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Journal of Applied Mathematics
Volumeย 2012ย (2012), Article IDย 538912, 29 pages
http://dx.doi.org/10.1155/2012/538912
Research Article

Iterative Algorithms for Solving the System of Mixed Equilibrium Problems, Fixed-Point Problems, and Variational Inclusions with Application to Minimization Problem

Department of Mathematics, Faculty of Science, King Mongkutโ€™s University of Technology Thonburi (KMUTT), Bangmod, Thrungkru, Bangkok 10140, Thailand

Received 7 October 2011; Accepted 1 November 2011

Academic Editor: Yeong-Chengย Liou

Copyright ยฉ 2012 Tanom Chamnarnpan and Poom Kumam. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We introduce a new iterative algorithm for solving a common solution of the set of solutions of fixed point for an infinite family of nonexpansive mappings, the set of solution of a system of mixed equilibrium problems, and the set of solutions of the variational inclusion for a ๐›ฝ-inverse-strongly monotone mapping in a real Hilbert space. We prove that the sequence converges strongly to a common element of the above three sets under some mild conditions. Furthermore, we give a numerical example which supports our main theorem in the last part.

1. Introduction

Let ๐ถ be a closed convex subset of a real Hilbert space ๐ป with the inner product โŸจโ‹…,โ‹…โŸฉ and the norm โ€–โ‹…โ€–. Let ๐น be a bifunction of ๐ถร—๐ถ into โ„›, where โ„› is the set of real numbers, ๐œ‘โˆถ๐ถโ†’โ„› be a real-valued function. Let ฮ› be arbitrary index set. The system of mixed equilibrium problem is for finding ๐‘ฅโˆˆ๐ถ such that๐น๐‘˜(๐‘ฅ,๐‘ฆ)+๐œ‘(๐‘ฆ)โˆ’๐œ‘(๐‘ฅ)โ‰ฅ0,๐‘˜โˆˆฮ›,โˆ€๐‘ฆโˆˆ๐ถ.(1.1) The set of solutions of (1.1) is denoted by SMEP(๐น๐‘˜), that is, ๎€ท๐นSMEP๐‘˜๎€ธ=๎€ฝ๐‘ฅโˆˆ๐ถโˆถ=๐น๐‘˜๎€พ(๐‘ฅ,๐‘ฆ)+๐œ‘(๐‘ฆ)โˆ’๐œ‘(๐‘ฅ)โ‰ฅ0,๐‘˜โˆˆฮ›,โˆ€๐‘ฆโˆˆ๐ถ.(1.2) If ฮ› is a singleton, then problem (1.1) becomes the following mixed equilibrium problem: finding ๐‘ฅโˆˆ๐ถ such that๐น(๐‘ฅ,๐‘ฆ)+๐œ‘(๐‘ฆ)โˆ’๐œ‘(๐‘ฅ)โ‰ฅ0,โˆ€๐‘ฆโˆˆ๐ถ.(1.3) The set of solutions of (1.3) is denoted by MEP(๐น).

If ๐œ‘โ‰ก0, the problem (1.3) is reduced into the equilibrium problem [1] for finding ๐‘ฅโˆˆ๐ถ such that๐น(๐‘ฅ,๐‘ฆ)โ‰ฅ0,โˆ€๐‘ฆโˆˆ๐ถ.(1.4) The set of solutions of (1.4) is denoted by EP(๐น). This problem contains fixed-point problems, includes as special cases numerous problems in physics, optimization, and economics. Some methods have been proposed to solve the system of mixed equilibrium problem and the equilibrium problem, please consult [2โ€“19].

Recall that, a mapping ๐‘†โˆถ๐ถโ†’๐ถ is said to be nonexpansive if โ€–๐‘†๐‘ฅโˆ’๐‘†๐‘ฆโ€–โ‰คโ€–๐‘ฅโˆ’๐‘ฆโ€–,(1.5) for all ๐‘ฅ,๐‘ฆโˆˆ๐ถ. If ๐ถ is a bounded closed convex and ๐‘† is a nonexpansive mapping of ๐ถ into itself, then ๐น(๐‘†) is nonempty [20]. Let ๐ดโˆถ๐ถโ†’๐ป be a mapping, the Hartmann-Stampacchia variational inequality for finding ๐‘ฅโˆˆ๐ถ such thatโŸจ๐ด๐‘ฅ,๐‘ฆโˆ’๐‘ฅโŸฉโ‰ฅ0,โˆ€๐‘ฆโˆˆ๐ถ.(1.6) The set of solutions of (1.6) is denoted by VI(๐ถ,๐ด). The variational inequality has been extensively studied in the literature [21โ€“28].

Iterative methods for nonexpansive mappings have recently been applied to solve convex minimization problems. Convex minimization problems have a great impact and influence on the development of almost all branches of pure and applied sciences. 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:1๐œƒ(๐‘ฅ)=2โŸจ๐ด๐‘ฅ,๐‘ฅโŸฉโˆ’โŸจ๐‘ฅ,๐‘ฆโŸฉ,โˆ€๐‘ฅโˆˆ๐น(๐‘†),(1.7) where ๐ด is a linear bounded operator, ๐น(๐‘†) is the fixed point set of a nonexpansive mapping ๐‘†, and ๐‘ฆ is a given point in ๐ป [29].

We denote weak convergence and strong convergence by notations โ‡€ and โ†’, respectively. A mapping ๐ด of ๐ถ into ๐ป is called monotone if โŸจ๐ด๐‘ฅโˆ’๐ด๐‘ฆ,๐‘ฅโˆ’๐‘ฆโŸฉโ‰ฅ0,(1.8) for all ๐‘ฅ,๐‘ฆโˆˆ๐ถ. A mapping ๐ด of ๐ถ into ๐ป is called ๐›ผ-inverse-strongly monotone if there exists a positive real number ๐›ผ such that โŸจ๐ด๐‘ฅโˆ’๐ด๐‘ฆ,๐‘ฅโˆ’๐‘ฆโŸฉโ‰ฅ๐›ผโ€–๐ด๐‘ฅโˆ’๐ด๐‘ฆโ€–2,(1.9) for all ๐‘ฅ,๐‘ฆโˆˆ๐ถ. It is obvious that any ๐›ผ-inverse-strongly monotone mappings ๐ด are monotone and Lipschitz continuous mapping. A linear bounded operator ๐ด is strongly positive if there exists a constant ๐›พ>0 with the property โŸจ๐ด๐‘ฅ,๐‘ฅโŸฉโ‰ฅ๐›พโ€–๐‘ฅโ€–2,(1.10) for all ๐‘ฅโˆˆ๐ป. A self-mapping ๐‘“โˆถ๐ถโ†’๐ถ is a contraction on ๐ถ if there exists a constant ๐›ผโˆˆ(0,1) such that โ€–๐‘“(๐‘ฅ)โˆ’๐‘“(๐‘ฆ)โ€–โ‰ค๐›ผโ€–๐‘ฅโˆ’๐‘ฆโ€–,(1.11) for all ๐‘ฅ,๐‘ฆโˆˆ๐ถ. We use ฮ ๐ถ to denote the collection of all contraction on ๐ถ. Note that each ๐‘“โˆˆฮ ๐ถ has a unique fixed point in ๐ถ.

Let ๐ตโˆถ๐ปโ†’๐ป be a single-valued nonlinear mapping and ๐‘€โˆถ๐ปโ†’2๐ป be a set-valued mapping. The variational inclusion problem is to find ๐‘ฅโˆˆ๐ป such that๐œƒโˆˆ๐ต(๐‘ฅ)+๐‘€(๐‘ฅ),(1.12) where ๐œƒ is the zero vector in ๐ป. The set of solutions of problem (1.12) is denoted by ๐ผ(๐ต,๐‘€). The variational inclusion has been extensively studied in the literature, see, for example, [30โ€“32] and the reference therein.

A set-valued mapping ๐‘€โˆถ๐ปโ†’2๐ป is called monotone if for all ๐‘ฅ,๐‘ฆโˆˆ๐ป, ๐‘“โˆˆ๐‘€(๐‘ฅ), and ๐‘”โˆˆ๐‘€(๐‘ฆ) impling โŸจ๐‘ฅโˆ’๐‘ฆ,๐‘“โˆ’๐‘”โŸฉโ‰ฅ0. A monotone mapping ๐‘€ is maximal if its graph ๐บ(๐‘€)โˆถ={(๐‘“,๐‘ฅ)โˆˆ๐ปร—๐ปโˆถ๐‘“โˆˆ๐‘€(๐‘ฅ)} of ๐‘€ is not properly contained in the graph of any other monotone mapping. It is known that a monotone mapping ๐‘€ is maximal if and only if for (๐‘ฅ,๐‘“)โˆˆ๐ปร—๐ป,โŸจ๐‘ฅโˆ’๐‘ฆ,๐‘“โˆ’๐‘”โŸฉโ‰ฅ0 for all (๐‘ฆ,๐‘”)โˆˆ๐บ(๐‘€) impling ๐‘“โˆˆ๐‘€(๐‘ฅ).

Let ๐ต be an inverse-strongly monotone mapping of ๐ถ into ๐ป, and let ๐‘๐ถ๐‘ฃ be normal cone to ๐ถ at ๐‘ฃโˆˆ๐ถ, that is, ๐‘๐ถ๐‘ฃ={๐‘คโˆˆ๐ปโˆถโŸจ๐‘ฃโˆ’๐‘ข,๐‘คโŸฉโ‰ฅ0,forall๐‘ขโˆˆ๐ถ}, and define๎‚ป๐‘‡๐‘ฃ=๐ต๐‘ฃ+๐‘๐ถ๐‘ฃ,if๐‘ฃโˆˆ๐ถ,โˆ…,if๐‘ฃโˆ‰๐ถ.(1.13) Then, ๐‘‡ is a maximal monotone and ๐œƒโˆˆ๐‘‡๐‘ฃ if and only if ๐‘ฃโˆˆVI(๐ถ,๐ต) (see [33]).

Let ๐‘€โˆถ๐ปโ†’2๐ป be a set-valued maximal monotone mapping, then the single-valued mapping ๐ฝ๐‘€,๐œ†โˆถ๐ปโ†’๐ป defined by๐ฝ๐‘€,๐œ†(๐‘ฅ)=(๐ผ+๐œ†๐‘€)โˆ’1(๐‘ฅ),๐‘ฅโˆˆ๐ป,(1.14) is called the resolvent operator associated with ๐‘€, where ๐œ† is any positive number and ๐ผ is the identity mapping. It is worth mentioning that the resolvent operator is nonexpansive, 1-inverse-strongly monotone, and that a solution of problem (1.12) is a fixed point of the operator ๐ฝ๐‘€,๐œ†(๐ผโˆ’๐œ†๐ต) for all ๐œ†>0, (for more details see [34]).

In 2000, Moudafi [35] introduced the viscosity approximation method for nonexpansive mappings and proved that if ๐ป is a real Hilbert space, the sequence {๐‘ฅ๐‘›} defined by the iterative method below, with the initial guess ๐‘ฅ0โˆˆ๐ถ is chosen arbitrarily,๐‘ฅ๐‘›+1=๐›ผ๐‘›๐‘“๎€ท๐‘ฅ๐‘›๎€ธ+๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐‘†๐‘ฅ๐‘›,๐‘›โ‰ฅ0,(1.15) where {๐›ผ๐‘›}โŠ‚(0,1) satisfies certain conditions and converges strongly to a fixed point of ๐‘† (say ๐‘ฅโˆˆ๐ถ), which is then a unique solution of the following variational inequality:๎ซ(๐ผโˆ’๐‘“)๐‘ฅ,๐‘ฅโˆ’๐‘ฅ๎ฌโ‰ฅ0,โˆ€๐‘ฅโˆˆ๐น(๐‘†).(1.16)

In 2006, Marino and Xu [29] introduced a general iterative method for nonexpansive mapping. They defined the sequence {๐‘ฅ๐‘›} generated by the algorithm ๐‘ฅ0โˆˆ๐ถ,๐‘ฅ๐‘›+1=๐›ผ๐‘›๎€ท๐‘ฅ๐›พ๐‘“๐‘›๎€ธ+๎€ท๐ผโˆ’๐›ผ๐‘›๐ด๎€ธ๐‘†๐‘ฅ๐‘›,๐‘›โ‰ฅ0,(1.17) where {๐›ผ๐‘›}โŠ‚(0,1), and ๐ด is a strongly positive linear bounded operator. They proved that if ๐ถ=๐ป, and the sequence {๐›ผ๐‘›} satisfies appropriate conditions, then the sequence {๐‘ฅ๐‘›} generated by (1.17) converges strongly to a fixed point of ๐‘† (say ๐‘ฅโˆˆ๐ป) which is the unique solution of the following variational inequality:๎ซ(๐ดโˆ’๐›พ๐‘“)๐‘ฅ,๐‘ฅโˆ’๐‘ฅ๎ฌโ‰ฅ0,โˆ€๐‘ฅโˆˆ๐น(๐‘†),(1.18) which is the optimality condition for the minimization problemmin๐‘ฅโˆˆ๐น(๐‘†)โˆฉEP(๐น)12โŸจ๐ด๐‘ฅ,๐‘ฅโŸฉโˆ’โ„Ž(๐‘ฅ),(1.19) where โ„Ž is a potential function for ๐›พ๐‘“ (i.e., โ„Ž๎…ž(๐‘ฅ)=๐›พ๐‘“(๐‘ฅ) for ๐‘ฅโˆˆ๐ป).

For finding a common element of the set of fixed points of nonexpansive mappings and the set of solution of the variational inequalities. Let ๐‘ƒ๐ถ be the projection of ๐ป onto ๐ถ. In 2005, Iiduka and Takahashi [36] introduced the following iterative process for ๐‘ฅ0โˆˆ๐ถ,๐‘ฅ๐‘›+1=๐›ผ๐‘›๎€ท๐‘ข+1โˆ’๐›ผ๐‘›๎€ธ๐‘†๐‘ƒ๐ถ๎€ท๐‘ฅ๐‘›โˆ’๐œ†๐‘›๐ด๐‘ฅ๐‘›๎€ธ,โˆ€๐‘›โ‰ฅ0,(1.20) where ๐‘ขโˆˆ๐ถ, {๐›ผ๐‘›}โŠ‚(0,1), and {๐œ†๐‘›}โŠ‚[๐‘Ž,๐‘] for some ๐‘Ž,๐‘ with 0<๐‘Ž<๐‘<2๐›ฝ. They proved that under certain appropriate conditions imposed on {๐›ผ๐‘›} and {๐œ†๐‘›}, the sequence {๐‘ฅ๐‘›} generated by (1.20) converges strongly to a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality for an inverse-strongly monotone mapping (say ๐‘ฅโˆˆ๐ถ) which solve some variational inequality๎ซ๐‘ฅโˆ’๐‘ข,๐‘ฅโˆ’๐‘ฅ๎ฌโ‰ฅ0,โˆ€๐‘ฅโˆˆ๐น(๐‘†)โˆฉVI(๐ถ,A).(1.21)

In 2008, Su et al. [37] introduced the following iterative scheme by the viscosity approximation method in a real Hilbert space: ๐‘ฅ1,๐‘ข๐‘›โˆˆ๐ป๐น๎€ท๐‘ข๐‘›๎€ธ+1,๐‘ฆ๐‘Ÿ๐‘›โŸจ๐‘ฆโˆ’๐‘ข๐‘›,๐‘ข๐‘›โˆ’๐‘ฅ๐‘›๐‘ฅโŸฉโ‰ฅ0,โˆ€๐‘ฆโˆˆ๐ถ,๐‘›+1=๐›ผ๐‘›๐‘“๎€ท๐‘ฅ๐‘›๎€ธ+๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐‘†๐‘ƒ๐ถ๎€ท๐‘ข๐‘›โˆ’๐œ†๐‘›๐ด๐‘ข๐‘›๎€ธ,(1.22) for all ๐‘›โˆˆโ„•, where {๐›ผ๐‘›}โŠ‚[0,1) and {๐‘Ÿ๐‘›}โŠ‚(0,โˆž) satisfing some appropriate conditions. Furthermore, they proved that {๐‘ฅ๐‘›} and {๐‘ข๐‘›} converge strongly to the same point ๐‘งโˆˆ๐น(๐‘†)โˆฉVI(๐ถ,๐ด)โˆฉEP(๐น), where ๐‘ง=๐‘ƒ๐น(๐‘†)โˆฉVI(๐ถ,๐ด)โˆฉEP(๐น)๐‘“(๐‘ง).

Let {๐‘‡๐‘–} be an infinite family of nonexpansive mappings of ๐ป into itself, and let {๐œ†๐‘–} be a real sequence such that 0โ‰ค๐œ†๐‘–โ‰ค1 for every ๐‘–โˆˆ๐‘. For ๐‘›โ‰ฅ1, we defined a mapping ๐‘Š๐‘› of ๐ป into itself as follows:๐‘ˆ๐‘›,๐‘›+1๐‘ˆโˆถ=๐ผ,๐‘›,๐‘›โˆถ=๐œ†๐‘›๐‘‡๐‘›๐‘ˆ๐‘›,๐‘›+1+๎€ท1โˆ’๐œ†๐‘›๎€ธโ‹ฎ๐‘ˆ๐ผ,๐‘›,๐‘˜โˆถ=๐œ†๐‘˜๐‘‡๐‘˜๐‘ˆ๐‘›,๐‘˜+1+๎€ท1โˆ’๐œ†๐‘˜๎€ธโ‹ฎ๐‘ˆ๐ผ,๐‘›,2โˆถ=๐œ†2๐‘‡2๐‘ˆ๐‘›,3+๎€ท1โˆ’๐œ†2๎€ธ๐‘Š๐ผ,๐‘›โˆถ=๐‘ˆ๐‘›,1โˆถ=๐œ†1๐‘‡1๐‘ˆ๐‘›,2+๎€ท1โˆ’๐œ†1๎€ธ๐ผ.(1.23)

In 2011, He et al. [38] introduced the following iterative process for {๐‘‡๐‘›โˆถ๐ถโ†’๐ถ} which is a sequence of nonexpansive mappings. Let {๐‘ง๐‘›} be the sequence defined by๐‘ง๐‘›+1=๐œ–๐‘›๎€ท๐‘ง๐›พ๐‘“๐‘›๎€ธ+๎€ท๐ผโˆ’๐œ–๐‘›๎€ธ๐‘Š๐‘›๐พ1๐‘Ÿ1,๐‘›๐พ2๐‘Ÿ2,๐‘›โ‹…โ‹ฏโ‹…๐พ๐พ๐‘Ÿ๐พ,๐‘›๐‘ง๐‘›,โˆ€๐‘›โˆˆ๐‘.(1.24) The sequence {๐‘ง๐‘›} defined by (1.24) converges strongly to a common element of the set of fixed points of nonexpansive mappings, the set of solutions of the variational inequality, and the generalized equilibrium problem. Recently, Jitpeera and Kumam [39] introduced the following new general iterative method for finding a common element of the set of solutions of fixed point for nonexpansive mappings, the set of solution of generalized mixed equilibrium problems, and the set of solutions of the variational inclusion for a ๐›ฝ-inverse-strongly monotone mapping in a real Hilbert space.

In this paper, we modify the iterative methods (1.17), (1.22), and (1.24) by purposing the following new general viscosity iterative method: ๐‘ฅ0,๐‘ข๐‘›โˆˆ๐ถ,๐‘ข๐‘›=๐พ๐น๐‘๐‘Ÿ๐‘›,๐‘›โ‹…๐พ๐น๐‘โˆ’1๐‘Ÿ๐‘›โˆ’1,๐‘›โ‹…๐พ๐น๐‘โˆ’2๐‘Ÿ๐‘›โˆ’2,๐‘›โ‹…โ‹ฏโ‹…๐พ๐น2๐‘Ÿ2,๐‘›โ‹…๐พ๐น1๐‘Ÿ1,๐‘›โ‹…๐‘ฅ๐‘›๐‘ฅ,โˆ€๐‘›โˆˆ๐‘๐‘›+1=๐‘ƒ๐ถ๎€บ๐œ–๐‘›๎€ท๐‘ฅ๐›พ๐‘“๐‘›๎€ธ+๎€ท๐ผโˆ’๐œ–๐‘›๐ด๎€ธ๐‘Š๐‘›๐ฝ๐‘€,๐œ†๎€ท๐‘ข๐‘›โˆ’๐œ†๐ต๐‘ข๐‘›,๎€ธ๎€ป(1.25) for all ๐‘›โˆˆโ„•, where {๐›ผ๐‘›}โŠ‚(0,1), {๐‘Ÿ๐‘›}โŠ‚(0,2๐œŽ), and ๐œ†โˆˆ(0,2๐›ฝ) satisfy some appropriate conditions. The purpose of this paper shows that under some control conditions the sequence {๐‘ฅ๐‘›} converges strongly to a common element of the set of common fixed points of nonexpansive mappings, the solution of the system of mixed equilibrium problems, and the set of solutions of the variational inclusion in a real Hilbert space. Moreover, we apply our results to the class of strictly pseudocontractive mappings. Finally, we give a numerical example which supports our main theorem in the last part. Our results improve and extend the corresponding results of Marino and Xu [29], Su et al. [37], He et al. [38], and some authors.

2. Preliminaries

Let ๐ป be a real Hilbert space and ๐ถ be a nonempty closed and convex subset of ๐ป. Recall that the (nearest point) projection ๐‘ƒ๐ถ from ๐ป onto ๐ถ assigns to each ๐‘ฅโˆˆ๐ป and the unique point in ๐‘ƒ๐ถ๐‘ฅโˆˆ๐ถ satisfies the property โ€–โ€–๐‘ฅโˆ’๐‘ƒ๐ถ๐‘ฅโ€–โ€–=min๐‘ฆโˆˆ๐ถโ€–๐‘ฅโˆ’๐‘ฆโ€–,(2.1) which is equivalent to the following inequalityโŸจ๐‘ฅโˆ’๐‘ƒ๐ถ๐‘ฅ,๐‘ƒ๐ถ๐‘ฅโˆ’๐‘ฆโŸฉโ‰ฅ0,โˆ€๐‘ฆโˆˆ๐ถ.(2.2) The following characterizes the projection ๐‘ƒ๐ถ. We recall some lemmas which will be needed in the rest of this paper.

Lemma 2.1. The function ๐‘ขโˆˆ๐ถ is a solution of the variational inequality if and only if ๐‘ขโˆˆ๐ถ satisfies the relation ๐‘ข=๐‘ƒ๐ถ(๐‘ขโˆ’๐œ†๐ต๐‘ข) for all ๐œ†>0.

Lemma 2.2. For a given ๐‘งโˆˆ๐ป, ๐‘ขโˆˆ๐ถ, ๐‘ข=๐‘ƒ๐ถ๐‘งโ‡”โŸจ๐‘ขโˆ’๐‘ง,๐‘ฃโˆ’๐‘ขโŸฉโ‰ฅ0,โˆ€๐‘ฃโˆˆ๐ถ.
It is well known that ๐‘ƒ๐ถ is a firmly nonexpansive mapping of ๐ป onto ๐ถ and satisfies โ€–โ€–๐‘ƒ๐ถ๐‘ฅโˆ’๐‘ƒ๐ถ๐‘ฆโ€–โ€–2โ‰คโŸจ๐‘ƒ๐ถ๐‘ฅโˆ’๐‘ƒ๐ถ๐‘ฆ,๐‘ฅโˆ’๐‘ฆโŸฉ,โˆ€๐‘ฅ,๐‘ฆโˆˆ๐ป.(2.3) Moreover, ๐‘ƒ๐ถ๐‘ฅ is characterized by the following properties: ๐‘ƒ๐ถ๐‘ฅโˆˆ๐ถ and for all ๐‘ฅโˆˆ๐ป,๐‘ฆโˆˆ๐ถ, โŸจ๐‘ฅโˆ’๐‘ƒ๐ถ๐‘ฅ,๐‘ฆโˆ’๐‘ƒ๐ถ๐‘ฅโŸฉโ‰ค0.(2.4)

Lemma 2.3 (see [40]). Let ๐‘€โˆถ๐ปโ†’2๐ป be a maximal monotone mapping, and let ๐ตโˆถ๐ปโ†’๐ป be a monotone and Lipshitz continuous mapping. Then the mapping ๐ฟ=๐‘€+๐ตโˆถ๐ปโ†’2๐ป is a maximal monotone mapping.

Lemma 2.4 (see [41]). Each Hilbert space ๐ป satisfies Opial's condition, that is, for any sequence {๐‘ฅ๐‘›}โŠ‚๐ป with ๐‘ฅ๐‘›โ‡€๐‘ฅ, the inequality liminf๐‘›โ†’โˆžโ€–๐‘ฅ๐‘›โˆ’๐‘ฅโ€–<liminf๐‘›โ†’โˆžโ€–๐‘ฅ๐‘›โˆ’๐‘ฆโ€–, hold for each ๐‘ฆโˆˆ๐ป with ๐‘ฆโ‰ ๐‘ฅ.

Lemma 2.5 (see [42]). Assume {๐‘Ž๐‘›} is a sequence of nonnegative real numbers such that ๐‘Ž๐‘›+1โ‰ค๎€ท1โˆ’๐›พ๐‘›๎€ธ๐‘Ž๐‘›+๐›ฟ๐‘›,โˆ€๐‘›โ‰ฅ0,(2.5) where {๐›พ๐‘›}โŠ‚(0,1) and {๐›ฟ๐‘›} is a sequence in โ„› such that(i)โˆ‘โˆž๐‘›=1๐›พ๐‘›=โˆž, (ii)limsup๐‘›โ†’โˆž๐›ฟ๐‘›/๐›พ๐‘›โ‰ค0 or โˆ‘โˆž๐‘›=1|๐›ฟ๐‘›|<โˆž.Then lim๐‘›โ†’โˆž๐‘Ž๐‘›=0.

Lemma 2.6 (see [43]). Let ๐ถ be a closed convex subset of a real Hilbert space ๐ป, and let ๐‘‡โˆถ๐ถโ†’๐ถ be a nonexpansive mapping. Then ๐ผโˆ’๐‘‡ is demiclosed at zero, that is, ๐‘ฅ๐‘›โŸถ๐‘ฅ,๐‘ฅ๐‘›โˆ’๐‘‡๐‘ฅ๐‘›โŸถ0,(2.6) implying ๐‘ฅ=๐‘‡๐‘ฅ.

For solving the mixed equilibrium problem, let us assume that the bifunction ๐นโˆถ๐ถร—๐ถโ†’โ„› and the nonlinear mapping ๐œ‘โˆถ๐ถโ†’โ„› satisfy the following conditions: (A1)๐น(๐‘ฅ,๐‘ฅ)=0 for all ๐‘ฅโˆˆ๐ถ; (A2)๐น is monotone, that is, ๐น(๐‘ฅ,๐‘ฆ)+๐น(๐‘ฆ,๐‘ฅ)โ‰ค0 for any ๐‘ฅ,๐‘ฆโˆˆ๐ถ; (A3)for each fixed ๐‘ฆโˆˆ๐ถ, ๐‘ฅโ†ฆ๐น(๐‘ฅ,๐‘ฆ) is weakly upper semicontinuous; (A4)for each fixed ๐‘ฅโˆˆ๐ถ, ๐‘ฆโ†ฆ๐น(๐‘ฅ,๐‘ฆ) is convex and lower semicontinuous; (B1)for each ๐‘ฅโˆˆ๐ถ and ๐‘Ÿ>0, there exist a bounded subset ๐ท๐‘ฅโŠ†๐ถ and ๐‘ฆ๐‘ฅโˆˆ๐ถ such that for any ๐‘งโˆˆ๐ถโงต๐ท๐‘ฅ,๐น๎€ท๐‘ง,๐‘ฆ๐‘ฅ๎€ธ๎€ท๐‘ฆ+๐œ‘๐‘ฅ๎€ธ1โˆ’๐œ‘(๐‘ง)+๐‘ŸโŸจ๐‘ฆ๐‘ฅโˆ’๐‘ง,๐‘งโˆ’๐‘ฅโŸฉ<0,(2.7)(B2)๐ถ is a bounded set.

Lemma 2.7 (see [44]). Let ๐ถ be a nonempty closed and convex subset of a real Hilbert space ๐ป. Let ๐นโˆถ๐ถร—๐ถโ†’โ„› be a bifunction mapping satisfying (A1)โ€“(A4), and let ๐œ‘โˆถ๐ถโ†’โ„› be a convex and lower semicontinuous function such that ๐ถโˆฉdom๐œ‘โ‰ โˆ…. Assume that either (B1) or (B2) holds. For ๐‘Ÿ>0 and ๐‘ฅโˆˆ๐ป, then there exists ๐‘ขโˆˆ๐ถ such that 1๐น(๐‘ข,๐‘ฆ)+๐œ‘(๐‘ฆ)โˆ’๐œ‘(๐‘ข)+๐‘ŸโŸจ๐‘ฆโˆ’๐‘ข,๐‘ขโˆ’๐‘ฅโŸฉโ‰ฅ0.(2.8) Define a mapping ๐พ๐‘Ÿโˆถ๐ปโ†’๐ถ as follows: ๐พ๐‘Ÿ๎‚†1(๐‘ฅ)=๐‘ขโˆˆ๐ถโˆถ๐น(๐‘ข,๐‘ฆ)+๐œ‘(๐‘ฆ)โˆ’๐œ‘(๐‘ข)+๐‘Ÿ๎‚‡,โŸจ๐‘ฆโˆ’๐‘ข,๐‘ขโˆ’๐‘ฅโŸฉโ‰ฅ0,โˆ€๐‘ฆโˆˆ๐ถ(2.9) for all ๐‘ฅโˆˆ๐ป. Then, the following hold: (i)๐พ๐‘Ÿ is single-valued; (ii)๐พ๐‘Ÿ is firmly nonexpansive, that is, for any ๐‘ฅ,๐‘ฆโˆˆ๐ป, โ€–๐พ๐‘Ÿ๐‘ฅโˆ’๐พ๐‘Ÿ๐‘ฆโ€–2โ‰คโŸจ๐พ๐‘Ÿ๐‘ฅโˆ’๐พ๐‘Ÿ๐‘ฆ,๐‘ฅโˆ’๐‘ฆโŸฉ;(iii)๐น(๐พ๐‘Ÿ)=MEP(๐น); (iv)MEP(๐น) is closed and convex.

Lemma 2.8 (see [29]). Assume ๐ด is a strongly positive linear bounded operator on a Hilbert space ๐ป with coefficient ๐›พ>0 and 0<๐œŒโ‰คโ€–๐ดโ€–โˆ’1, then โ€–๐ผโˆ’๐œŒ๐ดโ€–โ‰ค1โˆ’๐œŒ๐›พ.

Lemma 2.9 (see [38]). Let C be a nonempty closed and convex subset of a strictly convex Banach space. Let {๐‘‡๐‘–}๐‘–โˆˆ๐‘ be an infinite family of nonexpansive mappings of C into itself such that โˆฉ๐‘–โˆˆ๐‘๐น(๐‘‡๐‘–)โ‰ โˆ…, and let {๐œ†๐‘–} be a real sequence such that 0โ‰ค๐œ†๐‘–โ‰คb<1 for every ๐‘–โˆˆ๐‘. Then F(W)=โˆฉ๐‘–โˆˆ๐‘๐น(๐‘‡๐‘–)โ‰ โˆ….

Lemma 2.10 (see [38]). Let C be a nonempty closed and convex subset of a strictly convex Banach space. Let {๐‘‡๐‘–} be an infinite family of nonexpansive mappings of C into itself, and let {๐œ†๐‘–} be a real sequence such that 0โ‰ค๐œ†๐‘–โ‰คb<1 for every ๐‘–โˆˆ๐‘. Then, for every ๐‘ฅโˆˆ๐ถ and ๐‘˜โˆˆ๐‘, the limit lim๐‘›โ†’โˆž๐‘ˆ๐‘›,๐‘˜ exist.
In view of the previous lemma, we define ๐‘Š๐‘ฅโˆถ=lim๐‘›โ†’โˆž๐‘ˆ๐‘›,1๐‘ฅ=lim๐‘›โ†’โˆž๐‘Š๐‘›๐‘ฅ.(2.10)

3. Strong Convergence Theorems

In this section, we show a strong convergence theorem which solves the problem of finding a common element of the common fixed points, the common solution of a system of mixed equilibrium problems and variational inclusion of inverse-strongly monotone mappings in a Hilbert space.

Theorem 3.1. Let ๐ป be a real Hilbert space and ๐ถ a nonempty close and convex subset of ๐ป, and let ๐ต be a ๐›ฝ-inverse-strongly monotone mapping. Let ๐œ‘โˆถ๐ถโ†’๐‘… be a convex and lower semicontinuous function, ๐‘“โˆถ๐ถโ†’๐ถ a contraction mapping with coefficient ๐›ผ(0<๐›ผ<1), and ๐‘€โˆถ๐ปโ†’2๐ป a maximal monotone mapping. Let ๐ด be a strongly positive linear bounded operator of ๐ป into itself with coefficient ๐›พ>0. Assume that 0<๐›พ<๐›พ/๐›ผ and ๐œ†โˆˆ(0,2๐›ฝ). Let {๐‘‡๐‘›} be a family of nonexpansive mappings of ๐ป into itself such that ๐œƒโˆถ=โˆž๎™๐‘›=1๐น๎€ท๐‘‡๐‘›๎€ธโˆฉ๎ƒฉ๐‘๎™๐‘˜=1๎€ท๐นSMEP๐‘˜๎€ธ๎ƒชโˆฉ๐ผ(๐ต,๐‘€)โ‰ โˆ….(3.1) Suppose that {๐‘ฅ๐‘›} is a sequence generated by the following algorithm for ๐‘ฅ0โˆˆ๐ถ arbitrarily and ๐‘ข๐‘›=๐พ๐น๐‘๐‘Ÿ๐‘›,๐‘›โ‹…๐พ๐น๐‘โˆ’1๐‘Ÿ๐‘›โˆ’1,๐‘›โ‹…๐พ๐น๐‘โˆ’2๐‘Ÿ๐‘›โˆ’2,๐‘›โ‹…โ‹ฏโ‹…๐พ๐น2๐‘Ÿ2,๐‘›โ‹…๐พ๐น1๐‘Ÿ1,๐‘›โ‹…๐‘ฅ๐‘›๐‘ฅ,โˆ€๐‘›โˆˆ๐‘๐‘›+1=๐‘ƒ๐ถ๎€บ๐œ–๐‘›๎€ท๐‘ฅ๐›พ๐‘“๐‘›๎€ธ+๎€ท๐ผโˆ’๐œ–๐‘›๐ด๎€ธ๐‘Š๐‘›๐ฝ๐‘€,๐œ†๎€ท๐‘ข๐‘›โˆ’๐œ†๐ต๐‘ข๐‘›,๎€ธ๎€ป(3.2) for all ๐‘›=1,2,3,โ€ฆ, where K๐น๐‘–๐‘Ÿ๐‘–,๐‘›๎‚ป๐‘ข(๐‘ฅ)=๐‘›โˆˆ๐ถโˆถ๐น๐‘–๎€ท๐‘ข๐‘›๎€ธ๎€ท๐‘ข,๐‘ฆ+๐œ‘(๐‘ฆ)โˆ’๐œ‘๐‘›๎€ธ+1๐‘Ÿ๐‘–,๐‘›โŸจ๐‘ฆโˆ’๐‘ข๐‘›,๐‘ข๐‘›โˆ’๐‘ฅ๐‘›๎‚ผ,โŸฉโ‰ฅ0,โˆ€๐‘ฆโˆˆ๐ถ๐‘–=1,2,3,โ€ฆ,๐‘,(3.3) and the following conditions are satisfied(C1):{๐œ–๐‘›}โŠ‚(0,1),lim๐‘›โ†’0๐œ–๐‘›โˆ‘=0,โˆž๐‘›=1๐œ–๐‘›โˆ‘=โˆž,โˆž๐‘›=1|๐œ–๐‘›+1โˆ’๐œ–๐‘›|<โˆž; (C2):{๐‘Ÿ๐‘›}โŠ‚[๐‘,๐‘‘] with ๐‘,๐‘‘โˆˆ(0,2๐œŽ) and โˆ‘โˆž๐‘›=1|๐‘Ÿ๐‘›+1โˆ’๐‘Ÿ๐‘›|<โˆž.
Then, the sequence {๐‘ฅ๐‘›} converges strongly to ๐‘žโˆˆ๐œƒ, where ๐‘ž=๐‘ƒ๐œƒ(๐›พ๐‘“+๐ผโˆ’๐ด)(๐‘ž) which solves the following variational inequality: โŸจ(๐›พ๐‘“โˆ’๐ด)๐‘ž,๐‘โˆ’๐‘žโŸฉโ‰ค0,โˆ€๐‘โˆˆ๐œƒ,(3.4) which is the optimality condition for the minimization problem min๐‘žโˆˆ๐œƒ12โŸจ๐ด๐‘ž,๐‘žโŸฉโˆ’โ„Ž(๐‘ž),(3.5) where h is a potential function for ๐›พ๐‘“ (i.e., โ„Ž๎…ž(๐‘ž)=๐›พ๐‘“(๐‘ž) for ๐‘žโˆˆ๐ป).

Proof. For condition (C1), we may assume without loss of generality, and ๐œ–๐‘›โˆˆ(0,โ€–๐ดโ€–โˆ’1) for all ๐‘›. By Lemma 2.8, we have โ€–๐ผโˆ’๐œ–๐‘›๐ดโ€–โ‰ค1โˆ’๐œ–๐‘›๐›พ. Next, we will assume that โ€–๐ผโˆ’๐ดโ€–โ‰คโ€–1โˆ’๐›พโ€–.
Next, we will divide the proof into six steps.
Step 1. First, we will show that {๐‘ฅ๐‘›} and {๐‘ข๐‘›} are bounded. Since ๐ต is ๐›ฝ-inverse-strongly monotone mappings, we have โ€–(๐ผโˆ’๐œ†๐ต)๐‘ฅโˆ’(๐ผโˆ’๐œ†๐ต)๐‘ฆโ€–2=โ€–๐ผ๐‘ฅโˆ’๐œ†๐ต๐‘ฅโˆ’๐ผ๐‘ฆ+๐œ†๐ต๐‘ฆโ€–2=โ€–๐‘ฅโˆ’๐‘ฆโˆ’๐œ†๐ต๐‘ฅ+๐œ†๐ต๐‘ฆโ€–2=โ€–(๐‘ฅโˆ’๐‘ฆ)โˆ’๐œ†(๐ต๐‘ฅ+๐ต๐‘ฆ)โ€–2โ‰คโ€–๐‘ฅโˆ’๐‘ฆโ€–2โˆ’2๐œ†โŸจ๐‘ฅโˆ’๐‘ฆโŸฉโŸจ๐ต๐‘ฅ+๐ต๐‘ฆโŸฉ+๐œ†2โ€–๐ต๐‘ฅโˆ’๐ต๐‘ฆโ€–2โ‰คโ€–๐‘ฅโˆ’๐‘ฆโ€–2โˆ’2๐œ†๐›ฝโ€–๐ต๐‘ฅ+๐ต๐‘ฆโ€–2+๐œ†2โ€–๐ต๐‘ฅโˆ’๐ต๐‘ฆโ€–2โ‰คโ€–๐‘ฅโˆ’๐‘ฆโ€–2+๐œ†(๐œ†โˆ’2๐›ฝ)โ€–๐ต๐‘ฅ+๐ต๐‘ฆโ€–2,(3.6) if 0<๐œ†<2๐›ฝ, then ๐ผโˆ’๐œ†๐ต is nonexpansive.
Put ๐‘ฆ๐‘›โˆถ=๐ฝ๐‘€,๐œ†(๐‘ข๐‘›โˆ’๐œ†๐ต๐‘ข๐‘›),๐‘›โ‰ฅ0. Since ๐ฝ๐‘€,๐œ† and ๐ผโˆ’๐œ†๐ต are nonexpansive mapping, it follows that โ€–โ€–๐‘ฆ๐‘›โ€–โ€–=โ€–โ€–๐ฝโˆ’๐‘ž๐‘€,๐œ†๎€ท๐‘ข๐‘›โˆ’๐œ†๐ต๐‘ข๐‘›๎€ธโˆ’๐ฝ๐‘€,๐œ†โ€–โ€–โ‰คโ€–โ€–๎€ท๐‘ข(๐‘žโˆ’๐œ†๐ต๐‘ž)๐‘›โˆ’๐œ†๐ต๐‘ข๐‘›๎€ธโ€–โ€–โ‰คโ€–โ€–๐‘ขโˆ’(๐‘žโˆ’๐œ†๐ต๐‘ž)๐‘›โ€–โ€–.โˆ’๐‘ž(3.7) By Lemma 2.7, we have ๐‘ข๐‘›=๐พ๐น๐‘๐‘Ÿ๐‘›,๐‘›โ‹…๐พ๐น๐‘โˆ’1๐‘Ÿ๐‘›โˆ’1,๐‘›โ‹…๐พ๐น๐‘โˆ’2๐‘Ÿ๐‘›โˆ’2,๐‘›โ‹…โ‹ฏโ‹…๐พ๐น2๐‘Ÿ2,๐‘›โ‹…๐พ๐น1๐‘Ÿ1,๐‘›โ‹…๐‘ฅ๐‘›๐œ,for๐‘›โ‰ฅ0๐‘˜๐‘›=๐พ๐น๐‘˜๐‘Ÿ๐‘˜,๐‘›โ‹…๐พ๐น๐‘˜โˆ’1๐‘Ÿ๐‘˜โˆ’1,๐‘›โ‹…โ‹ฏโ‹…๐พ๐น2๐‘Ÿ2,๐‘›โ‹…๐พ๐น1๐‘Ÿ1,๐‘›,for๐‘˜โˆˆ{0,1,2,โ€ฆ,๐‘},(3.8) and ๐œ0๐‘›=๐ผforall๐‘›โˆˆ๐‘,๐‘ž=๐œ๐น๐‘˜๐‘Ÿ๐‘˜,๐‘›๐‘ž,๐‘ข๐‘›=๐œ๐‘๐‘Ÿ๐‘˜,๐‘๐‘ฅ๐‘› Then, we have โ€–โ€–๐‘ข๐‘›โ€–โ€–โˆ’๐‘ž2=โ€–โ€–๐œ๐‘๐‘Ÿ๐‘˜,๐‘›๐‘ฅ๐‘›โˆ’๐œ๐น๐‘˜๐‘Ÿ๐‘˜,๐‘›๐‘žโ€–โ€–2=โ€–โ€–๐‘ฅ๐‘›โ€–โ€–โˆ’๐‘ž2.(3.9) Hence, we get โ€–โ€–๐‘ฆ๐‘›โ€–โ€–โ‰คโ€–โ€–๐‘ฅโˆ’๐‘ž๐‘›โ€–โ€–.โˆ’๐‘ž(3.10) From (3.2), we deduce that โ€–โ€–๐‘ฅ๐‘›+1โ€–โ€–=โ€–โ€–๐‘ƒโˆ’๐‘ž๐ถ๎€ท๐œ–๐‘›๎€ท๐‘ฅ๐›พ๐‘“๐‘›๎€ธ+๎€ท๐ผโˆ’๐œ–๐‘›๐ด๎€ธ๐‘Š๐‘›๐‘ฆ๐‘›๎€ธโˆ’๐‘ƒ๐ถ๐‘žโ€–โ€–โ‰คโ€–โ€–๐œ–๐‘›๎€ท๎€ท๐‘ฅ๐›พ๐‘“๐‘›๎€ธ๎€ธ+๎€ทโˆ’๐ด๐‘ž๐ผโˆ’๐œ–๐‘›๐ด๐‘Š๎€ธ๎€ท๐‘›๐‘ฆ๐‘›๎€ธโ€–โ€–โˆ’๐‘žโ‰ค๐œ–๐‘›โ€–โ€–๎€ท๐‘ฅ๐›พ๐‘“๐‘›๎€ธโ€–โ€–+๎€ทโˆ’๐ด๐‘ž1โˆ’๐œ–๐‘›๐›พ๎€ธโ€–โ€–๐‘ฆ๐‘›โ€–โ€–โˆ’๐‘žโ‰ค๐œ–๐›พ๐œ–๐‘›โ€–โ€–๐‘ฅ๐‘›โ€–โ€–โˆ’๐‘ž+๐œ–๐‘›(+๎€ทโ€–๐›พ๐‘“๐‘”)โˆ’๐ด๐‘žโ€–1โˆ’๐œ–๐‘›๐›พ๎€ธโ€–โ€–๐‘ฅ๐‘›โ€–โ€–=๎€ท๎€ทโˆ’๐‘ž1โˆ’๎€ธ๐œ–๐›พโˆ’๐›พ๐œ–๐‘›๎€ธโ€–โ€–๐‘ฅ๐‘›โ€–โ€–โˆ’๐‘žโˆ’๐œ–๐‘›=๎€ท๎€ทโ€–๐›พ๐‘“(๐‘ž)โˆ’๐ด๐‘žโ€–1โˆ’๎€ธ๐œ–๐›พโˆ’๐›พ๐œ–๐‘›๎€ธโ€–โ€–๐‘ฅ๐‘›โ€–โ€–+๎€ทโˆ’๐‘ž๎€ธ๐œ–๐›พโˆ’๐›พ๐œ–๐‘›โ€–๐›พ๐‘“(๐‘ž)โˆ’๐ด๐‘žโ€–โ‹ฎ๎‚ปโ€–โ€–๐‘ฅ๐›พโˆ’๐›พ๐œ–โ‰คmax๐‘›โ€–โ€–,โ€–โˆ’๐‘ž๐›พ๐‘“(๐‘ž)โˆ’๐ด๐‘žโ€–๎‚ผ.๐›พโˆ’๐›พ๐œ–(3.11) It follows by induction that โ€–โ€–๐‘ฅ๐‘›โ€–โ€–๎‚ปโ€–โ€–๐‘ฅโˆ’๐‘žโ‰คmax0โ€–โ€–,(โˆ’๐‘žโ€–๐›พ๐‘“๐‘ž)โˆ’๐ด๐‘žโ€–๎‚ผ๐›พโˆ’๐›พ๐œ–,๐‘›โ‰ฅ0.(3.12) Therefore {๐‘ฅ๐‘›} is bounded, so are {๐‘ฆ๐‘›},{๐ต๐‘ข๐‘›},{๐‘“(๐‘ฅ๐‘›)}, and {๐ด๐‘Š๐‘›๐‘ฆ๐‘›}.
Step 2. We claim that lim๐‘›โ†’โˆžโ€–๐‘ฅ๐‘›+1โˆ’๐‘ฅ๐‘›โ€–=0 and lim๐‘›โ†’โˆžโ€–๐‘ฆ๐‘›+1โˆ’๐‘ฆ๐‘›โ€–=0. From (3.2), we have โ€–โ€–๐‘ฅ๐‘›+1โˆ’๐‘ฅ๐‘›โ€–โ€–=โ€–โ€–๐‘ƒ๐ถ๎€ท๐œ–๐‘›๎€ท๐‘ฅ๐›พ๐‘“๐‘›๎€ธ+๎€ท๐ผโˆ’๐œ–๐‘›๐ด๎€ธ๐‘Š๐‘›๐‘ฆ๐‘›๎€ธโˆ’๐‘ƒ๐ถ๎€ท๐œ–๐‘›โˆ’1๎€ท๐‘ฅ๐›พ๐‘“๐‘›โˆ’1๎€ธ+๎€ท๐ผโˆ’๐œ–๐‘›โˆ’1๐ด๎€ธ๐‘Š๐‘›๐‘ฆ๐‘›โˆ’1๎€ธโ€–โ€–โ‰คโ€–โ€–๎€ท๐ผโˆ’๐œ–๐‘›๐ด๐‘Š๎€ธ๎€ท๐‘›๐‘ฆ๐‘›โˆ’๐‘Š๐‘›๐‘ฆ๐‘›โˆ’1๎€ธโˆ’๎€ท๐œ–๐‘›โˆ’๐œ–๐‘›โˆ’1๎€ธ๐ด๐‘Š๐‘›๐‘ฆ๐‘›โˆ’1+๐›พ๐œ–๐‘›๎€ท๐‘“๎€ท๐‘ฅ๐‘›๎€ธ๎€ท๐‘ฅโˆ’๐‘“๐‘›โˆ’1๎€ท๐œ–๎€ธ๎€ธ+๐›พ๐‘›โˆ’๐œ–๐‘›โˆ’1๎€ธ๐‘“๎€ท๐‘ฅ๐‘›โˆ’1๎€ธโ€–โ€–โ‰ค๎€ท1โˆ’๐œ–๐‘›๐›พ๎€ธโ€–โ€–๐‘ฆ๐‘›โˆ’๐‘ฆ๐‘›โˆ’1โ€–โ€–+||๐œ–๐‘›โˆ’๐œ–๐‘›โˆ’1||โ€–โ€–๐ด๐‘Š๐‘›๐‘ฆ๐‘›โ€–โ€–+๐›พ๐œ–๐œ–๐‘›โ€–โ€–๐‘ฅ๐‘›โˆ’๐‘ฅ๐‘›โˆ’1โ€–โ€–||๐œ–+๐›พ๐‘›โˆ’๐œ–๐‘›โˆ’1||โ€–โ€–๐‘“๎€ท๐‘ฅ๐‘›โˆ’1๎€ธโ€–โ€–.(3.13) Since ๐ฝ๐‘€,๐œ† and ๐ผโˆ’๐œ†๐ต are nonexpansive, we also have โ€–โ€–๐‘ฆ๐‘›โˆ’๐‘ฆ๐‘›โˆ’1โ€–โ€–=โ€–โ€–๐ฝ๐‘€,๐œ†๎€ท๐‘ข๐‘›โˆ’๐œ†๐ต๐‘ข๐‘›๎€ธโˆ’๐ฝ๐‘€,๐œ†๎€ท๐‘ข๐‘›โˆ’1โˆ’๐œ†๐ต๐‘ข๐‘›โˆ’1๎€ธโ€–โ€–โ‰คโ€–โ€–๎€ท๐‘ข๐‘›โˆ’๐œ†๐ต๐‘ข๐‘›๎€ธโˆ’๎€ท๐‘ข๐‘›โˆ’1โˆ’๐œ†๐ต๐‘ข๐‘›โˆ’1๎€ธโ€–โ€–โ‰คโ€–โ€–๐‘ข๐‘›โˆ’๐‘ข๐‘›โˆ’1โ€–โ€–.(3.14) On the other hand, from ๐‘ข๐‘›โˆ’1=๐œ๐‘๐‘Ÿ๐‘˜,๐‘›โˆ’1๐‘ฅ๐‘›โˆ’1 and ๐‘ข๐‘›=๐œ๐‘๐‘Ÿ๐‘˜,๐‘›๐‘ฅ๐‘›, it follows that ๐น๎€ท๐‘ข๐‘›โˆ’1๎€ธ๎€ท๐‘ข,๐‘ฆ+๐œ‘(๐‘ฆ)โˆ’๐œ‘๐‘›โˆ’1๎€ธ+1๐‘Ÿ๐‘›โˆ’1โŸจ๐‘ฆโˆ’๐‘ข๐‘›โˆ’1,๐‘ข๐‘›โˆ’1โˆ’๐‘ฅ๐‘›โˆ’1๐น๎€ท๐‘ขโŸฉโ‰ฅ0,โˆ€๐‘ฆโˆˆ๐ถ,(3.15)๐‘›๎€ธ๎€ท๐‘ข,๐‘ฆ+๐œ‘(๐‘ฆ)โˆ’๐œ‘๐‘›๎€ธ+1๐‘Ÿ๐‘›โŸจ๐‘ฆโˆ’๐‘ข๐‘›,๐‘ข๐‘›โˆ’๐‘ฅ๐‘›โŸฉโ‰ฅ0,โˆ€๐‘ฆโˆˆ๐ถ.(3.16) Substituting ๐‘ฆ=๐‘ข๐‘› into (3.15) and ๐‘ฆ=๐‘ข๐‘›โˆ’1 into (3.16), we get ๐น๎€ท๐‘ข๐‘›โˆ’1,๐‘ข๐‘›๎€ธ๎€ท๐‘ข+๐œ‘๐‘›๎€ธ๎€ท๐‘ขโˆ’๐œ‘๐‘›โˆ’1๎€ธ+1๐‘Ÿ๐‘›โˆ’1โŸจ๐‘ข๐‘›โˆ’๐‘ข๐‘›โˆ’1,๐‘ขnโˆ’1โˆ’๐‘ฅ๐‘›โˆ’1๐น๎€ท๐‘ขโŸฉโ‰ฅ0,๐‘›,๐‘ข๐‘›+1๎€ธ๎€ท๐‘ข+๐œ‘๐‘›+1๎€ธ๎€ท๐‘ขโˆ’๐œ‘๐‘›๎€ธ+1๐‘Ÿ๐‘›๎ซ๐‘ข๐‘›+1โˆ’๐‘ข๐‘›,๐‘ข๐‘›โˆ’๐‘ฅ๐‘›๎ฌโ‰ฅ0.(3.17) From (A2), we obtain ๎ƒก๐‘ข๐‘›โˆ’๐‘ข๐‘›โˆ’1,๐‘ข๐‘›โˆ’1โˆ’๐‘ฅ๐‘›โˆ’1๐‘Ÿ๐‘›โˆ’1โˆ’๐‘ข๐‘›โˆ’๐‘ฅ๐‘›๐‘Ÿ๐‘›๎ƒข๎ƒก๐‘ขโ‰ฅ0,๐‘›โˆ’๐‘ข๐‘›โˆ’1,๐‘ข๐‘›โˆ’1โˆ’๐‘ฅ๐‘›โˆ’1โˆ’๐‘Ÿ๐‘›โˆ’1๐‘Ÿ๐‘›๎€ท๐‘ข๐‘›โˆ’๐‘ฅ๐‘›๎€ธ๎ƒขโ‰ฅ0,(3.18) so, ๎ƒก๐‘ข๐‘›โˆ’๐‘ข๐‘›โˆ’1,๐‘ข๐‘›โˆ’1โˆ’๐‘ข๐‘›+๐‘ข๐‘›โˆ’๐‘ฅ๐‘›โˆ’1โˆ’๐‘Ÿ๐‘›โˆ’1๐‘Ÿ๐‘›๎€ท๐‘ข๐‘›โˆ’๐‘ฅ๐‘›๎€ธ๎ƒขโ‰ฅ0.(3.19) It follows that ๎ƒก๐‘ข๐‘›โˆ’๐‘ข๐‘›โˆ’1,๐‘ข๐‘›โˆ’1โˆ’๐‘ข๐‘›+๐‘ข๐‘›โˆ’๐‘ฅ๐‘›โˆ’๐‘Ÿ๐‘›โˆ’1๐‘Ÿ๐‘›๎€ท๐‘ข๐‘›โˆ’๐‘ฅ๐‘›๎€ธ๎ƒขโ‰ฅ0,โŸจ๐‘ข๐‘›โˆ’๐‘ข๐‘›โˆ’1,๐‘ข๐‘›โˆ’1โˆ’๐‘ข๐‘›๎ƒก๐‘ขโŸฉ+๐‘›โˆ’๐‘ข๐‘›โˆ’1,๎‚ต๐‘Ÿ1โˆ’๐‘›โˆ’1๐‘Ÿ๐‘›๎‚ถ๎€ท๐‘ข๐‘›โˆ’๐‘ฅ๐‘›๎€ธ๎ƒขโ‰ฅ0.(3.20) Without loss of generality, let us assume that there exists a real number ๐‘ such that ๐‘Ÿ๐‘›โˆ’1>๐‘>0, for all ๐‘›โˆˆโ„•. Then, we have โ€–โ€–๐‘ข๐‘›โˆ’๐‘ข๐‘›โˆ’1โ€–โ€–2โ‰ค๎ƒก๐‘ข๐‘›โˆ’๐‘ข๐‘›โˆ’1,๎‚ต๐‘Ÿ1โˆ’๐‘›โˆ’1๐‘Ÿ๐‘›๎‚ถ๎€ท๐‘ข๐‘›โˆ’๐‘ฅ๐‘›๎€ธ๎ƒขโ‰คโ€–โ€–๐‘ข๐‘›โˆ’๐‘ข๐‘›โˆ’1โ€–โ€–๎ƒฏ||||๐‘Ÿ1โˆ’๐‘›โˆ’1๐‘Ÿ๐‘›||||โ€–โ€–๐‘ข๐‘›โˆ’๐‘ฅ๐‘›โ€–โ€–๎ƒฐ,(3.21) and hence โ€–โ€–๐‘ข๐‘›โˆ’๐‘ข๐‘›โˆ’1โ€–โ€–โ‰คโ€–โ€–๐‘ฅ๐‘›โˆ’๐‘ฅ๐‘›โˆ’1โ€–โ€–+1๐‘Ÿ๐‘›||๐‘Ÿ๐‘›โˆ’๐‘Ÿ๐‘›โˆ’1||โ€–โ€–๐‘ข๐‘›โˆ’๐‘ฅ๐‘›โ€–โ€–โ‰คโ€–โ€–๐‘ฅ๐‘›โˆ’๐‘ฅ๐‘›โˆ’1โ€–โ€–+๐‘€1๐‘||๐‘Ÿ๐‘›โˆ’๐‘Ÿ๐‘›โˆ’1||,(3.22) where ๐‘€1=sup{โ€–๐‘ข๐‘›โˆ’๐‘ฅ๐‘›โ€–โˆถ๐‘›โˆˆโ„•}. Substituting (3.22) into (3.14), we have โ€–โ€–๐‘ฆ๐‘›โˆ’๐‘ฆ๐‘›โˆ’1โ€–โ€–โ‰คโ€–โ€–๐‘ฅ๐‘›โˆ’๐‘ฅ๐‘›โˆ’1โ€–โ€–+๐‘€1๐‘||๐‘Ÿ๐‘›โˆ’๐‘Ÿ๐‘›โˆ’1||.(3.23) Substituting (3.23) into (3.13), we get โ€–โ€–๐‘ฅ๐‘›+1โˆ’๐‘ฅ๐‘›โ€–โ€–โ‰ค๎€ท1โˆ’๐œ–๐‘›๐›พ๎€ธ๎‚ตโ€–โ€–๐‘ฅ๐‘›โˆ’๐‘ฅ๐‘›โˆ’1โ€–โ€–+๐‘€1๐‘||๐‘Ÿ๐‘›โˆ’๐‘Ÿ๐‘›โˆ’1||๎‚ถ+||๐œ–๐‘›โˆ’๐œ–๐‘›โˆ’1||โ€–โ€–๐ด๐‘Š๐‘›๐‘ฆ๐‘›โˆ’1โ€–โ€–+๐›พ๐œ–๐œ–๐‘›โ€–โ€–๐‘ฅ๐‘›โˆ’๐‘ฅ๐‘›โˆ’1โ€–โ€–||๐œ–+๐›พ๐‘›โˆ’๐œ–๐‘›โˆ’1||โ€–โ€–๐‘“๎€ท๐‘ฅ๐‘›โˆ’1๎€ธโ€–โ€–=๎€ท1โˆ’๐œ–๐‘›๐›พ๎€ธโ€–โ€–๐‘ฅ๐‘›โˆ’๐‘ฅ๐‘›โˆ’1โ€–โ€–+๎€ท1โˆ’๐œ–๐‘›๐›พ๎€ธ๐‘€1๐‘||๐‘Ÿ๐‘›โˆ’๐‘Ÿ๐‘›โˆ’1||+||๐œ–๐‘›โˆ’๐œ–๐‘›โˆ’1||โ€–โ€–๐ด๐‘Š๐‘›๐‘ฆ๐‘›โˆ’1โ€–โ€–+๐›พ๐œ–๐œ–๐‘›โ€–โ€–๐‘ฅ๐‘›โˆ’๐‘ฅ๐‘›โˆ’1โ€–โ€–||๐œ–+๐›พ๐‘›โˆ’๐œ–๐‘›โˆ’1||โ€–โ€–๐‘“๎€ท๐‘ฅnโˆ’1๎€ธโ€–โ€–โ‰ค๎€ท๎€ท1โˆ’๎€ธ๐œ–๐›พโˆ’๐›พ๐œ–๐‘›๎€ธโ€–โ€–๐‘ฅ๐‘›โˆ’๐‘ฅ๐‘›โˆ’1โ€–โ€–+๐‘€1๐‘||๐‘Ÿ๐‘›โˆ’๐‘Ÿ๐‘›โˆ’1||+||๐œ–๐‘›โˆ’๐œ–๐‘›โˆ’1||โ€–โ€–๐ด๐‘Š๐‘›๐‘ฆ๐‘›โˆ’1โ€–โ€–||๐œ–+๐›พ๐‘›โˆ’๐œ–๐‘›โˆ’1||โ€–โ€–๐‘“๎€ท๐‘ฅ๐‘›โˆ’1๎€ธโ€–โ€–โ‰ค๎€ท๎€ท1โˆ’๎€ธ๐œ–๐›พโˆ’๐›พ๐œ–๐‘›๎€ธโ€–โ€–๐‘ฅ๐‘›โˆ’๐‘ฅ๐‘›โˆ’1โ€–โ€–+๐‘€1๐‘||๐‘Ÿ๐‘›โˆ’๐‘Ÿ๐‘›โˆ’1||+๐‘€2||๐œ–๐‘›โˆ’๐œ–๐‘›โˆ’1||,(3.24) where ๐‘€2=sup{max{โ€–๐ด๐‘Š๐‘›๐‘ฆ๐‘›โˆ’1โ€–,โ€–๐‘“(๐‘ฅ๐‘›โˆ’1)โ€–โˆถ๐‘›โˆˆโ„•}}. Since conditions (C1)-(C2) and by Lemma 2.5, we have โ€–๐‘ฅ๐‘›+1โˆ’๐‘ฅ๐‘›โ€–โ†’0 as ๐‘›โ†’โˆž. From (3.23), we also have โ€–๐‘ฆ๐‘›+1โˆ’๐‘ฆ๐‘›โ€–โ†’0 as ๐‘›โ†’โˆž.Step 3. Next, we show that lim๐‘›โ†’โˆžโ€–๐ต๐‘ข๐‘›โˆ’๐ต๐‘žโ€–=0.
For ๐‘žโˆˆ๐œƒ hence ๐‘ž=๐ฝ๐‘€,๐œ†(๐‘žโˆ’๐œ†๐ต๐‘ž). By (3.6) and (3.9), we get โ€–โ€–๐‘ฆ๐‘›โ€–โ€–โˆ’๐‘ž2=โ€–โ€–๐ฝ๐‘€,๐œ†๎€ท๐‘ข๐‘›โˆ’๐œ†๐ต๐‘ข๐‘›๎€ธโˆ’๐ฝ๐‘€,๐œ†โ€–โ€–(๐‘žโˆ’๐œ†๐ต๐‘ž)2โ‰คโ€–โ€–๎€ท๐‘ข๐‘›โˆ’๐œ†๐ต๐‘ข๐‘›๎€ธโ€–โ€–โˆ’(๐‘žโˆ’๐œ†๐ต๐‘ž)2โ‰คโ€–โ€–๐‘ข๐‘›โ€–โ€–โˆ’๐‘ž2โ€–โ€–+๐œ†(๐œ†โˆ’2๐›ฝ)๐ต๐‘ข๐‘›โ€–โ€–โˆ’๐ต๐‘ž2โ‰คโ€–โ€–๐‘ฅ๐‘›โ€–โ€–โˆ’๐‘ž2โ€–โ€–+๐œ†(๐œ†โˆ’2๐›ฝ)๐ต๐‘ข๐‘›โ€–โ€–โˆ’๐ต๐‘ž2.(3.25) It follows that โ€–โ€–๐‘ฅ๐‘›+1โ€–โ€–โˆ’๐‘ž2=โ€–โ€–๐‘ƒ๐ถ๎€ท๐œ–๐‘›๎€ท๐‘ฅ๐›พ๐‘“๐‘›๎€ธ+๎€ท๐ผโˆ’๐œ–๐‘›๐ด๎€ธ๐‘Š๐‘›๐‘ฆ๐‘›๎€ธโˆ’๐‘ƒ๐ถโ€–โ€–(๐‘ž)2โ‰คโ€–โ€–๐œ–๐‘›๎€ท๎€ท๐‘ฅ๐›พ๐‘“๐‘›๎€ธ๎€ธ+๎€ทโˆ’๐ด๐‘ž๐ผโˆ’๐œ–๐‘›๐ด๐‘Š๎€ธ๎€ท๐‘›๐‘ฆ๐‘›๎€ธโ€–โ€–โˆ’๐‘ž2โ‰ค๎€ท๐œ–๐‘›โ€–โ€–๎€ท๐‘ฅ๐›พ๐‘“๐‘›๎€ธโ€–โ€–+๎€ทโˆ’๐ด๐‘ž1โˆ’๐œ–๐‘›๐›พ๎€ธโ€–โ€–๐‘ฆ๐‘›โ€–โ€–๎€ธโˆ’๐‘ž2โ‰ค๐œ–๐‘›โ€–โ€–๎€ท๐‘ฅ๐›พ๐‘“๐‘›๎€ธโ€–โ€–โˆ’๐ด๐‘ž2+๎€ท1โˆ’๐œ–๐‘›๐›พ๎€ธโ€–โ€–๐‘ฆ๐‘›โ€–โ€–โˆ’๐‘ž2+2๐œ–๐‘›๎€ท1โˆ’๐œ–๐‘›๐›พ๎€ธโ€–โ€–๎€ท๐‘ฅ๐›พ๐‘“๐‘›๎€ธโ€–โ€–โ€–โ€–๐‘ฆโˆ’๐ด๐‘ž๐‘›โ€–โ€–โˆ’๐‘žโ‰ค๐œ–๐‘›โ€–โ€–๎€ท๐‘ฅ๐›พ๐‘“๐‘›๎€ธโ€–โ€–โˆ’๐ด๐‘ž2+2๐œ–๐‘›๎€ท1โˆ’๐œ–๐‘›๐›พ๎€ธโ€–โ€–๎€ท๐‘ฅ๐›พ๐‘“๐‘›๎€ธโ€–โ€–โ€–โ€–๐‘ฆโˆ’๐ด๐‘ž๐‘›โ€–โ€–+๎€ทโˆ’๐‘ž1โˆ’๐œ–๐‘›๐›พ๎€ธ๎‚€โ€–โ€–๐‘ฅ๐‘›โ€–โ€–โˆ’๐‘ž2โ€–โ€–+๐œ†(๐œ†โˆ’2๐›ฝ)๐ต๐‘ข๐‘›โ€–โ€–โˆ’๐ต๐‘ž2๎‚โ‰ค๐œ–๐‘›โ€–โ€–๎€ท๐‘ฅ๐›พ๐‘“๐‘›๎€ธโ€–โ€–โˆ’๐ด๐‘ž2+2๐œ–๐‘›๎€ท1โˆ’๐œ–๐‘›๐›พ๎€ธโ€–โ€–๎€ท๐‘ฅ๐›พ๐‘“๐‘›๎€ธโ€–โ€–โ€–โ€–๐‘ฆโˆ’๐ด๐‘ž๐‘›โ€–โ€–+โ€–โ€–๐‘ฅโˆ’๐‘ž๐‘›โ€–โ€–โˆ’๐‘ž2+๎€ท1โˆ’๐œ–๐‘›๐›พ๎€ธ๐œ†โ€–โ€–(๐œ†โˆ’2๐›ฝ)๐ต๐‘ข๐‘›โ€–โ€–โˆ’๐ต๐‘ž2.(3.26) So, we obtain ๎€ท1โˆ’๐œ–๐‘›๐›พ๎€ธโ€–โ€–๐œ†(2๐›ฝโˆ’๐œ†)๐ต๐‘ข๐‘›โ€–โ€–โˆ’๐ต๐‘ž2โ‰ค๐œ–๐‘›โ€–โ€–๎€ท๐‘ฅ๐›พ๐‘“๐‘›๎€ธโ€–โ€–โˆ’๐ด๐‘ž2+โ€–โ€–๐‘ฅ๐‘›โˆ’๐‘ฅ๐‘›+1โ€–โ€–๎€ทโ€–โ€–๐‘ฅ๐‘›โ€–โ€–+โ€–โ€–๐‘ฅโˆ’๐‘ž๐‘›+1โ€–โ€–๎€ธโˆ’๐‘ž+๐œ‰๐‘›,(3.27) where ๐œ‰๐‘›=2๐œ–๐‘›(1โˆ’๐œ–๐‘›๐›พ)โ€–๐›พ๐‘“(๐‘ฅ๐‘›)โˆ’๐ด๐‘žโ€–โ€–๐‘ฆ๐‘›โˆ’๐‘žโ€–. By conditions (C1), (C3) and lim๐‘›โ†’โˆžโ€–๐‘ฅ๐‘›+1โˆ’๐‘ฅ๐‘›โ€–=0, then, we obtain that โ€–๐ต๐‘ข๐‘›โˆ’๐ต๐‘žโ€–โ†’0 as ๐‘›โ†’โˆž.
Step 4. We show the following: (i)lim๐‘›โ†’โˆžโ€–๐‘ฅ๐‘›โˆ’๐‘ข๐‘›โ€–=0; (ii)lim๐‘›โ†’โˆžโ€–๐‘ข๐‘›โˆ’๐‘ฆ๐‘›โ€–=0; (iii)lim๐‘›โ†’โˆžโ€–๐‘ฆ๐‘›โˆ’๐‘Š๐‘›๐‘ฆ๐‘›โ€–=0. Since ๐พ๐‘Ÿ๐‘›(๐‘ฅ) is firmly nonexpansive and (2.3), we observe that โ€–โ€–๐‘ข๐‘›โ€–โ€–โˆ’๐‘ž2=โ€–โ€–๐œ๐‘๐‘Ÿ๐‘›,๐‘›๐‘ฅ๐‘›โˆ’๐œ๐‘๐‘Ÿ๐‘›,๐‘›๐‘žโ€–โ€–2โ‰คโŸจ๐‘ฅ๐‘›โˆ’๐‘ž,๐‘ข๐‘›=1โˆ’๐‘žโŸฉ2๎‚€โ€–โ€–๐‘ฅ๐‘›โ€–โ€–โˆ’๐‘ž2+โ€–โ€–๐‘ข๐‘›โ€–โ€–โˆ’๐‘ž2โˆ’โ€–โ€–๐‘ฅ๐‘›โˆ’๐‘žโˆ’๐‘ข๐‘›โ€–โ€–โˆ’๐‘ž2๎‚โ‰ค12๎‚€โ€–โ€–๐‘ฅ๐‘›โ€–โ€–โˆ’๐‘ž2+โ€–โ€–๐‘ข๐‘›โ€–โ€–โˆ’๐‘ž2โˆ’โ€–โ€–๐‘ฅ๐‘›โˆ’๐‘ข๐‘›โ€–โ€–2๎‚,(3.28) it follows that โ€–โ€–๐‘ข๐‘›โ€–โ€–โˆ’๐‘ž2โ‰คโ€–โ€–๐‘ฅ๐‘›โ€–โ€–โˆ’๐‘ž2โˆ’โ€–โ€–๐‘ฅ๐‘›โˆ’๐‘ข๐‘›โ€–โ€–2.(3.29)
Since ๐ฝ๐‘€,๐œ† is 1-inverse-strongly monotone and by (2.3), we compute โ€–โ€–๐‘ฆ๐‘›โ€–โ€–โˆ’๐‘ž2=โ€–โ€–๐ฝ๐‘€,๐œ†๎€ท๐‘ข๐‘›โˆ’๐œ†๐ต๐‘ข๐‘›๎€ธโˆ’๐ฝ๐‘€,๐œ†โ€–โ€–(๐‘žโˆ’๐œ†๐ต๐‘ž)2โ‰ค๐‘ข๎ซ๎€ท๐‘›โˆ’๐œ†๐ต๐‘ข๐‘›๎€ธโˆ’(๐‘žโˆ’๐œ†๐ต๐‘ž),๐‘ฆ๐‘›๎ฌ=1โˆ’๐‘ž2๎‚€โ€–โ€–๎€ท๐‘ข๐‘›โˆ’๐œ†๐ต๐‘ข๐‘›๎€ธโ€–โ€–โˆ’(๐‘žโˆ’๐œ†๐ต๐‘ž)2+โ€–โ€–๐‘ฆ๐‘›โ€–โ€–โˆ’๐‘ž2โˆ’โ€–โ€–๎€ท๐‘ข๐‘›โˆ’๐œ†๐ต๐‘ข๐‘›๎€ธ๎€ท๐‘ฆโˆ’(๐‘žโˆ’๐œ†๐ต๐‘ž)โˆ’๐‘›๎€ธโ€–โ€–โˆ’๐‘ž2๎‚โ‰ค12๎‚€โ€–โ€–๐‘ข๐‘›โ€–โ€–โˆ’๐‘ž2+โ€–โ€–๐‘ฆ๐‘›โ€–โ€–โˆ’๐‘ž2โˆ’โ€–โ€–๎€ท๐‘ข๐‘›โˆ’๐‘ฆ๐‘›๎€ธ๎€ทโˆ’๐œ†๐ต๐‘ข๐‘›๎€ธโ€–โ€–โˆ’๐ต๐‘ž2๎‚=12๎‚€โ€–โ€–๐‘ข๐‘›โ€–โ€–โˆ’๐‘ž2+โ€–โ€–๐‘ฆ๐‘›โ€–โ€–โˆ’๐‘ž2โˆ’โ€–โ€–๐‘ข๐‘›โˆ’๐‘ฆ๐‘›โ€–โ€–2+2๐œ†โŸจ๐‘ข๐‘›โˆ’๐‘ฆ๐‘›,๐ต๐‘ข๐‘›โˆ’๐ต๐‘žโŸฉโˆ’๐œ†2โ€–โ€–๐ต๐‘ข๐‘›โ€–โ€–โˆ’๐ต๐‘ž2๎‚,(3.30) which implies that โ€–โ€–๐‘ฆ๐‘›โ€–โ€–โˆ’๐‘ž2โ‰คโ€–โ€–๐‘ข๐‘›โ€–โ€–โˆ’๐‘ž2โˆ’โ€–โ€–๐‘ข๐‘›โˆ’๐‘ฆ๐‘›โ€–โ€–2โ€–โ€–๐‘ข+2๐œ†๐‘›โˆ’๐‘ฆ๐‘›โ€–โ€–โ€–โ€–๐ต๐‘ข๐‘›โ€–โ€–.โˆ’๐ต๐‘ž(3.31) Substituting (3.31) into (3.26), we have โ€–โ€–๐‘ฅ๐‘›+1โ€–โ€–โˆ’๐‘ž2โ‰ค๐œ–๐‘›โ€–โ€–๎€ท๐‘ฅ๐›พ๐‘“๐‘›๎€ธโ€–โ€–โˆ’๐ด๐‘ž2+โ€–โ€–๐‘ฆ๐‘›โ€–โ€–โˆ’๐‘ž2+2๐œ–๐‘›๎€ท1โˆ’๐œ–๐‘›๐›พ๎€ธโ€–โ€–๎€ท๐‘ฅ๐›พ๐‘“๐‘›๎€ธโ€–โ€–โ€–โ€–๐‘ฆโˆ’๐ด๐‘ž๐‘›โ€–โ€–โˆ’๐‘žโ‰ค๐œ–๐‘›โ€–โ€–๎€ท๐‘ฅ๐›พ๐‘“๐‘›๎€ธโ€–โ€–โˆ’๐ด๐‘ž2+๎‚€โ€–โ€–๐‘ข๐‘›โ€–โ€–โˆ’๐‘ž2โˆ’โ€–โ€–๐‘ข๐‘›โˆ’๐‘ฆ๐‘›โ€–โ€–2+2๐œ†๐‘›โ€–โ€–๐‘ข๐‘›โˆ’๐‘ฆ๐‘›โ€–โ€–โ€–โ€–๐ต๐‘ข๐‘›โ€–โ€–๎‚โˆ’๐ต๐‘ž+2๐œ–๐‘›๎€ท1โˆ’๐œ–๐‘›๐›พ๎€ธโ€–โ€–๎€ท๐‘ฅ๐›พ๐‘“๐‘›๎€ธโ€–โ€–โ€–โ€–๐‘ฆโˆ’๐ด๐‘ž๐‘›โ€–โ€–.โˆ’๐‘ž(3.32) Then, we derive โ€–โ€–๐‘ฅ๐‘›โˆ’๐‘ข๐‘›โ€–โ€–2+โ€–โ€–๐‘ข๐‘›โˆ’๐‘ฆ๐‘›โ€–โ€–2โ‰ค๐œ–๐‘›โ€–โ€–๎€ท๐‘ฅ๐›พ๐‘“๐‘›๎€ธโ€–โ€–โˆ’๐ด๐‘ž2+โ€–โ€–๐‘ฅ๐‘›โ€–โ€–โˆ’๐‘ž2โˆ’โ€–โ€–๐‘ฅ๐‘›+1โ€–โ€–โˆ’๐‘ž2โ€–โ€–๐‘ข+2๐œ†๐‘›โˆ’๐‘ฆ๐‘›โ€–โ€–โ€–โ€–๐ต๐‘ข๐‘›โ€–โ€–โˆ’๐ต๐‘ž+2๐œ–๐‘›๎€ท1โˆ’๐œ–๐‘›๐›พ๎€ธโ€–โ€–๎€ท๐‘ฅ๐›พ๐‘“๐‘›๎€ธโ€–โ€–โ€–โ€–๐‘ฆโˆ’๐ด๐‘ž๐‘›โ€–โ€–.โˆ’๐‘ž=๐œ–๐‘›โ€–โ€–๎€ท๐‘ฅ๐›พ๐‘“๐‘›๎€ธโ€–โ€–โˆ’๐ด๐‘ž2+โ€–โ€–๐‘ฅ๐‘›โˆ’๐‘ฅ๐‘›+1โ€–โ€–๎€ทโ€–โ€–๐‘ฅ๐‘›โ€–โ€–+โ€–โ€–๐‘ฅโˆ’๐‘ž๐‘›+1โ€–โ€–๎€ธโ€–โ€–๐‘ขโˆ’๐‘ž+2๐œ†๐‘›โˆ’๐‘ฆ๐‘›โ€–โ€–โ€–โ€–๐ต๐‘ข๐‘›โ€–โ€–โˆ’๐ต๐‘ž+2๐œ–๐‘›๎€ท1โˆ’๐œ–๐‘›๐›พ๎€ธโ€–โ€–๎€ท๐‘ฅ๐›พ๐‘“๐‘›๎€ธโ€–โ€–โ€–โ€–๐‘ฆโˆ’๐ด๐‘ž๐‘›โ€–โ€–.โˆ’๐‘ž(3.33) By condition (C1), lim๐‘›โ†’โˆžโ€–๐‘ฅ๐‘›โˆ’๐‘ฅ๐‘›+1โ€–=0 and lim๐‘›โ†’โˆžโ€–๐ต๐‘ข๐‘›โˆ’๐ต๐‘žโ€–=0.
So, we have โ€–๐‘ฅ๐‘›โˆ’๐‘ข๐‘›โ€–โ†’0,โ€–๐‘ข๐‘›โˆ’๐‘ฆ๐‘›โ€–โ†’0 as ๐‘›โ†’โˆž. It follows that โ€–โ€–๐‘ฅ๐‘›โˆ’๐‘ฆ๐‘›โ€–โ€–โ‰คโ€–โ€–๐‘ฅ๐‘›โˆ’๐‘ข๐‘›โ€–โ€–+โ€–โ€–๐‘ข๐‘›โˆ’๐‘ฆ๐‘›โ€–โ€–โŸถ0,as๐‘›โŸถโˆž.(3.34) From (3.2), we have โ€–โ€–๐‘ฅ๐‘›โˆ’๐‘Š๐‘›๐‘ฆ๐‘›โ€–โ€–โ‰คโ€–โ€–๐‘ฅ๐‘›โˆ’๐‘Š๐‘›๐‘ฆ๐‘›โˆ’1โ€–โ€–+โ€–โ€–๐‘Š๐‘›๐‘ฆ๐‘›โˆ’1โˆ’๐‘Š๐‘›๐‘ฆ๐‘›โ€–โ€–โ‰คโ€–โ€–๐‘ƒ๐ถ๎€ท๐œ–๐‘›โˆ’1๎€ท๐‘ฅ๐›พ๐‘“๐‘›โˆ’1๎€ธ+๎€ท๐ผโˆ’๐›ผ๐‘›โˆ’1๐ด๎€ธ๐‘Š๐‘›๐‘ฆ๐‘›โˆ’1๎€ธโˆ’๐‘ƒ๐ถ๎€ท๐‘Š๐‘›๐‘ฆ๐‘›โˆ’1๎€ธโ€–โ€–+โ€–โ€–๐‘ฆ๐‘›โˆ’1โˆ’๐‘ฆ๐‘›โ€–โ€–โ‰ค๐œ–๐‘›โˆ’1โ€–โ€–๐›พ๐‘“๐‘ฅ๐‘›โˆ’1โˆ’๐ด๐‘Š๐‘›๐‘ฆ๐‘›โˆ’1โ€–โ€–+โ€–โ€–๐‘ฆ๐‘›โˆ’1โˆ’๐‘ฆ๐‘›โ€–โ€–.(3.35) By condition (C1) and lim๐‘›โ†’โˆžโ€–๐‘ฆ๐‘›โˆ’1โˆ’๐‘ฆ๐‘›โ€–=0, we obtain that โ€–๐‘ฅ๐‘›โˆ’๐‘Š๐‘›๐‘ฆ๐‘›โ€–โ†’0 as ๐‘›โ†’โˆž.
Hence, we have โ€–โ€–๐‘ฅ๐‘›โˆ’๐‘Š๐‘›๐‘ฅ๐‘›โ€–โ€–โ‰คโ€–โ€–๐‘ฅ๐‘›โˆ’๐‘Š๐‘›๐‘ฆ๐‘›โ€–โ€–+โ€–โ€–๐‘Š๐‘›๐‘ฆ๐‘›โˆ’๐‘Š๐‘›๐‘ฅ๐‘›โ€–โ€–โ‰คโ€–โ€–๐‘ฅ๐‘›โˆ’๐‘Š๐‘›๐‘ฆ๐‘›โ€–โ€–+โ€–โ€–๐‘ฆ๐‘›โˆ’๐‘ฅ๐‘›โ€–โ€–.(3.36) By (3.34) and lim๐‘›โ†’โˆžโ€–๐‘ฅ๐‘›โˆ’๐‘Š๐‘›๐‘ฆ๐‘›โ€–=0, we obtain โ€–๐‘ฅ๐‘›โˆ’๐‘Š๐‘›๐‘ฅ๐‘›โ€–โ†’0 as ๐‘›โ†’โˆž.
Moreover, we also have โ€–โ€–๐‘ฆ๐‘›โˆ’๐‘Š๐‘›๐‘ฆ๐‘›โ€–โ€–โ‰คโ€–โ€–๐‘ฆ๐‘›โˆ’๐‘ฅ๐‘›โ€–โ€–+โ€–โ€–๐‘ฅ๐‘›โˆ’๐‘Š๐‘›๐‘ฆ๐‘›โ€–โ€–.(3.37) By (3.34) and lim๐‘›โ†’โˆžโ€–๐‘ฅ๐‘›โˆ’๐‘Š๐‘›๐‘ฆ๐‘›โ€–=0, we obtain โ€–๐‘ฆ๐‘›โˆ’๐‘Š๐‘›๐‘ฆ๐‘›โ€–โ†’0 as ๐‘›โ†’โˆž.
Step 5. We show that โ‹‚๐‘žโˆˆ๐œƒโˆถ=โˆž๐‘›=1๐น(๐‘‡๐‘›โ‹‚)โˆฉ(๐‘๐‘˜=1SMEP(๐น๐‘˜))โˆฉ๐ผ(๐ต,๐‘€) and limsup๐‘›โ†’โˆžโŸจ(๐›พ๐‘“โˆ’๐ด)๐‘ž,๐‘Š๐‘›๐‘ฆ๐‘›โˆ’๐‘žโŸฉโ‰ค0. It is easy to see that ๐‘ƒ๐œƒ(๐›พ๐‘“+(๐ผโˆ’๐ด)) is a contraction of ๐ป into itself.
Indeed, since 0<๐›พ<๐›พ/๐œ–, we have โ€–โ€–๐‘ƒ๐œƒ(๐›พ๐‘“+(๐ผโˆ’๐ด))๐‘ฅโˆ’๐‘ƒ๐œƒโ€–โ€–๎€ท(๐›พ๐‘“+(๐ผโˆ’๐ด))๐‘ฆโ‰ค๐›พโ€–๐‘“(๐‘ฅ)โˆ’๐‘“(๐‘ฆ)โ€–+โ€–๐ผโˆ’๐ดโ€–โ€–๐‘ฅโˆ’yโ€–โ‰ค๐›พ๐œ–โ€–๐‘ฅโˆ’๐‘ฆโ€–+1โˆ’๐›พ๎€ธโ‰ค๎€ทโ€–๐‘ฅโˆ’๐‘ฆโ€–1โˆ’๎€ธ๐›พ+๐›พ๐œ–โ€–๐‘ฅโˆ’๐‘ฆโ€–.(3.38) Since ๐ป is complete, then there exists a unique fixed point ๐‘žโˆˆ๐ป such that ๐‘ž=๐‘ƒ๐œƒ(๐›พ๐‘“+(๐ผโˆ’๐ด))(๐‘ž). By Lemma 2.2, we obtain that โŸจ(๐›พ๐‘“โˆ’๐ด)๐‘ž,๐‘คโˆ’๐‘žโŸฉโ‰ค0 for all ๐‘คโˆˆ๐œƒ.
Next, we show that limsup๐‘›โ†’โˆžโŸจ(๐›พ๐‘“โˆ’๐ด)๐‘ž,๐‘Š๐‘›๐‘ฆ๐‘›โˆ’๐‘žโŸฉโ‰ค0, where ๐‘ž=๐‘ƒ๐œƒ(๐›พ๐‘“+๐ผโˆ’๐ด)(๐‘ž) is the unique solution of the variational inequality โŸจ(๐›พ๐‘“โˆ’๐ด)๐‘ž,๐‘คโˆ’๐‘žโŸฉโ‰ฅ0 for all ๐‘คโˆˆ๐œƒ. We can choose a subsequence {๐‘ฆ๐‘›๐‘–} of {๐‘ฆ๐‘›} such that limsup๐‘›โ†’โˆžโŸจ(๐›พ๐‘“โˆ’๐ด)๐‘ž,๐‘Š๐‘›๐‘ฆ๐‘›โˆ’๐‘žโŸฉ=lim๐‘–โ†’โˆž๎ซ(๐›พ๐‘“โˆ’๐ด)๐‘ž,๐‘Š๐‘›๐‘ฆ๐‘›๐‘–๎ฌ.โˆ’๐‘ž(3.39) As {๐‘ฆ๐‘›๐‘–} is bounded, there exists a subsequence {๐‘ฆ๐‘›๐‘–๐‘—} of {๐‘ฆ๐‘›๐‘–} which converges weakly to ๐‘ค. We may assume without loss of generality that ๐‘ฆ๐‘›๐‘–โ‡€๐‘ค.
Next we claim that ๐‘คโˆˆ๐œƒ. Since โ€–๐‘ฆ๐‘›โˆ’๐‘Š๐‘›๐‘ฆ๐‘›โ€–โ†’0,โ€–๐‘ฅ๐‘›โˆ’๐‘Š๐‘›๐‘ฅ๐‘›โ€–โ†’0, and โ€–๐‘ฅ๐‘›โˆ’๐‘ฆ๐‘›โ€–โ†’0, and by Lemma 2.6, we have โ‹‚๐‘คโˆˆโˆž๐‘›=1๐น(๐‘‡๐‘›).
Next, we show that โ‹‚๐‘คโˆˆโˆž๐‘˜=1SMEP(๐น๐‘˜). Since ๐‘ข๐‘›=๐œ๐‘๐‘Ÿ๐‘˜,๐‘›๐‘ฅ๐‘›, for ๐‘˜=1,2,3,โ€ฆ,๐‘, we know that ๐น๐‘˜๎€ท๐‘ข๐‘›๎€ธ๎€ท๐‘ข,๐‘ฆ+๐œ‘(๐‘ฆ)โˆ’๐œ‘๐‘›๎€ธ+1๐‘Ÿ๐‘›โŸจ๐‘ฆโˆ’๐‘ข๐‘›,๐‘ข๐‘›โˆ’๐‘ฅ๐‘›โŸฉโ‰ฅ0,โˆ€๐‘ฆโˆˆ๐ถ.(3.40) It follows by (A2) that ๎€ท๐‘ข๐œ‘(๐‘ฆ)โˆ’๐œ‘๐‘›๎€ธ+1๐‘Ÿ๐‘›โŸจ๐‘ฆโˆ’๐‘ข๐‘›,๐‘ข๐‘›โˆ’๐‘ฅ๐‘›โŸฉโ‰ฅ๐น๐‘˜๎€ท๐‘ฆ,๐‘ข๐‘›๎€ธ,โˆ€๐‘ฆโˆˆ๐ถ.(3.41) Hence, for ๐‘˜=1,2,3,โ€ฆ,๐‘, we get ๎€ท๐‘ข๐œ‘(๐‘ฆ)โˆ’๐œ‘๐‘›๐‘–๎€ธ+1๐‘Ÿ๐‘›๐‘–๎ซ๐‘ฆโˆ’๐‘ข๐‘›๐‘–,๐‘ข๐‘›๐‘–โˆ’๐‘ฅ๐‘›๐‘–๎ฌโ‰ฅ๐น๐‘˜๎€ท๐‘ฆ,๐‘ข๐‘›๐‘–๎€ธ,โˆ€๐‘ฆโˆˆ๐ถ.(3.42) For ๐‘กโˆˆ(0,1] and ๐‘ฆโˆˆ๐ป, let ๐‘ฆ๐‘ก=๐‘ก๐‘ฆ+(1โˆ’๐‘ก)๐‘ค. From (3.42), we have ๎€ท๐‘ฆ0โ‰ฅ๐œ‘๐‘ก๎€ธ๎€ท๐‘ข+๐œ‘๐‘›๐‘–๎€ธโˆ’1๐‘Ÿ๐‘›๐‘–๎ซ๐‘ฆ๐‘กโˆ’๐‘ข๐‘›๐‘–,๐‘ข๐‘›๐‘–โˆ’๐‘ฅ๐‘›๐‘–๎ฌ+๐น๐‘˜๎€ท๐‘ฆ๐‘ก,๐‘ข๐‘›๐‘–๎€ธ.(3.43)
Since โ€–๐‘ข๐‘›๐‘–โˆ’๐‘ฅ๐‘›๐‘–โ€–โ†’0, from (A4) and the weakly lower semicontinuity of ๐œ‘, (๐‘ข๐‘›๐‘–โˆ’๐‘ฅ๐‘›๐‘–)/๐‘Ÿ๐‘›๐‘–โ†’0 and ๐‘ข๐‘›๐‘–โ‡€๐‘ค. From (A1) and (A4), we have 0=๐น๐‘˜๎€ท๐‘ฆ๐‘ก,๐‘ฆ๐‘ก๎€ธ๎€ท๐‘ฆโˆ’๐œ‘๐‘ก๎€ธ๎€ท๐‘ฆ+๐œ‘๐‘ก๎€ธโ‰ค๐‘ก๐น๐‘˜๎€ท๐‘ฆ๐‘ก๎€ธ,๐‘ฆ+(1โˆ’๐‘ก)๐น๐‘˜๎€ท๐‘ฆ๐‘ก๎€ธ๎€ท๐‘ฆ,๐‘ค+๐‘ก๐œ‘(๐‘ฆ)+(1โˆ’๐‘ก)๐œ‘(๐‘ค)โˆ’๐œ‘๐‘ก๎€ธ๎€บ๐นโ‰ค๐‘ก๐‘˜๎€ท๐‘ฆ๐‘ก๎€ธ๎€ท๐‘ฆ,๐‘ฆ+๐œ‘(๐‘ฆ)โˆ’๐œ‘๐‘ก.๎€ธ๎€ป(3.44) Dividing by ๐‘ก, we get ๐น๐‘˜๎€ท๐‘ฆ๐‘ก๎€ธ๎€ท๐‘ฆ,๐‘ฆ+๐œ‘(๐‘ฆ)โˆ’๐œ‘๐‘ก๎€ธโ‰ฅ0.(3.45) The weakly lower semicontinuity of ๐œ‘ for ๐‘˜=1,2,3,โ€ฆ,๐‘, we get ๐น๐‘˜(๐‘ค,๐‘ฆ)+๐œ‘(๐‘ฆ)โ‰ฅ๐œ‘(๐‘ค).(3.46) So, we have ๐น๐‘˜(๐‘ค,๐‘ฆ)+๐œ‘(๐‘ฆ)โˆ’๐œ‘(๐‘ค)โ‰ฅ0,โˆ€๐‘˜=1,2,3,โ€ฆ,๐‘.(3.47) This implies that โ‹‚๐‘คโˆˆ๐‘๐‘˜=1SMEP(๐น๐‘˜).
Lastly, we show that ๐‘คโˆˆ๐ผ(๐ต,๐‘€). In fact, since B is ๐›ฝ-inverse strongly monotone, hence ๐ต is a monotone and Lipschitz continuous mapping. It follows from Lemma 2.3 that ๐‘€+๐ต is a maximal monotone. Let (๐‘ฃ,๐‘”)โˆˆ๐บ(๐‘€+๐ต), since ๐‘”โˆ’๐ต๐‘ฃโˆˆ๐‘€(๐‘ฃ). Again since ๐‘ฆ๐‘›๐‘–=๐ฝ๐‘€,๐œ†(๐‘ข๐‘›๐‘–โˆ’๐œ†๐ต๐‘ข๐‘›๐‘–), we have ๐‘ข๐‘›๐‘–โˆ’๐œ†๐ต๐‘ข๐‘›๐‘–โˆˆ(๐ผ+๐œ†๐‘€)(๐‘ฆ๐‘›๐‘–), that is, (1/๐œ†)(๐‘ข๐‘›๐‘–โˆ’๐‘ฆ๐‘›๐‘–โˆ’๐œ†๐ต๐‘ข๐‘›๐‘–)โˆˆ๐‘€(๐‘ฆ๐‘›๐‘–). By virtue of the maximal monotonicity of ๐‘€+๐ต, we have ๎‚ฌ๐‘ฃโˆ’๐‘ฆ๐‘›๐‘–1,๐‘”โˆ’๐ต๐‘ฃโˆ’๐œ†๎€ท๐‘ข๐‘›๐‘–โˆ’๐‘ฆ๐‘›๐‘–โˆ’๐œ†๐ต๐‘ข๐‘›๐‘–๎€ธ๎‚ญโ‰ฅ0,(3.48) and hence ๎ซ๐‘ฃโˆ’๐‘ฆ๐‘›๐‘–๎ฌโ‰ฅ๎‚ฌ,๐‘”๐‘ฃโˆ’๐‘ฆ๐‘›๐‘–1,๐ต๐‘ฃ+๐œ†๎€ท๐‘ข๐‘›๐‘–โˆ’๐‘ฆ๐‘›๐‘–โˆ’๐œ†๐ต๐‘ข๐‘›๐‘–๎€ธ๎‚ญ=๎ซ๐‘ฃโˆ’๐‘ฆ๐‘›๐‘–,๐ต๐‘ฃโˆ’๐ต๐‘ฆ๐‘›๐‘–๎ฌ+๎ซ๐‘ฃโˆ’๐‘ฆ๐‘›๐‘–,๐ต๐‘ฆ๐‘›๐‘–โˆ’๐ต๐‘ข๐‘›๐‘–๎ฌ+๎‚ฌ๐‘ฃโˆ’๐‘ฆ๐‘›๐‘–,1๐œ†๎€ท๐‘ข๐‘›๐‘–โˆ’๐‘ฆ๐‘›๐‘–๎€ธ๎‚ญ.(3.49) It follows from lim๐‘›โ†’โˆžโ€–๐‘ข๐‘›โˆ’๐‘ฆ๐‘›โ€–=0, we have lim๐‘›โ†’โˆžโ€–๐ต๐‘ข๐‘›โˆ’๐ต๐‘ฆ๐‘›โ€–=0 and ๐‘ฆ๐‘›๐‘–โ‡€๐‘ค, it follows that limsup๐‘›โ†’โˆžโŸจ๐‘ฃโˆ’๐‘ฆ๐‘›,๐‘”โŸฉ=โŸจ๐‘ฃโˆ’๐‘ค,๐‘”โŸฉโ‰ฅ0.(3.50) It follows from the maximal monotonicity of ๐ต+๐‘€ that ๐œƒโˆˆ(๐‘€+๐ต)(๐‘ค), that is, ๐‘คโˆˆ๐ผ(๐ต,๐‘€). Therefore, ๐‘คโˆˆ๐œƒ. We observe that limsup๐‘›โ†’โˆžโŸจ(๐›พ๐‘“โˆ’๐ด)๐‘ž,๐‘Š๐‘›๐‘ฆ๐‘›โˆ’๐‘žโŸฉ=lim๐‘–โ†’โˆž๎ซ(๐›พ๐‘“โˆ’๐ด)๐‘ž,๐‘Š๐‘›๐‘ฆ๐‘›๐‘–๎ฌโˆ’๐‘ž=โŸจ(๐›พ๐‘“โˆ’๐ด)๐‘ž,๐‘คโˆ’๐‘žโŸฉโ‰ค0.(3.51)
Step 6. Finally, we prove ๐‘ฅ๐‘›โ†’๐‘ž. By using (3.2) and together with Schwarz inequality, we have โ€–โ€–๐‘ฅ๐‘›+1โ€–โ€–โˆ’๐‘ž2=โ€–โ€–๐‘ƒ๐ถ๎€ท๐œ–๐‘›๎€ท๐‘ฅ๐›พ๐‘“๐‘›๎€ธ+๎€ท๐ผโˆ’๐œ–๐‘›๐ด๎€ธ๐‘Š๐‘›๐‘ฆ๐‘›๎€ธโˆ’๐‘ƒ๐ถโ€–โ€–(๐‘ž)2โ‰คโ€–โ€–๐œ–๐‘›๎€ท๎€ท๐‘ฅ๐›พ๐‘“๐‘›๎€ธ๎€ธ+๎€ทโˆ’๐ด๐‘ž๐ผโˆ’๐œ–๐‘›๐ด๐‘Š๎€ธ๎€ท๐‘›๐‘ฆ๐‘›๎€ธโ€–โ€–โˆ’๐‘ž2โ‰ค๎€ท๐ผโˆ’๐œ–๐‘›๐ด๎€ธ2โ€–โ€–๎€ท๐‘Š๐‘›๐‘ฆ๐‘›๎€ธโ€–โ€–โˆ’๐‘ž2+๐œ–2๐‘›โ€–โ€–๎€ท๐‘ฅ๐›พ๐‘“๐‘›๎€ธโ€–โ€–โˆ’๐ด๐‘ž2+2๐œ–๐‘›๎ซ๎€ท๐ผโˆ’๐œ–๐‘›๐ด๐‘Š๎€ธ๎€ท๐‘›๐‘ฆ๐‘›๎€ธ๎€ท๐‘ฅโˆ’๐‘ž,๐›พ๐‘“๐‘›๎€ธ๎ฌโ‰ค๎€ทโˆ’๐ด๐‘ž1โˆ’๐œ–๐‘›๐›พ๎€ธ2โ€–โ€–๐‘ฆ๐‘›โ€–โ€–โˆ’๐‘ž2+๐œ–2๐‘›โ€–โ€–๎€ท๐‘ฅ๐›พ๐‘“๐‘›๎€ธโ€–โ€–โˆ’๐ด๐‘ž2+2๐œ–๐‘›๎ซ๐‘Š๐‘›๐‘ฆ๐‘›๎€ท๐‘ฅโˆ’๐‘ž,๐›พ๐‘“๐‘›๎€ธ๎ฌโˆ’๐ด๐‘žโˆ’2๐œ–2๐‘›๎ซ๐ด๎€ท๐‘Š๐‘›๐‘ฆ๐‘›๎€ธ๎€ท๐‘ฅโˆ’๐‘ž,๐›พ๐‘“๐‘›๎€ธ๎ฌโ‰ค๎€ทโˆ’๐ด๐‘ž1โˆ’๐œ–๐‘›๐›พ๎€ธ2โ€–โ€–๐‘ฅ๐‘›โ€–โ€–โˆ’๐‘ž2+๐œ–2๐‘›โ€–โ€–๎€ท๐‘ฅ๐›พ๐‘“๐‘›๎€ธโ€–โ€–โˆ’๐ด๐‘ž2+2๐œ–๐‘›๎ซ๐‘Š๐‘›๐‘ฆ๐‘›๎€ท๐‘ฅโˆ’๐‘ž,๐›พ๐‘“๐‘›๎€ธ๎ฌโˆ’๐›พ๐‘“(๐‘ž)+2๐œ–๐‘›โŸจ๐‘Š๐‘›๐‘ฆ๐‘›โˆ’๐‘ž,๐›พ๐‘“(๐‘ž)โˆ’๐ด๐‘žโŸฉโˆ’2๐œ–2๐‘›๎ซ๐ด๎€ท๐‘Š๐‘›๐‘ฆ๐‘›๎€ธ๎€ท๐‘ฅโˆ’๐‘ž,๐›พ๐‘“๐‘›๎€ธ๎ฌโ‰ค๎€ทโˆ’๐ด๐‘ž1โˆ’๐œ–๐‘›๐›พ๎€ธ2โ€–โ€–๐‘ฅ๐‘›โ€–โ€–โˆ’๐‘ž2+๐œ–2๐‘›โ€–โ€–๎€ท๐‘ฅ๐›พ๐‘“๐‘›๎€ธโ€–โ€–โˆ’๐ด๐‘ž2+2๐œ–๐‘›โ€–โ€–๐‘Š๐‘›๐‘ฆ๐‘›โ€–โ€–โ€–โ€–๎€ท๐‘ฅโˆ’๐‘ž๐›พ๐‘“๐‘›๎€ธโ€–โ€–โˆ’๐›พ๐‘“(๐‘ž)+2๐œ–๐‘›โŸจ๐‘Š๐‘›๐‘ฆ๐‘›โˆ’๐‘ž,๐›พ๐‘“(๐‘ž)โˆ’๐ด๐‘žโŸฉโˆ’2๐œ–2๐‘›๎ซ๐ด๎€ท๐‘Š๐‘›๐‘ฆ๐‘›๎€ธ๎€ท๐‘ฅโˆ’๐‘ž,๐›พ๐‘“๐‘›๎€ธ๎ฌโ‰ค๎€ทโˆ’๐ด๐‘ž1โˆ’๐œ–๐‘›๐›พ๎€ธ2โ€–โ€–๐‘ฅ๐‘›โ€–โ€–โˆ’๐‘ž2+๐œ–2๐‘›โ€–โ€–๎€ท๐‘ฅ๐›พ๐‘“๐‘›๎€ธโ€–โ€–โˆ’๐ด๐‘ž2+2๐›พ๐œ–๐œ–๐‘›โ€–โ€–๐‘ฆ๐‘›โ€–โ€–โ€–โ€–๐‘ฅโˆ’๐‘ž๐‘›โ€–โ€–โˆ’๐‘ž+2๐œ–๐‘›โŸจ๐‘Š๐‘›๐‘ฆ๐‘›โˆ’๐‘ž,๐›พ๐‘“(๐‘ž)โˆ’๐ด๐‘žโŸฉโˆ’2๐œ–2๐‘›๎ซ๐ด๎€ท๐‘Š๐‘›๐‘ฆ๐‘›๎€ธ๎€ท๐‘ฅโˆ’๐‘ž,๐›พ๐‘“๐‘›๎€ธ๎ฌโ‰ค๎€ทโˆ’๐ด๐‘ž1โˆ’๐œ–๐‘›๐›พ๎€ธ2โ€–โ€–๐‘ฅ๐‘›โ€–โ€–โˆ’๐‘ž2+๐œ–2๐‘›โ€–โ€–๎€ท๐‘ฅ๐›พ๐‘“๐‘›๎€ธโ€–โ€–โˆ’๐ด๐‘ž2+2๐›พ๐œ–๐œ–๐‘›โ€–โ€–๐‘ฅ๐‘›โ€–โ€–โˆ’๐‘ž2+2๐œ–๐‘›โŸจ๐‘Š๐‘›๐‘ฆ๐‘›โˆ’๐‘ž,๐›พ๐‘“(๐‘ž)โˆ’๐ด๐‘žโŸฉโˆ’2๐œ–2๐‘›๎ซ๐ด๎€ท๐‘Š๐‘›๐‘ฆ๐‘›๎€ธ๎€ท๐‘ฅโˆ’๐‘ž,๐›พ๐‘“๐‘›๎€ธ๎ฌโ‰ค๎‚€๎€ทโˆ’๐ด๐‘ž1โˆ’๐œ–๐‘›๐›พ๎€ธ2+2๐›พ๐œ–๐œ–๐‘›๎‚โ€–โ€–๐‘ฅ๐‘›โ€–โ€–โˆ’๐‘ž2+๐œ–๐‘›๎‚†๐œ–๐‘›โ€–โ€–๎€ท๐‘ฅ๐›พ๐‘“๐‘›๎€ธโ€–โ€–โˆ’๐ด๐‘ž2+2โŸจ๐‘Š๐‘›๐‘ฆ๐‘›โˆ’๐‘ž,๐›พ๐‘“(๐‘ž)โˆ’๐ด๐‘žโŸฉโˆ’2๐œ–๐‘›โ€–โ€–๐ด๎€ท๐‘Š๐‘›๐‘ฆ๐‘›๎€ธโ€–โ€–โ€–โ€–๎€ท๐‘ฅโˆ’๐‘ž๐›พ๐‘“๐‘›๎€ธโ€–โ€–๎‚‡=๎€ท๎€ทโˆ’๐ด๐‘ž1โˆ’2๎€ธ๐œ–๐›พโˆ’๐›พ๐œ–๐‘›๎€ธโ€–โ€–๐‘ฅ๐‘›โ€–โ€–โˆ’๐‘ž2+๐œ–๐‘›๎‚†๐œ–๐‘›โ€–โ€–๎€ท๐‘ฅ๐›พ๐‘“๐‘›๎€ธโ€–โ€–โˆ’๐ด๐‘ž2+2โŸจ๐‘Š๐‘›๐‘ฆ๐‘›โˆ’๐‘ž,๐›พ๐‘“(๐‘ž)โˆ’๐ด๐‘žโŸฉโˆ’2๐œ–๐‘›โ€–โ€–๐ด๎€ท๐‘Š๐‘›๐‘ฆ๐‘›๎€ธโ€–โ€–โ€–โ€–๎€ท๐‘ฅโˆ’๐‘ž๐›พ๐‘“๐‘›๎€ธโ€–โ€–โˆ’๐ด๐‘ž+๐œ–๐‘›๐›พ2โ€–โ€–๐‘ฅ๐‘›โ€–โ€–โˆ’๐‘ž2๎‚‡.(3.52)
Since {๐‘ฅ๐‘›} is bounded, where ๐œ‚โ‰ฅโ€–๐›พ๐‘“(๐‘ฅ๐‘›)โˆ’๐ด๐‘žโ€–2โˆ’2โ€–๐ด(๐‘Š๐‘›๐‘ฆ๐‘›โˆ’๐‘ž)โ€–โ€–๐›พ๐‘“(๐‘ฅ๐‘›)โˆ’๐ด๐‘žโ€–+๐›พ2โ€–๐‘ฅ๐‘›โˆ’๐‘žโ€–2 for all ๐‘›โ‰ฅ0. It follows that โ€–โ€–๐‘ฅ๐‘›+1โ€–โ€–โˆ’๐‘ž2โ‰ค๎€ท๎€ท1โˆ’2๎€ธ๐œ–๐›พโˆ’๐›พ๐œ–๐‘›๎€ธโ€–โ€–๐‘ฅ๐‘›โ€–โ€–โˆ’๐‘ž2+๐œ–๐‘›๐›ฟ๐‘›,(3.53) where ๐›ฟ๐‘›=2โŸจ๐‘Š๐‘›๐‘ฆ๐‘›โˆ’๐‘ž,๐›พ๐‘“(๐‘ž)โˆ’๐ด๐‘žโŸฉ+๐œ‚๐›ผ๐‘›. Since limsup๐‘›โ†’โˆžโŸจ(๐›พ๐‘“โˆ’๐ด)๐‘ž,๐‘Š๐‘›๐‘ฆ๐‘›โˆ’๐‘žโŸฉโ‰ค0, we get limsup๐‘›โ†’โˆž๐›ฟ๐‘›โ‰ค0. Applying Lemma 2.5, we can conclude that ๐‘ฅ๐‘›โ†’๐‘ž. This completes the proof.

Corollary 3.2. Let ๐ป be a real Hilbert space and ๐ถ a nonempty closed and convex subset of ๐ป. Let ๐ต be ๐›ฝ-inverse-strongly monotone and ๐œ‘โˆถ๐ถโ†’โ„› a convex and lower semicontinuous function. Let ๐‘“โˆถ๐ถโ†’๐ถ be a contraction with coefficient ๐›ผ(0<๐›ผ<1), ๐‘€โˆถ๐ปโ†’2๐ป a maximal monotone mapping, and {๐‘‡๐‘›} a family of nonexpansive mappings of ๐ป into itself such that ๐œƒโˆถ=โˆž๎™๐‘›=1๐น๎€ท๐‘‡๐‘›๎€ธโˆฉ๎ƒฉ๐‘๎™๐‘˜=1๎€ท๐นSMEP๐‘˜๎€ธ๎ƒชโˆฉ๐ผ(๐ต,๐‘€)โ‰ 0.(3.54) Suppose that {๐‘ฅ๐‘›} is a sequence generated by the following algorithm for ๐‘ฅ0,๐‘ข๐‘›โˆˆ๐ถ arbitrarily: ๐‘ข๐‘›=๐พ๐น๐‘๐‘Ÿ๐‘›,๐‘›โ‹…๐พ๐น๐‘โˆ’1๐‘Ÿ๐‘›โˆ’1,๐‘›โ‹…๐พ๐น๐‘โˆ’2๐‘Ÿ๐‘›โˆ’2,๐‘›โ‹…โ‹ฏโ‹…๐พ๐น2๐‘Ÿ2,๐‘›โ‹…๐พ๐น1๐‘Ÿ1,๐‘›โ‹…๐‘ฅ๐‘›๐‘ฅ,โˆ€๐‘›โˆˆ๐‘๐‘›+1=๐‘ƒ๐ถ๎€บ๐œ–๐‘›๐‘“๎€ท๐‘ฅ๐‘›๎€ธ+๎€ท๐ผโˆ’๐œ–๐‘›๎€ธ๐‘Š๐‘›๐ฝ๐‘€,๐œ†๎€ท๐‘ข๐‘›โˆ’๐œ†๐ต๐‘ข๐‘›,๎€ธ๎€ป(3.55) for all ๐‘›=0,1,2,โ€ฆ, and the conditions (C1)โ€“(C3) in Theorem 3.1 are satisfied.
Then, the sequence {๐‘ฅ๐‘›} converges strongly to ๐‘žโˆˆ๐œƒ, where ๐‘ž=๐‘ƒ๐œƒ(๐‘“+๐ผ)(๐‘ž) which solves the following variational inequality: โŸจ(๐‘“โˆ’๐ผ)๐‘ž,๐‘โˆ’๐‘žโŸฉโ‰ค0,โˆ€๐‘โˆˆ๐œƒ.(3.56)

Proof. Putting ๐ดโ‰ก๐ผ and ๐›พโ‰ก1 in Theorem 3.1, we can obtain the desired conclusion immediately.

Corollary 3.3. Let ๐ป be a real Hilbert space and ๐ถ a nonempty closed and convex subset of ๐ป. Let ๐ต be ๐›ฝ-inverse-strongly monotone, ๐œ‘โˆถ๐ถโ†’โ„› a convex and lower semicontinuous function, and ๐‘€โˆถ๐ปโ†’2๐ป a maximal monotone mapping. Let {๐‘‡๐‘›} be a family of nonexpansive mappings of ๐ป into itself such that ๐œƒโˆถ=โˆž๎™๐‘›=1๐น๎€ท๐‘‡๐‘›๎€ธโˆฉ๎ƒฉ๐‘๎™๐‘˜=1๎€ท๐นSMEP๐‘˜๎€ธ๎ƒชโˆฉ๐ผ(๐ต,๐‘€)โ‰ 0