An Iterative Algorithm for a Hierarchical Problem

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 ⟨⋅,⋅⟩ and norm ‖⋅‖, respectively. Let 𝐶 be a nonempty closed convex subset of 𝐻. The hierarchical problem is of finding 𝑥̃∈Fix(𝑇) such that⟨𝑆𝑥̃−𝑥̃,𝑥−𝑥̃⟩≤0,∀𝑥∈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:𝑥_𝑡,𝑠=𝑠𝑄𝑥_𝑡,𝑠+(1−𝑠)𝑡𝑆𝑥_𝑡,𝑠+(1−𝑡)𝑇𝑥_𝑡,𝑠(1.2) and proved that the net {𝑥_𝑡,𝑠} defined by (1.2) strongly converges to 𝑥_𝑡 as 𝑠→0, where 𝑥_𝑡 satisfies 𝑥_𝑡=proj_{Fix(𝑃𝑡)}𝑄(𝑥_𝑡), where 𝑃_𝑡∶𝐶→𝐶 is a mapping defined by 𝑃_𝑡(𝑥)=𝑡𝑆(𝑥)+(1−𝑡)𝑇(𝑥),∀𝑥∈𝐶,𝑡∈(0,1),(1.3) or, equivalently, 𝑥_𝑡 is the unique solution of the quasivariational inequality 0∈(𝐼−𝑄)𝑥_𝑡+𝑁_{Fix(𝑃𝑡)}𝑥_𝑡,(1.4) where the normal cone to Fix(𝑃_𝑡), 𝑁_{Fix(𝑃𝑡)}, is defined as follows: 𝑁_{Fix(𝑃𝑡)}∶𝑥⟶{∅𝑢,∈𝐻∶⟨𝑦−𝑥,𝑢⟩≤0},iofth𝑥er∈wiFsiex.𝑃^𝑡,(1.5)

Moreover, as 𝑡→0, the net {𝑥_𝑡} in turn weakly converges to the unique solution 𝑥_∞ of the fixed point equation 𝑥_∞=proj_Ω𝑄(𝑥_∞) or, equivalently, 𝑥_∞ is the unique solution of the variational inequality 0∈(𝐼−𝑄)𝑥_∞+𝑁_Ω𝑥_∞.(1.6)

Recently, Moudafi [2] constructed an explicit iterative algorithm:𝑥_𝑛+1=1−𝛿_𝑛𝑥_𝑛+𝛿_𝑛𝜎_𝑛𝑆𝑥_𝑛+1−𝜎_𝑛𝑇𝑥_𝑛,∀𝑛≥0,(1.7) where {𝛿_𝑛} and {𝜎_𝑛} 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:𝑥_𝑛+1=1−𝛿_𝑛𝑄𝑥_𝑛+𝛿_𝑛𝜎_𝑛𝑆𝑥_𝑛+1−𝜎_𝑛𝑇𝑥_𝑛,∀𝑛≥0,(1.8) where {𝛿_𝑛} and {𝜎_𝑛} are two real numbers in (0,1), and proved that, under appropriate conditions, the iterative sequence {𝑥_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 𝑥̃∈Fix(𝑇) such that, for any 𝑛≥1,⟨𝑊_𝑛𝑥̃−𝑥̃,𝑥−𝑥̃⟩≤0,∀𝑥∈Fix(𝑇),(1.9) where 𝑊_𝑛 is the 𝑊-mapping defined by (2.3) below and 𝑇 is a nonexpansive mapping, and introduce an explicit iterative algorithm which converges strongly to a solution 𝑥̃ of the hierarchical problem (1.9).

2. Preliminaries

Let 𝐶 a nonempty closed convex subset of a real Hilbert space 𝐻. Recall that a mapping 𝑄∶𝐶→𝐶 is said to be contractive if there exists a constant 𝛾∈(0,1) such that ‖𝑄𝑥−𝑄𝑦‖≤𝛾‖𝑥−𝑦‖,∀𝑥,𝑦∈𝐶.(2.1)

A mapping 𝑇∶𝐶→𝐶 is called nonexpansive if‖𝑇𝑥−𝑇𝑦‖≤‖𝑥−𝑦‖,∀𝑥,𝑦∈𝐶.(2.2)

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

Let {𝑇_𝑖}^∞𝑖=1∶𝐶→𝐶 be an infinite family of nonexpansive mappings and {𝜉_𝑖}^∞𝑖=1 a real number sequence such that 0≤𝜉_𝑖≤1 for each 𝑖≥1.

For each 𝑛≥1, define a mapping 𝑊_𝑛∶𝐶→𝐶 as follows:𝑈𝑈_𝑛,𝑛+1=𝐼,_𝑛,𝑛𝑈=𝜉_𝑛𝑇_𝑛𝑈_𝑛,𝑛+1+1−𝜉_𝑛𝐼,_{𝑛,𝑛−1}𝑈⋯=𝜉_𝑛−1𝑇_𝑛−1𝑈_𝑛,𝑛+1−𝜉_𝑛−1𝐼,_𝑛,𝑘𝑈=𝜉_𝑘𝑇_𝑘𝑈_𝑛,𝑘+1+1−𝜉_𝑘𝐼,_{𝑛,𝑘−1}𝑈⋯=𝜉_𝑘−1𝑇_𝑘−1𝑈_𝑛,𝑘+1−𝜉_𝑘−1𝐼,_𝑛,2𝑊=𝜉₂𝑇₂𝑈_𝑛,3+1−𝜉₂𝐼,_𝑛=𝑈_𝑛,1=𝜉₁𝑇₁𝑈_𝑛,2+1−𝜉₁𝐼.(2.3)

Such 𝑊_𝑛 is called the 𝑊-mapping generated by {𝑇_𝑖}^∞𝑖=1 and {𝜉_𝑖}^∞𝑖=1.

Lemma 2.1 (see [20]). Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let {𝑇_𝑖}^∞𝑖=1 be an infinite family of nonexpansive mappings of 𝐶 into itself with ⋂^∞𝑛=1Fix(𝑇_𝑛)≠∅. Let 𝜉₁,𝜉₂,… be real numbers such that 0<𝜉_𝑖≤𝑏<1 for each 𝑖≥1. Then one has the following results: (1)for any 𝑥∈𝐶 and 𝑘≥1, the limit lim_𝑛→∞𝑈_𝑛,𝑘𝑥 exists; (2)Fix(𝑊)=⋂^∞𝑛=1Fix(𝑇_𝑛).

Using Lemma  3.1 in [21], we can define a mapping 𝑊 of 𝐶 into itself by 𝑊𝑥=lim_𝑛→∞𝑊_𝑛𝑥=lim_𝑛→∞𝑈_𝑛,1𝑥 for all 𝑥∈𝐶. Thus we have the following.

Lemma 2.2 (see [21]). If {𝑥_𝑛} is a bounded sequence in 𝐶, then one has lim_𝑛→∞‖‖𝑊𝑥_𝑛−𝑊_𝑛𝑥_𝑛‖‖=0.(2.4)

Lemma 2.3 (see [22]). Let 𝐶 be a nonempty closed convex of a real Hilbert space 𝐻 and 𝑇∶𝐶→𝐶 be nonexpansive mapping. Then 𝑇 is demiclosed on 𝐶, that is, if 𝑥_𝑛⇀𝑥∈𝐶 and 𝑥_𝑛−𝑇𝑥_𝑛→0, then 𝑥=𝑇𝑥.

Lemma 2.4 (see [23]). Assume {𝑎_𝑛} is a sequence of nonnegative real numbers such that 𝑎_𝑛+1≤1−𝛾_𝑛𝑎_𝑛+𝛿_𝑛𝛾_𝑛+𝜂_𝑛,∀𝑛≥1,(2.5) where {𝛾_𝑛} is a sequence in (0,1) and {𝛿_𝑛},{𝜂_𝑛} are two sequences such that (i)∑^∞𝑛=1𝛾_𝑛=∞;(ii)limsup_𝑛→∞𝛿_𝑛≤0 or ∑^∞𝑛=1|𝛿_𝑛𝛾_𝑛|<∞;(iii)∑^∞𝑛=1|𝜂_𝑛|<∞.
Then lim_𝑛→∞𝑎_𝑛=0.

3. Main Results

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

Algorithm 3.1. Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻 and {𝑇_𝑛}^∞𝑛=1 be infinite family of nonexpansive mappings of 𝐶 into itself. Let 𝑄∶𝐶→𝐶 be a contraction with coefficient 𝛾∈[0,1). For any 𝑥₀∈𝐶, let {𝑥_𝑛} the sequence generated iteratively by 𝑥_𝑛+1=𝛼_𝑛𝑊_𝑛𝑥_𝑛+1−𝛼_𝑛𝑇𝛽_𝑛𝑄𝑥_𝑛+1−𝛽_𝑛𝑥_𝑛,∀𝑛≥0,(3.1) where {𝛼_𝑛},{𝛽_𝑛} are two real numbers in (0,1) and 𝑊_𝑛 is the 𝑊-mapping defined by (2.3).

Now, we give the convergence analysis of the algorithm.

Theorem 3.2. Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻 and {𝑇_𝑛}^∞𝑛=1 be an infinite family of nonexpansive mappings of 𝐶 into itself. Let 𝑄∶𝐶→𝐶 be a contraction with coefficient 𝛾∈[0,1). Assume that the set Ω of solutions of the hierarchical problem (1.9) is nonempty. Let {𝛼_𝑛},{𝛽_𝑛} be two real numbers in (0,1) and {𝑥_𝑛} the sequence generated by (3.1). Assume that the sequence {𝑥_𝑛} is bounded and (i)lim_𝑛→∞𝛼_𝑛=0 and lim_𝑛→∞(𝛽_𝑛/𝛼_𝑛)=0; (ii)∑^∞𝑛=0𝛽_𝑛=∞; (iii)lim_𝑛→∞(1/𝛽_𝑛)|(1/𝛼_𝑛)−(1/𝛼_𝑛−1)|=0 and lim_𝑛→∞(∏^{𝑛−1𝑖=1}𝜉_𝑖/𝛼_𝑛𝛽_𝑛)=lim_𝑛→∞(1/𝛼_𝑛)|1−(𝛽_𝑛−1/𝛽_𝑛)|=0.
Then lim_𝑛→∞(‖𝑥_𝑛+1−𝑥_𝑛‖/𝛼_𝑛)=0 and every weak cluster point of the sequence {𝑥_𝑛} solves the following variational inequality⟨(𝐼−𝑄)𝑥̃,𝑥−𝑥̃∈𝑥̃⟩Ω≥,0,∀𝑥∈Ω.(3.2)

Proof. Set 𝑦_𝑛=𝛽_𝑛𝑄𝑥_𝑛+(1−𝛽_𝑛)𝑥_𝑛 for each 𝑛≥0. Then we have 𝑦_𝑛−𝑦_𝑛−1=𝛽_𝑛𝑄𝑥_𝑛+1−𝛽_𝑛𝑥_𝑛−𝛽_𝑛−1𝑄𝑥_𝑛−1−1−𝛽_𝑛−1𝑥_𝑛−1=+𝛽_𝑛𝛽𝑄𝑥_𝑛𝑛−1−𝑄𝑥_𝑛−1+𝛽_𝑛−𝛽_𝑛−1𝑄𝑥_𝑛−1+1−𝛽_𝑛𝑥_𝑛−𝑥_𝑛−1−𝛽_𝑛𝑥_𝑛−1.(3.3) It follows that ‖‖𝑦_𝑛−𝑦_𝑛−1‖‖≤=𝛾1𝛽_𝑛−‖‖(𝑥1_𝑛−−𝛾𝑥)𝛽_𝑛−1𝑛‖‖+1−𝛽_𝑛‖‖𝑥_𝑛−𝑥_𝑛−1‖‖+||𝛽_𝑛−𝛽_𝑛−1||‖‖𝑄𝑥_𝑛−1‖‖+‖‖𝑥_𝑛−1‖‖‖‖𝑥_𝑛−𝑥_𝑛−1‖‖+||𝛽_𝑛−𝛽_𝑛−1||‖‖𝑄𝑥_𝑛−1‖‖+‖‖𝑥_𝑛−1‖‖.(3.4) From (3.1), we have 𝑥_𝑛+1−𝑥_𝑛=𝛼_𝑛𝑊_𝑛𝑥_𝑛+1−𝛼_𝑛𝑇𝑦_𝑛−𝛼_𝑛−1𝑊_𝑛−1𝑥_𝑛−1−1−𝛼_𝑛−1𝑇𝑦_𝑛−1=+𝛼_𝑛1𝑊−_𝑛𝛼𝑥_𝑛𝑛−𝑊_𝑛𝑥_𝑛−1+𝛼_𝑛−𝛼_𝑛−1𝑊_𝑛𝑥_𝑛−1+𝛼_𝑛−1𝑊_𝑛𝑥_𝑛−1−𝑊_𝑛−1𝑥_𝑛−1𝑇𝑦_𝑛−𝑇𝑦_𝑛−1+𝛼_𝑛−1−𝛼_𝑛𝑇𝑦_𝑛−1.(3.5) Then we obtain ‖‖𝑥_𝑛+1−𝑥_𝑛‖‖≤+𝛼_𝑛||𝛼‖‖_𝑛𝑊_𝑛𝑥_𝑛−𝑊_𝑛𝑥_𝑛−1‖‖+1−𝛼_𝑛‖‖𝑇𝑦_𝑛−𝑇𝑦_𝑛−1‖‖≤𝛼_𝑛−𝛼_𝑛−1||‖‖𝑊_𝑛𝑥_𝑛−1‖‖+‖‖𝑇𝑦_𝑛−1‖‖+𝛼_𝑛−1‖‖𝑊_𝑛𝑥_𝑛−1−𝑊_𝑛−1𝑥_𝑛−1‖‖+𝛼‖‖𝑥_𝑛𝑛−1−𝑥_𝑛−1‖‖+1−𝛼_𝑛‖‖𝑦_𝑛−𝑦_𝑛−1‖‖+||𝛼_𝑛−𝛼_𝑛−1||‖‖𝑊_𝑛𝑥_𝑛−1‖‖+‖‖𝑇𝑦_𝑛−1‖‖‖‖𝑊_𝑛𝑥_𝑛−1−𝑊_𝑛−1𝑥_𝑛−1‖‖.(3.6) From (2.3), since 𝑇_𝑖 and 𝑈_𝑛,𝑖 are nonexpansive, we have ‖‖𝑊_𝑛𝑥_𝑛−1−𝑊_𝑛−1𝑥_𝑛−1‖‖=≤𝜉‖‖₁𝜉₁𝑇₁U_𝑛,2𝑥_𝑛−1−𝜉₁𝑇₁𝑈_𝑛−1,2𝑥_𝑛−1‖‖=𝜉₁‖‖𝑈_𝑛,2𝑥_𝑛−1−𝑈_𝑛−1,2𝑥_𝑛−1‖‖≤𝜉₁‖‖𝜉₂𝑇₂𝑈_𝑛,3𝑥_𝑛−1−𝜉₂𝑇₂𝑈_𝑛−1,3𝑥_𝑛−1‖‖≤≤𝜉⋯₁𝜉₂‖‖𝑈_𝑛,3𝑥_𝑛−1−𝑈_𝑛−1,3𝑥_𝑛−1‖‖≤𝑀𝜉₂⋯_1𝑛−1𝜉_𝑛−1‖‖𝑈_𝑛,𝑛𝑥_𝑛−1−𝑈_{𝑛−1,𝑛}𝑥_𝑛−1‖‖_𝑖=1𝜉_𝑖,(3.7) where 𝑀₁ is a constant such that sup_𝑛≥1{‖𝑈_𝑛,𝑛𝑥_𝑛−1−𝑈_{𝑛−1,𝑛}𝑥_𝑛−1‖}≤𝑀₁. Substituting (3.4) and (3.7) into (3.6), we get ‖‖𝑥_𝑛+1−𝑥_𝑛‖‖≤+𝛼_𝑛||𝛽‖‖_𝑛𝑥_𝑛−𝑥_𝑛−1‖‖+1−𝛼_𝑛1−(1−𝛾)𝛽_𝑛‖‖𝑥_𝑛−𝑥_𝑛−1‖‖+||𝛼_𝑛−𝛽_𝑛−1||‖‖𝑄𝑥_𝑛−1‖‖+‖‖𝑥_𝑛−1‖‖−𝛼_𝑛−1||‖‖𝑊_𝑛𝑥_𝑛−1‖‖+‖‖𝑇𝑦_𝑛−1‖‖+𝛼_𝑛−1𝑀_1𝑛−1_𝑖=1𝜉_𝑖=+1||𝛽−_𝑛(1−𝛾)𝛽_𝑛1−𝛼_𝑛‖‖𝑥_𝑛−𝑥_𝑛−1‖‖+||𝛼_𝑛−𝛽_𝑛−1||‖‖𝑄𝑥_𝑛−1‖‖+‖‖𝑥_𝑛−1‖‖−𝛼_𝑛−1||‖‖𝑊_𝑛𝑥_𝑛−1‖‖+‖‖𝑇𝑦_𝑛−1‖‖+𝛼_𝑛−1𝑀_1𝑛−1_𝑖=1𝜉_𝑖.(3.8) Therefore, it follows that ‖‖𝑥_𝑛+1−𝑥_𝑛‖‖𝛼_𝑛≤1−(1−𝛾)𝛽_𝑛1−𝛼_𝑛‖‖𝑥^𝑛−𝑥_𝑛−1‖‖𝛼_𝑛+||𝛽^𝑛−𝛽_𝑛−1||𝛼_𝑛‖‖𝑄𝑥_𝑛−1+||𝛼_𝑛‖‖+‖‖𝑥_𝑛−1‖‖−𝛼_𝑛−1||𝛼_𝑛‖‖𝑊_𝑛𝑥_𝑛−1‖‖+‖‖𝑇𝑦_𝑛−1‖‖+𝛼_𝑛−1𝑀₁∏^{𝑛−1𝑖=1}𝜉_𝑖𝛼_𝑛=1−(1−𝛾)𝛽_𝑛1−𝛼_𝑛‖‖𝑥^𝑛−𝑥_𝑛−1‖‖𝛼_𝑛−1+1−(1−𝛾)𝛽_𝑛1−𝛼_𝑛‖‖𝑥^𝑛−𝑥_𝑛−1‖‖𝛼_𝑛−‖‖𝑥^𝑛−𝑥_𝑛−1‖‖+||𝛽_𝑛𝛼_𝑛−1−𝛽_𝑛−1||𝛼_𝑛‖‖𝑄𝑥_𝑛−1+||𝛼_𝑛‖‖+‖‖𝑥_𝑛−1‖‖−𝛼_𝑛−1||𝛼_𝑛‖‖𝑊_𝑛𝑥_𝑛−1‖‖+‖‖𝑇𝑦_𝑛−1‖‖+𝛼_𝑛−1𝑀₁∏^{𝑛−1𝑖=1}𝜉_𝑖𝛼_𝑛≤1−(1−𝛾)𝛽_𝑛1−𝛼_𝑛‖‖𝑥^𝑛−𝑥_𝑛−1‖‖𝛼_𝑛−1+||||1𝛼_𝑛−1𝛼_𝑛−1||||+||𝛼_𝑛−𝛼_𝑛−1||𝛼_𝑛+||𝛽^𝑛−𝛽_𝑛−1||𝛼_𝑛+∏^{𝑛−1𝑖=1}𝜉_𝑖=1−(1−𝛾)𝛽_𝑛𝛼_𝑛𝑀1−𝛼_𝑛‖‖𝑥^𝑛−𝑥_𝑛−1‖‖𝛼_𝑛−1×𝑀+(1−𝛾)𝛽_𝑛1−𝛼_𝑛(1−𝛾)1−𝛼_𝑛𝛽1^𝑛||||1𝛼_𝑛−1𝛼_𝑛−1||||+1𝛽_𝑛||𝛼_𝑛−𝛼_𝑛−1||𝛼_𝑛+1𝛽_𝑛||𝛽_𝑛−𝛽_𝑛−1||𝛼_𝑛+∏^{𝑛−1𝑖=1}𝜉_𝑖𝛼_𝑛𝛽_𝑛,(3.9) where 𝑀 is a constant such that sup_𝑛≥1𝑀₁,‖‖𝑥_𝑛−𝑥_𝑛−1‖‖,‖‖𝑊_𝑛𝑥_𝑛−1‖‖+‖‖𝑇𝑦_𝑛−1‖‖,‖‖𝑄𝑥_𝑛−1‖‖+‖‖𝑥_𝑛−1‖‖≤𝑀.(3.10) From (iii), we note that lim_𝑛→∞(1/𝛼_𝑛−1)|𝛼_𝑛−𝛼_𝑛−1/𝛽_𝑛𝛼_𝑛|=0, which implies that lim_𝑛→∞1𝛽_𝑛||𝛼_𝑛−𝛼_𝑛−1||𝛼_𝑛=0.(3.11) Thus it follows from (iii) and (3.11) that lim_𝑛→∞𝛽1_𝑛||||1𝛼_𝑛−1𝛼_𝑛−1||||+1𝛽_𝑛||𝛼_𝑛−𝛼_𝑛−1||𝛼_𝑛+1𝛽_𝑛||𝛽_𝑛−𝛽_𝑛−1||𝛼_𝑛+∏^{𝑛−1𝑖=1}𝜉_𝑖𝛼_𝑛𝛽_𝑛=0.(3.12) Hence, applying Lemma 2.4 to (3.9), we immediately conclude that lim_𝑛→∞‖‖𝑥_𝑛+1−𝑥_𝑛‖‖𝛼_𝑛=0.(3.13) This implies that lim_𝑛→∞‖‖𝑥_𝑛+1−𝑥_𝑛‖‖=0.(3.14) Thus, from (3.1) and (3.14), we have lim_𝑛→∞‖‖𝑥_𝑛−𝑇𝑦_𝑛‖‖=0.(3.15) At the same time, we note that 𝑦_𝑛−𝑥_𝑛=𝛽_𝑛𝑄𝑥_𝑛−𝑥_𝑛⟶0.(3.16) Hence we get lim_𝑛→∞‖‖𝑦_𝑛−𝑇𝑦_𝑛‖‖=0.(3.17) Since the sequence {𝑥_𝑛} is bounded, {𝑦_𝑛} is also bounded. Thus there exists a subsequence of {𝑦_𝑛}, which is still denoted by {𝑦_𝑛} which converges weakly to a point 𝑥̃∈𝐻. Therefore, 𝑥̃∈Fix(𝑇) by (3.17) and Lemma 2.3. By (3.1), we observe that 𝑥_𝑛+1−𝑥_𝑛=𝛼_𝑛𝑊_𝑛𝑥_𝑛−𝑥_𝑛+1−𝛼_𝑛𝑇𝑦_𝑛−𝑦_𝑛+1−𝛼_𝑛𝛽_𝑛𝑄𝑥_𝑛−𝑥_𝑛,(3.18) that is, 𝑥_𝑛−𝑥_𝑛+1𝛼_𝑛=𝐼−𝑊_𝑛𝑥_𝑛+1−𝛼^𝑛𝛼_𝑛(𝐼−𝑇)𝑦_𝑛+𝛽^𝑛1−𝛼_𝑛𝛼_𝑛(𝐼−𝑄)𝑥_𝑛.(3.19) Set 𝑧_𝑛=(𝑥_𝑛−𝑥_𝑛+1)/𝛼_𝑛 for each 𝑛≥1, that is, 𝑧_𝑛=𝐼−𝑊_𝑛𝑥_𝑛+1−𝛼^𝑛𝛼_𝑛(𝐼−𝑇)𝑦_𝑛+𝛽^𝑛1−𝛼_𝑛𝛼_𝑛(𝐼−𝑄)𝑥_𝑛.(3.20) Using monotonicity of 𝐼−𝑇 and 𝐼−𝑊_𝑛, we derive that, for all 𝑢∈Fix(𝑇), ⟨𝑧_𝑛,𝑥=_𝑛−𝑢𝐼⟩−𝑊_𝑛+1−𝛼_𝑛𝑥_𝑛,𝑥_𝑛−𝑢+1−𝛼_𝑛𝛼^𝑛⟨(𝐼−𝑇)𝑦_𝑛−(𝐼−𝑇)𝑢,𝑦_𝑛−𝑢⟩𝛼_𝑛⟨(𝐼−𝑇)𝑦_𝑛,𝑥_𝑛−𝑦_𝑛⟩+𝛽^𝑛1−𝛼_𝑛𝛼_𝑛≥𝐼−𝑊_𝑛⟨(𝐼−𝑄)𝑥_𝑛,𝑥_𝑛−𝑢⟩𝑢,𝑥_𝑛−𝑢+𝛽^𝑛1−𝛼_𝑛𝛼_𝑛⟨(𝐼−𝑄)𝑥_𝑛,𝑥_𝑛−𝑢⟩+1−𝛼^𝑛𝛽_𝑛𝛼_𝑛=⟨(𝐼−𝑊)𝑢,𝑥_𝑛⟨(𝐼−𝑇)𝑦_𝑛,𝑥_𝑛−𝑄𝑥_𝑛⟩−𝑢⟩+𝑊−𝑊_𝑛𝑢,𝑥_𝑛−𝑢+𝛽^𝑛1−𝛼_𝑛𝛼_𝑛+1−𝛼_𝑛⟨(𝐼−𝑄)𝑥_𝑛,𝑥_𝑛−𝑢⟩𝛽_𝑛𝛼_𝑛⟨(𝐼−𝑇)𝑦_𝑛,𝑥_𝑛−𝑄𝑥_𝑛⟩.(3.21) But, since 𝑧_𝑛→0,𝛽_𝑛/𝛼_𝑛→0 and lim_𝑛→∞‖𝑊_𝑛𝑢−𝑊𝑢‖=0 (by Lemma 2.2), it follows from the above inequality that limsup_𝑛→∞⟨(𝐼−𝑊)𝑢,𝑥_𝑛−𝑢⟩≤0,∀𝑢∈Fix(𝑇).(3.22) This suffices to guarantee that 𝜔_𝑤(𝑥_𝑛)⊂Ω. As a matter of fact, if we take any 𝑥^∗∈𝜔_𝑤(𝑥_𝑛), then there exists a subsequence {𝑥_𝑛𝑗} of {𝑥_𝑛} such that 𝑥_𝑛𝑗⇀𝑥^∗. Therefore, we have ⟨(𝐼−𝑊)𝑢,𝑥^∗−𝑢⟩=lim_𝑗→∞(𝐼−𝑊)𝑢,𝑥^𝑛_𝑗−𝑢≤0,∀𝑢∈Fix(𝑇).(3.23) Note that 𝑥^∗∈Fix(𝑇). Hence 𝑥^∗ solves the following problem: ⟨(𝐼−𝑊)𝑢,𝑥_∗𝑥^∗∈Fix(𝑇),−𝑢⟩≤0,∀𝑢∈Fix(𝑇).(3.24) It is obvious that this is equivalent to the problem (1.9) since 𝑊_𝑛→𝑊 uniformly in any bounded set (by Lemma 2.2). Thus 𝑥^∗∈Ω.
Let 𝑥̃ be the unique solution of the variational inequality (3.2). Now, take a subsequence {𝑥_𝑛𝑖} of {𝑥_𝑛} such thatlimsup_𝑛→∞⟨(𝐼−𝑄)𝑥̃,𝑥_𝑛−𝑥̃⟩=lim_𝑖→∞(𝐼−𝑄)𝑥̃,𝑥_𝑛𝑖−𝑥̃.(3.25) Without loss of generality, we may further assume that 𝑥_𝑛𝑖⇀𝑥. Then 𝑥∈Ω. Therefore, we have limsup_𝑛→∞⟨(𝐼−𝑄)𝑥̃,𝑥_𝑛−𝑥̃⟩=(𝐼−𝑄)𝑥̃,𝑥−𝑥̃≥0.(3.26) This completes the proof.

Theorem 3.3. Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let {𝑇_𝑛}^∞𝑛=1 be infinite family of nonexpansive mappings of 𝐶 into itself. Let 𝑄∶𝐶→𝐶 be a contraction with coefficient 𝛾∈[0,1). Assume that the set Ω of solutions of the hierarchical problem (1.9) is nonempty. Let {𝛼_𝑛},{𝛽_𝑛} be two real numbers in (0,1) and {𝑥_𝑛} the sequence generated by (3.1). Assume that the sequence {𝑥_𝑛} is bounded and (i)lim_𝑛→∞𝛼_𝑛=0, lim_𝑛→∞𝛽_𝑛/𝛼_𝑛=0 and lim_𝑛→∞𝛼^2𝑛/𝛽_𝑛=0; (ii)∑^∞𝑛=0𝛽_𝑛=∞; (iii)lim_𝑛→∞(1/𝛽_𝑛)|(1/𝛼_𝑛)−(1/𝛼_𝑛−1)|=0 and lim_𝑛→∞∏^{𝑛−1𝑖=1}𝜉_𝑖/𝛼_𝑛𝛽_𝑛 = lim_𝑛→∞(1/𝛼_𝑛)|1−(𝛽_𝑛−1/𝛽_𝑛)| = 0; (iv)there exists a constant 𝑘>0 such that ‖𝑥−𝑇𝑥‖≥𝑘Dist(𝑥,Fix(𝑇)), where Dist(𝑥,Fix(𝑇))=inf_{𝑦∈Fix(𝑇)}‖𝑥−𝑦‖.(3.27) Then the sequence {𝑥_𝑛} defined by (3.1) converges strongly to a point 𝑥̃∈Fix(𝑇), which solves the variational inequality problem (3.2).

Proof. From (3.1), we have 𝑥_𝑛+1−𝑥̃=𝛼_𝑛𝑊_𝑛𝑥_𝑛−𝑊_𝑛𝑥̃+𝛼_𝑛𝑊_𝑛𝑥̃−𝑥̃+1−𝛼_𝑛𝑇𝑦_𝑛−𝑥̃.(3.28) Thus we have ‖‖𝑥_𝑛+1−𝑥̃‖‖²≤≤‖‖𝛼1_𝑛−𝑊𝛼_𝑛𝑛𝑥_𝑛−𝑊_𝑛𝑥̃+1−𝛼_𝑛𝑇𝑦_𝑛−𝑥̃‖‖²+2𝛼_𝑛𝑊_𝑛𝑥̃−𝑥̃,𝑥_𝑛+1−𝑥̃≤1−𝛼_𝑛‖‖𝑇𝑦_𝑛−𝑥̃‖‖²+𝛼_𝑛‖‖𝑊_𝑛𝑥_𝑛−𝑊_𝑛𝑥̃‖‖²+2𝛼_𝑛𝑊_𝑛𝑥̃−𝑥̃,𝑥_𝑛+1−𝑥̃‖‖𝑦_𝑛−𝑥̃‖‖²+𝛼_𝑛‖‖𝑥_𝑛−𝑥̃‖‖²+2𝛼_𝑛𝑊_𝑛𝑥̃−𝑥̃,𝑥_𝑛+1−𝑥̃.(3.29) At the same time, we observe that ‖‖𝑦_𝑛−𝑥̃‖‖²=‖‖1−𝛽_𝑛𝑥_𝑛−𝑥̃+𝛽_𝑛𝑄𝑥_𝑛−𝑄𝑥̃+𝛽_𝑛(𝑄𝑥̃−𝑥̃)‖‖²≤≤‖‖11−−𝛽𝛽_𝑛𝑛𝑥_𝑛−𝑥̃+𝛽_𝑛𝑄𝑥_𝑛−𝑄𝑥̃‖‖²+2𝛽_𝑛⟨𝑄𝑥̃−𝑥̃,𝑦_𝑛−𝑥̃⟩≤1−𝛽_𝑛‖‖𝑥_𝑛−𝑥̃‖‖²+𝛽_𝑛‖‖𝑄𝑥_𝑛−𝑄𝑥̃‖‖²+2𝛽_𝑛⟨𝑄𝑥̃−𝑥̃,𝑦_𝑛−𝑥̃⟩=1−1−‖‖𝛾𝑥_𝑛2−𝑥̃‖‖²+𝛽_𝑛𝛾²‖‖𝑥_𝑛−𝑥̃‖‖²+2𝛽_𝑛⟨𝑄𝑥̃−𝑥̃,𝑦_𝑛−𝑥̃⟩𝛽_𝑛‖‖𝑥_𝑛−𝑥̃‖‖²+2𝛽_𝑛⟨𝑄𝑥̃−𝑥̃,𝑦_𝑛−𝑥̃⟩.(3.30) Substituting (3.30) into (3.29), we get ‖‖𝑥_𝑛+1−𝑥̃‖‖²≤𝛼_𝑛‖‖𝑥_𝑛−𝑥̃‖‖²+1−𝛼_𝑛1−1−𝛾²𝛽_𝑛‖‖𝑥_𝑛−𝑥̃‖‖²=+12−𝛽_𝑛11−−𝛾𝛼_𝑛2⟨𝑄𝑥̃−𝑥̃,𝑦_𝑛−𝑥̃⟩+2𝛼_𝑛𝑊_𝑛𝑥̃−𝑥̃,𝑥_𝑛+1−𝑥̃+2𝛼_𝑛𝛽_𝑛1−𝛼_𝑛‖‖𝑥_𝑛−𝑥̃‖‖²+2𝛽_𝑛1−𝛼_𝑛⟨𝑄𝑥̃−𝑥̃,𝑦_𝑛−𝑥̃⟩=1−1𝑊−_𝑛𝛾𝑥₂̃−𝑥̃,𝑥_𝑛+1−𝑥̃×2𝛽_𝑛1−𝛼_𝑛‖‖𝑥_𝑛−𝑥̃‖‖²+1−𝛾²𝛽_𝑛1−𝛼_𝑛1−𝛾²⟨𝑄𝑥̃−𝑥̃,𝑦_𝑛−𝑥̃⟩+21−𝛾²1−𝛼_𝑛×𝛼𝛽^𝑛_𝑛𝑊_𝑛𝑥̃−𝑥̃,𝑥_𝑛+1−𝑥̃.(3.31) By Theorem 3.2, we note that every weak cluster point of the sequence {𝑥_𝑛} is in Ω. Since 𝑦_𝑛−𝑥_𝑛→0, then every weak cluster point of {𝑦_𝑛} is also in Ω. Consequently, since 𝑥̃=proj_Ω(𝑄𝑥̃), we easily have limsup_𝑛→∞⟨𝑄𝑥̃−𝑥̃,𝑦_𝑛−𝑥̃⟩≤0.(3.32)
On the other hand, we observe that𝑊_𝑛𝑥̃−𝑥̃,𝑥_𝑛+1−𝑥̃=𝑊_𝑛𝑥̃−𝑥̃,proj_Fix(𝑇)𝑥_𝑛+1−𝑥̃+𝑊_𝑛𝑥̃−𝑥̃,𝑥_𝑛+1−proj_Fix(𝑇)𝑥_𝑛+1.(3.33) Since 𝑥̃ is a solution of the problem (1.9) and proj_Fix(𝑇)𝑥_𝑛+1∈Fix(𝑇), we have 𝑊_𝑛𝑥̃−𝑥̃,proj_Fix(𝑇)𝑥_𝑛+1−𝑥̃≤0.(3.34) Thus it follows that 𝑊_𝑛𝑥̃−𝑥̃,𝑥_𝑛+1−𝑥̃≤≤‖‖𝑊𝑊_𝑛𝑛𝑥̃−𝑥̃,𝑥_𝑛+1−proj_Fix(𝑇)𝑥_𝑛+1=‖‖𝑊_𝑛𝑥̃−𝑥̃‖‖‖‖𝑥_𝑛+1−proj_Fix(𝑇)𝑥_𝑛+1‖‖≤1𝑥̃−𝑥̃‖‖×Dist𝑥_𝑛+1,Fix(𝑇)𝑘‖‖𝑊^𝑛𝑥̃−𝑥̃‖‖‖‖𝑥_𝑛+1−𝑇𝑥_𝑛+1‖‖.(3.35) We note that ‖‖𝑥_𝑛+1−𝑇𝑥_𝑛+1‖‖≤≤𝛼‖‖_𝑛𝑥_𝑛+1−𝑇𝑥_𝑛‖‖+‖‖𝑇𝑥_𝑛−𝑇𝑥_𝑛+1‖‖≤𝛼_𝑛‖‖𝑊_𝑛𝑥_𝑛−𝑇𝑥_𝑛‖‖+1−𝛼_𝑛‖‖𝑇𝑦_𝑛−𝑇𝑥_𝑛‖‖+‖‖𝑥_𝑛+1−𝑥_𝑛‖‖≤𝛼_𝑛‖‖𝑊_𝑛𝑥_𝑛−𝑇𝑥_𝑛‖‖+‖‖𝑦_𝑛−𝑥_𝑛‖‖+‖‖𝑥_𝑛+1−𝑥_𝑛‖‖‖‖𝑊_𝑛𝑥_𝑛−𝑇𝑥_𝑛‖‖+𝛽_𝑛‖‖𝑄𝑥_𝑛−𝑥_𝑛‖‖+‖‖𝑥_𝑛+1−𝑥_𝑛‖‖.(3.36) Hence we have 𝛼_𝑛𝛽_𝑛𝑊_𝑛≤𝛼𝑥̃−𝑥̃,𝑥_𝑛+1−𝑥_2𝑛̃𝛽_𝑛𝑘1‖‖𝑊^𝑛𝑥̃−𝑥̃‖‖‖‖𝑊_𝑛𝑥_𝑛−𝑇𝑥_𝑛‖‖+𝛼_𝑛1𝑘‖‖𝑊^𝑛+𝛼_2𝑛𝑥̃−𝑥̃‖‖‖‖𝑄𝑥_𝑛−𝑥_𝑛‖‖𝛽_𝑛‖‖𝑥_𝑛+1−𝑥_𝑛‖‖𝛼_𝑛𝑘1‖‖𝑊^𝑛𝑥̃−𝑥̃‖‖.(3.37) From Theorem 3.2, we have lim_𝑛→∞‖𝑥_𝑛+1−𝑥_𝑛‖/𝛼_𝑛=0. At the same time, we note that {(1/𝑘)‖𝑊_𝑛𝑥̃−𝑥̃‖‖𝑊_𝑛𝑥_𝑛−𝑇𝑥_𝑛‖}, {(1/𝑘)‖𝑊_𝑛𝑥̃−𝑥̃‖‖𝑄𝑥_𝑛−𝑥_𝑛‖}, and {(1/𝑘)‖𝑊_𝑛𝑥̃−𝑥̃‖} are all bounded. Hence it follows from (i) and the above inequality that limsup_𝑛→∞𝛼_𝑛𝛽_𝑛𝑊_𝑛𝑥̃−𝑥̃,𝑥_𝑛+1−𝑥̃≤0.(3.38)
Finally, by (3.31)–(3.38) and Lemma 2.4, we conclude that the sequence {𝑥_𝑛} converges strongly to a point 𝑥̃∈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 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻 and 𝑆 a nonexpansive mapping of 𝐶 into itself. Let 𝑄∶𝐶→𝐶 be a contraction with coefficient 𝛾∈[0,1). For any 𝑥₀∈𝐶, let{𝑥_𝑛} the sequence generated iteratively by 𝑥_𝑛+1=𝛼_𝑛𝑆𝑥_𝑛+1−𝛼_𝑛𝑇𝛽_𝑛𝑄𝑥_𝑛+1−𝛽_𝑛𝑥_𝑛,∀𝑛≥0,(3.39) where {𝛼_𝑛},{𝛽_𝑛} are two real numbers in (0,1).

Corollary 3.6. Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. Let S∶𝐶→𝐶 be a nonexpansive mapping. Let 𝑄∶𝐶→𝐶 be a contraction with coefficient 𝛾∈[0,1). Assume that the set Ω^ of solutions of the hierarchical problem (1.1) is nonempty. Let {𝛼_𝑛},{𝛽_𝑛} be two real numbers in (0,1) and {𝑥_𝑛} the sequence generated by (3.1). Assume that the sequence {𝑥_𝑛} is bounded and (i)lim_𝑛→∞𝛼_𝑛=0, lim_𝑛→∞𝛽_𝑛/𝛼_𝑛=0 and lim_𝑛→∞𝛼^2𝑛/𝛽_𝑛=0; (ii)∑^∞𝑛=0𝛽_𝑛=∞; (iii)lim_𝑛→∞(1/𝛽_𝑛)|(1/𝛼_𝑛)−(1/𝛼_𝑛−1)|=0 and lim_𝑛→∞(1/𝛼_𝑛)|1−(𝛽_𝑛−1/𝛽_𝑛)|=0; (iv)there exists a constant 𝑘>0 such that ‖𝑥−𝑇𝑥‖≥𝑘Dist(𝑥,Fix(𝑇)), where Dist(𝑥,Fix(𝑇))=inf_{𝑦∈Fix(𝑇)}‖𝑥−𝑦‖.(3.40) Then the sequence {𝑥_𝑛} defined by (3.39) converges strongly to a point 𝑥̃∈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).