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Existence of Positive Solutions for Multipoint Boundary Value Problem on the Half-Line with Impulses

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Abstract

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.

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 effects (see [13]). 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 problems [424].

In this paper, we consider the existence of multiple positive solutions of the following impulsive boundary value problem (for short BVP) on a half-line:

(11)

where , , , , and , and satisfy

();

(), , and when is bounded, and are bounded on ;

() and is not identically zero on any compact subinterval of . Furthermore satisfies

(12)

where

(13)

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, see [8, 13, 14, 20, 2527], for example.

Lian et al. [25] studied the following boundary value problem of second-order differential equation with a -Laplacian operator on a half-line:

(14)

They showed the existence at least three positive solutions for (1.4) by using a fixed point theorem in a cone due to Avery-Peterson [28].

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 by [20, 25], in this paper, we aim to investigate the existence of triple positive solutions for BVP (1.1). The method chosen in this paper is a fixed point technique due to Avery and Peterson [28].

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 is a cone in a Banach. The map is a nonnegative continuous concave functional on provided is continuous and

(21)

for all , and . Similarly, the map is a nonnegative continuous convex functional on provided is continuous and

(22)

for all , and .

Let be nonnegative, continuous, convex functionals on and be a nonnegative, continuous, concave functionals on , and be a nonnegative continuous functionals on . Then, for positive real numbers , and , we define the convex sets

(23)

and the closed set

(24)

To prove our main results, we need the following fixed point theorem due to Avery and Peterson in [28].

Theorem 2.2.

Let be a cone in a real Banach space . Let and be nonnegative continuous convex functionals on a cone , be a nonnegative continuous concave functional on , and be a nonnegative continuous functional on satisfying for , such that for some positive numbers and

(25)

for all . Suppose

(26)

is completely continuous and there exist positive numbers , and with such that

(i) and for ;

(ii) for with ;

(iii) and for , with

Then has at least three fixed points such that

(27)

3. Some Lemmas

Define is continuous at each , left continuous at , exists, .

By a solution of (1.1) we mean a function in satisfying the relations in (1.1).

Lemma 3.1.

is a solution of (1.1) if and only if is a solution of the following equation:

(31)

where is defined as (1.3).

The proof is similar to Lemma in [9], and here we omit it.

For , let . Then

(32)

It is clear that . Consider the space defined by

(33)

is a Banach space, equipped with the norm . Define the cone by

(34)

Lemma 3.2 (see [20, Theorem ]).

Let . Then is compact in , if the following conditions hold:

(a) is bounded in ;

(b)the functions belonging to are piecewise equicontinuous on any interval of ;

(c)the functions from are equiconvergent, that is, given , there corresponds such that for any and .

Lemma 3.3.

is completely continuous.

Proof.

Firstly, for , from , it is easy to check that is well defined, and for all . For

(35)

so

(36)

which shows .

Now we prove that is continuous and compact, respectively. Let as in . Then there exists such that . By we have is bounded on . Set , and we have

(37)

Therefore by the Lebesgue dominated convergence theorem and continuity of and , one arrives at

(38)

Therefore is continuous.

Let be any bounded subset of . Then there exists such that for all . Set , , then

(39)

So is bounded.

Moreover, for any and , and , then

(310)

So is quasi-equicontinuous on any compact interval of .

Finally, we prove for any , there exists sufficiently large such that

(311)

Since , we can choose such that

(312)

For , it follows that

(313)

That is (3.11) holds. By Lemma 3.2, is relatively compact. In sum, is completely continuous.

4. Existence of Three Positive Solutions

Let the nonnegative continuous concave functional , the nonnegative continuous convex functionals and , and the nonnegative continuous functionals be defined on the cone by

(41)

For notational convenience, we denote by

(42)

The main result of this paper is the following.

Theorem 4.1.

Assume hold. Let , , and suppose that satisfy the following conditions:

()

() for ,

(),

where . Then (1.1) has at least three positive solutions and such that

(43)

Proof.

Step 1.

From the definition , and , we easily show that

(44)

Next we will show that

(45)

In fact, for , then

(46)

From condition , we obtain

(47)

It follows that

(48)

Thus (4.5) holds.

Step 2.

We show that condition (i) in Theorem 2.2 holds. Taking , then and , which shows . Thus for , there is

(49)

Hence by , we have

(410)

Therefore we have

(411)

This shows the condition (i) in Theorem 2.2 is satisfied.

Step 3.

We now prove (ii) in Theorem 2.2 holds. For with , we have

(412)

Hence, condition (ii) in Theorem 2.2 is satisfied.

Step 4.

Finally, we prove (iii) in Theorem 2.2 is satisfied. Since , so . Suppose that with , then

(413)

by the condition of this theorem,

(414)

Thus condition (iii) in Theorem 2.2 holds. Therefore an application of Theorem 2.2 implies the boundary value problem (1.1) has at least three positive solutions such that

(415)

5. An Example

Now we consider the following boundary value problem

(51)

. Choose , , , . If taking , then , and . Consequently, satisfies the following:

(1), , for ;

(2), for ;

(3), , for .

Then all conditions of Theorem 4.1 hold, so by Theorem 4.1, boundary value problem (5.1) has at least three positive solutions.

References

  1. 1.

    Benchohra M, Henderson J, Ntouyas S: Impulsive Differential Equations and Inclusions, Contemporary Mathematics and Its Applications. Volume 2. Hindawi Publishing Corporation, New York, NY, USA; 2006:xiv+366.

  2. 2.

    Lakshmikantham V, Baĭnov DD, Simeonov PS: Theory of Impulsive Differential Equations, Series in Modern Applied Mathematics. Volume 6. World Scientific, Teaneck, NJ, USA; 1989:xii+273.

  3. 3.

    Zavalishchin ST, Sesekin AN: Dynamic Impulse Systems: Theory and Application, Mathematics and Its Applications. Volume 394. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1997:xii+256.

  4. 4.

    Belley J-M, Virgilio M: Periodic Liénard-type delay equations with state-dependent impulses. Nonlinear Analysis: Theory, Methods & Applications 2006, 64(3):568–589. 10.1016/j.na.2005.06.025

  5. 5.

    Chu J, Nieto JJ: Impulsive periodic solutions of first-order singular differential equations. Bulletin of the London Mathematical Society 2008, 40(1):143–150. 10.1112/blms/bdm110

  6. 6.

    Cardinali T, Rubbioni P: Impulsive semilinear differential inclusions: topological structure of the solution set and solutions on non-compact domains. Nonlinear Analysis: Theory, Methods & Applications 2008, 69(1):73–84. 10.1016/j.na.2007.05.001

  7. 7.

    Di Piazza L, Satco B: A new result on impulsive differential equations involving non-absolutely convergent integrals. Journal of Mathematical Analysis and Applications 2009, 352(2):954–963. 10.1016/j.jmaa.2008.11.048

  8. 8.

    Guo D: Existence of positive solutions for th-order nonlinear impulsive singular integro-differential equations in Banach spaces. Nonlinear Analysis: Theory, Methods & Applications 2008, 68(9):2727–2740. 10.1016/j.na.2007.02.019

  9. 9.

    Gao S, Chen L, Nieto JJ, Torres A: Analysis of a delayed epidemic model with pulse vaccination and saturation incidence. Vaccine 2006, 24(35–36):6037–6045. 10.1016/j.vaccine.2006.05.018

  10. 10.

    Jiao J, Chen L, Nieto JJ, Torres A: Permanence and global attractivity of stage-structured predator-prey model with continuous harvesting on predator and impulsive stocking on prey. Applied Mathematics and Mechanics 2008, 29(5):653–663. 10.1007/s10483-008-0509-x

  11. 11.

    Li J, Nieto JJ, Shen J: Impulsive periodic boundary value problems of first-order differential equations. Journal of Mathematical Analysis and Applications 2007, 325(1):226–236. 10.1016/j.jmaa.2005.04.005

  12. 12.

    Luo Z, Shen J: Stability of impulsive functional differential equations via the Liapunov functional. Applied Mathematics Letters 2009, 22(2):163–169. 10.1016/j.aml.2008.03.004

  13. 13.

    Li J, Shen J: Existence of positive solution for second-order impulsive boundary value problems on infinity intervals. Boundary Value Problems 2006, 2006:-11.

  14. 14.

    Liang S, Zhang J: The existence of three positive solutions for some nonlinear boundary value problems on the half-line. Positivity 2009, 13(2):443–457. 10.1007/s11117-008-2213-z

  15. 15.

    Nieto JJ, O'Regan D: Variational approach to impulsive differential equations. Nonlinear Analysis: Real World Applications 2009, 10(2):680–690. 10.1016/j.nonrwa.2007.10.022

  16. 16.

    Nieto JJ: Impulsive resonance periodic problems of first order. Applied Mathematics Letters 2002, 15(4):489–493. 10.1016/S0893-9659(01)00163-X

  17. 17.

    Stamov GTr: On the existence of almost periodic solutions for the impulsive Lasota-Wazewska model. Applied Mathematics Letters 2009, 22(4):516–520. 10.1016/j.aml.2008.07.002

  18. 18.

    Wang JR, Xiang X, Wei W, Chen Q: Bounded and periodic solutions of semilinear impulsive periodic system on Banach spaces. Fixed Point Theory and Applications 2008, 2008:-15.

  19. 19.

    Xian X, O'Regan D, Agarwal RP: Multiplicity results via topological degree for impulsive boundary value problems under non-well-ordered upper and lower solution conditions. Boundary Value Problems 2008, 2008:-21.

  20. 20.

    Yan B: Boundary value problems on the half-line with impulses and infinite delay. Journal of Mathematical Analysis and Applications 2001, 259(1):94–114. 10.1006/jmaa.2000.7392

  21. 21.

    Yan J, Zhao A, Nieto JJ: Existence and global attractivity of positive periodic solution of periodic single-species impulsive Lotka-Volterra systems. Mathematical and Computer Modelling 2004, 40(5–6):509–518. 10.1016/j.mcm.2003.12.011

  22. 22.

    Zhang H, Chen L, Nieto JJ: A delayed epidemic model with stage-structure and pulses for pest management strategy. Nonlinear Analysis: Real World Applications 2008, 9(4):1714–1726. 10.1016/j.nonrwa.2007.05.004

  23. 23.

    Zhang X, Shuai Z, Wang K: Optimal impulsive harvesting policy for single population. Nonlinear Analysis: Real World Applications 2003, 4(4):639–651. 10.1016/S1468-1218(02)00084-6

  24. 24.

    Zeng G, Wang F, Nieto JJ: Complexity of a delayed predator-prey model with impulsive harvest and Holling type II functional response. Advances in Complex Systems 2008, 11(1):77–97. 10.1142/S0219525908001519

  25. 25.

    Lian H, Pang H, Ge W: Triple positive solutions for boundary value problems on infinite intervals. Nonlinear Analysis: Theory, Methods & Applications 2007, 67(7):2199–2207. 10.1016/j.na.2006.09.016

  26. 26.

    Liu Y: Existence and unboundedness of positive solutions for singular boundary value problems on half-line. Applied Mathematics and Computation 2003, 144(2–3):543–556. 10.1016/S0096-3003(02)00431-9

  27. 27.

    O'Regan D: Theory of Singular Boundary Value Problems. World Scientific, River Edge, NJ, USA; 1994:xii+154.

  28. 28.

    Avery RI, Peterson AC: Three positive fixed points of nonlinear operators on ordered Banach spaces. Computers & Mathematics with Applications 2001, 42(3–5):313–322.

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Acknowledgments

This work is supported by the NNSF of China (no. 60671066), A project supported by Scientific Research Fund of Hunan Provincial Education Department (07B041) and 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.

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Correspondence to Jianli Li.

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Keywords

  • Fixed Point Theorem
  • Point Index
  • Nonlinear Elliptic Equation
  • Impulsive Differential Equation
  • Lebesgue Dominate Convergence Theorem