Existence of Weak Solutions for Second Order Boundary Value Problem of Impulsive Dynamic Equations on Time Scales

doi:10.1155/2009/907368

Research Article

Existence of Weak Solutions for

Second-Order Boundary Value Problem of

Impulsive Dynamic Equations on Time Scales

Hongbo Duan and Hui Fang

Department of Applied Mathematics, Kunming University of Science and Technology, Kunming, Yunnan 650093, China

Correspondence should be addressed to Hui Fang,[email protected]

Received 9 April 2009; Accepted 28 June 2009

Recommended by Victoria Otero-Espinar

We study the existence of weak solutions for second-order boundary value problem of impulsive dynamic equations on time scales by employing critical point theory.

Copyrightq2009 H. Duan and H. Fang. 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.

1. Introduction

Consider the following boundary value problem:

−uΔΔt fσt, uσt, t∈0, T_T, t /tj, j 1,2, . . . , p, 1.1

ut_j−ut−_jAju

t−_j, j 1,2, . . . , p, 1.2

uΔt_j−uΔt−_jBjuΔ

t−_j Ij

ut−_j, j1,2, . . . , p, 1.3

u0 0uT, 1.4

whereTis a time scale,0, T_T : 0, T∩T, σ0 0 andσT T, f : 0, T_T×R → R is a given function,Ij ∈CR,R, {Aj}, {Bj}are real sequences withBj 1 Aj−1−1 and

p

k1|Ak|< 1,the impulsive pointstj ∈0, TTare right-dense and 0 t0 < t1 < · · · < tp <

tp 1 T, limh→0 uΔtj hand limh→0 uΔtj−hrepresent the right and left limits ofuΔt

atttjin the sense of the time scale, that is, in terms ofh >0 for whichtj h, tj−h∈0, T_T,

The theory of time scales, which unifies continuous and discrete analysis, was first introduced by Hilger1. The study of boundary value problems for dynamic equations on time scales has recently received a lot of attention, see2–16. At the same time, there have been significant developments in impulsive differential equations, see the monographs of Lakshmikantham et al.17and Samo˘ılenko and Perestyuk18. Recently, Benchohra and Ntouyas19obtained some existence results for second-order boundary value problem of impulsive differential equations on time scales by using Schaefer’s fixed point theorem and nonlinear alternative of Leray-Schauder type. However, to the best of our knowledge, few papers have been published on the existence of solutions for second-order boundary value problem of impulsive dynamic equations on time scales via critical point theory. Inspired and motivated by Jiang and Zhou10, Nieto and O’Regan20, and Zhang and Li21, we study the existence of weak solutions for boundary value problems of impulsive dynamic equations on time scales1.1–1.4via critical point theory.

This paper is organized as follows. InSection 2, we present some preliminary results concerning the time scales calculus and Sobolev’s spaces on time scales. In Section 3, we construct a variational framework for1.1–1.4and present some basic notation and results. Finally,Section 4is devoted to the main results and their proof.

2. Preliminaries about Time Scales

In this section, we briefly present some fundamental definitions and results from the calculus on time scales and Sobolev’s spaces on time scales so that the paper is self-contained. For more details, one can see22–25.

Definition 2.1. A time scaleTis an arbitrary nonempty closed subset ofR,equipped with the topology induced from the standard topology onR.

Fora, b∈T, a < b, a, b_T: a, b∩T,a, b_T: a, b∩T.

Definition 2.2. One defines the forward jump operator σ : T → T, the backward jump operatorρ:T → T,and the graininessμ:T → R 0,∞by

σt:inf{s∈T:s > t}, ρt:sup{s∈T:s < t}, μt σt−t fort∈T, 2.1

respectively. Ifσt t,thentis called right-denseotherwise: right-scattered, and ifρt t, thentis called left-denseotherwise: left-scattered. Denoteyσt yσt.

Definition 2.3. Assumef:T → Ris a function and lett∈T.Then one definesfΔtto be the numberprovided it existswith the property that given anyε >0,there is a neighborhood Uofti.e.,U t−δ, t δ∩Tfor someδ >0such that

fσt−fs−fΔtσt−s≤ε|σt−s| ∀s∈U. 2.2

Definition 2.4. A functionf :T → Ris said to be rd-continuous if it is continuous at right-dense points inT and its left-sided limits existfiniteat left-dense points inT.The set of rd-continuous functionsf:T → Rwill be denoted byCrdT.

As mentioned in24, the LebesgueμΔ-measure can be characterized as follows:

μΔλ

i∈I

σti−tiδti, 2.3

where λ is the Lebesgue measure on R, and {ti}i∈I is the at most countable set of all

right-scattered points of T.A functionf which is measurable with respect toμ_Δ is called

Δ-measurable, and the Lebesgue integral overa, b_Tis denoted by

b

a

ftΔt:

a,bT

ftdμ_Δ. 2.4

The Lebesgue integral associated with the measure μ_Δ on T is called the Lebesgue delta integral.

Lemma 2.5see24, Theorem 2.11. Iff, g :a, b_T → Rare absolutely continuous functions ona, b_T, thenf·gis absolutely continuous ona, b_Tand the following equality is valid:

a,bT

ftgΔtΔt ftgtb_a−

a,bT

fΔtgσtΔt. 2.5

For 1< p <∞, the Banach spaceLp_Δmay be defined in the standard way, namely,

Lp_Δa, b_T:

f :a, b_T−→R|f isΔ-measurable and

b

a

ftpΔt <∞

, 2.6

equipped with the norm

f_Lp

Δ :

b

a

ftpΔt 1/p

. 2.7

LetH1

Δa, bTbe the space of the form

H_Δ1a, b_T:W_Δ1,2a, b_T

:f :0, T_T−→R|f is absolutely continuous ona, b_T, fΔ∈L2_Δa, b_T

its norm is induced by the inner product given by

f, g_H1

Δ:

b

a

fΔtgΔtΔt

b

a

ftgtΔt, 2.9

for allf, g∈H1

Δa, bT.

LetCa, b_Tdenote the linear space of all continuous functionf :a, b_T → Rwith the maximum normfCmaxt∈a,bT|ft|.

Lemma 2.6see24, Corollary 3.8. Let{xm} ⊂ H_Δ1a, b_T, and x ∈ H_Δ1a, b_T.If {xm}

converges weakly inH_Δ1a, b_Ttox, then{xm}converges strongly inCa, bTtox.

Lemma 2.7H ¨older inequality25, Theorem 3.1. Letf, g ∈ Crda, b, p > 1andqbe the

conjugate number ofp.Then

b

a

_ftgtΔt≤ b a

_ftp_Δ

t

1/p _b

a

_gtq_Δ

t 1/q

. 2.10

Whenpq2,we obtain the Cauchy-Schwarz inequality.

For more basic properties of Sobolev’s spaces on time scales, one may refer to Agarwal et al.24.

3. Variational Framework

In this section, we will establish the corresponding variational framework for problem1.1–

1.4.

LetΓj tj, tj 1T,and

fΓjt:

⎧ ⎨ ⎩

ft, t∈tj, tj 1

T,

ft_j, ttj,

3.1

forj0,1, . . . , p.

Now we consider the following space:

H:f:0, T_T−→R|f is continuous from left at eachtj, f

t_jexists,

f_Γj is absolutely continuous onΓj, the delta derivative offΓj∈L

2

Δ

tj, tj 1

T

,

f satisfies the condition1.2for allj0,1, . . . , p, f0 fT 0,

its norm is induced by the inner product given by

First, we give some lemmas which are useful in the proof of theorems.

Lemma 3.1. If p_k1|Ak| < 1, then for any x ∈ H, supt∈0,TT|xt| ≤ R0xH, where R0

Then we have

ut_jujt_jujtj

1 Aj

uj−1tj

1 Aj

utj

1 Aj

ut−_j,

uΓj u

j_{, j0,1, . . . , p.}

3.13

Thusu∈H.Noting that

uk−uH

⎡ ⎣p

j0

tj1

tj

uj_kΔt−uΔ_Γ

jt

2Δt

⎤ ⎦

1/2

≤

⎡ ⎣p

j0

uj

k−u j2

H1

Δtj,tj1T ⎤ ⎦

1/2

,

3.14

we haveukconverges touinHask → ∞. The proof is complete.

Lemma 3.3. Ifp_k1|Ak|<1,then for anyu∈H,

p

j0

tj1

tj

uσ

Γjt

2Δt≤R2

0Tu2H, 3.15

whereR0is given inLemma 3.1.

Proof. For anyu∈H, t∈tj, tj 1_T,byLemma 3.1, we have

uσ

Γjt

≤R0uH, j 0,1, . . . , p, 3.16

which implies that

p

j0

tj1

tj

uσ_Γ

jt

2Δt≤R2₀Tu2_H. 3.17

The proof is complete.

For any u ∈ H satisfying 1.1–1.4, take v ∈ H and multiply 1.1 byvσ_Γ

j, then

integrate it betweentjandtj 1:

− tj1

tj

uΔΔtvσ_Γ

jtΔt tj1

tj

fσt, uσtv_Γσ

By a standard argument, one can prove that the functional ϕ is continuously differentiable at anyu∈Hand

ϕu, v

p

j0

tj1

tj

uΔ_Γ

jtv

Δ ΓjtΔt−

p

j0

tj1

tj

fσt, uσ_Γ

jt

v_Γσ

jtΔt p

j1

Ij

utj

vtj

,

3.23

for allu, v∈H.

We call such critical points weak solutions of problem1.1–1.4.

LetEbe a Banach space,ϕ∈C1_{E,R,which means thatϕis a continuously}

Fr´echet-differentiable functional onE.ϕis said to satisfy the Palais-Smale conditionP-S condition if any sequence{xn} ⊂ Esuch that{ϕxn}is bounded and ϕxn → 0 as n → ∞,has a

convergent subsequence inE.

Lemma 3.4 Mountain pass theorem26, Theorem 2.2,27. Let E be a real Hilbert space. Supposeϕ∈C1_{E,R,satisfies the P-S condition and the following assumptions:}

l1there exist constantsρ >0anda >0such thatϕx≥afor allx∈∂Bρ,whereBρ{x∈

E| xE< ρ}which will be the open ball inEwith radiusρand centered at0;

l2ϕ0≤0and there existsx0/∈Bρsuch thatϕx0≤0.

Thenϕpossesses a critical valuec≥a.Moreover,ccan be characterized as

cinf

h∈Γsmax∈0,1ϕhs, 3.24

where

Γ {h∈C0,1;E|h0 0, h1 x0}. 3.25

4. Main Results

Now we introduce some assumptions, which are used hereafter:

H1the functionf:0, T_T×R → Ris continuous;

H2limx→0ft, x/x 0 holds uniformly fort∈0, TT; H3there exist constantsμ >2 andL >0 such that

0< μFt, x≤xft, x, ∀|x| ≥L; 4.1

H4there exist constantsMj,with 0< M <min{1/2R2₀,μ−2/R2₀μ 2}such that

1 Aj

Ijx≤Mj|x|, ∀x∈R, j 1,2, . . . , p, 4.2

Remark 4.1. H3is the well-known Ambrosetti-Rabinowitz condition from the paper27.

Lemma 4.2. Suppose that the conditions (H1)–(H4) are satisfied, thenϕsatisfies the Palais-Smale condition.

Proof. Let{uk}be the sequence inHsatisfying that{ϕuk}is bounded andϕuk → 0 as

k → ∞.Then there exists a constantβ >0 such that

ϕuk≤β, 4.3

for everyk∈N.ByH3,we know that there exist constantsc1>0, c2>0 such that

Ft, x≥c1|x|μ−c2, 4.4

for allx∈R. ByH4andLemma 3.1, we have

p

j1

utj

0

Ijsds≥ − p

j1

max{0,utj}

min{0,utj}

Ijsds

≥ −1

2

p

j1

Mju

tj2

≥ −1

2MR

2 0u2H,

4.5

p

j1

Ij

utj

utj

≤

p

j1

Ij

utju

tj

≤p

j1

Mju

tj2

≤MR2₀u2_H,

4.6

for allu∈H. Set

Ωj k

t∈tj, tj 1

T|u

j

kσt≥L

,

uj_kt: uk_Γj:

⎧ ⎨ ⎩

ukt, t∈

tj, tj 1

T,

uk

t_j, ttj,

4.7

For anyv∈H,we have

which implies that{uk}converges weakly touinH.

that is,

uk−uH−→0 ask−→ ∞. 4.13

Thus,{uk}possesses a convergent subsequence inH.Then, the P-S condition is now satisfied.

Theorem 4.3. Suppose that (H1)–(H4) hold. Then problem1.1–1.4has at least one nontrivial weak solution onH.

Proof. In order to show thatϕhas at least one nonzero critical point, it suffices to check the conditionsl1andl2. It follows fromH2that there is a constantδ >0 such that

ft, s≤ 1 2R2

0T

|s|, 4.14

for all 0<|s| ≤δandt∈0, T_T.Hence, we have

Fσt, uσt

uσ_t

0

fσt, sds

≤ max{0,u

σ_t}

min{0,uσ_t}

_{fσt, sds}

≤ 1

2R2 0T

max{0,uσ_t}

min{0,uσ_t}|

s|ds

≤ 1

4R2 0T

|uσt|2,

4.15

for all sup_t∈0,T

T|ut| ≤δandt∈0, TT.ByLemma 3.3, we obtain

p

j0

tj1

tj

Fσt, uσ_ΓjtΔt≤ 1 4R2₀T

p

j0

tj1

tj

uσ

Γjt

2Δt

≤ 1

4u

2

H,

4.16

for all sup_t∈0,T

T|ut| ≤δandt∈0, TT.It follows from4.5and4.16that

ϕu≥ 1 2u

2

H−

1 4u

2

H−

1 2MR

2 0u2H

1

2 $

1 2 −MR

2 0

%

u2_H,

Example 4.4. Let0, T_T 0,0.01∪0.02,0.05, t10.02, T 0.05.Then the system

−uΔΔt 4uσt3expuσt4, t∈0, T_T, t /t1,

ut₁−ut−₁ 1 2u

t−₁,

uΔt₁−uΔt−₁ut−₁−1 3u

Δ_t−

1

,

u0 uT 0

4.23

is solvable.

Acknowledgment

This research is supported by the National Natural Science Foundation of China no. 10561004.

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