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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,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F867136/MediaObjects/13662_2010_Article_75_Equ1_HTML.gif)
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),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F867136/MediaObjects/13662_2010_Article_75_Equ2_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F867136/MediaObjects/13662_2010_Article_75_Equ3_HTML.gif)
for (1.1), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F867136/MediaObjects/13662_2010_Article_75_Equ4_HTML.gif)
By letting ,
and
, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F867136/MediaObjects/13662_2010_Article_75_Equ5_HTML.gif)
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
:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F867136/MediaObjects/13662_2010_Article_75_Equ6_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F867136/MediaObjects/13662_2010_Article_75_Equ7_HTML.gif)
This implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F867136/MediaObjects/13662_2010_Article_75_Equ8_HTML.gif)
Then is bounded and this proves necessity. Now, we prove sufficiency. By monotonicity of
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F867136/MediaObjects/13662_2010_Article_75_Equ9_HTML.gif)
for all . Multiplying both sides of the above inequality by
and using (1.5), we deduce
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F867136/MediaObjects/13662_2010_Article_75_Equ10_HTML.gif)
Summing both sides of this inequality from to
, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F867136/MediaObjects/13662_2010_Article_75_Equ11_HTML.gif)
Divide both sides of the above inequality by and suppose that
and
as
. By assumptions on
,
and Lemma 2.1, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F867136/MediaObjects/13662_2010_Article_75_Equ12_HTML.gif)
This implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F867136/MediaObjects/13662_2010_Article_75_Equ13_HTML.gif)
From (1.6), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F867136/MediaObjects/13662_2010_Article_75_Equ14_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F867136/MediaObjects/13662_2010_Article_75_Equ15_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F867136/MediaObjects/13662_2010_Article_75_Equ16_HTML.gif)
(The last inequality follows from Assumption (1.6)). Summing both sides of this inequality from to
and letting
, since
satisfies (1.6), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F867136/MediaObjects/13662_2010_Article_75_Equ17_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F867136/MediaObjects/13662_2010_Article_75_Equ18_HTML.gif)
Multiplying both sides of (2.12) by and summing from
to
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F867136/MediaObjects/13662_2010_Article_75_Equ19_HTML.gif)
(The last inequality follows from Assumption (1.6)). Letting , we get:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F867136/MediaObjects/13662_2010_Article_75_Equ20_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F867136/MediaObjects/13662_2010_Article_75_Equ21_HTML.gif)
Multiplying both sides of the above inequality by and summing from
to
and letting
, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F867136/MediaObjects/13662_2010_Article_75_Equ22_HTML.gif)
By assumption on , we deduce
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F867136/MediaObjects/13662_2010_Article_75_Equ23_HTML.gif)
By convexity of , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F867136/MediaObjects/13662_2010_Article_75_Equ24_HTML.gif)
Therefore,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F867136/MediaObjects/13662_2010_Article_75_Equ25_HTML.gif)
From (2.19), by Assumption (1.6), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F867136/MediaObjects/13662_2010_Article_75_Equ26_HTML.gif)
Again by (2.19), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F867136/MediaObjects/13662_2010_Article_75_Equ27_HTML.gif)
for all . By (2.20) and (2.21), we have that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F867136/MediaObjects/13662_2010_Article_75_Equ28_HTML.gif)
exists. From Assumptions (1.6), (2.17), and (2.21), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F867136/MediaObjects/13662_2010_Article_75_Equ29_HTML.gif)
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]).
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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