Volume 2009, Article ID 834158,12pages doi:10.1155/2009/834158
Research Article
Existence of Positive Solutions for
Multipoint Boundary Value Problem on
the Half-Line with Impulses
Jianli Li
1and Juan J. Nieto
21Department of Mathematics, Hunan Normal University, Changsha, Hunan 410081, China
2Departamento de An´alisis Matem´atico, Facultad de Matem´aticas, Universidad de Santiago de Compostela,
15782 Santiago de Compostela, Spain
Correspondence should be addressed to Jianli Li,[email protected]
Received 7 March 2009; Revised 18 April 2009; Accepted 25 April 2009
Recommended by Donal O’Regan
We consider a multi-point boundary value problem on the half-line with impulses. By using a fixed-point theorem due to Avery and Peterson, the existence of at least three positive solutions is obtained.
Copyrightq2009 J. Li and J. J. Nieto. 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
Impulsive differential equations are a basic tool to study evolution processes that are subjected to abrupt changes in their state. For instance, many biological, physical, and engineering applications exhibit impulsive effectssee1–3. It should be noted that recent progress in the development of the qualitative theory of impulsive differential equations has been stimulated primarily by a number of interesting applied problems4–24.
In this paper, we consider the existence of multiple positive solutions of the following impulsive boundary value problemfor short BVPon a half-line:
ut qtft, u 0, 0< t <∞, t /tk,
Δutk Ikutk, k1, . . . , p,
u0
m−2
i1
αiuξi, u∞ 0,
whereu∞ limt→∞ut, 0 < ξ1 < ξ2 < · · · < ξm−2 < ∞, 0 < t1 < t2 < · · · < tp < ∞,
Δutk utk−ut−k, andαi, f, q, andIksatisfy
H10<mi1−2αi<1;
H2ft, u∈C0,∞×0,∞,0,∞,Iku∈C0,∞,0,∞, and whenu/1t
is bounded,ft, uandIkuare bounded on0,∞;
H3qt∈C0,∞,0,∞andqtis not identically zero on any compact subinterval of0,∞. Furthermoreqtsatisfies
sup
t∈0,∞
∞
0
Gt, sqsds <∞, 1.2
where
Gt, s
⎧ ⎨ ⎩
s, 0≤s≤t <∞,
t, 0≤t≤s <∞.
1.3
Boundary value problems on the half-line arise quite naturally in the study of radially symmetric solutions of nonlinear elliptic equations and there are many results in this area, see8,13,14,20,25–27, for example.
Lian et al. 25 studied the following boundary value problem of second-order differential equation with ap-Laplacian operator on a half-line:
ϕput φtf
t, u, u 0, 0< t <∞,
αu0−βu0 0, u∞ 0. 1.4
They showed the existence at least three positive solutions for1.4 by using a fixed point theorem in a cone due to Avery-Peterson28.
Yan 20, by using Leray-Schauder theorem and fixed point index theory presents some results on the existence for the boundary value problems on the half-line with impulses and infinite delay.
However to the best knowledge of the authors, there is no paper concerned with the existence of three positive solutions to multipoint boundary value problems of impulsive differential equation on infinite interval so far. Motivated by20,25, in this paper, we aim to investigate the existence of triple positive solutions for BVP1.1. The method chosen in this paper is a fixed point technique due to Avery and Peterson28.
2. Preliminaries
In this section, we give some definitions and results that we will use in the rest of the paper.
Definition 2.1. Suppose P is a cone in a Banach. The map α is a nonnegative continuous concave functional onP providedα:P → 0,∞is continuous and
for all x, y ∈ P, and t ∈ 0,1. Similarly, the map β is a nonnegative continuous convex functional onPprovidedβ:P → 0,∞is continuous and
βtx 1−ty ≤tβx 1−tβy 2.2
for allx, y∈P, andt∈0,1.
Letγ, θbe nonnegative, continuous, convex functionals onP andαbe a nonnegative, continuous, concave functionals onP, andψ be a nonnegative continuous functionals onP. Then, for positive real numbersa, b, c, andd, we define the convex sets
Pγ, d x∈P :γx< d,
Pγ, α, b, d x∈P:b≤αx, γx≤d,
Pγ, θ, α, b, c, d x∈P :b≤αx, θx≤c, γx≤d,
2.3
and the closed set
Rγ, ψ, a, d x∈P :a≤ψx, γx≤d. 2.4
To prove our main results, we need the following fixed point theorem due to Avery and Peterson in28.
Theorem 2.2. Let P be a cone in a real Banach spaceE. Let γ and θ be nonnegative continuous convex functionals on a coneP,αbe a nonnegative continuous concave functional onP, andψ be a nonnegative continuous functional onPsatisfyingψλx≤λψxfor0≤λ≤1, such that for some positive numbersMandd
αx≤ψx, x ≤Mγx 2.5
for allx∈Pγ, d. Suppose
Φ:Pγ, d−→Pγ, d 2.6
is completely continuous and there exist positive numbersa, d, andcwitha < bsuch that
i{x∈Pγ, θ, α, b, c, d:αx> b}/∅andαΦx> bforx∈Pγ, θ, α, b, c, d;
iiαΦx> bforx∈Pγ, α, b, dwithθΦx> c;
iii0/∈Rγ, ψ, a, dandψTx< aforx∈Rγ, ψ, a, d, withψΦx a.
ThenΦhas at least three fixed pointsx1, x2, x3 ∈Pγ, dsuch that
γxi≤d, fori1,2,3, ψx1< a,
a < ψx2 withαx2< b, αx3> b.
3. Some Lemmas
DefineP C0,∞ {u: 0,∞ → R | utis continuous at eacht /tk, left continuous at
ttk,utkexists,k1, . . . , p}.
By a solution of1.1we mean a functionuinP C0,∞satisfying the relations in1.1.
Lemma 3.1. utis a solution of1.1if and only ifutis a solution of the following equation:
ut
∞
0
Gt, sqsfs, usds 0<tk<t
Iku
m−2
i1 αi
1−mi1−2αi
⎡ ⎣∞
0
Gξi, sqsfs, usds
0<tk<ξi
Iku
⎤ ⎦
:Tut,
3.1
whereGt, sis defined as1.3.
The proof is similar to Lemma 3 in9, and here we omit it. Fortp< a∗< b∗<∞, letc∗min{a∗/1a∗,1/1b∗}. Then
Gt, s
1t ≥c
∗Gr, s
1r ,
1 1t ≥
c∗
1r, fort∈a
∗, b∗, r∈0,∞, s∈0,∞. 3.2
It is clear that 0< c∗<1. Consider the spaceEdefined by
E
u∈P C0,∞: sup
t∈0,∞
|ut|
1t <∞
. 3.3
Eis a Banach space, equipped with the normu sup0≤t<∞|ut|/1t < ∞. Define the coneP ⊂Eby
P
u∈E:ut≥0, t∈0,∞, min
t∈a∗,b∗
ut
1t ≥c
∗u. 3.4
Lemma 3.2see20, Theorem 2.2. LetM ⊂P C0,∞. ThenMis compact inP C0,∞, if the following conditions hold:
aMis bounded inP C0,∞;
bthe functions belonging toMare piecewise equicontinuous on any interval of0,∞;
cthe functions fromMare equiconvergent, that is, givenε >0, there correspondsτε>0 such that|ft−f∞|< εfor anyt≥τεandf ∈M.
Proof. Firstly, for u ∈ P, from H1–H3, it is easy to check that Tu is well defined, and Tut≥0 for allt∈0,∞. Fort∈a∗, b∗
1
1tTut
1 1t
∞
0
Gt, sqsfs, usds 1
1t
p
k1
Iku
1
1t
m−2
i1 αi
1−mi1−2 αi
⎡ ⎣∞
0
Gξi, sqsfs, usds
0<tk<ξi
Iku
⎤ ⎦
≥c∗
∞
0
Gr, s
1r qsfs, usds
1 1r
p
k1
Iku
c∗
m−2
i1 αi
1−mi1−2 αi
⎡ ⎣∞
0
Gξi, s
1r qsfs, usds
1 1r
0<tk<ξi
Iku
⎤ ⎦
≥c∗Tur
1r , forr ∈0,∞
3.5
so
min
t∈a∗,b∗ Tut
1t ≥c
∗Tu, 3.6
which showsTP ⊆P.
Now we prove thatTis continuous and compact, respectively. Letun → uasn → ∞
inP. Then there existsr0such that supn∈N\{0}un < r0. ByH2we haveft, uis bounded on0,∞×0, r0. SetB0sup{ft, u:t, u/1t∈0,∞×0, r0}, and we have
∞
0
Gt, s
1t qsfs, un−fs, uds≤2B0
∞
0
Gt, s
1t qsds. 3.7
Therefore by the Lebesgue dominated convergence theorem and continuity offandIk, one
arrives at
Tun−Tu
≤ sup
t∈0,∞ 1 1t
∞
0
Gt, sqsfs, un−fs, uds
0<tk<t
|Ikun−Iku|
m−2
i1 αi
1−mi1−2αi
×
⎡ ⎣∞
0
Gξi, sqsfs, un−fs, uds
0<tk<ξi
|Ikun−Iku|
⎤ ⎦
⎫ ⎬ ⎭
−→0 asn−→∞.
3.8
LetΩbe any bounded subset ofP. Then there exists r > 0 such thatu ≤r for all
u∈Ω. SetB1 sup{ft, u:t, u/1t∈0,∞×0, r,B2ksup{Iku:u/1t∈0, r},
then
Tu sup
t∈0,∞ 1 1t
∞
0
Gt, sqsfs, uds 0<tk<t
|Iku|
m−2
i1 αi
1−mi1−2αi
⎡ ⎣∞
0
Gξi, sqsfs, uds
0<tk<ξi
|Iku|
⎤ ⎦
⎫ ⎬ ⎭
≤B1
∞
0
Gt, sqsds
m−2
i1 αi
1−mi1−2 αi
∞
0
Gξi, sqsds
1
m−2
i1 αi
1−mi1−2αi
p
k1
B2k.
3.9
SoTΩis bounded.
Moreover, for anyν∈0,∞andt, t ∈tk, tk1⊂0, νt< t, andu∈Ω, then
Tu1tt − Tut
1t
≤
m−2
i1 αi
1−mi1−2αi
⎡ ⎣B1∞
0
Gξi, sqsds
0<tk<ξi
B2k
⎤ ⎦ 1
1t −
1 1t
B1 ∞ 0
G1t, st− Gt, s
1t
qsds 0<tk<t
B2k
11t −
1 1t
−→0, uniformly ast−→t.
3.10
SoTΩis quasi-equicontinuous on any compact interval of0,∞.
Finally, we prove for anyε, there exists sufficiently largeN1>0 such that
Tu1tt− Tut
1t
< ε, ∀t, t≥N1, u∈Ω. 3.11
Since0∞Gt, sqsds <∞, we can chooseN1 >0 such that
m−2
i1 αi
N1
1−mi1−2αi
⎡ ⎣B1∞
0
Gξi, sqsds
0<tk<ξi
B2k
⎤ ⎦< ε
6,
B1
∞
0 Gt, sqsds N1 <
ε
6,
p
k1
B2k
N1 ≤ ε
6.
Fort, t≥N1, it follows that
Tu1tt−
Tut
1t
≤
m−2
i1 αi
1−mi1−2αi
⎡ ⎣B1∞
0
Gξi, sqsds
0<tk<ξi
B2k
⎤ ⎦ 1
1t
1 1t
B1
∞
0
Gt, s
1t qsdsB1
∞
0
Gt, s
1t qsds
p
k1
B2k
1 1t
1 1t
< ε
3
ε
6
ε
6
ε
3 ε.
3.13
That is3.11holds. ByLemma 3.2,TΩis relatively compact. In sum,T :P → Pis completely continuous.
4. Existence of Three Positive Solutions
Let the nonnegative continuous concave functional α, the nonnegative continuous convex functionalsγandθ, and the nonnegative continuous functionalsψbe defined on the coneP
by
γu ψu θu sup
t∈0,∞
ut
1t, αu t∈mina∗,b∗ ut
1t. 4.1
For notational convenience, we denote by
M min
t∈a∗,b∗
∞
0
Gt, s
1t qsds,
M1
m−2
i1 αi
1−mi1−2αi
∞
0
Gξi, sqsds.
4.2
The main result of this paper is the following.
Theorem 4.1. AssumeH1–H3hold. Letak≥0,0< a < b/c∗< cd,b/M < c∗d/2MM1 and suppose thatf, Iksatisfy the following conditions:
A1ft, u≤c∗d/2MM1, Iku≤dc∗/2M2 fort, u/1t∈0,∞×0, d,
A2ft, u> b/Mfort, u/1t∈a∗, b∗×b, c,
whereM2 pk1ak/1−mi1−2αi. Then1.1has at least three positive solutionsu1, u2 andu3 such that
γui≤d, fori1,2,3, ψu1< a, a < ψu2 withαu2< b, αu3> b. 4.3
Proof.
Step 1. From the definitionα, ψ, andγ, we easily show that
αu≤ψu, u ≤γu foru∈Pγ, d. 4.4
Next we will show that
T :Pγ, d−→Pγ, d. 4.5
In fact, foru∈Pγ, d, then
sup
t∈0,∞
ut
1t ≤d. 4.6
From conditionA1, we obtain
ft, u≤ dc ∗
2MM1, Iku≤ dc∗
2M2. 4.7
It follows that
γTu sup
t∈0,∞
Tut
1t ≤
1
c∗t∈mina∗,b∗
Tut
1t
≤ 1
c∗t∈mina∗,b∗
1 1t
∞
0
Gt, sqsfs, uds 1
1t
p
k1
Iku
1
1t·
m−2
i1 αi
1−mi1−2 αi
⎛ ⎝∞
0
Gξi, sqsfs, uds
0<tk<ξi
Iku
⎞ ⎠ ⎤ ⎦
≤ 1
c∗ ·
c∗d
2MM1
min
t∈a∗,b∗
∞
0
Gt, s
1t qsds
m−2
i1 αi
1−mi1−2αi
∞
0
Gξi, sqsds
1
c∗ ·
c∗dpk1 ak
2M2
1
m−2
i1 αi
1−mi1−2 αi
≤ d
2
d
2 d.
4.8
Step 2. We show that conditioniinTheorem 2.2holds. Takingut 1tbd/2, then
u∈Pγ, θ, α, b, c, dandαu> b, which shows{u∈Pγ, θ, α, b, c, d|αu> b}/∅. Thus for
u∈Pγ, θ, α, b, c, d, there is
b≤ ut
1t ≤ fort∈a
∗, b∗. 4.9
Hence byA2, we have
αTu min
t∈a∗,b∗ Tut
1t
≥ min
t∈a∗,b∗
∞
0
Gt, s
1t qsfs, uds
> b
M·t∈mina∗,b∗
∞
0
Gt, s
1t qsdsb.
4.10
Therefore we have
αTu> b, ∀u∈Pγ, θ, α, b, c, d . 4.11
This shows the conditioniinTheorem 2.2is satisfied.
Step 3. We now proveiiinTheorem 2.2holds. Foru∈Pγ, α, b, dwithθTu> c, we have
αTu min
t∈a∗,b∗ Tut
1t ≥c
∗Tuc∗θTu> c∗c > b. 4.12
Hence, conditioniiinTheorem 2.2is satisfied.
Step 4. Finally, we proveiiiinTheorem 2.2is satisfied. Sinceψ0 0< a, so 0/∈Rγ, ψ, a, d. Suppose thatu∈Rγ, θ, a, dwithψu a, then
0≤ ut
1t≤a, 4.13
by the conditionA3of this theorem,
ψTu sup
t∈0,∞ Tut
1t ≤
1
c∗t∈mina∗,b∗
Tut
1t
≤ 1
c∗ ·
c∗a
2MM1
min
t∈a∗,b∗
∞
0
Gt, s
1t qsds
m−2
i1 αi
1−mi1−2αi
∞
0
Gξi, sqsds
1
c∗ ·
c∗apk1ak
2M2
1
m−2
i1 αi
1−mi1−2αi
≤ a
2
a
2 a.
Thus conditioniiiin Theorem 2.2holds. Therefore an application ofTheorem 2.2implies the boundary value problem1.1has at least three positive solutions such that
sup
t∈0,∞ uit
1t ≤d, i1,2,3,
sup
t∈0,∞
u1t
1t < a, a <t∈sup0,∞ u2t
1t with mint∈a∗,b∗ u2t
1t < b,
min
t∈a∗,b∗ u3t
1t > b.
4.15
5. An Example
Now we consider the following boundary value problem
ut qtft, u 0, 0< t <∞, t /t1,
Δut1 I1ut1, t11,
u0 1 4u
1
4
1
4u4, u
∞ 0
ft, u
⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩
1 100e−
t4 u
1t
7 , u
1t ≤1,
1 100e−
t4, u
1t ≥1,
I1u
⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩
1 2
u
1t
4 , u
1t ≤1,
1 2,
u
1t ≥1,
5.1
qt e−t. Choosea
11/2,a1/2,b3/4,cd48. If takinga∗2, b∗3, thenc∗1/4, andM 1−e−3/4, M
1 2−e−1/4−e−4, M2 1. Consequently,ft, u, Ikusatisfies the
following:
1ft, u≤1/1004< c∗d/2MM1,I1u≤1/2<3c∗a1d/2M2, fort, u/1t∈
0,∞×0,48;
2ft, u>4> b/M, fort, u/1t∈2,3×3/4,48;
3ft, u<1/100 1/25≤c∗a/2MM1,I1u≤1/32c∗a1a/2M2, fort, u/1 t∈0,∞×0,1/2.
Then all conditions ofTheorem 4.1hold, so byTheorem 4.1, boundary value problem5.1
Acknowledgments
This work is supported by the NNSF of China no. 60671066, A project supported by Scientific Research Fund of Hunan Provincial Education Department07B041and Program for Young Excellent Talents in Hunan Normal University, The research of J. J. Nieto has been partially supported by Ministerio de Educacion y Ciencia and FEDER, Project MTM2007-61724, and by Xunta de Galicia and FEDER, Project PGIDIT06PXIB207023PR.
References
1 M. Benchohra, J. Henderson, and S. Ntouyas,Impulsive Differential Equations and Inclusions, vol. 2 of
Contemporary Mathematics and Its Applications, Hindawi Publishing Corporation, New York, NY, USA, 2006.
2 V. Lakshmikantham, D. D. Ba˘ınov, and P. S. Simeonov,Theory of Impulsive Differential Equations, vol. 6 ofSeries in Modern Applied Mathematics, World Scientific, Teaneck, NJ, USA, 1989.
3 S. T. Zavalishchin and A. N. Sesekin, Dynamic Impulse Systems: Theory and Application, vol. 394 of
Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1997. 4 J.-M. Belley and M. Virgilio, “Periodic Li´enard-type delay equations with state-dependent impulses,”
Nonlinear Analysis: Theory, Methods & Applications, vol. 64, no. 3, pp. 568–589, 2006.
5 J. Chu and J. J. Nieto, “Impulsive periodic solutions of first-order singular differential equations,”
Bulletin of the London Mathematical Society, vol. 40, no. 1, pp. 143–150, 2008.
6 T. Cardinali and P. Rubbioni, “Impulsive semilinear differential inclusions: topological structure of the solution set and solutions on non-compact domains,”Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 1, pp. 73–84, 2008.
7 L. Di Piazza and B. Satco, “A new result on impulsive differential equations involving non-absolutely convergent integrals,”Journal of Mathematical Analysis and Applications, vol. 352, no. 2, pp. 954–963, 2009.
8 D. Guo, “Existence of positive solutions for nth-order nonlinear impulsive singular integro-differential equations in Banach spaces,”Nonlinear Analysis: Theory, Methods & Applications, vol. 68, no. 9, pp. 2727–2740, 2008.
9 S. Gao, L. Chen, J. J. Nieto, and A. Torres, “Analysis of a delayed epidemic model with pulse vaccination and saturation incidence,”Vaccine, vol. 24, no. 35-36, pp. 6037–6045, 2006.
10 J. Jiao, L. Chen, J. J. Nieto, and A. Torres, “Permanence and global attractivity of stage-structured predator-prey model with continuous harvesting on predator and impulsive stocking on prey,”
Applied Mathematics and Mechanics, vol. 29, no. 5, pp. 653–663, 2008.
11 J. Li, J. J. Nieto, and J. Shen, “Impulsive periodic boundary value problems of first-order differential equations,”Journal of Mathematical Analysis and Applications, vol. 325, no. 1, pp. 226–236, 2007. 12 Z. Luo and J. Shen, “Stability of impulsive functional differential equations via the Liapunov
functional,”Applied Mathematics Letters, vol. 22, no. 2, pp. 163–169, 2009.
13 J. Li and J. Shen, “Existence of positive solution for second-order impulsive boundary value problems on infinity intervals,”Boundary Value Problems, vol. 2006, Article ID 14594, 11 pages, 2006.
14 S. Liang and J. Zhang, “The existence of three positive solutions for some nonlinear boundary value problems on the half-line,”Positivity, vol. 13, no. 2, pp. 443–457, 2009.
15 J. J. Nieto and D. O’Regan, “Variational approach to impulsive differential equations,” Nonlinear Analysis: Real World Applications, vol. 10, no. 2, pp. 680–690, 2009.
16 J. J. Nieto, “Impulsive resonance periodic problems of first order,”Applied Mathematics Letters, vol. 15, no. 4, pp. 489–493, 2002.
17 G. Tr. Stamov, “On the existence of almost periodic solutions for the impulsive Lasota-Wazewska model,”Applied Mathematics Letters, vol. 22, no. 4, pp. 516–520, 2009.
18 J. R. Wang, X. Xiang, W. Wei, and Q. Chen, “Bounded and periodic solutions of semilinear impulsive periodic system on Banach spaces,”Fixed Point Theory and Applications, vol. 2008, Article ID 401947, 15 pages, 2008.
20 B. Yan, “Boundary value problems on the half-line with impulses and infinite delay,”Journal of Mathematical Analysis and Applications, vol. 259, no. 1, pp. 94–114, 2001.
21 J. Yan, A. Zhao, and J. J. Nieto, “Existence and global attractivity of positive periodic solution of periodic single-species impulsive Lotka-Volterra systems,”Mathematical and Computer Modelling, vol. 40, no. 5-6, pp. 509–518, 2004.
22 H. Zhang, L. Chen, and J. J. Nieto, “A delayed epidemic model with stage-structure and pulses for pest management strategy,”Nonlinear Analysis: Real World Applications, vol. 9, no. 4, pp. 1714–1726, 2008.
23 X. Zhang, Z. Shuai, and K. Wang, “Optimal impulsive harvesting policy for single population,”
Nonlinear Analysis: Real World Applications, vol. 4, no. 4, pp. 639–651, 2003.
24 G. Zeng, F. Wang, and J. J. Nieto, “Complexity of a delayed predator-prey model with impulsive harvest and Holling type II functional response,”Advances in Complex Systems, vol. 11, no. 1, pp. 77– 97, 2008.
25 H. Lian, H. Pang, and W. Ge, “Triple positive solutions for boundary value problems on infinite intervals,”Nonlinear Analysis: Theory, Methods & Applications, vol. 67, no. 7, pp. 2199–2207, 2007. 26 Y. Liu, “Existence and unboundedness of positive solutions for singular boundary value problems on
half-line,”Applied Mathematics and Computation, vol. 144, no. 2-3, pp. 543–556, 2003.
27 D. O’Regan,Theory of Singular Boundary Value Problems, World Scientific, River Edge, NJ, USA, 1994. 28 R. I. Avery and A. C. Peterson, “Three positive fixed points of nonlinear operators on ordered Banach
