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Journal of Applied Mathematics
VolumeΒ 2012, Article IDΒ 638632, 11 pages
http://dx.doi.org/10.1155/2012/638632
Research Article

Iterative Methods for the Sum of Two Monotone Operators

Department of Information Management, Cheng Shiu University, Kaohsiung 833, Taiwan

Received 3 October 2011; Accepted 7 October 2011

Academic Editor: YonghongΒ Yao

Copyright Β© 2012 Yeong-Cheng Liou. 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 an iterative for finding the zeros point of the sum of two monotone operators. We prove that the suggested method converges strongly to the zeros point of the sum of two monotone operators.

1. Introduction

Let 𝐢 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let π΄βˆΆπΆβ†’π» be a single-valued nonlinear mapping and let π΅βˆΆπ»β†’2𝐻 be a multivalued mapping. The β€œso-called” quasi-variational inclusion problem is to find a π‘’βˆˆ2𝐻 such that0∈𝐴π‘₯+𝐡π‘₯.(1.1) The set of solutions of (1.1) is denoted by (𝐴+𝐡)βˆ’1(0). A number of problems arising in structural analysis, mechanics, and economics can be studied in the framework of this kind of variational inclusions; see, for instance, [1–4]. The problem (1.1) includes many problems as special cases.(1)If 𝐡=πœ•πœ™βˆΆπ»β†’2𝐻, where πœ™βˆΆπ»β†’π‘…βˆͺ+∞ is a proper convex lower semicontinuous function and πœ•πœ™ is the subdif and if onlyerential of πœ™, then the variational inclusion problem (1.1) is equivalent to find π‘’βˆˆπ» such that βŸ¨π΄π‘’,π‘¦βˆ’π‘’βŸ©+πœ™(𝑦)βˆ’πœ™(𝑒)β‰₯0,βˆ€π‘¦βˆˆπ»,(1.2) which is called the mixed quasi-variational inequality (see, Noor [5]).(2)If 𝐡=πœ•π›ΏπΆ, where 𝐢 is a nonempty closed convex subset of 𝐻 and π›ΏπΆβˆΆπ»β†’[0,∞] is the indicator function of 𝐢, that is, 𝛿𝐢=ξƒ―0,π‘₯∈𝐢,+∞,π‘₯βˆ‰πΆ,(1.3) then the variational inclusion problem (1.1) is equivalent to find π‘’βˆˆπΆ such that βŸ¨π΄π‘’,π‘£βˆ’π‘’βŸ©β‰₯0,βˆ€π‘£βˆˆπΆ.(1.4)

This problem is called Hartman-Stampacchia variational inequality (see, e.g., [6]).

Recently, Zhang et al. [7] introduced a new iterative scheme for finding a common element of the set of solutions to the inclusion problem, and the set of fixed points of nonexpansive mappings in Hilbert spaces. Peng et al. [8] introduced another iterative scheme by the viscosity approximate method for finding a common element of the set of solutions of a variational inclusion with set-valued maximal monotone mapping and inverse strongly monotone mappings, the set of solutions of an equilibrium problem, and the set of fixed points of a nonexpansive mapping. For some related works, please see [9–27] and the references therein.

Inspired and motivated by the works in the literature, in this paper, we introduce an iterative for solving the problem (1.1). We prove that the suggested method converges strongly to the zeros point of the sum of two monotone operators 𝐴+𝐡.

2. Preliminaries

Let 𝐻 be a real Hilbert space with inner product βŸ¨β‹…,β‹…βŸ© and norm β€–β‹…β€–, respectively. Let 𝐢 be a nonempty closed convex subset of 𝐻. Recall that a mapping π΄βˆΆπΆβ†’π» is said to be 𝛼-inverse strongly-monotone if and if only ⟨𝐴π‘₯βˆ’π΄π‘¦,π‘₯βˆ’π‘¦βŸ©β‰₯𝛼‖𝐴π‘₯βˆ’π΄π‘¦β€–2(2.1) for some 𝛼>0 and for all π‘₯,π‘¦βˆˆπΆ. It is known that if 𝐴 is 𝛼-inverse strongly monotone, then 1‖𝐴π‘₯βˆ’π΄π‘¦β€–β‰€π›Όβ€–π‘₯βˆ’π‘¦β€–(2.2) for all π‘₯,π‘¦βˆˆπΆ.

Let 𝐡 be a mapping of 𝐻 into 2𝐻. The effective domain of 𝐡 is denoted by dom(𝐡), that is,dom(𝐡)={π‘₯∈𝐻∢𝐡π‘₯β‰ βˆ…}.(2.3) A multivalued mapping 𝐡 is said to be a monotone operator on 𝐻 if and if only ⟨π‘₯βˆ’π‘¦,π‘’βˆ’π‘£βŸ©β‰₯0(2.4) for all π‘₯,π‘¦βˆˆdom(𝐡), π‘’βˆˆπ΅π‘₯, and π‘£βˆˆπ΅π‘¦. A monotone operator 𝐡 on 𝐻 is said to be maximal if and if only its graph is not strictly contained in the graph of any other monotone operator on 𝐻. Let 𝐡 be a maximal monotone operator on 𝐻 and let π΅βˆ’10={π‘₯∈𝐻∢0∈𝐡π‘₯}.

For a maximal monotone operator 𝐡 on 𝐻 and πœ†>0, we may define a single-valued operator: π½π΅πœ†=(𝐼+πœ†π΅)βˆ’1βˆΆπ»β†’dom(𝐡),(2.5) which is called the resolvent of 𝐡 for πœ†. It is known that the resolvent π½π΅πœ† is firmly nonexpansive, that is, β€–β€–π½π΅πœ†π‘₯βˆ’π½π΅πœ†π‘¦β€–β€–2β‰€ξ«π½π΅πœ†π‘₯βˆ’π½π΅πœ†ξ¬π‘¦,π‘₯βˆ’π‘¦(2.6) for all π‘₯,π‘¦βˆˆπΆ and π΅βˆ’10=𝐹(π½π΅πœ†) for all πœ†>0.

The following resolvent identity is well known: for πœ†>0 and πœ‡>0, there holds the following identity:π½π΅πœ†π‘₯=π½π΅πœ‡ξ‚€πœ‡πœ†ξ‚€πœ‡π‘₯+1βˆ’πœ†ξ‚π½π΅πœ†π‘₯,π‘₯∈𝐻.(2.7)

We use the following notation: (i)π‘₯𝑛⇀π‘₯ stands for the weak convergence of (π‘₯𝑛) to π‘₯; (ii)π‘₯𝑛→π‘₯ stands for the strong convergence of (π‘₯𝑛) to π‘₯.

We need the following lemmas for the next section.

Lemma 2.1 (see [28]). Let 𝐢 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let the mapping π΄βˆΆπΆβ†’π» be 𝛼-inverse strongly monotone and let πœ†>0 be a constant. Then, one has β€–β€–(πΌβˆ’πœ†π΄)π‘₯βˆ’(πΌβˆ’πœ†π΄)𝑦2≀‖π‘₯βˆ’π‘¦β€–2+πœ†(πœ†βˆ’2𝛼)‖𝐴π‘₯βˆ’π΄π‘¦β€–2,βˆ€π‘₯,π‘¦βˆˆπΆ.(2.8) In particular, if 0β‰€πœ†β‰€2𝛼, then πΌβˆ’πœ†π΄ is nonexpansive.

Lemma 2.2 (see [29]). 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.(2.9) Suppose that π‘₯𝑛+1=ξ€·1βˆ’π›½π‘›ξ€Έπ‘¦π‘›+𝛽𝑛π‘₯𝑛(2.10) for all 𝑛β‰₯0 and limsupπ‘›β†’βˆžξ€·β€–β€–π‘¦π‘›+1βˆ’π‘¦π‘›β€–β€–βˆ’β€–β€–π‘₯𝑛+1βˆ’π‘₯𝑛‖‖≀0.(2.11) Then, limπ‘›β†’βˆžβ€–π‘¦π‘›βˆ’π‘₯𝑛‖=0.

Lemma 2.3 (see [30]). Assume that {π‘Žπ‘›} is a sequence of nonnegative real numbers such that π‘Žπ‘›+1≀1βˆ’π›Ύπ‘›ξ€Έπ‘Žπ‘›+𝛿𝑛𝛾𝑛,(2.12) where {𝛾𝑛} is a sequence in (0,1) and {𝛿𝑛} is a sequence such that (1)βˆ‘βˆžπ‘›=1𝛾𝑛=∞; (2)limsupπ‘›β†’βˆžπ›Ώπ‘›β‰€0 or βˆ‘βˆžπ‘›=1|𝛿𝑛𝛾𝑛|<∞. Then limπ‘›β†’βˆžπ‘Žπ‘›=0.

3. Main Results

In this section, we will prove our main result.

Theorem 3.1. Let 𝐢 be a nonempty closed and convex subset of a real Hilbert space 𝐻. Let 𝐴 be an 𝛼-inverse strongly monotone mapping of 𝐢 into H and let 𝐡 be a maximal monotone operator on 𝐻, such that the domain of 𝐡 is included in 𝐢. Let π½π΅πœ†=(𝐼+πœ†π΅)βˆ’1 be the resolvent of 𝐡 for πœ†>0. Suppose that (𝐴+𝐡)βˆ’10β‰ βˆ…. For π‘’βˆˆπΆ and given π‘₯0∈𝐢, let {π‘₯𝑛}βŠ‚πΆ be a sequence generated by π‘₯𝑛+1=𝛽𝑛π‘₯𝑛+ξ€·1βˆ’π›½π‘›ξ€Έπ½π΅πœ†π‘›ξ€·π›Όπ‘›ξ€·π‘’+1βˆ’π›Όπ‘›π‘₯ξ€Έξ€·π‘›βˆ’πœ†π‘›π΄π‘₯𝑛(3.1) for all 𝑛β‰₯0, where {πœ†π‘›}βŠ‚(0,2𝛼), {𝛼𝑛}βŠ‚(0,1), and {𝛽𝑛}βŠ‚(0,1) satisfy (i)limπ‘›β†’βˆžπ›Όπ‘›=0 and βˆ‘π‘›π›Όπ‘›=∞; (ii)0<liminfπ‘›β†’βˆžπ›½π‘›β‰€limsupπ‘›β†’βˆžπ›½π‘›<1; (iii)π‘Žβ‰€πœ†π‘›β‰€π‘ where [π‘Ž,𝑏]βŠ‚(0,2𝛼) and limπ‘›β†’βˆž(πœ†π‘›+1βˆ’πœ†π‘›)=0. Then {π‘₯𝑛} generated by (3.1) converges strongly to Μƒπ‘₯=𝑃(𝐴+𝐡)βˆ’10(𝑒).

Proof. First, we choose any π‘§βˆˆ(𝐴+𝐡)βˆ’10. Note that 𝑧=π½π΅πœ†π‘›ξ€·π‘§βˆ’πœ†π‘›ξ€·1βˆ’π›Όπ‘›ξ€Έξ€Έπ΄π‘§=π½π΅πœ†π‘›ξ€·π›Όπ‘›ξ€·π‘§+1βˆ’π›Όπ‘›ξ€Έξ€·π‘§βˆ’πœ†π‘›π΄π‘§ξ€Έξ€Έ(3.2) for all 𝑛β‰₯0. Since π½π΅πœ† is nonexpansive for all πœ†>0, we have β€–β€–π½π΅πœ†π‘›ξ€·π›Όπ‘›ξ€·π‘’+1βˆ’π›Όπ‘›π‘₯ξ€Έξ€·π‘›βˆ’πœ†π‘›π΄π‘₯π‘›β€–β€–ξ€Έξ€Έβˆ’π‘§2=β€–β€–π½π΅πœ†π‘›ξ€·π›Όπ‘›ξ€·π‘’+1βˆ’π›Όπ‘›π‘₯ξ€Έξ€·π‘›βˆ’πœ†π‘›π΄π‘₯π‘›ξ€Έξ€Έβˆ’π½π΅πœ†π‘›ξ€·π›Όπ‘›ξ€·π‘§+1βˆ’π›Όπ‘›ξ€Έξ€·π‘§βˆ’πœ†π‘›β€–β€–π΄π‘§ξ€Έξ€Έ2≀‖‖𝛼𝑛𝑒+1βˆ’π›Όπ‘›π‘₯ξ€Έξ€·π‘›βˆ’πœ†π‘›π΄π‘₯π‘›βˆ’ξ€·π›Όξ€Έξ€Έπ‘›ξ€·π‘§+1βˆ’π›Όπ‘›ξ€Έξ€·π‘§βˆ’πœ†π‘›β€–β€–π΄π‘§ξ€Έξ€Έ2=β€–β€–ξ€·1βˆ’π›Όπ‘›π‘₯ξ€Έξ€·ξ€·π‘›βˆ’πœ†π‘›π΄π‘₯π‘›ξ€Έβˆ’ξ€·π‘§βˆ’πœ†π‘›π΄π‘§ξ€Έξ€Έ+𝛼𝑛‖‖(π‘’βˆ’π‘§)2.(3.3) Since 𝐴 is 𝛼-inverse strongly monotone, we get β€–β€–ξ€·1βˆ’π›Όπ‘›π‘₯ξ€Έξ€·ξ€·π‘›βˆ’πœ†π‘›π΄π‘₯π‘›ξ€Έβˆ’ξ€·π‘§βˆ’πœ†π‘›π΄π‘§ξ€Έξ€Έ+𝛼𝑛‖‖(π‘’βˆ’π‘§)2≀1βˆ’π›Όπ‘›ξ€Έβ€–β€–ξ€·π‘₯π‘›βˆ’πœ†π‘›π΄π‘₯π‘›ξ€Έβˆ’ξ€·π‘§βˆ’πœ†π‘›ξ€Έβ€–β€–π΄π‘§2+π›Όπ‘›β€–π‘’βˆ’π‘§β€–2=ξ€·1βˆ’π›Όπ‘›ξ€Έβ€–β€–ξ€·π‘₯π‘›ξ€Έβˆ’π‘§βˆ’πœ†π‘›ξ€·π΄π‘₯π‘›ξ€Έβ€–β€–βˆ’π΄π‘§2+π›Όπ‘›β€–π‘’βˆ’π‘§β€–2=ξ€·1βˆ’π›Όπ‘›ξ€Έξ‚€β€–β€–π‘₯π‘›β€–β€–βˆ’π‘§2βˆ’2πœ†π‘›βŸ¨π΄π‘₯π‘›βˆ’π΄π‘§,π‘₯π‘›βˆ’π‘§βŸ©+πœ†2𝑛‖‖𝐴π‘₯π‘›β€–β€–βˆ’π΄π‘§2+π›Όπ‘›β€–π‘’βˆ’π‘§β€–2≀1βˆ’π›Όπ‘›ξ€Έξ‚€β€–β€–π‘₯π‘›β€–β€–βˆ’π‘§2βˆ’2π›Όπœ†π‘›β€–β€–π΄π‘₯π‘›β€–β€–βˆ’π΄π‘§2+πœ†2𝑛‖‖𝐴π‘₯π‘›β€–β€–βˆ’π΄π‘§2+π›Όπ‘›β€–π‘’βˆ’π‘§β€–2=ξ€·1βˆ’π›Όπ‘›ξ€Έξ‚€β€–β€–π‘₯π‘›β€–β€–βˆ’π‘§2+πœ†π‘›ξ€·πœ†π‘›ξ€Έβ€–β€–βˆ’2𝛼𝐴π‘₯π‘›β€–β€–βˆ’π΄π‘§2+π›Όπ‘›β€–π‘’βˆ’π‘§β€–2.(3.4) By (3.3) and (3.4), we obtain β€–β€–π½π΅πœ†π‘›ξ€·π›Όπ‘›ξ€·π‘’+1βˆ’π›Όπ‘›π‘₯ξ€Έξ€·π‘›βˆ’πœ†π‘›π΄π‘₯π‘›β€–β€–ξ€Έξ€Έβˆ’π‘§2≀1βˆ’π›Όπ‘›ξ€Έξ‚€β€–β€–π‘₯π‘›β€–β€–βˆ’π‘§2+πœ†π‘›ξ€·πœ†π‘›ξ€Έβ€–β€–βˆ’2𝛼𝐴π‘₯π‘›β€–β€–βˆ’π΄π‘§2+π›Όπ‘›β€–π‘’βˆ’π‘§β€–2≀1βˆ’π›Όπ‘›ξ€Έβ€–β€–π‘₯π‘›β€–β€–βˆ’π‘§2+π›Όπ‘›β€–π‘’βˆ’π‘§β€–2.(3.5) It follows from (3.1) and (3.5) that β€–β€–π‘₯𝑛+1β€–β€–βˆ’π‘§2=‖‖𝛽𝑛π‘₯𝑛+ξ€·βˆ’π‘§1βˆ’π›½π‘›ξ€Έξ‚€π½π΅πœ†π‘›ξ€·π›Όπ‘›ξ€·π‘’+1βˆ’π›Όπ‘›π‘₯ξ€Έξ€·π‘›βˆ’πœ†π‘›π΄π‘₯π‘›ξ‚β€–β€–ξ€Έξ€Έβˆ’π‘§2≀𝛽𝑛‖‖π‘₯π‘›β€–β€–βˆ’π‘§2+ξ€·1βˆ’π›½π‘›ξ€Έβ€–β€–π½π΅πœ†π‘›ξ€·π›Όπ‘›ξ€·π‘’+1βˆ’π›Όπ‘›π‘₯ξ€Έξ€·π‘›βˆ’πœ†π‘›π΄π‘₯π‘›β€–β€–ξ€Έξ€Έβˆ’π‘§2≀𝛽𝑛‖‖π‘₯π‘›β€–β€–βˆ’π‘§2+ξ€·1βˆ’π›½π‘›ξ€Έξ‚€ξ€·1βˆ’π›Όπ‘›ξ€Έβ€–β€–π‘₯π‘›β€–β€–βˆ’π‘§2+π›Όπ‘›β€–π‘’βˆ’π‘§β€–2=ξ€Ίξ€·1βˆ’1βˆ’π›½π‘›ξ€Έπ›Όπ‘›ξ€»β€–β€–π‘₯π‘›β€–β€–βˆ’π‘§2+ξ€·1βˆ’π›½π‘›ξ€Έπ›Όπ‘›β€–π‘’βˆ’π‘§β€–2‖‖π‘₯≀maxπ‘›β€–β€–βˆ’π‘§2,β€–π‘’βˆ’π‘§β€–2.(3.6) By induction, we have β€–β€–π‘₯𝑛+1β€–β€–ξ€½β€–β€–π‘₯βˆ’π‘§β‰€max0β€–β€–ξ€Ύβˆ’π‘§,β€–π‘’βˆ’π‘§β€–.(3.7) Therefore, {π‘₯𝑛} is bounded. We deduce immediately that {𝐴π‘₯𝑛} is also bounded. Set 𝑒𝑛=𝛼𝑛𝑒+(1βˆ’π›Όπ‘›)(π‘₯π‘›βˆ’πœ†π‘›π΄π‘₯𝑛) for all 𝑛. Then {𝑒𝑛} and {π½π΅πœ†π‘›π‘’π‘›} are bounded.
Next, we estimate β€–π½π΅πœ†π‘›+1𝑒𝑛+1βˆ’π½π΅πœ†π‘›π‘’π‘›β€–. In fact, we have β€–β€–π½π΅πœ†π‘›+1𝑒𝑛+1βˆ’π½π΅πœ†π‘›π‘’π‘›β€–β€–=β€–β€–π½π΅πœ†π‘›+1𝛼𝑛+1𝑒+1βˆ’π›Όπ‘›+1π‘₯𝑛+1βˆ’πœ†π‘›+1𝐴π‘₯𝑛+1ξ€Έξ€Έβˆ’π½π΅πœ†π‘›ξ€·π›Όπ‘›ξ€·π‘’+1βˆ’π›Όπ‘›π‘₯ξ€Έξ€·π‘›βˆ’πœ†π‘›π΄π‘₯π‘›β€–β€–β‰€β€–β€–π½ξ€Έξ€Έπ΅πœ†π‘›+1𝛼𝑛+1𝑒+1βˆ’π›Όπ‘›+1π‘₯𝑛+1βˆ’πœ†π‘›+1𝐴π‘₯𝑛+1ξ€Έξ€Έβˆ’π½π΅πœ†π‘›+1𝛼𝑛𝑒+1βˆ’π›Όπ‘›π‘₯ξ€Έξ€·π‘›βˆ’πœ†π‘›π΄π‘₯𝑛‖‖+β€–β€–π½ξ€Έξ€Έπ΅πœ†π‘›+1𝛼𝑛𝑒+1βˆ’π›Όπ‘›π‘₯ξ€Έξ€·π‘›βˆ’πœ†π‘›π΄π‘₯π‘›ξ€Έξ€Έβˆ’π½π΅πœ†π‘›ξ€·π›Όπ‘›ξ€·π‘’+1βˆ’π›Όπ‘›π‘₯ξ€Έξ€·π‘›βˆ’πœ†π‘›π΄π‘₯𝑛‖‖≀‖‖𝛼𝑛+1𝑒+1βˆ’π›Όπ‘›+1π‘₯𝑛+1βˆ’πœ†π‘›+1𝐴π‘₯𝑛+1βˆ’ξ€·π›Όξ€Έξ€Έπ‘›ξ€·π‘’+1βˆ’π›Όπ‘›π‘₯ξ€Έξ€·π‘›βˆ’πœ†π‘›π΄π‘₯𝑛‖‖+β€–β€–π½ξ€Έξ€Έπ΅πœ†π‘›+1π‘’π‘›βˆ’π½π΅πœ†π‘›π‘’π‘›β€–β€–β‰€β€–β€–ξ€·πΌβˆ’πœ†π‘›+1𝐴π‘₯𝑛+1βˆ’ξ€·πΌβˆ’πœ†π‘›+1𝐴π‘₯𝑛‖‖+||πœ†π‘›+1βˆ’πœ†π‘›||‖‖𝐴π‘₯𝑛‖‖+𝛼𝑛+1ξ€·β€–β€–π‘₯‖𝑒‖+𝑛+1β€–β€–+πœ†π‘›+1‖‖𝐴π‘₯𝑛+1β€–β€–ξ€Έ+𝛼𝑛‖‖π‘₯‖𝑒‖+𝑛‖‖+πœ†π‘›β€–β€–π΄π‘₯𝑛‖‖+β€–β€–π½π΅πœ†π‘›+1π‘’π‘›βˆ’π½π΅πœ†π‘›π‘’π‘›β€–β€–.(3.8)
Since πΌβˆ’πœ†π‘›+1𝐴 is nonexpansive for πœ†π‘›+1∈(0,2𝛼), we have β€–(πΌβˆ’πœ†π‘›+1𝐴)π‘₯𝑛+1βˆ’(πΌβˆ’πœ†π‘›+1𝐴)π‘₯𝑛‖≀‖π‘₯𝑛+1βˆ’π‘₯𝑛‖. By the resolvent identity (2.7), we have π½π΅πœ†π‘›+1𝑒𝑛=π½π΅πœ†π‘›ξ‚΅πœ†π‘›πœ†π‘›+1𝑒𝑛+ξ‚΅πœ†1βˆ’π‘›πœ†π‘›+1ξ‚Άπ½π΅πœ†π‘›+1𝑒𝑛.(3.9) It follows that β€–β€–π½π΅πœ†π‘›+1π‘’π‘›βˆ’π½π΅πœ†π‘›π‘’π‘›β€–β€–=β€–β€–β€–π½π΅πœ†π‘›ξ‚΅πœ†π‘›πœ†π‘›+1𝑒𝑛+ξ‚΅πœ†1βˆ’π‘›πœ†π‘›+1ξ‚Άπ½π΅πœ†π‘›+1π‘’π‘›ξ‚Άβˆ’π½π΅πœ†π‘›π‘’π‘›β€–β€–β€–β‰€β€–β€–β€–ξ‚΅πœ†π‘›πœ†π‘›+1𝑒𝑛+ξ‚΅πœ†1βˆ’π‘›πœ†π‘›+1ξ‚Άπ½π΅πœ†π‘›+1π‘’π‘›ξ‚Άβˆ’π‘’π‘›β€–β€–β€–β‰€||πœ†π‘›+1βˆ’πœ†π‘›||πœ†π‘›+1β€–β€–π‘’π‘›βˆ’π½π΅πœ†π‘›+1𝑒𝑛‖‖.(3.10) So, β€–β€–π½π΅πœ†π‘›+1𝑒𝑛+1βˆ’π½π΅πœ†π‘›π‘’π‘›β€–β€–β‰€β€–β€–π‘₯𝑛+1βˆ’π‘₯𝑛‖‖+||πœ†π‘›+1βˆ’πœ†π‘›||‖‖𝐴π‘₯𝑛‖‖+𝛼𝑛+1ξ€·β€–β€–π‘₯‖𝑒‖+𝑛+1β€–β€–+πœ†π‘›+1‖‖𝐴π‘₯𝑛+1β€–β€–ξ€Έ+𝛼𝑛‖‖π‘₯‖𝑒‖+𝑛‖‖+πœ†π‘›β€–β€–π΄π‘₯𝑛‖‖+||πœ†π‘›+1βˆ’πœ†π‘›||πœ†π‘›+1β€–β€–π‘’π‘›βˆ’π½π΅πœ†π‘›+1𝑒𝑛‖‖.(3.11) Thus, limsupπ‘›β†’βˆžξ‚€β€–β€–π½π΅πœ†π‘›+1𝑒𝑛+1βˆ’π½π΅πœ†π‘›π‘’π‘›β€–β€–βˆ’β€–β€–π‘₯𝑛+1βˆ’π‘₯𝑛‖‖≀0.(3.12) From Lemma 2.2, we get limπ‘›β†’βˆžβ€–β€–π½π΅πœ†π‘›π‘’π‘›βˆ’π‘₯𝑛‖‖=0.(3.13) Consequently, we obtain limπ‘›β†’βˆžβ€–β€–π‘₯𝑛+1βˆ’π‘₯𝑛‖‖=limπ‘›β†’βˆžξ€·1βˆ’π›½π‘›ξ€Έβ€–β€–π½π΅πœ†π‘›π‘’π‘›βˆ’π‘₯𝑛‖‖=0.(3.14) From (3.5) and (3.6), we have β€–β€–π‘₯𝑛+1β€–β€–βˆ’π‘§2≀𝛽𝑛‖‖π‘₯π‘›β€–β€–βˆ’π‘§2+ξ€·1βˆ’π›½π‘›ξ€Έβ€–β€–π½π΅πœ†π‘›ξ€·π›Όπ‘›ξ€·π‘’+1βˆ’π›Όπ‘›π‘₯ξ€Έξ€·π‘›βˆ’πœ†π‘›π΄π‘₯π‘›β€–β€–ξ€Έξ€Έβˆ’π‘§2≀1βˆ’π›½π‘›ξ€Έξ‚†ξ€·1βˆ’π›Όπ‘›ξ€Έξ‚€β€–β€–π‘₯π‘›β€–β€–βˆ’π‘§2+πœ†π‘›ξ€·πœ†π‘›ξ€Έβ€–β€–βˆ’2𝛼𝐴π‘₯π‘›β€–β€–βˆ’π΄π‘§2+π›Όπ‘›β€–π‘’βˆ’π‘§β€–2+𝛽𝑛‖‖π‘₯π‘›β€–β€–βˆ’π‘§2=ξ€Ίξ€·1βˆ’1βˆ’π›½π‘›ξ€Έπ›Όπ‘›ξ€»β€–β€–π‘₯π‘›β€–β€–βˆ’π‘§2+ξ€·1βˆ’π›½π‘›ξ€Έπœ†π‘›ξ€·πœ†π‘›ξ€Έβ€–β€–βˆ’2𝛼𝐴π‘₯π‘›β€–β€–βˆ’π΄π‘§2+ξ€·1βˆ’π›½π‘›ξ€Έπ›Όπ‘›β€–π‘’βˆ’π‘§β€–2≀‖‖π‘₯π‘›β€–β€–βˆ’π‘§2+ξ€·1βˆ’π›½π‘›ξ€Έπœ†π‘›ξ€·πœ†π‘›ξ€Έβ€–β€–βˆ’2𝛼𝐴π‘₯π‘›β€–β€–βˆ’π΄π‘§2+ξ€·1βˆ’π›½π‘›ξ€Έπ›Όπ‘›β€–π‘’βˆ’π‘§β€–2.(3.15) It follows that ξ€·1βˆ’π›½π‘›ξ€Έπœ†π‘›ξ€·2π›Όβˆ’πœ†π‘›ξ€Έβ€–β€–π΄π‘₯π‘›β€–β€–βˆ’π΄π‘§2≀‖‖π‘₯π‘›β€–β€–βˆ’π‘§2βˆ’β€–β€–π‘₯𝑛+1β€–β€–βˆ’π‘§2+ξ€·1βˆ’π›½π‘›ξ€Έπ›Όπ‘›β€–π‘’βˆ’π‘§β€–2≀‖‖π‘₯π‘›β€–β€–βˆ’β€–β€–π‘₯βˆ’π‘§π‘›+1β€–β€–ξ€Έβ€–β€–π‘₯βˆ’π‘§π‘›+1βˆ’π‘₯𝑛‖‖+ξ€·1βˆ’π›½π‘›ξ€Έπ›Όπ‘›β€–π‘’βˆ’π‘§β€–2.(3.16) Since limπ‘›β†’βˆžπ›Όπ‘›=0, limπ‘›β†’βˆžβ€–π‘₯𝑛+1βˆ’π‘₯𝑛‖=0, and liminfπ‘›β†’βˆž(1βˆ’π›½π‘›)πœ†π‘›(2π›Όβˆ’πœ†π‘›)>0, we have limπ‘›β†’βˆžβ€–β€–π΄π‘₯π‘›β€–β€–βˆ’π΄π‘§=0.(3.17) Put Μƒπ‘₯=𝑃(𝐴+𝐡)βˆ’10(𝑒). Set 𝑣𝑛=π‘₯π‘›βˆ’(πœ†π‘›/(1βˆ’π›Όπ‘›))(𝐴π‘₯π‘›βˆ’π΄Μƒπ‘₯) for all 𝑛. Take 𝑧=Μƒπ‘₯ in (3.17) to get ‖𝐴π‘₯π‘›βˆ’π΄Μƒπ‘₯β€–β†’0. First, we prove limsupπ‘›β†’βˆžβŸ¨π‘’βˆ’Μƒπ‘₯,π‘£π‘›βˆ’Μƒπ‘₯βŸ©β‰€0. We take a subsequence {𝑣𝑛𝑖} of {𝑣𝑛} such that limsupπ‘›β†’βˆžβŸ¨π‘’βˆ’Μƒπ‘₯,π‘£π‘›βˆ’Μƒπ‘₯⟩=limπ‘–β†’βˆžξ«π‘’βˆ’Μƒπ‘₯,𝑣𝑛𝑖.βˆ’Μƒπ‘₯(3.18) It is clear that {𝑣𝑛𝑖} is bounded due to the boundedness of {π‘₯𝑛} and ‖𝐴π‘₯π‘›βˆ’π΄Μƒπ‘₯β€–β†’0. Then, there exists a subsequence {𝑣𝑛𝑖𝑗} of {𝑣𝑛𝑖} which converges weakly to some point π‘€βˆˆπΆ. Hence, {π‘₯𝑛𝑖𝑗} also converges weakly to 𝑀 because of β€–π‘£π‘›π‘–π‘—βˆ’π‘₯𝑛𝑖𝑗‖→0. By the similar argument as that in [31], we can show that π‘€βˆˆ(𝐴+𝐡)βˆ’10. This implies that limsupπ‘›β†’βˆžβŸ¨π‘’βˆ’Μƒπ‘₯,π‘£π‘›βˆ’Μƒπ‘₯⟩=limπ‘—β†’βˆžξ‚¬π‘’βˆ’Μƒπ‘₯,π‘£π‘›π‘–π‘—ξ‚­βˆ’Μƒπ‘₯=βŸ¨π‘’βˆ’Μƒπ‘₯,π‘€βˆ’Μƒπ‘₯⟩.(3.19) Note that Μƒπ‘₯=𝑃(𝐴+𝐡)βˆ’10(𝑒). Then, βŸ¨π‘’βˆ’Μƒπ‘₯,π‘€βˆ’Μƒπ‘₯βŸ©β‰€0,π‘€βˆˆ(𝐴+𝐡)βˆ’10. Therefore, limsupπ‘›β†’βˆžβŸ¨π‘’βˆ’Μƒπ‘₯,π‘£π‘›βˆ’Μƒπ‘₯βŸ©β‰€0.(3.20) Finally, we prove that π‘₯𝑛→̃π‘₯. From (3.1), we have β€–β€–π‘₯𝑛+1β€–β€–βˆ’Μƒπ‘₯2≀𝛽𝑛‖‖π‘₯π‘›β€–β€–βˆ’Μƒπ‘₯2+ξ€·1βˆ’π›½π‘›ξ€Έβ€–β€–π½π΅πœ†π‘›π‘’π‘›β€–β€–βˆ’Μƒπ‘₯2=𝛽𝑛‖‖π‘₯π‘›β€–β€–βˆ’Μƒπ‘₯2+ξ€·1βˆ’π›½π‘›ξ€Έβ€–β€–π½π΅πœ†π‘›π‘’π‘›βˆ’π½π΅πœ†π‘›ξ€·ξ€·Μƒπ‘₯βˆ’1βˆ’π›Όπ‘›ξ€Έπœ†π‘›ξ€Έβ€–β€–π΄Μƒπ‘₯2≀𝛽𝑛‖‖π‘₯π‘›β€–β€–βˆ’Μƒπ‘₯2+ξ€·1βˆ’π›½π‘›ξ€Έβ€–β€–π‘’π‘›βˆ’ξ€·ξ€·Μƒπ‘₯βˆ’1βˆ’π›Όπ‘›ξ€Έπœ†π‘›ξ€Έβ€–β€–π΄Μƒπ‘₯2=𝛽𝑛‖‖π‘₯π‘›β€–β€–βˆ’Μƒπ‘₯2+ξ€·1βˆ’π›½π‘›ξ€Έβ€–β€–π›Όπ‘›ξ€·π‘’+1βˆ’π›Όπ‘›π‘₯ξ€Έξ€·π‘›βˆ’πœ†π‘›π΄π‘₯π‘›ξ€Έβˆ’ξ€·ξ€·Μƒπ‘₯βˆ’1βˆ’π›Όπ‘›ξ€Έπœ†π‘›ξ€Έβ€–β€–π΄Μƒπ‘₯2=𝛽𝑛‖‖π‘₯π‘›β€–β€–βˆ’Μƒπ‘₯2+ξ€·1βˆ’π›½π‘›ξ€Έβ€–β€–ξ€·1βˆ’π›Όπ‘›π‘₯ξ€Έξ€·ξ€·π‘›βˆ’πœ†π‘›π΄π‘₯π‘›ξ€Έβˆ’ξ€·Μƒπ‘₯βˆ’πœ†π‘›π΄Μƒπ‘₯ξ€Έξ€Έ+𝛼𝑛‖‖(π‘’βˆ’Μƒπ‘₯)2=𝛽𝑛‖‖π‘₯π‘›β€–β€–βˆ’Μƒπ‘₯2+ξ€·1βˆ’π›½π‘›ξ€ΈΓ—ξ‚€ξ€·1βˆ’π›Όπ‘›ξ€Έ2β€–β€–ξ€·π‘₯π‘›βˆ’πœ†π‘›π΄π‘₯π‘›ξ€Έβˆ’ξ€·Μƒπ‘₯βˆ’πœ†π‘›ξ€Έβ€–β€–π΄Μƒπ‘₯2+2𝛼𝑛1βˆ’π›Όπ‘›ξ€·π‘₯ξ€Έξ«π‘’βˆ’Μƒπ‘₯,π‘›βˆ’πœ†π‘›π΄π‘₯π‘›ξ€Έβˆ’ξ€·Μƒπ‘₯βˆ’πœ†π‘›π΄Μƒπ‘₯+𝛼2π‘›β€–π‘’βˆ’Μƒπ‘₯β€–2≀𝛽𝑛‖‖π‘₯π‘›β€–β€–βˆ’Μƒπ‘₯2+ξ€·1βˆ’π›½π‘›ξ€ΈΓ—ξ‚€ξ€·1βˆ’π›Όπ‘›ξ€Έβ€–β€–π‘₯π‘›β€–β€–βˆ’Μƒπ‘₯2+2𝛼𝑛1βˆ’π›Όπ‘›ξ€Έξ«π‘’βˆ’Μƒπ‘₯,π‘₯π‘›βˆ’πœ†π‘›ξ€·π΄π‘₯π‘›ξ€Έξ¬βˆ’π΄Μƒπ‘₯βˆ’Μƒπ‘₯+𝛼2π‘›β€–π‘’βˆ’Μƒπ‘₯β€–2≀1βˆ’1βˆ’π›½π‘›ξ€Έπ›Όπ‘›ξ€»β€–β€–π‘₯π‘›β€–β€–βˆ’Μƒπ‘₯2+ξ€·1βˆ’π›½π‘›ξ€Έπ›Όπ‘›ξ€½2ξ€·1βˆ’π›Όπ‘›ξ€ΈβŸ¨π‘’βˆ’Μƒπ‘₯,π‘£π‘›βˆ’Μƒπ‘₯⟩+π›Όπ‘›β€–π‘’βˆ’Μƒπ‘₯β€–2ξ€Ύ.(3.21) It is clear that βˆ‘π‘›(1βˆ’π›½π‘›)𝛼𝑛=∞ and limsupπ‘›β†’βˆž(2(1βˆ’π›Όπ‘›)βŸ¨π‘’βˆ’Μƒπ‘₯,π‘£π‘›βˆ’Μƒπ‘₯⟩+π›Όπ‘›β€–π‘’βˆ’Μƒπ‘₯β€–2)≀0. We can therefore apply Lemma 2.3 to conclude that π‘₯𝑛→̃π‘₯. This completes the proof.

4. Applications

Next, we consider the problem for finding the minimum norm solution of a mathematical model related to equilibrium problems. Let 𝐢 be a nonempty, closed, and convex subset of a Hilbert space and let πΊβˆΆπΆΓ—πΆβ†’π‘… be a bifunction satisfying the following conditions:(E1)𝐺(π‘₯,π‘₯)=0 for all π‘₯∈𝐢;(E2)𝐺 is monotone, that is, 𝐺(π‘₯,𝑦)+𝐺(𝑦,π‘₯)≀0 for all π‘₯,π‘¦βˆˆπΆ;(E3)for all π‘₯,𝑦,π‘§βˆˆπΆ, limsup𝑑↓0𝐺(𝑑𝑧+(1βˆ’π‘‘)π‘₯,𝑦)≀𝐺(π‘₯,𝑦);(E4)for all π‘₯∈𝐢, 𝐺(π‘₯,β‹…) is convex and lower semicontinuous.

Then, the mathematical model related to equilibrium problems (with respect to 𝐢) is to find Μƒπ‘₯∈𝐢 such that𝐺(Μƒπ‘₯,𝑦)β‰₯0(4.1) for all π‘¦βˆˆπΆ. The set of such solutions Μƒπ‘₯ is denoted by 𝐸𝑃(𝐺). The following lemma appears implicitly in Blum and Oettli [32].

Lemma 4.1. Let 𝐢 be a nonempty, closed, and convex subset of 𝐻 and let 𝐺 be a bifunction of 𝐢×𝐢 into 𝑅 satisfying (E1)–(E4). Let π‘Ÿ>0 and π‘₯∈𝐻. Then, there exists π‘§βˆˆπΆ such that 1𝐺(𝑧,𝑦)+π‘ŸβŸ¨π‘¦βˆ’π‘§,π‘§βˆ’π‘₯⟩β‰₯0,βˆ€π‘¦βˆˆC.(4.2)

The following lemma was given by Combettes and Hirstoaga [33].

Lemma 4.2. Assume that πΊβˆΆπΆΓ—πΆβ†’π‘… satisfies (E1)–(E4). For π‘Ÿ>0 and π‘₯∈𝐻, define a mapping π‘‡π‘ŸβˆΆπ»β†’πΆ as follows: π‘‡π‘Ÿξ‚†1(π‘₯)=π‘§βˆˆπΆβˆΆπΊ(𝑧,𝑦)+π‘Ÿξ‚‡βŸ¨π‘¦βˆ’π‘§,π‘§βˆ’π‘₯⟩β‰₯0,βˆ€π‘¦βˆˆπΆ(4.3) for all π‘₯∈𝐻. Then, the following holds: (1)π‘‡π‘Ÿ is single valued; (2)π‘‡π‘Ÿ is a firmly nonexpansive mapping, that is, for all π‘₯,π‘¦βˆˆπ», β€–β€–π‘‡π‘Ÿπ‘₯βˆ’π‘‡π‘Ÿπ‘¦β€–β€–2β‰€βŸ¨π‘‡π‘Ÿπ‘₯βˆ’π‘‡π‘Ÿπ‘¦,π‘₯βˆ’π‘¦βŸ©;(4.4)(3)𝐹(π‘‡π‘Ÿ)=𝐸𝑃(𝐺); (4)𝐸𝑃(𝐺) is closed and convex.

We call such π‘‡π‘Ÿ the resolvent of 𝐺 for π‘Ÿ>0. Using Lemmas 4.1 and 4.2, we have the following lemma. See [34] for a more general result.

Lemma 4.3. Let 𝐻 be a Hilbert space and let 𝐢 be a nonempty, closed, and convex subset of 𝐻. Let πΊβˆΆπΆΓ—πΆβ†’π‘… satisfy (E1)–(E4). Let 𝐴𝐺 be a multivalued mapping of 𝐻 into itself defined by 𝐴𝐺π‘₯={π‘§βˆˆπ»βˆΆπΊ(π‘₯,𝑦)β‰₯βŸ¨π‘¦βˆ’π‘₯,π‘§βŸ©,βˆ€π‘¦βˆˆπΆ},βˆ…,π‘₯∈𝐢,π‘₯βˆ‰πΆ.(4.5) Then, 𝐸𝑃(𝐺)=π΄πΊβˆ’1(0) and 𝐴𝐺 is a maximal monotone operator with dom(𝐴𝐺)βŠ‚πΆ. Further, for any π‘₯∈𝐻 and π‘Ÿ>0, the resolvent π‘‡π‘Ÿ of 𝐺 coincides with the resolvent of 𝐴𝐺; that is, π‘‡π‘Ÿπ‘₯=(𝐼+π‘Ÿπ΄πΊ)βˆ’1π‘₯.(4.6)

Form Lemma 4.3 and Theorems 3.1, we have the following result.

Theorem 4.4. Let 𝐢 be a nonempty, closed, and convex subset of a real Hilbert space 𝐻. Let 𝐺 be a bifunction from 𝐢×𝐢→𝑅 satisfying (E1)–(E4) and let π‘‡πœ† be the resolvent of 𝐺 for πœ†>0. Suppose 𝐸𝑃(𝐺)β‰ βˆ…. For π‘’βˆˆπΆ and given π‘₯0∈𝐢, let {π‘₯𝑛}βŠ‚πΆ be a sequence generated by π‘₯𝑛+1=𝛽𝑛π‘₯𝑛+ξ€·1βˆ’π›½π‘›ξ€Έπ‘‡πœ†π‘›ξ€·π›Όπ‘›ξ€·π‘’+1βˆ’π›Όπ‘›ξ€Έπ‘₯𝑛(4.7) for all 𝑛β‰₯0, where {πœ†π‘›}βŠ‚(0,∞), {𝛼𝑛}βŠ‚(0,1), and {𝛽𝑛}βŠ‚(0,1) satisfy (i)limπ‘›β†’βˆžπ›Όπ‘›=0 and βˆ‘π‘›π›Όπ‘›=∞;(ii)0<liminfπ‘›β†’βˆžπ›½π‘›β‰€limsupπ‘›β†’βˆžπ›½π‘›<1;(iii)π‘Žβ‰€πœ†π‘›β‰€π‘ where [π‘Ž,𝑏]βŠ‚(0,∞) and limπ‘›β†’βˆž(πœ†π‘›+1βˆ’πœ†π‘›)=0.Then {π‘₯𝑛} converges strongly to a point Μƒπ‘₯=𝑃𝐸𝑃(𝐺)(𝑒).

Acknowledgment

The author was supported in part by NSC 100-2221-E-230-012.

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