐‘ฅ๐‘›+1=๐‘ƒ๐ถ๎€บ๐›ผ๐‘›๐‘“๎€ท๐‘ฅ๐‘›๎€ธ+๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐‘‡๐‘ฆ๐‘›๎€ป,โˆ€๐‘›โ‰ฅ1.(3.23) Suppose that the following conditions are satisfied: (C1)โ€‰โ€‰lim๐‘›โ†’โˆž๐›ผ๐‘›=0 and โˆ‘โˆž๐‘›=1๐›ผ๐‘›=โˆž, (C2)โ€‰โ€‰lim๐‘›โ†’โˆž(๐›ฝ๐‘›/๐›ผ๐‘›)=0, (C3)โ€‰โ€‰lim๐‘›โ†’โˆž(|๐›ผ๐‘›โˆ’๐›ผ๐‘›โˆ’1|/๐›ผ๐‘›)=0 and lim๐‘›โ†’โˆž(|๐›ฝ๐‘›โˆ’๐›ฝ๐‘›โˆ’1|/๐›ฝ๐‘›)=0, or (C4)โ€‰โ€‰โˆ‘โˆž๐‘›=1|๐›ผ๐‘›โˆ’๐›ผ๐‘›โˆ’1|<โˆž and โˆ‘โˆž๐‘›=1|๐›ฝ๐‘›โˆ’๐›ฝ๐‘›โˆ’1|<โˆž. Then the sequence {๐‘ฅ๐‘›} converges strongly to a point ฬƒ๐‘ฅโˆˆ๐ป, which is the unique solution of the variational inequality: ฬƒ๐‘ฅโˆˆ๐น(๐‘‡),โŸจ(๐ผโˆ’๐‘“)ฬƒ๐‘ฅ,๐‘ฅโˆ’ฬƒ๐‘ฅโŸฉโ‰ฅ0,โˆ€๐‘ฅโˆˆ๐น(๐‘‡).(3.24) Equivalently, one has ๐‘ƒ๐น(๐‘‡)(๐‘“)ฬƒ๐‘ฅ=ฬƒ๐‘ฅ. In particular, if one takes ๐‘“=0, then the sequence {๐‘ฅ๐‘›} converges in norm to the Minimum norm fixed point ฬƒ๐‘ฅ of ๐‘‡, namely, the point ฬƒ๐‘ฅ is the unique solution to the quadratic minimization problem: ๐‘ง=argmin๐‘ฅโˆˆ๐น(๐‘‡)โ€–๐‘ฅโ€–2.(3.25)

Proof. As a matter of fact, if we take ๐ด=๐ผ and ๐›พ=1 in Theorem 3.1. This completes the proof.

Under different conditions on data we obtain the following result.

Theorem 3.3. Let ๐ถ be a nonempty closed convex subset of a real Hilbert space ๐ป. Let ๐‘“โˆถ๐ถโ†’๐ป be a ๐œŒ-contraction (possibly nonself) with ๐œŒโˆˆ(0,1). Let ๐‘†,๐‘‡โˆถ๐ถโ†’๐ถ be two nonexpansive mappings with ๐น(๐‘‡)โ‰ โˆ…. Let ๐ด be a strongly positive linear bounded operator on a Hilbert space ๐ป with coefficient โˆ’๐›พ>0 and 0<๐›พ<โˆ’๐›พ/๐œŒ. {๐›ผ๐‘›} and {๐›ฝ๐‘›} are two sequences in (0,1). Starting with an arbitrary initial guess ๐‘ฅ0โˆˆ๐ถ and {๐‘ฅ๐‘›} is a sequence generated by ๐‘ฆ๐‘›=๐›ฝ๐‘›๐‘†๐‘ฅ๐‘›+๎€ท1โˆ’๐›ฝ๐‘›๎€ธ๐‘ฅ๐‘›,๐‘ฅ๐‘›+1=๐‘ƒ๐ถ๎€บ๐›ผ๐‘›๎€ท๐‘ฅ๐›พ๐‘“๐‘›๎€ธ+๎€ท๐ผโˆ’๐›ผ๐‘›๐ด๎€ธ๐‘‡๐‘ฆ๐‘›๎€ป,โˆ€๐‘›โ‰ฅ1.(3.26) Suppose that the following conditions are satisfied: (C1)โ€‰โ€‰lim๐‘›โ†’โˆž๐›ผ๐‘›=0 and โˆ‘โˆž๐‘›=1๐›ผ๐‘›=โˆž, (C2)โ€‰โ€‰lim๐‘›โ†’โˆž(๐›ฝ๐‘›/๐›ผ๐‘›)=๐œโˆˆ(0,โˆž), (C5)โ€‰โ€‰lim๐‘›โ†’โˆž((|๐›ผ๐‘›โˆ’๐›ผ๐‘›โˆ’1|+|๐›ฝ๐‘›โˆ’๐›ฝ๐‘›โˆ’1|)/๐›ผ๐‘›๐›ฝ๐‘›)=0, (C6)โ€‰โ€‰there exists a constant ๐พ>0 such that (1/๐›ผ๐‘›)|1/๐›ฝ๐‘›โˆ’1/๐›ฝ๐‘›โˆ’1|โ‰ค๐พ. Then the sequence {๐‘ฅ๐‘›} converges strongly to a point ฬƒ๐‘ฅโˆˆ๐ป, which is the unique solution of the variational inequality: ๎‚ฌ1ฬƒ๐‘ฅโˆˆ๐น(๐‘‡),๐œ๎‚ญ(๐ดโˆ’๐›พ๐‘“)ฬƒ๐‘ฅ+(๐ผโˆ’๐‘†)ฬƒ๐‘ฅ,๐‘ฅโˆ’ฬƒ๐‘ฅโ‰ฅ0,โˆ€๐‘ฅโˆˆ๐น(๐‘‡).(3.27)

Proof. First of all, we show that (3.27) has the unique solution. Indeed, let โˆ’๐‘ฅ and ฬƒ๐‘ฅ be two solutions. Then ๎‚ฌ(๐ดโˆ’๐›พ๐‘“)ฬƒ๐‘ฅ,ฬƒ๐‘ฅโˆ’โˆ’๐‘ฅ๎‚ญ๎‚ฌโ‰ค๐œ(๐ผโˆ’๐‘†)ฬƒ๐‘ฅ,โˆ’๐‘ฅ๎‚ญ.โˆ’ฬƒ๐‘ฅ(3.28) Analogously, we have ๎‚ฌ(๐ดโˆ’๐›พ๐‘“)โˆ’๐‘ฅ,โˆ’๐‘ฅ๎‚ญ๎‚ฌโˆ’ฬƒ๐‘ฅโ‰ค๐œ(๐ผโˆ’๐‘†)โˆ’๐‘ฅ,ฬƒ๐‘ฅโˆ’โˆ’๐‘ฅ๎‚ญ.(3.29) Adding (3.28) and (3.29), by Lemma 2.3, we obtain ๎‚€โˆ’๎‚โ€–โ€–๐›พโˆ’๐›พ๐œŒฬƒ๐‘ฅโˆ’โˆ’๐‘ฅโ€–โ€–2โ‰ค๎‚ฌ(๐ดโˆ’๐›พ๐‘“)ฬƒ๐‘ฅโˆ’(๐ดโˆ’๐›พ๐‘“)โˆ’๐‘ฅ,ฬƒ๐‘ฅโˆ’โˆ’๐‘ฅ๎‚ญ๎‚ฌโ‰คโˆ’๐œ(๐ผโˆ’๐‘†)ฬƒ๐‘ฅโˆ’(๐ผโˆ’๐‘†)โˆ’๐‘ฅ,ฬƒ๐‘ฅโˆ’โˆ’๐‘ฅ๎‚ญโ‰ค0,(3.30) and so ฬƒ๐‘ฅ=โˆ’๐‘ฅ. From (C2), we can assume, without loss of generality, that ๐›ฝ๐‘›โ‰ค(๐œ+1)๐›ผ๐‘› for all ๐‘›โ‰ฅ1. By a similar argument in Theorem 3.1, we have โ€–โ€–๐‘ฅ๐‘›+1โ€–โ€–โˆ’๐‘ขโ‰ค๐›ผ๐‘›โ€–โ€–๐‘ฅ๐›พ๐œŒ๐‘›โ€–โ€–โˆ’๐‘ข+๐›ผ๐‘›โ€–+๎‚€โ€–๐›พ๐‘“(๐‘ข)โˆ’๐ด๐‘ข1โˆ’๐›ผ๐‘›โˆ’๐›พ๎‚๎€บโ€–โ€–๐‘ฅ๐‘›โ€–โ€–โˆ’๐‘ข+๐›ฝ๐‘›๎€ทโ€–๐‘†๐‘ขโˆ’๐‘ขโ€–+1โˆ’๐›ฝ๐‘›๎€ธโ€–โ€–๐‘ฅ๐‘›โ€–โ€–๎€ป=๎‚€โˆ’๐‘ข1โˆ’๐›ผ๐‘›๎‚€โˆ’โ€–โ€–๐‘ฅ๐›พโˆ’๐›พ๐œŒ๎‚๎‚๐‘›โ€–โ€–โˆ’๐‘ข+๐›ผ๐‘›๎‚€โ€–๐›พ๐‘“(๐‘ข)โˆ’๐ด๐‘ขโ€–+1โˆ’๐›ผ๐‘›โˆ’๐›พ๎‚๐›ฝ๐‘›โ‰ค๎‚€โ€–๐‘†๐‘ขโˆ’๐‘ขโ€–1โˆ’๐›ผ๐‘›๎‚€โˆ’โ€–โ€–๐‘ฅ๐›พโˆ’๐›พ๐œŒ๎‚๎‚๐‘›โ€–โ€–โˆ’๐‘ข+๐›ผ๐‘›(โ€–๐›พ๐‘“๐‘ข)โˆ’๐ด๐‘ขโ€–+๐›ฝ๐‘›โ‰ค๎‚€โ€–๐‘†๐‘ขโˆ’๐‘ขโ€–1โˆ’๐›ผ๐‘›๎‚€โˆ’โ€–โ€–๐‘ฅ๐›พโˆ’๐›พ๐œŒ๎‚๎‚๐‘›โ€–โ€–โˆ’๐‘ข+๐›ผ๐‘›โ€–๐›พ๐‘“(๐‘ข)โˆ’๐ด๐‘ขโ€–+(๐œ+1)๐›ผ๐‘›=๎‚€โ€–๐‘†๐‘ขโˆ’๐‘ขโ€–1โˆ’๐›ผ๐‘›๎‚€โˆ’โ€–โ€–๐‘ฅ๐›พโˆ’๐›พ๐œŒ๎‚๎‚๐‘›โ€–โ€–โˆ’๐‘ข+๐›ผ๐‘›[(]=๎‚€โ€–๐›พ๐‘“๐‘ข)โˆ’๐ด๐‘ขโ€–+(๐œ+1)โ€–๐‘†๐‘ขโˆ’๐‘ขโ€–1โˆ’๐›ผ๐‘›๎‚€โˆ’โ€–โ€–๐‘ฅ๐›พโˆ’๐›พ๐œŒ๎‚๎‚๐‘›โ€–โ€–โˆ’๐‘ข+๐›ผ๐‘›๎‚€โˆ’๎‚๐›พโˆ’๐›พ๐œŒโ€–๐›พ๐‘“(๐‘ข)โˆ’๐ด๐‘ขโ€–+(๐œ+1)โ€–๐‘†๐‘ขโˆ’๐‘ขโ€–๎‚€โˆ’๎‚.๐›พโˆ’๐›พ๐œŒ(3.31) By induction, we obtain โ€–โ€–๐‘ฅ๐‘›โ€–โ€–๎ƒฏโ€–โ€–๐‘ฅโˆ’๐‘ขโ‰คmax0โ€–โ€–,1โˆ’๐‘ขโˆ’[โ€–]๎ƒฐ,๐›พโˆ’๐›พ๐œŒ๐›พ๐‘“(๐‘ข)โˆ’๐ด๐‘ขโ€–+(๐œ+1)โ€–๐‘†๐‘ขโˆ’๐‘ขโ€–(3.32) which implies that the sequence {๐‘ฅ๐‘›} is bounded. Since (C5) implies (C4) then, from Theorem 3.1, we can deduce โ€–๐‘ฅ๐‘›+1โˆ’๐‘ฅ๐‘›โ€–โ†’0.
From (3.1), we note that ๐‘ฅ๐‘›+1=๐‘ƒ๐ถ๎€บ๐‘ค๐‘›๎€ปโˆ’๐‘ค๐‘›+๐‘ค๐‘›+๐‘ฆ๐‘›โˆ’๐‘ฆ๐‘›=๐‘ƒ๐ถ๎€บ๐‘ค๐‘›๎€ปโˆ’๐‘ค๐‘›+๐›ผ๐‘›๎€ท๐‘ฅ๐›พ๐‘“๐‘›๎€ธ+๎€ท๐‘‡๐‘ฆ๐‘›โˆ’๐‘ฆ๐‘›๎€ธ+๎€ท๐‘ฆ๐‘›โˆ’๐›ผ๐‘›๐ด๐‘‡๐‘ฆ๐‘›๎€ธ.(3.33) Hence, it follows that ๐‘ฅ๐‘›โˆ’๐‘ฅ๐‘›+1=๎€ท๐‘ค๐‘›โˆ’๐‘ƒ๐ถ๎€บ๐‘ค๐‘›๎€ป๎€ธ+๐›ผ๐‘›๎€ท๐ด๐‘ฅ๐‘›๎€ท๐‘ฅโˆ’๐›พ๐‘“๐‘›+๎€ท๐‘ฆ๎€ธ๎€ธ๐‘›โˆ’๐‘‡๐‘ฆ๐‘›๎€ธ+๎€ท๐‘ฅ๐‘›โˆ’๐‘ฆ๐‘›๎€ธ+๐›ผ๐‘›๎€ท๐ด๐‘‡๐‘ฆ๐‘›โˆ’๐ด๐‘ฅ๐‘›๎€ธ=๎€ท๐‘ค๐‘›โˆ’๐‘ƒ๐ถ๎€บ๐‘ค๐‘›๎€ป๎€ธ+๐›ผ๐‘›(๐ดโˆ’๐›พ๐‘“)๐‘ฅ๐‘›+(๐ผโˆ’๐‘‡)๐‘ฆ๐‘›+๐›ฝ๐‘›(๐ผโˆ’๐‘†)๐‘ฅ๐‘›+๐›ผ๐‘›๐ด๎€ท๐‘‡๐‘ฆ๐‘›โˆ’๐‘ฅ๐‘›๎€ธ,(3.34) and so ๐‘ฅ๐‘›โˆ’๐‘ฅ๐‘›+1๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐›ฝ๐‘›=1๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐›ฝ๐‘›๎€ท๐‘ค๐‘›โˆ’๐‘ƒ๐ถ๎€บ๐‘ค๐‘›+๐›ผ๎€ป๎€ธ๐‘›๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐›ฝ๐‘›(๐ดโˆ’๐›พ๐‘“)๐‘ฅ๐‘›+1๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐›ฝ๐‘›(๐ผโˆ’๐‘‡)๐‘ฆ๐‘›+1๎€ท1โˆ’๐›ผ๐‘›๎€ธ(๐ผโˆ’๐‘†)๐‘ฅ๐‘›+๐›ผ๐‘›๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐›ฝ๐‘›๐ด๎€ท๐‘‡๐‘ฆ๐‘›โˆ’๐‘ฅ๐‘›๎€ธ.(3.35) Set ๐‘ฃ๐‘›โˆถ=(๐‘ฅ๐‘›โˆ’๐‘ฅ๐‘›+1)/(1โˆ’๐›ผ๐‘›)๐›ฝ๐‘›. Then, we have ๐‘ฃ๐‘›=1๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐›ฝ๐‘›๎€ท๐‘ค๐‘›โˆ’๐‘ƒ๐ถ๎€บ๐‘ค๐‘›+๐›ผ๎€ป๎€ธ๐‘›๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐›ฝ๐‘›(๐ดโˆ’๐›พ๐‘“)๐‘ฅ๐‘›+1๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐›ฝ๐‘›(๐ผโˆ’๐‘‡)๐‘ฆ๐‘›+1๎€ท1โˆ’๐›ผ๐‘›๎€ธ(๐ผโˆ’๐‘†)๐‘ฅ๐‘›+๐›ผ๐‘›๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐›ฝ๐‘›๐ด๎€ท๐‘‡๐‘ฆ๐‘›โˆ’๐‘ฅ๐‘›๎€ธ.(3.36) From (3.12) in Theorem 3.1 and (C6), we obtain โ€–โ€–๐‘ฅ๐‘›+1โˆ’๐‘ฅ๐‘›โ€–โ€–๐›ฝ๐‘›โ‰ค๎‚€1โˆ’๐›ผ๐‘›๎‚€โˆ’โ€–โ€–๐‘ฅ๐›พโˆ’๐›พ๐œŒ๎‚๎‚๐‘›โˆ’๐‘ฅ๐‘›โˆ’1โ€–โ€–๐›ฝ๐‘›๎‚ธ||๐›ผ+๐‘€๐‘›โˆ’๐›ผ๐‘›โˆ’1||๐›ฝ๐‘›+||๐›ฝ๐‘›โˆ’๐›ฝ๐‘›โˆ’1||๐›ฝ๐‘›๎‚น=๎‚€1โˆ’๐›ผ๐‘›๎‚€โˆ’โ€–โ€–๐‘ฅ๐›พโˆ’๐›พ๐œŒ๎‚๎‚๐‘›โˆ’๐‘ฅ๐‘›โˆ’1โ€–โ€–๐›ฝ๐‘›+๎‚€1โˆ’๐›ผ๐‘›๎‚€โˆ’โ€–โ€–๐‘ฅ๐›พโˆ’๐›พ๐œŒ๎‚๎‚๐‘›โˆ’๐‘ฅ๐‘›โˆ’1โ€–โ€–๐›ฝ๐‘›โˆ’1โˆ’๎‚€1โˆ’๐›ผ๐‘›๎‚€โˆ’โ€–โ€–๐‘ฅ๐›พโˆ’๐›พ๐œŒ๎‚๎‚๐‘›โˆ’๐‘ฅ๐‘›โˆ’1โ€–โ€–๐›ฝ๐‘›โˆ’1๎‚ธ||๐›ผ+๐‘€๐‘›โˆ’๐›ผ๐‘›โˆ’1||๐›ฝ๐‘›+||๐›ฝ๐‘›โˆ’๐›ฝ๐‘›โˆ’1||๐›ฝ๐‘›๎‚น=๎‚€1โˆ’๐›ผ๐‘›๎‚€โˆ’โ€–โ€–๐‘ฅ๐›พโˆ’๐›พ๐œŒ๎‚๎‚๐‘›โˆ’๐‘ฅ๐‘›โˆ’1โ€–โ€–๐›ฝ๐‘›โˆ’1+๎‚€1โˆ’๐›ผ๐‘›๎‚€โˆ’โ€–โ€–๐‘ฅ๐›พโˆ’๐›พ๐œŒ๎‚๎‚๐‘›โˆ’๐‘ฅ๐‘›โˆ’1โ€–โ€–๎‚ธ1๐›ฝ๐‘›โˆ’1๐›ฝ๐‘›โˆ’1๎‚น๎‚ธ||๐›ผ+๐‘€๐‘›โˆ’๐›ผ๐‘›โˆ’1||๐›ฝ๐‘›+||๐›ฝ๐‘›โˆ’๐›ฝ๐‘›โˆ’1||๐›ฝ๐‘›๎‚นโ‰ค๎‚€1โˆ’๐›ผ๐‘›๎‚€โˆ’โ€–โ€–๐‘ฅ๐›พโˆ’๐›พ๐œŒ๎‚๎‚๐‘›โˆ’๐‘ฅ๐‘›โˆ’1โ€–โ€–๐›ฝ๐‘›โˆ’1+โ€–โ€–๐‘ฅ๐‘›โˆ’๐‘ฅ๐‘›โˆ’1โ€–โ€–||||1๐›ฝ๐‘›โˆ’1๐›ฝ๐‘›โˆ’1||||๎‚ธ||๐›ผ+๐‘€๐‘›โˆ’๐›ผ๐‘›โˆ’1||๐›ฝ๐‘›+||๐›ฝ๐‘›โˆ’๐›ฝ๐‘›โˆ’1||๐›ฝ๐‘›๎‚นโ‰ค๎‚€1โˆ’๐›ผ๐‘›๎‚€โˆ’โ€–โ€–๐‘ฅ๐›พโˆ’๐›พ๐œŒ๎‚๎‚๐‘›โˆ’๐‘ฅ๐‘›โˆ’1โ€–โ€–๐›ฝ๐‘›โˆ’1+๐›ผ๐‘›๐พโ€–โ€–๐‘ฅ๐‘›โˆ’๐‘ฅ๐‘›โˆ’1โ€–โ€–๎‚ธ||๐›ผ+๐‘€๐‘›โˆ’๐›ผ๐‘›โˆ’1||๐›ฝ๐‘›+||๐›ฝ๐‘›โˆ’๐›ฝ๐‘›โˆ’1||๐›ฝ๐‘›๎‚นโ‰ค๎‚€1โˆ’๐›ผ๐‘›๎‚€โˆ’โ€–โ€–๐‘ค๐›พโˆ’๐›พ๐œŒ๎‚๎‚๐‘›โˆ’๐‘ค๐‘›โˆ’1โ€–โ€–๐›ฝ๐‘›โˆ’1+๐›ผ๐‘›๐พโ€–โ€–๐‘ฅ๐‘›โˆ’๐‘ฅ๐‘›โˆ’1โ€–โ€–๎‚ธ||๐›ผ+๐‘€๐‘›โˆ’๐›ผ๐‘›โˆ’1||๐›ฝ๐‘›+||๐›ฝ๐‘›โˆ’๐›ฝ๐‘›โˆ’1||๐›ฝ๐‘›๎‚น.(3.37) This together with Lemma 2.4 and (C2) imply that lim๐‘›โ†’โˆžโ€–โ€–๐‘ฅ๐‘›+1โˆ’๐‘ฅ๐‘›โ€–โ€–๐›ฝ๐‘›=lim๐‘›โ†’โˆžโ€–โ€–๐‘ค๐‘›+1โˆ’๐‘ค๐‘›โ€–โ€–๐›ฝ๐‘›=lim๐‘›โ†’โˆžโ€–โ€–๐‘ค๐‘›+1โˆ’๐‘ค๐‘›โ€–โ€–๐›ผ๐‘›=0.(3.38) From (3.36), for ๐‘งโˆˆ๐น(๐‘‡), we have โŸจ๐‘ฃ๐‘›,๐‘ฅ๐‘›1โˆ’๐‘งโŸฉ=๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐›ฝ๐‘›๎ซ๐‘ค๐‘›โˆ’๐‘ƒ๐ถ๎€บ๐‘ค๐‘›๎€ป,๐‘ƒ๐ถ๎€บ๐‘ค๐‘›โˆ’1๎€ป๎ฌ+๐›ผโˆ’๐‘ง๐‘›๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐›ฝ๐‘›โŸจ(๐ดโˆ’๐›พ๐‘“)๐‘ฅ๐‘›,๐‘ฅ๐‘›+1โˆ’๐‘งโŸฉ๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐›ฝ๐‘›โŸจ(๐ผโˆ’๐‘‡)๐‘ฆ๐‘›,๐‘ฅ๐‘›1โˆ’๐‘งโŸฉ+๎€ท1โˆ’๐›ผ๐‘›๎€ธโŸจ(๐ผโˆ’๐‘†)๐‘ฅ๐‘›,๐‘ฅ๐‘›+๐›ผโˆ’๐‘งโŸฉ๐‘›๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐›ฝ๐‘›๎ซ๐ด๎€ท๐‘‡๐‘ฆ๐‘›โˆ’๐‘ฅ๐‘›๎€ธ,๐‘ฅ๐‘›๎ฌ=1โˆ’๐‘ง๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐›ฝ๐‘›๎ซ๐‘ค๐‘›โˆ’๐‘ƒ๐ถ๎€บ๐‘ค๐‘›๎€ป,๐‘ƒ๐ถ๎€บ๐‘ค๐‘›๎€ป๎ฌ+1โˆ’๐‘ง๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐›ฝ๐‘›๎ซ๐‘ค๐‘›โˆ’๐‘ƒ๐ถ๎€บ๐‘ค๐‘›๎€ป,๐‘ƒ๐ถ๎€บ๐‘ค๐‘›โˆ’1๎€ปโˆ’๐‘ƒ๐ถ๎€บ๐‘ค๐‘›+๐›ผ๎€ป๎ฌ๐‘›๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐›ฝ๐‘›โŸจ(๐ดโˆ’๐›พ๐‘“)๐‘ฅ๐‘›โˆ’(๐ดโˆ’๐›พ๐‘“)๐‘ง,๐‘ฅ๐‘›๐›ผโˆ’๐‘งโŸฉ+๐‘›๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐›ฝ๐‘›โŸจ(๐ดโˆ’๐›พ๐‘“)๐‘ง,๐‘ฅ๐‘›+1โˆ’๐‘งโŸฉ๎€ท1โˆ’๐›ผ๐‘›๎€ธโŸจ(๐ผโˆ’๐‘†)๐‘ฅ๐‘›โˆ’(๐ผโˆ’๐‘†)๐‘ง,๐‘ฅ๐‘›1โˆ’๐‘งโŸฉ+๎€ท1โˆ’๐›ผ๐‘›๎€ธโŸจ(๐ผโˆ’๐‘†)๐‘ง,๐‘ฅ๐‘›+1โˆ’๐‘งโŸฉ๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐›ฝ๐‘›โŸจ(๐ผโˆ’๐‘‡)๐‘ฆ๐‘›,๐‘ฅ๐‘›๐›ผโˆ’๐‘งโŸฉ+๐‘›๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐›ฝ๐‘›๎ซ๐ด๎€ท๐‘‡๐‘ฆ๐‘›โˆ’๐‘ฅ๐‘›๎€ธ,๐‘ฅ๐‘›๎ฌ.โˆ’๐‘ง(3.39) By Lemmas 2.2 and 2.3, we obtain โŸจ๐‘ฃ๐‘›,๐‘ฅ๐‘›1โˆ’๐‘งโŸฉโ‰ฅ๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐›ฝ๐‘›๎ซ๐‘ค๐‘›โˆ’๐‘ƒ๐ถ๎€บ๐‘ค๐‘›๎€ป,๐‘ƒ๐ถ๎€บ๐‘ค๐‘›โˆ’1๎€ปโˆ’๐‘ƒ๐ถ๎€บ๐‘ค๐‘›+๎‚€๎€ป๎ฌโˆ’๎‚๐›ผ๐›พโˆ’๐›พ๐œŒ๐‘›๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐›ฝ๐‘›โ€–โ€–๐‘ฅ๐‘›โ€–โ€–โˆ’๐‘ง2+๐›ผ๐‘›๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐›ฝ๐‘›โŸจ(๐ดโˆ’๐›พ๐‘“)๐‘ง,๐‘ฅ๐‘›1โˆ’๐‘งโŸฉ+๎€ท1โˆ’๐›ผ๐‘›๎€ธโŸจ(๐ผโˆ’๐‘†)๐‘ง,๐‘ฅ๐‘›+1โˆ’๐‘งโŸฉ๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐›ฝ๐‘›โŸจ(๐ผโˆ’๐‘‡)๐‘ฆ๐‘›,๐‘ฅ๐‘›๐›ผโˆ’๐‘งโŸฉ+๐‘›๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐›ฝ๐‘›๎ซ๐ด๎€ท๐‘‡๐‘ฆ๐‘›โˆ’๐‘ฅ๐‘›๎€ธ,๐‘ฅ๐‘›๎ฌ.โˆ’๐‘ง(3.40) Now, we observe that โ€–โ€–๐‘ฅ๐‘›โ€–โ€–โˆ’๐‘ง2โ‰ค๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐›ฝ๐‘›๎‚€โˆ’๎‚๐›ผ๐›พโˆ’๐›พ๐œŒ๐‘›โŸจ๐‘ฃ,๐‘ฅ๐‘›๐›ฝโˆ’๐‘งโŸฉโˆ’๐‘›๎‚€โˆ’๎‚๐›ผ๐›พโˆ’๐›พ๐œŒ๐‘›โŸจ(๐ผโˆ’๐‘†)๐‘ง,๐‘ฅ๐‘›โˆ’1โˆ’๐‘งโŸฉ๎‚€โˆ’๎‚๐›พโˆ’๐›พ๐œŒโŸจ(๐ดโˆ’๐›พ๐‘“)๐‘ง,๐‘ฅ๐‘›1โˆ’๐‘งโŸฉโˆ’๎‚€โˆ’๎‚๐›ผ๐›พโˆ’๐›พ๐œŒ๐‘›โŸจ(๐ผโˆ’๐‘‡)๐‘ฆ๐‘›,๐‘ฅ๐‘›โˆ’1โˆ’๐‘งโŸฉ๎‚€โˆ’๎‚๎ซ๐ด๎€ท๐›พโˆ’๐›พ๐œŒ๐‘‡๐‘ฆ๐‘›โˆ’๐‘ฅ๐‘›๎€ธ,๐‘ฅ๐‘›๎ฌโˆ’1โˆ’๐‘ง๎‚€โˆ’๎‚๐›ผ๐›พโˆ’๐›พ๐œŒ๐‘›๎ซ๐‘ค๐‘›โˆ’๐‘ƒ๐ถ๎€บ๐‘ค๐‘›๎€ป,๐‘ƒ๐ถ๎€บ๐‘ค๐‘›โˆ’1๎€ปโˆ’๐‘ƒ๐ถ๎€บ๐‘ค๐‘›โ‰ค๎€ท๎€ป๎ฌ1โˆ’๐›ผ๐‘›๎€ธ๐›ฝ๐‘›๎‚€โˆ’๎‚๐›ผ๐›พโˆ’๐›พ๐œŒ๐‘›โŸจ๐‘ฃ,๐‘ฅ๐‘›๐›ฝโˆ’๐‘งโŸฉโˆ’๐‘›๎‚€โˆ’๎‚๐›ผ๐›พโˆ’๐›พ๐œŒ๐‘›โŸจ(๐ผโˆ’๐‘†)๐‘ง,๐‘ฅ๐‘›โˆ’1โˆ’๐‘งโŸฉ๎‚€โˆ’๎‚๐›พโˆ’๐›พ๐œŒโŸจ(๐ดโˆ’๐›พ๐‘“)๐‘ง,๐‘ฅ๐‘›1โˆ’๐‘งโŸฉโˆ’๎‚€โˆ’๎‚๐›ผ๐›พโˆ’๐›พ๐œŒ๐‘›โŸจ(๐ผโˆ’๐‘‡)๐‘ฆ๐‘›,๐‘ฅ๐‘›โˆ’1โˆ’๐‘งโŸฉ๎‚€โˆ’๎‚๎ซ๐ด๎€ท๐›พโˆ’๐›พ๐œŒ๐‘‡๐‘ฆ๐‘›โˆ’๐‘ฅ๐‘›๎€ธ,๐‘ฅ๐‘›๎ฌ+โ€–โ€–๐‘คโˆ’๐‘ง๐‘›โˆ’๐‘ค๐‘›โˆ’1โ€–โ€–๎‚€โˆ’๎‚โ€–โ€–๐‘ค๐›พโˆ’๐›พ๐œŒ๐‘›โˆ’๐‘ƒ๐ถ๎€บ๐‘ค๐‘›๎€ปโ€–โ€–.(3.41) From (C1) and (C2), we have ๐›ฝ๐‘›โ†’0. Hence, from (3.1), we deduce โ€–๐‘ฆ๐‘›โˆ’๐‘ฅ๐‘›โ€–โ†’0 and โ€–๐‘ฅ๐‘›+1โˆ’๐‘‡๐‘ฆ๐‘›โ€–โ†’0. Therefore, โ€–โ€–๐‘ฆ๐‘›โˆ’๐‘‡๐‘ฆ๐‘›โ€–โ€–โ‰คโ€–โ€–๐‘ฆ๐‘›โˆ’๐‘ฅ๐‘›โ€–โ€–+โ€–โ€–๐‘ฅ๐‘›โˆ’๐‘ฅ๐‘›+1โ€–โ€–+โ€–โ€–๐‘ฅ๐‘›+1โˆ’๐‘‡๐‘ฆ๐‘›โ€–โ€–โ†’0.(3.42)
Since ๐‘ฃ๐‘›โ†’0, (๐ผโˆ’๐‘‡)๐‘ฆ๐‘›โ†’0, ๐ด(๐‘‡๐‘ฆ๐‘›โˆ’๐‘ฅ๐‘›)โ†’0, and โ€–๐‘ค๐‘›โˆ’๐‘ค๐‘›โˆ’1โ€–/(โˆ’๐›พโˆ’๐›พ๐œŒ)โ†’0, every weak cluster point of {๐‘ฅ๐‘›} is also a strong cluster point. Note that the sequence {๐‘ฅ๐‘›} is bounded, thus there exists a subsequence {๐‘ฅ๐‘›๐‘˜} converging to a point ฬƒ๐‘ฅโˆˆ๐ป. For all ๐‘งโˆˆ๐น(๐‘‡), it follows from (3.39) that ๎ซ(๐ดโˆ’๐›พ๐‘“)๐‘ฅ๐‘›๐‘˜,๐‘ฅ๐‘›๐‘˜๎ฌ=๎€ทโˆ’๐‘ง1โˆ’๐›ผ๐‘›๐‘˜๎€ธ๐›ฝ๐‘›๐‘˜๐›ผ๐‘›๐‘˜๎ซ๐‘ฃ๐‘›๐‘˜,๐‘ฅ๐‘›๐‘˜๎ฌโˆ’1โˆ’๐‘ง๐›ผ๐‘›๐‘˜๎ซ(๐ผโˆ’๐‘‡)๐‘ฆ๐‘›๐‘˜,๐‘ฅ๐‘›๐‘˜๎ฌโˆ’๐›ฝโˆ’๐‘ง๐‘›๐‘˜๐›ผ๐‘›๐‘˜๎ซ(๐ผโˆ’๐‘†)๐‘ฅ๐‘›๐‘˜,๐‘ฅ๐‘›๐‘˜๎ฌโˆ’๎ซ๐ด๎€ทโˆ’๐‘ง๐‘‡๐‘ฆ๐‘›๐‘˜โˆ’๐‘ฅ๐‘›๐‘˜๎€ธ,๐‘ฅ๐‘›๐‘˜๎ฌโˆ’1โˆ’๐‘ง๐›ผ๐‘›๐‘˜๎ซ๐‘ค๐‘›๐‘˜โˆ’๐‘ƒ๐ถ๎€บ๐‘ค๐‘›๐‘˜๎€ป,๐‘ƒ๐ถ๎€บ๐‘ค๐‘›๐‘˜โˆ’1๎€ป๎ฌโ‰ค๎€ทโˆ’๐‘ง1โˆ’๐›ผ๐‘›๐‘˜๎€ธ๐›ฝ๐‘›๐‘˜๐›ผ๐‘›๐‘˜๎ซ๐‘ฃ๐‘›๐‘˜,๐‘ฅ๐‘›๐‘˜๎ฌโˆ’1โˆ’๐‘ง๐›ผ๐‘›๐‘˜๎ซ(๐ผโˆ’๐‘‡)๐‘ฆ๐‘›๐‘˜,๐‘ฅ๐‘›๐‘˜๎ฌโˆ’๐›ฝโˆ’๐‘ง๐‘›๐‘˜๐›ผ๐‘›๐‘˜๎ซ(๐ผโˆ’๐‘†)๐‘ฅ๐‘›๐‘˜,๐‘ฅ๐‘›๐‘˜๎ฌโˆ’๎ซ๐ด๎€ทโˆ’๐‘ง๐‘‡๐‘ฆ๐‘›๐‘˜โˆ’๐‘ฅ๐‘›๐‘˜๎€ธ,๐‘ฅ๐‘›๐‘˜๎ฌโˆ’1โˆ’๐‘ง๐›ผ๐‘›๐‘˜๎ซ๐‘ค๐‘›๐‘˜โˆ’๐‘ƒ๐ถ๎€บ๐‘ค๐‘›๐‘˜๎€ป,๐‘ƒ๐ถ๎€บ๐‘ค๐‘›๐‘˜โˆ’1๎€ปโˆ’๐‘ƒ๐ถ๎€บ๐‘ค๐‘›๐‘˜โˆ’๎ซ๐ด๎€ท๎€ป๎ฌ๐‘‡๐‘ฆ๐‘›๐‘˜โˆ’๐‘ฅ๐‘›๐‘˜๎€ธ,๐‘ฅ๐‘›๐‘˜๎ฌโˆ’1โˆ’๐‘ง๐›ผ๐‘›๐‘˜๎ซ๐‘ค๐‘›๐‘˜โˆ’๐‘ƒ๐ถ๎€บ๐‘ค๐‘›๐‘˜๎€ป,๐‘ƒ๐ถ๎€บ๐‘ค๐‘›๐‘˜โˆ’1๎€ป๎ฌโ‰ค๎€ทโˆ’๐‘ง1โˆ’๐›ผ๐‘›๐‘˜๎€ธ๐›ฝ๐‘›๐‘˜๐›ผ๐‘›๐‘˜๎ซ๐‘ฃ๐‘›๐‘˜,๐‘ฅ๐‘›๐‘˜๎ฌโˆ’1โˆ’๐‘ง๐›ผ๐‘›๐‘˜๎ซ(๐ผโˆ’๐‘‡)๐‘ฆ๐‘›๐‘˜,๐‘ฅ๐‘›๐‘˜๎ฌโˆ’๐›ฝโˆ’๐‘ง๐‘›๐‘˜๐›ผ๐‘›๐‘˜๎ซ(๐ผโˆ’๐‘†)๐‘ฅ๐‘›๐‘˜,๐‘ฅ๐‘›๐‘˜๎ฌโˆ’๎ซ๐ด๎€ทโˆ’๐‘ง๐‘‡๐‘ฆ๐‘›๐‘˜โˆ’๐‘ฅ๐‘›๐‘˜๎€ธ,๐‘ฅ๐‘›๐‘˜๎ฌ+โ€–โ€–๐‘คโˆ’๐‘ง๐‘›๐‘˜โˆ’๐‘ค๐‘›๐‘˜โˆ’1โ€–โ€–๐›ผ๐‘›๐‘˜โ€–โ€–๐‘ค๐‘›๐‘˜โˆ’๐‘ƒ๐ถ๎€บ๐‘ค๐‘›๐‘˜๎€ปโ€–โ€–.(3.43) Letting ๐‘˜โ†’โˆž, we obtain โŸจ(๐ดโˆ’๐›พ๐‘“)ฬƒ๐‘ฅ,ฬƒ๐‘ฅโˆ’๐‘งโŸฉโ‰คโˆ’๐œโŸจ(๐ผโˆ’๐‘†)ฬƒ๐‘ฅ,ฬƒ๐‘ฅโˆ’๐‘งโŸฉ,โˆ€๐‘งโˆˆ๐น(๐‘‡).(3.44) By Lemma 2.6 and (3.27) having the unique solution, it follows that ๐œ”๐‘ค(๐‘ฅ๐‘›)={ฬƒ๐‘ฅ}. Therefore, ๐‘ฅ๐‘›โ†’ฬƒ๐‘ฅ as ๐‘›โ†’โˆž. This completes the proof.

From Theorem 3.3, we can deduce the following interesting corollary.

Corollary 3.4 (Yao et al. [9]). Let ๐ถ be a nonempty closed convex subset of a real Hilbert space ๐ป. Let ๐‘“โˆถ๐ถโ†’๐ป be a ๐œŒ-contraction (possibly nonself) with ๐œŒโˆˆ(0,1). Let ๐‘†,๐‘‡โˆถ๐ถโ†’๐ถ be two nonexpansive mappings with ๐น(๐‘‡)โ‰ โˆ…. {๐›ผ๐‘›} and {๐›ฝ๐‘›} are two sequences in (0,1) Starting with an arbitrary initial guess ๐‘ฅ0โˆˆ๐ถ and {๐‘ฅ๐‘›} is a sequence generated by ๐‘ฆ๐‘›=๐›ฝ๐‘›๐‘†๐‘ฅ๐‘›+๎€ท1โˆ’๐›ฝ๐‘›๎€ธ๐‘ฅ๐‘›,๐‘ฅ๐‘›+1=๐‘ƒ๐ถ๎€บ๐›ผ๐‘›๐‘“๎€ท๐‘ฅ๐‘›๎€ธ+๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐‘‡๐‘ฆ๐‘›๎€ป,โˆ€๐‘›โ‰ฅ1.(3.45) Suppose that the following conditions are satisfied: (C1)โ€‰โ€‰lim๐‘›โ†’โˆž๐›ผ๐‘›=0 and โˆ‘โˆž๐‘›=1๐›ผ๐‘›=โˆž, (C2)โ€‰โ€‰lim๐‘›โ†’โˆž(๐›ฝ๐‘›/๐›ผ๐‘›)=๐œโˆˆ(0,โˆž), (C5)โ€‰โ€‰lim๐‘›โ†’โˆž((|๐›ผ๐‘›โˆ’๐›ผ๐‘›โˆ’1|+|๐›ฝ๐‘›โˆ’๐›ฝ๐‘›โˆ’1|)/๐›ผ๐‘›๐›ฝ๐‘›)=0, (C6)โ€‰โ€‰there exists a constant ๐พ>0 such that (1/๐›ผ๐‘›)|1/๐›ฝ๐‘›โˆ’1/๐›ฝ๐‘›โˆ’1|โ‰ค๐พ. Then the sequence {๐‘ฅ๐‘›} converges strongly to a point ฬƒ๐‘ฅโˆˆ๐ป, which is the unique solution of the variational inequality: ๎‚ฌ1ฬƒ๐‘ฅโˆˆ๐น(๐‘‡),๐œ๎‚ญ(๐ผโˆ’๐‘“)ฬƒ๐‘ฅ+(๐ผโˆ’๐‘†)ฬƒ๐‘ฅ,๐‘ฅโˆ’ฬƒ๐‘ฅโ‰ฅ0,โˆ€๐‘ฅโˆˆ๐น(๐‘‡).(3.46)

Proof. As a matter of fact, if we take ๐ด=๐ผ and ๐›พ=1 in Theorem 3.3 then this completes the proof.

Corollary 3.5 (Yao et al. [9]). Let ๐ถ be a nonempty closed convex subset of a real Hilbert space ๐ป. Let ๐‘†,๐‘‡โˆถ๐ถโ†’๐ถ be two nonexpansive mappings with ๐น(๐‘‡)โ‰ โˆ…. {๐›ผ๐‘›} and {๐›ฝ๐‘›} are two sequences in (0,1). Starting with an arbitrary initial guess ๐‘ฅ0โˆˆ๐ถ and suppose {๐‘ฅ๐‘›} is a sequence generated by ๐‘ฆ๐‘›=๐›ฝ๐‘›๐‘†๐‘ฅ๐‘›+๎€ท1โˆ’๐›ฝ๐‘›๎€ธ๐‘ฅ๐‘›,๐‘ฅ๐‘›+1=๐‘ƒ๐ถ๎€บ๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐‘‡๐‘ฆ๐‘›๎€ป,โˆ€๐‘›โ‰ฅ1.(3.47) Suppose that the following conditions are satisfied: (C1)โ€‰โ€‰lim๐‘›โ†’โˆž๐›ผ๐‘›=0 and โˆ‘โˆž๐‘›=1๐›ผ๐‘›=โˆž, (C2)โ€‰โ€‰lim๐‘›โ†’โˆž(๐›ฝ๐‘›/๐›ผ๐‘›)=1, (C5)โ€‰โ€‰lim๐‘›โ†’โˆž((|๐›ผ๐‘›โˆ’๐›ผ๐‘›โˆ’1|+|๐›ฝ๐‘›โˆ’๐›ฝ๐‘›โˆ’1|)/๐›ผ๐‘›๐›ฝ๐‘›)=0, (C6)โ€‰โ€‰there exists a constant ๐พ>0 such that (1/๐›ผ๐‘›)|1/๐›ฝ๐‘›โˆ’1/๐›ฝ๐‘›โˆ’1|โ‰ค๐พ. Then the sequence {๐‘ฅ๐‘›} converges strongly to a point ฬƒ๐‘ฅโˆˆ๐ป, which is the unique solution of the variational inequality: ๐‘†ฬƒ๐‘ฅโˆˆ๐น(๐‘‡),๎‚ฌ๎‚€๐ผโˆ’2๎‚๎‚ญฬƒ๐‘ฅ,๐‘ฅโˆ’ฬƒ๐‘ฅโ‰ฅ0,โˆ€๐‘ฅโˆˆ๐น(๐‘‡).(3.48)

Proof. As a matter of fact, if we take ๐ด=๐ผ, ๐‘“=0, and ๐›พ=1 in Theorem 3.3 then this is completes the proof.

Remark 3.6. Prototypes for the iterative parameters are, for example, ๐›ผ๐‘›=๐‘›โˆ’๐œƒ and ๐›ฝ๐‘›=๐‘›โˆ’๐œ” (with ๐œƒ,๐œ”>0). Since |๐›ผ๐‘›โˆ’๐›ผ๐‘›โˆ’1|โ‰ˆ๐‘›โˆ’๐œƒ and |๐›ฝ๐‘›โˆ’๐›ฝ๐‘›โˆ’1|โ‰ˆ๐‘›โˆ’๐œ”, it is not difficult to prove that (C5) is satisfied for 0<๐œƒ,๐œ”<1 and (C6) is satisfied if ๐œƒ+๐œ”โ‰ค1.

Remark 3.7. Our results improve and extend the results of Yao et al. [9] by taking ๐ด=๐ผ and ๐›พ=1 in Theorems 3.1 and 3.3.

The following is an example to support Theorem 3.3.

Example 3.8. Let ๐ป=โ„,๐ถ=[โˆ’1/4,1/4],๐‘‡=๐ผ,๐‘†=โˆ’๐ผ,๐ด=๐ผ, ๐‘“(๐‘ฅ)=๐‘ฅ2, ๐‘ƒ๐ถ=๐ผ,๐›ฝ๐‘›โˆš=1/๐‘›, ๐›ผ๐‘›โˆš=1/๐‘› for every ๐‘›โˆˆโ„•, we have ๐œ=1 and choose โˆ’๐›พ=1/2, ๐œŒ=1/3 and ๐›พ=1. Then {๐‘ฅ๐‘›} is the sequence ๐‘ฅ๐‘›+1=๐‘ฅ2๐‘›โˆš๐‘›+๎ƒฉ11โˆ’โˆš๐‘›2๎ƒช๎ƒฉ1โˆ’โˆš๐‘›๎ƒช๐‘ฅ๐‘›,(3.49) and ๐‘ฅ๐‘›โ†’ฬƒ๐‘ฅ=0 as ๐‘›โ†’โˆž, where ฬƒ๐‘ฅ=0 is the unique solution of the variational inequality ๎‚ƒโˆ’1ฬƒ๐‘ฅโˆˆ๐น(๐‘‡)=4,14๎‚„,๎ซ๎€ท3ฬƒ๐‘ฅโˆ’ฬƒ๐‘ฅ2๎€ธ๎ฌ๎‚ƒโˆ’1,๐‘ฅโˆ’ฬƒ๐‘ฅโ‰ฅ0,โˆ€๐‘ฅโˆˆ๐น(๐‘‡)=4,14๎‚„.(3.50)

Acknowledgments

The authors would like to thank the National Research University Project of Thailandโ€™s Office of the Higher Education Commission under the Project NRU-CSEC no. 55000613 for financial support.