Periodic boundary value problem for second-order impulsive functional differential equations☆
Introduction
In the mathematical simulation in various fields of science and technology, impulsive differential equations are often used [1], [2]. This leads to necessity of justification of methods for their approximate solution, in particular for periodic solution [19], [20], [21], [22]. The existence analysis of periodic boundary value problems of ordinary or functional differential equations with impulsive effects has been the subject of many investigations in recent years, and various interesting results have been reported, we refer the readers to the articles [3], [4], [5], [6], [8], [10], [11], [12], [13], [14], [15], [16], [17], [18]. Recently, Nieto and Rodriguez-Lopez [7], [9] consider periodic boundary value problem for the following first-order functional differential equationsIn their paper, they introduce a new concept of lower and upper solutions. Motivated by Nieto and Rodriguez-Lopez [7], [9], [12], in this paper, we present the similar definition of lower and upper solutions for the second-order impulsive functional differential equations,where . In recent paper [11], the authors also investigated the PBVP (1), they claimed the following assertion: Lemma 1 [11] Assume that satisfieswhere constants , , and they satisfyThen, u(t) ⩽ 0 for all t ∈ J.
Unfortunately this statement is wrong. Lemma 1 means , but in the proof of the lemma, it must satisfy . Therefore the new lower and upper solutions in their paper are not valid. The aim of this paper is to introduce a new definition of lower and upper solutions.
Denote . Let is continuous everywhere except for some tk at which and exist and ; is continuous everywhere except for some tk at which and exist and .
Let with normwhere , then E is a Banach space. A function is called a solution of PBVP (1) if it satisfies (1).
This paper is organized as follows. In Section 2, we present new comparison results. In Section 3, we introduce a new more general concept of upper and lower solutions relative to problem (1). These results are an important tool to develop the monotone iterative technique for (1) and to obtain two sequences approximating the extremal solutions of this problem between lower and upper solutions (see section 4).
Section snippets
Maximum principles
In the following, we denote Theorem 1 Assume that ,where , for , andholds, then u(t) ⩽ 0 on J. Proof Suppose, to the contrary, that u(t) > 0 for some t ∈ J. It is enough to consider the following cases. There exists a , such that , and u(t) ⩾ 0 for
Lower and upper solutions
Definition 1 A function α0 is called a lower solution of PBVP (1) ifwhere , , Lk ⩾ 0 are constants, and expressions are given byfor some with g(0) = g(T), and g ⩾ 0 in , Definition 2 A
Existence results
Consider the linear PBVP,where , are constants and . Lemma 2 is a solution of (4) if and only if is a solution of the following impulsive integral equation,here[10], [11]
References (22)
Impulsive resonance periodic problems of first order
Appl. Math. Lett.
(2002)Periodic boundary value problem for first order impulsive ordinary differential equations
Nonlinear Anal.
(2002)- et al.
Periodic boundary value problems for delay differential equations with impulses
J. Comput. Appl. Math.
(2006) - et al.
Periodic boundary value problem for the first order impulsive functional differential equations
J. Comput. Appl. Math.
(2007) - et al.
Existence and approximation of solutions for nonlinear differential equations with periodic boundary value conditions
Comput. Math. Appl
(2000) - et al.
Periodic boundary value problem for the second order impulsive functional differential equations
Appl. Math. Comput.
(2004) - et al.
Periodic boundary value problem for the second-order impulsive functional differential equations
Comput. Math. Appl.
(2005) - et al.
New comparison for impulsive integro-differential equations and applications
J. Math. Anal. Appl.
(2007) - et al.
Periodic boundary value problem for first-order impulsive functional differential equations
J. Comput. Appl. Math.
(2002) - et al.
Monotone iterative technique for impulsive inter-differential equations with periodic boundary conditions
Comput. Math. Appl.
(2004)
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2016, Advances in Difference EquationsEigenvalues of a class of singular boundary value problems of impulsive differential equations in banach spaces
2014, Journal of Function SpacesExtremely solutions for the BVP of the impulsive differential equations
2013, International Journal of Applied Mathematics and Statistics
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Supported by the NNSF of China (No. 10571050) and the Key Project of Chinese Ministry of Education.