- Research
- Open access
- Published:
The method of upper and lower solutions to impulsive differential equations with integral boundary conditions
Advances in Difference Equations volume 2014, Article number: 183 (2014)
Abstract
This paper considers a second-order impulsive differential equation with integral boundary conditions. Some sufficient conditions for the existence of solutions are proposed by using the method of upper and lower solutions and Leray-Schauder degree theory.
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. For an introduction of the basic theory of impulsive differential equations in , see Lakshmikantham et al. [1], Bainov and Simeonov [2], Samoilenko and Perestyuk [3], and the references therein. The theory of impulsive differential equations has become an important area of investigation in recent years, and it is much richer than the corresponding theory of differential equations (see, for instance, [4–6] and the references therein).
On the other hand, the theory of boundary value problems with integral boundary conditions for ordinary differential equations arises in different areas of applied mathematics and physics. For example, heat conduction, chemical engineering, underground water flow, thermo-elasticity, and plasma physics can be reduced to nonlocal problems with integral boundary conditions. For boundary value problems with integral boundary conditions and comments on their importance, we refer the reader to the papers in Gallardo [7–9] and the references therein. For more information about the general theory of integral equations and their relation with boundary value problems, we refer to the book of Corduneanu [10] and Agarwal and O’Regan [11].
Recently, some well-known works, such as Hao et al. [12], Zhang et al. [13] and Ding and Wang [14], deal with impulsive differential equations with integral boundary conditions. However, most of these results are obtained by using the fixed point theorem in cones. It is well known that the method of upper and lower solutions is a powerful tool for proving the existence results for a large class of boundary value problems, see [15–17].
In [17], Shen and Wang applied the method of upper and lower solutions to solve impulsive differential equations with nonlinear boundary conditions as follows:
Motivated by the works mentioned above, in this paper, we shall employ the method of upper and lower solutions together with Leray-Schauder degree theory to study the existence of a solution of the impulsive BVP
where , , is continuous, for , is nonnegative, , denotes the jump of at , and represent the right and left limits of at , respectively, has a similar meaning for .
2 Preliminaries
Define = {, and exist, and , }. = {, and exist, and , }. Note that and are Banach spaces with the respective norms
A function is called a solution of (1.1) if it satisfies the differential equation
the impulsive conditions , and the boundary conditions .
Definition 2.1 The function is said to be a lower solution for boundary value problem (1.1) if
The function is said to be an upper solution for boundary value problem (1.1) if
For , we write if for all .
Definition 2.2 Let be such that on J. We say that f satisfies the Nagumo condition relative to α, β if for
there exists a constant D such that
a continuous function , and constants , such that
and
In addition, we assume that the following conditions hold:
(H1) f satisfies the Nagumo condition relative to α, β;
(H2) . is nondecreasing in for all ;
(H3) .
We consider the modified problem
where
and
One can find the next result, with its proof, in [18].
Lemma 2.1 For each , the following two properties hold:
-
(i)
exists for a.e. ;
-
(ii)
if and in E, then for a.e. .
Lemma 2.2 For any , the following boundary value problem
has a unique solution as follows:
where
and
Lemma 2.3 If x is a solution of BVP (2.1), and are lower and upper solutions of (1.1), respectively, , and
for
then
Proof Denote , we will only see that for every . An analogous reasoning shows that for all . Otherwise, if , , does not hold, then , there are three cases.
Case 1. Suppose that , or , we only see that . Easily, it holds that . From Definition 2.1 and (H3), we have
which is a contradiction.
Case 2. Suppose that there exist and such that
Then and . On the other hand,
which is a contradiction. Hence the function y cannot have any positive maximum interior to the interval for .
Case 3. According to Case 2, if , then , or , we only prove that , . Suppose that , easily, .
From (2.1) and (2.3), we have
Hence
Suppose that and y is nonincreasing on some interval , where is sufficiently small such that on . For ,
which contradicts the assumption of monotonicity of y. Thus, we obtain
We use the preceding procedure and deduce by induction that
which contradicts that cannot be the maximum point. By the same analysis, we can get that , , cannot hold. □
Lemma 2.4 on J, where is the solution of (2.1).
Proof Here we only show . Suppose that there exists with , , and as a result,
Therefore, there exist such that , and either
or
We only consider the first case, since the other case can be handled similarly. It follows from the assumption that
This implies that
which yields
This, obviously, contradicts the choice of D. The proof is complete. □
Lemma 2.5 (Schauder’s fixed point theorem)
Let K be a convex subset of a normed linear space E. Each continuous, compact map has a fixed point.
3 Existence results
Theorem 3.1 Suppose that conditions (H1)-(H3) hold. Then BVP (1.1) has at least one solution such that
Proof Solving (2.1) is equivalent to finding which satisfies
where , .
Now, define the following operator by
It is obvious that is completely continuous.
By the Schauder fixed point theorem, we can easily obtain that T has a fixed point , which is a solution of BVP (2.1). And by Lemma 2.3 and Lemma 2.4, we know that , , then BVP (2.1) becomes BVP (1.1), therefore is a solution of BVP (1.1). The proof is complete. □
Theorem 3.2 Suppose that conditions (H1)-(H3) hold. Assume that there exist two lower solutions and and two upper solutions and for problem (1.1), satisfying the following:
-
(i)
;
-
(ii)
;
-
(iii)
, which means that there exists such that ;
-
(iv)
if x is a solution of (1.1) with , then on ;
-
(v)
if x is a solution of (1.1) with , then on .
If f satisfies the Nagumo condition with respect to , , then problem (1.1) has at least three solutions , and satisfying
Proof We consider the following modified problem:
where , .
Now define the following operator by
Since the functions and n are continuous and bounded, we obtain that there exists such that
Let
where and
It is standard that is completely continuous. It is immediate from the argument above that .
Thus,
Let
Since , , (i.e., choose such that ). It follows that , , and . By assumptions (iv) and (v), there are no solutions in . Thus,
We show that , then
and there are solutions in , , , as required.
We now show . The proof that is similar and hence omitted. We define , the extension to of the restriction of to as follows.
Let
Thus, w is a continuous function on and satisfies
for some constants . Moreover, we may choose so that .
Consider the following problem:
where , . Now define the following operator:
Again, it is easy to check that x is a solution of (3.3) if and (note that is compact). Thus, . Moreover, it is easy to see that . By assumptions (iv) and (v), there are no solutions in . So,
Thus there are three solutions, as required. The proof is complete. □
Theorem 3.3 Suppose that conditions (H1)-(H3) hold. Assume that there exist two lower solutions and and two upper solutions and for problem (1.1), satisfying
-
(i)
;
-
(ii)
;
-
(iii)
;
-
(iv)
there exists such that all , the functions and are, respectively, lower and upper solutions of (1.1);
-
(v)
.
If f satisfies the Nagumo condition with respect to , , then problem (1.1) has at least three solutions , , and satisfying
Proof In the proof of Theorem 3.3, define
where Ω is defined in Theorem 3.2. □
References
Lakshmikantham V, Bainov DD, Simeonov PS Series in Modern Applied Mathematics 6. In Theory of Impulsive Differential Equations. World Scientific, Singapore; 1989.
Bainov DD, Simeonov PS Ellis Horwood Series: Mathematics and Its Applications. In Systems with Impulse Effect. Ellis Horwood, Chichester; 1989.
Samoilenko AM, Perestyuk NA World Scientific Series on Nonlinear Science: Series A: Monographs and Treatises 14. In Impulsive Differential Equations. World Scientific, Singapore; 1995.
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
Ding W, Han M: Periodic boundary value problem for the second order impulsive functional differential equations. Appl. Math. Comput. 2004, 155: 709-726. 10.1016/S0096-3003(03)00811-7
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
Gallardo JM: Second order differential operators with integral boundary conditions and generation of semigroups. Rocky Mt. J. Math. 2000, 30: 1265-1292. 10.1216/rmjm/1021477351
Karakostas GL, Tsamatos PC: Multiple positive solutions of some Fredholm integral equations arisen from nonlocal boundary-value problems. Electron. J. Differ. Equ. 2002, 30: 1-17.
Lomtatidze A, Malaguti L: On a nonlocal boundary-value problems for second order nonlinear singular differential equations. Georgian Math. J. 2000, 7: 133-154.
Corduneanu C: Integral Equations and Applications. Cambridge University Press, Cambridge; 1991.
Agarwal RP, O’Regan D: Infinite Interval Problems for Differential, Difference and Integral Equations. Kluwer Academic, Dordrecht; 2001.
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
Zhang X, Feng M, Ge W: Existence of solutions of boundary value problems with integral boundary conditions for second-order impulsive integro-differential equations in Banach spaces. J. Comput. Appl. Math. 2010, 233: 1915-1926. 10.1016/j.cam.2009.07.060
Ding W, Wang Y: New result for a class of impulsive differential equation with integral boundary conditions. Commun. Nonlinear Sci. Numer. Simul. 2013, 18: 1095-1105. 10.1016/j.cnsns.2012.09.021
Cabada A, Pouso RL:Existence results for the problem with nonlinear boundary conditions. Nonlinear Anal. 1999, 35: 221-231. 10.1016/S0362-546X(98)00009-1
Lee Y, Liu X: Study of singular boundary value problems for second order impulsive differential equations. J. Math. Anal. Appl. 2007, 331: 159-176. 10.1016/j.jmaa.2006.07.106
Shen J, Wang W: Impulsive boundary value problems with nonlinear boundary conditions. Nonlinear Anal. 2008, 69: 4055-4062. 10.1016/j.na.2007.10.036
Wang MX, Cabada A, Nieto JJ: Monotone method for nonlinear second order periodic boundary value problems with Carathéodory functions. Ann. Pol. Math. 1993, 58: 221-235.
Acknowledgements
The work is supported by Beijing Higher Education Young Elite Teacher Project (Project No. YETP0322) and Chinese Universities Scientific Fund (Project No. 2013QJ004).
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The authors declare that the study was realized in collaboration with the same responsibility. All authors read and approved the final manuscript.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Pang, H., Lu, M. & Cai, C. The method of upper and lower solutions to impulsive differential equations with integral boundary conditions. Adv Differ Equ 2014, 183 (2014). https://doi.org/10.1186/1687-1847-2014-183
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1687-1847-2014-183