Asymptotic Behavior of a Discrete Nonlinear Oscillator with Damping Dynamical System

Abstract

We propose a new discrete version of nonlinear oscillator with damping dynamical system governed by a general maximal monotone operator. We show the weak convergence of solutions and their weighted averages to a zero of a maximal monotone operator . We also prove some strong convergence theorems with additional assumptions on . This iterative scheme gives also an extension of the proximal point algorithm for the approximation of a zero of a maximal monotone operator. These results extend previous results by Brézis and Lions (1978), Lions (1978) as well as Djafari Rouhani and H. Khatibzadeh (2008).

1. Introduction

Let be a real Hilbert space with inner product and norm . We denote weak convergence in by and strong convergence by . Let be a nonempty subset of which we will refer to as a (nonlinear) possibly multivalued operator in . is called monotone (resp. strongly monotone) if (resp. for some ) for all , . is maximal monotone if is monotone and is surjective, where is the identity operator on .

Nonlinear oscillator with damping dynamical system,

(1.1)

where is a maximal monotone operator and , has been investigated by many authors specially for asymptotic behavior. We refer the reader to [1–6] and references in there. Following discrete version of (1.1),

(1.2)

is called inertial proximal method and has been studied in [3]. This iterative algorithm gives a method for approximation of a zero of a maximal monotone operator. In this paper, we propose another discrete version of (1.1) and study asymptotic behavior of its solutions. By using approximations

(1.3)

for (1.1), we get

(1.4)

By letting , and , we get

(1.5)

where (resp. ) is nonnegative (resp. positive) sequence and . This discrete version gives also an algorithm for approximation of a zero of maximal monotone operator . This algorithm extends proximal point algorithm which was introduced by Martinet in [7] with and and then generalized by Rockafellar [8]. We investigate asymptotic behavior of solutions of (1.5) as discrete version of (1.1) which also extend previous results of [9–11] on proximal point algorithm.

Let . Under suitable assumptions, we investigate weak and strong convergence of and to an element of if and only if is bounded. Therefore, if and only if is bounded provided . Our results extend previous results in [2, 3, 5].

Throughout the paper, we denote , and we assume the following assumptions on the sequence : 

(1.6)

2. Main Results

In this section, we establish convergence of the sequence or its weighted average to an element of . First we recall the following elementary lemma without proof.

Lemma 2.1.

Suppose that is a nonnegative sequence and is a positive sequence such that . If as , then as .

We start with a weak ergodic theorem which extends a theorem of Lions [11] (see also [12]  page 139 Theorem  3.1 as well as [10] Theorem  2.1).

Theorem 2.2.

Assume that is a solution to (1.5) and satisfies (1.6). If and , then as if and only if is bounded.

Proof.

Suppose that by (1.5); we get

(2.1)

This implies that

(2.2)

Then is bounded and this proves necessity. Now, we prove sufficiency. By monotonicity of , we have

(2.3)

for all . Multiplying both sides of the above inequality by and using (1.5), we deduce

(2.4)

Summing both sides of this inequality from to , we get

(2.5)

Divide both sides of the above inequality by and suppose that and as . By assumptions on , and Lemma 2.1, we have

(2.6)

This implies that

(2.7)

From (1.6), we get

(2.8)

By (1.6) and boundedness of , we get exists. If , we obtain again exists. Therefore, , and hence exists. This follows that exists. It implies that and hence and as . Now we prove . Suppose that . By monotonicity of and Assumption (1.6), we get

(2.9)

Letting , we get: . By maximality of , we get .

Remark 2.3.

Since range of is (the domain of ), as a trivial consequence of Theorem 2.2, we have that If is bounded then .

In the following, we prove a weak convergence theorem. Since the necessity is obvious, we omit the proof of necessity in the next theorems.

Theorem 2.4.

Let be a solution to (1.5) and . If satisfies (1.6), then as if and only if is bounded.

Proof.

Since assumption on implies that , from (1.5) and (2.7), we get

(2.10)

(The last inequality follows from Assumption (1.6)). Summing both sides of this inequality from to and letting , since satisfies (1.6), we have

(2.11)

By assumption on , we have as . Assume as , by the monotonicity of , we have . Letting , we get . Similar to the proof of Theorem 2.2, exists. This implies that as .

In two following, theorems we show strong convergence of under suitable assumptions on operator and the sequence .

Theorem 2.5.

Assume that is compact and . If satisfies (1.6), then as if and only if is bounded.

Proof.

By (2.11) and assumption on , we get and as . Therefore, there exists a subsequence of such that as and is bounded. The compacity of implies that has a strongly convergent subsequence (we denote again by ) to . By the monotonicity of , we have . Letting , we obtain . Now, the proof of Theorem 2.2 shows that exists. This implies that as .

Theorem 2.6.

Assume that is strongly monotone operator and . If satisfies (1.6), then as if and only if is bounded.

Proof.

By the proof of Theorem 2.2,   as , and exists. Since is strongly monotone, we have

(2.12)

Multiplying both sides of (2.12) by and summing from to , we have

(2.13)

(The last inequality follows from Assumption (1.6)). Letting , we get:

(2.14)

So, . This implies that as .

In the following theorem, we assume that , where is a proper, lower semicontinuous and convex function and .

Theorem 2.7.

Let , where is a proper, lower semicontinuous, and convex function. Assume that is nonempty (i.e., has at least one minimum point) and . If satisfies (1.6), then as .

Proof.

Since is subdifferential of and , by Assumption (1.6), we have

(2.15)

Multiplying both sides of the above inequality by and summing from to and letting , we get

(2.16)

By assumption on , we deduce

(2.17)

By convexity of , we have

(2.18)

Therefore,

(2.19)

From (2.19), by Assumption (1.6), we get

(2.20)

Again by (2.19), we get

(2.21)

for all . By (2.20) and (2.21), we have that

(2.22)

exists. From Assumptions (1.6), (2.17), and (2.21), we get

(2.23)

If , then . This implies that . On the other hand, for each by (1.5), we get (2.7). The proof of Theorem 2.2 implies that there exists . Then the theorem is concluded by Opial's Lemma (see [13]).