International Scholarly Research Notices

International Scholarly Research Notices / 2011 / Article

Research Article | Open Access

Volume 2011 |Article ID 795379 | 16 pages | https://doi.org/10.5402/2011/795379

Shrinking Projection Method for Fixed Point Problems of an Infinite Family of Strictly Pseudocontractive Mappings and the System of Cocoercive Quasivariational Inclusions Problems in Hilbert Spaces

Academic Editor: C. Zhu
Received12 Apr 2011
Accepted03 May 2011
Published30 Jun 2011

Abstract

This paper is concerned with a common element of the set of common fixed points for an infinite family of strictly pseudocontractive mappings and the set of solutions of a system of cocoercive quasivariational inclusions problems in Hilbert spaces. The strong convergence theorem for the above two sets is obtained by a general iterative scheme based on the shrinking projection method, and the applicability of the results is shown to extend and improve some well-known results existing in the current literature.

1. Introduction

Throughout this paper, we always assume that ๐ถ is a nonempty closed convex subset of a real Hilbert space ๐ป with inner product and norm denoted by โŸจโ‹…,โ‹…โŸฉ and โ€–โ‹…โ€–, respectively, 2๐ป denoting the family of all the nonempty subsets of ๐ป.

Let ๐ตโˆถ๐ปโ†’๐ป be a single-valued nonlinear mapping and ๐‘€โˆถ๐ปโ†’2๐ป a set-valued mapping. We consider the following quasivariational inclusion problem, which is to find a point ๐‘ฅโˆˆ๐ป,๐œƒโˆˆ๐ต๐‘ฅ+๐‘€๐‘ฅ,(1.1) where ๐œƒ is the zero vector in ๐ป. The set of solutions of problem (1.1) is denoted by VI(๐ป,๐ต,๐‘€).

Recall that ๐‘ƒ๐ถ is the metric projection of ๐ป onto ๐ถ; that is, for each ๐‘ฅโˆˆ๐ป, there exists the unique point in ๐‘ƒ๐ถ๐‘ฅโˆˆ๐ถ such that โ€–๐‘ฅโˆ’๐‘ƒ๐ถ๐‘ฅโ€–=min๐‘ฆโˆˆ๐ถโ€–๐‘ฅโˆ’๐‘ฆโ€–. A mapping ๐‘‡โˆถ๐ถโ†’๐ถ is called nonexpansive if โ€–๐‘‡๐‘ฅโˆ’๐‘‡๐‘ฆโ€–โ‰คโ€–๐‘ฅโˆ’๐‘ฆโ€– for all ๐‘ฅ,๐‘ฆโˆˆ๐ถ. A point ๐‘ฅโˆˆ๐ถ is a fixed point of ๐‘‡ provided ๐‘‡๐‘ฅ=๐‘ฅ. We denote by ๐น(๐‘‡) the set of fixed points of ๐‘‡; that is, ๐น(๐‘‡)={๐‘ฅโˆˆ๐ถโˆถ๐‘‡๐‘ฅ=๐‘ฅ}. If ๐ถ is nonempty bounded closed convex subset of ๐ป and ๐‘‡ is a nonexpansive mapping of ๐ถ into itself, then ๐น(๐‘‡) is nonempty (see [1]). Recall that a mapping ๐ดโˆถ๐ถโ†’๐ถ is said to be (i)monotone if โŸจ๐ด๐‘ฅโˆ’๐ด๐‘ฆ,๐‘ฅโˆ’๐‘ฆโŸฉโ‰ฅ0,โˆ€๐‘ฅ,๐‘ฆโˆˆ๐ถ,(1.2)(ii)๐‘˜-Lipschitz continuous if there exists a constant ๐‘˜>0 such that โ€–๐ด๐‘ฅโˆ’๐ด๐‘ฆโ€–โ‰ค๐‘˜โ€–๐‘ฅโˆ’๐‘ฆโ€–,โˆ€๐‘ฅ,๐‘ฆโˆˆ๐ถ,(1.3) if ๐‘˜=1, then ๐ด is a nonexpansive, (iii)pseudocontractive if โ€–๐ด๐‘ฅโˆ’๐ด๐‘ฆโ€–2โ‰คโ€–๐‘ฅโˆ’๐‘ฆโ€–2โ€–+โ€–(๐ผโˆ’๐ด)๐‘ฅโˆ’(๐ผโˆ’๐ด)๐‘ฆ2,โˆ€๐‘ฅ,๐‘ฆโˆˆ๐ถ,(1.4)(iv)๐‘˜-strictly pseudocontractive if there exists a constant ๐‘˜โˆˆ[0,1) such that โ€–๐ด๐‘ฅโˆ’๐ด๐‘ฆโ€–2โ‰คโ€–๐‘ฅโˆ’๐‘ฆโ€–2โ€–+๐‘˜โ€–(๐ผโˆ’๐ด)๐‘ฅโˆ’(๐ผโˆ’๐ด)๐‘ฆ2,โˆ€๐‘ฅ,๐‘ฆโˆˆ๐ถ,(1.5) it is obvious that ๐ด is a nonexpansive if and only if ๐ด is 0-strictly pseudocontractive, (v)๐›ผ-strongly monotone if there exists a constant ๐›ผ>0 such that โŸจ๐ด๐‘ฅโˆ’๐ด๐‘ฆ,๐‘ฅโˆ’๐‘ฆโŸฉโ‰ฅ๐›ผโ€–๐‘ฅโˆ’๐‘ฆโ€–2,โˆ€๐‘ฅ,๐‘ฆโˆˆ๐ถ,(1.6)(vi)๐›ผ-inverse-strongly monotone (or ๐›ผ-cocoercive) if there exists a constant ๐›ผ>0 such that โŸจ๐ด๐‘ฅโˆ’๐ด๐‘ฆ,๐‘ฅโˆ’๐‘ฆโŸฉโ‰ฅ๐›ผโ€–๐ด๐‘ฅโˆ’๐ด๐‘ฆโ€–2,โˆ€๐‘ฅ,๐‘ฆโˆˆ๐ถ,(1.7) if ๐›ผ=1, then ๐ด is said to be firmly nonexpansive; it is obvious that any ๐›ผ-inverse-strongly monotone mapping ๐ด is monotone and (1/๐›ผ)-Lipschitz continuous.

The existence of common fixed points for a finite family of nonexpansive mappings has been considered by many authors (see [2โ€“5] and the references therein).

In this paper, we study the mapping ๐‘Š๐‘› defined by๐‘ˆ๐‘›,๐‘›+1๐‘ˆ=๐ผ,๐‘›,๐‘›=๐œ‡๐‘›๐‘†๐‘›๐‘ˆ๐‘›,๐‘›+1+๎€ท1โˆ’๐œ‡๐‘›๎€ธ๐‘ˆ๐ผ,๐‘›,๐‘›โˆ’1=๐œ‡๐‘›โˆ’1๐‘†๐‘›โˆ’1๐‘ˆ๐‘›,๐‘›+๎€ท1โˆ’๐œ‡๐‘›โˆ’1๎€ธโ‹ฎ๐‘ˆ๐ผ,๐‘›,๐‘˜=๐œ‡๐‘˜๐‘†๐‘˜๐‘ˆ๐‘›,๐‘˜+1+๎€ท1โˆ’๐œ‡๐‘˜๎€ธ๐‘ˆ๐ผ,๐‘›,๐‘˜โˆ’1=๐œ‡๐‘˜โˆ’1๐‘†๐‘˜โˆ’1๐‘ˆ๐‘›,๐‘˜+๎€ท1โˆ’๐œ‡๐‘˜โˆ’1๎€ธโ‹ฎ๐‘ˆ๐ผ,๐‘›,2=๐œ‡2๐‘†2๐‘ˆ๐‘›,3+๎€ท1โˆ’๐œ‡2๎€ธ๐‘Š๐ผ,๐‘›=๐‘ˆ๐‘›,1=๐œ‡1๐‘†1๐‘ˆ๐‘›,2+๎€ท1โˆ’๐œ‡1๎€ธ๐ผ,(1.8) where {๐œ‡๐‘–} is nonnegative real sequence in (0,1), for all ๐‘–โˆˆโ„•, ๐‘†1,๐‘†2,โ€ฆ from a family of infinitely nonexpansive mappings of ๐ถ into itself. It is obvious that ๐‘Š๐‘› is a nonexpansive mapping of ๐ถ into itself; such a mapping ๐‘Š๐‘› is called a ๐‘Š-mapping generated by ๐‘†1,๐‘†2,โ€ฆ,๐‘†๐‘› and ๐œ‡1,๐œ‡2,โ€ฆ,๐œ‡๐‘›.

Definition 1.1 (see [6]). Let ๐‘€โˆถ๐ปโ†’2๐ป be a multivalued maximal monotone mapping. Then, the single-valued mapping ๐ฝ๐‘€,๐œ†โˆถ๐ปโ†’๐ป defined by ๐ฝ๐‘€,๐œ†(๐‘ข)=(๐ผ+๐œ†๐‘€)โˆ’1(๐‘ข), for all ๐‘ขโˆˆ๐ป, is called the resolvent operator associated with ๐‘€, where ๐œ† is any positive number and ๐ผ is the identity mapping.

Recently, Zhang et al. [6] considered the problem (1.1) and the problem of a fixed point of nonexpansive mapping. To be more precise, they proved the following theorem.

Theorem ZLC. Let ๐ป be a real Hilbert space, ๐ตโˆถ๐ปโ†’๐ป an ๐›ผ-inverse-strongly monotone mapping, ๐‘€โˆถ๐ปโ†’2๐ป a maximal monotone mapping, and ๐‘‡โˆถ๐ปโ†’๐ป a nonexpansive mapping. Suppose that the set ๐น(๐‘‡)โˆฉVI(๐ป,๐ต,๐‘€)โ‰ โˆ…, where VI(๐ป,๐ต,๐‘€) is the set of solutions of quasivariational inclusion (1.1). Suppose that ๐‘ฅ1=๐‘ฅโˆˆ๐ป and {๐‘ฅ๐‘›} is the sequence defined by ๐‘ฆ๐‘›=๐ฝ๐‘€,๐œ†๎€ท๐‘ฅ๐‘›โˆ’๐œ†๐ต๐‘ฅ๐‘›๎€ธ,๐‘ฅ๐‘›+1=๐›ผ๐‘›๎€ท๐‘ฅ+1โˆ’๐›ผ๐‘›๎€ธ๐‘‡๐‘ฆ๐‘›,(1.9) for all ๐‘›โˆˆโ„•, where ๐œ†โˆˆ(0,2๐›ผ) and {๐›ผ๐‘›}โŠ‚(0,1) satisfying the following conditions:
(C1) lim๐‘›โ†’โˆž๐›ผ๐‘›=0 and โˆ‘โˆž๐‘›=1๐›ผ๐‘›=โˆž,
(C2) โˆ‘โˆž๐‘›=1|๐›ผ๐‘›+1โˆ’๐›ผ๐‘›|<โˆž.
Then, {๐‘ฅ๐‘›} converges strongly to ๐‘ƒ๐น(๐‘‡)โˆฉVI(๐ป,๐ต,๐‘€)(๐‘ฅ).

Nakajo and Takahashi [7] introduced an iterative scheme for finding a fixed point of a nonexpansive mapping by a hybrid method which is called that shrinking projection method (or ๐ถ๐‘„ method) as in the following theorem.

Theorem NT. Let ๐ถ be a nonempty closed convex subset of a real Hilbert space ๐ป. Let ๐‘‡ be a nonexpansive mapping of ๐ถ into itself such that ๐น(๐‘‡)โ‰ โˆ…. Suppose that ๐‘ฅ1=๐‘ฅโˆˆ๐ถ and {๐‘ฅ๐‘›} is the sequence defined by ๐‘ฆ๐‘›=๐›ผ๐‘›๐‘ฅ๐‘›+๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐‘‡๐‘ฅ๐‘›,๐ถ๐‘›=๎€ฝโ€–โ€–๐‘ฆ๐‘งโˆˆ๐ถโˆถ๐‘›โ€–โ€–โ‰คโ€–โ€–๐‘ฅโˆ’๐‘ง๐‘›โ€–โ€–๎€พ,๐‘„โˆ’๐‘ง๐‘›=๎€ฝ๐‘งโˆˆ๐ถโˆถโŸจ๐‘ฅ๐‘›โˆ’๐‘ง,๐‘ฅ1โˆ’๐‘ฅ๐‘›๎€พ,๐‘ฅโŸฉโ‰ฅ0๐‘›+1=๐‘ƒ๐ถ๐‘›โˆฉ๐‘„๐‘›๎€ท๐‘ฅ1๎€ธ,โˆ€๐‘›โˆˆโ„•,(1.10) where 0โ‰ค๐›ผ๐‘›โ‰ค๐›ผ<1. Then, {๐‘ฅ๐‘›} converges strongly to ๐‘ƒ๐น(๐‘‡)(๐‘ฅ1).

In the same way, Kikkawa and Takahashi [8] introduced an iterative scheme for finding a common fixed point of an infinite family of nonexpansive mappings as follows:๐‘ฆ๐‘›=๐‘Š๐‘›๐‘ฅ๐‘›,๐ถ๐‘›=๎€ฝโ€–โ€–๐‘ฆ๐‘งโˆˆ๐ถโˆถ๐‘›โ€–โ€–โ‰คโ€–โ€–๐‘ฅโˆ’๐‘ง๐‘›โ€–โ€–๎€พ,๐‘„โˆ’๐‘ง๐‘›=๎€ฝ๐‘งโˆˆ๐ถโˆถโŸจ๐‘ฅ๐‘›โˆ’๐‘ง,๐‘ฅ1โˆ’๐‘ฅ๐‘›๎€พ,๐‘ฅโŸฉโ‰ฅ0๐‘›+1=๐‘ƒ๐ถ๐‘›โˆฉ๐‘„๐‘›๎€ท๐‘ฅ1๎€ธ,โˆ€๐‘›โˆˆโ„•,(1.11) where ๐‘ฅ1=๐‘ฅโˆˆ๐ถ and ๐‘Š๐‘› is a ๐‘Š-mapping of ๐ถ into itself generated by {๐‘‡๐‘›โˆถ๐ถโ†’๐ถ} and {๐œ‡๐‘›}. They prove that, if โ‹‚ฮฉ=โˆž๐‘›=1๐น(๐‘‡๐‘›)โ‰ โˆ…, then the sequence {๐‘ฅ๐‘›} generated by (1.11) converges strongly to ๐‘ƒฮฉ(๐‘ฅ1).

Recently, Su and Qin [9] modified the shrinking projection method for finding a fixed point of a nonexpansive mapping, for which the convergence rate of the iterative scheme is faster than that of the iterative scheme of Nakajo and Takahashi [7] as follows:๐‘ฆ๐‘›=๐›ผ๐‘›๐‘ฅ๐‘›+๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐‘‡๐‘ฅ๐‘›,๐ถ๐‘›=๎€ฝ๐‘งโˆˆ๐ถ๐‘›โˆ’1โˆฉ๐‘„๐‘›โˆ’1โˆถโ€–โ€–๐‘ฆ๐‘›โ€–โ€–โ‰คโ€–โ€–๐‘ฅโˆ’๐‘ง๐‘›โ€–โ€–๎€พ๐‘„โˆ’z,๐‘›โ‰ฅ1,๐‘›=๎€ฝ๐‘งโˆˆ๐ถ๐‘›โˆ’1โˆฉ๐‘„๐‘›โˆ’1โˆถโŸจ๐‘ฅ๐‘›โˆ’๐‘ง,๐‘ฅ0โˆ’๐‘ฅ๐‘›๎€พ๐ถโŸฉโ‰ฅ0,๐‘›โ‰ฅ1,0=๎€ฝโ€–โ€–๐‘ฆ๐‘งโˆˆ๐ถโˆถ0โ€–โ€–โ‰คโ€–โ€–๐‘ฅโˆ’๐‘ง0โ€–โ€–๎€พ,๐‘„โˆ’๐‘ง0๐‘ฅ=๐ถ,๐‘›+1=๐‘ƒ๐ถ๐‘›โˆฉ๐‘„๐‘›๎€ท๐‘ฅ0๎€ธ,โˆ€๐‘›โˆˆโ„•โˆช{0},(1.12) where ๐‘ฅ0=๐‘ฅโˆˆ๐ถ and ๐‘‡ is a nonexpansive mapping of ๐ถ into itself. They prove that, under the parameter 0โ‰ค๐›ผ๐‘›โ‰ค๐›ผ<1, if ๐น(๐‘‡)โ‰ โˆ…, then the sequence {๐‘ฅ๐‘›} generated by (1.12) converges strongly to ๐‘ƒ๐น(๐‘‡)(๐‘ฅ0).

On the other hand, Tada and Takahashi [10] introduced an iterative scheme for finding a common element of the set of solutions of an equilibrium problem and the set of solutions of a fixed point problem of a nonexpansive mapping as follows:๐‘ข๐‘›๎€ท๐‘ขโˆˆ๐ถsuchthat๐น๐‘›๎€ธ+1,๐‘ฆ๐‘Ÿ๐‘›โŸจ๐‘ฆโˆ’๐‘ข๐‘›,๐‘ข๐‘›โˆ’๐‘ฅ๐‘›๐‘ฆโŸฉโ‰ฅ0,โˆ€๐‘ฆโˆˆ๐ถ,๐‘›=๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐‘ฅ๐‘›+๐›ผ๐‘›๐‘‡๐‘ข๐‘›,๐ถ๐‘›=๎€ฝโ€–โ€–๐‘ฆ๐‘งโˆˆ๐ปโˆถ๐‘›โ€–โ€–โ‰คโ€–โ€–๐‘ฅโˆ’๐‘ง๐‘›โ€–โ€–๎€พ,๐‘„โˆ’๐‘ง๐‘›=๎€ฝ๐‘งโˆˆ๐ปโˆถโŸจ๐‘ฅ๐‘›โˆ’๐‘ง,๐‘ฅ1โˆ’๐‘ฅ๐‘›๎€พ,๐‘ฅโŸฉโ‰ฅ0๐‘›+1=๐‘ƒ๐ถ๐‘›โˆฉ๐‘„๐‘›๎€ท๐‘ฅ1๎€ธ,โˆ€๐‘›โˆˆโ„•,(1.13) where ๐‘ฅ1=๐‘ฅโˆˆ๐ป, ๐‘‡ is a nonexpansive mapping of ๐ถ into ๐ป and ๐น is a bifunction from ๐ถร—๐ถ into โ„. They prove that, under the sequences {๐›ผ๐‘›}โŠ‚[๐›ผ,1] for some ๐›ผโˆˆ(0,1) and {๐‘Ÿ๐‘›}โŠ‚[๐‘Ÿ,โˆž) for some ๐‘Ÿ>0, if ฮฉ=๐น(๐‘‡)โˆฉEP(๐น)โ‰ โˆ…, then the sequence {๐‘ฅ๐‘›} generated by (1.13) converges strongly to ๐‘ƒฮฉ(๐‘ฅ1) such that EP(๐น) is the set of solutions of equilibrium problem defined by EP(๐น)={๐‘ฅโˆˆ๐ถโˆถ๐น(๐‘ฅ,๐‘ฆ)โ‰ฅ0,โˆ€๐‘ฆโˆˆ๐ถ}.(1.14)

In this paper, we introduce an iterative scheme (1.15) for finding a common element of the set of common fixed points for an infinite family of strictly pseudocontractive mappings and the set of solutions of a system of cocoercive quasivariational inclusions problems by the shrinking projection method in Hilbert spaces as follows:๐‘ฆ๐‘›=๐›ผ๐‘›๐‘Š๐‘›๐‘ฅ๐‘›+๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐‘๎“๐‘–=1๐›ฝ๐‘–๐ฝ๐‘€๐‘–,๐œ†๐‘–๎€ท๐‘ฅ๐‘›โˆ’๐œ†๐‘–๐ต๐‘–๐‘ฅ๐‘›๎€ธ,๐œ–๐‘›=๐›ผ๐‘›๎€ท1โˆ’๐›ผ๐‘›๎€ธโ€–โ€–โ€–โ€–๐‘Š๐‘›๐‘ฅ๐‘›โˆ’๐‘๎“๐‘–=1๐›ฝ๐‘–๐ฝ๐‘€๐‘–,๐œ†๐‘–๎€ท๐‘ฅ๐‘›โˆ’๐œ†๐‘–๐ต๐‘–๐‘ฅ๐‘›๎€ธโ€–โ€–โ€–โ€–2,๐ถ๐‘›+1=๎‚†๐‘งโˆˆ๐ถ๐‘›โˆฉ๐‘„๐‘›โˆถโ€–โ€–๐‘ฆ๐‘›โ€–โ€–โˆ’๐‘ง2โ‰คโ€–โ€–๐‘ฅ๐‘›โ€–โ€–โˆ’๐‘ง2โˆ’๐œ–๐‘›๎‚‡,๐‘„๐‘›+1=๎€ฝ๐‘งโˆˆ๐ถ๐‘›โˆฉQ๐‘›โˆถโŸจ๐‘ฅ๐‘›โˆ’๐‘ง,๐‘ฅ1โˆ’๐‘ฅ๐‘›๎€พ,๐ถโŸฉโ‰ฅ01=๐‘„1๐‘ฅ=๐ป,๐‘›+1=๐‘ƒ๐ถ๐‘›+1โˆฉ๐‘„๐‘›+1๎€ท๐‘ฅ1๎€ธ,โˆ€๐‘›โˆˆโ„•,(1.15) where ๐‘ฅ1=๐‘ขโˆˆ๐ป chosen arbitrarily, ๐‘€๐‘–โˆถ๐ปโ†’2๐ป is a maximal monotone mapping, ๐ต๐‘–โˆถ๐ปโ†’๐ป is a ๐œ‰๐‘–-cocoercive mapping for each ๐‘–=1,2,โ€ฆ,๐‘, and ๐‘Š๐‘› is a ๐‘Š-mapping on ๐ป generated by {๐‘†๐‘›} and {๐œ‡๐‘›} such that the mapping ๐‘†๐‘›โˆถ๐ปโ†’๐ป defined by ๐‘†๐‘›๐‘ฅ=๐›ผ๐‘ฅ+(1โˆ’๐›ผ)๐‘‡๐‘›๐‘ฅ for all ๐‘ฅโˆˆ๐ป, where {๐‘‡๐‘›โˆถ๐ปโ†’๐ป} is an infinite family of ๐‘˜-strictly pseudocontractive mappings with a fixed point.

It is well known that the class of strictly pseudocontractive mappings contains the class of nonexpansive mappings, and it follows that, if ๐‘˜=0, then the iterative scheme (1.15) is reduced to find a common element of the set of common fixed points for an infinite family of nonexpansive mappings and the set of solutions of a system of cocoercive quasivariational inclusions problems in Hilbert spaces.

Furthermore, if ๐‘€๐‘–โ‰ก๐ต๐‘–โ‰ก0 for all ๐‘–=1,2,โ€ฆ,๐‘ and โˆ‘๐‘๐‘–=1๐›ฝ๐‘–=1, then the iterative scheme (1.15) is reduced to extend and improve the results of Kikkawa and Takahashi [8] for finding a common fixed point of an infinite family of ๐‘˜-strictly pseudocontractive mappings as follows:๐‘ฅ1๐‘ฆ=๐‘ขโˆˆ๐ถchosenarbitrarily,๐‘›=๐›ผ๐‘›๐‘Š๐‘›๐‘ฅ๐‘›+๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐‘ฅ๐‘›,๐œ–๐‘›=๐›ผ๐‘›๎€ท1โˆ’๐›ผ๐‘›๎€ธโ€–โ€–๐‘Š๐‘›๐‘ฅ๐‘›โˆ’๐‘ฅ๐‘›โ€–โ€–2,๐ถ๐‘›+1=๎‚†๐‘งโˆˆ๐ถ๐‘›โˆฉ๐‘„๐‘›โˆถโ€–โ€–๐‘ฆ๐‘›โ€–โ€–โˆ’๐‘ง2โ‰คโ€–โ€–๐‘ฅ๐‘›โ€–โ€–โˆ’๐‘ง2โˆ’๐œ–๐‘›๎‚‡,๐‘„๐‘›+1=๎€ฝ๐‘งโˆˆ๐ถ๐‘›โˆฉ๐‘„๐‘›โˆถโŸจ๐‘ฅ๐‘›โˆ’๐‘ง,๐‘ฅ1โˆ’๐‘ฅ๐‘›๎€พ,๐ถโŸฉโ‰ฅ01=๐‘„1๐‘ฅ=๐ถ,๐‘›+1=๐‘ƒ๐ถ๐‘›+1โˆฉ๐‘„๐‘›+1๎€ท๐‘ฅ1๎€ธ,โˆ€๐‘›โˆˆโ„•,(1.16) and if ๐‘˜=๐›ผ=0 and setting ๐‘‡1โ‰ก๐‘‡, ๐‘‡๐‘›โ‰ก๐ผ for all ๐‘›=2,3,โ€ฆ, then the iterative scheme (1.16) is reduced to find a fixed point of a nonexpansive mapping, for which the convergence rate of the iterative scheme is faster than that of the iterative scheme of Su and Qin [9] as follows:๐‘ฅ1๐‘ฆ=๐‘ขโˆˆ๐ถchosenarbitrarily,๐‘›=๐œŽ๐‘›๐‘‡๐‘ฅ๐‘›+๎€ท1โˆ’๐œŽ๐‘›๎€ธ๐‘ฅ๐‘›,๐›ฟ๐‘›=๐œŽ๐‘›๎€ท1โˆ’๐œŽ๐‘›๎€ธโ€–โ€–๐‘‡๐‘ฅ๐‘›โˆ’๐‘ฅ๐‘›โ€–โ€–2,๐ท๐‘›+1=๎‚†๐‘งโˆˆ๐ท๐‘›โˆฉ๐‘„nโˆถโ€–โ€–๐‘ฆ๐‘›โ€–โ€–โˆ’๐‘ง2โ‰คโ€–โ€–๐‘ฅ๐‘›โ€–โ€–โˆ’๐‘ง2โˆ’๐›ฟ๐‘›๎‚‡,๐‘„๐‘›+1=๎€ฝ๐‘งโˆˆ๐ท๐‘›โˆฉ๐‘„๐‘›โˆถโŸจ๐‘ฅ๐‘›โˆ’๐‘ง,๐‘ฅ1โˆ’๐‘ฅ๐‘›๎€พ,๐ทโŸฉโ‰ฅ01=๐‘„1๐‘ฅ=๐ถ,๐‘›+1=๐‘ƒ๐ท๐‘›+1โˆฉ๐‘„๐‘›+1๎€ท๐‘ฅ1๎€ธ,โˆ€๐‘›โˆˆโ„•.(1.17)

We suggest and analyze the iterative scheme (1.15) under some appropriate conditions imposed on the parameters. The strong convergence theorem for the above two sets is obtained, and the applicability of the results is shown to extend and improve some well-known results existing in the current literature.

2. Preliminaries

We collect the following lemmas which will be used in the proof of the main results in the next section.

Lemma 2.1 (see [11]). Let ๐ป be a Hilbert space. For any ๐‘ฅ,๐‘ฆโˆˆ๐ป and ๐œ†โˆˆโ„, one has โ€–โ€–๐œ†๐‘ฅ+(1โˆ’๐œ†)๐‘ฆ2=๐œ†โ€–๐‘ฅโ€–2+(1โˆ’๐œ†)โ€–๐‘ฆโ€–2โˆ’๐œ†(1โˆ’๐œ†)โ€–๐‘ฅโˆ’๐‘ฆโ€–2.(2.1)

Lemma 2.2 (see [1]). Let ๐ถ be a nonempty closed convex subset of a Hilbert space ๐ป. Then the following inequality holds: โŸจ๐‘ฅโˆ’๐‘ƒ๐ถ๐‘ฅ,๐‘ƒ๐ถ๐‘ฅโˆ’๐‘ฆโŸฉโ‰ฅ0,โˆ€๐‘ฅโˆˆ๐ป,๐‘ฆโˆˆ๐ถ.(2.2)

Lemma 2.3 (see [5]). Let ๐ถ be a nonempty closed convex subset of a Hilbert space ๐ป, define mapping ๐‘Š๐‘› as (1.8), let ๐‘†๐‘–โˆถ๐ถโ†’๐ถ be a family of infinitely nonexpansive mappings with โ‹‚โˆž๐‘–=1๐น(๐‘†๐‘–)โ‰ โˆ…, and let {๐œ‡๐‘–} be a sequence such that 0<๐œ‡๐‘–โ‰ค๐œ‡<1, for all ๐‘–โ‰ฅ1. Then
(1)๐‘Š๐‘› is nonexpansive and ๐น(๐‘Š๐‘›โ‹‚)=๐‘›๐‘–=1๐น(๐‘†๐‘–) for each ๐‘›โ‰ฅ1, (2)for each ๐‘ฅโˆˆ๐ถ and for each positive integer ๐‘˜, lim๐‘›โ†’โˆž๐‘ˆ๐‘›,๐‘˜๐‘ฅ exists, (3)the mapping ๐‘Šโˆถ๐ถโ†’๐ถ defined by ๐‘Š๐‘ฅโˆถ=lim๐‘›โ†’โˆž๐‘Š๐‘›๐‘ฅ=lim๐‘›โ†’โˆž๐‘ˆ๐‘›,1๐‘ฅ,๐‘ฅโˆˆ๐ถ,(2.3) is a nonexpansive mapping satisfying โ‹‚๐น(๐‘Š)=โˆž๐‘–=1๐น(๐‘†๐‘–) and it is called the ๐‘Š-mapping generated by ๐‘†1,๐‘†2,โ€ฆ and ๐œ‡1,๐œ‡2,โ€ฆ.

Lemma 2.4 (see [6]). The resolvent operator ๐ฝ๐‘€,๐œ† associated with ๐‘€ is single valued and nonexpansive for all ๐œ†>0.

Lemma 2.5 (see [6]). ๐‘ขโˆˆ๐ป is a solution of quasivariational inclusion (1.1) if and only if ๐‘ข=๐ฝ๐‘€,๐œ†(๐‘ขโˆ’๐œ†๐ต๐‘ข), for all ๐œ†>0, that is, ๎€ท๐ฝVI(๐ป,๐ต,๐‘€)=๐น๐‘€,๐œ†๎€ธ(๐ผโˆ’๐œ†๐ต),โˆ€๐œ†>0.(2.4)

Lemma 2.6 (see [12]). Let ๐ถ be a nonempty closed convex subset of a strictly convex Banach space ๐‘‹. Let {๐‘‡๐‘›โˆถ๐‘›โˆˆโ„•} be a sequence of nonexpansive mappings on ๐ถ. Suppose that โ‹‚โˆž๐‘›=1๐น(๐‘‡๐‘›)โ‰ โˆ…. Let {๐›ผ๐‘›} be a sequence of positive real numbers such that โˆ‘โˆž๐‘›=1๐›ผ๐‘›=1. Then a mapping ๐‘† on ๐ถ defined by ๐‘†๐‘ฅ=โˆž๎“๐‘›=1๐›ผ๐‘›๐‘‡๐‘›๐‘ฅ,(2.5) for ๐‘ฅโˆˆ๐ถ, is well defined, nonexpansive and โ‹‚๐น(๐‘†)=โˆž๐‘›=1๐น(๐‘‡๐‘›) holds.

Lemma 2.7 (see [13]). Let ๐ถ be a nonempty closed convex subset of a Hilbert space ๐ป and ๐‘†โˆถ๐ถโ†’๐ถ a nonexpansive mapping. Then ๐ผโˆ’๐‘† is demiclosed at zero. That is, whenever {๐‘ฅ๐‘›} is a sequence in ๐ถ weakly converging to some ๐‘ฅโˆˆ๐ถ and the sequence {(๐ผโˆ’๐‘†)๐‘ฅ๐‘›} strongly converges to some ๐‘ฆ, it follows that (๐ผโˆ’๐‘†)๐‘ฅ=๐‘ฆ.

Lemma 2.8 (see [14]). Let ๐ถ be a nonempty closed convex subset of a real Hilbert space ๐ป and ๐‘‡โˆถ๐ถโ†’๐ถ a ๐‘˜-strict pseudocontraction. Define ๐‘†โˆถ๐ถโ†’๐ถ by ๐‘†๐‘ฅ=๐›ผ๐‘ฅ+(1โˆ’๐›ผ)๐‘‡๐‘ฅ for each ๐‘ฅโˆˆ๐ถ. Then, as ๐›ผโˆˆ[๐‘˜,1), S is nonexpansive such that ๐น(๐‘†)=๐น(๐‘‡).

Lemma 2.9 (see [1]). Every Hilbert space ๐ป has Radon-Riesz property or Kadec-Klee property, that is, for a sequence {๐‘ฅ๐‘›}โŠ‚๐ป with ๐‘ฅ๐‘›โ‡€๐‘ฅ and โ€–๐‘ฅ๐‘›โ€–โ†’โ€–๐‘ฅโ€– then ๐‘ฅ๐‘›โ†’๐‘ฅ.

3. Main Results

Theorem 3.1. Let ๐ป be a real Hilbert space, ๐‘€๐‘–โˆถ๐ปโ†’2๐ป a maximal monotone mapping, and ๐ต๐‘–โˆถ๐ปโ†’๐ป a ๐œ‰๐‘–-cocoercive mapping for each ๐‘–=1,2,โ€ฆ,๐‘. Let {๐‘‡๐‘›โˆถ๐ปโ†’๐ป} be an infinite family of ๐‘˜-strictly pseudocontractive mappings with a fixed point such that ๐‘˜โˆˆ[0,1). Define a mapping ๐‘†๐‘›โˆถ๐ปโ†’๐ป by ๐‘†๐‘›๐‘ฅ=๐›ผ๐‘ฅ+(1โˆ’๐›ผ)๐‘‡๐‘›๐‘ฅ,โˆ€๐‘ฅโˆˆ๐ป,(3.1) for all ๐‘›โˆˆโ„•, where ๐›ผโˆˆ[๐‘˜,1). Let ๐‘Š๐‘›โˆถ๐ปโ†’๐ป be a ๐‘Š-mapping generated by {๐‘†๐‘›} and {๐œ‡๐‘›} such that {๐œ‡๐‘›}โŠ‚(0,๐œ‡], for some ๐œ‡โˆˆ(0,1). Assume that โ‹‚ฮฉโˆถ=(โˆž๐‘›=1๐น(๐‘‡๐‘›โ‹‚))โˆฉ(๐‘๐‘–=1VI(๐ป,๐ต๐‘–,๐‘€๐‘–))โ‰ โˆ…. For ๐‘ฅ1=๐‘ขโˆˆ๐ป chosen arbitrarily, suppose that {๐‘ฅ๐‘›} is generated iteratively by ๐‘ฆ๐‘›=๐›ผ๐‘›๐‘Š๐‘›๐‘ฅ๐‘›+๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐‘๎“๐‘–=1๐›ฝ๐‘–๐ฝ๐‘€๐‘–,๐œ†๐‘–๎€ท๐‘ฅ๐‘›โˆ’๐œ†๐‘–๐ต๐‘–๐‘ฅ๐‘›๎€ธ,๐œ–๐‘›=๐›ผ๐‘›๎€ท1โˆ’๐›ผ๐‘›๎€ธโ€–โ€–โ€–โ€–๐‘Š๐‘›๐‘ฅ๐‘›โˆ’๐‘๎“๐‘–=1๐›ฝ๐‘–๐ฝ๐‘€๐‘–,๐œ†๐‘–๎€ท๐‘ฅ๐‘›โˆ’๐œ†๐‘–๐ต๐‘–๐‘ฅ๐‘›๎€ธโ€–โ€–โ€–โ€–2,๐ถ๐‘›+1=๎‚†๐‘งโˆˆ๐ถ๐‘›โˆฉ๐‘„๐‘›โˆถโ€–โ€–๐‘ฆ๐‘›โ€–โ€–โˆ’๐‘ง2โ‰คโ€–โ€–๐‘ฅ๐‘›โ€–โ€–โˆ’๐‘ง2โˆ’๐œ–๐‘›๎‚‡,๐‘„๐‘›+1=๎€ฝ๐‘งโˆˆ๐ถ๐‘›โˆฉ๐‘„๐‘›โˆถโŸจ๐‘ฅ๐‘›โˆ’๐‘ง,๐‘ฅ1โˆ’๐‘ฅ๐‘›๎€พ,๐ถโŸฉโ‰ฅ01=๐‘„1๐‘ฅ=๐ป,๐‘›+1=๐‘ƒ๐ถ๐‘›+1โˆฉ๐‘„๐‘›+1๎€ท๐‘ฅ1๎€ธ,โˆ€๐‘›โˆˆโ„•,(3.2) where
(C1) {๐›ผ๐‘›}โŠ‚[๐‘Ž,๐‘] such that 0<๐‘Ž<๐‘<1,
(C2) ๐›ฝ๐‘–โˆˆ(0,1) and ๐œ†๐‘–โˆˆ(0,2๐œ‰๐‘–] for each ๐‘–=1,2,โ€ฆ,๐‘,
(C3) โˆ‘๐‘๐‘–=1๐›ฝ๐‘–=1.
Then, the sequences {๐‘ฅ๐‘›} and {๐‘ฆ๐‘›} converge strongly to ๐‘ค=๐‘ƒฮฉ(๐‘ฅ1).

Proof. For any ๐‘ฅ,๐‘ฆโˆˆ๐ป and for each ๐‘–=1,2,โ€ฆ,๐‘, by the ๐œ‰๐‘–-cocoercivity of ๐ต๐‘–, we have โ€–โ€–๎€ท๐ผโˆ’๐œ†๐‘–๐ต๐‘–๎€ธ๎€ท๐‘ฅโˆ’๐ผโˆ’๐œ†๐‘–๐ต๐‘–๎€ธ๐‘ฆโ€–โ€–2=โ€–โ€–(๐‘ฅโˆ’๐‘ฆ)โˆ’๐œ†๐‘–๎€ท๐ต๐‘–๐‘ฅโˆ’๐ต๐‘–๐‘ฆ๎€ธโ€–โ€–2=โ€–๐‘ฅโˆ’๐‘ฆโ€–2โˆ’2๐œ†๐‘–โŸจ๐‘ฅโˆ’๐‘ฆ,๐ต๐‘–๐‘ฅโˆ’๐ต๐‘–๐‘ฆโŸฉ+๐œ†2๐‘–โ€–โ€–๐ต๐‘–๐‘ฅโˆ’๐ต๐‘–๐‘ฆโ€–โ€–2โ‰คโ€–๐‘ฅโˆ’๐‘ฆโ€–2โˆ’๎€ท2๐œ‰๐‘–โˆ’๐œ†๐‘–๎€ธ๐œ†๐‘–โ€–โ€–๐ต๐‘–๐‘ฅโˆ’๐ต๐‘–๐‘ฆโ€–โ€–2โ‰คโ€–๐‘ฅโˆ’๐‘ฆโ€–2,(3.3) which implies that ๐ผโˆ’๐œ†๐‘–๐ต๐‘– is nonexpansive. Pick ๐‘โˆˆฮฉ. Therefore, by Lemma 2.5, we have ๐‘=๐ฝ๐‘€๐‘–,๐œ†๐‘–๎€ท๐ผโˆ’๐œ†๐‘–๐ต๐‘–๎€ธ๐‘,(3.4) for each ๐‘–=1,2,โ€ฆ,๐‘. Since ๐‘†๐‘›๐‘ฅ=๐›ผ๐‘ฅ+(1โˆ’๐›ผ)๐‘‡๐‘›๐‘ฅ, where ๐›ผโˆˆ[๐‘˜,1) and {๐‘‡๐‘›} is a family of ๐‘˜-strict pseudocontraction, therefore, by Lemma 2.8, we have that ๐‘†๐‘› is nonexpansive and ๐น(๐‘†๐‘›)=๐น(๐‘‡๐‘›). It follows from Lemma 2.3(1) that ๐น(๐‘Š๐‘›โ‹‚)=๐‘›๐‘–=1๐น(๐‘†๐‘–โ‹‚)=๐‘›๐‘–=1๐น(๐‘‡๐‘–), which implies that ๐‘Š๐‘›๐‘=๐‘. Therefore, by (C3), (3.4), Lemma 2.1, and the nonexpansiveness of ๐‘Š๐‘›,๐ฝ๐‘€๐‘–,๐œ†๐‘–, and ๐ผโˆ’๐œ†๐‘–๐ต๐‘–, we have โ€–โ€–๐‘ฆ๐‘›โ€–โ€–โˆ’๐‘2=โ€–โ€–โ€–โ€–๐›ผ๐‘›๐‘Š๐‘›๐‘ฅ๐‘›+๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐‘๎“๐‘–=1๐›ฝ๐‘–๐ฝ๐‘€๐‘–,๐œ†๐‘–๎€ท๐‘ฅ๐‘›โˆ’๐œ†๐‘–๐ต๐‘–๐‘ฅ๐‘›๎€ธโ€–โ€–โ€–โ€–โˆ’๐‘2=โ€–โ€–โ€–โ€–๐›ผ๐‘›๎€ท๐‘Š๐‘›๐‘ฅ๐‘›๎€ธ+๎€ทโˆ’๐‘1โˆ’๐›ผ๐‘›๎€ธ๐‘๎“๐‘–=1๐›ฝ๐‘–๎€ท๐ฝ๐‘€๐‘–,๐œ†๐‘–๎€ท๐‘ฅ๐‘›โˆ’๐œ†๐‘–๐ต๐‘–๐‘ฅ๐‘›๎€ธ๎€ธโ€–โ€–โ€–โ€–โˆ’๐‘2=๐›ผ๐‘›โ€–โ€–๐‘Š๐‘›๐‘ฅ๐‘›โˆ’๐‘Š๐‘›๐‘โ€–โ€–2+๎€ท1โˆ’๐›ผ๐‘›๎€ธโ€–โ€–โ€–โ€–๐‘๎“๐‘–=1๐›ฝ๐‘–๎€ท๐ฝ๐‘€๐‘–,๐œ†๐‘–๎€ท๐‘ฅ๐‘›โˆ’๐œ†๐‘–๐ต๐‘–๐‘ฅ๐‘›๎€ธโˆ’๐ฝ๐‘€๐‘–,๐œ†๐‘–๎€ท๐‘โˆ’๐œ†๐‘–๐ต๐‘–๐‘โ€–โ€–โ€–โ€–๎€ธ๎€ธ2โˆ’๐›ผ๐‘›๎€ท1โˆ’๐›ผ๐‘›๎€ธโ€–โ€–โ€–โ€–๐‘Š๐‘›๐‘ฅ๐‘›โˆ’๐‘๎“๐‘–=1๐›ฝ๐‘–๐ฝ๐‘€๐‘–,๐œ†๐‘–๎€ท๐‘ฅ๐‘›โˆ’๐œ†๐‘–๐ต๐‘–๐‘ฅ๐‘›๎€ธโ€–โ€–โ€–โ€–2โ‰ค๐›ผ๐‘›โ€–โ€–๐‘ฅ๐‘›โ€–โ€–โˆ’๐‘2+๎€ท1โˆ’๐›ผ๐‘›๎€ธ๎ƒฉ๐‘๎“๐‘–=1๐›ฝ๐‘–โ€–โ€–๎€ท๐‘ฅ๐‘›โˆ’๐œ†๐‘–๐ต๐‘–๐‘ฅ๐‘›๎€ธโˆ’๎€ท๐‘โˆ’๐œ†๐‘–๐ต๐‘–๐‘๎€ธโ€–โ€–๎ƒช2โˆ’๐œ–๐‘›โ‰ค๐›ผ๐‘›โ€–โ€–๐‘ฅ๐‘›โ€–โ€–โˆ’๐‘2+๎€ท1โˆ’๐›ผ๐‘›๎€ธโ€–โ€–๐‘ฅ๐‘›โ€–โ€–โˆ’๐‘2โˆ’๐œ–๐‘›=โ€–โ€–๐‘ฅ๐‘›โ€–โ€–โˆ’๐‘2โˆ’๐œ–๐‘›,(3.5) for all ๐‘›โˆˆโ„•. Firstly, we prove that ๐ถ๐‘›โˆฉ๐‘„๐‘› is closed and convex for all ๐‘›โˆˆโ„•. It is obvious that ๐ถ1โˆฉ๐‘„1 is closed and, by mathematical induction, that ๐ถ๐‘›โˆฉ๐‘„๐‘› is closed for all ๐‘›โ‰ฅ2, that is ๐ถ๐‘›โˆฉ๐‘„๐‘› is closed for all ๐‘›โˆˆโ„•. Since โ€–๐‘ฆ๐‘›โˆ’๐‘งโ€–2โ‰คโ€–๐‘ฅ๐‘›โˆ’๐‘งโ€–2โˆ’๐œ–๐‘› is equivalent to โ€–โ€–๐‘ฆ๐‘›โˆ’๐‘ฅ๐‘›โ€–โ€–2+2โŸจ๐‘ฆ๐‘›โˆ’๐‘ฅ๐‘›,๐‘ฅ๐‘›โˆ’zโŸฉ+๐œ–๐‘›โ‰ค0,(3.6) for all ๐‘›โˆˆโ„•, therefore, for any ๐‘ง1,๐‘ง2โˆˆ๐ถ๐‘›+1โˆฉ๐‘„๐‘›+1โŠ‚๐ถ๐‘›โˆฉ๐‘„๐‘› and ๐œ–โˆˆ(0,1), we have โ€–โ€–๐‘ฆ๐‘›โˆ’๐‘ฅ๐‘›โ€–โ€–2๎ซ๐‘ฆ+2๐‘›โˆ’๐‘ฅ๐‘›,๐‘ฅ๐‘›โˆ’๎€ท๐œ–๐‘ง1+(1โˆ’๐œ–)๐‘ง2๎€ธ๎ฌ+๐œ–๐‘›๎‚€โ€–โ€–๐‘ฆ=๐œ–๐‘›โˆ’๐‘ฅ๐‘›โ€–โ€–2+2โŸจ๐‘ฆ๐‘›โˆ’๐‘ฅ๐‘›,๐‘ฅ๐‘›โˆ’๐‘ง1โŸฉ+๐œ–๐‘›๎‚๎‚€โ€–โ€–๐‘ฆ+(1โˆ’๐œ–)๐‘›โˆ’๐‘ฅ๐‘›โ€–โ€–2+2โŸจ๐‘ฆ๐‘›โˆ’๐‘ฅ๐‘›,๐‘ฅ๐‘›โˆ’๐‘ง2โŸฉ+๐œ–๐‘›๎‚โ‰ค0,(3.7) for all ๐‘›โˆˆโ„•, and we have ๎ซ๐‘ฅ๐‘›โˆ’๎€ท๐œ–๐‘ง1+(1โˆ’๐œ–)๐‘ง2๎€ธ,๐‘ฅ1โˆ’๐‘ฅ๐‘›๎ฌ=๐œ–โŸจ๐‘ฅ๐‘›โˆ’๐‘ง1,๐‘ฅ1โˆ’๐‘ฅ๐‘›โŸฉ+(1โˆ’๐œ–)โŸจ๐‘ฅ๐‘›โˆ’๐‘ง2,๐‘ฅ1โˆ’๐‘ฅ๐‘›โŸฉโ‰ฅ0,(3.8) for all ๐‘›โˆˆโ„•. Since ๐ถ1โˆฉ๐‘„1 is convex, and by putting ๐‘›=1 in (3.6), (3.7), and (3.8), we have that ๐ถ2โˆฉ๐‘„2 is convex. Suppose that ๐‘ฅ๐‘˜ is given and ๐ถ๐‘˜โˆฉ๐‘„๐‘˜ is convex for some ๐‘˜โ‰ฅ2. It follows by putting ๐‘›=๐‘˜ in (3.6), (3.7), and (3.8) that ๐ถ๐‘˜+1โˆฉ๐‘„๐‘˜+1 is convex. Therefore, by mathematical induction, we have that ๐ถ๐‘›โˆฉ๐‘„๐‘› is convex for all ๐‘›โ‰ฅ2, that is, ๐ถ๐‘›โˆฉ๐‘„๐‘› is convex for all ๐‘›โˆˆโ„•. Hence, we obtain that ๐ถ๐‘›โˆฉ๐‘„๐‘› is closed and convex for all ๐‘›โˆˆโ„•.
Next, we prove that ฮฉโŠ‚๐ถ๐‘›โˆฉ๐‘„๐‘› for all ๐‘›โˆˆโ„•. It is obvious that ๐‘โˆˆฮฉโŠ‚๐ป=๐ถ1โˆฉ๐‘„1. Therefore, by (3.2) and (3.5), we have ๐‘โˆˆ๐ถ2 and note that ๐‘โˆˆ๐ป=๐‘„2, and so ๐‘โˆˆ๐ถ2โˆฉ๐‘„2. Hence, we have ฮฉโŠ‚๐ถ2โˆฉ๐‘„2. Since ๐ถ2โˆฉ๐‘„2 is a nonempty closed convex subset of ๐ป, there exists a unique element ๐‘ฅ2โˆˆ๐ถ2โˆฉ๐‘„2 such that ๐‘ฅ2=๐‘ƒ๐ถ2โˆฉ๐‘„2(๐‘ฅ1). Suppose that ๐‘ฅ๐‘˜โˆˆ๐ถ๐‘˜โˆฉ๐‘„๐‘˜ is given such that ๐‘ฅ๐‘˜=๐‘ƒ๐ถ๐‘˜โˆฉ๐‘„๐‘˜(๐‘ฅ1), and ๐‘โˆˆฮฉโŠ‚๐ถ๐‘˜โˆฉ๐‘„๐‘˜ for some ๐‘˜โ‰ฅ2. Therefore, by (3.2) and (3.5), we have ๐‘โˆˆ๐ถ๐‘˜+1. Since ๐‘ฅ๐‘˜=๐‘ƒ๐ถ๐‘˜โˆฉ๐‘„๐‘˜(๐‘ฅ1), therefore, by Lemma 2.2, we have โŸจ๐‘ฅ๐‘˜โˆ’๐‘ง,๐‘ฅ1โˆ’๐‘ฅ๐‘˜โŸฉโ‰ฅ0(3.9) for all ๐‘งโˆˆ๐ถ๐‘˜โˆฉ๐‘„๐‘˜. Thus, by (3.2), we have ๐‘โˆˆ๐‘„๐‘˜+1, and so ๐‘โˆˆ๐ถ๐‘˜+1โˆฉ๐‘„๐‘˜+1. Hence, we have ฮฉโŠ‚๐ถ๐‘˜+1โˆฉ๐‘„๐‘˜+1. Since ๐ถ๐‘˜+1โˆฉ๐‘„๐‘˜+1 is a nonempty closed convex subset of ๐ป, there exists a unique element ๐‘ฅ๐‘˜+1โˆˆ๐ถ๐‘˜+1โˆฉ๐‘„๐‘˜+1 such that ๐‘ฅ๐‘˜+1=๐‘ƒ๐ถ๐‘˜+1โˆฉ๐‘„๐‘˜+1(๐‘ฅ1). Therefore, by mathematical induction, we obtain ฮฉโŠ‚๐ถ๐‘›โˆฉ๐‘„๐‘› for all ๐‘›โ‰ฅ2, and so ฮฉโŠ‚๐ถ๐‘›โˆฉ๐‘„๐‘› for all ๐‘›โˆˆโ„•, and we can define ๐‘ฅ๐‘›+1=๐‘ƒ๐ถ๐‘›+1โˆฉ๐‘„๐‘›+1(๐‘ฅ1) for all ๐‘›โˆˆโ„•. Hence, we obtain that the iteration (3.2) is well defined.
Next, we prove that {๐‘ฅ๐‘›} is bounded. Since ๐‘ฅ๐‘›=๐‘ƒ๐ถ๐‘›โˆฉ๐‘„๐‘›(๐‘ฅ1) for all ๐‘›โˆˆโ„•, we have โ€–โ€–๐‘ฅ๐‘›โˆ’๐‘ฅ1โ€–โ€–โ‰คโ€–โ€–๐‘งโˆ’๐‘ฅ1โ€–โ€–,(3.10) for all ๐‘งโˆˆ๐ถ๐‘›โˆฉ๐‘„๐‘›. It follows by ๐‘โˆˆฮฉโŠ‚๐ถ๐‘›โˆฉ๐‘„๐‘› that โ€–๐‘ฅ๐‘›โˆ’๐‘ฅ1โ€–โ‰คโ€–๐‘โˆ’๐‘ฅ1โ€– for all ๐‘›โˆˆโ„•. This implies that {๐‘ฅ๐‘›} is bounded, and so is {๐‘ฆ๐‘›}.
Next, we prove that โ€–๐‘ฆ๐‘›โˆ’๐‘ฅ๐‘›โ€–โ†’0 as ๐‘›โ†’โˆž. Since ๐‘ฅ๐‘›+1=๐‘ƒ๐ถ๐‘›+1โˆฉ๐‘„๐‘›+1(๐‘ฅ1)โˆˆ๐ถ๐‘›+1โˆฉ๐‘„๐‘›+1โŠ‚๐ถ๐‘›โˆฉ๐‘„๐‘›, therefore, by (3.10), we have โ€–๐‘ฅ๐‘›โˆ’๐‘ฅ1โ€–โ‰คโ€–๐‘ฅ๐‘›+1โˆ’๐‘ฅ1โ€– for all ๐‘›โˆˆโ„•. This implies that {โ€–๐‘ฅ๐‘›โˆ’๐‘ฅ1โ€–} is a bounded nondecreasing sequence and there exists the limit of โ€–๐‘ฅ๐‘›โˆ’๐‘ฅ1โ€–, that is, lim๐‘›โ†’โˆžโ€–โ€–๐‘ฅ๐‘›โˆ’๐‘ฅ1โ€–โ€–=๐‘š,(3.11) for some ๐‘šโ‰ฅ0. Since ๐‘ฅ๐‘›+1โˆˆ๐‘„๐‘›+1, therefore, by (3.2), we have ๎ซ๐‘ฅ๐‘›โˆ’๐‘ฅ๐‘›+1,๐‘ฅ1โˆ’x๐‘›๎ฌโ‰ฅ0.(3.12) It follows by (3.12) that โ€–โ€–๐‘ฅ๐‘›โˆ’๐‘ฅ๐‘›+1โ€–โ€–2=โ€–โ€–๎€ท๐‘ฅ๐‘›โˆ’๐‘ฅ1๎€ธ+๎€ท๐‘ฅ1โˆ’๐‘ฅ๐‘›+1๎€ธโ€–โ€–2=โ€–โ€–๐‘ฅ๐‘›โˆ’๐‘ฅ1โ€–โ€–2+2โŸจ๐‘ฅ๐‘›โˆ’๐‘ฅ1,๐‘ฅ1โˆ’๐‘ฅ๐‘›โŸฉ๎ซ๐‘ฅ+2๐‘›โˆ’๐‘ฅ1,๐‘ฅ๐‘›โˆ’๐‘ฅ๐‘›+1๎ฌ+โ€–โ€–๐‘ฅ๐‘›+1โˆ’๐‘ฅ1โ€–โ€–2โ‰คโ€–โ€–๐‘ฅ๐‘›+1โˆ’๐‘ฅ1โ€–โ€–2โˆ’โ€–โ€–๐‘ฅ๐‘›โˆ’๐‘ฅ1โ€–โ€–2.(3.13) Therefore, by (3.11), we have โ€–โ€–๐‘ฅ๐‘›โˆ’๐‘ฅ๐‘›+1โ€–โ€–โŸถ0as๐‘›โŸถโˆž.(3.14) Since ๐‘ฅ๐‘›+1โˆˆ๐ถ๐‘›+1, therefore, by (3.2), we have โ€–โ€–๐‘ฆ๐‘›โˆ’๐‘ฅ๐‘›+1โ€–โ€–2โ‰คโ€–โ€–๐‘ฅ๐‘›โˆ’๐‘ฅ๐‘›+1โ€–โ€–2โˆ’๐œ–๐‘›โ‰คโ€–โ€–๐‘ฅ๐‘›โˆ’๐‘ฅ๐‘›+1โ€–โ€–2.(3.15) It follows by (3.15) that โ€–โ€–๐‘ฆ๐‘›โˆ’๐‘ฅ๐‘›โ€–โ€–โ‰คโ€–โ€–๐‘ฆ๐‘›โˆ’๐‘ฅ๐‘›+1โ€–โ€–+โ€–โ€–๐‘ฅ๐‘›+1โˆ’๐‘ฅ๐‘›โ€–โ€–โ‰คโ€–โ€–๐‘ฅ๐‘›โˆ’๐‘ฅ๐‘›+1โ€–โ€–+โ€–โ€–๐‘ฅ๐‘›+1โˆ’๐‘ฅ๐‘›โ€–โ€–โ€–โ€–๐‘ฅ=2๐‘›โˆ’๐‘ฅ๐‘›+1โ€–โ€–.(3.16) Therefore, by (3.14), we obtain โ€–โ€–๐‘ฆ๐‘›โˆ’๐‘ฅ๐‘›โ€–โ€–โŸถ0as๐‘›โŸถโˆž.(3.17)
Since {๐‘ฅ๐‘›} is bounded, there exists a subsequence {๐‘ฅ๐‘›๐‘–} of {๐‘ฅ๐‘›} which converges weakly to ๐‘ค. Next, we prove that ๐‘คโˆˆฮฉ. Define the sequence of mappings {๐‘„๐‘›โˆถ๐ปโ†’๐ป} and the mapping ๐‘„โˆถ๐ปโ†’๐ป by ๐‘„๐‘›๐‘ฅ=๐›ผ๐‘›๐‘Š๐‘›๎€ท๐‘ฅ+1โˆ’๐›ผ๐‘›๎€ธ๐‘๎“๐‘–=1๐›ฝ๐‘–๐ฝ๐‘€๐‘–,๐œ†๐‘–๎€ท๐ผโˆ’๐œ†๐‘–๐ต๐‘–๎€ธ๐‘ฅ,โˆ€๐‘ฅโˆˆ๐ป,๐‘„๐‘ฅ=lim๐‘›โ†’โˆž๐‘„๐‘›๐‘ฅ,(3.18) for all ๐‘›โˆˆโ„•. Therefore, by (C1) and Lemma 2.3(3), we have ๐‘„๐‘ฅ=๐‘๐‘Š๐‘ฅ+(1โˆ’๐‘)๐‘๎“๐‘–=1๐›ฝ๐‘–๐ฝ๐‘€๐‘–,๐œ†๐‘–๎€ท๐ผโˆ’๐œ†๐‘–๐ต๐‘–๎€ธ๐‘ฅ,โˆ€๐‘ฅโˆˆ๐ป,(3.19) where ๐‘Žโ‰ค๐‘=lim๐‘›โ†’โˆž๐›ผ๐‘›โ‰ค๐‘. From (C3) and Lemma 2.3(3), we have that ๐‘Š and โˆ‘๐‘๐‘–=1๐›ฝ๐‘–๐ฝ๐‘€๐‘–,๐œ†๐‘–(๐ผโˆ’๐œ†๐‘–๐ต๐‘–) are nonexpansive. Therefore, by (C2), (C3), Lemmas 2.3(3), 2.5, 2.6, and 2.8, we have ๎ƒฉ๐น(๐‘„)=๐น(๐‘Š)โˆฉ๐น๐‘๎“๐‘–=1๐›ฝ๐‘–๐ฝ๐‘€๐‘–,๐œ†๐‘–๎€ท๐ผโˆ’๐œ†๐‘–๐ต๐‘–๎€ธ๎ƒช=๎ƒฉโˆž๎™๐‘–=1๐น๎€ท๐‘†๐‘–๎€ธ๎ƒชโˆฉ๎ƒฉ๐‘๎™๐‘–=1๐น๎€ท๐ฝ๐‘€๐‘–,๐œ†๐‘–๎€ท๐ผโˆ’๐œ†๐‘–๐ต๐‘–๎ƒช=๎ƒฉ๎€ธ๎€ธโˆž๎™๐‘–=1๐น๎€ท๐‘‡๐‘–๎€ธ๎ƒชโˆฉ๎ƒฉ๐‘๎™๐‘–=1๎€ทVI๐ป,๐ต๐‘–,๐‘€๐‘–๎€ธ๎ƒช,(3.20) that is, ๐น(๐‘„)=ฮฉ. From (3.17), we have โ€–๐‘ฆ๐‘›๐‘–โˆ’๐‘ฅ๐‘›๐‘–โ€–โ†’0as๐‘–โ†’โˆž. Thus, from (3.2) and (3.18), we get โ€–๐‘„๐‘ฅ๐‘›๐‘–โˆ’๐‘ฅ๐‘›๐‘–โ€–โ†’0as๐‘–โ†’โˆž. It follows from ๐‘ฅ๐‘›๐‘–โ‡€๐‘ค and Lemma 2.7 that ๐‘คโˆˆ๐น(๐‘„), that is, ๐‘คโˆˆฮฉ.
Since ฮฉ is a nonempty closed convex subset of ๐ป, there exists a unique ๐‘คโˆˆฮฉ such that ๐‘ค=๐‘ƒฮฉ(๐‘ฅ1). Next, we prove that ๐‘ฅ๐‘›โ†’๐‘ค as ๐‘›โ†’โˆž. Since ๐‘ค=๐‘ƒฮฉ(๐‘ฅ1), we have โ€–๐‘ฅ1โˆ’๐‘คโ€–โ‰คโ€–๐‘ฅ1โˆ’๐‘งโ€– for all ๐‘งโˆˆฮฉ, and it follows that โ€–โ€–๐‘ฅ1โ€–โ€–โ‰คโ€–โ€–๐‘ฅโˆ’๐‘ค1โˆ’๐‘คโ€–โ€–.(3.21) Since ๐‘คโˆˆฮฉโŠ‚๐ถ๐‘›โˆฉ๐‘„๐‘›, therefore, by (3.10), we have โ€–โ€–๐‘ฅ1โˆ’๐‘ฅ๐‘›โ€–โ€–โ‰คโ€–โ€–๐‘ฅ1โ€–โ€–.โˆ’๐‘ค(3.22) Therefore, by (3.21), (3.22), and the weak lower semicontinuity of norm, we have โ€–โ€–๐‘ฅ1โ€–โ€–โ‰คโ€–โ€–๐‘ฅโˆ’๐‘ค1โˆ’๐‘คโ€–โ€–โ‰คliminf๐‘–โ†’โˆžโ€–โ€–๐‘ฅ1โˆ’๐‘ฅ๐‘›๐‘–โ€–โ€–โ‰คlimsup๐‘–โ†’โˆžโ€–โ€–๐‘ฅ1โˆ’๐‘ฅ๐‘›๐‘–โ€–โ€–โ‰คโ€–โ€–๐‘ฅ1โ€–โ€–.โˆ’๐‘ค(3.23) It follows that โ€–โ€–๐‘ฅ1โ€–โ€–โˆ’๐‘ค=lim๐‘–โ†’โˆžโ€–โ€–๐‘ฅ1โˆ’๐‘ฅ๐‘›๐‘–โ€–โ€–=โ€–โ€–๐‘ฅ1โˆ’๐‘คโ€–โ€–.(3.24) Since ๐‘ฅ๐‘›๐‘–โ‡€๐‘ค as ๐‘–โ†’โˆž, therefore, we have ๎€ท๐‘ฅ1โˆ’๐‘ฅ๐‘›๐‘–๎€ธโ‡€๎€ท๐‘ฅ1โˆ’๐‘ค๎€ธas๐‘–โŸถโˆž.(3.25) Hence, from (3.24), (3.25), the Kadec-Klee property, and the uniqueness of ๐‘ค=๐‘ƒฮฉ(๐‘ฅ1), we obtain ๐‘ฅ๐‘›๐‘–โŸถ๐‘ค=๐‘คas๐‘–โŸถโˆž.(3.26) It follows that {๐‘ฅ๐‘›} converges strongly to ๐‘ค, and so is {๐‘ฆ๐‘›}. This completes the proof.

Remark 3.2. The iteration (3.2) is the difference with some well known results as the following.
(1)The sequence {๐‘ฅ๐‘›} is the projection sequence of ๐‘ฅ1 onto ๐ถ๐‘›โˆฉ๐‘„๐‘› for all ๐‘›โˆˆโ„• such that ๐ถ1โˆฉ๐‘„1โŠƒ๐ถ2โˆฉ๐‘„2โŠƒโ‹ฏโŠƒ๐ถ๐‘›โˆฉ๐‘„๐‘›โŠƒโ‹ฏโŠƒฮฉ.(3.27)(2)The proof of ๐‘คโˆˆฮฉ is simple by the demiclosedness principle because the sequence {๐‘ฆ๐‘›} is a linear nonexpansive mapping form of the mappings ๐‘Š๐‘› and ๐ฝ๐‘€๐‘–,๐œ†๐‘–(๐ผโˆ’๐œ†๐‘–๐ต๐‘–).(3)Solving a common fixed point for an infinite family of strictly pseudocontractive mappings and a system of cocoercive quasivariational inclusions problems by iteration is obtained.

4. Applications

Theorem 4.1. Let ๐ป be a real Hilbert space, ๐‘€๐‘–โˆถ๐ปโ†’2๐ป a maximal monotone mapping, and ๐ต๐‘–โˆถ๐ปโ†’๐ป a ๐œ‰๐‘–-cocoercive mapping for each ๐‘–=1,2,โ€ฆ,๐‘. Let {๐‘‡๐‘›โˆถ๐ปโ†’๐ป} be an infinite family of nonexpansive mappings. Define a mapping ๐‘†๐‘›โˆถ๐ปโ†’๐ป by ๐‘†๐‘›๐‘ฅ=๐›ผ๐‘ฅ+(1โˆ’๐›ผ)๐‘‡๐‘›๐‘ฅ,โˆ€๐‘ฅโˆˆ๐ป,(4.1) for all ๐‘›โˆˆโ„•, where ๐›ผโˆˆ[0,1). Let ๐‘Š๐‘›โˆถ๐ปโ†’๐ป be a ๐‘Š-mapping generated by {๐‘†๐‘›} and {๐œ‡๐‘›} such that {๐œ‡๐‘›}โŠ‚(0,๐œ‡], for some ๐œ‡โˆˆ(0,1). Assume that โ‹‚ฮฉโˆถ=(โˆž๐‘›=1๐น(๐‘‡๐‘›โ‹‚))โˆฉ(๐‘๐‘–=1VI(๐ป,๐ต๐‘–,๐‘€๐‘–))โ‰ โˆ…. For ๐‘ฅ1=๐‘ขโˆˆ๐ป chosen arbitrarily, suppose that {๐‘ฅ๐‘›} is generated iteratively by ๐‘ฆ๐‘›=๐›ผ๐‘›๐‘Š๐‘›๐‘ฅ๐‘›+๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐‘๎“๐‘–=1๐›ฝ๐‘–๐ฝ๐‘€๐‘–,๐œ†๐‘–๎€ท๐‘ฅ๐‘›โˆ’๐œ†๐‘–๐ต๐‘–๐‘ฅ๐‘›๎€ธ,๐œ–๐‘›=๐›ผ๐‘›๎€ท1โˆ’๐›ผ๐‘›๎€ธโ€–โ€–โ€–โ€–๐‘Š๐‘›๐‘ฅ๐‘›โˆ’๐‘๎“๐‘–=1๐›ฝ๐‘–๐ฝ๐‘€๐‘–,๐œ†๐‘–๎€ท๐‘ฅ๐‘›โˆ’๐œ†๐‘–๐ต๐‘–๐‘ฅ๐‘›๎€ธโ€–โ€–โ€–โ€–2,๐ถ๐‘›+1=๎‚†๐‘งโˆˆ๐ถ๐‘›โˆฉ๐‘„๐‘›โˆถโ€–โ€–๐‘ฆ๐‘›โ€–โ€–โˆ’๐‘ง2โ‰คโ€–โ€–๐‘ฅ๐‘›โ€–โ€–โˆ’๐‘ง2โˆ’๐œ–๐‘›๎‚‡,๐‘„๐‘›+1=๎€ฝ๐‘งโˆˆ๐ถ๐‘›โˆฉ๐‘„๐‘›โˆถโŸจ๐‘ฅ๐‘›โˆ’๐‘ง,๐‘ฅ1โˆ’๐‘ฅ๐‘›๎€พ,๐ถโŸฉโ‰ฅ01=๐‘„1๐‘ฅ=๐ป,๐‘›+1=๐‘ƒ๐ถ๐‘›+1โˆฉ๐‘„๐‘›+1๎€ท๐‘ฅ1๎€ธ,โˆ€๐‘›โˆˆโ„•,(4.2) where
(C1) {๐›ผ๐‘›}โŠ‚[๐‘Ž,๐‘] such that 0<๐‘Ž<๐‘<1,
(C2) ๐›ฝ๐‘–โˆˆ(0,1) and ๐œ†๐‘–โˆˆ(0,2๐œ‰๐‘–] for each ๐‘–=1,2,โ€ฆ,๐‘,
(C3) โˆ‘๐‘๐‘–=1๐›ฝ๐‘–=1.
Then the sequences {๐‘ฅ๐‘›} and {๐‘ฆ๐‘›} converge strongly to ๐‘ค=๐‘ƒฮฉ(๐‘ฅ1).

Proof. It is concluded from Theorem 3.1 immediately, by putting ๐‘˜=0.

Theorem 4.2. Let ๐ถ be a nonempty closed convex subset of a real Hilbert space ๐ป and {๐‘‡๐‘›โˆถ๐ถโ†’๐ถ} an infinite family of ๐‘˜-strictly pseudocontractive mappings with a fixed point such that ๐‘˜โˆˆ[0,1). Define a mapping ๐‘†๐‘›โˆถ๐ถโ†’๐ถ by ๐‘†๐‘›๐‘ฅ=๐›ผ๐‘ฅ+(1โˆ’๐›ผ)๐‘‡๐‘›๐‘ฅ,โˆ€๐‘ฅโˆˆ๐ถ,(4.3) for all ๐‘›โˆˆโ„•, where ๐›ผโˆˆ[๐‘˜,1). Let ๐‘Š๐‘›โˆถ๐ถโ†’๐ถ be a ๐‘Š-mapping generated by {๐‘†๐‘›} and {๐œ‡๐‘›} such that {๐œ‡๐‘›}โŠ‚(0,๐œ‡], for some ๐œ‡โˆˆ(0,1). Assume that โ‹‚ฮฉโˆถ=โˆž๐‘›=1๐น(๐‘‡๐‘›)โ‰ โˆ…. For ๐‘ฅ1=๐‘ขโˆˆ๐ถ chosen arbitrarily, suppose that {๐‘ฅ๐‘›} is generated iteratively by ๐‘ฆ๐‘›=๐›ผ๐‘›๐‘Š๐‘›๐‘ฅ๐‘›+๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐‘ฅ๐‘›,๐œ–๐‘›=๐›ผ๐‘›๎€ท1โˆ’๐›ผ๐‘›๎€ธโ€–โ€–๐‘Š๐‘›๐‘ฅ๐‘›โˆ’๐‘ฅ๐‘›โ€–โ€–2,๐ถ๐‘›+1=๎‚†๐‘งโˆˆ๐ถ๐‘›โˆฉ๐‘„๐‘›โˆถโ€–โ€–๐‘ฆ๐‘›โ€–โ€–โˆ’๐‘ง2โ‰คโ€–โ€–๐‘ฅ๐‘›โ€–โ€–โˆ’๐‘ง2โˆ’๐œ–๐‘›๎‚‡,๐‘„๐‘›+1=๎€ฝ๐‘งโˆˆ๐ถ๐‘›โˆฉ๐‘„๐‘›โˆถโŸจ๐‘ฅ๐‘›โˆ’๐‘ง,๐‘ฅ1โˆ’๐‘ฅ๐‘›๎€พ,๐ถโŸฉโ‰ฅ01=๐‘„1๐‘ฅ=๐ถ,๐‘›+1=๐‘ƒ๐ถ๐‘›+1โˆฉ๐‘„๐‘›+1๎€ท๐‘ฅ1๎€ธ,โˆ€๐‘›โˆˆโ„•,(4.4) where {๐›ผ๐‘›}โŠ‚[๐‘Ž,๐‘] such that 0<๐‘Ž<๐‘<1. Then the sequences {๐‘ฅ๐‘›} and {๐‘ฆ๐‘›} converge strongly to ๐‘ค=๐‘ƒฮฉ(๐‘ฅ1).

Proof. It is concluded from Theorem 3.1 immediately, by putting ๐‘€๐‘–โ‰ก๐ต๐‘–โ‰ก0 for all ๐‘–=1,2,โ€ฆ,๐‘.

Theorem 4.3. Let ๐ถ be a nonempty closed convex subset of a real Hilbert space ๐ป and ๐‘‡โˆถ๐ถโ†’๐ถ a nonexpansive mapping. Assume that ๐น(๐‘‡)โ‰ โˆ…. For ๐‘ฅ1=๐‘ขโˆˆ๐ถ chosen arbitrarily, suppose that {๐‘ฅ๐‘›} is generated iteratively by ๐‘ฆ๐‘›=๐œŽ๐‘›๐‘‡๐‘ฅ๐‘›+๎€ท1โˆ’๐œŽ๐‘›๎€ธ๐‘ฅ๐‘›๐›ฟ๐‘›=๐œŽ๐‘›๎€ท1โˆ’๐œŽ๐‘›๎€ธโ€–โ€–๐‘‡๐‘ฅ๐‘›โˆ’๐‘ฅ๐‘›โ€–โ€–2,๐ท๐‘›+1=๎‚†๐‘งโˆˆ๐ท๐‘›โˆฉ๐‘„๐‘›โˆถโ€–โ€–๐‘ฆ๐‘›โ€–โ€–โˆ’๐‘ง2โ‰คโ€–โ€–๐‘ฅ๐‘›โ€–โ€–โˆ’๐‘ง2โˆ’๐›ฟ๐‘›๎‚‡,๐‘„๐‘›+1=๎€ฝ๐‘งโˆˆ๐ท๐‘›โˆฉ๐‘„๐‘›โˆถโŸจ๐‘ฅ๐‘›โˆ’๐‘ง,๐‘ฅ1โˆ’๐‘ฅ๐‘›๎€พ,๐ทโŸฉโ‰ฅ01=๐‘„1๐‘ฅ=๐ถ,๐‘›+1=๐‘ƒ๐ท๐‘›+1โˆฉ๐‘„๐‘›+1๎€ท๐‘ฅ1๎€ธ,โˆ€๐‘›โˆˆโ„•,(4.5) where {๐œŽ๐‘›}โŠ‚[๐‘Ž,๐‘] such that 0<๐‘Ž<๐‘<1. Then the sequences {x๐‘›} and {๐‘ฆ๐‘›} converge strongly to ๐‘ค=๐‘ƒ๐น(๐‘‡)(๐‘ฅ1).

Proof. It is concluded from Theorem 4.2, by putting ๐›ผ=0. Setting ๐‘‡1โ‰ก๐‘‡, ๐‘‡๐‘›โ‰ก๐ผ for all ๐‘›=2,3,โ€ฆ and leting ๐œ‡๐‘›โŠ‚(0,๐œ‡] for some ๐œ‡โˆˆ(0,1), therefore, from the definition of ๐‘†๐‘› in Theorem 4.2, we have ๐‘†1=๐‘‡1=๐‘‡ and ๐‘†๐‘›=๐ผ for all ๐‘›=2,3,โ€ฆ. Since ๐‘Š๐‘› is a ๐‘Š-mapping generated by {๐‘†๐‘›} and {๐œ‡๐‘›}, therefore, by the definition of ๐‘ˆ๐‘›,๐‘– and ๐‘Š๐‘› in (1.8), we have ๐‘ˆ๐‘›,๐‘–=๐ผ for all ๐‘–=2,3,โ€ฆ and ๐‘Š๐‘›=๐‘ˆ๐‘›,1=๐œ‡1๐‘†1๐‘ˆ๐‘›,2+(1โˆ’๐œ‡1)๐ผ=๐œ‡1๐‘‡+(1โˆ’๐œ‡1)๐ผ. Hence, by Theorem 4.2, we obtain ๐‘ฆ๐‘›=๐›ผ๐‘›๐‘Š๐‘›๐‘ฅ๐‘›+๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐‘ฅ๐‘›=๐›ผ๐‘›๎€ท๐œ‡1๐‘‡๐‘ฅ๐‘›+๎€ท1โˆ’๐œ‡1๎€ธ๐‘ฅ๐‘›๎€ธ+๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐‘ฅ๐‘›=๐œŽ๐‘›๐‘‡๐‘ฅ๐‘›+๎€ท1โˆ’๐œŽ๐‘›๎€ธ๐‘ฅ๐‘›,(4.6) where ๐œŽ๐‘›โˆถ=๐›ผ๐‘›๐œ‡1. Since, the same as in the proof of Theorem 3.1, we have that ๐ท๐‘›โˆฉ๐‘„๐‘› is a nonempty closed convex subset of ๐ถ for all ๐‘›โˆˆโ„• and by Theorem 4.2, we have ๐œ–๐‘›=๐›ผ๐‘›๎€ท1โˆ’๐›ผ๐‘›๎€ธโ€–โ€–๐‘Š๐‘›๐‘ฅ๐‘›โˆ’๐‘ฅ๐‘›โ€–โ€–2=๐›ผ๐‘›๎€ท1โˆ’๐›ผ๐‘›๎€ธโ€–โ€–๐œ‡1๐‘‡๐‘ฅ๐‘›+(1โˆ’๐œ‡1)๐‘ฅ๐‘›โˆ’๐‘ฅ๐‘›โ€–โ€–2=๎€ท๐›ผ๐‘›๐œ‡1๐œ‡๎€ธ๎€ท1โˆ’๐œ‡1๐›ผ๐‘›๎€ธโ€–โ€–๐‘‡๐‘ฅ๐‘›โˆ’๐‘ฅ๐‘›โ€–โ€–2=๐œŽ๐‘›๎€ท๐œ‡1โˆ’๐œŽ๐‘›๎€ธโ€–โ€–๐‘‡๐‘ฅ๐‘›โˆ’๐‘ฅ๐‘›โ€–โ€–2โ‰ค๐œŽ๐‘›๎€ท1โˆ’๐œŽ๐‘›๎€ธโ€–โ€–๐‘‡๐‘ฅ๐‘›โˆ’๐‘ฅ๐‘›โ€–โ€–2=๐›ฟ๐‘›,(4.7) for all ๐‘›โˆˆโ„•. It follows that ๐ท๐‘›โŠ‚๐ถ๐‘› for all ๐‘›โˆˆโ„•, where ๐ถ๐‘› is defined as in Theorem 4.2. Hence, by Theorem 4.2, we obtain the desired result. This completes the proof.

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Copyright © 2011 Pattanapong Tianchai. 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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