Discrete Dynamics in Nature and Society

Discrete Dynamics in Nature and Society / 2011 / Article

Research Article | Open Access

Volume 2011 |Article ID 976505 | 23 pages | https://doi.org/10.1155/2011/976505

Generalized Systems of Variational Inequalities and Projection Methods for Inverse-Strongly Monotone Mappings

Academic Editor: Jianshe Yu
Received20 Feb 2011
Accepted03 May 2011
Published14 Jul 2011

Abstract

We introduce an iterative sequence for finding a common element of the set of fixed points of a nonexpansive mapping and the solutions of the variational inequality problem for three inverse-strongly monotone mappings. Under suitable conditions, some strong convergence theorems for approximating a common element of the above two sets are obtained. Moreover, using the above theorem, we also apply to find solutions of a general system of variational inequality and a zero of a maximal monotone operator in a real Hilbert space. As applications, at the end of the paper we utilize our results to study some convergence problem for strictly pseudocontractive mappings. Our results include the previous results as special cases extend and improve the results of Ceng et al., (2008) and many others.

1. Introduction

Variational inequalities are known to play a crucial role in mathematics as a unified framework for studying a large variety of problems arising, for instance, in structural analysis, engineering sciences and others. Roughly speaking, they can be recast as fixed-point problems, and most of the numerical methods related to this topic are based on projection methods. Let 𝐻 be a real Hilbert space with inner product βŸ¨β‹…,β‹…βŸ© and β€–β‹…β€–, and let 𝐸 be a nonempty, closed, convex subset of 𝐻. A mapping π΄βˆΆπΈβ†’π» is called 𝛼-inverse-strongly monotone if there exists a positive real number 𝛼>0 such that ⟨𝐴π‘₯βˆ’π΄π‘¦,π‘₯βˆ’π‘¦βŸ©β‰₯𝛼‖𝐴π‘₯βˆ’π΄π‘¦β€–2,βˆ€π‘₯,π‘¦βˆˆπΈ(1.1) (see [1, 2]). It is obvious that every 𝛼-inverse-strongly monotone mapping 𝐴 is monotone and Lipschitz continuous. A mapping π‘†βˆΆπΈβ†’πΈ is called nonexpansive if ‖𝑆π‘₯βˆ’π‘†π‘¦β€–β‰€β€–π‘₯βˆ’π‘¦β€–,βˆ€π‘₯,π‘¦βˆˆπΈ.(1.2) We denote by 𝐹(𝑆) the set of fixed points of 𝑆 and by 𝑃𝐸 the metric projection of 𝐻 onto 𝐸. Recall that the classical variational inequality, denoted by 𝑉𝐼(𝐴,𝐸), is to find an π‘₯βˆ—βˆˆπΈ such that ⟨𝐴π‘₯βˆ—,π‘₯βˆ’π‘₯βˆ—βŸ©β‰₯0,βˆ€π‘₯∈𝐸.(1.3) The set of solutions of 𝑉𝐼(𝐴,𝐸) is denoted by Ξ“. The variational inequality has been widely studied in the literature; see, for example, [3–6] and the references therein.

For finding an element of 𝐹(𝑆)βˆ©Ξ“, Takahashi and Toyoda [7] introduced the following iterative scheme:π‘₯𝑛+1=𝛼𝑛π‘₯𝑛+ξ€·1βˆ’π›Όπ‘›ξ€Έπ‘†π‘ƒπΈξ€·π‘₯π‘›βˆ’πœ†π‘›π΄π‘₯𝑛,(1.4) for every 𝑛=0,1,2,…, where π‘₯0=π‘₯∈𝐸,{𝛼𝑛} is a sequence in (0,1), and {πœ†π‘›} is a sequence in (0,2𝛼). On the other hand, for solving the variational inequality problem in the finite-dimensional Euclidean space 𝐑𝑛, Korpelevich (1976) [8] introduced the following so-called extragradient method:π‘₯0𝑦=π‘₯∈𝐸,𝑛=𝑃𝐸π‘₯π‘›βˆ’πœ†π‘›π΄π‘₯𝑛,π‘₯𝑛+1=𝑃𝐸π‘₯nβˆ’πœ†π‘›π΄π‘¦π‘›ξ€Έ,(1.5) for every 𝑛=0,1,2,…, where πœ†π‘›βˆˆ(0,1/π‘˜). Many authors using extragradient method for approximating common fixed points and variational inequality problems (see also [9, 10]). Recently, Nadezhkina and Takahashi [11] and Zeng et al. [12] proposed some iterative schemes for finding elements in 𝐹(𝑆)βˆ©Ξ“ by combining (1.4) and (1.5). Further, these iterative schemes are extended in Y. Yao and J. C. Yao [13] to develop a new iterative scheme for finding elements in 𝐹(𝑆)βˆ©Ξ“.

Consider the following problem of finding (π‘₯βˆ—,π‘¦βˆ—)βˆˆπΈΓ—πΈ such that (see cf. Ceng et al. [14]):βŸ¨πœ†π΄π‘¦βˆ—+π‘₯βˆ—βˆ’π‘¦βˆ—,π‘₯βˆ’π‘₯βˆ—βŸ©β‰₯0,βˆ€π‘₯∈𝐸,βŸ¨πœ‡π΅π‘₯βˆ—+π‘¦βˆ—βˆ’π‘₯βˆ—,π‘₯βˆ’π‘¦βˆ—βŸ©β‰₯0,βˆ€π‘₯∈𝐸,(1.6) which is called general system of variational inequalities (GSVI), where πœ†>0 and πœ‡>0 are two constants. In particular, if 𝐴=𝐡, then problem (1.6) reduces to finding (π‘₯βˆ—,π‘¦βˆ—)βˆˆπΈΓ—πΈ such thatβŸ¨πœ†π΄π‘¦βˆ—+π‘₯βˆ—βˆ’π‘¦βˆ—,π‘₯βˆ’π‘₯βˆ—βŸ©β‰₯0,βˆ€π‘₯∈𝐸,βŸ¨πœ‡π΄π‘₯βˆ—+π‘¦βˆ—βˆ’π‘₯βˆ—,π‘₯βˆ’π‘¦βˆ—βŸ©β‰₯0,βˆ€π‘₯∈𝐸,(1.7) which is defined by [15, 16], and is called the new system of variational inequalities. Further, if π‘₯βˆ—=π‘¦βˆ—, then problem (1.7) reduces to the classical variational inequality VI(𝐴,𝐸), that is, find π‘₯βˆ—βˆˆπΈ such that ⟨𝐴π‘₯βˆ—,π‘₯βˆ’π‘₯βˆ—βŸ©β‰₯0, forallπ‘₯∈𝐸.

We can characteristic problem, if π‘₯βˆ—βˆˆπΉ(𝑆)βˆ©π‘‰πΌ(𝐴,𝐸), then it follows that π‘₯βˆ—=𝑆π‘₯βˆ—=𝑃𝐸[π‘₯βˆ—βˆ’πœŒπ΄π‘₯βˆ—], where 𝜌>0 is a constant.

In 2008, Ceng et al. [14] introduced a relaxed extragradient method for finding solutions of problem (1.6). Let the mappings 𝐴,π΅βˆΆπΈβ†’π» be 𝛼-inverse-strongly monotone and 𝛽-inverse-strongly monotone, respectively. Let π‘†βˆΆπΈβ†’πΈ be a nonexpansive mapping. Suppose π‘₯1=π‘’βˆˆπΈ and {π‘₯𝑛} is generated by𝑦𝑛=𝑃𝐸π‘₯π‘›βˆ’πœ‡π΅π‘₯𝑛,π‘₯𝑛+1=𝛼𝑛𝑒+𝛽𝑛π‘₯𝑛+π›Ύπ‘›π‘†π‘ƒπΈξ€·π‘¦π‘›βˆ’πœ†π‘›A𝑦𝑛,(1.8) where πœ†βˆˆ(0,2𝛼),πœ‡βˆˆ(0,2𝛽), and {𝛼𝑛},{𝛽𝑛},{𝛾𝑛} are three sequences in [0,1] such that 𝛼𝑛+𝛽𝑛+𝛾𝑛=1,forall𝑛β‰₯1. First, problem (1.6) is proven to be equivalent to a fixed point problem of a nonexpansive mapping.

In this paper, motivated by what is mentioned above, we consider generalized system of variational inequalities as follows.

Let 𝐸 be a nonempty, closed, convex subset of a real Hilbert space 𝐻. Let 𝐴,𝐡,πΆβˆΆπΈβ†’π» be three mappings. We consider the following problem of finding (π‘₯βˆ—,π‘¦βˆ—,π‘§βˆ—)βˆˆπΈΓ—πΈΓ—πΈ such thatβŸ¨πœ†π΄π‘¦βˆ—+π‘₯βˆ—βˆ’π‘¦βˆ—,π‘₯βˆ’π‘₯βˆ—βŸ©β‰₯0,βˆ€π‘₯∈𝐸,βŸ¨πœ‡π΅π‘§βˆ—+π‘¦βˆ—βˆ’π‘§βˆ—,π‘₯βˆ’π‘¦βˆ—βŸ©β‰₯0,βˆ€π‘₯∈𝐸,⟨𝜏𝐢π‘₯βˆ—+π‘§βˆ—βˆ’π‘₯βˆ—,π‘₯βˆ’π‘§βˆ—βŸ©β‰₯0,βˆ€π‘₯∈𝐸,(1.9) which is called a general system of variational inequalities where πœ†>0,πœ‡>0 and 𝜏>0 are three constants.

In particular, if 𝐴=𝐡=𝐢, then problem (1.9) reduces to finding (π‘₯βˆ—,π‘¦βˆ—,π‘§βˆ—)βˆˆπΈΓ—πΈΓ—πΈ such thatβŸ¨πœ†π΄π‘¦βˆ—+π‘₯βˆ—βˆ’π‘¦βˆ—,π‘₯βˆ’π‘₯βˆ—βŸ©β‰₯0,βˆ€π‘₯∈𝐸,βŸ¨πœ‡π΄π‘§βˆ—+π‘¦βˆ—βˆ’π‘§βˆ—,π‘₯βˆ’π‘¦βˆ—βŸ©β‰₯0,βˆ€π‘₯∈𝐸,⟨𝜏𝐴π‘₯βˆ—+π‘§βˆ—βˆ’π‘₯βˆ—,π‘₯βˆ’π‘§βˆ—βŸ©β‰₯0,βˆ€π‘₯∈𝐸.(1.10) Next, we consider some special classes of the GSVI problem (1.9) reduce to the following GSVI.(i)If 𝜏=0, then the GSVI problems (1.9) reduce to GSVI problem (1.6).(ii)If 𝜏=πœ‡=0, then the GSVI problems (1.9) reduce to classical variational inequality VI(A,E) problem.

The above system enters a class of more general problems which originated mainly from the Nash equilibrium points and was treated from a theoretical viewpoint in [17, 18]. Observe at the same time that, to construct a mathematical model which is as close as possible to a real complex problem, we often have to use constraints which can be expressed as one several subproblems of a general problem. These constrains can be given, for instance, by variational inequalities, by fixed point problems, or by problems of different types.

This paper deals with a relaxed extragradient approximation method for solving a system of variational inequalities over the fixed-point sets of nonexpansive mapping. Under classical conditions, we prove a strong convergence theorem for this method. Moreover, the proposed algorithm can be applied for instance to solving the classical variational inequality problems.

2. Preliminaries

Let 𝐸 be a nonempty, closed, convex subset of a real Hilbert space 𝐻. For every point π‘₯∈𝐻, there exists a unique nearest point in 𝐸, denoted by 𝑃𝐸π‘₯, such that β€–β€–π‘₯βˆ’π‘ƒπΈπ‘₯‖‖≀‖π‘₯βˆ’π‘¦β€–,βˆ€π‘¦βˆˆπΈ.(2.1)𝑃𝐸 is call the metric projection of 𝐻 onto 𝐸.

Recall that, 𝑃𝐸π‘₯ is characterized by following properties: 𝑃𝐸π‘₯∈𝐸 and ⟨π‘₯βˆ’π‘ƒπΈπ‘₯,π‘¦βˆ’π‘ƒπΈπ‘₯βŸ©β‰€0,β€–π‘₯βˆ’π‘¦β€–2β‰₯β€–β€–π‘₯βˆ’π‘ƒπΈπ‘₯β€–β€–2+‖‖𝑃𝐸‖‖π‘₯βˆ’π‘¦2,(2.2) for all π‘₯∈𝐻 and π‘¦βˆˆπΈ.

Lemma 2.1 (see cf. Zhang et al. [19]). The metric projection 𝑃𝐸 has the following properties: (i)π‘ƒπΈβˆΆπ»β†’πΈ is nonexpansive;(ii)π‘ƒπΈβˆΆπ»β†’πΈ is firmly nonexpansive, that is, ‖‖𝑃𝐸π‘₯βˆ’π‘ƒπΈπ‘¦β€–β€–2β‰€βŸ¨π‘ƒπΈπ‘₯βˆ’π‘ƒπΈπ‘¦,π‘₯βˆ’π‘¦βŸ©,βˆ€π‘₯,π‘¦βˆˆπ»;(2.3)(iii)for each π‘₯∈𝐻, 𝑧=𝑃𝐸(π‘₯)⟺⟨π‘₯βˆ’π‘§,π‘§βˆ’π‘¦βŸ©β‰₯0,βˆ€π‘¦βˆˆπΈ.(2.4)

Lemma 2.2 (see Osilike and Igbokwe [20]). Let (𝐸,βŸ¨β‹…,β‹…βŸ©) be an inner product space. Then for all π‘₯,𝑦,π‘§βˆˆπΈ and 𝛼,𝛽,π›Ύβˆˆ[0,1] with 𝛼+𝛽+𝛾=1, one has ‖𝛼π‘₯+𝛽𝑦+𝛾𝑧‖2=𝛼‖π‘₯β€–2+𝛽‖𝑦‖2+𝛾‖𝑧‖2βˆ’π›Όπ›½β€–π‘₯βˆ’π‘¦β€–2βˆ’π›Όπ›Ύβ€–π‘₯βˆ’π‘§β€–2βˆ’π›½π›Ύβ€–π‘¦βˆ’π‘§β€–2.(2.5)

Lemma 2.3 (see Suzuki [21]). Let {π‘₯𝑛} and {𝑦𝑛} be bounded sequences in a Banach space 𝑋 and let {𝛽𝑛} be a sequence in [0,1] with 0<liminfπ‘›β†’βˆžπ›½π‘›β‰€limsupπ‘›β†’βˆžπ›½π‘›<1. Suppose π‘₯𝑛+1=(1βˆ’π›½π‘›)𝑦𝑛+𝛽𝑛π‘₯𝑛 for all integers 𝑛β‰₯0 and limsupπ‘›β†’βˆž(‖𝑦𝑛+1βˆ’yπ‘›β€–βˆ’β€–π‘₯𝑛+1βˆ’π‘₯𝑛‖)≀0. Then, limπ‘›β†’βˆžβ€–π‘¦π‘›βˆ’π‘₯𝑛‖=0.

Lemma 2.4 (see Xu [22]). Assume {π‘Žπ‘›} is a sequence of nonnegative real numbers such that π‘Žπ‘›+1≀1βˆ’π›Όπ‘›ξ€Έπ‘Žπ‘›+𝛿𝑛,𝑛β‰₯0,(2.6) where {𝛼𝑛} is a sequence in (0,1) and {𝛿𝑛} is a sequence in 𝐑 such that (i)βˆ‘βˆžπ‘›=1𝛼𝑛=∞, (ii)limsupπ‘›β†’βˆž(𝛿𝑛/𝛼𝑛)≀0 or βˆ‘βˆžπ‘›=1|𝛿𝑛|<∞. Then, limπ‘›β†’βˆžπ‘Žπ‘›=0.

Lemma 2.5 (Goebel and Kirk [23]). Demiclosedness Principle. Assume that 𝑇 is a nonexpansive self-mapping of a nonempty, closed, convex subset 𝐸 of a real Hilbert space 𝐻. If 𝑇 has a fixed point, then πΌβˆ’π‘‡ is demiclosed; that is, whenever {π‘₯𝑛} is a sequence in 𝐸 converging weakly to some π‘₯∈𝐸 (for short, π‘₯𝑛⇀π‘₯∈𝐸), and the sequence {(πΌβˆ’π‘‡)π‘₯𝑛} converges strongly to some 𝑦 (for short, (πΌβˆ’π‘‡)π‘₯𝑛→𝑦), it follows that (πΌβˆ’π‘‡)π‘₯=𝑦. Here, 𝐼 is the identity operator of 𝐻.

The following lemma is an immediate consequence of an inner product.

Lemma 2.6. In a real Hilbert space 𝐻, there holds the inequality β€–π‘₯+𝑦‖2≀‖π‘₯β€–2+2βŸ¨π‘¦,π‘₯+π‘¦βŸ©,βˆ€π‘₯,π‘¦βˆˆπ».(2.7)

Remark 2.7. We also have that, for all 𝑒,π‘£βˆˆπΈ and πœ†>0, β€–β€–(πΌβˆ’πœ†π΄)π‘’βˆ’(πΌβˆ’πœ†π΄)𝑣2β€–=β€–(π‘’βˆ’π‘£)βˆ’πœ†(π΄π‘’βˆ’π΄π‘£)2=β€–π‘’βˆ’π‘£β€–2βˆ’2πœ†βŸ¨π‘’βˆ’π‘£,π΄π‘’βˆ’π΄π‘£βŸ©+πœ†2β€–π΄π‘’βˆ’π΄π‘£β€–2β‰€β€–π‘’βˆ’π‘£β€–2+πœ†(πœ†βˆ’2𝛼)β€–π΄π‘’βˆ’π΄π‘£β€–2.(2.8) So, if πœ†β‰€2𝛼, then πΌβˆ’πœ†π΄ is a nonexpansive mapping from 𝐸 to 𝐻.

3. Main Results

In this section, we introduce an iterative precess by the relaxed extragradient approximation method for finding a common element of the set of fixed points of a nonexpansive mapping and the solution set of the variational inequality problem for three inverse-strongly monotone mappings in a real Hilbert space. We prove that the iterative sequence converges strongly to a common element of the above two sets.

In order to prove our main result, the following lemmas are needed.

Lemma 3.1. For given π‘₯βˆ—,π‘¦βˆ—,π‘§βˆ—βˆˆπΈΓ—πΈΓ—πΈ,(π‘₯βˆ—,yβˆ—,π‘§βˆ—) is a solution of problem (1.9) if and only if π‘₯βˆ— is a fixed point of the mapping πΊβˆΆπΈβ†’πΈ defined by 𝐺(π‘₯)=𝑃𝐸𝑃𝐸𝑃𝐸(π‘₯βˆ’πœπΆπ‘₯)βˆ’πœ‡π΅π‘ƒπΈξ€»(π‘₯βˆ’πœπΆπ‘₯)βˆ’πœ†π΄π‘ƒπΈξ€Ίπ‘ƒπΈ(π‘₯βˆ’πœπΆπ‘₯)βˆ’πœ‡π΅π‘ƒπΈ(π‘₯βˆ’πœπΆπ‘₯)ξ€»ξ€Ύ,βˆ€π‘₯∈𝐸,(3.1) where π‘¦βˆ—=𝑃𝐸(π‘§βˆ—βˆ’πœ‡π΅π‘§βˆ—) and π‘§βˆ—=𝑃𝐸(π‘₯βˆ—βˆ’πœπΆπ‘₯βˆ—).

Proof. βŸ¨πœ†π΄π‘¦βˆ—+π‘₯βˆ—βˆ’π‘¦βˆ—,π‘₯βˆ’π‘₯βˆ—βŸ©β‰₯0,βˆ€π‘₯∈𝐸,βŸ¨πœ‡π΅π‘§βˆ—+π‘¦βˆ—βˆ’π‘§βˆ—,π‘₯βˆ’π‘¦βˆ—βŸ©β‰₯0,βˆ€π‘₯∈𝐸,⟨𝜏𝐢π‘₯βˆ—+π‘§βˆ—βˆ’π‘₯βˆ—,π‘₯βˆ’π‘§βˆ—βŸ©β‰₯0,βˆ€π‘₯∈𝐸,(3.2)β‡”ξ«ξ€·βˆ’π‘¦βˆ—+πœ†π΄π‘¦βˆ—ξ€Έ+π‘₯βˆ—,π‘₯βˆ’π‘₯βˆ—ξ¬β‰₯0,βˆ€π‘₯∈𝐸,ξ«ξ€·βˆ’π‘§βˆ—+πœ‡π΅π‘§βˆ—ξ€Έ+π‘¦βˆ—,π‘₯βˆ’π‘¦βˆ—ξ¬β‰₯0,βˆ€π‘₯∈𝐸,ξ«ξ€·βˆ’π‘₯βˆ—+𝜏𝐢π‘₯βˆ—ξ€Έ+π‘§βˆ—,π‘₯βˆ’π‘§βˆ—ξ¬β‰₯0,βˆ€π‘₯∈𝐸,(3.3)β‡”π‘¦ξ«ξ€·βˆ—βˆ’πœ†π΄π‘¦βˆ—ξ€Έβˆ’π‘₯βˆ—,π‘₯βˆ—ξ¬π‘§βˆ’π‘₯β‰₯0,βˆ€π‘₯∈𝐸,ξ«ξ€·βˆ—βˆ’πœ‡π΅π‘§βˆ—ξ€Έβˆ’π‘¦βˆ—,π‘¦βˆ—ξ¬π‘₯βˆ’π‘₯β‰₯0,βˆ€π‘₯∈𝐸,ξ«ξ€·βˆ—βˆ’πœπΆπ‘₯βˆ—ξ€Έβˆ’π‘§βˆ—,π‘§βˆ—ξ¬βˆ’π‘₯β‰₯0,βˆ€π‘₯∈𝐸,(3.4)⇔xβˆ—=π‘ƒπΈξ€·π‘¦βˆ—βˆ’πœ†π΄π‘¦βˆ—ξ€Έ,π‘¦βˆ—=π‘ƒπΈξ€·π‘§βˆ—βˆ’πœ‡π΅π‘§βˆ—ξ€Έ,π‘§βˆ—=𝑃𝐸π‘₯βˆ—βˆ’πœπΆπ‘₯βˆ—ξ€Έ,(3.5)⇔π‘₯βˆ—=𝑃𝐸[𝑃𝐸(π‘§βˆ—βˆ’πœ‡π΅π‘§βˆ—)βˆ’πœ†π΄π‘ƒπΈ(π‘§βˆ—βˆ’πœ‡π΅π‘§βˆ—)].
Thus, π‘₯βˆ—=𝑃𝐸𝑃𝐸𝑃𝐸π‘₯βˆ—βˆ’πœπΆπ‘₯βˆ—ξ€Έβˆ’πœ‡π΅π‘ƒπΈξ€·π‘₯βˆ—βˆ’πœπΆπ‘₯βˆ—ξ€Έξ€»βˆ’πœ†π΄π‘ƒπΈξ€Ίπ‘ƒπΈξ€·π‘₯βˆ—βˆ’πœπΆπ‘₯βˆ—ξ€Έβˆ’πœ‡π΅π‘ƒπΈξ€·π‘₯βˆ—βˆ’πœπΆπ‘₯βˆ—.ξ€Έξ€»ξ€Ύ(3.6)

Lemma 3.2. The mapping 𝐺 defined by Lemma 3.1 is nonexpansive mappings.

Proof. For all π‘₯,π‘¦βˆˆπΈ, ‖‖𝑃‖𝐺(π‘₯)βˆ’πΊ(𝑦)β€–=𝐸𝑃𝐸𝑃𝐸(π‘₯βˆ’πœπΆπ‘₯)βˆ’πœ‡π΅π‘ƒπΈξ€»(π‘₯βˆ’πœπΆπ‘₯)βˆ’πœ†π΄π‘ƒπΈξ€Ίπ‘ƒπΈ(π‘₯βˆ’πœπΆπ‘₯)βˆ’πœ‡π΅π‘ƒπΈ(π‘₯βˆ’πœπΆπ‘₯)ξ€»ξ€Ύβˆ’π‘ƒπΈξ€½π‘ƒπΈξ€Ίπ‘ƒπΈ(π‘¦βˆ’πœπΆπ‘¦)βˆ’πœ‡π΅π‘ƒπΈξ€»(π‘¦βˆ’πœπΆπ‘¦)βˆ’πœ†π΄π‘ƒπΈξ€Ίπ‘ƒπΈ(π‘¦βˆ’πœπΆπ‘¦)βˆ’πœ‡π΅π‘ƒπΈ(β‰€β€–β€–ξ€Ίπ‘ƒπ‘¦βˆ’πœπΆπ‘¦)𝐸(π‘₯βˆ’πœπΆπ‘₯)βˆ’πœ‡π΅π‘ƒπΈξ€»(π‘₯βˆ’πœπΆπ‘₯)βˆ’πœ†π΄π‘ƒπΈξ€Ίπ‘ƒπΈ(π‘₯βˆ’πœπΆπ‘₯)βˆ’πœ‡π΅π‘ƒπΈ(ξ€»βˆ’ξ€Ίπ‘ƒπ‘₯βˆ’πœπΆπ‘₯)𝐸(π‘¦βˆ’πœπΆπ‘¦)βˆ’πœ‡π΅π‘ƒπΈξ€»(π‘¦βˆ’πœπΆπ‘¦)βˆ’πœ†π΄π‘ƒπΈξ€Ίπ‘ƒπΈ(π‘¦βˆ’πœπΆπ‘¦)βˆ’πœ‡π΅π‘ƒπΈξ€»β€–β€–=‖‖𝑃(π‘¦βˆ’πœπΆπ‘¦)(πΌβˆ’πœ†π΄)𝐸(π‘₯βˆ’πœπΆπ‘₯)βˆ’πœ‡π΅π‘ƒπΈξ€»ξ€Ίπ‘ƒ(π‘₯βˆ’πœπΆπ‘₯)βˆ’(πΌβˆ’πœ†π΄)𝐸(π‘¦βˆ’πœπΆπ‘¦)βˆ’πœ‡π΅π‘ƒπΈξ€»β€–β€–β‰€β€–β€–ξ€Ίπ‘ƒ(π‘¦βˆ’πœπΆπ‘¦)𝐸(π‘₯βˆ’πœπΆπ‘₯)βˆ’πœ‡π΅π‘ƒπΈξ€»βˆ’ξ€Ίπ‘ƒ(π‘₯βˆ’πœπΆπ‘₯)𝐸(π‘¦βˆ’πœπΆπ‘¦)βˆ’πœ‡π΅π‘ƒπΈξ€»β€–β€–=‖‖𝑃(π‘¦βˆ’πœπΆπ‘¦)(πΌβˆ’πœ‡π΅)𝐸𝑃(π‘₯βˆ’πœπΆπ‘₯)βˆ’(πΌβˆ’πœ‡π΅)𝐸‖‖≀‖‖𝑃(π‘¦βˆ’πœπΆπ‘¦)𝐸(π‘₯βˆ’πœπΆπ‘₯)βˆ’π‘ƒπΈβ€–β€–(π‘¦βˆ’πœπΆπ‘¦)≀‖(π‘₯βˆ’πœπΆπ‘₯)βˆ’(π‘¦βˆ’πœπΆπ‘¦)β€–=β€–(πΌβˆ’πœπΆ)(π‘₯)βˆ’(πΌβˆ’πœπΆ)(𝑦)‖≀‖π‘₯βˆ’π‘¦β€–.(3.7) This shows that πΊβˆΆπΈβ†’πΈ is a nonexpansive mapping.

Throughout this paper, the set of fixed points of the mapping 𝐺 is denoted by Ξ₯.

Now, we are ready to proof our main results in this paper.

Theorem 3.3. Let 𝐸 be a nonempty, closed, convex subset of a real Hilbert space 𝐻. Let the mapping 𝐴,𝐡,πΆβˆΆπΈβ†’π» be 𝛼-inverse-strongly monotone, 𝛽-inverse-strongly monotone, and 𝛾-inverse-strongly monotone, respectively. Let 𝑆 be a nonexpansive mapping of 𝐸 into itself such that 𝐹(𝑆)∩Ξ₯β‰ βˆ…. Let 𝑓 be a contraction of 𝐻 into itself and given π‘₯1∈𝐻 arbitrarily and {π‘₯𝑛} is generated by 𝑧𝑛=𝑃𝐸π‘₯π‘›βˆ’πœπΆπ‘₯𝑛,𝑦𝑛=π‘ƒπΈξ€·π‘§π‘›βˆ’πœ‡π΅π‘§π‘›ξ€Έ,π‘₯𝑛+1=𝛼𝑛𝑓π‘₯𝑛+𝛽𝑛π‘₯𝑛+π›Ύπ‘›π‘†π‘ƒπΈξ€·π‘¦π‘›βˆ’πœ†π΄π‘¦π‘›ξ€Έ,𝑛β‰₯0,(3.8) where πœ†βˆˆ(0,2𝛼),πœ‡βˆˆ(0,2𝛽),𝜏∈(0,2𝛾), and {𝛼𝑛},{𝛽𝑛},{𝛾𝑛} are three sequences in [0,1] such that (i)𝛼𝑛+𝛽𝑛+𝛾𝑛=1,(ii)limπ‘›β†’βˆžπ›Όπ‘›=0 and βˆ‘βˆžπ‘›=1𝛼𝑛=∞,(iii)0<liminfπ‘›β†’βˆžπ›½π‘›β‰€limsupπ‘›β†’βˆžπ›½π‘›<1. Then, {π‘₯𝑛} converges strongly to π‘₯∈𝐹(𝑆)βˆ©Ξ“, where π‘₯=𝑃𝐹(𝑆)βˆ©Ξ“π‘“(π‘₯) and (π‘₯,𝑦,𝑧) is a solution of problem (1.9), where 𝑦=π‘ƒπΈξ€·π‘§βˆ’πœ‡π΅π‘§ξ€Έ,𝑧=𝑃𝐸π‘₯βˆ’πœπΆπ‘₯ξ€Έ.(3.9)

Proof. Let π‘₯βˆ—βˆˆπΉ(𝑆)βˆ©Ξ“. Then, π‘₯βˆ—=𝑆π‘₯βˆ— and π‘₯βˆ—=𝐺π‘₯βˆ—, that is, π‘₯βˆ—=𝑃𝐸𝑃𝐸𝑃𝐸π‘₯βˆ—βˆ’πœπΆπ‘₯βˆ—ξ€Έβˆ’πœ‡π΅π‘ƒπΈξ€·π‘₯βˆ—βˆ’πœπΆπ‘₯βˆ—ξ€Έξ€»βˆ’πœ†π΄π‘ƒπΈξ€Ίπ‘ƒπΈξ€·π‘₯βˆ—βˆ’πœπΆπ‘₯βˆ—ξ€Έβˆ’πœ‡π΅π‘ƒπΈξ€·π‘₯βˆ—βˆ’πœπΆπ‘₯βˆ—.ξ€Έξ€»ξ€Ύ(3.10) Put π‘₯βˆ—=𝑃𝐸(yβˆ—βˆ’πœ†π΄π‘¦βˆ—) and 𝑑𝑛=𝑃𝐸(π‘¦π‘›βˆ’πœ†π΄π‘¦π‘›). Then, π‘₯βˆ—=𝑃𝐸[𝑃𝐸(π‘§βˆ—βˆ’πœ‡π΅π‘§βˆ—)βˆ’πœ†π΄π‘ƒπΈ(π‘§βˆ—βˆ’πœ‡π΅π‘§βˆ—)] implies that π‘¦βˆ—=𝑃𝐸(π‘§βˆ—βˆ’πœ‡π΅π‘§βˆ—), where π‘§βˆ—=𝑃𝐸(π‘₯βˆ—βˆ’πœπΆπ‘₯βˆ—). Since πΌβˆ’πœ†π΄, πΌβˆ’πœ‡π΅ and πΌβˆ’πœπΆ are nonexpansive mappings. We obtain that β€–β€–π‘‘π‘›βˆ’π‘₯βˆ—β€–β€–=β€–β€–π‘ƒπΈξ€·π‘¦π‘›βˆ’πœ†π΄π‘¦π‘›ξ€Έβˆ’π‘₯βˆ—β€–β€–=β€–β€–π‘ƒπΈξ€·π‘¦π‘›βˆ’πœ†π΄π‘¦π‘›ξ€Έβˆ’π‘ƒπΈξ€·π‘¦βˆ—βˆ’πœ†π΄π‘¦βˆ—ξ€Έβ€–β€–β‰€β€–β€–ξ€·π‘¦π‘›βˆ’πœ†π΄π‘¦π‘›ξ€Έβˆ’ξ€·π‘¦βˆ—βˆ’πœ†π΄π‘¦βˆ—ξ€Έβ€–β€–=β€–β€–(πΌβˆ’πœ†π΄)π‘¦π‘›βˆ’(πΌβˆ’πœ†π΄)π‘¦βˆ—β€–β€–β‰€β€–β€–π‘¦π‘›βˆ’π‘¦βˆ—β€–β€–=‖‖𝑦(3.11)π‘›βˆ’π‘ƒπΈξ€·π‘§βˆ—βˆ’πœ‡π΅π‘§βˆ—ξ€Έβ€–β€–=β€–β€–π‘ƒπΈξ€·π‘§π‘›βˆ’πœ‡π΅π‘§π‘›ξ€Έβˆ’π‘ƒπΈξ€·π‘§βˆ—βˆ’πœ‡π΅π‘§βˆ—ξ€Έβ€–β€–β‰€β€–β€–(πΌβˆ’πœ‡π΅)π‘§π‘›βˆ’(πΌβˆ’πœ‡π΅)π‘§βˆ—β€–β€–β‰€β€–β€–π‘§π‘›βˆ’π‘§βˆ—β€–β€–,‖‖𝑧(3.12)π‘›βˆ’π‘§βˆ—β€–β€–=‖‖𝑃𝐸π‘₯π‘›βˆ’πœπΆπ‘₯π‘›ξ€Έβˆ’π‘ƒπΈξ€·π‘₯βˆ—βˆ’πœπΆπ‘₯βˆ—ξ€Έβ€–β€–β‰€β€–β€–ξ€·π‘₯π‘›βˆ’πœπΆπ‘₯π‘›ξ€Έβˆ’ξ€·π‘₯βˆ—βˆ’πœπΆπ‘₯βˆ—ξ€Έβ€–β€–=β€–β€–(πΌβˆ’πœπΆ)π‘₯π‘›βˆ’(πΌβˆ’πœπΆ)π‘₯βˆ—β€–β€–β‰€β€–β€–π‘₯π‘›βˆ’π‘₯βˆ—β€–β€–.(3.13) Substituting (3.13) into (3.12), we have β€–β€–π‘‘π‘›βˆ’π‘₯βˆ—β€–β€–β‰€β€–β€–π‘₯π‘›βˆ’π‘₯βˆ—β€–β€–,(3.14) and by (3.11) we also have β€–β€–π‘¦π‘›βˆ’π‘¦βˆ—β€–β€–β‰€β€–β€–π‘₯π‘›βˆ’π‘₯βˆ—β€–β€–.(3.15) Since π‘₯𝑛+1=𝛼𝑛𝑓(π‘₯𝑛)+𝛽𝑛π‘₯𝑛+𝛾𝑛𝑆𝑑𝑛 and by Lemma 2.2, we compute β€–β€–π‘₯𝑛+1βˆ’π‘₯βˆ—β€–β€–=‖‖𝛼𝑛𝑓π‘₯𝑛+𝛽𝑛π‘₯𝑛+π›Ύπ‘›π‘†π‘‘π‘›βˆ’π‘₯βˆ—β€–β€–=‖‖𝛼𝑛𝑓π‘₯π‘›ξ€Έβˆ’π‘₯βˆ—ξ€Έ+𝛽𝑛π‘₯π‘›βˆ’π‘₯βˆ—ξ€Έ+π›Ύπ‘›ξ€·π‘†π‘‘π‘›βˆ’π‘₯βˆ—ξ€Έβ€–β€–β‰€π›Όπ‘›β€–β€–π‘“ξ€·π‘₯π‘›ξ€Έβˆ’π‘₯βˆ—β€–β€–+𝛽𝑛‖‖π‘₯π‘›βˆ’π‘₯βˆ—β€–β€–+π›Ύπ‘›β€–β€–π‘†π‘‘π‘›βˆ’π‘₯βˆ—β€–β€–β‰€π›Όπ‘›β€–β€–π‘“ξ€·π‘₯π‘›ξ€Έβˆ’π‘₯βˆ—β€–β€–+𝛽𝑛‖‖π‘₯π‘›βˆ’π‘₯βˆ—β€–β€–+π›Ύπ‘›β€–β€–π‘‘π‘›βˆ’π‘₯βˆ—β€–β€–β‰€π›Όπ‘›β€–β€–π‘“ξ€·π‘₯π‘›ξ€Έβˆ’π‘₯βˆ—β€–β€–+𝛽𝑛‖‖π‘₯π‘›βˆ’π‘₯βˆ—β€–β€–+𝛾𝑛‖‖π‘₯π‘›βˆ’π‘₯βˆ—β€–β€–=𝛼𝑛‖‖𝑓π‘₯π‘›ξ€Έβˆ’π‘₯βˆ—β€–β€–+ξ€·1βˆ’π›Όπ‘›ξ€Έβ€–β€–π‘₯π‘›βˆ’π‘₯βˆ—β€–β€–=𝛼𝑛‖‖𝑓π‘₯𝑛π‘₯βˆ’π‘“βˆ—ξ€Έξ€·π‘₯+π‘“βˆ—ξ€Έβˆ’π‘₯βˆ—β€–β€–+ξ€·1βˆ’π›Όπ‘›ξ€Έβ€–β€–π‘₯π‘›βˆ’π‘₯βˆ—β€–β€–β‰€π›Όπ‘›β€–β€–π‘“ξ€·π‘₯𝑛π‘₯βˆ’π‘“βˆ—ξ€Έβ€–β€–+𝛼𝑛‖‖𝑓π‘₯βˆ—ξ€Έβˆ’π‘₯βˆ—β€–β€–+ξ€·1βˆ’π›Όπ‘›ξ€Έβ€–β€–π‘₯π‘›βˆ’π‘₯βˆ—β€–β€–β‰€π›Όπ‘›π‘˜β€–β€–π‘₯π‘›βˆ’π‘₯βˆ—β€–β€–+𝛼𝑛‖‖𝑓π‘₯βˆ—ξ€Έβˆ’π‘₯βˆ—β€–β€–+ξ€·1βˆ’π›Όπ‘›ξ€Έβ€–β€–π‘₯π‘›βˆ’π‘₯βˆ—β€–β€–=ξ€·π›Όπ‘›ξ€·π‘˜+1βˆ’π›Όπ‘›β€–β€–π‘₯ξ€Έξ€Έπ‘›βˆ’π‘₯βˆ—β€–β€–+𝛼𝑛‖‖𝑓π‘₯βˆ—ξ€Έβˆ’π‘₯βˆ—β€–β€–=ξ€·1βˆ’π›Όπ‘›ξ€Έβ€–β€–π‘₯(1βˆ’π‘˜)π‘›βˆ’π‘₯βˆ—β€–β€–+𝛼𝑛‖‖𝑓π‘₯βˆ—ξ€Έβˆ’π‘₯βˆ—β€–β€–=ξ€·1βˆ’π›Όπ‘›(ξ€Έβ€–β€–π‘₯1βˆ’π‘˜)π‘›βˆ’π‘₯βˆ—β€–β€–+𝛼𝑛(‖‖𝑓π‘₯1βˆ’π‘˜)βˆ—ξ€Έβˆ’π‘₯βˆ—β€–β€–.(1βˆ’π‘˜)(3.16) By induction, we get β€–β€–π‘₯𝑛+1βˆ’π‘₯βˆ—β€–β€–β‰€π‘€,(3.17) where 𝑀=max{β€–π‘₯0βˆ’π‘₯βˆ—β€–+(1/(1βˆ’π‘˜))‖𝑓(π‘₯βˆ—)βˆ’π‘₯βˆ—β€–},𝑛β‰₯0. Therefore, {π‘₯𝑛}is bounded. Consequently, by (3.11), (3.12) and (3.13), the sequences {𝑑𝑛},{𝑆𝑑𝑛},{𝑦𝑛},{𝐴𝑦𝑛},{𝑧𝑛},{𝐡𝑧𝑛},{𝐢π‘₯𝑛}, and {𝑓(π‘₯𝑛)} are also bounded. Also, we observe that ‖‖𝑧𝑛+1βˆ’π‘§π‘›β€–β€–=‖‖𝑃𝐸π‘₯𝑛+1βˆ’πœπΆπ‘₯𝑛+1ξ€Έβˆ’π‘ƒπΈξ€·π‘₯π‘›βˆ’πœπΆπ‘₯𝑛‖‖≀‖‖(πΌβˆ’πœπΆ)π‘₯𝑛+1βˆ’(πΌβˆ’πœπΆ)π‘₯𝑛‖‖≀‖‖π‘₯𝑛+1βˆ’π‘₯𝑛‖‖,‖‖𝑑(3.18)𝑛+1βˆ’π‘‘π‘›β€–β€–=‖‖𝑃𝐸𝑦𝑛+1βˆ’πœ†π΄π‘¦π‘›+1ξ€Έβˆ’π‘ƒπΈξ€·π‘¦π‘›βˆ’πœ†π΄π‘¦π‘›ξ€Έβ€–β€–β‰€β€–β€–ξ€·π‘¦π‘›+1βˆ’πœ†π΄π‘¦π‘›+1ξ€Έβˆ’ξ€·π‘¦π‘›βˆ’πœ†π΄π‘¦π‘›ξ€Έβ€–β€–=β€–β€–(πΌβˆ’πœ†π΄)𝑦𝑛+1βˆ’(πΌβˆ’πœ†π΄)𝑦𝑛‖‖≀‖‖𝑦𝑛+1βˆ’π‘¦π‘›β€–β€–=‖‖𝑃𝐸𝑧𝑛+1βˆ’πœ‡π΅π‘§π‘›+1ξ€Έβˆ’π‘ƒπΈξ€·π‘§π‘›βˆ’πœ‡π΅π‘§π‘›ξ€Έβ€–β€–β‰€β€–β€–π‘§π‘›+1βˆ’π‘§π‘›β€–β€–β‰€β€–β€–π‘₯𝑛+1βˆ’π‘₯𝑛‖‖.(3.19) Let π‘₯𝑛+1=(1βˆ’π›½π‘›)𝑀𝑛+𝛽𝑛π‘₯𝑛. Thus, we get 𝑀𝑛=π‘₯𝑛+1βˆ’π›½π‘›π‘₯𝑛1βˆ’π›½π‘›=𝛼𝑛𝑓π‘₯𝑛+π›Ύπ‘›π‘†π‘ƒπΆξ€·π‘¦π‘›βˆ’πœ†π‘›π΄π‘¦π‘›ξ€Έ1βˆ’π›½π‘›=𝛼𝑛𝑒+𝛾𝑛𝑆𝑑𝑛1βˆ’π›½π‘›(3.20) It follows that 𝑀𝑛+1βˆ’π‘€π‘›=𝛼𝑛+1𝑓π‘₯𝑛+1ξ€Έ+𝛾𝑛+1𝑆𝑑𝑛+11βˆ’π›½π‘›+1βˆ’π›Όπ‘›π‘“ξ€·π‘₯𝑛+𝛾𝑛𝑆𝑑𝑛1βˆ’π›½π‘›=𝛼𝑛+1𝑓π‘₯𝑛+1ξ€Έ1βˆ’π›½π‘›+1+𝛾𝑛+1𝑆𝑑𝑛+11βˆ’π›½π‘›+1βˆ’π›Όπ‘›+1𝑓π‘₯𝑛1βˆ’π›½π‘›+1+𝛼𝑛+1𝑓π‘₯𝑛1βˆ’π›½π‘›+1βˆ’π›Όπ‘›π‘“ξ€·π‘₯𝑛1βˆ’π›½π‘›βˆ’π›Ύπ‘›π‘†π‘‘π‘›1βˆ’π›½π‘›=𝛼𝑛+11βˆ’π›½π‘›+1𝑓π‘₯𝑛+1ξ€Έξ€·π‘₯βˆ’π‘“π‘›+𝛼𝑛+11βˆ’π›½π‘›+1βˆ’π›Όπ‘›1βˆ’π›½π‘›ξ‚Άπ‘“ξ€·π‘₯𝑛+𝛾𝑛+1𝑆𝑑𝑛+11βˆ’π›½π‘›+1βˆ’π›Ύπ‘›π‘†π‘‘π‘›1βˆ’π›½π‘›=𝛼𝑛+11βˆ’π›½π‘›+1𝑓π‘₯𝑛+1ξ€Έξ€·π‘₯βˆ’π‘“π‘›+𝛼𝑛+11βˆ’π›½π‘›+1βˆ’π›Όπ‘›1βˆ’π›½π‘›ξ‚Άπ‘“ξ€·π‘₯𝑛+𝛾𝑛+1𝑆𝑑𝑛+11βˆ’π›½π‘›+1βˆ’π›Ύπ‘›+1𝑆𝑑𝑛1βˆ’π›½π‘›+1+𝛾𝑛+1𝑆𝑑𝑛1βˆ’π›½π‘›+1βˆ’π›Ύπ‘›π‘†π‘‘π‘›1βˆ’π›½π‘›=𝛼𝑛+11βˆ’π›½π‘›+1𝑓π‘₯𝑛+1ξ€Έξ€·π‘₯βˆ’π‘“π‘›+𝛼𝑛+11βˆ’π›½π‘›+1βˆ’π›Όπ‘›1βˆ’π›½π‘›ξ‚Άπ‘“ξ€·π‘₯𝑛+𝛾𝑛+11βˆ’π›½π‘›+1𝑆𝑑𝑛+1βˆ’π‘†π‘‘π‘›ξ€Έ+𝛾𝑛+11βˆ’π›½π‘›+1βˆ’π›Ύπ‘›1βˆ’π›½π‘›ξ‚Άπ‘†π‘‘π‘›=𝛼𝑛+11βˆ’π›½π‘›+1𝑓π‘₯𝑛+1ξ€Έξ€·π‘₯βˆ’π‘“π‘›+𝛼𝑛+11βˆ’π›½π‘›+1βˆ’π›Όπ‘›1βˆ’π›½π‘›ξ‚Άπ‘“ξ€·π‘₯𝑛+𝛼𝑛+11βˆ’π›½π‘›+1βˆ’π›Όπ‘›1βˆ’π›½π‘›ξ‚Άπ‘†π‘‘π‘›+𝛾𝑛+11βˆ’π›½π‘›+1𝑆𝑑𝑛+1βˆ’π‘†π‘‘π‘›ξ€Έ=𝛼𝑛+11βˆ’π›½π‘›+1𝑓π‘₯𝑛+1ξ€Έξ€·π‘₯βˆ’π‘“π‘›+𝛼𝑛+11βˆ’π›½π‘›+1βˆ’π›Όπ‘›1βˆ’π›½π‘›ξ‚Άξ€·π‘“ξ€·π‘₯𝑛+𝑆𝑑𝑛+𝛾𝑛+11βˆ’π›½π‘›+1𝑆𝑑𝑛+1βˆ’π‘†π‘‘π‘›ξ€Έ.(3.21) Combining (3.19) and (3.21), we obtain ‖‖𝑀𝑛+1βˆ’π‘€π‘›β€–β€–βˆ’β€–β€–π‘₯𝑛+1βˆ’π‘₯𝑛‖‖≀||||𝛼𝑛+11βˆ’π›½π‘›+1||||‖‖𝑓π‘₯𝑛+1ξ€Έξ€·π‘₯βˆ’π‘“π‘›ξ€Έβ€–β€–+||||𝛼𝑛+11βˆ’π›½π‘›+1βˆ’π›Όπ‘›1βˆ’π›½π‘›||||‖‖𝑓π‘₯𝑛+𝑆𝑑𝑛‖‖+||||𝛾𝑛+11βˆ’π›½π‘›+1||||‖‖𝑆𝑑𝑛+1βˆ’π‘†π‘‘π‘›β€–β€–βˆ’β€–β€–π‘₯𝑛+1βˆ’π‘₯𝑛‖‖≀||||𝛼𝑛+11βˆ’π›½π‘›+1||||π‘˜β€–β€–π‘₯𝑛+1βˆ’π‘₯𝑛‖‖+||||𝛼𝑛+11βˆ’π›½π‘›+1βˆ’π›Όπ‘›1βˆ’π›½π‘›||||‖‖𝑓π‘₯𝑛+𝑆𝑑𝑛‖‖+||||𝛾𝑛+11βˆ’π›½π‘›+1||||‖‖𝑑𝑛+1βˆ’π‘‘π‘›β€–β€–βˆ’β€–β€–π‘₯𝑛+1βˆ’π‘₯𝑛‖‖≀||||𝛼𝑛+11βˆ’π›½π‘›+1||||π‘˜β€–β€–π‘₯𝑛+1βˆ’π‘₯𝑛‖‖+||||𝛼𝑛+11βˆ’π›½π‘›+1βˆ’π›Όπ‘›1βˆ’π›½π‘›||||‖‖𝑓π‘₯𝑛+𝑆𝑑𝑛‖‖+||||𝛾𝑛+11βˆ’π›½π‘›+1||||β€–β€–π‘₯𝑛+1βˆ’π‘₯π‘›β€–β€–βˆ’β€–β€–π‘₯𝑛+1βˆ’π‘₯𝑛‖‖=||||𝛼𝑛+11βˆ’π›½π‘›+1||||π‘˜β€–β€–π‘₯𝑛+1βˆ’π‘₯𝑛‖‖+||||𝛼𝑛+11βˆ’π›½π‘›+1βˆ’π›Όπ‘›1βˆ’π›½π‘›||||‖‖𝑓π‘₯𝑛+𝑆𝑑𝑛‖‖+||||𝛾𝑛+1βˆ’1+𝛽𝑛+11βˆ’π›½π‘›+1||||β€–β€–π‘₯𝑛+1βˆ’π‘₯𝑛‖‖=||||𝛼𝑛+11βˆ’π›½π‘›+1||||π‘˜β€–β€–π‘₯𝑛+1βˆ’π‘₯𝑛‖‖+||||𝛼𝑛+11βˆ’π›½π‘›+1βˆ’π›Όπ‘›1βˆ’π›½π‘›||||‖‖𝑓π‘₯𝑛+𝑆𝑑𝑛‖‖+||||𝛼𝑛+11βˆ’π›½π‘›+1||||β€–β€–π‘₯𝑛+1βˆ’π‘₯𝑛‖‖.(3.22) This together with (i), (ii), and (iii) implies that limsupπ‘›β†’βˆžξ€·β€–β€–π‘€π‘›+1βˆ’π‘€π‘›β€–β€–βˆ’β€–β€–π‘₯𝑛+1βˆ’π‘₯𝑛‖‖≀0.(3.23) Hence, by Lemma 2.3, we have limπ‘›β†’βˆžβ€–β€–π‘€π‘›βˆ’π‘₯𝑛‖‖=0.(3.24) Consequently, limπ‘›β†’βˆžβ€–β€–π‘₯𝑛+1βˆ’π‘₯𝑛‖‖=limπ‘›β†’βˆžξ€·1βˆ’π›½π‘›ξ€Έβ€–β€–π‘€π‘›βˆ’π‘₯𝑛‖‖=0.(3.25) From (3.18) and (3.19), we also have ‖𝑧𝑛+1βˆ’π‘§π‘›β€–β†’0‖𝑑𝑛+1βˆ’π‘‘π‘›β€–β†’0 and ‖𝑦𝑛+1βˆ’π‘¦π‘›β€–β†’0 as π‘›β†’βˆž. Since π‘₯𝑛+1βˆ’π‘₯𝑛=𝛼𝑛𝑓π‘₯𝑛+𝛽𝑛π‘₯𝑛+π›Ύπ‘›π‘†π‘‘π‘›βˆ’π‘₯𝑛=𝛼𝑛𝑓π‘₯π‘›ξ€Έβˆ’π‘₯𝑛+π›Ύπ‘›ξ€·π‘†π‘‘π‘›βˆ’π‘₯𝑛,(3.26) it follows by (ii) and (3.25) that limπ‘›β†’βˆžβ€–β€–π‘₯π‘›βˆ’π‘†π‘‘π‘›β€–β€–=0.(3.27) Since π‘₯βˆ—βˆˆπΉ(𝑆)βˆ©Ξ“, from (3.15) and Lemma 2.2, we get β€–β€–π‘₯𝑛+1βˆ’π‘₯βˆ—β€–β€–2=‖‖𝛼𝑛𝑓π‘₯𝑛+𝛽𝑛π‘₯𝑛+π›Ύπ‘›π‘†π‘‘π‘›βˆ’π‘₯βˆ—β€–β€–2≀𝛼𝑛‖‖𝑓π‘₯π‘›ξ€Έβˆ’π‘₯βˆ—β€–β€–2+𝛽𝑛‖‖π‘₯π‘›βˆ’π‘₯βˆ—β€–β€–2+π›Ύπ‘›β€–β€–π‘†π‘‘π‘›βˆ’π‘₯βˆ—β€–β€–2≀𝛼𝑛‖‖𝑓π‘₯π‘›ξ€Έβˆ’π‘₯βˆ—β€–β€–2+𝛽𝑛‖‖π‘₯π‘›βˆ’π‘₯βˆ—β€–β€–2+π›Ύπ‘›β€–β€–π‘‘π‘›βˆ’π‘₯βˆ—β€–β€–2=𝛼𝑛‖‖𝑓π‘₯π‘›ξ€Έβˆ’π‘₯βˆ—β€–β€–2+𝛽𝑛‖‖π‘₯π‘›βˆ’π‘₯βˆ—β€–β€–2+π›Ύπ‘›β€–β€–π‘ƒπΈξ€·π‘¦π‘›βˆ’πœ†π΄π‘¦π‘›ξ€Έβˆ’π‘ƒπΈξ€·π‘¦βˆ—βˆ’πœ†π΄π‘¦βˆ—ξ€Έβ€–β€–2≀𝛼𝑛‖‖𝑓π‘₯π‘›ξ€Έβˆ’π‘₯βˆ—β€–β€–2+𝛽𝑛‖‖π‘₯π‘›βˆ’π‘₯βˆ—β€–β€–2+π›Ύπ‘›β€–β€–ξ€·π‘¦π‘›βˆ’πœ†π΄π‘¦π‘›ξ€Έβˆ’ξ€·π‘¦βˆ—βˆ’πœ†π΄π‘¦βˆ—ξ€Έβ€–β€–2=𝛼𝑛‖‖𝑓π‘₯π‘›ξ€Έβˆ’π‘₯βˆ—β€–β€–2+𝛽𝑛‖‖π‘₯π‘›βˆ’π‘₯βˆ—β€–β€–2+π›Ύπ‘›β€–β€–ξ€·π‘¦π‘›βˆ’π‘¦βˆ—ξ€Έξ€·βˆ’πœ†π΄π‘¦π‘›βˆ’π΄π‘¦βˆ—ξ€Έβ€–β€–2=𝛼𝑛‖‖𝑓π‘₯π‘›ξ€Έβˆ’π‘₯βˆ—β€–β€–2+𝛽𝑛‖‖π‘₯π‘›βˆ’π‘₯βˆ—β€–β€–2+π›Ύπ‘›ξ‚ƒβ€–π‘¦π‘›βˆ’π‘¦βˆ—β€–2βˆ’2πœ†βŸ¨π‘¦π‘›βˆ’π‘¦βˆ—,π΄π‘¦π‘›βˆ’π΄π‘¦βˆ—βŸ©+πœ†2β€–β€–π΄π‘¦π‘›βˆ’π΄π‘¦βˆ—β€–β€–2≀𝛼𝑛‖‖𝑓π‘₯π‘›ξ€Έβˆ’π‘₯βˆ—β€–β€–2+𝛽𝑛‖‖π‘₯π‘›βˆ’π‘₯βˆ—β€–β€–2+π›Ύπ‘›ξ‚ƒβ€–β€–π‘¦π‘›βˆ’π‘¦βˆ—β€–β€–2β€–β€–βˆ’2πœ†π›Όπ΄π‘¦π‘›βˆ’π΄π‘¦βˆ—β€–β€–2+πœ†2β€–β€–π΄π‘¦π‘›βˆ’π΄π‘¦βˆ—β€–β€–2ξ‚„=𝛼𝑛‖‖𝑓π‘₯π‘›ξ€Έβˆ’π‘₯βˆ—β€–β€–2+𝛽𝑛‖‖π‘₯π‘›βˆ’π‘₯βˆ—β€–β€–2+π›Ύπ‘›ξ‚ƒβ€–β€–π‘¦π‘›βˆ’π‘¦βˆ—β€–β€–2β€–β€–+πœ†(πœ†βˆ’2𝛼)π΄π‘¦π‘›βˆ’π΄π‘¦βˆ—β€–β€–2≀𝛼𝑛‖‖𝑓π‘₯π‘›ξ€Έβˆ’π‘₯βˆ—β€–β€–2+𝛽𝑛‖‖π‘₯π‘›βˆ’π‘₯βˆ—β€–β€–2+𝛾𝑛‖‖π‘₯π‘›βˆ’π‘₯βˆ—β€–β€–2β€–β€–+πœ†(πœ†βˆ’2𝛼)π΄π‘¦π‘›βˆ’π΄π‘¦βˆ—β€–β€–2ξ‚„=𝛼𝑛‖‖𝑓π‘₯π‘›ξ€Έβˆ’π‘₯βˆ—β€–β€–2+𝛽𝑛‖‖π‘₯π‘›βˆ’π‘₯βˆ—β€–β€–2+𝛾𝑛‖‖π‘₯π‘›βˆ’π‘₯βˆ—β€–β€–2+π›Ύπ‘›β€–β€–πœ†(πœ†βˆ’2𝛼)π΄π‘¦π‘›βˆ’π΄π‘¦βˆ—β€–β€–2=𝛼𝑛‖‖𝑓π‘₯π‘›ξ€Έβˆ’π‘₯βˆ—β€–β€–2+𝛽𝑛+𝛾𝑛‖‖π‘₯π‘›βˆ’π‘₯βˆ—β€–β€–2+π›Ύπ‘›β€–β€–πœ†(πœ†βˆ’2𝛼)π΄π‘¦π‘›βˆ’π΄π‘¦βˆ—β€–β€–2=𝛼𝑛‖‖𝑓π‘₯π‘›ξ€Έβˆ’π‘₯βˆ—β€–β€–2+ξ€·1βˆ’π›Όπ‘›ξ€Έβ€–β€–π‘₯π‘›βˆ’π‘₯βˆ—β€–β€–2+π›Ύπ‘›β€–β€–πœ†(πœ†βˆ’2𝛼)π΄π‘¦π‘›βˆ’π΄π‘¦βˆ—β€–β€–2≀𝛼𝑛‖‖𝑓π‘₯π‘›ξ€Έβˆ’π‘₯βˆ—β€–β€–2+β€–β€–π‘₯π‘›βˆ’π‘₯βˆ—β€–β€–2+π›Ύπ‘›β€–β€–πœ†(πœ†βˆ’2𝛼)π΄π‘¦π‘›βˆ’π΄π‘¦βˆ—β€–β€–2.(3.28) Therefore, we have βˆ’π›Ύπ‘›β€–β€–πœ†(πœ†βˆ’2𝛼)π΄π‘¦π‘›βˆ’π΄π‘¦βˆ—β€–β€–2≀𝛼𝑛‖‖𝑓π‘₯π‘›ξ€Έβˆ’π‘₯βˆ—β€–β€–2+β€–β€–π‘₯π‘›βˆ’π‘₯βˆ—β€–β€–2βˆ’β€–β€–π‘₯𝑛+1βˆ’π‘₯βˆ—β€–β€–2=𝛼𝑛‖‖𝑓π‘₯π‘›ξ€Έβˆ’π‘₯βˆ—β€–β€–2+ξ€·β€–β€–π‘₯π‘›βˆ’π‘₯βˆ—β€–β€–+β€–β€–π‘₯𝑛+1βˆ’π‘₯βˆ—β€–β€–ξ€ΈΓ—ξ€·β€–β€–π‘₯π‘›βˆ’π‘₯βˆ—β€–β€–βˆ’β€–β€–π‘₯𝑛+1βˆ’π‘₯βˆ—β€–β€–ξ€Έβ‰€π›Όπ‘›β€–β€–π‘“ξ€·π‘₯π‘›ξ€Έβˆ’π‘₯βˆ—β€–β€–2+ξ€·β€–β€–π‘₯π‘›βˆ’π‘₯βˆ—β€–β€–+β€–β€–π‘₯𝑛+1βˆ’π‘₯βˆ—β€–β€–ξ€Έβ€–β€–π‘₯π‘›βˆ’π‘₯𝑛+1β€–β€–.(3.29) From (ii), (iii), and β€–π‘₯𝑛+1βˆ’π‘₯𝑛‖→0, as π‘›β†’βˆž, we get β€–π΄π‘¦π‘›βˆ’π΄π‘¦βˆ—β€–β†’0 as π‘›β†’βˆž.
Since π‘₯βˆ—βˆˆπΉ(𝑆)∩Ξ₯, from (3.11) and Lemma 2.2, we get β€–β€–π‘₯𝑛+1βˆ’π‘₯βˆ—β€–β€–2=‖‖𝛼𝑛𝑓π‘₯𝑛+𝛽𝑛π‘₯𝑛+π›Ύπ‘›π‘†π‘‘π‘›βˆ’π‘₯βˆ—β€–β€–2≀𝛼𝑛‖‖𝑓π‘₯π‘›ξ€Έβˆ’π‘₯βˆ—β€–β€–2+𝛽𝑛‖‖π‘₯π‘›βˆ’π‘₯βˆ—β€–β€–2+π›Ύπ‘›β€–β€–π‘‘π‘›βˆ’π‘₯βˆ—β€–β€–2≀𝛼𝑛‖‖𝑓π‘₯π‘›ξ€Έβˆ’π‘₯βˆ—β€–β€–2+𝛽𝑛‖‖π‘₯π‘›βˆ’π‘₯βˆ—β€–β€–2+π›Ύπ‘›β€–β€–π‘¦π‘›βˆ’π‘¦βˆ—β€–β€–2=𝛼𝑛‖‖𝑓π‘₯π‘›ξ€Έβˆ’π‘₯βˆ—β€–β€–2+𝛽𝑛‖‖π‘₯π‘›βˆ’π‘₯βˆ—β€–β€–2+π›Ύπ‘›β€–β€–π‘ƒπΈξ€·π‘§π‘›βˆ’πœ‡π΅π‘§π‘›ξ€Έβˆ’π‘ƒπΈξ€·π‘§βˆ—βˆ’πœ‡π΅π‘§βˆ—ξ€Έβ€–β€–2≀𝛼𝑛‖‖𝑓π‘₯π‘›ξ€Έβˆ’π‘₯βˆ—β€–β€–2+𝛽𝑛‖‖π‘₯π‘›βˆ’π‘₯βˆ—β€–β€–2+π›Ύπ‘›β€–β€–ξ€·π‘§π‘›βˆ’πœ‡π΅π‘§π‘›ξ€Έβˆ’ξ€·π‘§βˆ—βˆ’πœ‡π΅π‘§βˆ—ξ€Έβ€–β€–2=𝛼𝑛‖‖𝑓π‘₯π‘›ξ€Έβˆ’π‘₯βˆ—β€–β€–2+𝛽𝑛‖‖π‘₯π‘›βˆ’π‘₯βˆ—β€–β€–2+π›Ύπ‘›β€–β€–ξ€·π‘§π‘›βˆ’π‘§βˆ—ξ€Έβˆ’ξ€·πœ‡π΅π‘§π‘›βˆ’πœ‡π΅π‘§βˆ—ξ€Έβ€–β€–2≀𝛼𝑛‖‖𝑓π‘₯π‘›ξ€Έβˆ’π‘₯βˆ—β€–β€–2+𝛽𝑛‖‖π‘₯π‘›βˆ’π‘₯βˆ—β€–β€–2+π›Ύπ‘›ξ‚ƒβ€–β€–π‘§π‘›βˆ’π‘§βˆ—β€–β€–2β€–β€–+πœ‡(πœ‡βˆ’2𝛽)π΅π‘§π‘›βˆ’π΅π‘§βˆ—β€–β€–2≀𝛼𝑛‖‖𝑓π‘₯π‘›ξ€Έβˆ’π‘₯βˆ—β€–β€–2+β€–β€–π‘₯π‘›βˆ’π‘₯βˆ—β€–β€–2+π›Ύπ‘›β€–β€–πœ‡(πœ‡βˆ’2𝛽)π΅π‘§π‘›βˆ’π΅π‘§βˆ—β€–β€–2.(3.30)
Thus, we also have βˆ’π›Ύπ‘›β€–β€–πœ‡(πœ‡βˆ’2𝛽)π΅π‘§π‘›βˆ’π΅π‘§βˆ—β€–β€–2≀𝛼𝑛‖‖𝑓π‘₯π‘›ξ€Έβˆ’π‘₯βˆ—β€–β€–2+β€–β€–π‘₯π‘›βˆ’π‘₯βˆ—β€–β€–2βˆ’β€–β€–π‘₯𝑛+1βˆ’π‘₯βˆ—β€–β€–2=𝛼𝑛‖‖𝑓π‘₯π‘›ξ€Έβˆ’π‘₯βˆ—β€–β€–2+ξ€·β€–β€–π‘₯π‘›βˆ’π‘₯βˆ—β€–β€–+β€–β€–π‘₯𝑛+1βˆ’π‘₯βˆ—β€–β€–ξ€ΈΓ—ξ€·β€–β€–π‘₯π‘›βˆ’π‘₯βˆ—β€–β€–βˆ’β€–β€–π‘₯𝑛+1βˆ’π‘₯βˆ—β€–β€–ξ€Έβ‰€π›Όπ‘›β€–β€–π‘“ξ€·π‘₯π‘›ξ€Έβˆ’π‘₯βˆ—β€–β€–2+ξ€·β€–β€–π‘₯π‘›βˆ’π‘₯βˆ—β€–β€–+β€–β€–π‘₯𝑛+1βˆ’π‘₯βˆ—β€–β€–ξ€Έβ€–β€–π‘₯π‘›βˆ’π‘₯𝑛+1β€–β€–.(3.31) By again (ii), (iii), and (3.25), we also get β€–π΅π‘§π‘›βˆ’π΅π‘§βˆ—β€–β†’0 as π‘›β†’βˆž.
Let π‘₯βˆ—βˆˆπΉ(𝑆)∩Ξ₯; again from (3.12), (3.13) and Lemma 2.2, we get β€–β€–π‘₯𝑛+1βˆ’π‘₯βˆ—β€–β€–2=‖‖𝛼𝑛𝑓π‘₯𝑛+𝛽𝑛π‘₯𝑛+π›Ύπ‘›π‘†π‘‘π‘›βˆ’π‘₯βˆ—β€–β€–2≀𝛼𝑛‖‖𝑓π‘₯π‘›ξ€Έβˆ’π‘₯βˆ—β€–β€–2+𝛽𝑛‖‖π‘₯π‘›βˆ’π‘₯βˆ—β€–β€–2+π›Ύπ‘›β€–β€–π‘‘π‘›βˆ’π‘₯βˆ—β€–β€–2≀𝛼𝑛‖‖𝑓π‘₯π‘›ξ€Έβˆ’π‘₯βˆ—β€–β€–2+𝛽𝑛‖‖π‘₯π‘›βˆ’π‘₯βˆ—β€–β€–2+π›Ύπ‘›β€–β€–π‘§π‘›βˆ’π‘§βˆ—β€–β€–2≀𝛼𝑛‖‖𝑓π‘₯π‘›ξ€Έβˆ’π‘₯βˆ—β€–β€–2+𝛽𝑛‖‖π‘₯π‘›βˆ’π‘₯βˆ—β€–β€–2+𝛾𝑛‖‖π‘₯π‘›βˆ’πœπΆπ‘₯π‘›ξ€Έβˆ’ξ€·π‘₯βˆ—βˆ’πœπΆπ‘₯βˆ—ξ€Έβ€–β€–2≀𝛼𝑛‖‖𝑓π‘₯π‘›ξ€Έβˆ’π‘₯βˆ—β€–β€–2+𝛽𝑛‖‖π‘₯π‘›βˆ’π‘₯βˆ—β€–β€–2+𝛾𝑛‖‖π‘₯π‘›βˆ’π‘₯βˆ—β€–β€–2β€–β€–+𝜏(πœβˆ’2𝛾)𝐢π‘₯π‘›βˆ’πΆπ‘₯βˆ—β€–β€–2≀𝛼𝑛‖‖𝑓π‘₯π‘›ξ€Έβˆ’π‘₯βˆ—β€–β€–2+β€–β€–π‘₯π‘›βˆ’π‘₯βˆ—β€–β€–2+π›Ύπ‘›β€–β€–πœ(πœβˆ’2𝛾)𝐢π‘₯π‘›βˆ’πΆπ‘₯βˆ—β€–β€–2.(3.32)
Again, we have βˆ’π›Ύπ‘›β€–β€–πœ(πœβˆ’2𝛾)𝐢π‘₯π‘›βˆ’πΆπ‘₯βˆ—β€–β€–2≀𝛼𝑛‖‖𝑓π‘₯π‘›ξ€Έβˆ’π‘₯βˆ—β€–β€–2+β€–β€–π‘₯π‘›βˆ’π‘₯βˆ—β€–β€–2βˆ’β€–β€–π‘₯𝑛+1βˆ’π‘₯βˆ—β€–β€–2=𝛼𝑛‖‖𝑓π‘₯π‘›ξ€Έβˆ’π‘₯βˆ—β€–β€–2+ξ€·β€–β€–π‘₯π‘›βˆ’π‘₯βˆ—β€–β€–+β€–β€–π‘₯𝑛+1βˆ’π‘₯βˆ—β€–β€–ξ€ΈΓ—ξ€·β€–β€–π‘₯π‘›βˆ’π‘₯βˆ—β€–β€–βˆ’β€–β€–π‘₯𝑛+1βˆ’π‘₯βˆ—β€–β€–ξ€Έβ‰€π›Όπ‘›β€–β€–π‘“ξ€·π‘₯π‘›ξ€Έβˆ’π‘₯βˆ—β€–β€–2+ξ€·β€–β€–π‘₯π‘›βˆ’π‘₯βˆ—β€–β€–+β€–β€–π‘₯𝑛+1βˆ’π‘₯βˆ—β€–β€–ξ€Έβ€–β€–π‘₯π‘›βˆ’π‘₯𝑛+1β€–β€–.(3.33) Similarly again by (ii), (iii), and β€–π‘₯π‘›βˆ’π‘₯𝑛+1β€–β†’0 as π‘›β†’βˆž, and from (3.33), we also that ‖𝐢π‘₯π‘›βˆ’πΆπ‘₯βˆ—β€–β†’0.
On the other hand, we compute that β€–β€–π‘§π‘›βˆ’π‘§βˆ—β€–β€–2=‖‖𝑃𝐸π‘₯π‘›βˆ’πœπΆπ‘₯π‘›ξ€Έβˆ’π‘ƒπΈξ€·π‘₯βˆ—βˆ’πœπΆπ‘₯βˆ—ξ€Έβ€–β€–2≀π‘₯ξ«ξ€·π‘›βˆ’πœπΆπ‘₯π‘›ξ€Έβˆ’ξ€·π‘₯βˆ—βˆ’πœπΆπ‘₯βˆ—ξ€Έ,𝑃𝐸π‘₯π‘›βˆ’πœπΆπ‘₯π‘›ξ€Έβˆ’π‘ƒπΈξ€·π‘₯βˆ—βˆ’πœπΆπ‘₯βˆ—=π‘₯ξ€Έξ¬ξ«ξ€·π‘›βˆ’πœπΆπ‘₯π‘›ξ€Έβˆ’ξ€·π‘₯βˆ—βˆ’πœπΆπ‘₯βˆ—ξ€Έ,π‘§π‘›βˆ’π‘§βˆ—ξ¬=12‖‖π‘₯π‘›βˆ’πœπΆπ‘₯π‘›ξ€Έβˆ’ξ€·π‘₯βˆ—βˆ’πœπΆπ‘₯βˆ—ξ€Έβ€–β€–2+β€–β€–π‘§π‘›βˆ’π‘§βˆ—β€–β€–2βˆ’β€–β€–ξ€·π‘₯π‘›βˆ’πœπΆπ‘₯π‘›ξ€Έβˆ’ξ€·π‘₯βˆ—βˆ’πœπΆπ‘₯βˆ—ξ€Έβˆ’ξ€·π‘§π‘›βˆ’π‘§βˆ—ξ€Έβ€–β€–2ξ‚„=12‖‖(πΌβˆ’πœπΆ)π‘₯π‘›βˆ’(πΌβˆ’πœπΆ)π‘₯βˆ—β€–β€–2+β€–β€–π‘§π‘›βˆ’π‘§βˆ—β€–β€–2βˆ’β€–β€–ξ€·π‘₯π‘›βˆ’πœπΆπ‘₯π‘›ξ€Έβˆ’ξ€·π‘₯βˆ—βˆ’πœπΆπ‘₯βˆ—ξ€Έβˆ’ξ€·π‘§π‘›βˆ’π‘§βˆ—ξ€Έβ€–β€–2≀12ξ€Ίβ€–π‘₯π‘›βˆ’π‘₯βˆ—β€–2+β€–π‘§π‘›βˆ’π‘§βˆ—β€–2ξ€·π‘₯βˆ’β€–π‘›βˆ’π‘§π‘›ξ€Έξ€·βˆ’πœπΆπ‘₯π‘›βˆ’πΆπ‘₯βˆ—ξ€Έβˆ’ξ€·π‘₯βˆ—βˆ’π‘§βˆ—ξ€Έβ€–2ξ€»=12‖‖π‘₯π‘›βˆ’π‘₯βˆ—β€–β€–2+β€–β€–π‘§π‘›βˆ’π‘§βˆ—β€–β€–2βˆ’β€–β€–π‘₯ξ€Ίξ€·π‘›βˆ’π‘§π‘›ξ€Έβˆ’ξ€·π‘₯βˆ—βˆ’π‘§βˆ—ξ€·ξ€Έξ€»βˆ’πœπΆπ‘₯π‘›βˆ’πΆπ‘₯βˆ—ξ€Έβ€–β€–2ξ‚„=12‖‖π‘₯π‘›βˆ’π‘₯βˆ—β€–β€–2+β€–β€–π‘§π‘›βˆ’π‘§βˆ—β€–β€–2βˆ’β€–β€–ξ€·π‘₯π‘›βˆ’π‘§π‘›ξ€Έβˆ’ξ€·π‘₯βˆ—βˆ’π‘§βˆ—ξ€Έβ€–β€–2π‘₯+2πœξ«ξ€·π‘›βˆ’π‘§π‘›ξ€Έβˆ’ξ€·π‘₯βˆ—βˆ’π‘§βˆ—ξ€Έ,𝐢π‘₯π‘›βˆ’πΆπ‘₯βˆ—ξ¬βˆ’πœ2‖‖𝐢π‘₯π‘›βˆ’πΆπ‘₯βˆ—β€–β€–2ξ‚„.(3.34)
So, we obtain β€–β€–π‘§π‘›βˆ’π‘§βˆ—β€–β€–2≀‖‖π‘₯π‘›βˆ’π‘₯βˆ—β€–β€–2βˆ’β€–β€–ξ€·π‘₯π‘›βˆ’zπ‘›ξ€Έβˆ’ξ€·π‘₯βˆ—βˆ’π‘§βˆ—ξ€Έβ€–β€–2π‘₯+2πœξ«ξ€·π‘›βˆ’π‘§π‘›ξ€Έβˆ’ξ€·π‘₯βˆ—βˆ’π‘§βˆ—ξ€Έ,𝐢π‘₯π‘›βˆ’πΆπ‘₯βˆ—ξ¬βˆ’πœ2‖‖𝐢π‘₯π‘›βˆ’πΆπ‘₯βˆ—β€–β€–2.(3.35) Hence, by (3.12), it follows that β€–β€–π‘₯𝑛+1βˆ’π‘₯βˆ—β€–β€–2=‖‖𝛼𝑛𝑓π‘₯𝑛+𝛽𝑛π‘₯𝑛+π›Ύπ‘›π‘†π‘‘π‘›βˆ’π‘₯βˆ—β€–β€–2≀𝛼𝑛‖‖𝑓π‘₯π‘›ξ€Έβˆ’π‘₯βˆ—β€–β€–2+𝛽𝑛‖‖π‘₯π‘›βˆ’π‘₯βˆ—β€–β€–2+π›Ύπ‘›β€–β€–π‘†π‘‘π‘›βˆ’π‘₯βˆ—β€–β€–2β‰€π›Όπ‘›π‘˜β€–β€–π‘₯π‘›βˆ’π‘₯βˆ—β€–β€–2+𝛽𝑛‖‖π‘₯π‘›βˆ’π‘₯βˆ—β€–β€–2+π›Ύπ‘›β€–β€–π‘‘π‘›βˆ’π‘₯βˆ—β€–β€–2β‰€π›Όπ‘›π‘˜β€–β€–π‘₯π‘›βˆ’π‘₯βˆ—β€–β€–2+𝛽𝑛‖‖π‘₯π‘›βˆ’π‘₯βˆ—β€–β€–2+π›Ύπ‘›β€–β€–π‘§π‘›βˆ’π‘§βˆ—β€–β€–2β‰€π›Όπ‘›π‘˜β€–β€–π‘₯π‘›βˆ’π‘₯βˆ—β€–β€–2+𝛽𝑛‖‖π‘₯π‘›βˆ’π‘₯βˆ—β€–β€–2+𝛾𝑛‖‖π‘₯π‘›βˆ’π‘₯βˆ—β€–β€–2βˆ’π›Ύπ‘›β€–β€–ξ€·π‘₯π‘›βˆ’π‘§π‘›ξ€Έβˆ’ξ€·π‘₯βˆ—βˆ’π‘§βˆ—ξ€Έβ€–β€–2+2πœπ›Ύπ‘›π‘₯ξ«ξ€·π‘›βˆ’π‘§π‘›ξ€Έβˆ’ξ€·π‘₯βˆ—βˆ’π‘§βˆ—ξ€Έ,𝐢π‘₯π‘›βˆ’πΆπ‘₯βˆ—ξ¬βˆ’πœ2𝛾𝑛‖‖𝐢π‘₯π‘›βˆ’πΆπ‘₯βˆ—β€–β€–2=π›Όπ‘›π‘˜β€–β€–π‘₯π‘›βˆ’π‘₯βˆ—β€–β€–2+ξ€·1βˆ’π›Όπ‘›ξ€Έβ€–β€–π‘₯π‘›βˆ’π‘₯βˆ—β€–β€–2βˆ’π›Ύπ‘›β€–β€–ξ€·π‘₯π‘›βˆ’π‘§π‘›ξ€Έβˆ’ξ€·π‘₯βˆ—βˆ’π‘§βˆ—ξ€Έβ€–β€–2+2πœπ›Ύπ‘›π‘₯ξ«ξ€·π‘›βˆ’π‘§π‘›ξ€Έβˆ’ξ€·π‘₯βˆ—βˆ’π‘§βˆ—ξ€Έ,𝐢π‘₯π‘›βˆ’πΆπ‘₯βˆ—ξ¬βˆ’πœ2𝛾𝑛‖‖𝐢π‘₯π‘›βˆ’πΆπ‘₯βˆ—β€–β€–2β‰€π›Όπ‘›π‘˜β€–β€–π‘₯π‘›βˆ’π‘₯βˆ—β€–β€–2+β€–β€–π‘₯π‘›βˆ’π‘₯βˆ—β€–β€–2βˆ’π›Ύπ‘›β€–β€–ξ€·π‘₯π‘›βˆ’π‘§π‘›ξ€Έβˆ’ξ€·π‘₯βˆ—βˆ’π‘§βˆ—ξ€Έβ€–β€–2+2πœπ›Ύπ‘›π‘₯ξ«ξ€·π‘›βˆ’π‘§π‘›ξ€Έβˆ’ξ€·π‘₯βˆ—βˆ’π‘§βˆ—ξ€Έ,𝐢π‘₯π‘›βˆ’πΆπ‘₯βˆ—ξ¬β‰€π›Όπ‘›π‘˜β€–β€–π‘₯π‘›βˆ’π‘₯βˆ—β€–β€–2+β€–β€–π‘₯π‘›βˆ’π‘₯βˆ—β€–β€–2βˆ’π›Ύπ‘›β€–β€–ξ€·π‘₯π‘›βˆ’π‘§π‘›ξ€Έβˆ’ξ€·π‘₯βˆ—βˆ’π‘§βˆ—ξ€Έβ€–β€–2+2πœπ›Ύπ‘›β€–β€–ξ€·π‘₯π‘›βˆ’π‘§π‘›ξ€Έβˆ’ξ€·π‘₯βˆ—βˆ’π‘§βˆ—ξ€Έβ€–β€–β€–β€–πΆπ‘₯π‘›βˆ’πΆπ‘₯βˆ—β€–β€–,(3.36) which implies that 𝛾𝑛‖‖π‘₯π‘›βˆ’π‘§π‘›ξ€Έβˆ’ξ€·π‘₯βˆ—βˆ’π‘§βˆ—ξ€Έβ€–β€–2β‰€π›Όπ‘›π‘˜β€–β€–π‘₯π‘›βˆ’π‘₯βˆ—β€–β€–2+β€–β€–π‘₯π‘›βˆ’π‘₯βˆ—β€–β€–2βˆ’β€–β€–π‘₯𝑛+1βˆ’π‘₯βˆ—β€–β€–2+2πœπ›Ύπ‘›β€–β€–ξ€·π‘₯π‘›βˆ’π‘§π‘›ξ€Έβˆ’ξ€·π‘₯βˆ—βˆ’π‘§βˆ—ξ€Έβ€–β€–β€–β€–πΆπ‘₯π‘›βˆ’πΆπ‘₯βˆ—β€–β€–β‰€π›Όπ‘›π‘˜β€–β€–π‘₯𝑛+1βˆ’π‘₯βˆ—β€–β€–2+2π›Ύπ‘›πœβ€–β€–ξ€·π‘₯π‘›βˆ’π‘§π‘›ξ€Έβˆ’ξ€·π‘₯βˆ—βˆ’π‘§βˆ—ξ€Έβ€–β€–β€–β€–πΆπ‘₯π‘›βˆ’πΆπ‘₯βˆ—β€–β€–+β€–β€–π‘₯π‘›βˆ’π‘₯𝑛+1β€–β€–ξ€·β€–β€–π‘₯π‘›βˆ’π‘₯βˆ—β€–β€–+β€–β€–π‘₯𝑛+1βˆ’π‘₯βˆ—β€–β€–ξ€Έ.(3.37) By (ii), (iii), β€–π‘₯π‘›βˆ’π‘₯𝑛+1β€–β†’0, and ‖𝐢π‘₯π‘›βˆ’πΆπ‘₯βˆ—β€–β†’0 as π‘›β†’βˆž, from (3.37) and we get β€–(π‘₯π‘›βˆ’π‘§π‘›)βˆ’(π‘₯βˆ—βˆ’π‘§βˆ—)β€–β†’0 as π‘›β†’βˆž. Now, observe that β€–β€–ξ€·π‘§π‘›βˆ’π‘‘π‘›ξ€Έ+ξ€·π‘₯βˆ—βˆ’π‘§βˆ—ξ€Έβ€–β€–2=β€–β€–π‘§π‘›βˆ’π‘ƒπΈξ€·π‘¦π‘›βˆ’πœ†π΄π‘¦π‘›ξ€Έ+π‘ƒπΈξ€·π‘¦βˆ—βˆ’πœ†π΄yβˆ—ξ€Έβˆ’π‘§βˆ—β€–β€–2=β€–β€–π‘§π‘›βˆ’π‘ƒπΈξ€·π‘¦π‘›βˆ’πœ†π΄π‘¦π‘›ξ€Έ+π‘ƒπΈξ€·π‘¦βˆ—βˆ’πœ†π΄π‘¦βˆ—ξ€Έβˆ’π‘§βˆ—+πœ‡π΅π‘§π‘›βˆ’πœ‡π΅π‘§π‘›+πœ‡π΅π‘§βˆ—βˆ’πœ‡π΅π‘§βˆ—β€–β€–2=β€–β€–π‘§π‘›βˆ’πœ‡π΅π‘§π‘›βˆ’ξ€·π‘§βˆ—βˆ’πœ‡π΅π‘§βˆ—ξ€Έβˆ’ξ€Ίπ‘ƒπΈξ€·π‘¦π‘›βˆ’πœ†π΄π‘¦π‘›ξ€Έβˆ’π‘ƒπΈξ€·π‘¦βˆ—βˆ’πœ†π΄π‘¦βˆ—ξ€·ξ€Έξ€»+πœ‡π΅π‘§π‘›βˆ’π΅π‘§βˆ—ξ€Έβ€–β€–2β‰€β€–β€–π‘§π‘›βˆ’πœ‡π΅π‘§π‘›βˆ’ξ€·π‘§βˆ—βˆ’πœ‡π΅π‘§βˆ—ξ€Έβˆ’ξ€Ίπ‘ƒπΈξ€·π‘¦π‘›βˆ’πœ†π΄π‘¦π‘›ξ€Έβˆ’π‘ƒπΈξ€·π‘¦βˆ—βˆ’πœ†π΄π‘¦βˆ—β€–β€–ξ€Έξ€»2+2πœ‡π΅π‘§π‘›βˆ’π΅π‘§βˆ—,π‘§π‘›βˆ’πœ‡π΅π‘§π‘›βˆ’ξ€·π‘§βˆ—βˆ’πœ‡π΅π‘§βˆ—ξ€Έβˆ’ξ€Ίπ‘ƒπΈξ€·π‘¦π‘›βˆ’πœ†π΄π‘¦π‘›ξ€Έβˆ’π‘ƒπΈξ€·π‘¦βˆ—βˆ’πœ†π΄π‘¦βˆ—ξ€·ξ€Έξ€»+πœ‡π΅π‘§π‘›βˆ’π΅π‘§βˆ—=β€–β€–π‘§ξ€Έξ¬π‘›βˆ’πœ‡π΅π‘§π‘›βˆ’ξ€·π‘§βˆ—βˆ’πœ‡π΅π‘§βˆ—ξ€Έβˆ’ξ€Ίπ‘ƒπΈξ€·π‘¦π‘›βˆ’πœ†π΄π‘¦π‘›ξ€Έβˆ’π‘ƒπΈξ€·π‘¦βˆ—βˆ’πœ†π΄π‘¦βˆ—