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 š» be a real Hilbert space with inner product āŸØā‹…,ā‹…āŸ© and norm ā€–ā‹…ā€–, respectively. Let š¶ be a nonempty closed convex subset of š». The hierarchical problem is of finding Ģƒš‘„āˆˆFix(š‘‡) such thatāŸØš‘†Ģƒš‘„āˆ’Ģƒš‘„,š‘„āˆ’Ģƒš‘„āŸ©ā‰¤0,āˆ€š‘„āˆˆFix(š‘‡),(1.1) where š‘†,š‘‡ are two nonexpansive mappings and Fix(š‘‡) is the set of fixed points of š‘‡. Recently, this problem has been studied by many authors (see, e.g., [1ā€“15]). The main reason is that this problem is closely associated with some monotone variational inequalities and convex programming problems (see [16ā€“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:š‘„š‘”,š‘ ī€·š‘„=š‘ š‘„š‘”,š‘ ī€ø+ī€ŗī€·š‘„(1āˆ’š‘ )š‘”š‘†š‘”,š‘ ī€ø+ī€·š‘„(1āˆ’š‘”)š‘‡š‘”,š‘ ī€øī€»(1.2) and proved that the net {š‘„š‘”,š‘ } defined by (1.2) strongly converges to š‘„š‘” as š‘ ā†’0, where š‘„š‘” satisfies š‘„š‘”=projFix(š‘ƒš‘”)š‘„(š‘„š‘”), where š‘ƒš‘”āˆ¶š¶ā†’š¶ is a mapping defined by š‘ƒš‘”(š‘„)=š‘”š‘†(š‘„)+(1āˆ’š‘”)š‘‡(š‘„),āˆ€š‘„āˆˆš¶,š‘”āˆˆ(0,1),(1.3) or, equivalently, š‘„š‘” is the unique solution of the quasivariational inequality 0āˆˆ(š¼āˆ’š‘„)š‘„š‘”+š‘Fix(š‘ƒš‘”)ī€·š‘„š‘”ī€ø,(1.4) where the normal cone to Fix(š‘ƒš‘”), š‘Fix(š‘ƒš‘”), is defined as follows: š‘Fix(š‘ƒš‘”)ī‚»ī€·š‘ƒāˆ¶š‘„āŸ¶{š‘¢āˆˆš»āˆ¶āŸØš‘¦āˆ’š‘„,š‘¢āŸ©ā‰¤0},ifš‘„āˆˆFixš‘”ī€ø,āˆ…,otherwise.(1.5)

Moreover, as š‘”ā†’0, the net {š‘„š‘”} in turn weakly converges to the unique solution š‘„āˆž of the fixed point equation š‘„āˆž=projĪ©š‘„(š‘„āˆž) or, equivalently, š‘„āˆž is the unique solution of the variational inequality 0āˆˆ(š¼āˆ’š‘„)š‘„āˆž+š‘Ī©ī€·š‘„āˆžī€ø.(1.6)

Recently, Moudafi [2] constructed an explicit iterative algorithm:š‘„š‘›+1=ī€·1āˆ’š›æš‘›ī€øš‘„š‘›+š›æš‘›ī€·šœŽš‘›š‘†š‘„š‘›+ī€·1āˆ’šœŽš‘›ī€øš‘‡š‘„š‘›ī€ø,āˆ€š‘›ā‰„0,(1.7) where {š›æš‘›} and {šœŽš‘›} 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:š‘„š‘›+1=ī€·1āˆ’š›æš‘›ī€øš‘„š‘„š‘›+š›æš‘›ī€ŗšœŽš‘›š‘†š‘„š‘›+ī€·1āˆ’šœŽš‘›ī€øš‘‡š‘„š‘›ī€»,āˆ€š‘›ā‰„0,(1.8) where {š›æš‘›} and {šœŽš‘›} are two real numbers in (0,1), and proved that, under appropriate conditions, the iterative sequence {š‘„š‘›} generated by (1.8) has strong convergence.

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

Motivated and inspired by the results in the literature, in this paper, we consider a general hierarchical problem of finding Ģƒš‘„āˆˆFix(š‘‡) such that, for any š‘›ā‰„1,āŸØš‘Šš‘›Ģƒš‘„āˆ’Ģƒš‘„,š‘„āˆ’Ģƒš‘„āŸ©ā‰¤0,āˆ€š‘„āˆˆFix(š‘‡),(1.9) where š‘Šš‘› is the š‘Š-mapping defined by (2.3) below and š‘‡ is a nonexpansive mapping, and introduce an explicit iterative algorithm which converges strongly to a solution Ģƒš‘„ of the hierarchical problem (1.9).

2. Preliminaries

Let š¶ a nonempty closed convex subset of a real Hilbert space š». Recall that a mapping š‘„āˆ¶š¶ā†’š¶ is said to be contractive if there exists a constant š›¾āˆˆ(0,1) such that ā€–š‘„š‘„āˆ’š‘„š‘¦ā€–ā‰¤š›¾ā€–š‘„āˆ’š‘¦ā€–,āˆ€š‘„,š‘¦āˆˆš¶.(2.1)

A mapping š‘‡āˆ¶š¶ā†’š¶ is called nonexpansive ifā€–š‘‡š‘„āˆ’š‘‡š‘¦ā€–ā‰¤ā€–š‘„āˆ’š‘¦ā€–,āˆ€š‘„,š‘¦āˆˆš¶.(2.2)

Forward, we use Fix(š‘‡) to denote the fixed points set of š‘‡.

Let {š‘‡š‘–}āˆžš‘–=1āˆ¶š¶ā†’š¶ be an infinite family of nonexpansive mappings and {šœ‰š‘–}āˆžš‘–=1 a real number sequence such that 0ā‰¤šœ‰š‘–ā‰¤1 for each š‘–ā‰„1.

For each š‘›ā‰„1, define a mapping š‘Šš‘›āˆ¶š¶ā†’š¶ as follows:š‘ˆš‘›,š‘›+1š‘ˆ=š¼,š‘›,š‘›=šœ‰š‘›š‘‡š‘›š‘ˆš‘›,š‘›+1+ī€·1āˆ’šœ‰š‘›ī€øš‘ˆš¼,š‘›,š‘›āˆ’1=šœ‰š‘›āˆ’1š‘‡š‘›āˆ’1š‘ˆš‘›,š‘›+ī€·1āˆ’šœ‰š‘›āˆ’1ī€øā‹Æš‘ˆš¼,š‘›,š‘˜=šœ‰š‘˜š‘‡š‘˜š‘ˆš‘›,š‘˜+1+ī€·1āˆ’šœ‰š‘˜ī€øš‘ˆš¼,š‘›,š‘˜āˆ’1=šœ‰š‘˜āˆ’1š‘‡š‘˜āˆ’1š‘ˆš‘›,š‘˜+ī€·1āˆ’šœ‰š‘˜āˆ’1ī€øā‹Æš‘ˆš¼,š‘›,2=šœ‰2š‘‡2š‘ˆš‘›,3+ī€·1āˆ’šœ‰2ī€øš‘Šš¼,š‘›=š‘ˆš‘›,1=šœ‰1š‘‡1š‘ˆš‘›,2+ī€·1āˆ’šœ‰1ī€øš¼.(2.3)

Such š‘Šš‘› is called the š‘Š-mapping generated by {š‘‡š‘–}āˆžš‘–=1 and {šœ‰š‘–}āˆžš‘–=1.

Lemma 2.1 (see [20]). Let š¶ be a nonempty closed convex subset of a real Hilbert space š». Let {š‘‡š‘–}āˆžš‘–=1 be an infinite family of nonexpansive mappings of š¶ into itself with ā‹‚āˆžš‘›=1Fix(š‘‡š‘›)ā‰ āˆ…. Let šœ‰1,šœ‰2,ā€¦ be real numbers such that 0<šœ‰š‘–ā‰¤š‘<1 for each š‘–ā‰„1. Then one has the following results: (1)for any š‘„āˆˆš¶ and š‘˜ā‰„1, the limit limš‘›ā†’āˆžš‘ˆš‘›,š‘˜š‘„ exists; (2)ā‹‚Fix(š‘Š)=āˆžš‘›=1Fix(š‘‡š‘›).

Using Lemma ā€‰3.1 in [21], we can define a mapping š‘Š of š¶ into itself by š‘Šš‘„=limš‘›ā†’āˆžš‘Šš‘›š‘„=limš‘›ā†’āˆžš‘ˆš‘›,1š‘„ for all š‘„āˆˆš¶. Thus we have the following.

Lemma 2.2 (see [21]). If {š‘„š‘›} is a bounded sequence in š¶, then one has limš‘›ā†’āˆžā€–ā€–š‘Šš‘„š‘›āˆ’š‘Šš‘›š‘„š‘›ā€–ā€–=0.(2.4)

Lemma 2.3 (see [22]). Let š¶ be a nonempty closed convex of a real Hilbert space š» and š‘‡āˆ¶š¶ā†’š¶ be nonexpansive mapping. Then š‘‡ is demiclosed on š¶, that is, if š‘„š‘›ā‡€š‘„āˆˆš¶ and š‘„š‘›āˆ’š‘‡š‘„š‘›ā†’0, then š‘„=š‘‡š‘„.

Lemma 2.4 (see [23]). Assume {š‘Žš‘›} is a sequence of nonnegative real numbers such that š‘Žš‘›+1ā‰¤ī€·1āˆ’š›¾š‘›ī€øš‘Žš‘›+š›æš‘›š›¾š‘›+šœ‚š‘›,āˆ€š‘›ā‰„1,(2.5) where {š›¾š‘›} is a sequence in (0,1) and {š›æš‘›},{šœ‚š‘›} are two sequences such that (i)āˆ‘āˆžš‘›=1š›¾š‘›=āˆž;(ii)limsupš‘›ā†’āˆžš›æš‘›ā‰¤0 or āˆ‘āˆžš‘›=1|š›æš‘›š›¾š‘›|<āˆž;(iii)āˆ‘āˆžš‘›=1|šœ‚š‘›|<āˆž.
Then limš‘›ā†’āˆžš‘Žš‘›=0.

3. Main Results

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

Algorithm 3.1. Let š¶ be a nonempty closed convex subset of a real Hilbert space š» and {š‘‡š‘›}āˆžš‘›=1 be infinite family of nonexpansive mappings of š¶ into itself. Let š‘„āˆ¶š¶ā†’š¶ be a contraction with coefficient š›¾āˆˆ[0,1). For any š‘„0āˆˆš¶, let {š‘„š‘›} the sequence generated iteratively by š‘„š‘›+1=š›¼š‘›š‘Šš‘›š‘„š‘›+ī€·1āˆ’š›¼š‘›ī€øš‘‡ī€·š›½š‘›š‘„š‘„š‘›+ī€·1āˆ’š›½š‘›ī€øš‘„š‘›ī€ø,āˆ€š‘›ā‰„0,(3.1) where {š›¼š‘›},{š›½š‘›} are two real numbers in (0,1) and š‘Šš‘› is the š‘Š-mapping defined by (2.3).

Now, we give the convergence analysis of the algorithm.

Theorem 3.2. Let š¶ be a nonempty closed convex subset of a real Hilbert space š» and {š‘‡š‘›}āˆžš‘›=1 be an infinite family of nonexpansive mappings of š¶ into itself. Let š‘„āˆ¶š¶ā†’š¶ be a contraction with coefficient š›¾āˆˆ[0,1). Assume that the set Ī© of solutions of the hierarchical problem (1.9) is nonempty. Let {š›¼š‘›},{š›½š‘›} be two real numbers in (0,1) and {š‘„š‘›} the sequence generated by (3.1). Assume that the sequence {š‘„š‘›} is bounded and (i)limš‘›ā†’āˆžš›¼š‘›=0 and limš‘›ā†’āˆž(š›½š‘›/š›¼š‘›)=0; (ii)āˆ‘āˆžš‘›=0š›½š‘›=āˆž; (iii)limš‘›ā†’āˆž(1/š›½š‘›)|(1/š›¼š‘›)āˆ’(1/š›¼š‘›āˆ’1)|=0 and limš‘›ā†’āˆž(āˆš‘›āˆ’1š‘–=1šœ‰š‘–/š›¼š‘›š›½š‘›)=limš‘›ā†’āˆž(1/š›¼š‘›)|1āˆ’(š›½š‘›āˆ’1/š›½š‘›)|=0.
Then limš‘›ā†’āˆž(ā€–š‘„š‘›+1āˆ’š‘„š‘›ā€–/š›¼š‘›)=0 and every weak cluster point of the sequence {š‘„š‘›} solves the following variational inequalityĢƒš‘„āˆˆĪ©,āŸØ(š¼āˆ’š‘„)Ģƒš‘„,š‘„āˆ’Ģƒš‘„āŸ©ā‰„0,āˆ€š‘„āˆˆĪ©.(3.2)

Proof. Set š‘¦š‘›=š›½š‘›š‘„š‘„š‘›+(1āˆ’š›½š‘›)š‘„š‘› for each š‘›ā‰„0. Then we have š‘¦š‘›āˆ’š‘¦š‘›āˆ’1=š›½š‘›š‘„š‘„š‘›+ī€·1āˆ’š›½š‘›ī€øš‘„š‘›āˆ’š›½š‘›āˆ’1š‘„š‘„š‘›āˆ’1āˆ’ī€·1āˆ’š›½š‘›āˆ’1ī€øš‘„š‘›āˆ’1=š›½š‘›ī€·š‘„š‘„š‘›āˆ’š‘„š‘„š‘›āˆ’1ī€ø+ī€·š›½š‘›āˆ’š›½š‘›āˆ’1ī€øš‘„š‘„š‘›āˆ’1+ī€·1āˆ’š›½š‘›š‘„ī€øī€·š‘›āˆ’š‘„š‘›āˆ’1ī€ø+ī€·š›½š‘›āˆ’1āˆ’š›½š‘›ī€øš‘„š‘›āˆ’1.(3.3) It follows that ā€–ā€–š‘¦š‘›āˆ’š‘¦š‘›āˆ’1ā€–ā€–ā‰¤š›¾š›½š‘›ā€–ā€–š‘„š‘›āˆ’š‘„š‘›āˆ’1ā€–ā€–+ī€·1āˆ’š›½š‘›ī€øā€–ā€–š‘„š‘›āˆ’š‘„š‘›āˆ’1ā€–ā€–+||š›½š‘›āˆ’š›½š‘›āˆ’1||ī€·ā€–ā€–š‘„š‘„š‘›āˆ’1ā€–ā€–+ā€–ā€–š‘„š‘›āˆ’1ā€–ā€–ī€ø=ī€ŗ1āˆ’(1āˆ’š›¾)š›½š‘›ī€»ā€–ā€–š‘„š‘›āˆ’š‘„š‘›āˆ’1ā€–ā€–+||š›½š‘›āˆ’š›½š‘›āˆ’1||ī€·ā€–ā€–š‘„š‘„š‘›āˆ’1ā€–ā€–+ā€–ā€–š‘„š‘›āˆ’1ā€–ā€–ī€ø.(3.4) From (3.1), we have š‘„š‘›+1āˆ’š‘„š‘›=š›¼š‘›š‘Šš‘›š‘„š‘›+ī€·1āˆ’š›¼š‘›ī€øš‘‡š‘¦š‘›āˆ’š›¼š‘›āˆ’1š‘Šš‘›āˆ’1š‘„š‘›āˆ’1āˆ’ī€·1āˆ’š›¼š‘›āˆ’1ī€øš‘‡š‘¦š‘›āˆ’1=š›¼š‘›ī€·š‘Šš‘›š‘„š‘›āˆ’š‘Šš‘›š‘„š‘›āˆ’1ī€ø+ī€·š›¼š‘›āˆ’š›¼š‘›āˆ’1ī€øš‘Šš‘›š‘„š‘›āˆ’1+š›¼š‘›āˆ’1ī€·š‘Šš‘›š‘„š‘›āˆ’1āˆ’š‘Šš‘›āˆ’1š‘„š‘›āˆ’1ī€ø+ī€·1āˆ’š›¼š‘›ī€øī€·š‘‡š‘¦š‘›āˆ’š‘‡š‘¦š‘›āˆ’1ī€ø+ī€·š›¼š‘›āˆ’1āˆ’š›¼š‘›ī€øš‘‡š‘¦š‘›āˆ’1.(3.5) Then we obtain ā€–ā€–š‘„š‘›+1āˆ’š‘„š‘›ā€–ā€–ā‰¤š›¼š‘›ā€–ā€–š‘Šš‘›š‘„š‘›āˆ’š‘Šš‘›š‘„š‘›āˆ’1ā€–ā€–+ī€·1āˆ’š›¼š‘›ī€øā€–ā€–š‘‡š‘¦š‘›āˆ’š‘‡š‘¦š‘›āˆ’1ā€–ā€–+||š›¼š‘›āˆ’š›¼š‘›āˆ’1||ī€·ā€–ā€–š‘Šš‘›š‘„š‘›āˆ’1ā€–ā€–+ā€–ā€–š‘‡š‘¦š‘›āˆ’1ā€–ā€–ī€ø+š›¼š‘›āˆ’1ā€–ā€–š‘Šš‘›š‘„š‘›āˆ’1āˆ’š‘Šš‘›āˆ’1š‘„š‘›āˆ’1ā€–ā€–ā‰¤š›¼š‘›ā€–ā€–š‘„š‘›āˆ’š‘„š‘›āˆ’1ā€–ā€–+ī€·1āˆ’š›¼š‘›ī€øā€–ā€–š‘¦š‘›āˆ’š‘¦š‘›āˆ’1ā€–ā€–+||š›¼š‘›āˆ’š›¼š‘›āˆ’1||ī€·ā€–ā€–š‘Šš‘›š‘„š‘›āˆ’1ā€–ā€–+ā€–ā€–š‘‡š‘¦š‘›āˆ’1ā€–ā€–ī€ø+š›¼š‘›āˆ’1ā€–ā€–š‘Šš‘›š‘„š‘›āˆ’1āˆ’š‘Šš‘›āˆ’1š‘„š‘›āˆ’1ā€–ā€–.(3.6) From (2.3), since š‘‡š‘– and š‘ˆš‘›,š‘– are nonexpansive, we have ā€–ā€–š‘Šš‘›š‘„š‘›āˆ’1āˆ’š‘Šš‘›āˆ’1š‘„š‘›āˆ’1ā€–ā€–=ā€–ā€–šœ‰1š‘‡1Uš‘›,2š‘„š‘›āˆ’1āˆ’šœ‰1š‘‡1š‘ˆš‘›āˆ’1,2š‘„š‘›āˆ’1ā€–ā€–ā‰¤šœ‰1ā€–ā€–š‘ˆš‘›,2š‘„š‘›āˆ’1āˆ’š‘ˆš‘›āˆ’1,2š‘„š‘›āˆ’1ā€–ā€–=šœ‰1ā€–ā€–šœ‰2š‘‡2š‘ˆš‘›,3š‘„š‘›āˆ’1āˆ’šœ‰2š‘‡2š‘ˆš‘›āˆ’1,3š‘„š‘›āˆ’1ā€–ā€–ā‰¤šœ‰1šœ‰2ā€–ā€–š‘ˆš‘›,3š‘„š‘›āˆ’1āˆ’š‘ˆš‘›āˆ’1,3š‘„š‘›āˆ’1ā€–ā€–ā‰¤ā‹Æā‰¤šœ‰1šœ‰2ā‹Æšœ‰š‘›āˆ’1ā€–ā€–š‘ˆš‘›,š‘›š‘„š‘›āˆ’1āˆ’š‘ˆš‘›āˆ’1,š‘›š‘„š‘›āˆ’1ā€–ā€–ā‰¤š‘€1š‘›āˆ’1ī‘š‘–=1šœ‰š‘–,(3.7) where š‘€1 is a constant such that supš‘›ā‰„1{ā€–š‘ˆš‘›,š‘›š‘„š‘›āˆ’1āˆ’š‘ˆš‘›āˆ’1,š‘›š‘„š‘›āˆ’1ā€–}ā‰¤š‘€1. Substituting (3.4) and (3.7) into (3.6), we get ā€–ā€–š‘„š‘›+1āˆ’š‘„š‘›ā€–ā€–ā‰¤š›¼š‘›ā€–ā€–š‘„š‘›āˆ’š‘„š‘›āˆ’1ā€–ā€–+ī€·1āˆ’š›¼š‘›ī€øī€ŗ1āˆ’(1āˆ’š›¾)š›½š‘›ī€»ā€–ā€–š‘„š‘›āˆ’š‘„š‘›āˆ’1ā€–ā€–+||š›½š‘›āˆ’š›½š‘›āˆ’1||ī€·ā€–ā€–š‘„š‘„š‘›āˆ’1ā€–ā€–+ā€–ā€–š‘„š‘›āˆ’1ā€–ā€–ī€ø+||š›¼š‘›āˆ’š›¼š‘›āˆ’1||ī€·ā€–ā€–š‘Šš‘›š‘„š‘›āˆ’1ā€–ā€–+ā€–ā€–š‘‡š‘¦š‘›āˆ’1ā€–ā€–ī€ø+š›¼š‘›āˆ’1š‘€1š‘›āˆ’1ī‘š‘–=1šœ‰š‘–=ī€ŗ1āˆ’(1āˆ’š›¾)š›½š‘›ī€·1āˆ’š›¼š‘›ā€–ā€–š‘„ī€øī€»š‘›āˆ’š‘„š‘›āˆ’1ā€–ā€–+||š›½š‘›āˆ’š›½š‘›āˆ’1||ī€·ā€–ā€–š‘„š‘„š‘›āˆ’1ā€–ā€–+ā€–ā€–š‘„š‘›āˆ’1ā€–ā€–ī€ø+||š›¼š‘›āˆ’š›¼š‘›āˆ’1||ī€·ā€–ā€–š‘Šš‘›š‘„š‘›āˆ’1ā€–ā€–+ā€–ā€–š‘‡š‘¦š‘›āˆ’1ā€–ā€–ī€ø+š›¼š‘›āˆ’1š‘€1š‘›āˆ’1ī‘š‘–=1šœ‰š‘–.(3.8) Therefore, it follows that ā€–ā€–š‘„š‘›+1āˆ’š‘„š‘›ā€–ā€–š›¼š‘›ā‰¤ī€ŗ1āˆ’(1āˆ’š›¾)š›½š‘›ī€·1āˆ’š›¼š‘›ā€–ā€–š‘„ī€øī€»š‘›āˆ’š‘„š‘›āˆ’1ā€–ā€–š›¼š‘›+||š›½š‘›āˆ’š›½š‘›āˆ’1||š›¼š‘›ī€·ā€–ā€–š‘„š‘„š‘›āˆ’1ā€–ā€–+ā€–ā€–š‘„š‘›āˆ’1ā€–ā€–ī€ø+||š›¼š‘›āˆ’š›¼š‘›āˆ’1||š›¼š‘›ī€·ā€–ā€–š‘Šš‘›š‘„š‘›āˆ’1ā€–ā€–+ā€–ā€–š‘‡š‘¦š‘›āˆ’1ā€–ā€–ī€ø+š›¼š‘›āˆ’1š‘€1āˆš‘›āˆ’1š‘–=1šœ‰š‘–š›¼š‘›=ī€ŗ1āˆ’(1āˆ’š›¾)š›½š‘›ī€·1āˆ’š›¼š‘›ā€–ā€–š‘„ī€øī€»š‘›āˆ’š‘„š‘›āˆ’1ā€–ā€–š›¼š‘›āˆ’1+ī€ŗ1āˆ’(1āˆ’š›¾)š›½š‘›ī€·1āˆ’š›¼š‘›ī‚µā€–ā€–š‘„ī€øī€»š‘›āˆ’š‘„š‘›āˆ’1ā€–ā€–š›¼š‘›āˆ’ā€–ā€–š‘„š‘›āˆ’š‘„š‘›āˆ’1ā€–ā€–š›¼š‘›āˆ’1ī‚¶+||š›½š‘›āˆ’š›½š‘›āˆ’1||š›¼š‘›ī€·ā€–ā€–š‘„š‘„š‘›āˆ’1ā€–ā€–+ā€–ā€–š‘„š‘›āˆ’1ā€–ā€–ī€ø+||š›¼š‘›āˆ’š›¼š‘›āˆ’1||š›¼š‘›ī€·ā€–ā€–š‘Šš‘›š‘„š‘›āˆ’1ā€–ā€–+ā€–ā€–š‘‡š‘¦š‘›āˆ’1ā€–ā€–ī€ø+š›¼š‘›āˆ’1š‘€1āˆš‘›āˆ’1š‘–=1šœ‰š‘–š›¼š‘›ā‰¤ī€ŗ1āˆ’(1āˆ’š›¾)š›½š‘›ī€·1āˆ’š›¼š‘›ā€–ā€–š‘„ī€øī€»š‘›āˆ’š‘„š‘›āˆ’1ā€–ā€–š›¼š‘›āˆ’1+īƒ©||||1š›¼š‘›āˆ’1š›¼š‘›āˆ’1||||+||š›¼š‘›āˆ’š›¼š‘›āˆ’1||š›¼š‘›+||š›½š‘›āˆ’š›½š‘›āˆ’1||š›¼š‘›+āˆš‘›āˆ’1š‘–=1šœ‰š‘–š›¼š‘›īƒŖš‘€=ī€ŗ1āˆ’(1āˆ’š›¾)š›½š‘›ī€·1āˆ’š›¼š‘›ā€–ā€–š‘„ī€øī€»š‘›āˆ’š‘„š‘›āˆ’1ā€–ā€–š›¼š‘›āˆ’1+(1āˆ’š›¾)š›½š‘›ī€·1āˆ’š›¼š‘›ī€øƗīƒÆš‘€ī€·(1āˆ’š›¾)1āˆ’š›¼š‘›ī€øīƒ©1š›½š‘›||||1š›¼š‘›āˆ’1š›¼š‘›āˆ’1||||+1š›½š‘›||š›¼š‘›āˆ’š›¼š‘›āˆ’1||š›¼š‘›+1š›½š‘›||š›½š‘›āˆ’š›½š‘›āˆ’1||š›¼š‘›+āˆš‘›āˆ’1š‘–=1šœ‰š‘–š›¼š‘›š›½š‘›,īƒŖīƒ°(3.9) where š‘€ is a constant such that supš‘›ā‰„1ī€½š‘€1,ā€–ā€–š‘„š‘›āˆ’š‘„š‘›āˆ’1ā€–ā€–,ī€·ā€–ā€–š‘Šš‘›š‘„š‘›āˆ’1ā€–ā€–+ā€–ā€–š‘‡š‘¦š‘›āˆ’1ā€–ā€–ī€ø,ī€·ā€–ā€–š‘„š‘„š‘›āˆ’1ā€–ā€–+ā€–ā€–š‘„š‘›āˆ’1ā€–ā€–ī€øī€¾ā‰¤š‘€.(3.10) From (iii), we note that limš‘›ā†’āˆž(1/š›¼š‘›āˆ’1)|š›¼š‘›āˆ’š›¼š‘›āˆ’1/š›½š‘›š›¼š‘›|=0, which implies that limš‘›ā†’āˆž1š›½š‘›||š›¼š‘›āˆ’š›¼š‘›āˆ’1||š›¼š‘›=0.(3.11) Thus it follows from (iii) and (3.11) that limš‘›ā†’āˆžīƒ©1š›½š‘›||||1š›¼š‘›āˆ’1š›¼š‘›āˆ’1||||+1š›½š‘›||š›¼š‘›āˆ’š›¼š‘›āˆ’1||š›¼š‘›+1š›½š‘›||š›½š‘›āˆ’š›½š‘›āˆ’1||š›¼š‘›+āˆš‘›āˆ’1š‘–=1šœ‰š‘–š›¼š‘›š›½š‘›īƒŖ=0.(3.12) Hence, applying Lemma 2.4 to (3.9), we immediately conclude that limš‘›ā†’āˆžā€–ā€–š‘„š‘›+1āˆ’š‘„š‘›ā€–ā€–š›¼š‘›=0.(3.13) This implies that limš‘›ā†’āˆžā€–ā€–š‘„š‘›+1āˆ’š‘„š‘›ā€–ā€–=0.(3.14) Thus, from (3.1) and (3.14), we have limš‘›ā†’āˆžā€–ā€–š‘„š‘›āˆ’š‘‡š‘¦š‘›ā€–ā€–=0.(3.15) At the same time, we note that š‘¦š‘›āˆ’š‘„š‘›=š›½š‘›ī€·š‘„š‘„š‘›āˆ’š‘„š‘›ī€øāŸ¶0.(3.16) Hence we get limš‘›ā†’āˆžā€–ā€–š‘¦š‘›āˆ’š‘‡š‘¦š‘›ā€–ā€–=0.(3.17) Since the sequence {š‘„š‘›} is bounded, {š‘¦š‘›} is also bounded. Thus there exists a subsequence of {š‘¦š‘›}, which is still denoted by {š‘¦š‘›} which converges weakly to a point Ģƒš‘„āˆˆš». Therefore, Ģƒš‘„āˆˆFix(š‘‡) by (3.17) and Lemma 2.3. By (3.1), we observe that š‘„š‘›+1āˆ’š‘„š‘›=š›¼š‘›ī€·š‘Šš‘›š‘„š‘›āˆ’š‘„š‘›ī€ø+ī€·1āˆ’š›¼š‘›ī€øī€·š‘‡š‘¦š‘›āˆ’š‘¦š‘›ī€ø+ī€·1āˆ’š›¼š‘›ī€øš›½š‘›ī€·š‘„š‘„š‘›āˆ’š‘„š‘›ī€ø,(3.18) that is, š‘„š‘›āˆ’š‘„š‘›+1š›¼š‘›=ī€·š¼āˆ’š‘Šš‘›ī€øš‘„š‘›+1āˆ’š›¼š‘›š›¼š‘›(š¼āˆ’š‘‡)š‘¦š‘›+š›½š‘›ī€·1āˆ’š›¼š‘›ī€øš›¼š‘›(š¼āˆ’š‘„)š‘„š‘›.(3.19) Set š‘§š‘›=(š‘„š‘›āˆ’š‘„š‘›+1)/š›¼š‘› for each š‘›ā‰„1, that is, š‘§š‘›=ī€·š¼āˆ’š‘Šš‘›ī€øš‘„š‘›+1āˆ’š›¼š‘›š›¼š‘›(š¼āˆ’š‘‡)š‘¦š‘›+š›½š‘›ī€·1āˆ’š›¼š‘›ī€øš›¼š‘›(š¼āˆ’š‘„)š‘„š‘›.(3.20) Using monotonicity of š¼āˆ’š‘‡ and š¼āˆ’š‘Šš‘›, we derive that, for all š‘¢āˆˆFix(š‘‡), āŸØš‘§š‘›,š‘„š‘›=āˆ’š‘¢āŸ©ī«ī€·š¼āˆ’š‘Šš‘›ī€øš‘„š‘›,š‘„š‘›ī¬+āˆ’š‘¢1āˆ’š›¼š‘›š›¼š‘›āŸØ(š¼āˆ’š‘‡)š‘¦š‘›āˆ’(š¼āˆ’š‘‡)š‘¢,š‘¦š‘›+āˆ’š‘¢āŸ©1āˆ’š›¼š‘›š›¼š‘›āŸØ(š¼āˆ’š‘‡)š‘¦š‘›,š‘„š‘›āˆ’š‘¦š‘›š›½āŸ©+š‘›ī€·1āˆ’š›¼š‘›ī€øš›¼š‘›āŸØ(š¼āˆ’š‘„)š‘„š‘›,š‘„š‘›ā‰„āˆ’š‘¢āŸ©ī«ī€·š¼āˆ’š‘Šš‘›ī€øš‘¢,š‘„š‘›ī¬+š›½āˆ’š‘¢š‘›ī€·1āˆ’š›¼š‘›ī€øš›¼š‘›āŸØ(š¼āˆ’š‘„)š‘„š‘›,š‘„š‘›ī€·āˆ’š‘¢āŸ©+1āˆ’š›¼š‘›ī€øš›½š‘›š›¼š‘›āŸØ(š¼āˆ’š‘‡)š‘¦š‘›,š‘„š‘›āˆ’š‘„š‘„š‘›āŸ©=āŸØ(š¼āˆ’š‘Š)š‘¢,š‘„š‘›āˆ’š‘¢āŸ©+ī«ī€·š‘Šāˆ’š‘Šš‘›ī€øš‘¢,š‘„š‘›ī¬+š›½āˆ’š‘¢š‘›ī€·1āˆ’š›¼š‘›ī€øš›¼š‘›āŸØ(š¼āˆ’š‘„)š‘„š‘›,š‘„š‘›+ī€·āˆ’š‘¢āŸ©1āˆ’š›¼š‘›ī€øš›½š‘›š›¼š‘›āŸØ(š¼āˆ’š‘‡)š‘¦š‘›,š‘„š‘›āˆ’š‘„š‘„š‘›āŸ©.(3.21) But, since š‘§š‘›ā†’0,š›½š‘›/š›¼š‘›ā†’0 and limš‘›ā†’āˆžā€–š‘Šš‘›š‘¢āˆ’š‘Šš‘¢ā€–=0 (by Lemma 2.2), it follows from the above inequality that limsupš‘›ā†’āˆžāŸØ(š¼āˆ’š‘Š)š‘¢,š‘„š‘›āˆ’š‘¢āŸ©ā‰¤0,āˆ€š‘¢āˆˆFix(š‘‡).(3.22) This suffices to guarantee that šœ”š‘¤(š‘„š‘›)āŠ‚Ī©. As a matter of fact, if we take any š‘„āˆ—āˆˆšœ”š‘¤(š‘„š‘›), then there exists a subsequence {š‘„š‘›š‘—} of {š‘„š‘›} such that š‘„š‘›š‘—ā‡€š‘„āˆ—. Therefore, we have āŸØ(š¼āˆ’š‘Š)š‘¢,š‘„āˆ—āˆ’š‘¢āŸ©=limš‘—ā†’āˆžī‚¬(š¼āˆ’š‘Š)š‘¢,š‘„š‘›š‘—ī‚­āˆ’š‘¢ā‰¤0,āˆ€š‘¢āˆˆFix(š‘‡).(3.23) Note that š‘„āˆ—āˆˆFix(š‘‡). Hence š‘„āˆ— solves the following problem: š‘„āˆ—āˆˆFix(š‘‡),āŸØ(š¼āˆ’š‘Š)š‘¢,š‘„āˆ—āˆ’š‘¢āŸ©ā‰¤0,āˆ€š‘¢āˆˆFix(š‘‡).(3.24) It is obvious that this is equivalent to the problem (1.9) since š‘Šš‘›ā†’š‘Š uniformly in any bounded set (by Lemma 2.2). Thus š‘„āˆ—āˆˆĪ©.
Let Ģƒš‘„ be the unique solution of the variational inequality (3.2). Now, take a subsequence {š‘„š‘›š‘–} of {š‘„š‘›} such thatlimsupš‘›ā†’āˆžāŸØ(š¼āˆ’š‘„)Ģƒš‘„,š‘„š‘›āˆ’Ģƒš‘„āŸ©=limš‘–ā†’āˆžī«(š¼āˆ’š‘„)Ģƒš‘„,š‘„š‘›š‘–ī¬.āˆ’Ģƒš‘„(3.25) Without loss of generality, we may further assume that š‘„š‘›š‘–ā‡€š‘„. Then š‘„āˆˆĪ©. Therefore, we have limsupš‘›ā†’āˆžāŸØ(š¼āˆ’š‘„)Ģƒš‘„,š‘„š‘›ī«āˆ’Ģƒš‘„āŸ©=(š¼āˆ’š‘„)Ģƒš‘„,ī¬š‘„āˆ’Ģƒš‘„ā‰„0.(3.26) This completes the proof.

Theorem 3.3. Let š¶ be a nonempty closed convex subset of a real Hilbert space š». Let {š‘‡š‘›}āˆžš‘›=1 be infinite family of nonexpansive mappings of š¶ into itself. Let š‘„āˆ¶š¶ā†’š¶ be a contraction with coefficient š›¾āˆˆ[0,1). Assume that the set Ī© of solutions of the hierarchical problem (1.9) is nonempty. Let {š›¼š‘›},{š›½š‘›} be two real numbers in (0,1) and {š‘„š‘›} the sequence generated by (3.1). Assume that the sequence {š‘„š‘›} is bounded and (i)limš‘›ā†’āˆžš›¼š‘›=0, limš‘›ā†’āˆžš›½š‘›/š›¼š‘›=0 and limš‘›ā†’āˆžš›¼2š‘›/š›½š‘›=0; (ii)āˆ‘āˆžš‘›=0š›½š‘›=āˆž; (iii)limš‘›ā†’āˆž(1/š›½š‘›)|(1/š›¼š‘›)āˆ’(1/š›¼š‘›āˆ’1)|=0 and limš‘›ā†’āˆžāˆš‘›āˆ’1š‘–=1šœ‰š‘–/š›¼š‘›š›½š‘› = limš‘›ā†’āˆž(1/š›¼š‘›)|1āˆ’(š›½š‘›āˆ’1/š›½š‘›)| = 0; (iv)there exists a constant š‘˜>0 such that ā€–š‘„āˆ’š‘‡š‘„ā€–ā‰„š‘˜Dist(š‘„,Fix(š‘‡)), where Dist(š‘„,Fix(š‘‡))=infš‘¦āˆˆFix(š‘‡)ā€–š‘„āˆ’š‘¦ā€–.(3.27) Then the sequence {š‘„š‘›} defined by (3.1) converges strongly to a point Ģƒš‘„āˆˆFix(š‘‡), which solves the variational inequality problem (3.2).

Proof. From (3.1), we have š‘„š‘›+1āˆ’Ģƒš‘„=š›¼š‘›ī€·š‘Šš‘›š‘„š‘›āˆ’š‘Šš‘›ī€øĢƒš‘„+š›¼š‘›ī€·š‘Šš‘›ī€ø+ī€·Ģƒš‘„āˆ’Ģƒš‘„1āˆ’š›¼š‘›ī€øī€·š‘‡š‘¦š‘›ī€øāˆ’Ģƒš‘„.(3.28) Thus we have ā€–ā€–š‘„š‘›+1ā€–ā€–āˆ’Ģƒš‘„2ā‰¤ā€–ā€–š›¼š‘›ī€·š‘Šš‘›š‘„š‘›āˆ’š‘Šš‘›ī€ø+ī€·Ģƒš‘„1āˆ’š›¼š‘›ī€øī€·š‘‡š‘¦š‘›ī€øā€–ā€–āˆ’Ģƒš‘„2+2š›¼š‘›ī«š‘Šš‘›Ģƒš‘„āˆ’Ģƒš‘„,š‘„š‘›+1ī¬ā‰¤ī€·āˆ’Ģƒš‘„1āˆ’š›¼š‘›ī€øā€–ā€–š‘‡š‘¦š‘›ā€–ā€–āˆ’Ģƒš‘„2+š›¼š‘›ā€–ā€–š‘Šš‘›š‘„š‘›āˆ’š‘Šš‘›ā€–ā€–Ģƒš‘„2+2š›¼š‘›ī«š‘Šš‘›Ģƒš‘„āˆ’Ģƒš‘„,š‘„š‘›+1ī¬ā‰¤ī€·āˆ’Ģƒš‘„1āˆ’š›¼š‘›ī€øā€–ā€–š‘¦š‘›ā€–ā€–āˆ’Ģƒš‘„2+š›¼š‘›ā€–ā€–š‘„š‘›ā€–ā€–āˆ’Ģƒš‘„2+2š›¼š‘›ī«š‘Šš‘›Ģƒš‘„āˆ’Ģƒš‘„,š‘„š‘›+1ī¬.āˆ’Ģƒš‘„(3.29) At the same time, we observe that ā€–ā€–š‘¦š‘›ā€–ā€–āˆ’Ģƒš‘„2=ā€–ā€–ī€·1āˆ’š›½š‘›š‘„ī€øī€·š‘›ī€øāˆ’Ģƒš‘„+š›½š‘›ī€·š‘„š‘„š‘›ī€øāˆ’š‘„Ģƒš‘„+š›½š‘›ā€–ā€–(š‘„Ģƒš‘„āˆ’Ģƒš‘„)2ā‰¤ā€–ā€–ī€·1āˆ’š›½š‘›š‘„ī€øī€·š‘›ī€øāˆ’Ģƒš‘„+š›½š‘›ī€·š‘„š‘„š‘›ī€øā€–ā€–āˆ’š‘„Ģƒš‘„2+2š›½š‘›āŸØš‘„Ģƒš‘„āˆ’Ģƒš‘„,š‘¦š‘›ā‰¤ī€·āˆ’Ģƒš‘„āŸ©1āˆ’š›½š‘›ī€øā€–ā€–š‘„š‘›ā€–ā€–āˆ’Ģƒš‘„2+š›½š‘›ā€–ā€–š‘„š‘„š‘›ā€–ā€–āˆ’š‘„Ģƒš‘„2+2š›½š‘›āŸØš‘„Ģƒš‘„āˆ’Ģƒš‘„,š‘¦š‘›ā‰¤ī€·āˆ’Ģƒš‘„āŸ©1āˆ’š›½š‘›ī€øā€–ā€–š‘„š‘›ā€–ā€–āˆ’Ģƒš‘„2+š›½š‘›š›¾2ā€–ā€–š‘„š‘›ā€–ā€–āˆ’Ģƒš‘„2+2š›½š‘›āŸØš‘„Ģƒš‘„āˆ’Ģƒš‘„,š‘¦š‘›=ī€ŗī€·āˆ’Ģƒš‘„āŸ©1āˆ’1āˆ’š›¾2ī€øš›½š‘›ī€»ā€–ā€–š‘„š‘›ā€–ā€–āˆ’Ģƒš‘„2+2š›½š‘›āŸØš‘„Ģƒš‘„āˆ’Ģƒš‘„,š‘¦š‘›āˆ’Ģƒš‘„āŸ©.(3.30) Substituting (3.30) into (3.29), we get ā€–ā€–š‘„š‘›+1ā€–ā€–āˆ’Ģƒš‘„2ā‰¤š›¼š‘›ā€–ā€–š‘„š‘›ā€–ā€–āˆ’Ģƒš‘„2+ī€·1āˆ’š›¼š‘›ī€·ī€øī€ŗ1āˆ’1āˆ’š›¾2ī€øš›½š‘›ī€»ā€–ā€–š‘„š‘›ā€–ā€–āˆ’Ģƒš‘„2+2š›½š‘›ī€·1āˆ’š›¼š‘›ī€øāŸØš‘„Ģƒš‘„āˆ’Ģƒš‘„,š‘¦š‘›āˆ’Ģƒš‘„āŸ©+2š›¼š‘›ī«š‘Šš‘›Ģƒš‘„āˆ’Ģƒš‘„,š‘„š‘›+1ī¬=ī€ŗī€·āˆ’Ģƒš‘„1āˆ’1āˆ’š›¾2ī€øš›½š‘›ī€·1āˆ’š›¼š‘›ā€–ā€–š‘„ī€øī€»š‘›ā€–ā€–āˆ’Ģƒš‘„2+2š›½š‘›ī€·1āˆ’š›¼š‘›ī€øāŸØš‘„Ģƒš‘„āˆ’Ģƒš‘„,š‘¦š‘›āˆ’Ģƒš‘„āŸ©+2š›¼š‘›ī«š‘Šš‘›Ģƒš‘„āˆ’Ģƒš‘„,š‘„š‘›+1ī¬=ī€ŗī€·āˆ’Ģƒš‘„1āˆ’1āˆ’š›¾2ī€øš›½š‘›ī€·1āˆ’š›¼š‘›ā€–ā€–š‘„ī€øī€»š‘›ā€–ā€–āˆ’Ģƒš‘„2+ī€·1āˆ’š›¾2ī€øš›½š‘›ī€·1āˆ’š›¼š‘›ī€øƗīƒÆ21āˆ’š›¾2āŸØš‘„Ģƒš‘„āˆ’Ģƒš‘„,š‘¦š‘›2āˆ’Ģƒš‘„āŸ©+ī€·1āˆ’š›¾2ī€øī€·1āˆ’š›¼š‘›ī€øƗš›¼š‘›š›½š‘›ī«š‘Šš‘›Ģƒš‘„āˆ’Ģƒš‘„,š‘„š‘›+1ī¬īƒ°.āˆ’Ģƒš‘„(3.31) By Theorem 3.2, we note that every weak cluster point of the sequence {š‘„š‘›} is in Ī©. Since š‘¦š‘›āˆ’š‘„š‘›ā†’0, then every weak cluster point of {š‘¦š‘›} is also in Ī©. Consequently, since Ģƒš‘„=projĪ©(š‘„Ģƒš‘„), we easily have limsupš‘›ā†’āˆžāŸØš‘„Ģƒš‘„āˆ’Ģƒš‘„,š‘¦š‘›āˆ’Ģƒš‘„āŸ©ā‰¤0.(3.32)
On the other hand, we observe thatī«š‘Šš‘›Ģƒš‘„āˆ’Ģƒš‘„,š‘„š‘›+1ī¬=ī«š‘Šāˆ’Ģƒš‘„š‘›Ģƒš‘„āˆ’Ģƒš‘„,projFix(š‘‡)š‘„š‘›+1ī¬+ī«š‘Šāˆ’Ģƒš‘„š‘›Ģƒš‘„āˆ’Ģƒš‘„,š‘„š‘›+1āˆ’projFix(š‘‡)š‘„š‘›+1ī¬.(3.33) Since Ģƒš‘„ is a solution of the problem (1.9) and projFix(š‘‡)š‘„š‘›+1āˆˆFix(š‘‡), we have ī«š‘Šš‘›Ģƒš‘„āˆ’Ģƒš‘„,projFix(š‘‡)š‘„š‘›+1ī¬āˆ’Ģƒš‘„ā‰¤0.(3.34) Thus it follows that ī«š‘Šš‘›Ģƒš‘„āˆ’Ģƒš‘„,š‘„š‘›+1ī¬ā‰¤ī«š‘Šāˆ’Ģƒš‘„š‘›Ģƒš‘„āˆ’Ģƒš‘„,š‘„š‘›+1āˆ’projFix(š‘‡)š‘„š‘›+1ī¬ā‰¤ā€–ā€–š‘Šš‘›ā€–ā€–ā€–ā€–š‘„Ģƒš‘„āˆ’Ģƒš‘„š‘›+1āˆ’projFix(š‘‡)š‘„š‘›+1ā€–ā€–=ā€–ā€–š‘Šš‘›ā€–ā€–ī€·š‘„Ģƒš‘„āˆ’Ģƒš‘„ƗDistš‘›+1ī€øā‰¤1,Fix(š‘‡)š‘˜ā€–ā€–š‘Šš‘›ā€–ā€–ā€–ā€–š‘„Ģƒš‘„āˆ’Ģƒš‘„š‘›+1āˆ’š‘‡š‘„š‘›+1ā€–ā€–.(3.35) We note that ā€–ā€–š‘„š‘›+1āˆ’š‘‡š‘„š‘›+1ā€–ā€–ā‰¤ā€–ā€–š‘„š‘›+1āˆ’š‘‡š‘„š‘›ā€–ā€–+ā€–ā€–š‘‡š‘„š‘›āˆ’š‘‡š‘„š‘›+1ā€–ā€–ā‰¤š›¼š‘›ā€–ā€–š‘Šš‘›š‘„š‘›āˆ’š‘‡š‘„š‘›ā€–ā€–+ī€·1āˆ’š›¼š‘›ī€øā€–ā€–š‘‡š‘¦š‘›āˆ’š‘‡š‘„š‘›ā€–ā€–+ā€–ā€–š‘„š‘›+1āˆ’š‘„š‘›ā€–ā€–ā‰¤š›¼š‘›ā€–ā€–š‘Šš‘›š‘„š‘›āˆ’š‘‡š‘„š‘›ā€–ā€–+ā€–ā€–š‘¦š‘›āˆ’š‘„š‘›ā€–ā€–+ā€–ā€–š‘„š‘›+1āˆ’š‘„š‘›ā€–ā€–ā‰¤š›¼š‘›ā€–ā€–š‘Šš‘›š‘„š‘›āˆ’š‘‡š‘„š‘›ā€–ā€–+š›½š‘›ā€–ā€–š‘„š‘„š‘›āˆ’š‘„š‘›ā€–ā€–+ā€–ā€–š‘„š‘›+1āˆ’š‘„š‘›ā€–ā€–.(3.36) Hence we have š›¼š‘›š›½š‘›ī«š‘Šš‘›Ģƒš‘„āˆ’Ģƒš‘„,š‘„š‘›+1ī¬ā‰¤š›¼āˆ’Ģƒš‘„2š‘›š›½š‘›ī‚€1š‘˜ā€–ā€–š‘Šš‘›ā€–ā€–ā€–ā€–š‘ŠĢƒš‘„āˆ’Ģƒš‘„š‘›š‘„š‘›āˆ’š‘‡š‘„š‘›ā€–ā€–ī‚+š›¼š‘›ī‚€1š‘˜ā€–ā€–š‘Šš‘›ā€–ā€–ā€–ā€–Ģƒš‘„āˆ’Ģƒš‘„š‘„š‘„š‘›āˆ’š‘„š‘›ā€–ā€–ī‚+š›¼2š‘›š›½š‘›ā€–ā€–š‘„š‘›+1āˆ’š‘„š‘›ā€–ā€–š›¼š‘›ī‚€1š‘˜ā€–ā€–š‘Šš‘›ā€–ā€–ī‚.Ģƒš‘„āˆ’Ģƒš‘„(3.37) From Theorem 3.2, we have limš‘›ā†’āˆžā€–š‘„š‘›+1āˆ’š‘„š‘›ā€–/š›¼š‘›=0. At the same time, we note that {(1/š‘˜)ā€–š‘Šš‘›Ģƒš‘„āˆ’Ģƒš‘„ā€–ā€–š‘Šš‘›š‘„š‘›āˆ’š‘‡š‘„š‘›ā€–}, {(1/š‘˜)ā€–š‘Šš‘›Ģƒš‘„āˆ’Ģƒš‘„ā€–ā€–š‘„š‘„š‘›āˆ’š‘„š‘›ā€–}, and {(1/š‘˜)ā€–š‘Šš‘›Ģƒš‘„āˆ’Ģƒš‘„ā€–} are all bounded. Hence it follows from (i) and the above inequality that limsupš‘›ā†’āˆžš›¼š‘›š›½š‘›ī«š‘Šš‘›Ģƒš‘„āˆ’Ģƒš‘„,š‘„š‘›+1ī¬āˆ’Ģƒš‘„ā‰¤0.(3.38)
Finally, by (3.31)ā€“(3.38) and Lemma 2.4, we conclude that the sequence {š‘„š‘›} converges strongly to a point Ģƒš‘„āˆˆFix(š‘‡). 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 š¶ be a nonempty closed convex subset of a real Hilbert space š» and š‘† a nonexpansive mapping of š¶ into itself. Let š‘„āˆ¶š¶ā†’š¶ be a contraction with coefficient š›¾āˆˆ[0,1). For any š‘„0āˆˆš¶, let{š‘„š‘›} the sequence generated iteratively by š‘„š‘›+1=š›¼š‘›š‘†š‘„š‘›+ī€·1āˆ’š›¼š‘›ī€øš‘‡ī€·š›½š‘›š‘„š‘„š‘›+ī€·1āˆ’š›½š‘›ī€øš‘„š‘›ī€ø,āˆ€š‘›ā‰„0,(3.39) where {š›¼š‘›},{š›½š‘›} are two real numbers in (0,1).

Corollary 3.6. Let š¶ be a nonempty closed convex subset of a real Hilbert space š». Let Sāˆ¶š¶ā†’š¶ be a nonexpansive mapping. Let š‘„āˆ¶š¶ā†’š¶ be a contraction with coefficient š›¾āˆˆ[0,1). Assume that the set Ī©ī…ž of solutions of the hierarchical problem (1.1) is nonempty. Let {š›¼š‘›},{š›½š‘›} be two real numbers in (0,1) and {š‘„š‘›} the sequence generated by (3.1). Assume that the sequence {š‘„š‘›} is bounded and (i)limš‘›ā†’āˆžš›¼š‘›=0, limš‘›ā†’āˆžš›½š‘›/š›¼š‘›=0 and limš‘›ā†’āˆžš›¼2š‘›/š›½š‘›=0; (ii)āˆ‘āˆžš‘›=0š›½š‘›=āˆž; (iii)limš‘›ā†’āˆž(1/š›½š‘›)|(1/š›¼š‘›)āˆ’(1/š›¼š‘›āˆ’1)|=0 and limš‘›ā†’āˆž(1/š›¼š‘›)|1āˆ’(š›½š‘›āˆ’1/š›½š‘›)|=0; (iv)there exists a constant š‘˜>0 such that ā€–š‘„āˆ’š‘‡š‘„ā€–ā‰„š‘˜Dist(š‘„,Fix(š‘‡)), where Dist(š‘„,Fix(š‘‡))=infš‘¦āˆˆFix(š‘‡)ā€–š‘„āˆ’š‘¦ā€–.(3.40) Then the sequence {š‘„š‘›} defined by (3.39) converges strongly to a point Ģƒš‘„āˆˆFix(š‘‡), 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).