Asymptotic Behavior of a Discrete Nonlinear Oscillator with Damping Dynamical System

doi:10.1155/2011/867136

Research Article

Asymptotic Behavior of a Discrete Nonlinear Oscillator with Damping Dynamical System

Hadi Khatibzadeh

1Department of Mathematics, Zanjan University, P.O. Box 45195-313, Zanjan, Iran

2School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5746, Tehran, Iran

Correspondence should be addressed to Hadi Khatibzadeh,[email protected] Received 24 December 2010; Accepted 10 February 2011

Academic Editor: Istvan Gyori

Copyrightq2011 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 operatorA. We also prove some strong convergence theorems with additional assumptions onA. 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 Br´ezis and Lions1978, Lions1978 as well as Djafari Rouhani and H. Khatibzadeh2008.

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×Hwhich we will refer to as anonlinearpossibly multivalued operator inH.Ais called monotoneresp. strongly monotoneify2−y1, x2−x1≥0resp.y2−y1, x2−x1≥α|x1−x2|² for someα >0for allx_i, y_i∈A,i1,2.Ais maximal monotone ifAis monotone andIA is surjective, whereIis the identity operator onH.

Nonlinear oscillator with damping dynamical system, ut γut Aut0,

u0 u0, u0 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 to1–6and references in there.

Following discrete version of1.1,

u_n1 Iλ_nA⁻¹u_nα_nu_n−u_n−1 1.2

is called inertial proximal method and has been studied in3. This iterative algorithm gives a method for approximation of a zero of a maximal monotone operator. In this paper, we propose another discrete version of1.1and study asymptotic behavior of its solutions. By using approximations

ut uth−ut−h

2h oh,

ut uth−2ut ut−h

h² oh,

1.3

for1.1, we get

un1−2unun−1

h²_n γun1−un−1

2h_n Au_n1 0. 1.4

By lettingβγ/2,λn1h²_n/1βhnandαn βhn−1/βhn1, we get

u_n1J_λn11−α_nu_nα_nu_n−1, n≥0,

u₋₁0, u0x∈H, 1.5

whereαnresp.λnis nonnegativeresp. positivesequence andJλ IλA⁻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 in7 withλn λandαn 0 and then generalized by Rockafellar8. We investigate asymptotic behavior of solutions of1.5as discrete version of1.1which also extend previous results of9–11on proximal point algorithm.

Letwn : _n

k1λk⁻¹_n

k1λkuk. Under suitable assumptions, we investigate weak and strong convergence ofw_nandu_nto an element ofA⁻¹0if and only if{u_n}is bounded.

Therefore, A⁻¹0/φ if and only if {u_n} is bounded provided_∞

n1λ_n ∞. Our results extend previous results in2,3,5.

Throughout the paper, we denoteAu_n1 1−α_nu_nα_nu_n−1−u_n1/λ_n1, and we assume the following assumptions on the sequence{α_n}:

0≤α_n ≤1, {α_n}is nonincreasing andα_n−→0 asn−→∞. 1.6

2. Main Results

In this section, we establish convergence of the sequence{u_n}or its weighted average to an element ofA⁻¹0. 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 _∞

n1λ_n ∞. Ifα_n/λ_n → 0 asn → ∞, then_n

k1α_k/_n

k1λ_k → 0 asn → ∞.

We start with a weak ergodic theorem which extends a theorem of Lions11 see also 12page 139 Theorem 3.1 as well as10Theorem 2.1.

Theorem 2.2. Assume thatu_nis a solution to1.5and{α_n}satisfies1.6. If_∞

k1λ_k ∞and αn/λn → 0, thenwn p∈A⁻¹0asn → ∞if and only ifunis bounded.

Proof. Suppose thatwn p∈A⁻¹0by1.5; we get

un1−p≤Jλn11−αnunαnun−1−p≤1−αnun−pαnun−1−p. 2.1

This implies that

u_n1−p≤maxu1−p,u0−p. 2.2

Then{u_n}is bounded and this proves necessity. Now, we prove sufficiency. By monotonicity ofA, we have

Au_n1, u_m1 Au_m1, u_n1≤Au_m1, u_m1 Au_n1, u_n1 2.3

for allm, n ≥0. Multiplying both sides of the above inequality byλ_m1λ_n1and using1.5, we deduce

1−α_nu_n−u_n1, λ_m1u_m1 α_nu_n−1−u_n1, λ_m1u_m1

1−α_mu_m−u_m1, λ_n1u_n1 α_mu_m−1−u_m1, λ_n1u_n1

≤λm11−αnun−un1, un1 λm1αnun−1−un1, un1 λn11−αmum−um1, um1 λn1αmum−1−um1, um1.

2.4

Summing both sides of this inequality fromm0 tomk−1, we get

1−αn

un−un1,^k−1

m0

λm1um1

αn

un−1−un1,^k−1

m0

λm1um1

≤λ_n1|u_n1|^k−1

m0

α_m|u_m−1−u_m|^k−1

m0

u_m1−u_m, λ_n1u_n1

_k−1

m0

λ_m1

1−α_nu_n−u_n1, u_n1

_k−1

m0

λ_m1

α_nu_n−1−u_n1, u_n1

λn1

k−1 m0

1−α_m

2 |um|²− 1−α_m 2 |um1|²

λn1

k−1 m0

α_m

2 |um−1|²−α_m

2 |um1|² λn1|un1|^k−1

m0

αm|um−1−um| uk−u0, λn1un1

_k−1

m0

λ_m1

1−α_nu_n−u_n1, u_n1

_k−1

m0

λ_m1

α_nu_n−1−u_n1, u_n1

λ_n1^k−1

m0

1

2|u_m|²− 1 2|u_m1|²

λ_n1^k−1

m0

α_m

2 |u_m−1|²− α_m 2 |u_m|²

.

2.5

Divide both sides of the above inequality by_k−1

m0λm1and suppose thatknjandwnj p asj → ∞. By assumptions on{αn},{λn}andLemma 2.1, we have

1−αn

un−un1, p αn

un−1−un1, p

≤1−αnun−un1, un1 αnun−1−un1, un1. 2.6

This implies that

1−α_nu_nα_nu_n−1−u_n1, u_n1−p

≥0. 2.7

From1.6, we get

u_n1−pα_nu_n−p≤u_n−pα_n−1u_n−1−p. 2.8 By 1.6 and boundedness of{un}, we get limn→∞|un −p| exists. If wnk q, we obtain again lim_n→∞|un − q| exists. Therefore, lim_n→∞1/2|un − p|² − |un − q|², and hence lim_n→∞u_n, p − q exists. This follows that lim_n→∞w_n, p − q exists. It implies that

q, p−q p, p−q and hence p q and wn p ∈ H asn → ∞. Now we prove p∈A⁻¹0. Suppose thatx, y∈A. By monotonicity ofAand Assumption1.6, we get

⎛

⎝x− _n−1

i0

λ_{i1 −1n−1}

i0

λ_i1u_i1, y

⎞

⎠

_n−1

i0

λ_{i1 −1n−1}

i0

λ_i1

x−u_i1, y

≥ _n−1

i0

λ_{i1 −1n−1}

i0

λ_i1x−u_i1, Au_i1

_n−1

i0

λi1

_−1n−1

i0

x−ui1,1−αiuiαiui−1−ui1

_n−1

i0

λi1

_−1n−1

i0

−1−αiui1−x, ui−x−αiui1−x, ui−1−x |ui1−x|²

≥ _n−1

i0

λ_{i1 −1n−1}

i0

1 2

|u_i1−x|²− |ui−x|² 1

2

α_i|u_i−x|²−α_i−1|u_i−1−x|² .

2.9

Lettingn → ∞, we get:x−p, y≥0. By maximality ofA, we getp∈A⁻¹0.

Remark 2.3. Since range of J_λn is DA the domain of A, as a trivial consequence of Theorem 2.2, we have that IfDAis bounded thenA⁻¹0/φ.

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. Letunbe a solution to1.5andλn ≥λ0 >0. If{αn}satisfies1.6, thenun p∈ A⁻¹0asn → ∞if and only if{un}is bounded.

Proof. Since assumption on{λ_n}implies that_∞

n1λ_n ∞, from1.5and2.7, we get λ²_n1|Aun1|²un1−pλn1Aun1²−un1−p²−2λn1

Aun1, un1−p

≤1−α_n u_n−p

α_n

u_n−1−p²−u_n1−p²

≤1−αnun−p²αnun−1−p²−un1−p²

≤α_n−1u_n−1−p²−α_nu_n−p²u_n−p²−u_n1−p².

2.10

The last inequality follows from Assumption1.6. Summing both sides of this inequality fromn1 tomand lettingm → ∞, since{α_n}satisfies1.6, we have

∞ n1

λ²_n1|Aun1|²<∞. 2.11

By assumption on {λn}, we have |Aun| → 0 asn → ∞. Assumeunj qasj → ∞, by the monotonicity of A, we haveAu_m−Au_nj, u_m −u_nj ≥ 0. Lettingj → ∞, we get Aum, um−q≥0. Similar to the proof ofTheorem 2.2, lim_m→∞|um−q|exists. This implies thatun qp∈A⁻¹0asn → ∞.

In two following, theorems we show strong convergence of {un} under suitable assumptions on operatorAand the sequence{λ_n}.

Theorem 2.5. Assume that I A⁻¹ is compact and _∞

n1λ²_n ∞. Ifαn satisfies 1.6, then un → p∈A⁻¹0asn → ∞if and only if{un}is bounded.

Proof. By2.11and assumption on{λ_n}, we get lim inf_n→∞|Au_n| 0 andu_n pasn →

∞. Therefore, there exists a subsequence{Aunj}of{Aun}such that|Aunj| → 0 asj → ∞ and{u_njAu_nj} is bounded. The compacity ofIA⁻¹ implies that{u_nj}has a strongly convergent subsequencewe denote again by{u_nj}top. By the monotonicity ofA, we have Aun−Aunj, un−unj≥ 0. Lettingj → ∞, we obtainAun, un−p ≥0. Now, the proof of Theorem 2.2shows that lim_n→∞|u_n−p|²exists. This implies thatu_n → pasn → ∞.

Theorem 2.6. Assume thatAis strongly monotone operator and_∞

n1λ_n ∞. If{α_n}satisfies 1.6, thenun → p∈A⁻¹0asn → ∞if and only if{un}is bounded.

Proof. By the proof ofTheorem 2.2,w_n p∈A⁻¹0asn → ∞, and lim_n→∞|u_n−p|²exists.

SinceAis strongly monotone, we have

Au_n1, u_n1−p

≥αu_n1−p². 2.12 Multiplying both sides of2.12byλ_n1and summing fromn1 tom, we have

α^m

n1

λn1un1−p²≤^m

n1

1−αnunαnun−1−un1, un1−p

^m

n1

1−αn

un−p, un1−p αn

un−1−p, un1−p

−un1−p²

≤ 1 2

m n1

1−α_nu_n−p²α_nu_n−1−p²−u_n1−p²

≤ 1 2

m n1

u_n−p²−u_n1−p²α_n−1u_n−1−p²−α_nu_n−p² .

2.13

The last inequality follows from Assumption1.6. Lettingm → ∞, we get:

∞ n1

λn1un1−p²<∞. 2.14 So, lim inf_n→∞|u_n−p|²0. This implies thatu_n → pasn → ∞.

In the following theorem, we assume that A ∂ϕ, where ϕ is a proper, lower semicontinuous and convex function and Argminϕ / φ.

Theorem 2.7. LetA∂ϕ, whereϕis a proper, lower semicontinuous, and convex function. Assume thatA⁻¹0is nonempty (i.e.,ϕhas at least one minimum point) and_∞

n1λn ∞. If{αn}satisfies 1.6, thenu_n p∈A⁻¹0asn → ∞.

Proof. SinceAis subdifferential ofϕandp∈A⁻¹0, by Assumption1.6, we have ϕu_n1−ϕ

p

≤ 1 λ_n1

1−α_nu_nα_nu_n−1−u_n1, u_n1−p

≤ 1 λ_n1

1−α_n 2

u_n−p²−u_n1−p² α_n

2

u_n−1−p²−u_n1−p²

≤ 1 λ_n1

1 2

un−p²−un1−p² 1

2

αn−1un−1−p²−αnun−p² . 2.15

Multiplying both sides of the above inequality byλ_n1 and summing fromn 1 to mand lettingm → ∞, we get

∞ n1

λn1

ϕun1−ϕ p

<∞. 2.16

By assumption on{λn}, we deduce lim inf

n→∞ϕu_n ϕ p

. 2.17

By convexity ofϕ, we have

ϕun1−1−αnϕun−αnϕun−1

≤ϕu_n1−ϕ1−α_nu_nα_nu_n−1

≤ 1

λ_n11−αnunαnun−1−un1, un1−1−αnun−αnun−1

≤0.

2.18

Therefore,

ϕu_n1≤1−α_nϕu_n α_nϕu_n−1. 2.19

From2.19, by Assumption1.6, we get

ϕun1 αnϕun≤ϕun αn−1ϕun−1. 2.20

Again by2.19, we get

ϕun≤max

ϕu0, ϕu1

2.21

for alln >1. By2.20and2.21, we have that

nlim→∞

ϕun1 αnϕun

2.22

exists. From Assumptions1.6,2.17, and2.21, we get

nlim→∞ϕun ϕ p

. 2.23

Ifu_nj q, thenϕp lim inf_j→∞ϕu_nj≥ ϕq. This implies thatq∈A⁻¹0. On the other hand, for eachp ∈A⁻¹0by1.5, we get2.7. The proof ofTheorem 2.2implies that there exists lim_n→∞|un−p|. Then the theorem is concluded by Opial’s Lemmasee13.

Acknowledgment

This research was in part supported by a Grant from IPMno. 89470017.

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