Boundary Value Problems
Volume 2011, Article ID 684542,15pages doi:10.1155/2011/684542
Research Article
Minimal Nonnegative Solution of
Nonlinear Impulsive Differential Equations on
Infinite Interval
Xuemei Zhang,
1Xiaozhong Yang,
1and Meiqiang Feng
21Department of Mathematics and Physics, North China Electric Power University, Beijing 102206, China 2School of Applied Science, Beijing Information Science and Technology University, Beijing 100192, China
Correspondence should be addressed to Xuemei Zhang,[email protected]
Received 20 May 2010; Accepted 19 July 2010
Academic Editor: Gennaro Infante
Copyrightq2011 Xuemei Zhang et al. 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 cone theory and monotone iterative technique are used to investigate the minimal nonnegative solution of nonlocal boundary value problems for second-order nonlinear impulsive differential equations on an infinite interval with an infinite number of impulsive times. All the existing results obtained in previous papers on nonlocal boundary value problems are under the case of the boundary conditions with no impulsive effects or the boundary conditions with impulsive effects on a finite interval with a finite number of impulsive times, so our work is new. Meanwhile, an example is worked out to demonstrate the main results.
1. Introduction
The theory of impulsive differential equations describes processes which experience a sudden change of their state at certain moments. Processes with such a character arise naturally and often, especially in phenomena studied in physics, chemical technology, population dynamics, biotechnology, and economics. The theory of impulsive differential equations has become an important area of investigation in the recent years and is much richer than the corresponding theory of differential equations. For an introduction of the basic theory of impulsive differential equations inRn; see Lakshmikantham et al.1, Bainov and Simeonov
2, and Samo˘ılenko and Perestyuk3and the references therein.
5and Guo6. In5, by using fixed-point index theory for cone mappings, Guo and Liu investigated the existence of multiple positive solutions of a boundary value problem for the following second-order impulsive differential equation:
−xt ft, xt t∈J, t /tk, k1,2, . . . , m,
Δx|ttk Ikxtk, k1,2, . . . , m,
ax0−bx0 θ, cx1 dx1 θ,
1.1
wheref ∈CJ×P, P, J 0,1, P is a cone in the real Banach spaceE,θdenotes the zero element ofE,ft, θ θfort∈J, Ikθ θ, k 1,2, . . . , m, 0< t1 < t2 <· · ·< tk <· · ·< tm < 1, a≥0, b≥0, c≥0, d≥0 andδac ad bc >0.
In 6, by using fixed-point theory, Guo established the existence of solutions of a boundary value problem for the following second-order impulsive differential equation in a Banach spaceE:
−xt ft, x, xt t∈J, t /tk, k1,2, . . . , m,
Δx|ttk Ikxtk, k1,2, . . . , m,
Δxtt k Nk
xtk, xtk
, k1,2, . . . , m,
ax0−bx0 x0, cx1 dx1 x0∗,
1.2
wheref∈CJ×E×E, E, J 0,1,Ik∈CE, E, Nk∈CE×E, E, x0, x∗0∈E,0< t1< t2 <
· · ·< tk<· · ·< tm<1, andpac ad bc /0.
On the other hand, the readers can also find some recent developments and applications of the case that impulse effects on a finite interval with a finite number of impulsive times to a variety of problems from Nieto and Rodr´ıguez-L ´opez 14–16, Jankowski17–19, Lin and Jiang20, Ma and Sun21, He and Yu22, Feng and Xie23, Yan24, Benchohra et al.25, and Benchohra et al.26.
Recently, in 27, Li and Nieto obtained some new results of the case that impulse effects on an infinite interval with a finite number of impulsive times. By using a fixed-point theorem due to Avery and Peterson28, Li and Nieto considered the existence of multiple positive solutions of the following impulsive boundary value problem on an infinite interval:
ut qtft, u 0, ∀0< t <∞, t /tk, k1,2, . . . , p
Δutk Ikutk, k1,2, . . . , p,
u0 m−2
i1
αiuξi, u∞ 0,
1.3
wheref ∈C0, ∞×0, ∞,0, ∞, Ik∈C0, ∞,0, ∞, u∞ limt→ ∞ut, 0<
At the same time, we also notice that there has been increasing interest in studying nonlinear differential equation and impulsive integrodifferential equation on an infinite interval with an infinite number of impulsive times; to identify a few, we refer the reader to Guo and Liu29, Guo30–32, and Li and Shen33. It is here worth mentioning the works by Guo31. In31, Guo investigated the minimal nonnegative solution of the following initial value problem for a second order nonlinear impulsive integrodifferential equation of Volterra type on an infinite interval with an infinite number of impulsive times in a Banach spaceE:
x ft, x, Tx, ∀t≥0, t /tk,
Δx|ttk Ikxtk,
Δxtt
k Nkxtk k1,2, . . ., x0 x0, x0 x0∗,
1.4
wheref ∈CJ×P×P, E, Ik,Nk ∈CP, P, J 0,∞, x0,x∗0 ∈P, 0 < t1 <· · · < tk <· · · <
· · ·, tk → ∞,ask → ∞, Pis a cone ofE.
However, the corresponding theory for nonlocal boundary value problems for impulsive differential equations on an infinite interval with an infinite number of impulsive times is not investigated till now. Now, in this paper, we will use the cone theory and monotone iterative technique to investigate the existence of minimal nonnegative solution for a class of second-order nonlinear impulsive differential equations on an infinite interval with an infinite number of impulsive times.
Consider the following boundary value problem for second-order nonlinear impulsive differential equation:
−xt ft, xt, xt t∈J, t /tk,
Δx|ttk Ikxtk, k1,2, . . . ,
Δxtt
k Ikxtk, k1,2, . . . , x0
∞
0
gtxtdt, x∞ 0,
1.5
where J 0,∞, f ∈ CJ × R ×R , R , R 0, ∞, 0 < t1 < t2 < · · · < tk <
· · ·, tk → ∞, Ik ∈CR , R , Ik∈CR , R , gt∈CR , R ,with ∞
0 gtdt <1. x∞
limt→ ∞xt.Δx|ttkdenotes the jump ofxtatttk, that is,
Δx|ttk xtk−xt−k, 1.6
Let
P CJ, R x : xis a map fromJ intoRsuch thatxtis continuous att /tk,
left continuous atttk andx
tkexist fork1,2, . . . ,
P C1J, R x∈P CJ, R: xtexists and is continuous att /tk,
left continuous atttk andx
tkexistfor k1,2, . . . .
1.7
LetE {x∈P C1J, R: sup
t∈J|xt|/1 t <∞, supt∈J|xt|< ∞}with the norm x max{ x 1, x ∞},where
x 1sup t∈J
|xt|
1 t, x
∞sup
t∈J
xt. 1.8
Define a coneP ⊂Eby
Px∈E : xt≥0, xt≥0 . 1.9
LetJ J\ {t1, t2, . . . , tk, . . . ,}, J0 0, t1,and Ji ti, ti 1 i 1,2,3, . . ..x ∈ E∩ C2J, Ris called a nonnegative solution of1.5, ifxt≥0, xt≥0 andxtsatisfies1.5.
IfIk0, Ik0, k1,2, . . . , gt 0, then boundary value problem1.5reduces to the following two point boundary value problem:
−xt ft, xt, xt t∈J,
x0 0, x∞ 0, 1.10
which has been intensively studied; see Ma34, Agarwal and O’Regan35, Constantin36, Liu37,38, and Yan and Liu39for some references along this line.
The organization of this paper is as follows. InSection 2, we provide some necessary background. In Section 3, the main result of problem1.5 will be stated and proved. In Section 4, we give an example to illustrate how the main results can be used in practice.
2. Preliminaries
H1Suppose thatf ∈CJ×R ×R , R , Ik ∈CR , R , Ik ∈CR , R ,and there existp,q,r∈CJ, R and nonnegative constantsck, dk, ek, fksuch that
ft, u, v≤ptu qtv rt, ∀t∈J, and∀u, v∈R ,
Iku≤cku dk, ∀u∈R k1,2,3. . .,
Iku≤eku fk, ∀u∈R k1,2,3. . .,
p∗ ∞
0
ptt 1dt <∞, q∗ ∞
0
qtdt <∞,
r∗ ∞
0
rtdt <∞, c∗
∞
k1
tk 1ck<∞,
d∗
∞
k1
dk<∞, e∗
∞
k1
tk 1ek<∞, f∗
∞
k1
fk<∞.
2.1
H2ft, u1, v1≤ ft, u2, v2, Iku1≤ Iku2, Iku1≤Iku2,fort∈J, u1 ≤u2, v1 ≤ v2k1,2,3. . ..
Lemma 2.1. Suppose thatH1holds. Then for allx∈P,0∞ft, xt, xtdt,∞k1Ikxtk,and ∞
k1Ikxtkare convergent. Proof. ByH1,we have
ft, xt, xt≤ptt 1xt
t 1 qtx
t rt,
Ikxtk≤cktk 1xtk tk 1 dk,
Ikxtk≤ektk 1xtk tk 1
fk.
2.2
Thus,
∞
0
fs, xs, xsds≤p∗||x||1 q∗x∞ r∗<∞,
∞
k1
Ikxtk≤c∗ x 1 d∗<∞,
∞
k1
Ikxtk≤e∗ x 1 f∗<∞.
2.3
Lemma 2.2. Suppose thatH1holds. If0 ≤0∞gtdt <1, thenx∈E∩C2J, Ris a solution of
problem1.5if and only ifx∈Eis a solution of the following impulsive integral equation:
xt ∞
0
Gt, sfs, xs, xsds
∞
k1
Gt, tkIkxtk
∞
k1
Gst, tkIkxtk
1 1−0∞gtdt
∞
0 gt
∞
0
Gt, sfs, xs, xsds
∞
k1
Gt, tkIkxtk
∞
k1
Gst, tkIkxtk
dt, ∀t∈J,
2.4
where
Gt, s ⎧ ⎨ ⎩
t, 0≤t≤s < ∞,
s, 0≤s≤t < ∞,
Gst, s ⎧ ⎨ ⎩
0, 0≤t≤s < ∞,
1, 0≤s≤t < ∞.
2.5
Proof. First, suppose thatx ∈E∩C2J, Ris a solution of problem1.5. It is easy to see by
integration of1.5that
−xt x0
t
0
fs, xs, xsds tk<t
Ikxtk. 2.6
Taking limit fort → ∞, byLemma 2.1and the boundary conditions, we have
x0
∞
0
fs, xs, xsds
∞
k1
Ikxtk. 2.7
Thus,
xt ∞
0
fs, xs, xsds
∞
k1
Ikxtk− t
0
fs, xs, xsds− tk<t
Integrating2.8, we can get
xt x0
∞
0
Gt, sfs, xs, xsds
∞
k1
Gt, tkIkxtk
∞
k1
Gst, tkIkxtk
∞
0
gtxtdt ∞
0
Gt, sfs, xs, xsds
∞
k1
Gt, tkIkxtk
∞
k1
Gst, tkIkxtk.
2.9
It follows that
∞
0
gtdt 1
1−0∞gtxtdt ∞
0 gt
∞
0
Gt, sfs, xs, xsds
∞
k1
Gt, tkIkxtk
∞
k1
Gst, tkIkxtk
dt.
2.10
So we have2.4.
Conversely, suppose thatx∈Eis a solution of2.4. Evidently,
Δx|ttk Ikxtk, k1,2, . . . ,. 2.11
Direct differentiation of2.4implies, fort /tk,
xt ∞
t
fs, xs, xsds tk≥t
Ikxtk,
Δxtt
k Ikxtk, k1,2, . . . ,, xt −ft, xt, xt.
2.12
Sox∈C2J, R. It is easy to verify thatx0 ∞
0 gtxtdt, x∞ 0. The proof of
Lemma 2.2is complete.
Define an operatorT :E → E,
Txt ∞
0
Gt, sfs, xs, xsds
∞
k1
Gt, tkIkxtk
∞
k1
Gst, tkIkxtk
1 1−0∞gtdt
∞
0 gt
∞
0
Gt, sfs, xs, xsds
∞
k1
Gt, tkIkxtk
∞
k1
Gst, tkIkxtk
dt, ∀t∈J.
Lemma 2.3. Assume thatH1andH2hold. Then operatorT mapsPintoP, and
Tx ≤β α x , ∀x∈P, 2.14
where
α 2− ∞
0 gtdt
1−0∞gtdt
p∗ q∗ c∗ e∗, β 2− ∞
0 gtdt
1−0∞gtdt
r∗ f∗ d∗. 2.15
Moreover, forx,y∈Pwithxt≤yt, xt≤yt, for allt∈J,one has
Txt≤Tyt, Txt≤Tyt, ∀t∈J. 2.16
Proof. Letx∈P. From the definition ofT andH1, we can obtain thatTis an operator from PintoP, and
|Txt|
1 t
≤
∞
0
fs, xs, xsds
∞
k1
Ikxtk
∞
k1
|Ikxtk| 1
1−0∞gtdt
∞
0 f
s, xs, xsds
∞
k1
Ikxtk
∞
k1
|Ikxtk|
≤ 2−
∞ 0 gtdt
1−0∞gtdt
p∗ q∗ c∗ e∗ x 2− ∞
0 gtdt
1−0∞gtdt
r∗ f∗ d∗
α x β, ∀t∈J.
2.17
Direct differentiation of2.13implies, fort /tk,
Txt ∞
t
fs, xs, xsds tk≥t
Ikxtk. 2.18
3. Main Result
In this section, we establish the existence of a minimal nonnegative solution for problem1.5.
Theorem 3.1. Let conditionsH1-H2be satisfied. Suppose further that
α 2− ∞
0 gtdt
1−0∞gtdt
p∗ q∗ c∗ e∗<1. 3.1
Then problem1.5has the minimal nonnegative solutionxwith x ≤β/1−α, whereβis defined as in Lemma 2.3. Here, the meaning of minimal nonnegative solution is that if x is an arbitrary nonnegative solution of 1.5, then xt ≥ xt, xt ≥ xt,for allt ∈ J. Moreover, if we let
x0t 0, xnt Txn−1t,for allt∈J n1,2, . . .,thenxn⊂Pwith
0x0t≤x1t≤ · · · ≤xnt≤ · · · ≤xt, ∀t∈J,
0x0t≤x1t≤ · · · ≤xnt≤ · · · ≤xt, ∀t∈J,
3.2
and{xnt}and{xnt}converge uniformly toxtandxtonJi,i0,1,2, . . ., respectively.
Proof. ByLemma 2.3and the definition of operatorT, we havexn⊂P, and
xn ≤β α xn−1 n1,2,3, . . ., 3.3
0x0t≤x1t≤ · · · ≤xnt≤ · · · , ∀t∈J, 3.4
0x0t≤x1t≤ · · · ≤xnt≤ · · · , ∀t∈J. 3.5
By3.3, we have
xn ≤β αβ α2β · · · αn−1β
β1−αn
1−α ≤
β
1−α, n1,2. . .. 3.6
From3.4,3.5, and3.6, we know that limn→ ∞xntand limn→ ∞xntexist. Suppose that
lim
n→ ∞xnt xt, nlim→ ∞x
By the definition ofxnt,we have
xnt ∞
t
fs, xn−1s, xn−1s
ds
tk≥t
Ikxntk, ∀t∈J, n1,2, . . ., 3.8
xnt −ft, xn−1t, xn−1t
, ∀t∈J,n1,2, . . .. 3.9
From3.6, we obtain
|xnt| t 1 ≤
β
1−α, x
nt≤ β
1−α, ∀t∈J
,n1,2, . . .. 3.10
It follows that {xnt}is equicontinuous on everyJi i 0,1,2, . . .. Combining this with Ascoli-Arzela theorem and diagonal process, there exists a subsequence which converges uniformly toxtonJi i0,1,2, . . .. Which together with3.4imply that{xnt}converges uniformly toxtonJi i0,1,2, . . ., andx∈P CJ, R, x 1≤β/1−α.On the other hand,
byH1,3.6, and3.9, we have
x
nt≤ptt 1 xn−1 1 qtxn−1∞ rt
≤ptt 1 β
1−α qt β
1−α rt st∈CJ, R , ∀t∈J
n1,2, . . .. 3.11
Sincestis bounded on0, M M is a finite positive number,{xnt}is equicontinuous on everyJi, i 1,2, . . . .Combining this with Ascoli-Arzela theorem and diagonal process, there exists a subsequence which converges uniformly to ytonJi i 0,1,2, . . ., which together with 3.5 imply that {xnt} converges uniformly to yt on Ji i 0,1,2, . . ., and y ∈ P CJ, R, y ∞ ≤ β/1−α. From above, we know thatxt exists and xt yt,for all t∈J.It follows thatx∈Pand
x ≤ β
1−α. 3.12
Now we prove thatxt Txt.
By the continuity offand the uniform convergence ofxnt, xnt, we know that
fs, xns, xns
−→fs, xs, xs, n−→ ∞,∀t∈J. 3.13
On the other hand, byH1and3.6and3.12, we have
fs, xns, xns
−fs, xs, xs
≤2pss 1 qs β
1−α 2rs zs∈LJ, R n1,2, . . ..
Combining this with the dominated convergence theorem, we have
lim n→ ∞
∞
t
fs, xns, xns
ds ∞
t
fs, xs, xsds, ∀t∈J,
lim n→ ∞
t
0
fs, xns, xns
ds t
0
fs, xs, xsds, ∀t∈J.
3.15
Moreover, we can see that
lim n→ ∞
0<tk<t
Ikxntk
0<tk<t
Ikxtk,
lim n→ ∞
0<tk<t
Ikxntk
0<tk<t
Ikxtk,
lim n→ ∞
tk≥t
Ikxntk
tk≥t
Ikxtk,
lim n→ ∞
tk≥t
Ikxntk
tk≥t
Ikxtk.
3.16
Now taking limits from two sides ofxnt Txn−1tand using3.15–3.16, we have
xt ∞
0
Gt, sfs, xs, xsds
∞
k1
Gt, tkIkxtk
∞
k1
Gst, tkIkxtk
1 1−0∞gtdt
∞
0 gt
∞
0
Gt, sfs, xs, xsds
∞
k1
Gt, tkIkxtk
∞
k1
Gst, tkIkxtk
dt, ∀t∈J.
3.17
ByLemma 2.2,xtis a nonnegative solution of1.5.
4. Example
To illustrate how our main results can be used in practice, we present an example.
Example 4.1. Consider the following boundary value problem of second-order impulsive differential equation on infinite interval
−x 1
100t 1
−9/41 x x2/3
, t∈J, t /k,
Δx|tk 1
8kxk 1
1/2 k1,2, . . .,
Δx|tk 1
10kxk 1
1/2 k1,2, . . .,
x0
∞
0
1 2e
−txtdt, x∞ 0,
4.1
where
ft, x, y 1
100t 1
−9/41 x y2/3 ,
Ikxtk 1
8kxk 1
1/2, k1,2, . . . ,
Ikxtk 1
10kxk 1
1/2, k1,2, . . . ,
gt 1
2e
−t.
4.2
Evidently,xt≡0 is not the solution of4.1.
Conclusion
Problem4.1has minimal positive solution.
Proof. It is clear that0∞1/2e−tdt <1 andH
2is satisfied.
By the inequality1 xγ ≤1 γx,for all 0≤x <∞,0< γ <1,we see that
ft, x, y≤ 1
100t 1
−9/4
1 2
3x 2 3y
,
Ikx≤ 1 8k
1 1
2x
, Ikx≤ 1 10k
1 1
2x
, k1,2, . . . .
Let
pt 1
150t 1
−9/4, qt 1
150t 1
−9/4, rt 1
100t 1
−9/4,
ck 1
2×8k, dk 1
8k, ek 1
2×10k, fk 1 10k.
4.4
Then, we easily obtain that
∞
0
pt1 tdt 4
150,
∞
0
qtdt 4
750,
∞
0
rtdt 2
250,
∞
k1
tk 1ck 15 98,
∞
k1 dk
1 7,
∞
k1
tk 1ek 19 162,
∞
k1 fk
1 9.
4.5
Thus,H1is satisfied andα150001/165375 <1. ByTheorem 3.1, it follows that problem
4.1has a minimal positive solution.
Acknowledgments
This work is supported by the National Natural Science Foundation of China10771065, the Natural Sciences Foundation of Hebei Province A2007001027, the Funding Project for Academic Human Resources Development in Institutions of Higher Learning Under the Jurisdiction of Beijing Municipality PHR201008430, the Scientific Research Common Program of Beijing Municipal Commission of EducationKM201010772018, the 2010 level of scientific research of improving project5028123900, and Beijing Municipal Education Commission71D0911003. The authors thank the referee for his/her careful reading of the paper and useful suggestions.
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