Journal of Applied Mathematics

Journal of Applied Mathematics / 2012 / Article
Special Issue

Applications of Fixed Point and Approximate Algorithms

View this Special Issue

Research Article | Open Access

Volume 2012 |Article ID 320421 | 13 pages | https://doi.org/10.1155/2012/320421

An Iterative Algorithm for a Hierarchical Problem

Academic Editor: Giuseppe Marino
Received29 Sep 2011
Accepted11 Nov 2011
Published29 Dec 2011

Abstract

A general hierarchical problem has been considered, and an explicit algorithm has been presented for solving this hierarchical problem. Also, it is shown that the suggested algorithm converges strongly to a solution of the hierarchical problem.

1. Introduction

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. The hierarchical problem is of finding Μƒπ‘₯∈Fix(𝑇) such thatβŸ¨π‘†Μƒπ‘₯βˆ’Μƒπ‘₯,π‘₯βˆ’Μƒπ‘₯βŸ©β‰€0,βˆ€π‘₯∈Fix(𝑇),(1.1) where 𝑆,𝑇 are two nonexpansive mappings and Fix(𝑇) is the set of fixed points of 𝑇. Recently, this problem has been studied by many authors (see, e.g., [1–15]). The main reason is that this problem is closely associated with some monotone variational inequalities and convex programming problems (see [16–19]).

Now, we briefly recall some historic results which relate to the problem (1.1).

For solving the problem (1.1), in 2006, Moudafi and Mainge [1] first introduced an implicit iterative algorithm:π‘₯𝑑,𝑠π‘₯=𝑠𝑄𝑑,𝑠+ξ€Ίξ€·π‘₯(1βˆ’π‘ )𝑑𝑆𝑑,𝑠+ξ€·π‘₯(1βˆ’π‘‘)𝑇𝑑,𝑠(1.2) and proved that the net {π‘₯𝑑,𝑠} defined by (1.2) strongly converges to π‘₯𝑑 as 𝑠→0, where π‘₯𝑑 satisfies π‘₯𝑑=projFix(𝑃𝑑)𝑄(π‘₯𝑑), where π‘ƒπ‘‘βˆΆπΆβ†’πΆ is a mapping defined by 𝑃𝑑(π‘₯)=𝑑𝑆(π‘₯)+(1βˆ’π‘‘)𝑇(π‘₯),βˆ€π‘₯∈𝐢,π‘‘βˆˆ(0,1),(1.3) or, equivalently, π‘₯𝑑 is the unique solution of the quasivariational inequality 0∈(πΌβˆ’π‘„)π‘₯𝑑+𝑁Fix(𝑃𝑑)ξ€·π‘₯𝑑,(1.4) where the normal cone to Fix(𝑃𝑑), 𝑁Fix(𝑃𝑑), is defined as follows: 𝑁Fix(𝑃𝑑)ξ‚»ξ€·π‘ƒβˆΆπ‘₯⟢{π‘’βˆˆπ»βˆΆβŸ¨π‘¦βˆ’π‘₯,π‘’βŸ©β‰€0},ifπ‘₯∈Fix𝑑,βˆ…,otherwise.(1.5)

Moreover, as 𝑑→0, the net {π‘₯𝑑} in turn weakly converges to the unique solution π‘₯∞ of the fixed point equation π‘₯∞=projΩ𝑄(π‘₯∞) or, equivalently, π‘₯∞ is the unique solution of the variational inequality 0∈(πΌβˆ’π‘„)π‘₯∞+𝑁Ωπ‘₯βˆžξ€Έ.(1.6)

Recently, Moudafi [2] constructed an explicit iterative algorithm:π‘₯𝑛+1=ξ€·1βˆ’π›Ώπ‘›ξ€Έπ‘₯𝑛+π›Ώπ‘›ξ€·πœŽπ‘›π‘†π‘₯𝑛+ξ€·1βˆ’πœŽπ‘›ξ€Έπ‘‡π‘₯𝑛,βˆ€π‘›β‰₯0,(1.7) where {𝛿𝑛} and {πœŽπ‘›} are two real numbers in (0,1). By using this iterative algorithm, Moudafi [2] only proved a weak convergence theorem for solving the problem (1.1).

In order to obtain a strong convergence result, Mainge and Moudafi [3] further introduced the following iterative algorithm:π‘₯𝑛+1=ξ€·1βˆ’π›Ώπ‘›ξ€Έπ‘„π‘₯𝑛+π›Ώπ‘›ξ€ΊπœŽπ‘›π‘†π‘₯𝑛+ξ€·1βˆ’πœŽπ‘›ξ€Έπ‘‡π‘₯𝑛,βˆ€π‘›β‰₯0,(1.8) where {𝛿𝑛} and {πœŽπ‘›} are two real numbers in (0,1), and proved that, under appropriate conditions, the iterative sequence {π‘₯𝑛} generated by (1.8) has strong convergence.

Subsequently, some authors have studied some algorithms on hierarchical fixed problems (see, e.g., [4–15]).

Motivated and inspired by the results in the literature, in this paper, we consider a general hierarchical problem of finding Μƒπ‘₯∈Fix(𝑇) such that, for any 𝑛β‰₯1,βŸ¨π‘Šπ‘›Μƒπ‘₯βˆ’Μƒπ‘₯,π‘₯βˆ’Μƒπ‘₯βŸ©β‰€0,βˆ€π‘₯∈Fix(𝑇),(1.9) where π‘Šπ‘› is the π‘Š-mapping defined by (2.3) below and 𝑇 is a nonexpansive mapping, and introduce an explicit iterative algorithm which converges strongly to a solution Μƒπ‘₯ of the hierarchical problem (1.9).

2. Preliminaries

Let 𝐢 a nonempty closed convex subset of a real Hilbert space 𝐻. Recall that a mapping π‘„βˆΆπΆβ†’πΆ is said to be contractive if there exists a constant π›Ύβˆˆ(0,1) such that ‖𝑄π‘₯βˆ’π‘„π‘¦β€–β‰€π›Ύβ€–π‘₯βˆ’π‘¦β€–,βˆ€π‘₯,π‘¦βˆˆπΆ.(2.1)

A mapping π‘‡βˆΆπΆβ†’πΆ is called nonexpansive if‖𝑇π‘₯βˆ’π‘‡π‘¦β€–β‰€β€–π‘₯βˆ’π‘¦β€–,βˆ€π‘₯,π‘¦βˆˆπΆ.(2.2)

Forward, we use Fix(𝑇) to denote the fixed points set of 𝑇.

Let {𝑇𝑖}βˆžπ‘–=1βˆΆπΆβ†’πΆ be an infinite family of nonexpansive mappings and {πœ‰π‘–}βˆžπ‘–=1 a real number sequence such that 0β‰€πœ‰π‘–β‰€1 for each 𝑖β‰₯1.

For each 𝑛β‰₯1, define a mapping π‘Šπ‘›βˆΆπΆβ†’πΆ as follows:π‘ˆπ‘›,𝑛+1π‘ˆ=𝐼,𝑛,𝑛=πœ‰π‘›π‘‡π‘›π‘ˆπ‘›,𝑛+1+ξ€·1βˆ’πœ‰π‘›ξ€Έπ‘ˆπΌ,𝑛,π‘›βˆ’1=πœ‰π‘›βˆ’1π‘‡π‘›βˆ’1π‘ˆπ‘›,𝑛+ξ€·1βˆ’πœ‰π‘›βˆ’1ξ€Έβ‹―π‘ˆπΌ,𝑛,π‘˜=πœ‰π‘˜π‘‡π‘˜π‘ˆπ‘›,π‘˜+1+ξ€·1βˆ’πœ‰π‘˜ξ€Έπ‘ˆπΌ,𝑛,π‘˜βˆ’1=πœ‰π‘˜βˆ’1π‘‡π‘˜βˆ’1π‘ˆπ‘›,π‘˜+ξ€·1βˆ’πœ‰π‘˜βˆ’1ξ€Έβ‹―π‘ˆπΌ,𝑛,2=πœ‰2𝑇2π‘ˆπ‘›,3+ξ€·1βˆ’πœ‰2ξ€Έπ‘ŠπΌ,𝑛=π‘ˆπ‘›,1=πœ‰1𝑇1π‘ˆπ‘›,2+ξ€·1βˆ’πœ‰1𝐼.(2.3)

Such π‘Šπ‘› is called the π‘Š-mapping generated by {𝑇𝑖}βˆžπ‘–=1 and {πœ‰π‘–}βˆžπ‘–=1.

Lemma 2.1 (see [20]). Let 𝐢 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let {𝑇𝑖}βˆžπ‘–=1 be an infinite family of nonexpansive mappings of 𝐢 into itself with β‹‚βˆžπ‘›=1Fix(𝑇𝑛)β‰ βˆ…. Let πœ‰1,πœ‰2,… be real numbers such that 0<πœ‰π‘–β‰€π‘<1 for each 𝑖β‰₯1. Then one has the following results: (1)for any π‘₯∈𝐢 and π‘˜β‰₯1, the limit limπ‘›β†’βˆžπ‘ˆπ‘›,π‘˜π‘₯ exists; (2)β‹‚Fix(π‘Š)=βˆžπ‘›=1Fix(𝑇𝑛).

Using Lemma  3.1 in [21], we can define a mapping π‘Š of 𝐢 into itself by π‘Šπ‘₯=limπ‘›β†’βˆžπ‘Šπ‘›π‘₯=limπ‘›β†’βˆžπ‘ˆπ‘›,1π‘₯ for all π‘₯∈𝐢. Thus we have the following.

Lemma 2.2 (see [21]). If {π‘₯𝑛} is a bounded sequence in 𝐢, then one has limπ‘›β†’βˆžβ€–β€–π‘Šπ‘₯π‘›βˆ’π‘Šπ‘›π‘₯𝑛‖‖=0.(2.4)

Lemma 2.3 (see [22]). Let 𝐢 be a nonempty closed convex of a real Hilbert space 𝐻 and π‘‡βˆΆπΆβ†’πΆ be nonexpansive mapping. Then 𝑇 is demiclosed on 𝐢, that is, if π‘₯𝑛⇀π‘₯∈𝐢 and π‘₯π‘›βˆ’π‘‡π‘₯𝑛→0, then π‘₯=𝑇π‘₯.

Lemma 2.4 (see [23]). Assume {π‘Žπ‘›} is a sequence of nonnegative real numbers such that π‘Žπ‘›+1≀1βˆ’π›Ύπ‘›ξ€Έπ‘Žπ‘›+𝛿𝑛𝛾𝑛+πœ‚π‘›,βˆ€π‘›β‰₯1,(2.5) where {𝛾𝑛} is a sequence in (0,1) and {𝛿𝑛},{πœ‚π‘›} are two sequences such that (i)βˆ‘βˆžπ‘›=1𝛾𝑛=∞;(ii)limsupπ‘›β†’βˆžπ›Ώπ‘›β‰€0 or βˆ‘βˆžπ‘›=1|𝛿𝑛𝛾𝑛|<∞;(iii)βˆ‘βˆžπ‘›=1|πœ‚π‘›|<∞.
Then limπ‘›β†’βˆžπ‘Žπ‘›=0.

3. Main Results

In this section, we introduce our algorithm and give its convergence analysis.

Algorithm 3.1. Let 𝐢 be a nonempty closed convex subset of a real Hilbert space 𝐻 and {𝑇𝑛}βˆžπ‘›=1 be infinite family of nonexpansive mappings of 𝐢 into itself. Let π‘„βˆΆπΆβ†’πΆ be a contraction with coefficient π›Ύβˆˆ[0,1). For any π‘₯0∈𝐢, let {π‘₯𝑛} the sequence generated iteratively by π‘₯𝑛+1=π›Όπ‘›π‘Šπ‘›π‘₯𝑛+ξ€·1βˆ’π›Όπ‘›ξ€Έπ‘‡ξ€·π›½π‘›π‘„π‘₯𝑛+ξ€·1βˆ’π›½π‘›ξ€Έπ‘₯𝑛,βˆ€π‘›β‰₯0,(3.1) where {𝛼𝑛},{𝛽𝑛} are two real numbers in (0,1) and π‘Šπ‘› is the π‘Š-mapping defined by (2.3).

Now, we give the convergence analysis of the algorithm.

Theorem 3.2. Let 𝐢 be a nonempty closed convex subset of a real Hilbert space 𝐻 and {𝑇𝑛}βˆžπ‘›=1 be an infinite family of nonexpansive mappings of 𝐢 into itself. Let π‘„βˆΆπΆβ†’πΆ be a contraction with coefficient π›Ύβˆˆ[0,1). Assume that the set Ξ© of solutions of the hierarchical problem (1.9) is nonempty. Let {𝛼𝑛},{𝛽𝑛} be two real numbers in (0,1) and {π‘₯𝑛} the sequence generated by (3.1). Assume that the sequence {π‘₯𝑛} is bounded and (i)limπ‘›β†’βˆžπ›Όπ‘›=0 and limπ‘›β†’βˆž(𝛽𝑛/𝛼𝑛)=0; (ii)βˆ‘βˆžπ‘›=0𝛽𝑛=∞; (iii)limπ‘›β†’βˆž(1/𝛽𝑛)|(1/𝛼𝑛)βˆ’(1/π›Όπ‘›βˆ’1)|=0 and limπ‘›β†’βˆž(βˆπ‘›βˆ’1𝑖=1πœ‰π‘–/𝛼𝑛𝛽𝑛)=limπ‘›β†’βˆž(1/𝛼𝑛)|1βˆ’(π›½π‘›βˆ’1/𝛽𝑛)|=0.
Then limπ‘›β†’βˆž(β€–π‘₯𝑛+1βˆ’π‘₯𝑛‖/𝛼𝑛)=0 and every weak cluster point of the sequence {π‘₯𝑛} solves the following variational inequalityΜƒπ‘₯∈Ω,⟨(πΌβˆ’π‘„)Μƒπ‘₯,π‘₯βˆ’Μƒπ‘₯⟩β‰₯0,βˆ€π‘₯∈Ω.(3.2)

Proof. Set 𝑦𝑛=𝛽𝑛𝑄π‘₯𝑛+(1βˆ’π›½π‘›)π‘₯𝑛 for each 𝑛β‰₯0. Then we have π‘¦π‘›βˆ’π‘¦π‘›βˆ’1=𝛽𝑛𝑄π‘₯𝑛+ξ€·1βˆ’π›½π‘›ξ€Έπ‘₯π‘›βˆ’π›½π‘›βˆ’1𝑄π‘₯π‘›βˆ’1βˆ’ξ€·1βˆ’π›½π‘›βˆ’1ξ€Έπ‘₯π‘›βˆ’1=𝛽𝑛𝑄π‘₯π‘›βˆ’π‘„π‘₯π‘›βˆ’1ξ€Έ+ξ€·π›½π‘›βˆ’π›½π‘›βˆ’1𝑄π‘₯π‘›βˆ’1+ξ€·1βˆ’π›½π‘›π‘₯ξ€Έξ€·π‘›βˆ’π‘₯π‘›βˆ’1ξ€Έ+ξ€·π›½π‘›βˆ’1βˆ’π›½π‘›ξ€Έπ‘₯π‘›βˆ’1.(3.3) It follows that β€–β€–π‘¦π‘›βˆ’π‘¦π‘›βˆ’1‖‖≀𝛾𝛽𝑛‖‖π‘₯π‘›βˆ’π‘₯π‘›βˆ’1β€–β€–+ξ€·1βˆ’π›½π‘›ξ€Έβ€–β€–π‘₯π‘›βˆ’π‘₯π‘›βˆ’1β€–β€–+||π›½π‘›βˆ’π›½π‘›βˆ’1||‖‖𝑄π‘₯π‘›βˆ’1β€–β€–+β€–β€–π‘₯π‘›βˆ’1β€–β€–ξ€Έ=ξ€Ί1βˆ’(1βˆ’π›Ύ)𝛽𝑛‖‖π‘₯π‘›βˆ’π‘₯π‘›βˆ’1β€–β€–+||π›½π‘›βˆ’π›½π‘›βˆ’1||‖‖𝑄π‘₯π‘›βˆ’1β€–β€–+β€–β€–π‘₯π‘›βˆ’1β€–β€–ξ€Έ.(3.4) From (3.1), we have π‘₯𝑛+1βˆ’π‘₯𝑛=π›Όπ‘›π‘Šπ‘›π‘₯𝑛+ξ€·1βˆ’π›Όπ‘›ξ€Έπ‘‡π‘¦π‘›βˆ’π›Όπ‘›βˆ’1π‘Šπ‘›βˆ’1π‘₯π‘›βˆ’1βˆ’ξ€·1βˆ’π›Όπ‘›βˆ’1ξ€Έπ‘‡π‘¦π‘›βˆ’1=π›Όπ‘›ξ€·π‘Šπ‘›π‘₯π‘›βˆ’π‘Šπ‘›π‘₯π‘›βˆ’1ξ€Έ+ξ€·π›Όπ‘›βˆ’π›Όπ‘›βˆ’1ξ€Έπ‘Šπ‘›π‘₯π‘›βˆ’1+π›Όπ‘›βˆ’1ξ€·π‘Šπ‘›π‘₯π‘›βˆ’1βˆ’π‘Šπ‘›βˆ’1π‘₯π‘›βˆ’1ξ€Έ+ξ€·1βˆ’π›Όπ‘›ξ€Έξ€·π‘‡π‘¦π‘›βˆ’π‘‡π‘¦π‘›βˆ’1ξ€Έ+ξ€·π›Όπ‘›βˆ’1βˆ’π›Όπ‘›ξ€Έπ‘‡π‘¦π‘›βˆ’1.(3.5) Then we obtain β€–β€–π‘₯𝑛+1βˆ’π‘₯π‘›β€–β€–β‰€π›Όπ‘›β€–β€–π‘Šπ‘›π‘₯π‘›βˆ’π‘Šπ‘›π‘₯π‘›βˆ’1β€–β€–+ξ€·1βˆ’π›Όπ‘›ξ€Έβ€–β€–π‘‡π‘¦π‘›βˆ’π‘‡π‘¦π‘›βˆ’1β€–β€–+||π›Όπ‘›βˆ’π›Όπ‘›βˆ’1||ξ€·β€–β€–π‘Šπ‘›π‘₯π‘›βˆ’1β€–β€–+β€–β€–π‘‡π‘¦π‘›βˆ’1β€–β€–ξ€Έ+π›Όπ‘›βˆ’1β€–β€–π‘Šπ‘›π‘₯π‘›βˆ’1βˆ’π‘Šπ‘›βˆ’1π‘₯π‘›βˆ’1‖‖≀𝛼𝑛‖‖π‘₯π‘›βˆ’π‘₯π‘›βˆ’1β€–β€–+ξ€·1βˆ’π›Όπ‘›ξ€Έβ€–β€–π‘¦π‘›βˆ’π‘¦π‘›βˆ’1β€–β€–+||π›Όπ‘›βˆ’π›Όπ‘›βˆ’1||ξ€·β€–β€–π‘Šπ‘›π‘₯π‘›βˆ’1β€–β€–+β€–β€–π‘‡π‘¦π‘›βˆ’1β€–β€–ξ€Έ+π›Όπ‘›βˆ’1β€–β€–π‘Šπ‘›π‘₯π‘›βˆ’1βˆ’π‘Šπ‘›βˆ’1π‘₯π‘›βˆ’1β€–β€–.(3.6) From (2.3), since 𝑇𝑖 and π‘ˆπ‘›,𝑖 are nonexpansive, we have β€–β€–π‘Šπ‘›π‘₯π‘›βˆ’1βˆ’π‘Šπ‘›βˆ’1π‘₯π‘›βˆ’1β€–β€–=β€–β€–πœ‰1𝑇1U𝑛,2π‘₯π‘›βˆ’1βˆ’πœ‰1𝑇1π‘ˆπ‘›βˆ’1,2π‘₯π‘›βˆ’1β€–β€–β‰€πœ‰1β€–β€–π‘ˆπ‘›,2π‘₯π‘›βˆ’1βˆ’π‘ˆπ‘›βˆ’1,2π‘₯π‘›βˆ’1β€–β€–=πœ‰1β€–β€–πœ‰2𝑇2π‘ˆπ‘›,3π‘₯π‘›βˆ’1βˆ’πœ‰2𝑇2π‘ˆπ‘›βˆ’1,3π‘₯π‘›βˆ’1β€–β€–β‰€πœ‰1πœ‰2β€–β€–π‘ˆπ‘›,3π‘₯π‘›βˆ’1βˆ’π‘ˆπ‘›βˆ’1,3π‘₯π‘›βˆ’1β€–β€–β‰€β‹―β‰€πœ‰1πœ‰2β‹―πœ‰π‘›βˆ’1β€–β€–π‘ˆπ‘›,𝑛π‘₯π‘›βˆ’1βˆ’π‘ˆπ‘›βˆ’1,𝑛π‘₯π‘›βˆ’1‖‖≀𝑀1π‘›βˆ’1𝑖=1πœ‰π‘–,(3.7) where 𝑀1 is a constant such that sup𝑛β‰₯1{β€–π‘ˆπ‘›,𝑛π‘₯π‘›βˆ’1βˆ’π‘ˆπ‘›βˆ’1,𝑛π‘₯π‘›βˆ’1β€–}≀𝑀1. Substituting (3.4) and (3.7) into (3.6), we get β€–β€–π‘₯𝑛+1βˆ’π‘₯𝑛‖‖≀𝛼𝑛‖‖π‘₯π‘›βˆ’π‘₯π‘›βˆ’1β€–β€–+ξ€·1βˆ’π›Όπ‘›ξ€Έξ€Ί1βˆ’(1βˆ’π›Ύ)𝛽𝑛‖‖π‘₯π‘›βˆ’π‘₯π‘›βˆ’1β€–β€–+||π›½π‘›βˆ’π›½π‘›βˆ’1||‖‖𝑄π‘₯π‘›βˆ’1β€–β€–+β€–β€–π‘₯π‘›βˆ’1β€–β€–ξ€Έ+||π›Όπ‘›βˆ’π›Όπ‘›βˆ’1||ξ€·β€–β€–π‘Šπ‘›π‘₯π‘›βˆ’1β€–β€–+β€–β€–π‘‡π‘¦π‘›βˆ’1β€–β€–ξ€Έ+π›Όπ‘›βˆ’1𝑀1π‘›βˆ’1𝑖=1πœ‰π‘–=ξ€Ί1βˆ’(1βˆ’π›Ύ)𝛽𝑛1βˆ’π›Όπ‘›β€–β€–π‘₯ξ€Έξ€»π‘›βˆ’π‘₯π‘›βˆ’1β€–β€–+||π›½π‘›βˆ’π›½π‘›βˆ’1||‖‖𝑄π‘₯π‘›βˆ’1β€–β€–+β€–β€–π‘₯π‘›βˆ’1β€–β€–ξ€Έ+||π›Όπ‘›βˆ’π›Όπ‘›βˆ’1||ξ€·β€–β€–π‘Šπ‘›π‘₯π‘›βˆ’1β€–β€–+β€–β€–π‘‡π‘¦π‘›βˆ’1β€–β€–ξ€Έ+π›Όπ‘›βˆ’1𝑀1π‘›βˆ’1𝑖=1πœ‰π‘–.(3.8) Therefore, it follows that β€–β€–π‘₯𝑛+1βˆ’π‘₯𝑛‖‖𝛼𝑛≀1βˆ’(1βˆ’π›Ύ)𝛽𝑛1βˆ’π›Όπ‘›β€–β€–π‘₯ξ€Έξ€»π‘›βˆ’π‘₯π‘›βˆ’1‖‖𝛼𝑛+||π›½π‘›βˆ’π›½π‘›βˆ’1||𝛼𝑛‖‖𝑄π‘₯π‘›βˆ’1β€–β€–+β€–β€–π‘₯π‘›βˆ’1β€–β€–ξ€Έ+||π›Όπ‘›βˆ’π›Όπ‘›βˆ’1||π›Όπ‘›ξ€·β€–β€–π‘Šπ‘›π‘₯π‘›βˆ’1β€–β€–+β€–β€–π‘‡π‘¦π‘›βˆ’1β€–β€–ξ€Έ+π›Όπ‘›βˆ’1𝑀1βˆπ‘›βˆ’1𝑖=1πœ‰π‘–π›Όπ‘›=ξ€Ί1βˆ’(1βˆ’π›Ύ)𝛽𝑛1βˆ’π›Όπ‘›β€–β€–π‘₯ξ€Έξ€»π‘›βˆ’π‘₯π‘›βˆ’1β€–β€–π›Όπ‘›βˆ’1+ξ€Ί1βˆ’(1βˆ’π›Ύ)𝛽𝑛1βˆ’π›Όπ‘›ξ‚΅β€–β€–π‘₯ξ€Έξ€»π‘›βˆ’π‘₯π‘›βˆ’1β€–β€–π›Όπ‘›βˆ’β€–β€–π‘₯π‘›βˆ’π‘₯π‘›βˆ’1β€–β€–π›Όπ‘›βˆ’1ξ‚Ά+||π›½π‘›βˆ’π›½π‘›βˆ’1||𝛼𝑛‖‖𝑄π‘₯π‘›βˆ’1β€–β€–+β€–β€–π‘₯π‘›βˆ’1β€–β€–ξ€Έ+||π›Όπ‘›βˆ’π›Όπ‘›βˆ’1||π›Όπ‘›ξ€·β€–β€–π‘Šπ‘›π‘₯π‘›βˆ’1β€–β€–+β€–β€–π‘‡π‘¦π‘›βˆ’1β€–β€–ξ€Έ+π›Όπ‘›βˆ’1𝑀1βˆπ‘›βˆ’1𝑖=1πœ‰π‘–π›Όπ‘›β‰€ξ€Ί1βˆ’(1βˆ’π›Ύ)𝛽𝑛1βˆ’π›Όπ‘›β€–β€–π‘₯ξ€Έξ€»π‘›βˆ’π‘₯π‘›βˆ’1β€–β€–π›Όπ‘›βˆ’1+||||1π›Όπ‘›βˆ’1π›Όπ‘›βˆ’1||||+||π›Όπ‘›βˆ’π›Όπ‘›βˆ’1||𝛼𝑛+||π›½π‘›βˆ’π›½π‘›βˆ’1||𝛼𝑛+βˆπ‘›βˆ’1𝑖=1πœ‰π‘–π›Όπ‘›ξƒͺ𝑀=ξ€Ί1βˆ’(1βˆ’π›Ύ)𝛽𝑛1βˆ’π›Όπ‘›β€–β€–π‘₯ξ€Έξ€»π‘›βˆ’π‘₯π‘›βˆ’1β€–β€–π›Όπ‘›βˆ’1+(1βˆ’π›Ύ)𝛽𝑛1βˆ’π›Όπ‘›ξ€ΈΓ—ξƒ―π‘€ξ€·(1βˆ’π›Ύ)1βˆ’π›Όπ‘›ξ€Έξƒ©1𝛽𝑛||||1π›Όπ‘›βˆ’1π›Όπ‘›βˆ’1||||+1𝛽𝑛||π›Όπ‘›βˆ’π›Όπ‘›βˆ’1||𝛼𝑛+1𝛽𝑛||π›½π‘›βˆ’π›½π‘›βˆ’1||𝛼𝑛+βˆπ‘›βˆ’1𝑖=1πœ‰π‘–π›Όπ‘›π›½π‘›,ξƒͺξƒ°(3.9) where 𝑀 is a constant such that sup𝑛β‰₯1𝑀1,β€–β€–π‘₯π‘›βˆ’π‘₯π‘›βˆ’1β€–β€–,ξ€·β€–β€–π‘Šπ‘›π‘₯π‘›βˆ’1β€–β€–+β€–β€–π‘‡π‘¦π‘›βˆ’1β€–β€–ξ€Έ,‖‖𝑄π‘₯π‘›βˆ’1β€–β€–+β€–β€–π‘₯π‘›βˆ’1‖‖≀𝑀.(3.10) From (iii), we note that limπ‘›β†’βˆž(1/π›Όπ‘›βˆ’1)|π›Όπ‘›βˆ’π›Όπ‘›βˆ’1/𝛽𝑛𝛼𝑛|=0, which implies that limπ‘›β†’βˆž1𝛽𝑛||π›Όπ‘›βˆ’π›Όπ‘›βˆ’1||𝛼𝑛=0.(3.11) Thus it follows from (iii) and (3.11) that limπ‘›β†’βˆžξƒ©1𝛽𝑛||||1π›Όπ‘›βˆ’1π›Όπ‘›βˆ’1||||+1𝛽𝑛||π›Όπ‘›βˆ’π›Όπ‘›βˆ’1||𝛼𝑛+1𝛽𝑛||π›½π‘›βˆ’π›½π‘›βˆ’1||𝛼𝑛+βˆπ‘›βˆ’1𝑖=1πœ‰π‘–π›Όπ‘›π›½π‘›ξƒͺ=0.(3.12) Hence, applying Lemma 2.4 to (3.9), we immediately conclude that limπ‘›β†’βˆžβ€–β€–π‘₯𝑛+1βˆ’π‘₯𝑛‖‖𝛼𝑛=0.(3.13) This implies that limπ‘›β†’βˆžβ€–β€–π‘₯𝑛+1βˆ’π‘₯𝑛‖‖=0.(3.14) Thus, from (3.1) and (3.14), we have limπ‘›β†’βˆžβ€–β€–π‘₯π‘›βˆ’π‘‡π‘¦π‘›β€–β€–=0.(3.15) At the same time, we note that π‘¦π‘›βˆ’π‘₯𝑛=𝛽𝑛𝑄π‘₯π‘›βˆ’π‘₯π‘›ξ€ΈβŸΆ0.(3.16) Hence we get limπ‘›β†’βˆžβ€–β€–π‘¦π‘›βˆ’π‘‡π‘¦π‘›β€–β€–=0.(3.17) Since the sequence {π‘₯𝑛} is bounded, {𝑦𝑛} is also bounded. Thus there exists a subsequence of {𝑦𝑛}, which is still denoted by {𝑦𝑛} which converges weakly to a point Μƒπ‘₯∈𝐻. Therefore, Μƒπ‘₯∈Fix(𝑇) by (3.17) and Lemma 2.3. By (3.1), we observe that π‘₯𝑛+1βˆ’π‘₯𝑛=π›Όπ‘›ξ€·π‘Šπ‘›π‘₯π‘›βˆ’π‘₯𝑛+ξ€·1βˆ’π›Όπ‘›ξ€Έξ€·π‘‡π‘¦π‘›βˆ’π‘¦π‘›ξ€Έ+ξ€·1βˆ’π›Όπ‘›ξ€Έπ›½π‘›ξ€·π‘„π‘₯π‘›βˆ’π‘₯𝑛,(3.18) that is, π‘₯π‘›βˆ’π‘₯𝑛+1𝛼𝑛=ξ€·πΌβˆ’π‘Šπ‘›ξ€Έπ‘₯𝑛+1βˆ’π›Όπ‘›π›Όπ‘›(πΌβˆ’π‘‡)𝑦𝑛+𝛽𝑛1βˆ’π›Όπ‘›ξ€Έπ›Όπ‘›(πΌβˆ’π‘„)π‘₯𝑛.(3.19) Set 𝑧𝑛=(π‘₯π‘›βˆ’π‘₯𝑛+1)/𝛼𝑛 for each 𝑛β‰₯1, that is, 𝑧𝑛=ξ€·πΌβˆ’π‘Šπ‘›ξ€Έπ‘₯𝑛+1βˆ’π›Όπ‘›π›Όπ‘›(πΌβˆ’π‘‡)𝑦𝑛+𝛽𝑛1βˆ’π›Όπ‘›ξ€Έπ›Όπ‘›(πΌβˆ’π‘„)π‘₯𝑛.(3.20) Using monotonicity of πΌβˆ’π‘‡ and πΌβˆ’π‘Šπ‘›, we derive that, for all π‘’βˆˆFix(𝑇), βŸ¨π‘§π‘›,π‘₯𝑛=βˆ’π‘’βŸ©ξ«ξ€·πΌβˆ’π‘Šπ‘›ξ€Έπ‘₯𝑛,π‘₯𝑛+βˆ’π‘’1βˆ’π›Όπ‘›π›Όπ‘›βŸ¨(πΌβˆ’π‘‡)π‘¦π‘›βˆ’(πΌβˆ’π‘‡)𝑒,𝑦𝑛+βˆ’π‘’βŸ©1βˆ’π›Όπ‘›π›Όπ‘›βŸ¨(πΌβˆ’π‘‡)𝑦𝑛,π‘₯π‘›βˆ’π‘¦π‘›π›½βŸ©+𝑛1βˆ’π›Όπ‘›ξ€Έπ›Όπ‘›βŸ¨(πΌβˆ’π‘„)π‘₯𝑛,π‘₯𝑛β‰₯βˆ’π‘’βŸ©ξ«ξ€·πΌβˆ’π‘Šπ‘›ξ€Έπ‘’,π‘₯𝑛+π›½βˆ’π‘’π‘›ξ€·1βˆ’π›Όπ‘›ξ€Έπ›Όπ‘›βŸ¨(πΌβˆ’π‘„)π‘₯𝑛,π‘₯π‘›ξ€·βˆ’π‘’βŸ©+1βˆ’π›Όπ‘›ξ€Έπ›½π‘›π›Όπ‘›βŸ¨(πΌβˆ’π‘‡)𝑦𝑛,π‘₯π‘›βˆ’π‘„π‘₯π‘›βŸ©=⟨(πΌβˆ’π‘Š)𝑒,π‘₯π‘›βˆ’π‘’βŸ©+ξ«ξ€·π‘Šβˆ’π‘Šπ‘›ξ€Έπ‘’,π‘₯𝑛+π›½βˆ’π‘’π‘›ξ€·1βˆ’π›Όπ‘›ξ€Έπ›Όπ‘›βŸ¨(πΌβˆ’π‘„)π‘₯𝑛,π‘₯𝑛+ξ€·βˆ’π‘’βŸ©1βˆ’π›Όπ‘›ξ€Έπ›½π‘›π›Όπ‘›βŸ¨(πΌβˆ’π‘‡)𝑦𝑛,π‘₯π‘›βˆ’π‘„π‘₯π‘›βŸ©.(3.21) But, since 𝑧𝑛→0,𝛽𝑛/𝛼𝑛→0 and limπ‘›β†’βˆžβ€–π‘Šπ‘›π‘’βˆ’π‘Šπ‘’β€–=0 (by Lemma 2.2), it follows from the above inequality that limsupπ‘›β†’βˆžβŸ¨(πΌβˆ’π‘Š)𝑒,π‘₯π‘›βˆ’π‘’βŸ©β‰€0,βˆ€π‘’βˆˆFix(𝑇).(3.22) This suffices to guarantee that πœ”π‘€(π‘₯𝑛)βŠ‚Ξ©. As a matter of fact, if we take any π‘₯βˆ—βˆˆπœ”π‘€(π‘₯𝑛), then there exists a subsequence {π‘₯𝑛𝑗} of {π‘₯𝑛} such that π‘₯𝑛𝑗⇀π‘₯βˆ—. Therefore, we have ⟨(πΌβˆ’π‘Š)𝑒,π‘₯βˆ—βˆ’π‘’βŸ©=limπ‘—β†’βˆžξ‚¬(πΌβˆ’π‘Š)𝑒,π‘₯π‘›π‘—ξ‚­βˆ’π‘’β‰€0,βˆ€π‘’βˆˆFix(𝑇).(3.23) Note that π‘₯βˆ—βˆˆFix(𝑇). Hence π‘₯βˆ— solves the following problem: π‘₯βˆ—βˆˆFix(𝑇),⟨(πΌβˆ’π‘Š)𝑒,π‘₯βˆ—βˆ’π‘’βŸ©β‰€0,βˆ€π‘’βˆˆFix(𝑇).(3.24) It is obvious that this is equivalent to the problem (1.9) since π‘Šπ‘›β†’π‘Š uniformly in any bounded set (by Lemma 2.2). Thus π‘₯βˆ—βˆˆΞ©.
Let Μƒπ‘₯ be the unique solution of the variational inequality (3.2). Now, take a subsequence {π‘₯𝑛𝑖} of {π‘₯𝑛} such thatlimsupπ‘›β†’βˆžβŸ¨(πΌβˆ’π‘„)Μƒπ‘₯,π‘₯π‘›βˆ’Μƒπ‘₯⟩=limπ‘–β†’βˆžξ«(πΌβˆ’π‘„)Μƒπ‘₯,π‘₯𝑛𝑖.βˆ’Μƒπ‘₯(3.25) Without loss of generality, we may further assume that π‘₯𝑛𝑖⇀π‘₯. Then π‘₯∈Ω. Therefore, we have limsupπ‘›β†’βˆžβŸ¨(πΌβˆ’π‘„)Μƒπ‘₯,π‘₯π‘›ξ«βˆ’Μƒπ‘₯⟩=(πΌβˆ’π‘„)Μƒπ‘₯,π‘₯βˆ’Μƒπ‘₯β‰₯0.(3.26) This completes the proof.

Theorem 3.3. Let 𝐢 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let {𝑇𝑛}βˆžπ‘›=1 be infinite family of nonexpansive mappings of 𝐢 into itself. Let π‘„βˆΆπΆβ†’πΆ be a contraction with coefficient π›Ύβˆˆ[0,1). Assume that the set Ξ© of solutions of the hierarchical problem (1.9) is nonempty. Let {𝛼𝑛},{𝛽𝑛} be two real numbers in (0,1) and {π‘₯𝑛} the sequence generated by (3.1). Assume that the sequence {π‘₯𝑛} is bounded and (i)limπ‘›β†’βˆžπ›Όπ‘›=0, limπ‘›β†’βˆžπ›½π‘›/𝛼𝑛=0 and limπ‘›β†’βˆžπ›Ό2𝑛/𝛽𝑛=0; (ii)βˆ‘βˆžπ‘›=0𝛽𝑛=∞; (iii)limπ‘›β†’βˆž(1/𝛽𝑛)|(1/𝛼𝑛)βˆ’(1/π›Όπ‘›βˆ’1)|=0 and limπ‘›β†’βˆžβˆπ‘›βˆ’1𝑖=1πœ‰π‘–/𝛼𝑛𝛽𝑛 = limπ‘›β†’βˆž(1/𝛼𝑛)|1βˆ’(π›½π‘›βˆ’1/𝛽𝑛)| = 0; (iv)there exists a constant π‘˜>0 such that β€–π‘₯βˆ’π‘‡π‘₯β€–β‰₯π‘˜Dist(π‘₯,Fix(𝑇)), where Dist(π‘₯,Fix(𝑇))=infπ‘¦βˆˆFix(𝑇)β€–π‘₯βˆ’π‘¦β€–.(3.27) Then the sequence {π‘₯𝑛} defined by (3.1) converges strongly to a point Μƒπ‘₯∈Fix(𝑇), which solves the variational inequality problem (3.2).

Proof. From (3.1), we have π‘₯𝑛+1βˆ’Μƒπ‘₯=π›Όπ‘›ξ€·π‘Šπ‘›π‘₯π‘›βˆ’π‘Šπ‘›ξ€ΈΜƒπ‘₯+π›Όπ‘›ξ€·π‘Šπ‘›ξ€Έ+ξ€·Μƒπ‘₯βˆ’Μƒπ‘₯1βˆ’π›Όπ‘›ξ€Έξ€·π‘‡π‘¦π‘›ξ€Έβˆ’Μƒπ‘₯.(3.28) Thus we have β€–β€–π‘₯𝑛+1β€–β€–βˆ’Μƒπ‘₯2β‰€β€–β€–π›Όπ‘›ξ€·π‘Šπ‘›π‘₯π‘›βˆ’π‘Šπ‘›ξ€Έ+ξ€·Μƒπ‘₯1βˆ’π›Όπ‘›ξ€Έξ€·π‘‡π‘¦π‘›ξ€Έβ€–β€–βˆ’Μƒπ‘₯2+2π›Όπ‘›ξ«π‘Šπ‘›Μƒπ‘₯βˆ’Μƒπ‘₯,π‘₯𝑛+1ξ¬β‰€ξ€·βˆ’Μƒπ‘₯1βˆ’π›Όπ‘›ξ€Έβ€–β€–π‘‡π‘¦π‘›β€–β€–βˆ’Μƒπ‘₯2+π›Όπ‘›β€–β€–π‘Šπ‘›π‘₯π‘›βˆ’π‘Šπ‘›β€–β€–Μƒπ‘₯2+2π›Όπ‘›ξ«π‘Šπ‘›Μƒπ‘₯βˆ’Μƒπ‘₯,π‘₯𝑛+1ξ¬β‰€ξ€·βˆ’Μƒπ‘₯1βˆ’π›Όπ‘›ξ€Έβ€–β€–π‘¦π‘›β€–β€–βˆ’Μƒπ‘₯2+𝛼𝑛‖‖π‘₯π‘›β€–β€–βˆ’Μƒπ‘₯2+2π›Όπ‘›ξ«π‘Šπ‘›Μƒπ‘₯βˆ’Μƒπ‘₯,π‘₯𝑛+1.βˆ’Μƒπ‘₯(3.29) At the same time, we observe that β€–β€–π‘¦π‘›β€–β€–βˆ’Μƒπ‘₯2=β€–β€–ξ€·1βˆ’π›½π‘›π‘₯ξ€Έξ€·π‘›ξ€Έβˆ’Μƒπ‘₯+𝛽𝑛𝑄π‘₯π‘›ξ€Έβˆ’π‘„Μƒπ‘₯+𝛽𝑛‖‖(𝑄̃π‘₯βˆ’Μƒπ‘₯)2≀‖‖1βˆ’π›½π‘›π‘₯ξ€Έξ€·π‘›ξ€Έβˆ’Μƒπ‘₯+𝛽𝑛𝑄π‘₯π‘›ξ€Έβ€–β€–βˆ’π‘„Μƒπ‘₯2+2π›½π‘›βŸ¨π‘„Μƒπ‘₯βˆ’Μƒπ‘₯,π‘¦π‘›β‰€ξ€·βˆ’Μƒπ‘₯⟩1βˆ’π›½π‘›ξ€Έβ€–β€–π‘₯π‘›β€–β€–βˆ’Μƒπ‘₯2+𝛽𝑛‖‖𝑄π‘₯π‘›β€–β€–βˆ’π‘„Μƒπ‘₯2+2π›½π‘›βŸ¨π‘„Μƒπ‘₯βˆ’Μƒπ‘₯,π‘¦π‘›β‰€ξ€·βˆ’Μƒπ‘₯⟩1βˆ’π›½π‘›ξ€Έβ€–β€–π‘₯π‘›β€–β€–βˆ’Μƒπ‘₯2+𝛽𝑛𝛾2β€–β€–π‘₯π‘›β€–β€–βˆ’Μƒπ‘₯2+2π›½π‘›βŸ¨π‘„Μƒπ‘₯βˆ’Μƒπ‘₯,𝑦𝑛=ξ€Ίξ€·βˆ’Μƒπ‘₯⟩1βˆ’1βˆ’π›Ύ2𝛽𝑛‖‖π‘₯π‘›β€–β€–βˆ’Μƒπ‘₯2+2π›½π‘›βŸ¨π‘„Μƒπ‘₯βˆ’Μƒπ‘₯,π‘¦π‘›βˆ’Μƒπ‘₯⟩.(3.30) Substituting (3.30) into (3.29), we get β€–β€–π‘₯𝑛+1β€–β€–βˆ’Μƒπ‘₯2≀𝛼𝑛‖‖π‘₯π‘›β€–β€–βˆ’Μƒπ‘₯2+ξ€·1βˆ’π›Όπ‘›ξ€·ξ€Έξ€Ί1βˆ’1βˆ’π›Ύ2𝛽𝑛‖‖π‘₯π‘›β€–β€–βˆ’Μƒπ‘₯2+2𝛽𝑛1βˆ’π›Όπ‘›ξ€ΈβŸ¨π‘„Μƒπ‘₯βˆ’Μƒπ‘₯,π‘¦π‘›βˆ’Μƒπ‘₯⟩+2π›Όπ‘›ξ«π‘Šπ‘›Μƒπ‘₯βˆ’Μƒπ‘₯,π‘₯𝑛+1=ξ€Ίξ€·βˆ’Μƒπ‘₯1βˆ’1βˆ’π›Ύ2𝛽𝑛1βˆ’π›Όπ‘›β€–β€–π‘₯ξ€Έξ€»π‘›β€–β€–βˆ’Μƒπ‘₯2+2𝛽𝑛1βˆ’π›Όπ‘›ξ€ΈβŸ¨π‘„Μƒπ‘₯βˆ’Μƒπ‘₯,π‘¦π‘›βˆ’Μƒπ‘₯⟩+2π›Όπ‘›ξ«π‘Šπ‘›Μƒπ‘₯βˆ’Μƒπ‘₯,π‘₯𝑛+1=ξ€Ίξ€·βˆ’Μƒπ‘₯1βˆ’1βˆ’π›Ύ2𝛽𝑛1βˆ’π›Όπ‘›β€–β€–π‘₯ξ€Έξ€»π‘›β€–β€–βˆ’Μƒπ‘₯2+ξ€·1βˆ’π›Ύ2𝛽𝑛1βˆ’π›Όπ‘›ξ€ΈΓ—ξƒ―21βˆ’π›Ύ2βŸ¨π‘„Μƒπ‘₯βˆ’Μƒπ‘₯,𝑦𝑛2βˆ’Μƒπ‘₯⟩+ξ€·1βˆ’π›Ύ2ξ€Έξ€·1βˆ’π›Όπ‘›ξ€ΈΓ—π›Όπ‘›π›½π‘›ξ«π‘Šπ‘›Μƒπ‘₯βˆ’Μƒπ‘₯,π‘₯𝑛+1.βˆ’Μƒπ‘₯(3.31) By Theorem 3.2, we note that every weak cluster point of the sequence {π‘₯𝑛} is in Ξ©. Since π‘¦π‘›βˆ’π‘₯𝑛→0, then every weak cluster point of {𝑦𝑛} is also in Ξ©. Consequently, since Μƒπ‘₯=projΞ©(𝑄̃π‘₯), we easily have limsupπ‘›β†’βˆžβŸ¨π‘„Μƒπ‘₯βˆ’Μƒπ‘₯,π‘¦π‘›βˆ’Μƒπ‘₯βŸ©β‰€0.(3.32)
On the other hand, we observe thatξ«π‘Šπ‘›Μƒπ‘₯βˆ’Μƒπ‘₯,π‘₯𝑛+1=ξ«π‘Šβˆ’Μƒπ‘₯𝑛̃π‘₯βˆ’Μƒπ‘₯,projFix(𝑇)π‘₯𝑛+1+ξ«π‘Šβˆ’Μƒπ‘₯𝑛̃π‘₯βˆ’Μƒπ‘₯,π‘₯𝑛+1βˆ’projFix(𝑇)π‘₯𝑛+1.(3.33) Since Μƒπ‘₯ is a solution of the problem (1.9) and projFix(𝑇)π‘₯𝑛+1∈Fix(𝑇), we have ξ«π‘Šπ‘›Μƒπ‘₯βˆ’Μƒπ‘₯,projFix(𝑇)π‘₯𝑛+1ξ¬βˆ’Μƒπ‘₯≀0.(3.34) Thus it follows that ξ«π‘Šπ‘›Μƒπ‘₯βˆ’Μƒπ‘₯,π‘₯𝑛+1ξ¬β‰€ξ«π‘Šβˆ’Μƒπ‘₯𝑛̃π‘₯βˆ’Μƒπ‘₯,π‘₯𝑛+1βˆ’projFix(𝑇)π‘₯𝑛+1ξ¬β‰€β€–β€–π‘Šπ‘›β€–β€–β€–β€–π‘₯Μƒπ‘₯βˆ’Μƒπ‘₯𝑛+1βˆ’projFix(𝑇)π‘₯𝑛+1β€–β€–=β€–β€–π‘Šπ‘›β€–β€–ξ€·π‘₯Μƒπ‘₯βˆ’Μƒπ‘₯Γ—Dist𝑛+1≀1,Fix(𝑇)π‘˜β€–β€–π‘Šπ‘›β€–β€–β€–β€–π‘₯Μƒπ‘₯βˆ’Μƒπ‘₯𝑛+1βˆ’π‘‡π‘₯𝑛+1β€–β€–.(3.35) We note that β€–β€–π‘₯𝑛+1βˆ’π‘‡π‘₯𝑛+1‖‖≀‖‖π‘₯𝑛+1βˆ’π‘‡π‘₯𝑛‖‖+‖‖𝑇π‘₯π‘›βˆ’π‘‡π‘₯𝑛+1β€–β€–β‰€π›Όπ‘›β€–β€–π‘Šπ‘›π‘₯π‘›βˆ’π‘‡π‘₯𝑛‖‖+ξ€·1βˆ’π›Όπ‘›ξ€Έβ€–β€–π‘‡π‘¦π‘›βˆ’π‘‡π‘₯𝑛‖‖+β€–β€–π‘₯𝑛+1βˆ’π‘₯π‘›β€–β€–β‰€π›Όπ‘›β€–β€–π‘Šπ‘›π‘₯π‘›βˆ’π‘‡π‘₯𝑛‖‖+β€–β€–π‘¦π‘›βˆ’π‘₯𝑛‖‖+β€–β€–π‘₯𝑛+1βˆ’π‘₯π‘›β€–β€–β‰€π›Όπ‘›β€–β€–π‘Šπ‘›π‘₯π‘›βˆ’π‘‡π‘₯𝑛‖‖+𝛽𝑛‖‖𝑄π‘₯π‘›βˆ’π‘₯𝑛‖‖+β€–β€–π‘₯𝑛+1βˆ’π‘₯𝑛‖‖.(3.36) Hence we have π›Όπ‘›π›½π‘›ξ«π‘Šπ‘›Μƒπ‘₯βˆ’Μƒπ‘₯,π‘₯𝑛+1ξ¬β‰€π›Όβˆ’Μƒπ‘₯2𝑛𝛽𝑛1π‘˜β€–β€–π‘Šπ‘›β€–β€–β€–β€–π‘ŠΜƒπ‘₯βˆ’Μƒπ‘₯𝑛π‘₯π‘›βˆ’π‘‡π‘₯𝑛‖‖+𝛼𝑛1π‘˜β€–β€–π‘Šπ‘›β€–β€–β€–β€–Μƒπ‘₯βˆ’Μƒπ‘₯𝑄π‘₯π‘›βˆ’π‘₯𝑛‖‖+𝛼2𝑛𝛽𝑛‖‖π‘₯𝑛+1βˆ’π‘₯𝑛‖‖𝛼𝑛1π‘˜β€–β€–π‘Šπ‘›β€–β€–ξ‚.Μƒπ‘₯βˆ’Μƒπ‘₯(3.37) From Theorem 3.2, we have limπ‘›β†’βˆžβ€–π‘₯𝑛+1βˆ’π‘₯𝑛‖/𝛼𝑛=0. At the same time, we note that {(1/π‘˜)β€–π‘Šπ‘›Μƒπ‘₯βˆ’Μƒπ‘₯β€–β€–π‘Šπ‘›π‘₯π‘›βˆ’π‘‡π‘₯𝑛‖}, {(1/π‘˜)β€–π‘Šπ‘›Μƒπ‘₯βˆ’Μƒπ‘₯‖‖𝑄π‘₯π‘›βˆ’π‘₯𝑛‖}, and {(1/π‘˜)β€–π‘Šπ‘›Μƒπ‘₯βˆ’Μƒπ‘₯β€–} are all bounded. Hence it follows from (i) and the above inequality that limsupπ‘›β†’βˆžπ›Όπ‘›π›½π‘›ξ«π‘Šπ‘›Μƒπ‘₯βˆ’Μƒπ‘₯,π‘₯𝑛+1ξ¬βˆ’Μƒπ‘₯≀0.(3.38)
Finally, by (3.31)–(3.38) and Lemma 2.4, we conclude that the sequence {π‘₯𝑛} converges strongly to a point Μƒπ‘₯∈Fix(𝑇). This completes the proof.

Remark 3.4. In the present paper, we consider the hierarchical problem (1.9) which includes the hierarchical problem (1.1) as a special case.
From the above discussion, we can easily deduce the following result.

Algorithm 3.5. Let 𝐢 be a nonempty closed convex subset of a real Hilbert space 𝐻 and 𝑆 a nonexpansive mapping of 𝐢 into itself. Let π‘„βˆΆπΆβ†’πΆ be a contraction with coefficient π›Ύβˆˆ[0,1). For any π‘₯0∈𝐢, let{π‘₯𝑛} the sequence generated iteratively by π‘₯𝑛+1=𝛼𝑛𝑆π‘₯𝑛+ξ€·1βˆ’π›Όπ‘›ξ€Έπ‘‡ξ€·π›½π‘›π‘„π‘₯𝑛+ξ€·1βˆ’π›½π‘›ξ€Έπ‘₯𝑛,βˆ€π‘›β‰₯0,(3.39) where {𝛼𝑛},{𝛽𝑛} are two real numbers in (0,1).

Corollary 3.6. Let 𝐢 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let SβˆΆπΆβ†’πΆ be a nonexpansive mapping. Let π‘„βˆΆπΆβ†’πΆ be a contraction with coefficient π›Ύβˆˆ[0,1). Assume that the set Ξ©ξ…ž of solutions of the hierarchical problem (1.1) is nonempty. Let {𝛼𝑛},{𝛽𝑛} be two real numbers in (0,1) and {π‘₯𝑛} the sequence generated by (3.1). Assume that the sequence {π‘₯𝑛} is bounded and (i)limπ‘›β†’βˆžπ›Όπ‘›=0, limπ‘›β†’βˆžπ›½π‘›/𝛼𝑛=0 and limπ‘›β†’βˆžπ›Ό2𝑛/𝛽𝑛=0; (ii)βˆ‘βˆžπ‘›=0𝛽𝑛=∞; (iii)limπ‘›β†’βˆž(1/𝛽𝑛)|(1/𝛼𝑛)βˆ’(1/π›Όπ‘›βˆ’1)|=0 and limπ‘›β†’βˆž(1/𝛼𝑛)|1βˆ’(π›½π‘›βˆ’1/𝛽𝑛)|=0; (iv)there exists a constant π‘˜>0 such that β€–π‘₯βˆ’π‘‡π‘₯β€–β‰₯π‘˜Dist(π‘₯,Fix(𝑇)), where Dist(π‘₯,Fix(𝑇))=infπ‘¦βˆˆFix(𝑇)β€–π‘₯βˆ’π‘¦β€–.(3.40) Then the sequence {π‘₯𝑛} defined by (3.39) converges strongly to a point Μƒπ‘₯∈Fix(𝑇), which solves the hierarchical problem (1.1).

Acknowledgment

This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant no. 2011-0021821).

References

  1. A. Moudafi and P. E. Mainge, β€œTowards viscosity approximations of hierarchical fixed-point problems,” Fixed Point Theory and Applications, vol. 2006, Article ID 95453, 10 pages, 2006. View at: Publisher Site | Google Scholar
  2. A. Moudafi, β€œKrasnosel'skii-Mann iteration for hierarchical fixed-point problems,” Inverse Problems, vol. 23, no. 4, pp. 1635–1640, 2007. View at: Publisher Site | Google Scholar
  3. P. E. Mainge and A. Moudafi, β€œStrong convergence of an iterative method for hierarchical fixed-point problems,” Pacific Journal of Optimization, vol. 3, no. 3, pp. 529–538, 2007. View at: Google Scholar
  4. X. Lu, H. K. Xu, and X. Yin, β€œHybrid methods for a class of monotone variational inequalities,” Nonlinear Analysis, vol. 71, no. 3-4, pp. 1032–1041, 2009. View at: Publisher Site | Google Scholar
  5. G. Marino and H. K. Xu, β€œExplicit hierarchical fixed point approach to variational inequalities,” Journal of Optimization Theory and Applications, vol. 149, no. 1, pp. 61–78, 2011. View at: Publisher Site | Google Scholar
  6. H. K. Xu, β€œViscosity method for hierarchical fixed point approach to variational inequalities,” Taiwanese Journal of Mathematics, vol. 14, no. 2, pp. 463–478, 2009. View at: Google Scholar
  7. F. Cianciaruso, V. Colao, L. Muglia, and H. K. Xu, β€œOn an implicit hierarchical fixed point approach to variational inequalities,” Bulletin of the Australian Mathematical Society, vol. 80, no. 1, pp. 117–124, 2009. View at: Publisher Site | Google Scholar
  8. Y. Yao and Y. C. Liou, β€œAn implicit extragradient method for hierarchical variational inequalities,” Fixed Point Theory and Applications, vol. 2011, Article ID 697248, 11 pages, 2011. View at: Publisher Site | Google Scholar
  9. Y. Yao, Y. C. Liou, and C. P. Chen, β€œHierarchical convergence of a double-net algorithm for equilibrium problems and variational inequality problems,” Fixed Point Theory and Applications, vol. 2010, Article ID 642584, 16 pages, 2010. View at: Publisher Site | Google Scholar
  10. Y. Yao, Y. J. Cho, and Y. C. Liou, β€œIterative algorithms for hierarchical fixed points problems and variational inequalities,” Mathematical and Computer Modelling, vol. 52, no. 9-10, pp. 1697–1705, 2010. View at: Publisher Site | Google Scholar
  11. F. Cianciaruso, G. Marino, L. Muglia, and Y. Yao, β€œOn a two-step algorithm for hierarchical fixed point problems and variational inequalities,” Journal of Inequalities and Applications, vol. 2009, Article ID 208692, 13 pages, 2009. View at: Publisher Site | Google Scholar
  12. G. Marino, V. Colao, L. Muglia, and Y. Yao, β€œKrasnosel'skii-Mann iteration for hierarchical fixed points and equilibrium problem,” Bulletin of the Australian Mathematical Society, vol. 79, no. 2, pp. 187–200, 2009. View at: Publisher Site | Google Scholar
  13. Y. Yao, Y. C. Liou, and G. Marino, β€œTwo-step iterative algorithms for hierarchical fixed point problems and variational inequality problems,” Journal of Applied Mathematics and Computing, vol. 31, no. 1-2, pp. 433–445, 2009. View at: Publisher Site | Google Scholar
  14. Y. Yao, Y. J. Cho, and Y. C. Liou, β€œHierarchical convergence of an implicit double-net algorithm for nonexpanseive semigroups and variational inequality problems,” Fixed Point Theory and Applications, vol. 2011, no. 101, 2011. View at: Publisher Site | Google Scholar
  15. G. Gou, S. Wang, and Y. J. Cho, β€œStrong convergence algorithms for hierarchical fixed point problems and variational inequalities,” Journal Applied Mathematics, vol. 2011, 17 pages, 2011. View at: Publisher Site | Google Scholar
  16. I. Yamada and N. Ogura, β€œHybrid steepest descent method for variational inequality problem over the fixed point set of certain quasi-nonexpansive mappings,” Numerical Functional Analysis and Optimization, vol. 25, no. 7-8, pp. 619–655, 2004. View at: Publisher Site | Google Scholar
  17. Z. Q. Luo, J. S. Pang, and D. Ralph, Mathematical Programs with Equilibrium Constraints, Cambridge University Press, Cambridge, UK, 1996.
  18. A. Cabot, β€œProximal point algorithm controlled by a slowly vanishing term: applications to hierarchical minimization,” SIAM Journal on Optimization, vol. 15, no. 2, pp. 555–572, 2005. View at: Publisher Site | Google Scholar
  19. M. Solodov, β€œAn explicit descent method for bilevel convex optimization,” Journal of Convex Analysis, vol. 14, no. 2, pp. 227–237, 2007. View at: Google Scholar
  20. K. Shimoji and W. Takahashi, β€œStrong convergence to common fixed points of infinite nonexpansive mappings and applications,” Taiwanese Journal of Mathematics, vol. 5, no. 2, pp. 387–404, 2001. View at: Google Scholar
  21. Y. Yao, Y. C. Liou, and J. C. Yao, β€œConvergence theorem for equilibrium problems and fixed point problems of infinite family of nonexpansive mappings,” Fixed Point Theory and Applications, vol. 2007, Article ID 64363, 12 pages, 2007. View at: Publisher Site | Google Scholar
  22. K. Goebel and W. A. Kirk, Topics in Metric Fixed Point Theory, vol. 28 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, UK, 1990. View at: Publisher Site
  23. H. K. Xu, β€œIterative algorithms for nonlinear operators,” Journal of the London Mathematical Society, vol. 66, no. 1, pp. 240–256, 2002. View at: Publisher Site | Google Scholar

Copyright Β© 2012 Yonghong Yao 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.


More related articles

646Β Views | 386Β Downloads | 3Β Citations
 PDF  Download Citation  Citation
 Download other formatsMore
 Order printed copiesOrder

Related articles

We are committed to sharing findings related to COVID-19 as quickly and safely as possible. Any author submitting a COVID-19 paper should notify us at help@hindawi.com to ensure their research is fast-tracked and made available on a preprint server as soon as possible. We will be providing unlimited waivers of publication charges for accepted articles related to COVID-19. Sign up here as a reviewer to help fast-track new submissions.