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Journal of Applied Mathematics
Volumeย 2012ย (2012), Article IDย 787419, 13 pages
http://dx.doi.org/10.1155/2012/787419
Research Article

Iterative Algorithm for Common Fixed Points of Infinite Family of Nonexpansive Mappings in Banach Spaces

College of Science, Civil Aviation University of China, Tianjin 300300, China

Received 9 January 2012; Accepted 18 January 2012

Academic Editor: Yonghongย Yao

Copyright ยฉ 2012 Songnian He and Jun Guo. 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

Let ๐ถ be a nonempty closed convex subset of a real uniformly smooth Banach space ๐‘‹, {๐‘‡๐‘˜}โˆž๐‘˜=1โˆถ๐ถโ†’๐ถ an infinite family of nonexpansive mappings with the nonempty set of common fixed points โ‹‚โˆž๐‘˜=1Fix(๐‘‡๐‘˜), and ๐‘“โˆถ๐ถโ†’๐ถ a contraction. We introduce an explicit iterative algorithm ๐‘ฅ๐‘›+1=๐›ผ๐‘›๐‘“(๐‘ฅ๐‘›)+(1โˆ’๐›ผ๐‘›)๐ฟ๐‘›๐‘ฅ๐‘›, where ๐ฟ๐‘›=โˆ‘๐‘›๐‘˜=1๎€ท๐œ”๐‘˜/s๐‘›๎€ธ๐‘‡๐‘˜,๐‘†๐‘›=โˆ‘๐‘›๐‘˜=1๐œ”๐‘˜, and ๐‘ค๐‘˜>0 with โˆ‘โˆž๐‘˜=1๐œ”๐‘˜=1. Under certain appropriate conditions on {๐›ผ๐‘›}, we prove that {๐‘ฅ๐‘›} converges strongly to a common fixed point ๐‘ฅโˆ— of {๐‘‡๐‘˜}โˆž๐‘˜=1, which solves the following variational inequality: โŸจ๐‘ฅโˆ—โˆ’๐‘“(๐‘ฅโˆ—),๐ฝ(๐‘ฅโˆ—โ‹‚โˆ’๐‘)โŸฉโ‰ค0,๐‘โˆˆโˆž๐‘˜=1Fix(๐‘‡๐‘˜), where ๐ฝ is the (normalized) duality mapping of ๐‘‹. This algorithm is brief and needs less computational work, since it does not involve ๐‘Š-mapping.

1. Introduction

Let ๐‘‹ be a real Banach space, ๐ถ a nonempty closed convex subset of ๐‘‹, and ๐‘‹โˆ— the dual space of ๐‘‹. The (normalized) duality mapping ๐ฝโˆถ๐‘‹โ†’2๐‘‹โˆ— is defined by ๐ฝ๎€ฝ๐‘ฅ(๐‘ฅ)=โˆ—โˆˆ๐‘‹โˆ—โˆถโŸจ๐‘ฅ,๐‘ฅโˆ—โŸฉ=โ€–๐‘ฅโ€–2,โ€–๐‘ฅโˆ—๎€พโ€–=โ€–๐‘ฅโ€–,โˆ€๐‘ฅโˆˆ๐‘‹.(1.1) If ๐‘‹ is a Hilbert space, then ๐ฝ=๐ผ, where ๐ผ is the identity mapping. It is well known that if ๐‘‹ is smooth, then ๐ฝ is single valued.

Recall that a mapping ๐‘“โˆถ๐ถโ†’๐ถ is a contraction, 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€–,โˆ€๐‘ฅ,๐‘ฆโˆˆ๐ถ.(1.2) We use ฮ ๐ถ to denote the collection of all contractions on ๐ถ, that is, ฮ ๐ถ={๐‘“โˆถ๐‘“isacontractionon๐ถ}.(1.3) 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€–,โˆ€๐‘ฅ,๐‘ฆโˆˆ๐ถ.(1.4) We use Fix(๐‘‡) to denote the set of fixed points of ๐‘‡, namely, Fix(๐‘‡)={๐‘ฅโˆˆ๐ถโˆถ๐‘‡๐‘ฅ=๐‘ฅ}. One classical way to study nonexpansive mappings is to use contractions to approximate a nonexpansive mapping ([1โ€“11]). Browder [1] first considered the following approximation in a Hilbert space. Fix ๐‘ขโˆˆ๐ถ and define a contraction ๐น๐‘ก from ๐ถ into itself by๐น๐‘ก๐‘ฅ=๐‘ก๐‘ข+(1โˆ’๐‘ก)๐‘‡๐‘ฅ,๐‘ฅโˆˆ๐ถ,(1.5) where ๐‘กโˆˆ(0,1). Banach contraction mapping principle guarantees that ๐น๐‘ก has a unique fixed point in ๐ถ. Denote by ๐‘ง๐‘กโˆˆ๐ถ the unique fixed point of ๐น๐‘ก, that is,๐‘ง๐‘ก=๐‘ก๐‘ข+(1โˆ’๐‘ก)๐‘‡๐‘ง๐‘ก.(1.6) In the case of ๐‘‡ having fixed points, Browder [1] proved the following.

Theorem 1.1. In a Hilbert space, as ๐‘กโ†’0, ๐‘ง๐‘ก defined in (1.6) converges strongly to a fixed point of ๐‘‡ that is closest to ๐‘ข, that is, the nearest point projection of ๐‘ข onto Fix(๐‘‡).

Halpern [3] introduced an iteration process (discretization of (1.6)) in a Hilbert as follows:๐‘ง๐‘›+1=๐›ผ๐‘›๎€ท๐‘ข+1โˆ’๐›ผ๐‘›๎€ธ๐‘‡๐‘ง๐‘›,๐‘›โ‰ฅ0,(1.7) where ๐‘ข,๐‘ง0โˆˆ๐ถ are arbitrary (but fixed) and {๐›ผ๐‘›} is a sequence in (0,1). Lions [4] proved the following.

Theorem 1.2. In a Hilbert space, if {๐›ผ๐‘›} satisfies the following conditions: (K1)lim๐‘›โ†’โˆž๐›ผ๐‘›=0;(K2)โˆ‘โˆž๐‘›=0๐›ผ๐‘›=โˆž;(K3)lim๐‘›โ†’โˆž|๐›ผ๐‘›โˆ’๐›ผ๐‘›โˆ’1|/๐›ผ2๐‘›+1=0. Then {๐‘ง๐‘›} converges strongly to the nearest point projection of ๐‘ข onto Fix(๐‘‡).

The Banach space versions of Theorems 1.1 and 1.2 were obtained by Reich [5]. He proved the following.

Theorem 1.3. In a uniformly smooth Banach space ๐‘‹, both ๐‘ง๐‘ก defined in (1.6) and {๐‘ง๐‘›} defined in (1.7) converge strongly to a same fixed point of T. If one defines ๐‘„โˆถ๐ถโ†’Fix(๐‘‡) by ๐‘„(๐‘ข)=lim๐‘กโ†’0๐‘ง๐‘ก,(1.8) then ๐‘„ is the sunny nonexpansive retraction from ๐ถ onto Fix(๐‘‡). Namely, Q satisfies the property: โ€–๐‘„๐‘ฅโˆ’๐‘„๐‘ฆโ€–2โ‰คโŸจ๐‘ฅโˆ’๐‘ฆ,๐ฝ(๐‘„๐‘ฅโˆ’๐‘„๐‘ฆ)โŸฉ,๐‘ฅ,๐‘ฆโˆˆ๐ถ,(1.9) where ๐ฝ is the duality mapping of ๐‘‹.

Moudafi [6] introduced a viscosity approximation method and proved the strong convergence of both the implicit and explicit methods in Hilbert spaces. Xu [7] extended Moudafi's results in Hilbert spaces. Given a real number ๐‘กโˆˆ(0,1) and a contraction ๐‘“โˆˆฮ ๐ถ, define a contraction ๐‘‡๐‘“๐‘กโˆถ๐ถโ†’๐ถ by๐‘‡๐‘“๐‘ก๐‘ฅ=๐‘ก๐‘“(๐‘ฅ)+(1โˆ’๐‘ก)๐‘‡๐‘ฅ,๐‘ฅโˆˆ๐ถ.(1.10) Let ๐‘ฅ๐‘กโˆˆ๐ถ be the unique fixed point of ๐‘‡๐‘“๐‘ก. Thus,๐‘ฅ๐‘ก๎€ท๐‘ฅ=๐‘ก๐‘“๐‘ก๎€ธ+(1โˆ’๐‘ก)๐‘‡๐‘ฅ๐‘ก.(1.11) Corresponding explicit iterative process is defined by๐‘ฅ๐‘›+1=๐›ผ๐‘›๐‘“๎€ท๐‘ฅ๐‘›๎€ธ+๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐‘‡๐‘ฅ๐‘›,(1.12) where ๐‘ฅ0โˆˆ๐ถ is arbitrary (but fixed) and {๐›ผ๐‘›} is a sequence in (0,1). It was proved by Xu [7] that under certain appropriate conditions on {๐›ผ๐‘›}, both ๐‘ฅ๐‘ก defined in (1.11) and {๐‘ฅ๐‘›} defined in (1.12) converged strongly to ๐‘ฅโˆ—โˆˆ๐ถ, which is the unique solution of the variational inequality: โŸจ(๐ผโˆ’๐‘“)๐‘ฅโˆ—,๐‘ฅโˆ’๐‘ฅโˆ—โŸฉโ‰ฅ0,๐‘ฅโˆˆFix(๐‘‡).(1.13) Xu [7] also extended Moudafi's results to the setting of Banach spaces and proved the strong convergence of both the implicit method (1.11) and explicit method (1.12) in uniformly smooth Banach spaces.

In order to deal with some problems involving the common fixed points of infinite family of nonexpansive mappings, ๐‘Š-mapping is often used, see [12โ€“20]. Let {๐‘‡๐‘˜}โˆž๐‘˜=1โˆถ๐ถโ†’๐ถ be an infinite family of nonexpansive mappings and let {๐œ‰๐‘˜}โˆž๐‘˜=1 be a real number sequence such that 0<๐œ‰๐‘˜<1 for every ๐‘˜โˆˆโ„•. For any ๐‘›โˆˆโ„•, we define a mapping ๐‘Š๐‘› of ๐ถ into itself as follows:๐‘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.(1.14) Such ๐‘Š๐‘› is called the ๐‘Š-mapping generated by {๐‘‡๐‘˜}โˆž๐‘˜=1 and {๐œ‰๐‘˜}โˆž๐‘˜=1, see [12, 13].

Yao et al. [10] introduced the following iterative algorithm for infinite family of nonexpansive mappings. Let ๐‘‹ be a uniformly convex Banach space with a uniformly Gรขteaux differentiable norm and ๐ถ a nonempty closed convex subset of ๐‘‹. Sequence {๐‘ฅ๐‘›} is defined by๐‘ฅ๐‘›+1=๐›ผ๐‘›๎€ท๐‘ข+1โˆ’๐›ผ๐‘›๎€ธ๐‘Š๐‘›๐‘ฅ๐‘›,๐‘›โ‰ฅ0,(1.15) where ๐‘ข,๐‘ฅ0โˆˆ๐ถ are arbitrary (but fixed) and {๐›ผ๐‘›}โŠ‚(0,1). It was proved that under certain appropriate conditions on {๐›ผ๐‘›}, the sequence {๐‘ฅ๐‘›} generated by (1.15) converges strongly to a common fixed point of {๐‘‡๐‘˜}โˆž๐‘˜=1 [13].

Since ๐‘Š-mapping contains many composite operations of {๐‘‡๐‘˜}, it is complicated and needs large computational work. In this paper, we introduce a new iterative algorithm for solving the common fixed point problem of infinite family of nonexpansive mappings. Let ๐‘‹ be a real uniformly smooth Banach space, ๐ถ a nonempty closed convex subset of ๐‘‹, {๐‘‡๐‘˜}โˆž๐‘˜=1โˆถ๐ถโ†’๐ถ an infinite family of nonexpansive mappings with the nonempty set of common fixed points โ‹‚โˆž๐‘˜=1Fix(๐‘‡๐‘˜), and ๐‘“โˆˆฮ ๐ถ. Given any ๐‘ฅ0โˆˆ๐ถ, define a sequence {๐‘ฅ๐‘›} by๐‘ฅ๐‘›+1=๐›ผ๐‘›๐‘“๎€ท๐‘ฅ๐‘›๎€ธ+๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐ฟ๐‘›๐‘ฅ๐‘›,๐‘›โ‰ฅ0,(1.16) where {๐›ผ๐‘›}โŠ‚(0,1), ๐ฟ๐‘›=โˆ‘๐‘›๐‘˜=1(๐œ”๐‘˜/๐‘ ๐‘›)๐‘‡๐‘˜, ๐‘†๐‘›=โˆ‘๐‘›๐‘˜=1๐œ”๐‘˜ and ๐‘ค๐‘˜>0 with โˆ‘โˆž๐‘˜=1๐œ”๐‘˜=1. Under certain appropriate conditions on {๐›ผ๐‘›}, we prove that {๐‘ฅ๐‘›} converges strongly to ๐‘ฅโˆ—โˆˆโ‹‚โˆž๐‘˜=1Fix(๐‘‡๐‘˜), which solves the following variational inequality: โŸจ๐‘ฅโˆ—๎€ท๐‘ฅโˆ’๐‘“โˆ—๎€ธ๎€ท๐‘ฅ,๐ฝโˆ—๎€ธโˆ’๐‘โŸฉโ‰ค0,๐‘โˆˆโˆž๎™๐‘˜=1๎€ท๐‘‡Fix๐‘˜๎€ธ,(1.17) where ๐ฝ is the duality mapping of ๐‘‹. Because ๐ฟ๐‘› doesn't contain many composite operations of {๐‘‡๐‘˜}, this algorithm is brief and needs less computational wok.

We will use ๐‘€ to denote a constant, which may be different in different places.

2. Preliminaries

Let ๐ต={๐‘ฅโˆˆ๐‘‹โˆถโ€–๐‘ฅโ€–=1} denotes the unit sphere of ๐‘‹. A Banach space ๐‘‹ is said to be strictly convex, if โ€–(๐‘ฅ+๐‘ฆ)/2โ€–<1 holds for all ๐‘ฅ,๐‘ฆโˆˆ๐ต, ๐‘ฅโ‰ ๐‘ฆ. A Banach space ๐‘‹ is said to be uniformly convex if for each ๐œ€โˆˆ(0,2], there exists a constant ๐›ฟ>0 such that for any x,๐‘ฆโˆˆ๐ต, โ€–๐‘ฅโˆ’๐‘ฆโ€–โ‰ฅ๐œ€ implies โ€–(๐‘ฅ+๐‘ฆ)/2โ€–โ‰ค1โˆ’๐›ฟ. It is known that a uniformly convex Banach space is reflexive and strictly convex, see [21].

The norm of ๐‘‹ is said to be Gรขteaux differentiable iflim๐‘กโ†’0โ€–๐‘ฅ+๐‘ก๐‘ฆโ€–โˆ’โ€–๐‘ฅโ€–๐‘ก(2.1) exists for each ๐‘ฅ,๐‘ฆโˆˆ๐ต and in this case ๐‘‹ is said to be smooth. The norm of ๐‘‹ is said to be uniformly Gรขteaux differentiable if for each ๐‘ฆโˆˆ๐ต, the limit (2.1) is attained uniformly for ๐‘ฅโˆˆ๐ต. The norm of ๐‘‹ is said to be Frรชchet differentiable, if for each ๐‘ฅโˆˆ๐ต, the limit (2.1) is attained uniformly for ๐‘ฆโˆˆ๐ต. The norm of ๐‘‹ is said to be uniformly Frรชchet differentiable, if the limit (2.1) is attained uniformly for ๐‘ฅ,๐‘ฆโˆˆ๐ต and in this case ๐‘‹ is said to be uniformly smooth.

Let ๐ท be a nonempty subset of ๐ถ. A mapping ๐‘„โˆถ๐ถโ†’๐ท is said to be sunny [22] if๐‘„(๐‘ฅ+๐‘ก(๐‘ฅโˆ’๐‘„(๐‘ฅ)))=๐‘„(๐‘ฅ),โˆ€๐‘ฅโˆˆ๐ถ,๐‘กโ‰ฅ0,(2.2) whenever ๐‘ฅ+๐‘ก(๐‘ฅโˆ’๐‘„(๐‘ฅ))โˆˆ๐ถ. A mapping ๐‘„โˆถ๐ถโ†’๐ท is called a retraction if ๐‘„๐‘ฅ=๐‘ฅ for all ๐‘ฅโˆˆ๐ท. Furthermore, ๐‘„ is sunny nonexpansive retraction from ๐ถ onto ๐ท if ๐‘„ is a retraction from ๐ถ onto ๐ท which is also sunny and nonexpansive.

A subset ๐ท of ๐ถ is called a sunny nonexpansive retraction of ๐ถ if there exits a sunny nonexpansive retraction from ๐ถ onto ๐ท.

Lemma 2.1 (see [22]). Let ๐ถ be a closed convex subset of a smooth Banach space ๐‘‹. Let ๐ท be a nonempty subset of ๐ถ and ๐‘„โˆถ๐ถโ†’๐ท be a retraction. Then the following are equivalent.(a)๐‘„ is sunny and nonexpansive.(b)โ€–๐‘„๐‘ฅโˆ’๐‘„๐‘ฆโ€–2โ‰คโŸจ๐‘ฅโˆ’๐‘ฆ,๐ฝ(๐‘„๐‘ฅโˆ’๐‘„๐‘ฆ)โŸฉ, for all ๐‘ฅ,๐‘ฆโˆˆ๐ถ.(c)โŸจ๐‘ฅโˆ’๐‘„๐‘ฅ,๐ฝ(๐‘ฆโˆ’๐‘„๐‘ฅ)โŸฉโ‰ค0, for all ๐‘ฅโˆˆ๐ถ,๐‘ฆโˆˆ๐ท.

Lemma 2.2 (see [23]). Let {๐‘ ๐‘›} be a sequence of nonnegative real numbers satisfying ๐‘ ๐‘›+1โ‰ค๎€ท1โˆ’๐›พ๐‘›๎€ธ๐‘ ๐‘›+๐›พ๐‘›๐›ฝ๐‘›+๐›ฟ๐‘›,๐‘›โ‰ฅ0,(2.3) where {๐›พ๐‘›}โŠ‚(0,1), {๐›ฝ๐‘›} and {๐›ฟ๐‘›} satisfy the following conditions: (A1)โˆ‘โˆž๐‘›=0๐›พ๐‘›=โˆž;(A2)limsup๐‘›โ†’โˆž๐›ฝ๐‘›โ‰ค0;(A3)๐›ฟ๐‘›โ‰ฅ0 (๐‘›โ‰ฅ0), โˆ‘โˆž๐‘›=1๐›ฟ๐‘›<โˆž. Then lim๐‘›โ†’โˆž๐‘ ๐‘›=0.

Lemma 2.3 (see [24]). In a Banach space ๐‘‹, the following inequality holds: โ€–๐‘ฅ+๐‘ฆโ€–2โ‰คโ€–๐‘ฅโ€–2+2โŸจ๐‘ฆ,๐‘—(๐‘ฅ+๐‘ฆ)โŸฉ,โˆ€๐‘ฅ,๐‘ฆโˆˆ๐‘‹,(2.4) where ๐‘—(๐‘ฅ+๐‘ฆ)โˆˆ๐ฝ(๐‘ฅ+๐‘ฆ).

Lemma 2.4 (see [25]). Let ๐ถ be a 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 โ‹‚โˆž๐‘›=1Fix(๐‘‡๐‘›) is nonempty. Let {๐œ†๐‘›} be a sequence of positive numbers with โˆ‘โˆž๐‘›=1๐œ†๐‘›=1. Then a mapping ๐‘† on ๐ถ defined by โˆ‘๐‘†๐‘ฅ=โˆž๐‘›=1๐œ†๐‘›๐‘‡๐‘›๐‘ฅ for ๐‘ฅโˆˆ๐ถ is well defined, nonexpansive and โ‹‚Fix(๐‘†)=โˆž๐‘›=1Fix(๐‘‡๐‘›) holds.

Lemma 2.5 (see [7]). Let ๐‘‹ be a uniformly smooth Banach space, ๐ถ a closed convex subset of ๐‘‹, ๐‘‡โˆถ๐ถโ†’๐ถ a nonexpansive mapping with Fix(๐‘‡)โ‰ โˆ…, and ๐‘“โˆˆฮ ๐ถ. Then {๐‘ฅ๐‘ก} defined by ๐‘ฅ๐‘ก๎€ท๐‘ฅ=๐‘ก๐‘“๐‘ก๎€ธ+(1โˆ’๐‘ก)๐‘‡๐‘ฅ๐‘ก(2.5) converges strongly to a point in Fix(๐‘‡). If we define a mapping ๐‘„โˆถฮ ๐ถโ†’Fix(๐‘‡) by ๐‘„(๐‘“)โˆถ=lim๐‘กโ†’0๐‘ฅ๐‘ก,๐‘“โˆˆฮ ๐ถ,(2.6) then ๐‘„(๐‘“) solves the variational inequality: โŸจ(๐ผโˆ’๐‘“)๐‘„(๐‘“),๐ฝ(๐‘„(๐‘“)โˆ’๐‘)โŸฉโ‰ค0,๐‘“โˆˆฮ ๐ถ,๐‘โˆˆFix(๐‘‡).(2.7)

Lemma 2.6. Let ๐‘‹ be a Banach space, {๐‘ฅ๐‘˜} a bounded sequence of ๐‘‹, and {๐œ”๐‘˜} a sequence of positive numbers with โˆ‘โˆž๐‘˜=1๐œ”๐‘˜=1. Then โˆ‘โˆž๐‘˜=1๐œ”๐‘˜๐‘ฅ๐‘˜ is convergent in ๐‘‹.

Lemma 2.7. Let X be Banach space, {๐‘‡๐‘˜โˆถ๐‘˜โˆˆโ„•} a sequence of nonexpansive mappings on X with โ‹‚โˆž๐‘˜=1Fix(๐‘‡๐‘˜)โ‰ โˆ…, and {๐œ”๐‘˜} a sequence of positive numbers with โˆ‘โˆž๐‘˜=1๐œ”๐‘˜=1. Let โˆ‘๐‘‡=โˆž๐‘˜=1๐œ”๐‘˜๐‘‡๐‘˜, ๐ฟ๐‘š=โˆ‘๐‘š๐‘˜=1(๐œ”๐‘˜/๐‘†๐‘š)๐‘‡๐‘˜, and ๐‘†๐‘š=โˆ‘๐‘š๐‘˜=1๐œ”๐‘˜. Then ๐ฟ๐‘š uniformly converges to ๐‘‡ in each bounded subset ๐‘† of ๐‘‹.

Proof. Forall๐‘ฅโˆˆ๐‘†, we observe that โ€–โ€–๐ฟ๐‘šโ€–โ€–=โ€–โ€–โ€–โ€–๐‘ฅโˆ’๐‘‡๐‘ฅ๐‘š๎“๐‘˜=1๐œ”๐‘˜๐‘†๐‘š๐‘‡๐‘˜๐‘ฅโˆ’โˆž๎“๐‘˜=1๐œ”๐‘˜๐‘‡๐‘˜๐‘ฅโ€–โ€–โ€–โ€–=โ€–โ€–โ€–โ€–๐‘š๎“๐‘˜=1๐œ”๐‘˜โˆ’๐œ”๐‘˜๐‘†๐‘š๐‘†๐‘š๐‘‡๐‘˜๐‘ฅโˆ’โˆž๎“๐‘˜=๐‘š+1๐œ”๐‘˜๐‘‡๐‘˜๐‘ฅโ€–โ€–โ€–โ€–โ‰คโ€–โ€–โ€–โ€–๐‘š๎“๐‘˜=11โˆ’๐‘†๐‘š๐‘†๐‘š๐œ”๐‘˜๐‘‡๐‘˜๐‘ฅโ€–โ€–โ€–โ€–+โ€–โ€–โ€–โ€–โˆž๎“๐‘˜=๐‘š+1๐œ”๐‘˜๐‘‡๐‘˜๐‘ฅโ€–โ€–โ€–โ€–โ‰ค1โˆ’๐‘†๐‘š๐‘†๐‘š๐‘š๎“๐‘˜=1๐œ”๐‘˜โ€–โ€–๐‘‡๐‘˜๐‘ฅโ€–โ€–+โˆž๎“๐‘˜=๐‘š+1๐œ”๐‘˜โ€–โ€–๐‘‡๐‘˜๐‘ฅโ€–โ€–โ‰ค1โˆ’๐‘†๐‘š๐‘†๐‘š๐‘€+๐‘€โˆž๎“๐‘˜=๐‘š+1๐œ”๐‘˜,(2.8) where ๐‘€=sup๐‘ฅโˆˆ๐‘†,๐‘˜โ‰ฅ1โ€–๐‘‡๐‘˜๐‘ฅโ€–<โˆž. Taking ๐‘šโ†’โˆž in above last inequality, we have that lim๐‘šโ†’โˆžโ€–โ€–๐ฟ๐‘šโ€–โ€–๐‘ฅโˆ’๐‘‡๐‘ฅ=0(2.9) holds uniformly for ๐‘ฅโˆˆ๐‘† and this completes the proof.

3. Main Results

Theorem 3.1. Let ๐ถ be a nonempty closed convex subset of a real uniformly smooth Banach space ๐‘‹, {๐‘‡๐‘˜}โˆž๐‘˜=1โˆถ๐ถโ†’๐ถ an infinite family of nonexpansive mappings with โ‹‚โˆž๐‘˜=1Fix(๐‘‡๐‘˜)โ‰ โˆ…, and {๐œ”๐‘˜} a sequence of positive numbers with โˆ‘โˆž๐‘˜=1๐œ”๐‘˜=1. Let ๐ฟ๐‘›=โˆ‘๐‘›๐‘˜=1(๐œ”๐‘˜/๐‘†๐‘›)๐‘‡๐‘˜, ๐‘†๐‘›=โˆ‘๐‘›๐‘˜=1๐œ”๐‘˜, and ๐‘“โˆˆฮ ๐ถ with coefficient ๐›ผโˆˆ[0,1). Given any ๐‘ฅ0โˆˆ๐ถ, let {๐‘ฅ๐‘›} be a sequence generated by ๐‘ฅ๐‘›+1=๐›ผ๐‘›๐‘“๎€ท๐‘ฅ๐‘›๎€ธ+๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐ฟ๐‘›๐‘ฅ๐‘›,๐‘›โ‰ฅ0,(3.1) where {๐›ผ๐‘›}โŠ‚(0,1) satisfies the following conditions: (A1)lim๐‘›โ†’โˆž๐›ผ๐‘›=0;(A2)โˆ‘โˆž๐‘›=0๐›ผ๐‘›=โˆž;(A3)either โˆ‘โˆž๐‘›=0|๐›ผ๐‘›+1โˆ’๐›ผ๐‘›|<โˆž or lim๐‘›โ†’โˆž(๐›ผ๐‘›+1/๐›ผ๐‘›)=1. Then {๐‘ฅ๐‘›} converges strongly to ๐‘ฅโˆ—โˆˆโ‹‚โˆž๐‘˜=1Fix(๐‘‡๐‘˜), which solves the following variational inequality: โŸจ๐‘ฅโˆ—๎€ท๐‘ฅโˆ’๐‘“โˆ—๎€ธ๎€ท๐‘ฅ,๐ฝโˆ—๎€ธโˆ’๐‘โŸฉโ‰ค0,๐‘โˆˆโˆž๎™๐‘˜=1๎€ท๐‘‡Fix๐‘˜๎€ธ.(3.2)

Proof. Step 1. We show that {๐‘ฅ๐‘›} is bounded.
Noticing nonexpansiveness of ๐ฟ๐‘›, take a โ‹‚๐‘โˆˆโˆž๐‘˜=1Fix(๐‘‡๐‘˜) to derive that โ€–โ€–๐‘ฅ๐‘›+1โ€–โ€–=โ€–โ€–๐›ผโˆ’p๐‘›๐‘“๎€ท๐‘ฅ๐‘›๎€ธ+๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐ฟ๐‘›๐‘ฅ๐‘›โ€–โ€–โˆ’๐‘โ‰ค๐›ผ๐‘›โ€–โ€–๐‘“๎€ท๐‘ฅ๐‘›๎€ธโ€–โ€–+๎€ทโˆ’๐‘1โˆ’๐›ผ๐‘›๎€ธโ€–โ€–๐ฟ๐‘›๐‘ฅ๐‘›โ€–โ€–โˆ’๐‘=๐›ผ๐‘›โ€–โ€–๐‘“๎€ท๐‘ฅ๐‘›๎€ธโ€–โ€–+๎€ทโˆ’๐‘“(๐‘)+๐‘“(๐‘)โˆ’๐‘1โˆ’๐›ผ๐‘›๎€ธโ€–โ€–๐ฟ๐‘›๐‘ฅ๐‘›โ€–โ€–โˆ’๐‘โ‰ค๐›ผ๐‘›โ€–โ€–๐‘“๎€ท๐‘ฅ๐‘›๎€ธโ€–โ€–โˆ’๐‘“(๐‘)+๐›ผ๐‘›(๎€ทโ€–๐‘“๐‘)โˆ’๐‘โ€–+1โˆ’๐›ผ๐‘›๎€ธโ€–โ€–๐ฟ๐‘›๐‘ฅ๐‘›โ€–โ€–โˆ’๐‘โ‰ค๐›ผ๐›ผ๐‘›โ€–โ€–๐‘ฅ๐‘›โ€–โ€–โˆ’๐‘+๐›ผ๐‘›๎€ทโ€–๐‘“(๐‘)โˆ’๐‘โ€–+1โˆ’๐›ผ๐‘›๎€ธโ€–โ€–๐‘ฅ๐‘›โ€–โ€–=๎€ทโˆ’๐‘1โˆ’(1โˆ’๐›ผ)๐›ผ๐‘›๎€ธโ€–โ€–๐‘ฅ๐‘›โ€–โ€–โˆ’๐‘+๐›ผ๐‘›โ€–(=๎€ท๐‘“(๐‘)โˆ’๐‘)โ€–1โˆ’(1โˆ’๐›ผ)๐›ผ๐‘›๎€ธโ€–โ€–๐‘ฅ๐‘›โ€–โ€–+โˆ’๐‘(1โˆ’๐›ผ)๐›ผ๐‘›โ€–(๐‘“(๐‘)โˆ’๐‘)โ€–๎‚ปโ€–(1โˆ’๐›ผโ‰คmax๐‘“(๐‘)โˆ’๐‘)โ€–,โ€–โ€–๐‘ฅ1โˆ’๐›ผ๐‘›โ€–โ€–๎‚ผ.โˆ’๐‘(3.3) By induction, we obtain โ€–โ€–๐‘ฅ๐‘›+1โ€–โ€–๎‚ปโ€–(โˆ’๐‘โ‰คmax๐‘“(๐‘)โˆ’๐‘)โ€–,โ€–โ€–๐‘ฅ1โˆ’๐›ผ0โ€–โ€–๎‚ผโˆ’๐‘,๐‘›โ‰ฅ0,(3.4) and {๐‘ฅ๐‘›} is bounded, so are {๐‘‡๐‘˜๐‘ฅ๐‘›}, {๐ฟ๐‘›๐‘ฅ๐‘›}, and {๐‘“(๐‘ฅ๐‘›)}.
Step 2. We prove that lim๐‘›โ†’โˆžโ€–๐‘ฅ๐‘›+1โˆ’๐‘ฅ๐‘›โ€–=0.
By (3.1), We have โ€–โ€–๐‘ฅ๐‘›+1โˆ’๐‘ฅ๐‘›โ€–โ€–=โ€–โ€–๐›ผ๐‘›๐‘“๎€ท๐‘ฅ๐‘›๎€ธ+๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐ฟ๐‘›๐‘ฅ๐‘›โˆ’๐›ผ๐‘›โˆ’1๐‘“๎€ท๐‘ฅ๐‘›โˆ’1๎€ธโˆ’๎€ท1โˆ’๐›ผ๐‘›โˆ’1๎€ธ๐ฟ๐‘›โˆ’1๐‘ฅ๐‘›โˆ’1โ€–โ€–=โ€–โ€–๐›ผ๐‘›๐‘“๎€ท๐‘ฅ๐‘›๎€ธโˆ’๐›ผ๐‘›๐‘“๎€ท๐‘ฅ๐‘›โˆ’1๎€ธ+๐›ผ๐‘›๐‘“๎€ท๐‘ฅ๐‘›โˆ’1๎€ธโˆ’๐›ผ๐‘›โˆ’1๐‘“๎€ท๐‘ฅ๐‘›โˆ’1๎€ธ+๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐ฟ๐‘›๐‘ฅ๐‘›โˆ’๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐ฟ๐‘›โˆ’1๐‘ฅ๐‘›โˆ’1+๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐ฟ๐‘›โˆ’1๐‘ฅ๐‘›โˆ’1โˆ’๎€ท1โˆ’๐›ผ๐‘›โˆ’1๎€ธ๐ฟ๐‘›โˆ’1๐‘ฅ๐‘›โˆ’1โ€–โ€–โ‰ค๐›ผ๐›ผ๐‘›โ€–โ€–๐‘ฅ๐‘›โˆ’๐‘ฅ๐‘›โˆ’1โ€–โ€–+||๐›ผ๐‘›โˆ’๐›ผ๐‘›โˆ’1||โ€–โ€–๐‘“๎€ท๐‘ฅ๐‘›โˆ’1๎€ธโ€–โ€–+๎€ท1โˆ’๐›ผ๐‘›๎€ธโ€–โ€–๐ฟ๐‘›๐‘ฅ๐‘›โˆ’๐ฟ๐‘›โˆ’1๐‘ฅ๐‘›โˆ’1โ€–โ€–+||๐›ผ๐‘›โˆ’๐›ผ๐‘›โˆ’1||โ€–โ€–๐ฟ๐‘›โˆ’1๐‘ฅ๐‘›โˆ’1โ€–โ€–=๐›ผ๐›ผ๐‘›โ€–โ€–๐‘ฅ๐‘›โˆ’๐‘ฅ๐‘›โˆ’1โ€–โ€–+||๐›ผ๐‘›โˆ’๐›ผ๐‘›โˆ’1||๎€ทโ€–โ€–๐‘“๎€ท๐‘ฅ๐‘›โˆ’1๎€ธโ€–โ€–+โ€–โ€–๐ฟ๐‘›โˆ’1๐‘ฅ๐‘›โˆ’1โ€–โ€–๎€ธ+๎€ท1โˆ’๐›ผ๐‘›๎€ธโ€–โ€–๐ฟ๐‘›๐‘ฅ๐‘›โˆ’๐ฟ๐‘›โˆ’1๐‘ฅ๐‘›โˆ’1โ€–โ€–โ‰ค๐›ผ๐›ผ๐‘›โ€–โ€–๐‘ฅ๐‘›โˆ’๐‘ฅ๐‘›โˆ’1โ€–โ€–+||๐›ผ๐‘›โˆ’๐›ผ๐‘›โˆ’1||๎€ทโ€–โ€–๐‘“๎€ท๐‘ฅ๐‘›โˆ’1๎€ธโ€–โ€–+โ€–โ€–๐ฟ๐‘›โˆ’1๐‘ฅ๐‘›โˆ’1โ€–โ€–๎€ธ+๎€ท1โˆ’๐›ผ๐‘›๎€ธโ€–โ€–๐ฟ๐‘›๐‘ฅ๐‘›โˆ’๐ฟ๐‘›๐‘ฅ๐‘›โˆ’1โ€–โ€–+๎€ท1โˆ’๐›ผ๐‘›๎€ธโ€–โ€–๐ฟ๐‘›๐‘ฅ๐‘›โˆ’1โˆ’๐ฟ๐‘›โˆ’1๐‘ฅ๐‘›โˆ’1โ€–โ€–โ‰ค๎€ท1โˆ’(1โˆ’๐›ผ)๐›ผ๐‘›๎€ธโ€–โ€–๐‘ฅ๐‘›โˆ’๐‘ฅ๐‘›โˆ’1โ€–โ€–+๎€ท1โˆ’๐›ผ๐‘›๎€ธโ€–โ€–๐ฟ๐‘›๐‘ฅ๐‘›โˆ’1โˆ’๐ฟ๐‘›โˆ’1๐‘ฅ๐‘›โˆ’1โ€–โ€–+||๐›ผ๐‘›โˆ’๐›ผ๐‘›โˆ’1||๎€ทโ€–โ€–๐‘“๎€ท๐‘ฅ๐‘›โˆ’1๎€ธโ€–โ€–+โ€–โ€–๐ฟ๐‘›โˆ’1๐‘ฅ๐‘›โˆ’1โ€–โ€–๎€ธโ‰ค๎€ท1โˆ’(1โˆ’๐›ผ)๐›ผ๐‘›๎€ธโ€–โ€–๐‘ฅ๐‘›โˆ’๐‘ฅ๐‘›โˆ’1โ€–โ€–+๎€ท1โˆ’๐›ผ๐‘›๎€ธโ€–โ€–๐ฟ๐‘›๐‘ฅ๐‘›โˆ’1โˆ’๐ฟ๐‘›โˆ’1๐‘ฅ๐‘›โˆ’1โ€–โ€–+||๐›ผ๐‘›โˆ’๐›ผ๐‘›โˆ’1||๐‘€,(3.5) where ๐‘€=sup๐‘›โ‰ฅ1(โ€–๐‘“(๐‘ฅ๐‘›โˆ’1)โ€–+โ€–๐ฟ๐‘›โˆ’1๐‘ฅ๐‘›โˆ’1โ€–). At the same time, we observe that โˆž๎“๐‘›=1โ€–โ€–๐ฟ๐‘›๐‘ฅ๐‘›โˆ’1โˆ’๐ฟ๐‘›โˆ’1๐‘ฅ๐‘›โˆ’1โ€–โ€–=โˆž๎“๐‘›=1โ€–โ€–โ€–โ€–๐‘›๎“๐‘˜=1๐œ”๐‘˜๐‘†๐‘›๐‘‡๐‘˜๐‘ฅ๐‘›โˆ’1โˆ’๐‘›โˆ’1๎“๐‘˜=1๐œ”๐‘˜๐‘†๐‘›โˆ’1๐‘‡๐‘˜๐‘ฅ๐‘›โˆ’1โ€–โ€–โ€–โ€–=โˆž๎“๐‘›=1โ€–โ€–โ€–โ€–๐œ”๐‘›๐‘†๐‘›๐‘‡๐‘›๐‘ฅ๐‘›โˆ’1+๐‘›โˆ’1๎“๐‘˜=1โˆ’๐œ”๐‘›๐œ”๐‘˜๐‘†๐‘›๐‘†๐‘›โˆ’1๐‘‡๐‘˜๐‘ฅ๐‘›โˆ’1โ€–โ€–โ€–โ€–โ‰คโˆž๎“๐‘›=1๎ƒฏโ€–โ€–โ€–๐œ”๐‘›๐‘†๐‘›๐‘‡๐‘›๐‘ฅ๐‘›โˆ’1โ€–โ€–โ€–+โ€–โ€–โ€–โ€–๐‘›โˆ’1๎“๐‘˜=1๐œ”๐‘›๐œ”๐‘˜๐‘†๐‘›๐‘†๐‘›โˆ’1๐‘‡๐‘˜๐‘ฅ๐‘›โˆ’1โ€–โ€–โ€–โ€–๎ƒฐโ‰คโˆž๎“๐‘›=1๐œ”๐‘›๐‘†๐‘›โ€–โ€–๐‘‡๐‘›๐‘ฅ๐‘›โˆ’1โ€–โ€–+โˆž๎“๐‘›=1๐‘›โˆ’1๎“๐‘˜=1๐œ”๐‘›๐œ”๐‘˜๐‘†๐‘›๐‘†๐‘›โˆ’1โ€–โ€–๐‘‡๐‘˜๐‘ฅ๐‘›โˆ’1โ€–โ€–โ‰คโˆž๎“๐‘›=1๐œ”๐‘›๐‘†๐‘›๐‘€+โˆž๎“๐‘›=1๐œ”๐‘›๐‘†๐‘›๐‘€=โˆž๎“๐‘›=12๐‘€๐‘†๐‘›๐œ”๐‘›,(3.6) where ๐‘€=sup๐‘˜โ‰ฅ1,๐‘›โ‰ฅ1โ€–๐‘‡๐‘˜๐‘ฅ๐‘›โˆ’1โ€–. Applying Lemma 2.6 and compatibility test of series, we have โˆž๎“๐‘›=1โ€–โ€–๐ฟ๐‘›๐‘ฅ๐‘›โˆ’1โˆ’๐ฟ๐‘›โˆ’1๐‘ฅ๐‘›โˆ’1โ€–โ€–<โˆž.(3.7) Put ๐›พ๐‘›=(1โˆ’๐›ผ)๐›ผ๐‘›,๐›ฝ๐‘›=||๐›ผ๐‘›โˆ’๐›ผ๐‘›โˆ’1||๐‘€(1โˆ’๐›ผ)๐›ผ๐‘›,๐›ฟ๐‘›=๎€ท1โˆ’๐›ผ๐‘›๎€ธโ€–โ€–๐ฟ๐‘›๐‘ฅ๐‘›โˆ’1โˆ’๐ฟ๐‘›โˆ’1๐‘ฅ๐‘›โˆ’1โ€–โ€–.(3.8) It follows that โ€–โ€–๐‘ฅ๐‘›+1โˆ’๐‘ฅ๐‘›โ€–โ€–โ‰ค๎€ท1โˆ’๐›พ๐‘›๎€ธโ€–โ€–๐‘ฅ๐‘›โˆ’๐‘ฅ๐‘›โˆ’1โ€–โ€–+๐›พ๐‘›๐›ฝ๐‘›+๐›ฟ๐‘›.(3.9) It is easily seen from (A2), (A3), and (3.7) that โˆž๎“๐‘›=1๐›พ๐‘›=โˆž,limsup๐‘›โ†’โˆž๐›ฝ๐‘›โ‰ค0,โˆž๎“๐‘›=1๐›ฟ๐‘›<โˆž.(3.10) Applying Lemma 2.2 to (3.9), we obtain lim๐‘›โ†’โˆžโ€–โ€–๐‘ฅ๐‘›+1โˆ’๐‘ฅ๐‘›โ€–โ€–=0.(3.11)
Step 3. We show that lim๐‘›โ†’โˆžโ€–๐‘ฅ๐‘›โˆ’๐‘‡๐‘ฅ๐‘›โ€–=0.
Indeed we observe that โ€–โ€–๐‘ฅ๐‘›โˆ’๐‘‡๐‘ฅ๐‘›โ€–โ€–โ‰คโ€–โ€–๐‘ฅ๐‘›โˆ’๐‘ฅ๐‘›+1โ€–โ€–+โ€–โ€–๐‘ฅ๐‘›+1โˆ’๐‘‡๐‘ฅ๐‘›โ€–โ€–=โ€–โ€–๐‘ฅ๐‘›โˆ’๐‘ฅ๐‘›+1โ€–โ€–+โ€–โ€–๐›ผ๐‘›๐‘“๎€ท๐‘ฅ๐‘›๎€ธ+๎€ท1โˆ’๐›ผ๐‘›๎€ธ๐ฟ๐‘›๐‘ฅ๐‘›โˆ’๐‘‡๐‘ฅ๐‘›โ€–โ€–โ‰คโ€–โ€–๐‘ฅ๐‘›โˆ’๐‘ฅ๐‘›+1โ€–โ€–+๐›ผ๐‘›โ€–โ€–๐‘“๎€ท๐‘ฅ๐‘›๎€ธโˆ’๐‘‡๐‘ฅ๐‘›โ€–โ€–+๎€ท1โˆ’๐›ผ๐‘›๎€ธโ€–โ€–๐ฟ๐‘›๐‘ฅ๐‘›โˆ’๐‘‡๐‘ฅ๐‘›โ€–โ€–.(3.12) Hence, by (3.11), (A1), and Lemma 2.7, we have lim๐‘›โ†’โˆžโ€–โ€–๐‘ฅ๐‘›โˆ’๐‘‡๐‘ฅ๐‘›โ€–โ€–=0.(3.13)
Step 4. We prove that limsup๐‘›โ†’โˆž๎ซ๐‘“๎€ท๐‘ฅโˆ—๎€ธโˆ’๐‘ฅโˆ—๎€ท๐‘ฅ,๐ฝ๐‘›โˆ’๐‘ฅโˆ—๎€ธ๎ฌโ‰ค0,(3.14) where ๐‘ฅโˆ—=lim๐‘กโ†’0๐‘ฅ๐‘ก with ๐‘ฅ๐‘ก being the fixed point of the contraction ๐‘ฅโŸผ๐‘ก๐‘“(๐‘ฅ)+(1โˆ’๐‘ก)๐‘‡๐‘ฅ.(3.15) From Lemma 2.5, we have ๐‘ฅโˆ—โˆˆFix(๐‘‡) and ๎ซ(๐ผโˆ’๐‘“)๐‘ฅโˆ—๎€ท๐‘ฅ,๐ฝโˆ—โˆ’๐‘๎€ธ๎ฌโ‰ค0,๐‘โˆˆFix(๐‘‡).(3.16) By Lemma 2.4, we have ๐‘ฅโˆ—โˆˆโ‹‚โˆž๐‘˜=1Fix(๐‘‡๐‘˜) and ๎ซ(๐ผโˆ’๐‘“)๐‘ฅโˆ—๎€ท๐‘ฅ,๐ฝโˆ—โˆ’๐‘๎€ธ๎ฌโ‰ค0,๐‘โˆˆโˆž๎™๐‘˜=1๎€ท๐‘‡Fix๐‘˜๎€ธ.(3.17) By ๐‘ฅ๐‘ก=๐‘ก๐‘“(๐‘ฅ๐‘ก)+(1โˆ’๐‘ก)๐‘‡๐‘ฅ๐‘ก, we have ๐‘ฅ๐‘กโˆ’๐‘ฅ๐‘›๎€ท๐‘“๎€ท๐‘ฅ=๐‘ก๐‘ก๎€ธโˆ’๐‘ฅ๐‘›๎€ธ+๎€ท(1โˆ’๐‘ก)๐‘‡๐‘ฅ๐‘กโˆ’๐‘ฅ๐‘›๎€ธ.(3.18) It follows from Lemma 2.3 that โ€–โ€–๐‘ฅ๐‘กโˆ’๐‘ฅ๐‘›โ€–โ€–2=โ€–โ€–๐‘ก(๐‘“(๐‘ฅ๐‘ก)โˆ’๐‘ฅ๐‘›)+(1โˆ’๐‘ก)(๐‘‡๐‘ฅ๐‘กโˆ’๐‘ฅ๐‘›)โ€–โ€–2โ‰ค(1โˆ’๐‘ก)2โ€–โ€–๐‘‡๐‘ฅ๐‘กโˆ’๐‘ฅ๐‘›โ€–โ€–2๎€ท๐‘ฅ+2๐‘กโŸจ๐‘“๐‘ก๎€ธโˆ’๐‘ฅ๐‘›๎€ท๐‘ฅ,๐ฝ๐‘กโˆ’๐‘ฅ๐‘›๎€ธโŸฉโ‰ค(1โˆ’๐‘ก)2๎€ทโ€–โ€–๐‘‡๐‘ฅ๐‘กโˆ’๐‘‡๐‘ฅ๐‘›โ€–โ€–+โ€–โ€–๐‘‡๐‘ฅ๐‘›โˆ’๐‘ฅ๐‘›โ€–โ€–๎€ธ2๎€ท๐‘ฅ+2๐‘กโŸจ๐‘“๐‘ก๎€ธโˆ’๐‘ฅ๐‘ก๎€ท๐‘ฅ,๐ฝ๐‘กโˆ’๐‘ฅ๐‘›๎€ธโŸฉ+2๐‘กโŸจ๐‘ฅ๐‘กโˆ’๐‘ฅ๐‘›๎€ท๐‘ฅ,๐ฝ๐‘กโˆ’๐‘ฅ๐‘›๎€ธโŸฉโ‰ค(1โˆ’๐‘ก)2๎€ทโ€–โ€–๐‘ฅ๐‘กโˆ’๐‘ฅ๐‘›โ€–โ€–+โ€–โ€–๐‘‡๐‘ฅ๐‘›โˆ’๐‘ฅ๐‘›โ€–โ€–๎€ธ2๎€ท๐‘ฅ+2๐‘กโŸจ๐‘“๐‘ก๎€ธโˆ’๐‘ฅ๐‘ก๎€ท๐‘ฅ,๐ฝ๐‘กโˆ’๐‘ฅ๐‘›๎€ธโŸฉ+2๐‘กโŸจ๐‘ฅ๐‘กโˆ’๐‘ฅ๐‘›๎€ท๐‘ฅ,๐ฝ๐‘กโˆ’๐‘ฅ๐‘›๎€ธโŸฉโ‰ค(1โˆ’๐‘ก)2โ€–โ€–๐‘ฅ๐‘กโˆ’๐‘ฅ๐‘›โ€–โ€–2+๐‘๐‘›๎€ท๐‘ฅ(๐‘ก)+2๐‘กโŸจ๐‘“๐‘ก๎€ธโˆ’๐‘ฅ๐‘ก๎€ท๐‘ฅ,๐ฝ๐‘กโˆ’๐‘ฅ๐‘›๎€ธโŸฉโ€–โ€–๐‘ฅ+2๐‘ก๐‘กโˆ’๐‘ฅ๐‘›โ€–โ€–2,(3.19) where ๐‘๐‘›(๐‘ก)=โ€–๐‘‡๐‘ฅ๐‘›โˆ’๐‘ฅ๐‘›โ€–(2โ€–๐‘ฅ๐‘กโˆ’๐‘ฅ๐‘›โ€–+โ€–๐‘‡๐‘ฅ๐‘›โˆ’๐‘ฅ๐‘›โ€–)โ†’0 (๐‘›โ†’โˆž). It follows from above last inequality that โŸจ๐‘ฅ๐‘ก๎€ท๐‘ฅโˆ’๐‘“๐‘ก๎€ธ๎€ท๐‘ฅ,๐ฝ๐‘กโˆ’๐‘ฅ๐‘›๎€ธ๐‘กโŸฉโ‰ค2โ€–โ€–๐‘ฅ๐‘กโˆ’๐‘ฅ๐‘›โ€–โ€–2+1๐‘2๐‘ก๐‘›(๐‘ก).(3.20) Taking ๐‘›โ†’โˆž in (3.20) yields limsup๐‘›โ†’โˆžโŸจ๐‘ฅ๐‘ก๎€ท๐‘ฅโˆ’๐‘“๐‘ก๎€ธ๎€ท๐‘ฅ,๐ฝ๐‘กโˆ’๐‘ฅ๐‘›๎€ธ๐‘กโŸฉโ‰ค2๐‘€,(3.21) where ๐‘€โ‰ฅโ€–๐‘ฅ๐‘กโˆ’๐‘ฅ๐‘›โ€–2 for all ๐‘›โ‰ฅ1 and ๐‘กโˆˆ(0,1). Taking ๐‘กโ†’0 in (3.21), we have limsup๐‘กโ†’0limsup๐‘›โ†’โˆžโŸจ๐‘ฅ๐‘ก๎€ท๐‘ฅโˆ’๐‘“๐‘ก๎€ธ๎€ท๐‘ฅ,๐ฝ๐‘กโˆ’๐‘ฅ๐‘›๎€ธโŸฉโ‰ค0.(3.22) Noticing the fact that two limits are interchangeable due to the fact the duality mapping ๐ฝ is norm-to-norm uniformly continuous on bounded sets, it follows from (3.22), we have limsup๐‘›โ†’โˆž๎ซ๐‘“๎€ท๐‘ฅโˆ—๎€ธโˆ’๐‘ฅโˆ—๎€ท๐‘ฅ,๐ฝ๐‘›โˆ’๐‘ฅโˆ—๎€ธ๎ฌ=limsup๐‘›โ†’โˆžlimsup๐‘กโ†’0โŸจ๐‘ฅ๐‘ก๎€ท๐‘ฅโˆ’๐‘“๐‘ก๎€ธ๎€ท๐‘ฅ,๐ฝ๐‘กโˆ’๐‘ฅ๐‘›๎€ธโŸฉ=limsup๐‘กโ†’0limsup๐‘›โ†’โˆžโŸจ๐‘ฅ๐‘ก๎€ท๐‘ฅโˆ’๐‘“๐‘ก๎€ธ๎€ท๐‘ฅ,๐ฝ๐‘กโˆ’๐‘ฅ๐‘›๎€ธโŸฉโ‰ค0.(3.23) Hence (3.14) holds.Step 5. Finally, we prove that ๐‘ฅ๐‘›โ†’๐‘ฅโˆ— (๐‘›โ†’โˆž).
Indeed we observe that โ€–โ€–๐‘ฅ๐‘›+1โˆ’๐‘ฅโˆ—โ€–โ€–2=โ€–โ€–๐›ผ๐‘›๐‘“(๐‘ฅ๐‘›)+(1โˆ’๐›ผ๐‘›)๐ฟ๐‘›๐‘ฅ๐‘›โˆ’๐‘ฅโˆ—โ€–โ€–2=โ€–โ€–๐›ผ๐‘›(๐‘“(๐‘ฅ๐‘›)โˆ’๐‘“(๐‘ฅโˆ—))+(1โˆ’๐›ผ๐‘›)(๐ฟ๐‘›๐‘ฅ๐‘›โˆ’๐‘ฅโˆ—)+๐›ผ๐‘›(๐‘“(๐‘ฅโˆ—)โˆ’๐‘ฅโˆ—)โ€–โ€–2โ‰คโ€–โ€–๐›ผ๐‘›(๐‘“(๐‘ฅ๐‘›)โˆ’๐‘“(๐‘ฅโˆ—))+(1โˆ’๐›ผ๐‘›)(๐ฟ๐‘›๐‘ฅ๐‘›โˆ’๐‘ฅโˆ—)โ€–โ€–2+2โŸจ๐›ผ๐‘›๎€ท๐‘“๎€ท๐‘ฅโˆ—๎€ธโˆ’๐‘ฅโˆ—๎€ธ๎€ท๐‘ฅ,๐ฝ๐‘›+1โˆ’๐‘ฅโˆ—๎€ธโŸฉโ‰ค๎€ฝ๐›ผ๐‘›โ€–โ€–๐‘“๎€ท๐‘ฅ๐‘›๎€ธ๎€ท๐‘ฅโˆ’๐‘“โˆ—๎€ธโ€–โ€–+๎€ท1โˆ’๐›ผ๐‘›๎€ธโ€–โ€–๐ฟ๐‘›๐‘ฅ๐‘›โˆ’๐‘ฅโˆ—โ€–โ€–๎€พ2+2๐›ผ๐‘›๎€ท๐‘ฅโŸจ๐‘“โˆ—๎€ธโˆ’๐‘ฅโˆ—๎€ท๐‘ฅ,๐ฝ๐‘›+1โˆ’๐‘ฅโˆ—๎€ธโŸฉโ‰ค๎€ท๐›ผ๐›ผ๐‘›โ€–โ€–๐‘ฅ๐‘›โˆ’๐‘ฅโˆ—โ€–โ€–+๎€ท1โˆ’๐›ผ๐‘›๎€ธโ€–โ€–๐‘ฅ๐‘›โˆ’๐‘ฅโˆ—โ€–โ€–๎€ธ2+2๐›ผ๐‘›๎€ท๐‘ฅโŸจ๐‘“โˆ—๎€ธโˆ’๐‘ฅโˆ—๎€ท๐‘ฅ,๐ฝ๐‘›+1โˆ’๐‘ฅโˆ—๎€ธโŸฉโ‰ค๎€ท1โˆ’(1โˆ’๐›ผ)๐›ผ๐‘›๎€ธ2โ€–โ€–๐‘ฅ๐‘›โˆ’๐‘ฅโˆ—โ€–โ€–2+2๐›ผ๐‘›๎€ท๐‘ฅโŸจ๐‘“โˆ—๎€ธโˆ’๐‘ฅโˆ—๎€ท๐‘ฅ,๐ฝ๐‘›+1โˆ’๐‘ฅโˆ—๎€ธโŸฉโ‰ค๎€ท1โˆ’(1โˆ’๐›ผ)๐›ผ๐‘›๎€ธโ€–โ€–๐‘ฅ๐‘›โˆ’๐‘ฅโˆ—โ€–โ€–2+2๐›ผ๐‘›๎€ท๐‘ฅโŸจ๐‘“โˆ—๎€ธโˆ’๐‘ฅโˆ—๎€ท๐‘ฅ,๐ฝ๐‘›+1โˆ’๐‘ฅโˆ—๎€ธโŸฉ.(3.24) By view of (3.14) and condition (A2), it follows from Lemma 2.2 that ๐‘ฅ๐‘›โ†’๐‘ฅโˆ—(๐‘›โ†’โˆž). This completes the proof.

Corollary 3.2. Let ๐ถ be a nonempty closed convex subset of a real uniformly smooth Banach space ๐‘‹, {๐‘‡๐‘˜}โˆž๐‘˜=1โˆถ๐ถโ†’๐ถ an infinite family of nonexpansive mappings with โ‹‚โˆž๐‘˜=1Fix(๐‘‡๐‘˜)โ‰ โˆ…, {๐œ”๐‘˜} a sequence of positive numbers with โˆ‘โˆž๐‘˜=1๐œ”๐‘˜=1. Let ๐ฟ๐‘›=โˆ‘๐‘›๐‘˜=1(๐œ”๐‘˜/๐‘†๐‘›)๐‘‡๐‘˜, ๐‘†๐‘›=โˆ‘๐‘›๐‘˜=1๐œ”๐‘˜, and ๐‘ขโˆˆ๐ถ. Given any ๐‘ฅ0โˆˆ๐ถ, let {๐‘ฅ๐‘›} be a sequence generated by ๐‘ฅ๐‘›+1=๐›ผ๐‘›๎€ท๐‘ข+1โˆ’๐›ผ๐‘›๎€ธ๐ฟ๐‘›๐‘ฅ๐‘›,๐‘›โ‰ฅ0,(3.25) where {๐›ผ๐‘›}โŠ‚(0,1) satisfies the following conditions: (A1)lim๐‘›โ†’โˆž๐›ผ๐‘›=0;(A2)โˆ‘โˆž๐‘›=0๐›ผ๐‘›=โˆž;(A3)eitherโˆ‘โˆž๐‘›=0|๐›ผ๐‘›+1โˆ’๐›ผ๐‘›|<โˆž or lim๐‘›โ†’โˆž(๐›ผ๐‘›+1/๐›ผ๐‘›=)1. Then {๐‘ฅ๐‘›} converges strongly to ๐‘ฅโˆ—โˆˆโ‹‚โˆž๐‘˜=1Fix(๐‘‡๐‘˜), which solves the following variational inequality: ๎ซ๐‘ฅโˆ—๎€ท๐‘ฅโˆ’๐‘ข,๐ฝโˆ—โˆ’๐‘๎€ธ๎ฌโ‰ค0,๐‘โˆˆโˆž๎™๐‘˜=1๎€ท๐‘‡Fix๐‘˜๎€ธ.(3.26)

Acknowledgment

This paper is supported by the Fundamental Research Funds for the Central Universities (ZXH2011D005).

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