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# Asymptotic Behavior of a Discrete Nonlinear Oscillator with Damping Dynamical System

*Advances in Difference Equations*
**volume 2011**, Article number: 867136 (2011)

## 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,

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),

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

for (1.1), we get

By letting , and , we get

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 :

## 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

This implies that

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

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

Summing both sides of this inequality from to , we get

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

This implies that

From (1.6), we get

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

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

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

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

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

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

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

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

By assumption on , we deduce

By convexity of , we have

Therefore,

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

Again by (2.19), we get

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

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

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]).

## References

Alvarez F:

**On the minimizing property of a second order dissipative system in Hilbert spaces.***SIAM Journal on Control and Optimization*2000,**38**(4):1102-1119. 10.1137/S0363012998335802Alvarez F, Attouch H:

**Convergence and asymptotic stabilization for some damped hyperbolic equations with non-isolated equilibria.***ESAIM. Control, Optimisation and Calculus of Variations*2001,**6:**539-552.Alvarez F, Attouch H:

**An inertial proximal method for maximal monotone operators via discretization of a nonlinear oscillator with damping.***Set-Valued Analysis*2001,**9**(1-2):3-11.Attouch H, Cabot A, Redont P:

**The dynamics of elastic shocks via epigraphical regularization of a differential inclusion. Barrier and penalty approximations.***Advances in Mathematical Sciences and Applications*2002,**12**(1):273-306.Attouch H, Alvarez F:

**The heavy ball with friction dynamical system for convex constrained minimization problems.**In*Optimization (Namur, 1998), Lecture Notes in Econom. and Math. Systems*.*Volume 481*. Springer, Berlin; 2000:25-35. 10.1007/978-3-642-57014-8_2Attouch H, Goudou X, Redont P:

**The heavy ball with friction method. I. The continuous dynamical system: global exploration of the local minima of a real-valued function by asymptotic analysis of a dissipative dynamical system.***Communications in Contemporary Mathematics*2000,**2**(1):1-34. 10.1142/S0219199700000025Martinet B:

**Régularisation d'inéquations variationnelles par approximations successives.***Revue Franćaise d'Informatique et de Recherche Opérationnelle*1970,**4**(Ser. R-3):154-158.Rockafellar RT:

**Monotone operators and the proximal point algorithm.***SIAM Journal on Control and Optimization*1976,**14**(5):877-898. 10.1137/0314056Brézis H, Lions P-L:

**Produits infinis de résolvantes.***Israel Journal of Mathematics*1978,**29**(4):329-345. 10.1007/BF02761171Djafari Rouhani B, Khatibzadeh H:

**On the proximal point algorithm.***Journal of Optimization Theory and Applications*2008,**137**(2):411-417. 10.1007/s10957-007-9329-3Lions P-L:

**Une méthode itérative de résolution d'une inéquation variationnelle.***Israel Journal of Mathematics*1978,**31**(2):204-208. 10.1007/BF02760552Moroşanu G:

*Nonlinear Evolution Equations and Applications, Mathematics and its Applications (East European Series)*.*Volume 26*. D. Reidel Publishing, Dordrecht, The Netherlands; 1988:xii+340.Opial Z:

**Weak convergence of the sequence of successive approximations for nonexpansive mappings.***Bulletin of the American Mathematical Society*1967,**73:**591-597. 10.1090/S0002-9904-1967-11761-0

## Acknowledgment

This research was in part supported by a Grant from IPM (no. 89470017).

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Khatibzadeh, H. Asymptotic Behavior of a Discrete Nonlinear Oscillator with Damping Dynamical System.
*Adv Differ Equ* **2011, **867136 (2011). https://doi.org/10.1155/2011/867136

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DOI: https://doi.org/10.1155/2011/867136

### Keywords

- Asymptotic Behavior
- Iterative Algorithm
- Convergence Theorem
- Weak Convergence
- Strong Convergence