Journal of Applied Mathematics

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

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Volume 2012 |Article ID 435676 | 15 pages | https://doi.org/10.1155/2012/435676

Cyclic Iterative Method for Strictly Pseudononspreading in Hilbert Space

Academic Editor: Hong-Kun Xu
Received17 Apr 2012
Revised01 Jun 2012
Accepted01 Jun 2012
Published18 Jul 2012

Abstract

Let {๐‘‡๐‘–}๐‘๐‘–=1 be ๐‘ strictly pseudononspreading mappings defined on closed convex subset ๐ถ of a real Hilbert space ๐ป. Consider the problem of finding a common fixed point of these mappings and introduce cyclic algorithms based on general viscosity iteration method for solving this problem. We will prove the strong convergence of these cyclic algorithm. Moreover, the common fixed point is the solution of the variational inequality โŸจ(๐›พ๐‘“โˆ’๐œ‡๐ต)๐‘ฅโˆ—,๐‘ฃโˆ’๐‘ฅโˆ—โ‹‚โŸฉโ‰ค0,โˆ€๐‘ฃโˆˆ๐‘๐‘–=1๐น๐‘–๐‘ฅ(๐‘‡๐‘–).

1. Introduction

Throughout this paper, we always assume that ๐ถ is a nonempty, closed, and convex subset of a real Hilbert space ๐ป. Let ๐ตโˆถ๐ถโ†’๐ป be a nonlinear mapping. Recall the following definitions.

Definition 1.1. ๐ต is said to be(i)monotone if โŸจ๐ต๐‘ฅโˆ’๐ต๐‘ฆ,๐‘ฅโˆ’๐‘ฆโŸฉโ‰ฅ0,โˆ€๐‘ฅ,๐‘ฆโˆˆ๐ถ,(1.1)(ii)strongly monotone if there exists a constant ๐›ผ>0 such that โŸจ๐ต๐‘ฅโˆ’๐ต๐‘ฆ,๐‘ฅโˆ’๐‘ฆโŸฉโ‰ฅ๐›ผโ€–๐‘ฅโˆ’๐‘ฆโ€–2,โˆ€๐‘ฅ,๐‘ฆโˆˆ๐ถ,(1.2) for such a case, ๐ต is said to be ๐›ผ-strongly-monotone, (iii)inverse-strongly monotone if there exists a constant ๐›ผ>0 such that โŸจ๐ต๐‘ฅโˆ’๐ต๐‘ฆ,๐‘ฅโˆ’๐‘ฆโŸฉโ‰ฅ๐›ผโ€–๐ต๐‘ฅโˆ’๐ต๐‘ฆโ€–2,โˆ€๐‘ฅ,๐‘ฆโˆˆ๐ถ,(1.3) for such a case, ๐ต is said to be ๐›ผ-inverse-strongly monotone, (iv)๐‘˜-Lipschitz continuous if there exists a constant ๐‘˜โ‰ฅ0 such that โ€–๐ต๐‘ฅโˆ’๐ต๐‘ฆโ€–โ‰ค๐‘˜โ€–๐‘ฅโˆ’๐‘ฆโ€–โˆ€๐‘ฅ,๐‘ฆโˆˆ๐ถ.(1.4)

Remark 1.2. Let ๐น=๐œ‡๐ตโˆ’๐›พ๐‘“, where ๐ต is a ๐‘˜-Lipschitz and ๐œ‚-strongly monotone operator on ๐ป with ๐‘˜>0 and ๐‘“ is a Lipschitz mapping on ๐ป with coefficient ๐ฟ>0, 0<๐›พโ‰ค๐œ‡๐œ‚/๐ฟ. It is a simple matter to see that the operator ๐น is (๐œ‡๐œ‚โˆ’๐›พ๐ฟ)-strongly monotone overโ€‰โ€‰๐ป,โ€‰โ€‰that is: โŸจ๐น๐‘ฅโˆ’๐น๐‘ฆ,๐‘ฅโˆ’๐‘ฆโŸฉโ‰ฅ(๐œ‡๐œ‚โˆ’๐›พ๐ฟ)โ€–๐‘ฅโˆ’๐‘ฆโ€–2,โˆ€(๐‘ฅ,๐‘ฆ)โˆˆ๐ปร—๐ป.(1.5)
Following the terminology of Browder-Petryshyn [1], we say that a mapping ๐‘‡โˆถ๐ท(๐‘‡)โŠ†๐ปโ†’๐ป is(1)๐‘˜-strict pseudocontraction if there exists ๐‘˜โˆˆ[0,1) such that โ€–๐‘‡๐‘ฅโˆ’๐‘‡๐‘ฆโ€–2โ‰คโ€–๐‘ฅโˆ’๐‘ฆโ€–2โ€–+๐‘˜โ€–๐‘ฅโˆ’๐‘‡๐‘ฅโˆ’(๐‘ฆโˆ’๐‘‡๐‘ฆ)2,โˆ€๐‘ฅ,๐‘ฆโˆˆ๐ท(๐‘‡),(1.6)(2)๐‘˜-strictly pseudononspreading if there exists ๐‘˜โˆˆ[0,1) such that โ€–๐‘‡๐‘ฅโˆ’๐‘‡๐‘ฆโ€–2โ‰คโ€–๐‘ฅโˆ’๐‘ฆโ€–2โ€–+๐‘˜โ€–๐‘ฅโˆ’๐‘‡๐‘ฅโˆ’(๐‘ฆโˆ’๐‘‡๐‘ฆ)2+2โŸจ๐‘ฅโˆ’๐‘‡๐‘ฅ,๐‘ฆโˆ’๐‘‡๐‘ฆโŸฉ,(1.7) for all ๐‘ฅ,๐‘ฆโˆˆ๐ท(๐‘‡),(3)nonspreading in [2] if โ€–๐‘‡๐‘ฅโˆ’๐‘‡๐‘ฆโ€–2โ‰คโ€–๐‘‡๐‘ฅโˆ’๐‘ฆโ€–2+โ€–๐‘‡๐‘ฆโˆ’๐‘ฅโ€–2,โˆ€๐‘ฅ,๐‘ฆโˆˆ๐ถ.(1.8) It is shown in [3] that (1.8) is equivalent to โ€–๐‘‡๐‘ฅโˆ’๐‘‡๐‘ฆโ€–2โ‰คโ€–๐‘ฅโˆ’๐‘ฆโ€–2+2โŸจ๐‘ฅโˆ’๐‘‡๐‘ฅ,๐‘ฆโˆ’๐‘‡๐‘ฆโŸฉ,โˆ€๐‘ฅ,๐‘ฆโˆˆ๐ถ.(1.9)

Clearly every nonspreading mapping is 0-strictly pseudononspreading. Iterative methods for strictly pseudononspreading mapping have been extensively investigated; see [2, 4โ€“6].

Let ๐ถ be a closed convex subset of ๐ป, and let {๐‘‡๐‘–}๐‘๐‘–=1 be ๐‘›โ€‰โ€‰๐‘˜๐‘–-strictly pseudocontractive mappings on ๐ถ such that โ‹‚๐‘๐‘–=1๐น๐‘–๐‘ฅ(๐‘‡๐‘–)โ‰ โˆ…. Let ๐‘ฅ1โˆˆ๐ถ and {๐›ผ๐‘›}โˆž๐‘›=1 be a sequence in (0,1). In [7], Acedo and Xu introduced an explicit iteration scheme called the followting cyclic algorithm for iterative approximation of common fixed points of {๐‘‡๐‘–}๐‘๐‘–=1 in Hilbert spaces. They define the sequence {๐‘ฅ๐‘›} cyclically by ๐‘ฅ1=๐›ผ0๐‘ฅ0+๎€ท1โˆ’๐›ผ0๎€ธ๐‘‡0๐‘ฅ0;๐‘ฅ2=๐›ผ1๐‘ฅ1+๎€ท1โˆ’๐›ผ1๎€ธ๐‘‡1๐‘ฅ1;โ‹ฎ๐‘ฅ๐‘=๐›ผ๐‘โˆ’1๐‘ฅ๐‘โˆ’1+๎€ท1โˆ’๐›ผ๐‘โˆ’1๎€ธ๐‘‡๐‘โˆ’1๐‘ฅ๐‘โˆ’1;๐‘ฅ๐‘+1=๐›ผ๐‘๐‘ฅ๐‘+๎€ท1โˆ’๐›ผ๐‘๎€ธ๐‘‡0๐‘ฅ๐‘;(1.10)โ‹ฎ.(1.11) In a more compact form, they rewrite ๐‘ฅ๐‘›+1 as ๐‘ฅ๐‘+1=๐›ผ๐‘๐‘ฅ๐‘+๎€ท1โˆ’๐›ผ๐‘๎€ธ๐‘‡๐‘๐‘ฅ๐‘›,(1.12) where ๐‘‡๐‘=๐‘‡๐‘–, with ๐‘–=๐‘› (mod ๐‘), 0โ‰ค๐‘–โ‰ค๐‘โˆ’1. Using the cyclic algorithm (1.12), Acedo and Xu [7] show that this cyclic algorithm (1.12) is weakly convergent if the sequence {๐›ผ๐‘›} of parameters is appropriately chosen.

Motivated and inspired by Acedo and Xu [7], we consider the following cyclic algorithm for finding a common element of the set of solutions of ๐‘˜๐‘–-strictly pseudononspreading mappings {๐‘‡๐‘–}๐‘๐‘–=1. The sequence {๐‘ฅ๐‘›}โˆž๐‘–=1 generated from an arbitrary ๐‘ฅ1โˆˆ๐ป as follows: ๐‘ฅ1=๐›ผ0๎€ท๐‘ฅ๐›พ๐‘“0๎€ธ+๎€ท๐ผโˆ’๐œ‡๐›ผ0๐ต๎€ธ๐‘‡๐œ”0๐‘ฅ0;๐‘ฅ2=๐›ผ1๎€ท๐‘ฅ๐›พ๐‘“1๎€ธ+๎€ท๐ผโˆ’๐œ‡๐›ผ1๐ต๎€ธ๐‘‡๐œ”1๐‘ฅ1;โ‹ฎ๐‘ฅ๐‘=๐›ผ๐‘โˆ’1๎€ท๐‘ฅ๐›พ๐‘“๐‘โˆ’1๎€ธ+๎€ท๐ผโˆ’๐œ‡๐›ผ๐‘โˆ’1๐ต๎€ธ๐‘‡๐œ”๐‘โˆ’1๐‘ฅ๐‘โˆ’1;๐‘ฅ๐‘+1=๐›ผ๐‘๎€ท๐‘ฅ๐›พ๐‘“๐‘๎€ธ+๎€ท๐ผโˆ’๐œ‡๐›ผ๐‘๐ต๎€ธ๐‘‡๐œ”0๐‘ฅ๐‘;(1.13)โ‹ฎ.(1.14) Indeed, the algorithm above can be rewritten as ๐‘ฅ๐‘›+1=๐›ผ๐‘›๎€ท๐‘ฅ๐›พ๐‘“๐‘›๎€ธ+๎€ท๐ผโˆ’๐œ‡๐›ผ๐‘›๐ต๎€ธ๐‘‡๐œ”[๐‘›]๐‘ฅ๐‘›,(1.15) where ๐‘‡๐œ”[๐‘›]=(๐ผโˆ’๐œ”[๐‘›])๐ผ+๐œ”[๐‘›]๐‘‡[๐‘›], ๐‘‡[๐‘›]=๐‘‡๐‘›mod๐‘; namely, ๐‘‡[๐‘] is one of ๐‘‡1,๐‘‡2,โ€ฆ,๐‘‡๐‘ circularly.

2. Preliminaries

Throughout this paper, we write ๐‘ฅ๐‘›โ‡€๐‘ฅ to indicate that the sequence {๐‘ฅ๐‘›} converges weakly to ๐‘ฅ. ๐‘ฅ๐‘›โ†’๐‘ฅ implies that {๐‘ฅ๐‘›} converges strongly to ๐‘ฅ. The following definitions and lemmas are useful for main results.

Definition 2.1. A mapping ๐‘‡ is said to be demiclosed, if for any sequence {๐‘ฅ๐‘›}which weakly converges to ๐‘ฆ, and if the sequence {๐‘‡๐‘ฅ๐‘›} strongly converges to ๐‘ง, then ๐‘‡(๐‘ฆ)=๐‘ง.

Definition 2.2. ๐‘‡โˆถ๐ปโ†’๐ป is called demicontractive on ๐ป, if there exists a constant ๐›ผ<1 such that โ€–๐‘‡๐‘ฅโˆ’๐‘žโ€–2โ‰คโ€–๐‘ฅโˆ’๐‘žโ€–2+๐›ผโ€–๐‘ฅโˆ’๐‘‡๐‘ฅโ€–2,โˆ€(๐‘ฅ,๐‘ž)โˆˆ๐ปร—๐น๐‘–๐‘ฅ(๐‘‡).(2.1)

Remark 2.3. Every ๐‘˜-strictly pseudononspreading mapping with a nonempty fixed point set ๐น๐‘–๐‘ฅ(๐‘‡) is demicontractive (see [8, 9]).

Remark 2.4 (see [10]). Let ๐‘‡ be a ๐›ผ-demicontractive mapping on ๐ป with ๐น๐‘–๐‘ฅ(๐‘‡)โ‰ โˆ… and ๐‘‡๐œ”=(1โˆ’๐œ”)๐ผ+๐œ”๐‘‡ for ๐œ”โˆˆ(0,โˆž): (A1)๐‘‡๐›ผ-demicontractive is equivalent to 1โŸจ๐‘ฅโˆ’๐‘‡๐‘ฅ,๐‘ฅโˆ’๐‘žโŸฉโ‰ฅ2(1โˆ’๐›ผ)โ€–๐‘ฅโˆ’๐‘‡๐‘ฅโ€–2,โˆ€(๐‘ฅ,๐‘ž)โˆˆ๐ปร—๐น๐‘–๐‘ฅ(๐‘‡),(2.2)(A2)๐น๐‘–๐‘ฅ(๐‘‡)=๐น๐‘–๐‘ฅ(๐‘‡๐œ”) if ๐œ”โ‰ 0.

Remark 2.5. According to ๐ผโˆ’๐‘‡๐œ”=๐œ”(๐ผโˆ’๐‘‡) with ๐‘‡ being a ๐‘˜-strictly pseudononspreading mapping, we obtain โŸจ๐‘ฅโˆ’๐‘‡๐œ”๐‘ฅ,๐‘ฅโˆ’๐‘žโŸฉโ‰ฅ๐œ”(1โˆ’๐‘˜)2โ€–๐‘ฅโˆ’๐‘‡๐‘ฅโ€–2,โˆ€(๐‘ฅ,๐‘ž)โˆˆ๐ปร—๐น๐‘–๐‘ฅ(๐‘‡).(2.3)

Proposition 2.6 (see [2]). Let ๐ถ be a nonempty closed convex subset of a real Hilbert space ๐ป, and let ๐‘‡โˆถ๐ถโ†’๐ถ be a ๐‘˜-strictly pseudononspreading mapping. If ๐น๐‘–๐‘ฅ(๐‘‡)โ‰ โˆ…, then it is closed and convex.

Proposition 2.7 (see [2]). Let ๐ถ be a nonempty closed convex subset of a real Hilbert space ๐ป, and let ๐‘‡โˆถ๐ถโ†’๐ถ be a ๐‘˜-strictly pseudononspreading mapping. Then (๐ผโˆ’๐‘‡) is demiclosed at 0.

Lemma 2.8 (see [11]). Let {๐’ฏ๐‘›} be a sequence of real numbers that does not decrease at infinity, in the sense that there exists a subsequence {๐’ฏ๐‘›๐‘—}๐‘—โ‰ฅ0 of {๐’ฏ๐‘›} which satisfies ๐’ฏ๐‘›๐‘—<๐’ฏ๐‘›๐‘—+1 for all ๐‘—โ‰ฅ0. Also consider the sequence of integers {๐›ฟ(๐‘›)}๐‘›โ‰ฅ๐‘›0 defined by ๐›ฟ๎€ฝ(๐‘›)=max๐‘˜โ‰ค๐‘›โˆฃ๐’ฏ๐‘˜<๐’ฏ๐‘˜+1๎€พ.(2.4) Then {๐›ฟ(๐‘›)}๐‘›โ‰ฅ๐‘›0 is a nondecreasing sequence verifying lim๐‘›โ†’โˆž๐›ฟ(๐‘›)=โˆž,โˆ€๐‘›โ‰ฅ๐‘›0; it holds that ๐’ฏ๐›ฟ(๐‘›)<๐’ฏ๐›ฟ(๐‘›)+1 and we have ๐’ฏ๐‘›<๐’ฏ๐›ฟ(๐‘›)+1.(2.5)

Lemma 2.9. Let ๐ป be a real Hilbert space. The following expressions hold: (i)โ€–๐‘ก๐‘ฅ+(1โˆ’๐‘ก)๐‘ฆโ€–2=๐‘กโ€–๐‘ฅโ€–2+(1โˆ’๐‘ก)โ€–๐‘ฆโ€–2โˆ’๐‘ก(1โˆ’๐‘ก)โ€–๐‘ฅโˆ’๐‘ฆโ€–2,โˆ€๐‘ฅ,๐‘ฆโˆˆ๐ป,โˆ€๐‘กโˆˆ[0,1], (ii)โ€–๐‘ฅ+๐‘ฆโ€–2โ‰คโ€–๐‘ฅโ€–2+2โŸจ๐‘ฆ,๐‘ฅ+๐‘ฆโŸฉ,โˆ€๐‘ฅ,๐‘ฆโˆˆ๐ป.

Lemma 2.10 (see [6]). Let ๐ถ be a closed convex subset of a Hilbert space ๐ป, and let ๐‘‡โˆถ๐ปโ†’๐ป be a ๐‘˜-strictly pseudononspreading mapping with a nonempty fixed point set. Let ๐‘˜โ‰ค๐œ”<1 be fixed and define ๐‘‡๐œ”๐ถโ†’๐ถ by ๐‘‡๐œ”(๐‘ฅ)=(1โˆ’๐œ”)(๐‘ฅ)+๐œ”๐‘‡(๐‘ฅ),โˆ€๐‘ฅโˆˆ๐ถ.(2.6) Then ๐น๐‘–๐‘ฅ(๐‘‡๐œ”)=๐น๐‘–๐‘ฅ(๐‘‡).

Lemma 2.11. Assume ๐ถ is a closed convex subset of a Hilbert space ๐ป. (a)Given an integer ๐‘โ‰ฅ1, assume, ๐‘‡๐‘–โˆถ๐ปโ†’๐ป is a ๐‘˜๐‘–-strictly pseudononspreading mapping for some ๐‘˜๐‘–โˆˆ[0,1), (๐‘–=1,2,โ€ฆ,๐‘). Let {๐œ†๐‘–}๐‘๐‘–=1 be a positive sequence such that โˆ‘๐‘๐‘–=1๐œ†๐‘–=1. Suppose that {๐‘‡๐‘–}๐‘๐‘–=1 has a common fixed point and โ‹‚๐‘๐‘–=1๐น๐‘–๐‘ฅ(๐‘‡๐‘–)โ‰ โˆ…. Then, ๐น๐‘–๐‘ฅ๎ƒฉ๐‘๎“๐‘–=1๐œ†๐‘–๐‘‡๐‘–๎ƒช=๐‘๎™๐‘–=1๐น๐‘–๐‘ฅ๎€ท๐‘‡๐‘–๎€ธ.(2.7)(b)Assuming ๐‘‡๐‘–โˆถ๐ปโ†’๐ป is a ๐‘˜๐‘–-strictly pseudononspreading mapping for some ๐‘˜๐‘–โˆˆ[0,1), (๐‘–=1,2,โ€ฆ,๐‘), let ๐‘‡๐œ”๐‘–=(1โˆ’๐œ”๐‘–)๐ผ+๐œ”๐‘–๐‘‡๐‘–, 1โ‰ค๐‘–โ‰ค๐‘. If โ‹‚๐‘๐‘–=1๐น๐‘–๐‘ฅ(๐‘‡๐‘–)โ‰ โˆ…, then ๐น๐‘–๐‘ฅ๎€ท๐‘‡๐œ”1๐‘‡๐œ”2โ‹ฏ๐‘‡๐œ”๐‘๎€ธ=๐‘๎™๐‘–=1๐น๐‘–๐‘ฅ๎€ท๐‘‡๐œ”๐‘–๎€ธ.(2.8)

Proof. To prove (a), we can assume ๐‘=2. It suffices to prove that ๐น๐‘–๐‘ฅ(๐น)โŠ‚๐น๐‘–๐‘ฅ(๐‘‡1)โˆฉ๐น๐‘–๐‘ฅ(๐‘‡2), where ๐น=(1โˆ’๐œ†)๐‘‡1+๐œ†๐‘‡2 with 0<๐œ†<1. Let ๐‘ฅโˆˆ๐น๐‘–๐‘ฅ(๐น) and write ๐‘‰1=๐ผโˆ’๐‘‡1 and ๐‘‰2=๐ผโˆ’๐‘‡2.
From Lemma 2.9 and taking ๐‘งโˆˆ๐น๐‘–๐‘ฅ(๐‘‡1)โˆฉ๐น๐‘–๐‘ฅ(๐‘‡2) to deduce that โ€–๐‘งโˆ’๐‘ฅโ€–2=โ€–โ€–(1โˆ’๐œ†)(๐‘งโˆ’๐‘‡1๐‘ฅ)+๐œ†(๐‘งโˆ’๐‘‡2โ€–โ€–๐‘ฅ)2=โ€–โ€–(1โˆ’๐œ†)๐‘งโˆ’๐‘‡1๐‘ฅโ€–โ€–2โ€–โ€–+๐œ†๐‘งโˆ’๐‘‡2๐‘ฅโ€–โ€–2โ€–โ€–๐‘‡โˆ’๐œ†(1โˆ’๐œ†)1๐‘ฅโˆ’๐‘‡2๐‘ฅโ€–โ€–2๎‚€โ‰ค(1โˆ’๐œ†)โ€–๐‘งโˆ’๐‘ฅโ€–2โ€–โ€–+๐‘˜๐‘ฅโˆ’๐‘‡1๐‘ฅโ€–โ€–2๎‚๎‚€+๐œ†โ€–๐‘งโˆ’๐‘ฅโ€–2โ€–โ€–+๐‘˜๐‘ฅโˆ’๐‘‡2๐‘ฅโ€–โ€–2๎‚โ€–โ€–๐‘‡โˆ’๐œ†(1โˆ’๐œ†)1๐‘ฅโˆ’๐‘‡2๐‘ฅโ€–โ€–2=โ€–๐‘งโˆ’๐‘ฅโ€–2๎‚ƒโ€–โ€–๐‘‰+๐‘˜(1โˆ’๐œ†)1๐‘ฅโ€–โ€–2โ€–โ€–๐‘‰+๐œ†2๐‘ฅโ€–โ€–2๎‚„โ€–โ€–๐‘‰โˆ’๐œ†(1โˆ’๐œ†)1๐‘ฅโˆ’๐‘‰2๐‘ฅโ€–โ€–2,(2.9) it follows that โ€–โ€–๐‘‰๐œ†(1โˆ’๐œ†)1๐‘ฅโˆ’๐‘‰2๐‘ฅโ€–โ€–2๎‚ƒโ€–โ€–๐‘‰โ‰ค๐‘˜(1โˆ’๐œ†)1๐‘ฅโ€–โ€–2โ€–โ€–๐‘‰+๐œ†2๐‘ฅโ€–โ€–2๎‚„.(2.10) Since (1โˆ’๐œ†)๐‘‰1๐‘ฅ+๐œ†๐‘‰2๐‘ฅ=0, we obtain โ€–โ€–๐‘‰(1โˆ’๐œ†)1๐‘ฅโ€–โ€–2โ€–โ€–๐‘‰+๐œ†2๐‘ฅโ€–โ€–2โ€–โ€–๐‘‰=๐œ†(1โˆ’๐œ†)1๐‘ฅโˆ’๐‘‰2๐‘ฅโ€–โ€–2.(2.11)
This together with (2.10) implies that โ€–โ€–๐‘‰(1โˆ’๐‘˜)๐œ†(1โˆ’๐œ†)1๐‘ฅโˆ’๐‘‰2๐‘ฅโ€–โ€–2โ‰ค0.(2.12) Since 0<๐œ†<1 and ๐‘˜<1, we get โ€–๐‘‰1๐‘ฅโˆ’๐‘‰2๐‘ฅโ€–=0 which implies ๐‘‡1๐‘ฅ=๐‘‡2๐‘ฅ which in turn implies that ๐‘‡1๐‘ฅ=๐‘‡2๐‘ฅ=๐‘ฅ since (1โˆ’๐œ†)๐‘‡1๐‘ฅ+๐œ†๐‘‡2๐‘ฅ=๐‘ฅ. Thus, ๐‘ฅโˆˆ๐น๐‘–๐‘ฅ(๐‘‡1)โˆฉ๐น๐‘–๐‘ฅ(๐‘‡2).
By induction, we also claim that ๐น๐‘–๐‘ฅ(โˆ‘๐‘๐‘–=1๐œ†๐‘–๐‘‡๐‘–โ‹‚)=๐‘๐‘–=1๐น๐‘–๐‘ฅ(๐‘‡๐‘–) with {๐œ†๐‘–}๐‘๐‘–=1 is a positive sequence such that โˆ‘๐‘๐‘–=1๐œ†๐‘–=1, (๐‘–=1,2,โ€ฆ,๐‘).
To prove (b), we can assume ๐‘=2. Set ๐‘‡๐œ”1=(1โˆ’๐œ”1)๐ผ+๐œ”1๐‘‡1 and ๐‘‡๐œ”2=(1โˆ’๐œ”2)๐ผ+๐œ”1๐‘‡2, 0<๐‘˜๐‘–<๐œ”๐‘–<1/2, ๐‘–=1,2. Obviously ๐น๐‘–๐‘ฅ๎€ท๐‘‡๐œ”1๎€ธโˆฉ๐น๐‘–๐‘ฅ๎€ท๐‘‡๐œ”2๎€ธโŠ‚๐น๐‘–๐‘ฅ๎€ท๐‘‡๐œ”1๐‘‡๐œ”2๎€ธ.(2.13) Now we prove ๐น๐‘–๐‘ฅ๎€ท๐‘‡๐œ”1๎€ธโˆฉ๐น๐‘–๐‘ฅ๎€ท๐‘‡๐œ”2๎€ธโŠƒ๐น๐‘–๐‘ฅ๎€ท๐‘‡๐œ”1๐‘‡๐œ”2๎€ธ,(2.14) for all ๐‘žโˆˆ๐น๐‘–๐‘ฅ(๐‘‡๐œ”1๐‘‡๐œ”2) and ๐‘‡๐œ”1๐‘‡๐œ”2๐‘ž=๐‘ž. If ๐‘‡๐œ”2๐‘ž=๐‘ž, then ๐‘‡๐œ”1๐‘ž=๐‘ž; the conclusion holds. From Lemma 2.10, we can know that ๐น๐‘–๐‘ฅ(๐‘‡๐œ”1)โˆฉ๐น๐‘–๐‘ฅ(๐‘‡๐œ”2)=๐น๐‘–๐‘ฅ(๐‘‡1)โˆฉ๐น๐‘–๐‘ฅ(๐‘‡2)โ‰ โˆ…. Taking ๐‘โˆˆ๐น๐‘–๐‘ฅ(๐‘‡๐œ”1)โˆฉ๐น๐‘–๐‘ฅ(๐‘‡๐œ”2), then โ€–๐‘โˆ’๐‘žโ€–2=โ€–โ€–๐‘โˆ’๐‘‡๐œ”1๐‘‡๐œ”2๐‘žโ€–โ€–2=โ€–โ€–๐‘โˆ’๎€บ๎€ท1โˆ’๐œ”1๐‘‡๎€ธ๎€ท๐œ”2๐‘ž๎€ธ+๐œ”1๐‘‡1๐‘‡๐œ”2๐‘ž๎€ปโ€–โ€–2=โ€–โ€–๎€ท1โˆ’๐œ”1๎€ธ๎€ท๐‘โˆ’๐‘‡๐œ”2๐‘ž๎€ธ+๐œ”1๎€ท๐‘โˆ’๐‘‡1๐‘‡๐œ”2๐‘ž๎€ธโ€–โ€–2=๎€ท1โˆ’๐œ”1๎€ธโ€–โ€–๐‘โˆ’๐‘‡๐œ”2๐‘žโ€–โ€–2+๐œ”1โ€–โ€–๐‘โˆ’๐‘‡1๐‘‡๐œ”2๐‘žโ€–โ€–2โˆ’๐œ”1๎€ท1โˆ’๐œ”1๎€ธโ€–โ€–๐‘‡๐œ”2๐‘žโˆ’๐‘‡1๐‘‡๐œ”2๐‘žโ€–โ€–2โ‰ค๎€ท1โˆ’๐œ”1๎€ธโ€–โ€–๐‘โˆ’๐‘‡๐œ”2๐‘žโ€–โ€–2โˆ’๐œ”1๎€ท1โˆ’๐œ”1๎€ธโ€–โ€–๐‘‡๐œ”2๐‘žโˆ’๐‘‡1๐‘‡๐œ”2๐‘žโ€–โ€–2+๐œ”1๎‚ƒโ€–โ€–๐‘โˆ’๐‘‡๐œ”2๐‘žโ€–โ€–2+๐‘˜1โ€–โ€–๐‘‡๐œ”2๐‘žโˆ’๐‘‡1๐‘‡๐œ”2๐‘žโ€–โ€–2๎ซ+2๐‘โˆ’๐‘‡1๐‘‡๐œ”2๐‘,๐‘‡๐œ”2๐‘žโˆ’๐‘‡1๐‘‡๐œ”2๐‘ž๎ฌ๎‚„=๎€ท1โˆ’๐œ”1๎€ธโ€–โ€–๐‘โˆ’๐‘‡๐œ”2๐‘žโ€–โ€–2โˆ’๐œ”1๎€ท1โˆ’๐œ”1๎€ธโ€–โ€–๐‘‡๐œ”2๐‘žโˆ’๐‘‡1๐‘‡๐œ”2๐‘žโ€–โ€–2+๐œ”1๎‚ƒโ€–โ€–๐‘โˆ’๐‘‡๐œ”2๐‘žโ€–โ€–2+๐‘˜1โ€–โ€–๐‘‡๐œ”2๐‘žโˆ’๐‘‡1๐‘‡๐œ”2๐‘žโ€–โ€–2๎‚„โ‰คโ€–โ€–๐‘โˆ’๐‘‡๐œ”2๐‘žโ€–โ€–2โˆ’๐œ”1๎€ท1โˆ’๐œ”1โˆ’๐‘˜1๎€ธโ€–โ€–๐‘‡๐œ”2๐‘žโˆ’๐‘‡1๐‘‡๐œ”2๐‘žโ€–โ€–2โ‰คโ€–๐‘โˆ’๐‘žโ€–2โˆ’๐œ”1๎€ท1โˆ’๐œ”1โˆ’๐‘˜1๎€ธโ€–โ€–๐‘‡๐œ”2๐‘žโˆ’๐‘‡1๐‘‡๐œ”2๐‘žโ€–โ€–2.(2.15) Since 0<๐‘˜1<๐œ”1<1/2, we obtain โ€–โ€–๐‘‡๐œ”2๐‘žโˆ’๐‘‡1๐‘‡๐œ”2๐‘žโ€–โ€–2โ‰ค0.(2.16) Namely, ๐‘‡๐œ”2๐‘ž=๐‘‡1๐‘‡๐œ”2๐‘ž, that is: ๐‘‡๐œ”2๐‘žโˆˆ๐น๐‘–๐‘ฅ๎€ท๐‘‡1๎€ธ=๐น๐‘–๐‘ฅ๎€ท๐‘‡๐œ”1๎€ธ,๐‘‡๐œ”2๐‘ž=๐‘‡๐œ”1๐‘‡๐œ”2๐‘ž.(2.17)
By induction, we also claim that the Lemma 2.11(b) holds.

Lemma 2.12. Let ๐พ be a closed convex subset of a real Hilbert space ๐ป, given ๐‘ฅโˆˆ๐ป and ๐‘ฆโˆˆ๐พ. Then ๐‘ฆ=๐‘ƒ๐พ๐‘ฅ if and only if there holds the inequality โŸจ๐‘ฅโˆ’๐‘ฆ,๐‘ฆโˆ’๐‘งโŸฉโ‰ฅ0,โˆ€๐‘งโˆˆ๐พ.(2.18)

3. Cyclic Algorithm

In this section, we are concerned with the problem of finding a point ๐‘ such that ๐‘โˆˆ๐‘๎™๐‘–=1๐น๐‘–๐‘ฅ๎€ท๐‘‡๐œ”๐‘–๎€ธ=๐‘๎™๐‘–=1๐น๐‘–๐‘ฅ๎€ท๐‘‡๐‘–๎€ธ,๐‘โ‰ฅ1,(3.1) where ๐‘‡๐œ”๐‘–=(1โˆ’๐œ”๐‘–)๐ผ+๐œ”๐‘–๐‘‡๐‘–, {๐œ”๐‘–}๐‘๐‘–=1โˆˆ(0,1/2] and {๐‘‡๐‘–}๐‘๐‘–=1 are ๐‘˜๐‘–-strictly pseudononspreading mappings with ๐‘˜๐‘–โˆˆ[0,๐œ”๐‘–), (๐‘–=1,2,โ€ฆ,๐‘), defined on a closed convex subset ๐ถ in Hilbert space ๐ป. Here ๐น๐‘–๐‘ฅ(๐‘‡๐œ”๐‘–)={๐‘žโˆˆ๐ถโˆถ๐‘‡๐œ”๐‘–๐‘ž=๐‘ž} is the set of fixed points of ๐‘‡๐‘–, 1โ‰ค๐‘–โ‰ค๐‘.

Let ๐ป be a real Hilbert space, and let ๐ตโˆถ๐ปโ†’๐ป be ๐œ‚-strongly monotone and ๐œŒ-Lipschitzian on ๐ป with ๐œŒ>0, ๐œ‚>0. Let 0<๐œ‡<2๐œ‚/๐œŒ2, 0<๐›พ<๐œ‡(๐œ‚โˆ’(๐œ‡๐œŒ2/2))/๐ฟ=๐œ/๐ฟ. Let ๐‘ be a positive integer, and let ๐‘‡๐‘–โˆถ๐ปโ†’๐ป be a ๐‘˜๐‘–-strictly pseudononspreading mapping for some ๐‘˜๐‘–โˆˆ[0,1), (๐‘–=1,2,โ€ฆ,๐‘), such that โ‹‚๐‘๐‘–=1๐น๐‘–๐‘ฅ(๐‘‡๐‘–)โ‰ โˆ…. We consider the problem of finding โ‹‚๐‘โˆˆ๐‘๐‘–=1๐น๐‘–๐‘ฅ(๐‘‡๐‘–) such that โŸจ(๐›พ๐‘“โˆ’๐œ‡๐ต)๐‘,๐‘ฃโˆ’๐‘โŸฉโ‰ค0,โˆ€๐‘ฃโˆˆ๐‘๎™๐‘–=1๐น๐‘–๐‘ฅ๎€ท๐‘‡๐‘–๎€ธ.(3.2)

Since โ‹‚๐‘๐‘–=1๐น๐‘–๐‘ฅ(๐‘‡๐‘–) is a nonempty closed convex subset of ๐ป, VI (3.2) has a unique solution. The variational inequality has been extensively studied in literature; see, for example, [12โ€“16].

Remark 3.1. Let ๐ป be a real Hilbert space. Let ๐ต be a ๐œŒ-Lipschitzian and ๐œ‚-strongly monotone operator on ๐ป with ๐œŒ>0,๐œ‚>0. Leting 0<๐œ‡<2๐œ‚/๐œŒ2 and leting ๐‘†=(๐ผโˆ’๐‘ก๐œ‡๐ต) and ๐œ‡(๐œ‚โˆ’(๐œ‡๐œŒ2/2))=๐œ, then for โˆ€๐‘ฅ,๐‘ฆโˆˆ๐ป and ๐‘กโˆˆ(0,min{1,1/๐œ}), ๐‘† is a contraction.

Proof. Consider โ€–๐‘†๐‘ฅโˆ’๐‘†๐‘ฆโ€–2=โ€–(๐ผโˆ’๐‘ก๐œ‡๐ต)๐‘ฅโˆ’(๐ผโˆ’๐‘ก๐œ‡๐ต)๐‘ฆโ€–2=โŸจ(๐ผโˆ’๐‘ก๐œ‡๐ต)๐‘ฅโˆ’(๐ผโˆ’๐‘ก๐œ‡๐ต)๐‘ฆ,(๐ผโˆ’๐‘ก๐œ‡๐ต)๐‘ฅโˆ’(๐ผโˆ’๐‘ก๐œ‡๐ต)๐‘ฆโŸฉ=โ€–๐‘ฅโˆ’๐‘ฆโ€–2+๐‘ก2๐œ‡2โ€–๐ต๐‘ฅโˆ’๐ต๐‘ฆโ€–2โˆ’2๐‘ก๐œ‡โŸจ๐‘ฅโˆ’๐‘ฆ,๐ต๐‘ฅโˆ’๐ต๐‘ฆโŸฉโ‰คโ€–๐‘ฅโˆ’๐‘ฆโ€–2+๐‘ก2๐œ‡2๐œŒ2โ€–๐‘ฅโˆ’๐‘ฆโ€–2โˆ’2๐‘ก๐œ‡๐œ‚โ€–๐‘ฅโˆ’๐‘ฆโ€–2โ‰ค๎‚ธ๎‚ต1โˆ’2๐‘ก๐œ‡๐œ‚โˆ’๐œ‡๐œŒ22๎‚ถ๎‚นโ€–๐‘ฅโˆ’๐‘ฆโ€–2=(1โˆ’2๐‘ก๐œ)โ€–๐‘ฅโˆ’๐‘ฆโ€–2โ‰ค(1โˆ’๐‘ก๐œ)2โ€–๐‘ฅโˆ’๐‘ฆโ€–2.(3.3) It follows that โ€–๐‘†๐‘ฅโˆ’๐‘†๐‘ฆโ€–โ‰ค(1โˆ’๐‘ก๐œ)โ€–๐‘ฅโˆ’๐‘ฆโ€–.(3.4) So ๐‘† is a contraction.

Next, we consider the cyclic algorithm (1.15), respectively, for solving the variational inequality over the set of the common fixed points of finite strictly pseudononspreading mappings.

Lemma 3.2. Assume that {๐‘ฅ๐‘›} is defined by (1.15); if ๐‘ is solution of (3.2) with ๐‘‡โˆถ๐ปโ†’๐ป being strictly pseudononspreading mapping and demiclosed and {๐‘ฆ๐‘›}โŠ‚๐ป is a bounded sequence such that โ€–๐‘‡๐‘ฆ๐‘›โˆ’๐‘ฆ๐‘›โ€–โ†’0, then liminf๐‘›โ†’โˆžโŸจ(๐›พ๐‘“โˆ’๐œ‡๐ต)๐‘,๐‘ฆ๐‘›โˆ’๐‘โŸฉโ‰ค0.(3.5)

Proof. By โ€–๐‘‡๐‘ฆ๐‘›โˆ’๐‘ฆ๐‘›โ€–โ†’0 and ๐‘‡โˆถ๐ปโ†’๐ป demi-closed, we know that any weak cluster point of {๐‘ฆ๐‘›} belongs to โ‹‚๐‘๐‘–=1๐น๐‘–๐‘ฅ(๐‘‡๐‘–). Furthermore, we can also obtain that there exists โˆ’๐‘ฆ and a subsequence {๐‘ฆ๐‘›๐‘—} of {๐‘ฆ๐‘›} such that ๐‘ฆ๐‘›๐‘—โ‡€โˆ’๐‘ฆ as ๐‘—โ†’โˆž (hence โˆ’๐‘ฆโˆˆโ‹‚๐‘๐‘–=1๐น๐‘–๐‘ฅ(๐‘‡๐‘–)) and liminf๐‘›โ†’โˆžโŸจ(๐›พ๐‘“โˆ’๐œ‡๐ต)๐‘,๐‘ฆ๐‘›โˆ’๐‘โŸฉ=lim๐‘—โ†’โˆž๎‚ฌ(๐›พ๐‘“โˆ’๐œ‡๐ต)๐‘,๐‘ฆ๐‘›๐‘—๎‚ญ.โˆ’๐‘(3.6) From (3.2), we can derive that liminf๐‘›โ†’โˆžโŸจ(๐›พ๐‘“โˆ’๐œ‡๐ต)๐‘,๐‘ฆ๐‘›๎‚ฌโˆ’๐‘โŸฉ=(๐›พ๐‘“โˆ’๐œ‡๐ต)๐‘,โˆ’๐‘ฆ๎‚ญโˆ’๐‘โ‰ค0.(3.7) It is the desired result. In addition, the variational inequality (3.7) can be written as ๎‚ฌ(๐ผโˆ’๐œ‡๐ต+๐›พ๐‘“)๐‘โˆ’๐‘,โˆ’๐‘ฆ๎‚ญโˆ’๐‘โ‰ค0,โˆ’๐‘ฆโˆˆ๐‘๎™๐‘–=1๐น๐‘–๐‘ฅ๎€ท๐‘‡๐‘–๎€ธ.(3.8) So, by Lemma 2.12, it is equivalent to the fixed point equation ๐‘ƒโ‹‚๐‘๐‘–=1๐น๐‘–๐‘ฅ(๐‘‡๐‘–)(๐ผโˆ’๐œ‡๐ต+๐›พ๐‘“)๐‘=๐‘.(3.9)

Theorem 3.3. Let ๐ถ be a nonempty closed convex subset of ๐ป and for 1โ‰ค๐‘–โ‰ค๐‘. Let ๐‘‡๐‘–โˆถ๐ปโ†’๐ป be ๐‘˜๐‘–-strictly pseudononspreading mappings for some ๐‘˜๐‘–โˆˆ[0,๐œ”๐‘–), ๐œ”๐‘–โˆˆ(0,1/2), (๐‘–=1,2,โ€ฆ,๐‘), and ๐‘˜=max{๐‘˜๐‘–โˆถ1โ‰ค๐‘–โ‰ค๐‘}. Let ๐‘“ be ๐ฟ-Lipschitz mapping on ๐ป with coefficient ๐ฟ>0, and let ๐ตโˆถ๐ปโ†’๐ป be ๐œ‚-strongly monotone and ๐œŒ-Lipschitzian on ๐ป with ๐œŒ>0, ๐œ‚>0. Let {๐›ผ๐‘›} being a sequence in (0,min{1,1/๐œ}) satisfying the following conditions: (c1)lim๐‘›โ†’โˆž๐›ผ๐‘›=0, (c2)โˆ‘โˆž๐‘›=0๐›ผ๐‘›=โˆž.
Given ๐‘ฅ0โˆˆ๐ถ, let {๐‘ฅ๐‘›}โˆž๐‘›=1 be the sequence generated by the cyclic algorithm (1.15). Then {๐‘ฅ๐‘›} converges strongly to the unique element ๐‘ in โ‹‚๐‘๐‘–=1๐น๐‘–๐‘ฅ(๐‘‡๐‘–) verifying ๐‘=๐‘ƒโ‹‚๐‘๐‘–=1๐น๐‘–๐‘ฅ(๐‘‡๐‘–)(๐ผโˆ’๐œ‡๐ต+๐›พ๐‘“)๐‘,(3.10) which equivalently solves the variational inequality problem (3.2).

Proof. Take a โ‹‚๐‘โˆˆ๐‘๐‘–=1๐น๐‘–๐‘ฅ(๐‘‡๐‘–). Let ๐‘‡๐œ”๐‘ฅ=(1โˆ’๐œ”)๐‘ฅ+๐œ”๐‘‡๐‘ฅ and 0<๐‘˜<๐œ”<1/2. Then โˆ€๐‘ฅ,๐‘ฆโˆˆ๐ถ, we have โ€–๐‘‡๐œ”๐‘ฅโˆ’๐‘‡๐œ”๐‘ฆโ€–2=๐œ”โ€–๐‘ฅโˆ’๐‘ฆโ€–2+๎€ท๎€ธ1โˆ’๐œ”โ€–๐‘‡๐‘ฅโˆ’๐‘‡๐‘ฆโ€–2๎€ท๎€ธโˆ’๐œ”1โˆ’๐œ”โ€–๐‘ฅโˆ’๐‘‡๐‘ฅโˆ’(๐‘ฆโˆ’๐‘‡๐‘ฆ)โ€–2โ‰ค๐œ”โ€–๐‘ฅโˆ’๐‘ฆโ€–2+๎€ท1โˆ’๐œ”๎€ธ๎€บโ€–๐‘ฅโˆ’๐‘ฆโ€–2+๐‘˜โ€–๐‘ฅโˆ’๐‘‡๐‘ฅโˆ’(๐‘ฆโˆ’๐‘‡๐‘ฆ)โ€–2๎€ป๎€ท๎€ธ+2โŸจ๐‘ฅโˆ’๐‘‡๐‘ฅ,๐‘ฆโˆ’๐‘‡๐‘ฆโŸฉโˆ’๐œ”1โˆ’๐œ”โ€–๐‘ฅโˆ’๐‘‡๐‘ฅโˆ’(๐‘ฆโˆ’๐‘‡๐‘ฆ)โ€–2=โ€–๐‘ฅโˆ’๐‘ฆโ€–2๎€ท๎€ธ๎€ท๎€ธ+21โˆ’๐œ”โŸจ๐‘ฅโˆ’๐‘‡๐‘ฅ,๐‘ฆโˆ’๐‘‡๐‘ฆโŸฉโˆ’1โˆ’๐œ”๎€ธ๎€ท๐œ”โˆ’๐‘˜โ€–๐‘ฅโˆ’๐‘‡๐‘ฅโˆ’(๐‘ฆโˆ’๐‘‡๐‘ฆ)โ€–2โ‰คโ€–๐‘ฅโˆ’๐‘ฆโ€–2๎€ท๎€ธ+21โˆ’๐œ”โŸจ๐‘ฅโˆ’๐‘‡๐‘ฅ,๐‘ฆโˆ’๐‘‡๐‘ฆโŸฉ=โ€–๐‘ฅโˆ’๐‘ฆโ€–2+2๎€ท๎€ธ1โˆ’๐œ”๐œ”2๎ซ๐‘ฅโˆ’๐‘‡๐œ”๐‘ฅ,๐‘ฆโˆ’๐‘‡๐œ”๐‘ฆ๎ฌ.(3.11)
From ๐‘โˆˆ๐น๐‘–๐‘ฅ(๐‘‡) and (3.11), we also have โ€–โ€–๐‘‡๐œ”๐‘ฅ๐‘›โ€–โ€–โ‰คโ€–โ€–๐‘ฅโˆ’๐‘๐‘›โ€–โ€–.โˆ’๐‘(3.12) Using (1.15) and (3.12), we obtain โ€–โ€–๐‘ฅ๐‘›+1โ€–โ€–=โ€–โ€–๐›ผโˆ’๐‘๐‘›๐›พ๎€ท๐‘“๎€ท๐‘ฅ๐‘›๎€ธ๎€ธโˆ’๐‘“(๐‘)+๐›ผ๐‘›๎€ท(๐›พ๐‘“(๐‘)โˆ’๐‘)+๐ผโˆ’๐œ‡๐›ผ๐‘›๐ต๎€ธ๎‚€๐‘‡๐œ”[๐‘›]๐‘ฅ๐‘›๎‚โ€–โ€–โˆ’๐‘โ‰ค๐›ผ๐‘›๐›พโ€–โ€–๐‘“๎€ท๐‘ฅ๐‘›๎€ธโ€–โ€–โˆ’๐‘“(๐‘)+๐›ผ๐‘›โ€–๎€ท๐›พ๐‘“(๐‘)โˆ’๐‘โ€–+1โˆ’๐›ผ๐‘›๐œ๎€ธโ€–โ€–๐‘ฅ๐‘›โ€–โ€–,โˆ’๐‘(3.13) which combined with (3.12) and โ€–๐‘“(๐‘ฅ๐‘›)โˆ’๐‘“(๐‘)โ€–โ‰ค๐ฟโ€–๐‘ฅ๐‘›โˆ’๐‘โ€– amounts to โ€–โ€–๐‘ฅ๐‘›+1โ€–โ€–โ‰ค๎€ทโˆ’๐‘1โˆ’๐›ผ๐‘›๎€ธโ€–โ€–๐‘ฅ(๐œโˆ’๐›พ๐ฟ)๐‘›โ€–โ€–โˆ’๐‘+๐›ผ๐‘›โ€–๐›พ๐‘“(๐‘)โˆ’๐‘โ€–.(3.14) Putting ๐‘€1=max{โ€–๐‘ฅ0โˆ’๐‘โ€–,โ€–๐›พ๐‘“(๐‘)โˆ’๐‘โ€–}, we clearly obtain โ€–๐‘ฅ๐‘›โˆ’๐‘โ€–โ‰ค๐‘€1. Hence {๐‘ฅ๐‘›} is bounded. We can also prove that the sequences {๐‘“(๐‘ฅ๐‘›)} and {๐‘‡๐œ”[๐‘›]๐‘ฅ๐‘›} are all bounded.
From (1.15) we obtain that ๐‘ฅ๐‘›+1โˆ’๐‘ฅ๐‘›+๐›ผ๐‘›๎€ท๐œ‡๐ต๐‘ฅ๐‘›๎€ท๐‘ฅโˆ’๐›พ๐‘“๐‘›=๎€ท๎€ธ๎€ธ๐ผโˆ’๐œ‡๐›ผ๐‘›๐ต๎€ธ๎‚€๐‘‡๐œ”[๐‘›]๐‘ฅ๐‘›โˆ’๐‘ฅ๐‘›๎‚,(3.15) hence ๎ซ๐‘ฅ๐‘›+1โˆ’๐‘ฅ๐‘›+๐›ผ๐‘›๎€ท๐œ‡๐ต๐‘ฅ๐‘›๎€ท๐‘ฅโˆ’๐›พ๐‘“๐‘›๎€ธ๎€ธ,๐‘ฅ๐‘›๎ฌ=๎€ทโˆ’๐‘1โˆ’๐›ผ๐‘›๎€ธ๎‚ฌ๐‘‡๐œ”[๐‘›]๐‘ฅ๐‘›โˆ’๐‘ฅ๐‘›,๐‘ฅ๐‘›๎‚ญโˆ’๐‘+๐›ผ๐‘›๎‚ฌ๎‚€๐‘‡(๐ผโˆ’๐œ‡๐ต)๐œ”[๐‘›]๐‘ฅ๐‘›โˆ’๐‘ฅ๐‘›๎‚,๐‘ฅ๐‘›๎‚ญ=๎€ทโˆ’๐‘1โˆ’๐›ผ๐‘›๎€ธ๎‚ฌ๐‘‡๐œ”[๐‘›]๐‘ฅ๐‘›โˆ’๐‘ฅ๐‘›,๐‘ฅ๐‘›๎‚ญโˆ’๐‘+๐›ผ๐‘›โ€–โ€–๎‚€๐‘‡(๐ผโˆ’๐œ‡๐ต)๐œ”[๐‘›]๐‘ฅ๐‘›โˆ’๐‘ฅ๐‘›๎‚โ€–โ€–โ€–โ€–๐‘ฅ๐‘›โ€–โ€–=๎€ทโˆ’๐‘1โˆ’๐›ผ๐‘›๎€ธ๎‚ฌ๐‘‡๐œ”[๐‘›]๐‘ฅ๐‘›โˆ’๐‘ฅ๐‘›,๐‘ฅ๐‘›๎‚ญโˆ’๐‘+๐›ผ๐‘›โ€–โ€–๐‘‡(1โˆ’๐œ)๐œ”[๐‘›]๐‘ฅ๐‘›โˆ’๐‘ฅ๐‘›โ€–โ€–โ€–โ€–๐‘ฅ๐‘›โ€–โ€–=๎€ทโˆ’๐‘1โˆ’๐›ผ๐‘›๎€ธ๎‚ฌ๐‘‡๐œ”[๐‘›]๐‘ฅ๐‘›โˆ’๐‘ฅ๐‘›,๐‘ฅ๐‘›๎‚ญโˆ’๐‘+๐œ”[๐‘›]๐›ผ๐‘›โ€–โ€–๐‘‡(1โˆ’๐œ)[๐‘›]๐‘ฅ๐‘›โˆ’๐‘ฅ๐‘›โ€–โ€–โ€–โ€–๐‘ฅ๐‘›โ€–โ€–.โˆ’๐‘(3.16) Moreover, by โ‹‚๐‘โˆˆ๐‘๐‘–=1๐น๐‘–๐‘ฅ(๐‘‡๐‘–) and using Remark 2.5, we obtain ๎‚ฌ๐‘ฅ๐‘›โˆ’๐‘‡๐œ”[๐‘›]๐‘ฅ๐‘›,๐‘ฅ๐‘›๎‚ญโ‰ฅ1โˆ’๐‘2๐œ”[๐‘›]๎€ท1โˆ’๐‘˜[๐‘›]๎€ธโ€–โ€–๐‘ฅ๐‘›โˆ’๐‘‡[๐‘›]๐‘ฅ๐‘›โ€–โ€–2,(3.17) which combined with (3.16) entails ๎ซ๐‘ฅ๐‘›+1โˆ’๐‘ฅ๐‘›+๐›ผ๐‘›(๐œ‡๐ตโˆ’๐›พ๐‘“)๐‘ฅ๐‘›,๐‘ฅ๐‘›๎ฌ1โˆ’๐‘โ‰คโˆ’2๐œ”[๐‘›]๎€ท1โˆ’๐‘˜[๐‘›]๎€ธ๎€ท1โˆ’๐›ผ๐‘›๎€ธโ€–โ€–๐‘ฅ๐‘›โˆ’๐‘‡[๐‘›]๐‘ฅ๐‘›โ€–โ€–2+๐œ”[๐‘›]๐›ผ๐‘›(โ€–โ€–๐‘‡1โˆ’๐œ)[๐‘›]๐‘ฅ๐‘›โˆ’๐‘ฅ๐‘›โ€–โ€–โ€–โ€–๐‘ฅ๐‘›โ€–โ€–,โˆ’๐‘(3.18) or equivalently โˆ’โŸจ๐‘ฅ๐‘›โˆ’๐‘ฅ๐‘›+1,๐‘ฅ๐‘›โˆ’๐‘โŸฉโ‰คโˆ’๐›ผ๐‘›โŸจ(๐œ‡๐ตโˆ’๐›พ๐‘“)๐‘ฅ๐‘›,๐‘ฅ๐‘›1โˆ’๐‘โŸฉโˆ’2๐œ”[๐‘›]๎€ท1โˆ’๐‘˜[๐‘›]๎€ธ๎€ท1โˆ’๐›ผ๐‘›๎€ธโ€–โ€–๐‘ฅ๐‘›โˆ’๐‘‡[๐‘›]๐‘ฅ๐‘›โ€–โ€–2+๐œ”[๐‘›]๐›ผ๐‘›โ€–โ€–๐‘‡(1โˆ’๐œ)[๐‘›]๐‘ฅ๐‘›โˆ’๐‘ฅ๐‘›โ€–โ€–โ€–โ€–๐‘ฅ๐‘›โ€–โ€–.โˆ’๐‘(3.19) Furthermore, using the following classical equality 1โŸจ๐‘ข,๐‘ฃโŸฉ=2โ€–๐‘ขโ€–2โˆ’12โ€–๐‘ขโˆ’๐‘ฃโ€–2+12โ€–๐‘ฃโ€–2,โˆ€๐‘ข,๐‘ฃโˆˆ๐ถ,(3.20) and setting ๐’ฏ๐‘›=1/2โ€–๐‘ฅ๐‘›โˆ’๐‘โ€–2, we have ๎ซ๐‘ฅ๐‘›โˆ’๐‘ฅ๐‘›+1,๐‘ฅ๐‘›๎ฌโˆ’๐‘=๐’ฏ๐‘›โˆ’๐’ฏ๐‘›+1+12โ€–โ€–๐‘ฅ๐‘›โˆ’๐‘ฅ๐‘›+1โ€–โ€–2.(3.21) So (3.19) can be equivalently rewritten as ๐’ฏ๐‘›+1โˆ’๐’ฏ๐‘›โˆ’12โ€–โ€–๐‘ฅ๐‘›โˆ’๐‘ฅ๐‘›+1โ€–โ€–2โ‰คโˆ’๐›ผ๐‘›โŸจ(๐œ‡๐ตโˆ’๐›พ๐‘“)๐‘ฅ๐‘›,๐‘ฅ๐‘›โˆ’1โˆ’๐‘โŸฉ2๐œ”[๐‘›]๎€ท1โˆ’๐‘˜[๐‘›]๎€ธ๎€ท1โˆ’๐›ผ๐‘›๎€ธโ€–โ€–๐‘ฅ๐‘›โˆ’๐‘‡[๐‘›]๐‘ฅ๐‘›โ€–โ€–2+๐œ”[๐‘›]๐›ผ๐‘›โ€–โ€–๐‘‡(1โˆ’๐œ)[๐‘›]๐‘ฅ๐‘›โˆ’๐‘ฅ๐‘›โ€–โ€–โ€–โ€–๐‘ฅ๐‘›โ€–โ€–.โˆ’๐‘(3.22) Now using (3.15) again, we have โ€–โ€–๐‘ฅ๐‘›+1โˆ’๐‘ฅ๐‘›โ€–โ€–2=โ€–โ€–๐›ผ๐‘›(๐›พ๐‘“(๐‘ฅ๐‘›)โˆ’๐œ‡๐ต๐‘ฅ๐‘›)+(๐ผโˆ’๐œ‡๐›ผ๐‘›๐ต)(๐‘‡๐œ”[๐‘›]๐‘ฅ๐‘›โˆ’๐‘ฅ๐‘›)โ€–โ€–2.(3.23) Since ๐ตโˆถ๐ปโ†’๐ป is ๐œ‚-strongly monotone and ๐‘˜-Lipschitzian on ๐ป, hence it is a classical matter to see that โ€–โ€–๐‘ฅ๐‘›+1โˆ’๐‘ฅ๐‘›โ€–โ€–2โ‰ค2๐›ผ2๐‘›โ€–โ€–๐›พ๐‘“(๐‘ฅ๐‘›)โˆ’๐œ‡๐ต๐‘ฅ๐‘›โ€–โ€–2๎€ท+21โˆ’๐›ผ๐‘›๐œ๎€ธ2โ€–โ€–๐‘‡๐œ”[๐‘›]๐‘ฅ๐‘›โˆ’๐‘ฅ๐‘›โ€–โ€–2,(3.24) which by โ€–๐‘‡๐œ”[๐‘›]๐‘ฅ๐‘›โˆ’๐‘ฅ๐‘›โ€–=๐œ”[๐‘›]โ€–๐‘ฅ๐‘›โˆ’๐‘‡[๐‘›]๐‘ฅ๐‘›โ€– and (1โˆ’๐›ผ๐‘›๐œ)2โ‰ค(1โˆ’๐›ผ๐‘›๐œ) yields 12โ€–โ€–๐‘ฅ๐‘›+1โˆ’๐‘ฅ๐‘›โ€–โ€–2โ‰ค๐›ผ2๐‘›โ€–โ€–๐›พ๐‘“(๐‘ฅ๐‘›)โˆ’๐œ‡๐ต๐‘ฅ๐‘›โ€–โ€–2+๎€ท1โˆ’๐›ผ๐‘›๐œ๎€ธ๐œ”2[๐‘›]โ€–โ€–๐‘ฅ๐‘›โˆ’๐‘‡[๐‘›]๐‘ฅ๐‘›โ€–โ€–2.(3.25) Then from (3.22) and (3.25), we have ๐’ฏ๐‘›+1โˆ’๐’ฏ๐‘›+๐œ”[๐‘›]๎‚€12๎€ท1โˆ’๐‘˜[๐‘›]๎€ธ๎€ท1โˆ’๐›ผ๐‘›๎€ธโˆ’๐œ”[๐‘›]๎€ท1โˆ’๐›ผ๐‘›๐œ๎€ธ๎‚โ€–โ€–๐‘ฅ๐‘›โˆ’๐‘‡[๐‘›]๐‘ฅ๐‘›โ€–โ€–2โ‰ค๐›ผ๐‘›๎‚€๐›ผ๐‘›โ€–โ€–๎€ท๐‘ฅ๐›พ๐‘“๐‘›๎€ธโˆ’๐œ‡๐ต๐‘ฅ๐‘›โ€–โ€–2โˆ’โŸจ(๐œ‡๐ตโˆ’๐›พ๐‘“)๐‘ฅ๐‘›,๐‘ฅ๐‘›โˆ’๐‘โŸฉ+๐œ”[๐‘›]โ€–โ€–๐‘‡(1โˆ’๐œ)[๐‘›]๐‘ฅ๐‘›โˆ’๐‘ฅ๐‘›โ€–โ€–โ€–โ€–๐‘ฅ๐‘›โ€–โ€–๎‚.โˆ’๐‘(3.26) The rest of the proof will be divided into two parts.
Caseโ€‰โ€‰1. Suppose that there exists ๐‘›0 such that {๐’ฏ๐‘›}๐‘›โ‰ฅ๐‘›0 is nonincreasing. In this situation, {๐’ฏ๐‘›} is then convergent because it is also nonnegative (hence it is bounded from below), so that lim๐‘›โ†’โˆž(๐’ฏ๐‘›+1โˆ’๐’ฏ๐‘›)=0; hence, in light of (3.26) together with lim๐‘›โ†’โˆž๐›ผ๐‘›=0 and the boundedness of {๐‘ฅ๐‘›}, we obtain lim๐‘›โ†’โˆžโ€–โ€–๐‘ฅ๐‘›โˆ’๐‘‡[๐‘›]๐‘ฅ๐‘›โ€–โ€–=0.(3.27) It also follows from (3.26) that ๐’ฏ๐‘›โˆ’๐’ฏ๐‘›+1โ‰ฅ๐›ผ๐‘›๎‚€โˆ’๐›ผ๐‘›โ€–โ€–๎€ท๐‘ฅ๐›พ๐‘“๐‘›๎€ธโˆ’๐œ‡๐ต๐‘ฅ๐‘›โ€–โ€–2+โŸจ(๐œ‡๐ตโˆ’๐›พ๐‘“)๐‘ฅ๐‘›,๐‘ฅ๐‘›โˆ’๐‘โŸฉ+๐œ”[๐‘›]โ€–โ€–๐‘‡(1โˆ’๐œ)[๐‘›]๐‘ฅ๐‘›โˆ’๐‘ฅ๐‘›โ€–โ€–โ€–โ€–๐‘ฅ๐‘›โ€–โ€–๎‚.โˆ’๐‘(3.28) Then, by โˆ‘โˆž๐‘›=0๐›ผ๐‘›=โˆž, we obviously deduce that liminf๐‘›โ†’โˆž๐›ผ๐‘›๎‚€โˆ’๐›ผ๐‘›โ€–โ€–๎€ท๐‘ฅ๐›พ๐‘“๐‘›๎€ธโˆ’๐œ‡๐ต๐‘ฅ๐‘›โ€–โ€–2+โŸจ(๐œ‡๐ตโˆ’๐›พ๐‘“)๐‘ฅ๐‘›,๐‘ฅ๐‘›โˆ’๐‘โŸฉ+๐œ”[๐‘›]โ€–โ€–๐‘‡(1โˆ’๐œ)[๐‘›]๐‘ฅ๐‘›โˆ’๐‘ฅ๐‘›โ€–โ€–โ€–โ€–๐‘ฅ๐‘›โ€–โ€–๎‚โˆ’๐‘โ‰ค0,(3.29) or equivalently (as ๐›ผ๐‘›โ€–๐›พ๐‘“(๐‘ฅ๐‘›)โˆ’๐œ‡๐ต๐‘ฅ๐‘›โ€–2โ†’0) liminf๐‘›โ†’โˆžโŸจ(๐œ‡๐ตโˆ’๐›พ๐‘“)๐‘ฅ๐‘›,๐‘ฅ๐‘›โˆ’๐‘โŸฉโ‰ค0.(3.30) Moreover, by Remark 1.2, we have 2(๐œ‡๐œ‚โˆ’๐›พ๐ฟ)๐’ฏ๐‘›+โŸจ(๐œ‡๐ตโˆ’๐›พ๐‘“)๐‘,๐‘ฅ๐‘›โˆ’๐‘โŸฉโ‰คโŸจ(๐œ‡๐ตโˆ’๐›พ๐‘“)๐‘ฅ๐‘›,๐‘ฅ๐‘›โˆ’๐‘โŸฉ,(3.31) which by (3.30) entails liminf๐‘›โ†’โˆžโŸจ2(๐œ‡๐œ‚โˆ’๐›พ๐ฟ)๐’ฏ๐‘›+(๐œ‡๐ตโˆ’๐›พ๐‘“)๐‘,๐‘ฅ๐‘›โˆ’๐‘โŸฉโ‰ค0,(3.32) hence, recalling that lim๐‘›โ†’โˆž๐’ฏ๐‘› exists, we equivalently obtain 2(๐œ‡๐œ‚โˆ’๐›พ๐ฟ)lim๐‘›โ†’โˆž๐’ฏ๐‘›+liminf๐‘›โ†’โˆžโŸจ(๐œ‡๐ตโˆ’๐›พ๐‘“)๐‘,๐‘ฅ๐‘›โˆ’๐‘โŸฉโ‰ค0,(3.33) namely 2(๐œ‡๐œ‚โˆ’๐›พ๐ฟ)lim๐‘›โ†’โˆž๐’ฏ๐‘›โ‰คโˆ’liminf๐‘›โ†’โˆžโŸจ(๐œ‡๐ตโˆ’๐›พ๐‘“)๐‘,๐‘ฅ๐‘›โˆ’๐‘โŸฉ.(3.34) From (3.27) and Lemma 3.2, we obtain liminf๐‘›โ†’โˆžโŸจ(๐œ‡๐ตโˆ’๐›พ๐‘“)๐‘,๐‘ฅ๐‘›โˆ’๐‘โŸฉโ‰ฅ0,(3.35) which yields lim๐‘›โ†’โˆž๐’ฏ๐‘›=0, so that {๐‘ฅ๐‘›} converges strongly to ๐‘.
Caseโ€‰โ€‰2. Suppose there exists a subsequence {๐’ฏ๐‘›๐‘˜}๐‘˜โ‰ฅ0 of {๐’ฏ๐‘›}๐‘›โ‰ฅ0 such that ๐’ฏ๐‘›๐‘˜โ‰ค๐’ฏ๐‘›๐‘˜+1 for all ๐‘˜โ‰ฅ0. In this situation, we consider the sequence of indices {๐›ฟ(๐‘›)} as defined in Lemma 2.8. It follows that ๐’ฏ๐›ฟ(๐‘›+1)โˆ’๐’ฏ๐›ฟ(๐‘›)>0, which by (3.26) amounts to ๐œ”[๐‘›]๎‚€12๎€ท1โˆ’๐‘˜[๐‘›]๎€ธ๎€ท1โˆ’๐›ผ๐›ฟ(๐‘›)๎€ธโˆ’๐œ”[๐‘›]๎€ท1โˆ’๐›ผ๐›ฟ(๐‘›)๐œ๎€ธ๎‚โ€–โ€–๐‘ฅ๐›ฟ(๐‘›)โˆ’๐‘‡[๐‘›]๐‘ฅ๐›ฟ(๐‘›)โ€–โ€–2<๐›ผ๐›ฟ(๐‘›)๎‚€๐›ผ๐›ฟ(๐‘›)โ€–โ€–๎€ท๐‘ฅ๐›พ๐‘“๐›ฟ(๐‘›)๎€ธโˆ’๐œ‡๐ต๐‘ฅ๐›ฟ(๐‘›)โ€–โ€–2โˆ’๎ซ(๐œ‡๐ตโˆ’๐›พ๐‘“)๐‘ฅ๐›ฟ(๐‘›),๐‘ฅ๐›ฟ(๐‘›)๎ฌโˆ’๐‘+๐œ”[๐‘›](โ€–โ€–๐‘‡1โˆ’๐œ)[๐‘›]๐‘ฅ๐›ฟ(๐‘›)โˆ’๐‘ฅ๐›ฟ(๐‘›)โ€–โ€–โ€–โ€–๐‘ฅ๐›ฟ(๐‘›)โ€–โ€–๎€ธ.โˆ’๐‘(3.36) By the boundedness of {๐‘ฅ๐‘›} and lim๐‘›โ†’โˆž๐›ผ๐‘›=0, we immediately obtain lim๐‘›โ†’โˆžโ€–โ€–๐‘ฅ๐›ฟ(๐‘›)โˆ’๐‘‡[๐‘›]๐‘ฅ๐›ฟ(๐‘›)โ€–โ€–=0.(3.37) Using (1.15), we have โ€–๐‘ฅ๐›ฟ(๐‘›)+1โˆ’๐‘ฅ๐›ฟ(๐‘›)โ€–โ‰ค๐›ผ๐›ฟ(๐‘›)โ€–โ€–๎€ท๐‘ฅ๐›พ๐‘“๐›ฟ(๐‘›)๎€ธโˆ’๐œ‡๐ต๐‘ฅ๐›ฟ(๐‘›)โ€–โ€–+๎€ท1โˆ’๐›ผ๐›ฟ(๐‘›)๐œ๎€ธโ€–โ€–๐‘‡๐œ”[๐‘›]๐‘ฅ๐›ฟ(๐‘›)โˆ’๐‘ฅ๐›ฟ(๐‘›)โ€–โ€–โ‰ค๐›ผ๐›ฟ(๐‘›)โ€–โ€–๎€ท๐‘ฅ๐›พ๐‘“๐›ฟ(๐‘›)๎€ธโˆ’๐œ‡๐ต๐‘ฅ๐›ฟ(๐‘›)โ€–โ€–+๐œ”[๐‘›]๎€ท1โˆ’๐›ผ๐›ฟ(๐‘›)๐œ๎€ธโ€–โ€–๐‘‡[๐‘›]๐‘ฅ๐›ฟ(๐‘›)โˆ’๐‘ฅ๐›ฟ(๐‘›)โ€–โ€–,(3.38) which together with (3.37) and lim๐‘›โ†’โˆž๐›ผ๐‘›=0 yields lim๐‘›โ†’โˆžโ€–โ€–๐‘ฅ๐›ฟ(๐‘›)+1โˆ’๐‘ฅ๐›ฟ(๐‘›)โ€–โ€–=0.(3.39)
Now by (3.36) we clearly have ๐›ผ๐›ฟ(๐‘›)โ€–โ€–๐›พ๐‘“(๐‘ฅ๐›ฟ(๐‘›))โˆ’๐œ‡๐ต๐‘ฅ๐›ฟ(๐‘›)โ€–โ€–2+๐œ”[๐‘›]โ€–โ€–๐‘‡(1โˆ’๐œ)[๐‘›]๐‘ฅ๐›ฟ(๐‘›)โˆ’๐‘ฅ๐›ฟ(๐‘›)โ€–โ€–โ€–โ€–๐‘ฅ๐›ฟ(๐‘›)โ€–โ€–โ‰ฅ๎ซโˆ’๐‘(๐œ‡๐ตโˆ’๐›พ๐‘“)๐‘ฅ๐›ฟ(๐‘›),๐‘ฅ๐›ฟ(๐‘›)๎ฌ,โˆ’๐‘(3.40) which in the light of (3.31) yields 2(๐œ‡๐œ‚โˆ’๐›พ๐ฟ)๐’ฏ๐›ฟ(๐‘›)+๎ซ(๐œ‡๐ตโˆ’๐›พ๐‘“)๐‘,๐‘ฅ๐›ฟ(๐‘›)๎ฌโˆ’๐‘โ‰ค๐›ผ๐›ฟ(๐‘›)โ€–โ€–๐›พ๐‘“(๐‘ฅ๐›ฟ(๐‘›))โˆ’๐œ‡๐ต๐‘ฅ๐›ฟ(๐‘›)โ€–โ€–2+๐œ”[๐‘›]โ€–โ€–๐‘‡(1โˆ’๐œ)[๐‘›]๐‘ฅ๐›ฟ(๐‘›)โˆ’๐‘ฅ[๐‘›]โ€–โ€–โ€–โ€–๐‘ฅ๐›ฟ(๐‘›)โ€–โ€–,โˆ’๐‘(3.41) hence (as lim๐‘›โ†’โˆž๐›ผ๐›ฟ(๐‘›)โ€–๐›พ๐‘“(๐‘ฅ๐›ฟ(๐‘›))โˆ’๐œ‡๐ต๐‘ฅ๐›ฟ(๐‘›)โ€–2=0 and (3.37)) it follows that 2(๐œ‡๐œ‚โˆ’๐›พ๐ฟ)limsup๐‘›โ†’โˆž๐’ฏ๐›ฟ(๐‘›)โ‰คโˆ’liminf๐‘›โ†’โˆž๎ซ(๐œ‡๐ตโˆ’๐›พ๐‘“)๐‘,๐‘ฅ๐›ฟ(๐‘›)๎ฌ.โˆ’๐‘(3.42) From (3.37) and Lemma 3.2, we obtain lim๐‘›โ†’โˆžโŸจ(๐œ‡๐ตโˆ’๐›พ๐‘“)๐‘,๐‘ฅ๐›ฟ(๐‘›)โˆ’๐‘โŸฉโ‰ฅ0,(3.43) which by (3.42) yields limsup๐‘›โ†’โˆž๐’ฏ๐›ฟ(๐‘›)=0, so that lim๐‘›โ†’โˆž๐’ฏ๐›ฟ(๐‘›)=0. Combining (3.39), we have lim๐‘›โ†’โˆž๐’ฏ๐›ฟ(๐‘›)+1=0. Then, recalling that ๐’ฏ๐‘›<๐’ฏ๐›ฟ(๐‘›)+1 (by Lemma 2.8), we get lim๐‘›โ†’โˆž๐’ฏ๐‘›=0, so that ๐‘ฅ๐‘›โ†’๐‘ strongly.

Taking ๐‘˜๐‘–=0, we know that ๐‘˜๐‘–-strictly pseudononspreading mapping is nonspreading mapping and ๐‘–=๐‘› (mod ๐‘), 0โ‰ค๐‘–โ‰ค๐‘โˆ’1. According to the proof Theorem 3.3, we obtain the following corollary.

Corollary 3.4. Let ๐ถ be a nonempty closed convex subset of ๐ป. Let ๐‘‡๐‘–โˆถ๐ถโ†’๐ถ be nonspreading mappings and ๐œ”๐‘–โˆˆ(0,1/2), (๐‘–=1,2,โ€ฆ,๐‘). Let ๐‘“ be ๐ฟ-Lipschitz mapping on ๐ป with coefficient ๐ฟ>0 and let ๐ตโˆถ๐ปโ†’๐ป be ๐œ‚-strongly monotone and ๐œŒ-Lipschitzian on ๐ป with ๐œŒ>0, ๐œ‚>0. Let {๐›ผ๐‘›} be a sequence in (0,min{1,1/๐œ}) satisfying the following conditions: (c1)lim๐‘›โ†’โˆž๐›ผ๐‘›=0, (c2)โˆ‘โˆž๐‘›=0๐›ผ๐‘›=โˆž.
Given ๐‘ฅ0โˆˆ๐ถ, let {๐‘ฅ๐‘›}โˆž๐‘›=1 be the sequence generated by the cyclic algorithm (1.15). Then {๐‘ฅ๐‘›} converges strongly to the unique element ๐‘ in โ‹‚๐‘๐‘–=1๐น๐‘–๐‘ฅ(๐‘‡๐‘–) verifying ๐‘=๐‘ƒโ‹‚๐‘๐‘–=1๐น๐‘–๐‘ฅ(๐‘‡๐‘–)(๐ผโˆ’๐œ‡๐ต+๐›พ๐‘“)๐‘,(3.44) which equivalently solves the variational inequality problem (3.2).

Acknowledgment

This work is supported in part by China Postdoctoral Science Foundation (Grant no. 20100470783).

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Copyright © 2012 Bin-Chao Deng et al. 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.

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