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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)
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).
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 . 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  with and and then generalized by Rockafellar . 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.
Suppose that is a nonnegative sequence and is a positive sequence such that . If as , then as .
Assume that is a solution to (1.5) and satisfies (1.6). If and , then as if and only if is bounded.
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 .
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.
Let be a solution to (1.5) and . If satisfies (1.6), then as if and only if is bounded.
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 .
Assume that is compact and . If satisfies (1.6), then as if and only if is bounded.
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 .
Assume that is strongly monotone operator and . If satisfies (1.6), then as if and only if is bounded.
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 .
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 .
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
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 ).
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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) doi:10.1155/2011/867136
- Asymptotic Behavior
- Iterative Algorithm
- Convergence Theorem
- Weak Convergence
- Strong Convergence