Hindawi Publishing Corporation
Advances in Difference Equations
Volume 2011, Article ID 867136, 9 pages
doi:10.1155/2011/867136
Research Article
Asymptotic Behavior of a Discrete Nonlinear
Oscillator with Damping Dynamical System
Hadi Khatibzadeh1, 2
1
2
Department of Mathematics, Zanjan University, P.O. Box 45195-313, Zanjan, Iran
School of Mathematics, Institute for Research in Fundamental Sciences (IPM),
P.O. Box 19395-5746, Tehran, Iran
Correspondence should be addressed to Hadi Khatibzadeh, hkhatibzadeh@znu.ac.ir
Received 24 December 2010; Accepted 10 February 2011
Academic Editor: Istvan Gyori
Copyright q 2011 Hadi Khatibzadeh. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
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 A. We also prove some
strong convergence theorems with additional assumptions on A. 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 H be a real Hilbert space with inner product ·, · and norm | · |. We denote weak
convergence in H by and strong convergence by → . Let A be a nonempty subset of
H × H which we will refer to as a nonlinear possibly multivalued operator in H. A is called
monotone resp. strongly monotone if y2 −y1 , x2 −x1 ≥ 0 resp. y2 −y1 , x2 −x1 ≥ α|x1 −x2 |2
for some α > 0 for all xi , yi ∈ A, i 1, 2. A is maximal monotone if A is monotone and I A
is surjective, where I is the identity operator on H.
Nonlinear oscillator with damping dynamical system,
u t γu t Aut 0,
u0 u0 , u 0 u1 ,
1.1
where A is a maximal monotone operator and γ > 0, has been investigated by many
authors specially for asymptotic behavior. We refer the reader to 1–6 and references in there.
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Following discrete version of 1.1,
un1 I λn A−1 un αn un − un−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
u t ut h − ut − h
oh,
2h
ut h − 2ut ut − h
u t oh,
h2
1.3
for 1.1, we get
un1 − 2un un−1
un1 − un−1
γ
Aun1 0.
2hn
h2n
1.4
By letting β γ/2, λn1 h2n /1 βhn and αn βhn − 1/βhn 1, we get
un1 Jλn1 1 − αn un αn un−1 ,
u−1 0,
n ≥ 0,
u0 x ∈ H,
1.5
where αn resp. λn is nonnegative resp. positive sequence and Jλ I λA−1 . This discrete
version gives also an algorithm for approximation of a zero of maximal monotone operator
A. This algorithm extends proximal point algorithm which was introduced by Martinet in 7
with λn λ and αn 0 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 wn : nk1 λk −1 nk1 λk uk . Under suitable assumptions, we investigate weak
and strong convergence of wn and un to an element of A−1 0 if and only if {un } is bounded.
φ if and only if {un } is bounded provided ∞
Therefore, A−1 0 /
n1 λn ∞. Our results
extend previous results in 2, 3, 5.
Throughout the paper, we denote Aun1 1 − αn un αn un−1 − un1 /λn1 , and we
assume the following assumptions on the sequence {αn }:
0 ≤ αn ≤ 1,
{αn } is nonincreasing and αn −→ 0 as n −→ ∞.
1.6
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2. Main Results
In this section, we establish convergence of the sequence {un } or its weighted average to an
element of A−1 0. First we recall the following elementary lemma without proof.
Lemma 2.1. Suppose that {αn } is a nonnegative sequence and {λn } is a positive sequence such that
n
n
∞
n1 λn ∞. If αn /λn → 0 as n → ∞, then
k1 αk /
k1 λk → 0 as n → ∞.
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 un is a solution to 1.5 and {αn } satisfies 1.6. If
αn /λn → 0, then wn p ∈ A−1 0 as n → ∞ if and only if un is bounded.
∞
k1
λk ∞ and
Proof. Suppose that wn p ∈ A−1 0 by 1.5; we get
un1 − p ≤ Jλ 1 − αn un αn un−1 − p ≤ 1 − αn un − p αn un−1 − p.
n1
2.1
This implies that
un1 − p ≤ max u1 − p, u0 − p .
2.2
Then {un } is bounded and this proves necessity. Now, we prove sufficiency. By monotonicity
of A, we have
Aun1 , um1 Aum1 , un1 ≤ Aum1 , um1 Aun1 , un1 2.3
for all m, n ≥ 0. Multiplying both sides of the above inequality by λm1 λn1 and using 1.5,
we deduce
1 − αn un − un1 , λm1 um1 αn un−1 − un1 , λm1 um1 1 − αm um − um1 , λn1 un1 αm um−1 − um1 , λn1 un1 ≤ λm1 1 − αn un − un1 , un1 λm1 αn un−1 − un1 , un1 λn1 1 − αm um − um1 , um1 λn1 αm um−1 − um1 , um1 .
2.4
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Summing both sides of this inequality from m 0 to m k − 1, we get
k−1
1 − αn un − un1 ,
λm1 um1
k−1
αn un−1 − un1 ,
m0
≤ λn1 |un1 |
k−1
k−1
αm |um−1 − um | k−1
um1 − um , λn1 un1 m0
λm1 1 − αn un − un1 , un1 m0
λn1
k−1
λm1 αn un−1 − un1 , un1 m0
k−1 1 − αm 2
m0
λn1 |un1 |
λm1 um1
m0
m0
k−1
|um |2 −
1 − αm |um1 |2
2
λn1
k−1 αm
m0
2
|um−1 |2 −
αm
|um1 |2 2.5
2
αm |um−1 − um | uk − u0 , λn1 un1 m0
k−1
λm1 1 − αn un − un1 , un1 m0
λn1
λm1 αn un−1 − un1 , un1 m0
k−1 1
m0
k−1
1
|um | − |um1 |2
2
2
2
λn1
k−1 αm
m0
2
|um−1 |2 −
αm
|um |2 .
2
Divide both sides of the above inequality by k−1
m0 λm1 and suppose that k nj and wnj p
as j → ∞. By assumptions on {αn }, {λn } and Lemma 2.1, we have
1 − αn un − un1 , p αn un−1 − un1 , p ≤ 1 − αn un − un1 , un1 αn un−1 − un1 , un1 .
2.6
This implies that
1 − αn un αn un−1 − un1 , un1 − p ≥ 0.
2.7
un1 − p αn un − p ≤ un − p αn−1 un−1 − p.
2.8
From 1.6, we get
By 1.6 and boundedness of {un }, we get limn → ∞ |un − p| exists. If wnk q, we obtain
again limn → ∞ |un − q| exists. Therefore, limn → ∞ 1/2|un − p|2 − |un − q|2 , and hence
limn → ∞ un , p − q exists. This follows that limn → ∞ wn , p − q exists. It implies that
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q, p − q p, p − q and hence p q and wn p ∈ H as n → ∞. Now we prove
p ∈ A−1 0. Suppose that x, y ∈ A. By monotonicity of A and Assumption 1.6, we get
⎞
⎛
−1
n−1
n−1
⎝x −
λi1
λi1 ui1 , y⎠
i0
n−1
i0
−1
λi1
i0
≥
n−1
i0
−1
λi1
i0
n−1
λi1
≥
i0
2.9
n−1
x − ui1 , 1 − αi ui αi ui−1 − ui1 i0
−1
λi1
i0
n−1
n−1
λi1 x − ui1 , Aui1 i0
−1
i0
n−1
n−1
λi1 x − ui1 , y
n−1 −1 − αi ui1 − x, ui − x − αi ui1 − x, ui−1 − x |ui1 − x|2
i0
−1
λi1
1
αi |ui − x|2 − αi−1 |ui−1 − x|2 .
|ui1 − x|2 − |ui − x|2 2
2
n−1 1
i0
Letting n → ∞, we get: x − p, y ≥ 0. By maximality of A, we get p ∈ A−1 0.
Remark 2.3. Since range of Jλn is DA the domain of A, as a trivial consequence of
φ.
Theorem 2.2, we have that If DA is bounded then A−1 0 /
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 un be a solution to 1.5 and λn ≥ λ0 > 0. If {αn } satisfies 1.6, then un p ∈
A−1 0 as n → ∞ if and only if {un } is bounded.
Proof. Since assumption on {λn } implies that
∞
n1
λn ∞, from 1.5 and 2.7, we get
2
2 λ2n1 |Aun1 |2 un1 − p λn1 Aun1 − un1 − p − 2λn1 Aun1 , un1 − p
2
2 ≤ 1 − αn un − p αn un−1 − p − un1 − p
2
2 2
≤ 1 − αn un − p αn un−1 − p − un1 − p
2
2 2 2
≤ αn−1 un−1 − p − αn un − p un − p − un1 − p .
2.10
The last inequality follows from Assumption 1.6. Summing both sides of this inequality
from n 1 to m and letting m → ∞, since {αn } satisfies 1.6, we have
∞
λ2n1 |Aun1 |2 < ∞.
n1
2.11
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By assumption on {λn }, we have |Aun | → 0 as n → ∞. Assume unj q as j → ∞,
by the monotonicity of A, we have Aum − Aunj , um − unj ≥ 0. Letting j → ∞, we get
Aum , um − q ≥ 0. Similar to the proof of Theorem 2.2, limm → ∞ |um − q| exists. This implies
that un q p ∈ A−1 0 as n → ∞.
In two following, theorems we show strong convergence of {un } under suitable
assumptions on operator A and the sequence {λn }.
2
Theorem 2.5. Assume that I A−1 is compact and ∞
n1 λn ∞. If αn satisfies 1.6, then
−1
un → p ∈ A 0 as n → ∞ if and only if {un } is bounded.
Proof. By 2.11 and assumption on {λn }, we get lim infn → ∞ |Aun | 0 and un p as n →
∞. Therefore, there exists a subsequence {Aunj } of {Aun } such that |Aunj | → 0 as j → ∞
and {unj Aunj } is bounded. The compacity of I A−1 implies that {unj } has a strongly
convergent subsequence we denote again by {unj } to p. By the monotonicity of A, we have
Aun − Aunj , un − unj ≥ 0. Letting j → ∞, we obtain Aun , un − p ≥ 0. Now, the proof of
Theorem 2.2 shows that limn → ∞ |un − p|2 exists. This implies that un → p as n → ∞.
Theorem 2.6. Assume that A is strongly monotone operator and ∞
n1 λn ∞. If {αn } satisfies
1.6, then un → p ∈ A−1 0 as n → ∞ if and only if {un } is bounded.
Proof. By the proof of Theorem 2.2, wn p ∈ A−1 0 as n → ∞, and limn → ∞ |un − p|2 exists.
Since A is strongly monotone, we have
2
Aun1 , un1 − p ≥ αun1 − p .
2.12
Multiplying both sides of 2.12 by λn1 and summing from n 1 to m, we have
m
m
2 α λn1 un1 − p ≤
1 − αn un αn un−1 − un1 , un1 − p
n1
n1
m 2 1 − αn un − p, un1 − p αn un−1 − p, un1 − p − un1 − p
n1
≤
m 2
2 2 1
1 − αn un − p αn un−1 − p − un1 − p
2 n1
≤
m 1
un − p2 − un1 − p2 αn−1 un−1 − p2 − αn un − p2 .
2 n1
2.13
The last inequality follows from Assumption 1.6. Letting m → ∞, we get:
∞
2
λn1 un1 − p < ∞.
n1
So, lim infn → ∞ |un − p|2 0. This implies that un → p as n → ∞.
2.14
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In the following theorem, we assume that A ∂ϕ, where ϕ is a proper, lower
semicontinuous and convex function and Argmin ϕ / φ.
Theorem 2.7. Let A ∂ϕ, where ϕ is a proper, lower semicontinuous, and convex function. Assume
that A−1 0 is nonempty (i.e., ϕ has at least one minimum point) and ∞
n1 λn ∞. If {αn } satisfies
1.6, then un p ∈ A−1 0 as n → ∞.
Proof. Since A is subdifferential of ϕ and p ∈ A−1 0, by Assumption 1.6, we have
1 1 − αn un αn un−1 − un1 , un1 − p
λn1
2 2 αn 1
1 − αn un−1 − p2 − un1 − p2
un − p − un1 − p ≤
λn1
2
2
2 2 1 2
2 1
1 ≤
un − p − un1 − p αn−1 un−1 − p − αn un − p
.
λn1 2
2
2.15
ϕun1 − ϕ p ≤
Multiplying both sides of the above inequality by λn1 and summing from n 1 to m and
letting m → ∞, we get
∞
λn1 ϕun1 − ϕ p < ∞.
2.16
n1
By assumption on {λn }, we deduce
lim inf ϕun ϕ p .
n → ∞
2.17
By convexity of ϕ, we have
ϕun1 − 1 − αn ϕun − αn ϕun−1 ≤ ϕun1 − ϕ1 − αn un αn un−1 ≤
1
λn1
1 − αn un αn un−1 − un1 , un1 − 1 − αn un − αn un−1 2.18
≤ 0.
Therefore,
ϕun1 ≤ 1 − αn ϕun αn ϕun−1 .
2.19
From 2.19, by Assumption 1.6, we get
ϕun1 αn ϕun ≤ ϕun αn−1 ϕun−1 .
2.20
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Again by 2.19, we get
ϕun ≤ max ϕu0 , ϕu1 2.21
for all n > 1. By 2.20 and 2.21, we have that
lim ϕun1 αn ϕun n → ∞
2.22
exists. From Assumptions 1.6, 2.17, and 2.21, we get
lim ϕun ϕ p .
n → ∞
2.23
If unj q, then ϕp lim infj → ∞ ϕunj ≥ ϕq. This implies that q ∈ A−1 0. On the other
hand, for each p ∈ A−1 0 by 1.5, we get 2.7. The proof of Theorem 2.2 implies that there
exists limn → ∞ |un − p|. Then the theorem is concluded by Opial’s Lemma see 13.
Acknowledgment
This research was in part supported by a Grant from IPM no. 89470017.
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