Positive solutions for second order impulsive differential equations with Stieltjes integral boundary conditions
© Jiang et al.; licensee Springer. 2012
Received: 14 April 2012
Accepted: 5 July 2012
Published: 24 July 2012
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.
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.
where , , , . The existence results of one and two positive solutions are obtained based on the fixed point theorems in a cone.
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.
Then is a Banach space. A function is called a positive solution of problem (1.1) if it satisfies (1.1).
Lemma 2.1 
respectively. Then ϕ is strictly increasing on J, ψ is strictly decreasing on J.
Throughout this paper, we adopt the following assumptions:
() 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 , , .
for , . Moreover, on J provided .
Proof By similar arguments in . So it is omitted. □
Clearly, K is a cone of . For any , let , and .
Lemma 2.3 Assume that ()-() hold. Then , are completely continuous.
This shows that .
This yields that .
Next, by similar arguments in , 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.
This implies that x also is a fixed point of the operator T.
This implies that x is also a fixed point of the operator S. The proof is completed. □
Lemma 2.5 
, and , , or
, 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 .
Remark 2.3 If there holds , then condition () of Lemma 2.6 implies condition () of Lemma 2.6.
3 Main results
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 .
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 .
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 . □
where γ, () and σ, τ are defined by (2.4), (3.1) and (3.2), respectively, , , and
for , .
for , .
for , .
Proof We shall show that all the conditions of Lemma 2.6 are satisfied.
which means that . Therefore, . By Lemma 2.3, we know that is completely continuous.
So the condition () of Lemma 2.6 holds.
Moreover, , . This proves that .
which implies that , for . This shows that condition () of Lemma 2.6 is also satisfied.
The proof of Theorem 3.3 is completed. □
We conclude that IBVP (4.1) has at least one positive solution.
So all the conditions of Theorem 3.2 are satisfied. By Theorem 3.2, IBVP (4.1) has at least one positive solution.
So all conditions of Theorem 3.2 are satisfied. By Theorem 3.2, IBVP (4.1) has at least one positive solution. □
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 , .
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.
- Bainov DD, Simeonov PS: Systems with Impulse Effect. Ellis Horwood, Chichester; 1989.MATHGoogle Scholar
- Lakshmikantham V, Bainov DD, Simeonov PS: Theory of Impulsive Differential Equations. World Scientific, Singapore; 1989.View ArticleMATHGoogle Scholar
- 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)00074-0MathSciNetView ArticleMATHGoogle Scholar
- 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.2003.12.096MathSciNetView ArticleMATHGoogle Scholar
- 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.2010.04.007MathSciNetView ArticleMATHGoogle Scholar
- 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.2008.02.040MathSciNetView ArticleMATHGoogle Scholar
- Jankowski T: Positive solutions for second order impulsive differential equations involving Stieltjes integral conditions. Nonlinear Anal. 2011, 74: 3775–3785. 10.1016/j.na.2011.03.022MathSciNetView ArticleMATHGoogle Scholar
- 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.2003.10.013MathSciNetView ArticleMATHGoogle Scholar
- 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.2005.07.076MathSciNetView ArticleMATHGoogle Scholar
- 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.2008.01.024MathSciNetView ArticleMATHGoogle Scholar
- 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.2006.01.095MathSciNetView ArticleMATHGoogle Scholar
- 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.2007.06.013MathSciNetView ArticleMATHGoogle Scholar
- 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)90009-7MathSciNetView ArticleMATHGoogle Scholar
- 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)00661-3MathSciNetView ArticleMATHGoogle Scholar
- 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.2003.09.019MathSciNetView ArticleMATHGoogle Scholar
- 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.2007.06.014MathSciNetView ArticleMATHGoogle Scholar
- Guo D, Lakshmikantham V: Nonlinear Problems in Abstract Cone. Academic Press, New York; 1988.MATHGoogle Scholar
- Guo D, Sun J, Liu Z: Functional Method for Nonlinear Ordinary Differential Equation. Shandong Science and Technology Press, Jinan; 1995. (in Chinese)Google Scholar
- 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.1979.28.28046MathSciNetView ArticleMATHGoogle Scholar
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