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Positive solutions for second order impulsive differential equations with Stieltjes integral boundary conditions
Advances in Difference Equations volume 2012, Article number: 124 (2012)
Abstract
In this paper, we study the existence of positive solutions for a singular second order impulsive differential equations with Stieltjes integral boundary conditions. By means of fixed point theorems, some results on the existence and multiplicity of positive solutions are obtained. Two examples are given to demonstrate the main results.
MSC:34B10, 34B15, 34B18, 34B37.
1 Introduction
In this paper, we consider the existence of positive solutions for the following second-order impulsive boundary value problem (IBVP for short)
where , , , , , , , . denotes the jump of at , i.e.,
where and represent the right-hand limit and the left-hand limit of at , respectively. , are linear functionals on given by
involving Stieltjes integrals with signed measures, that is, A, B are suitable functions of bounded variation.
Impulsive differential equations describe processes with sudden changes in their state at certain moments. The theory of impulse differential equations has been further developed significantly in recent years and has played a very important role in modern applied mathematical modeling of real world processes in physics, population dynamics, chemical technology, biotechnology and economics. For details, see [1–9] and references therein.
Recently, Feng and Xie [10] have dealt with the second order m-point boundary value problem with impulse effects
where , , , . The existence results of one and two positive solutions are obtained based on the fixed point theorems in a cone.
For the case of , , one of the special cases of problem (1.1) is the following multi-point boundary value problem
where . Boundary value problem (1.2) and related problems have been extensively studied in many papers in recent years (see [11–15] and references therein). The existence and multiplicity results of positive solutions are obtained by applying the Krasnosel’skii fixed-point theorem in cones, the Leggett-Williams fixed point theorem and the fixed point index theory. For example, Ma and Wang in [14] studied the existence of positive solutions to the nonlinear boundary-value problem
where , , and are given, ϕ is the unique solution of the linear boundary value problem
The authors established the existence of at least one positive solution of (1.3) if f is either superlinear or sublinear by applying the fixed point theorem in cones.
Inspired by the work of the above papers, the aim of this paper is to establish the existence and multiplicity of positive solutions for the IBVP (1.1). We discuss the boundary value problem with Stieltjes integral boundary conditions, i.e., the IBVP (1.1) which includes second order two-point, three-point, multi-point and nonlocal boundary value problems as special cases. Moreover, and are two linear functions on denoting the Stieltjes integrals, where are of bounded variation, that is dA and dB may change sign. By using the Krasnosel’skii fixed-point theorem and the Leggett-Williams fixed point theorem, some existence and multiplicity results of positive solutions are obtained.
This paper is organized as follows. In Section 2, we present some preliminaries and lemmas. Section 3 is devoted to the proof of the main results. In Section 4, two examples are given to demonstrate the validity of our main results.
2 Some preliminaries and lemmas
In this section, we first introduce some background definitions in a Banach space, present some basic lemmas, and then present the fixed point theorems that are to be used in the proof of the main results.
Let with the norm , where
Then is a Banach space. A function is called a positive solution of problem (1.1) if it satisfies (1.1).
Lemma 2.1 [14]
Assume that , . Let ϕ and ψ be the unique solution of the following boundary value problem
and
respectively. Then ϕ is strictly increasing on J, ψ is strictly decreasing on J.
Throughout this paper, we adopt the following assumptions:
() , , , , , and

where

() is a Lebesgue integral and , is continuous for .
() is continuous.
Remark 2.1 If dA and dB are two positive measures, then assumption () can be replaced by the weaker assumption
() , , and , , .
Lemma 2.2 Assume that () holds. Then for any , the problem
has a unique solution given by the following formula:
for , . Moreover, on J provided .
Proof By similar arguments in [5]. So it is omitted. □
Remark 2.2 If () holds, then for any , it is easy to testify that
where , . Let , , then
where .
Put
Clearly, K is a cone of . For any , let , and .
For , we define two operators T and S by
and
where

Lemma 2.3 Assume that ()-() hold. Then , are completely continuous.
Proof Problem (1.1) has a solution x if and only if x solves the operator equation in K. Let , by (2.3) and the monotonicity of ϕ, ψ, we have
Moreover, by (2.3) and the definition of γ, we have
On the other hand,
This shows that .
Now we consider the operator S. Similarly as for the operator T, we have
and
Moreover,
This yields that .
Next, by similar arguments in [16], one can prove that , are completely continuous. So we omit further details, and Lemma 2.3 is proved. □
Lemma 2.4 Assume that ()-() hold. Then operators T and S have the same fixed point in K.
Proof Let x be a fixed point of the operator S, i.e., . Then by (2.5) and (2.6), we have

So we have
Then
This implies that x also is a fixed point of the operator T.
On the other hand, let x be a fixed point of the operator T, i.e., . Then

So we have

Therefore,
This implies that x is also a fixed point of the operator S. The proof is completed. □
Lemma 2.5 [17]
Let X be a real Banach space, K is a cone in X. Assume that and are two bounded open sets of X with and . Let be a completely continuous operator such that either
-
(i)
, and , , or
-
(ii)
, and , .
Then T has a fixed point in .
Let K be a cone in a real Banach space X, , φ is a nonnegative continuous concave functional on K such that , for all , and . Suppose that is completely continuous and there exist positive constants such that
() and for ,
() for ,
() for with .
Then T has at least three fixed points , and with
Remark 2.3 If there holds , then condition () of Lemma 2.6 implies condition () of Lemma 2.6.
3 Main results
Let

where ω denotes 0 or ∞. Let


and let the nonnegative continuous concave functional φ on the cone K be defined by
In this section, we apply Lemmas 2.5 and 2.6 to establish the existence of positive solutions for IBVP (1.1). Since operators T and S have the same fixed points (see Lemma 2.4), to prove the following theorems we always use the operator S instead of T.
Theorem 3.1 Assume that ()-() hold. In addition, suppose and are satisfied, then IBVP (1.1) has at least one positive solution .
Proof From , there exists such that , () for all and , where satisfies
If , then . So we have , and






So, by (3.4)-(3.9), for any , , we have
which means that
Next, consider . Without loss of generality, we assume that , which means that there exists such that for all and , where satisfies
Let , then for any , , we have
Consequently, we have
Applying (i) of Lemma 2.5 to (3.11) and (3.13) yields that S has a fixed point . Thus it follows that IBVP (1.1) has a positive solution . □
Theorem 3.2 Assume that ()-() hold. In addition, suppose and are satisfied, then IBVP (1.1) has at least one positive solution .
Proof Consider . Without loss of generality, we assume that , which means that there exists such that for all and , where satisfies (3.12). Then for any , , we have
which yields that
On the other hand, it follows from that there exists such that , () for all and , where satisfies (3.3). Similar to (3.10), for any , , we have
which means that
Applying (ii) of Lemma 2.5 to (3.14) and (3.15) yields that S has a fixed point . Thus it follows that IBVP (1.1) has a positive solution . □
Theorem 3.3 Assume that ()-() hold. In addition, we suppose that there exist positive constants Γ, Λ and such that

where γ, () and σ, τ are defined by (2.4), (3.1) and (3.2), respectively, , , and
() , for , and
for , .
() , for , and
for , .
() , for , and
for , .
Then IBVP (1.1) has at least three positive solutions , and with
and
Proof We shall show that all the conditions of Lemma 2.6 are satisfied.
First, if , then and (3.4), (3.6), (3.7) are valid if is replaced by . From this and (3.5), (3.8) and (3.9), for any , we have
which means that . Therefore, . By Lemma 2.3, we know that is completely continuous.
Next, it follows from condition () that if then
So the condition () of Lemma 2.6 holds.
Now, we take , , then it is easy to see that , and hence
Moreover, , . This proves that .
On the other hand, if , then , . By condition (), we have
which implies that , for . This shows that condition () of Lemma 2.6 is also satisfied.
By Lemma 2.6 and Remark 2.3, IBVP (1.1) has at least three positive solutions and such that
and
The proof of Theorem 3.3 is completed. □
4 Examples
Example 4.1 Consider the following singular IBVP
We conclude that IBVP (4.1) has at least one positive solution.
Proof IBVP (4.1) can be regarded as a IBVP of the form (1.1), where , , , , . It is not difficult to see that is singular at and , , for , , for , . Choosing , , then

Let ϕ and ψ satisfy

Then

Case 1. Let , . By direct calculation, we have

So all the conditions of Theorem 3.2 are satisfied. By Theorem 3.2, IBVP (4.1) has at least one positive solution.
Case 2. Let , that is , so the measure dA changes sign on . For convenience, we still take . By direct calculation, we have

So all conditions of Theorem 3.2 are satisfied. By Theorem 3.2, IBVP (4.1) has at least one positive solution. □
Example 4.2 Consider the following singular IBVP
where
and
We conclude that IBVP (4.2) has at least three positive solutions.
Proof IBVP (4.2) can be regarded as a IBVP of the form (1.1), where , . It is not difficult to see that is singular at and , , for , for , .
Let ϕ and ψ satisfy

Then

By direct calculation, we have

Therefore the conditions ()-() hold. In addition, , ,

Let , , , . Then , , . Take , , , , . Then . Consequently, all the assumptions of Theorem 3.3 are satisfied, and thus, by Theorem 3.3, we infer that the singular IBVP (4.2) has at least three positive solutions , and satisfying

□
References
Bainov DD, Simeonov PS: Systems with Impulse Effect. Ellis Horwood, Chichester; 1989.
Lakshmikantham V, Bainov DD, Simeonov PS: Theory of Impulsive Differential Equations. World Scientific, Singapore; 1989.
Agarwal RP, O’Regan D: Multiple nonnegative solutions for second order impulsive differential equations. Appl. Math. Comput. 2000, 114: 51–59. 10.1016/S0096-3003(99)00074-0
Agarwal RP, O’Regan D: A multiplicity result for second order impulsive differential equations via the Leggett Williams fixed point theorem. Appl. Math. Comput. 2005, 161: 433–439. 10.1016/j.amc.2003.12.096
Hao X, Liu L, Wu Y: Positive solutions for second order impulsive differential equations with integral boundary conditions. Commun. Nonlinear Sci. Numer. Simul. 2011, 16: 101–111. 10.1016/j.cnsns.2010.04.007
Jankowski T: Positive solutions to second order four-point boundary value problems for impulsive differential equations. Appl. Math. Comput. 2008, 202: 550–561. 10.1016/j.amc.2008.02.040
Jankowski T: Positive solutions for second order impulsive differential equations involving Stieltjes integral conditions. Nonlinear Anal. 2011, 74: 3775–3785. 10.1016/j.na.2011.03.022
Lee EK, Lee YH: Multiple positive solutions of singular two point boundary value problems for second order impulsive differential equations. Appl. Math. Comput. 2004, 158: 745–759. 10.1016/j.amc.2003.10.013
Lin X, Jiang D: Multiple positive solutions of Dirichlet boundary value problems for second order impulsive differential equations. J. Math. Anal. Appl. 2006, 321: 501–514. 10.1016/j.jmaa.2005.07.076
Feng M, Xie D: Multiple positive solutions of multi-point boundary value problem for second-order impulsive differential equations. J. Comput. Appl. Math. 2009, 223: 438–448. 10.1016/j.cam.2008.01.024
Li J, Shen J: Multiple positive solutions for a second-order three-point boundary value problem. Appl. Math. Comput. 2006, 182: 258–268. 10.1016/j.amc.2006.01.095
Liu B, Liu L, Wu Y: Positive solutions for a singular second-order three-point boundary value problem. Appl. Math. Comput. 2008, 196: 532–541. 10.1016/j.amc.2007.06.013
Ma R, Ren L: Positive solutions for nonlinear m -point boundary value problems of Dirichlet type via fixed-point index theory. Appl. Math. Lett. 2003, 16: 863–869. 10.1016/S0893-9659(03)90009-7
Ma R, Wang H: Positive solutions of nonlinear three-point boundary-value problems. J. Math. Anal. Appl. 2003, 279: 216–227. 10.1016/S0022-247X(02)00661-3
Sun J, Li W, Zhao Y: Three positive solutions of a nonlinear three-point boundary value problem. J. Math. Anal. Appl. 2003, 288: 708–716. 10.1016/j.jmaa.2003.09.019
Hu L, Liu L, Wu Y: Positive solutions of nonlinear singular two-point boundary value problems for second-order impulsive differential equations. Appl. Math. Comput. 2008, 196: 550–562. 10.1016/j.amc.2007.06.014
Guo D, Lakshmikantham V: Nonlinear Problems in Abstract Cone. Academic Press, New York; 1988.
Guo D, Sun J, Liu Z: Functional Method for Nonlinear Ordinary Differential Equation. Shandong Science and Technology Press, Jinan; 1995. (in Chinese)
Leggett RW, Williams LR: Multiple positive fixed points of nonlinear operators on ordered Banach space. Indiana Univ. Math. J. 1979, 28: 673–688. 10.1512/iumj.1979.28.28046
Acknowledgements
The authors thank the referee for helpful comments and suggestions, which lead to an improvement of the paper. The first and second authors were supported financially by the National Natural Science Foundation of China (11071141, 11126231) and the Natural Science Foundation of Shandong Province of China (ZR2010AM017, ZR2011AQ008). The third author was supported financially by the Australia Research Council through an ARC Discovery Project Grant.
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The study was carried out in collaboration between all authors. JJ completed the main part of this paper and gave two examples; LL and YW corrected the main theorems and polished the manuscript. All authors read and approved the final manuscript.
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Jiang, J., Liu, L. & Wu, Y. Positive solutions for second order impulsive differential equations with Stieltjes integral boundary conditions. Adv Differ Equ 2012, 124 (2012). https://doi.org/10.1186/1687-1847-2012-124
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DOI: https://doi.org/10.1186/1687-1847-2012-124
Keywords
- impulsive differential equation
- positive solutions
- fixed point theorem
- integral boundary conditions