An Existence Result for Second-Order Impulsive Differential Equations with Nonlocal Conditions

Volume 2010, Article ID 293846,9pages doi:10.1155/2010/293846

Research Article

An Existence Result for Second-Order Impulsive Differential Equations with Nonlocal Conditions

Meili Li and Haiqiang Liu

Department of Applied Mathematics, Donghua University, Shanghai 201620, China

Correspondence should be addressed to Meili Li,[email protected] Received 2 May 2010; Accepted 17 June 2010

Academic Editor: Guang Y. Zhang

Copyrightq2010 M. Li and H. Liu. 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.

The Leray-Schauder alternative is used to investigate the existence of solutions for second-order impulsive differential equations with nonlocal conditions in Banach spaces. The results improve some recent results.

1. Introduction

The theory of impulsive differential equations is emerging as an important area of investigation since it is a lot richer than the corresponding theory of nonimpulsive differential equations. Many evolutionary processes in nature are characterized by the fact that at certain moments in time an abrupt change of state is experienced. That is the reason for the rapid development of the theory of impulsive differential equations; see the monographs1,2.

This paper is concerned with the study on existence of second-order impulsive differential equations with nonlocal conditions of the form

xt f

t, xt, xt

, t∈J 0, b, t /tk, Δx|_ttk Ikxtk, k1, . . . , m, Δx|_ttk Ik

xtk, xtk

, k1, . . . , m, x0 gx x₀, x0 η,

1.1

where the statex·takes values in Banach spaceX with the norm · , x0, η ∈X, 0 t₀ <

t1 < · · ·< tm < t_{m 1} b, Δx|_ttk xt_k−xt⁻_k,Δx|_ttk xt_k−xt⁻_k. f, g, andIk, Ik k 1,2, . . . , mare given functions to be specified later.

The nonlocal condition is a generalization of the classical initial condition. The first results concerning the existence and uniqueness of mild solutions to Cauchy problems with nonlocal conditions were studied by Byszewski 3. Recently, theorems about existence, uniqueness and continuous dependence of impulsive differential abstract evolution Cauchy problems with nonlocal conditions have been studied by Fu and Cao 4, Anguraj and Karthikeyan5, Abada et al.6, Li and Han7, and in the references therein.

Up to now there have been very few papers in this direction dealing with the existence of solutions for second-order impulsive differential equations with nonlocal conditions. Our purpose here is to extend the results of first-order impulsive differential equations to second- order impulsive differential equations with nonlocal conditions.

Our main results are based on the following lemma8.

Lemma 1.1Leray-Schauder alternative. LetSbe a convex subset of a normed linear spaceEand assume that 0∈S. LetG:S → Sbe a completely continuous operator, and let

ζG {x∈S:xλGx,for some 0< λ <1}. 1.2

Then eitherζGis unbounded orGhas a fixed point.

2. Preliminaries

In this section, we introduce notations, definitions, and preliminary facts which are used throughout this paper.

DenoteJ₀ 0, t1, Jk tk, t_{k 1}, JJ\ {tk}, k1,2, . . . , m.We define the following classes of functions:

PCJ, X {x:J → X :xk ∈CJk, X, k0,1, . . . , m, and there existxt_k, xt⁻_k, k 1, . . . , mwithxtk xt⁻_k},

PC¹J, X {x ∈ PCJ, X : x_k ∈ CJk, X, k 0,1, . . . , m, and there exist xt_k, xt⁻_k, k 1, . . . , m with xtk xt⁻_k},where xk and x_k represent the restriction ofxandxtoJ_k, respectivelyk0, . . . , m,andxkJk sup_s∈J

kxks.

Obviously, PCJ, X is a Banach space with the norm xPC max{xkJk, k 0, . . . , m},and PC¹J, Xis also a Banach space with the normx_PC¹ max{xPC,xPC}.

Definition 2.1. A mapf :J×X×X → Xis said to be anL¹-Carath´eodory if if :·, w, v:J → Xis measurable for everyw, v∈X,

iif :t,·,·:X×X → Xis continuous for almost allt∈J,

iiifor eachi >0, there existsαi∈L¹J, R such that for almost allt∈J

sup

w,v≤ift, w, v ≤αit. 2.1

Definition 2.2. A functionx∈PC¹J, X∩C²J, Xis said to be a solution of1.1ifxsatisfies the equationxt ft, xt, xta.e. on J, the conditionsΔx|ttk Ikxtk, Δx|ttk I_kxtk, xtk, k1, . . . , m, andx0 gx x₀, x0 η.

Lemma 2.3. Ifx∈PC¹J, X∩C²J, Xsatisfies

xt f

t, xt, xt

, t /t_k k1,2, . . . , m, 2.2

then

xt x0 _t

0

f

s, xs, xs

ds

0<tk<t

x t_k

−xtk

, ∀t∈J, 2.3

xt x0 x0t

0<tk<t

x t_k

−xtk

0<tk<t

x t_k

−xtk t−tk _t

0

t−sf

s, xs, xs

ds, ∀t∈J.

2.4

Proof. Assume thattk< t≤t_{k 1}heret00, t_{m 1}b. Then

xt1−x0 _t1

0

f

s, xs, xs ds,

xt2−x t₁

_t2

t1

f

s, xs, xs ds,

...

xtk−x t_k−1

_tk

t_k−1

f

s, xs, xs ds,

xt−x t_k

_t

t_k

f

s, xs, xs ds.

2.5

Adding these together, we get

xt x0 _t

0

f

s, xs, xs

ds

0<tk<t

x t_k

−xtk

, 2.6

that is,2.3holds.

Similarly, we have

xt x0

_t

0

xsds

0<tk<t

x t_k

−xtk

. 2.7

Substitution of2.3in2.7gives

xt x0 x0t

0<tk<t

x t_k

−xtk

0<tk<t

x t_k

−xtk t−t_{k t}

0

t−sf

s, xs, xs

ds, ∀t∈J,

2.8

that is,2.4holds.

We assume the following hypotheses:

H1f :J×X×X → Xis anL¹-Carath´eodory map;

H2I_k ∈C, X, Ik ∈CX×X, X, and there exist constantsd_k, d_ksuch thatIkw ≤ d_k, Ikw, v ≤d_kk1, . . . , mfor everyw, v∈X;

H3g : PCJ, X → Xis a continuous function and there exists a constantMsuch that

gx≤M, for eachx∈PCJ, X; 2.9

H4there exists a functionp∈L¹J, R such that

ft, w, v≤ptψw v, for a.e. t∈J and every w, v∈X, 2.10 whereψ:0,∞ → 0,∞is a continuous nondecreasing function with

b 1 _b

0

psds <

_∞

c

ds

ψs, 2.11

where

cx0 M b 1η ^m

k1

d_k b 1−t_kdk

; 2.12

H5for each boundedB⊆PC¹J, Xandt∈Jthe set

x₀−gx tη

0<tk<t

I_kxtk

0<tk<t

I_k

xtk, xtk t−t_k

_t

0

t−sf

s, xs, xs

ds:x∈B 2.13 is relatively compact inX.

3. Main Results

Theorem 3.1. If the hypothesesH1–H5are satisfied, then the second-order impulsive nonlocal initial value problem1.1has at least one solution onJ.

Proof. Consider the spaceBPC¹J, Xwith norm x_PC¹max

x_PC,x

PC

. 3.1

We will now show that the operatorGdefined by

Gxt x₀−gx tη

0<tk<t

I_kxtk

0<tk<t

I_k

xtk, xtk t−t_{k t}

0

t−sf

s, xs, xs

ds, t∈J

3.2

has a fixed point. This fixed point is then a solution of1.1.

First we obtain a priori bounds for the following equation:

xt λ

x₀−gx tη

λ

0<tk<t

I_kxtk λ

0<tk<t

I_k

xtk, xtk t−t_k λ

_t

0

t−sf

s, xs, xs

ds, t∈J.

3.3

We have

xt ≤ x0 M bη ^m

k1

d_k m k1

b−t_kdk b _t

0

psψ

xs xsds, t∈J.

3.4

Denoting byμtthe right-hand side of the above inequality, we have

μ0 x0 M bη ^m

k1

dk b−tkdk

, xt ≤μt, t∈J, μt bptψ

xt xt, t∈J.

3.5

But

xt λη λ _t

0

f

s, xs, xs

ds λ

0<tk<t

I_k

xtk, xtk

, t∈J. 3.6

Thus we have

xt≤η _t

0

psψ

xs xsds m k1

d_k, t∈J. 3.7

Denoting byrtthe right-hand side of the above inequality, we have

r0 η ^m

k1

d_k, xt≤rt, t∈J, rt ptψ

xt xt, t∈J.

3.8

Let

wt μt rt, t∈J. 3.9

Then

w0 μ0 r0 c, wt μt rt

≤bptψwt ptψwt

b 1ptψwt, t∈J.

3.10

This implies that _wt

w0

ds

ψs ≤b 1

_b

0

psds <

_∞

c

ds

ψs, t∈J. 3.11

This inequality implies that there is a constantKsuch that

wt μt rt≤K, t∈J. 3.12

Then

xt ≤μt, t∈J,

xt≤rt, t∈J, 3.13

and hence

x_PC¹ max

x_PC,x

PC

≤K. 3.14

Second, we must prove that the operator G : B → B is a completely continuous operator.

LetB_i {x ∈ B : x_PC¹ ≤ i}for somei ≥ 1.We first show that GmapsB_iinto an equicontinuous family. Letx∈Biandt, t∈J.Then for 0< t < t≤b,we have

Gxt−Gx

t≤tη−tη

t≤tk<t

I_kxtk

0<tk<t

t−t_k− t−t_k

I_k

xtk, xtk

t≤tk<t

t−t_k

I_k

xtk, xtk

_t

0

t−s− t−s

f

s, xs, xs ds

_t

t

t−s

f

s, xs, xs ds

≤

t−tη

t≤tk<t

dk

0<tk<t

t−t

dk

t≤tk<t

t−tk

dk

_t

0

t−t

α_isds _t

t

t−s

α_isds,

3.15

and similarly

Gxt−Gx t≤

_t

0

f

s, xs, xs ds−

_t

0

f

s, xs, xs ds

0<tk<t

I_k

xtk, xtk

−

0<tk<t

I_k

xtk, xtk

≤ _t

t

αisds

t≤tk<t

dk.

3.16

The right-hand sides are independent ofx ∈ Bi and tend to zero ast−t → 0.Thus GmapsB_iinto an equicontinuous family of functions. It is easy to see that the familyGB_iis uniformly bounded. And fromH5, we know thatGBi is compact. Then by Arzela-Ascoli theorem, we can conclude that the mapG:B → Bis compact.

Next, we show that G : B → Bis continuous. Let {un}^∞₀ Bwith u_n → uin B.

Then there is an integerqsuch thatunt,u_nt ≤ qfor allnandt ∈J, soun ∈ Bq and u∈ B_q.ByH1,ft, unt, u_nt → ft, ut, utn → ∞for almost allt ∈J, and since

ft, unt, u_nt−ft, ut, ut<2αqt, we have by the dominated convergence theorem that

Gun−Gu_PCsup

t∈J

gun−gu _t

0

t−s f

s, uns, u_ns

−f

s, us, us ds

0<tk<t

Ikuntk−

0<tk<t

Ikutk

0<tk<t

t−t_kIk

u_ntk, u_ntk

−

0<tk<t

t−t_kIk

utk, utk

≤gun−gu _b

0

b−sf

s, u_ns, u_ns

−f

s, us, usds m

k1

Ikuntk−Ikutk m

k1

b−t_kI_k

u_ntk, u_ntk

−I_k

utk, utk−→0,

3.17 Gun−Gu

PCsup

t∈J

_t

0

f

s, uns, u_ns

−f

s, us, us ds

0<tk<t

Ik

untk, u_ntk

−

0<tk<t

Ik

utk, utk

≤ _b

0

f

s, uns, u_ns

−f

s, us, usds m

k1

I_k

u_ntk, u_ntk

−I_k

utk, utk−→0.

3.18

ThusGis continuous. This completes the proof thatGis completely continuous.

Finally, the setζG {x∈B:x λGx, λ∈0,1}is bounded, as we proved in the first step. As a consequence ofLemma 1.1, we deduce thatGhas a fixed pointx∈Bwhich is a solution of1.1.

Acknowledgments

This work is supported by NNSF of Chinano. 10971139and Chinese Universities Scientific Fundno. B08-1.

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