- Research Article
- Open Access

# Existence of Weak Solutions for Second-Order Boundary Value Problem of Impulsive Dynamic Equations on Time Scales

- Hongbo Duan
^{1}and - Hui Fang
^{1}Email author

**2009**:907368

https://doi.org/10.1155/2009/907368

© H. Duan and H. Fang. 2009

**Received:**9 April 2009**Accepted:**28 June 2009**Published:**20 July 2009

## Abstract

We study the existence of weak solutions for second-order boundary value problem of impulsive dynamic equations on time scales by employing critical point theory.

## Keywords

- Weak Solution
- Cauchy Sequence
- Mountain Pass
- Critical Point Theory
- Impulsive Differential Equation

## 1. Introduction

where is a time scale, and is a given function, are real sequences with and the impulsive points are right-dense and and represent the right and left limits of at in the sense of the time scale, that is, in terms of for which whereas if is left-scattered, we interpret and .

The theory of time scales, which unifies continuous and discrete analysis, was first introduced by Hilger [1]. The study of boundary value problems for dynamic equations on time scales has recently received a lot of attention, see [2–16]. At the same time, there have been significant developments in impulsive differential equations, see the monographs of Lakshmikantham et al. [17] and Samoĭlenko and Perestyuk [18]. Recently, Benchohra and Ntouyas [19] obtained some existence results for second-order boundary value problem of impulsive differential equations on time scales by using Schaefer's fixed point theorem and nonlinear alternative of Leray-Schauder type. However, to the best of our knowledge, few papers have been published on the existence of solutions for second-order boundary value problem of impulsive dynamic equations on time scales via critical point theory. Inspired and motivated by Jiang and Zhou [10], Nieto and O'Regan [20], and Zhang and Li [21], we study the existence of weak solutions for boundary value problems of impulsive dynamic equations on time scales (1.1)–(1.4) via critical point theory.

This paper is organized as follows. In Section 2, we present some preliminary results concerning the time scales calculus and Sobolev's spaces on time scales. In Section 3, we construct a variational framework for (1.1)–(1.4) and present some basic notation and results. Finally, Section 4 is devoted to the main results and their proof.

## 2. Preliminaries about Time Scales

In this section, we briefly present some fundamental definitions and results from the calculus on time scales and Sobolev's spaces on time scales so that the paper is self-contained. For more details, one can see [22–25].

Definition 2.1.

A time scale is an arbitrary nonempty closed subset of equipped with the topology induced from the standard topology on

For

Definition 2.2.

respectively. If then is called right-dense (otherwise: right-scattered), and if then is called left-dense (otherwise: left-scattered). Denote

Definition 2.3.

In this case, is called the delta (or Hilger) derivative of at Moreover, is said to be delta or Hilger differentiable on if exists for all

Definition 2.4.

A function is said to be rd-continuous if it is continuous at right-dense points in and its left-sided limits exist (finite) at left-dense points in The set of rd-continuous functions will be denoted by

The Lebesgue integral associated with the measure on is called the Lebesgue delta integral.

Lemma 2.5 (see [24, Theorem 2.11]).

for all

Let denote the linear space of all continuous function with the maximum norm

Lemma 2.6 (see [24, Corollary 3.8]).

Let , and If converges weakly in to , then converges strongly in to .

Lemma 2.7 (Hölder inequality [25, Theorem 3.1]).

When we obtain the Cauchy-Schwarz inequality.

For more basic properties of Sobolev's spaces on time scales, one may refer to Agarwal et al. [24].

## 3. Variational Framework

In this section, we will establish the corresponding variational framework for problem (1.1)–(1.4).

for

for any

First, we give some lemmas which are useful in the proof of theorems.

Lemma 3.1.

If then for any , where

Proof.

Lemma 3.2.

is a Hilbert space.

Proof.

we have converges to in as . The proof is complete.

Lemma 3.3.

where is given in Lemma 3.1.

Proof.

The proof is complete.

for all

where and

for all

We call such critical points weak solutions of problem (1.1)–(1.4).

Let be a Banach space, which means that is a continuously Fréchet-differentiable functional on . is said to satisfy the Palais-Smale condition (P-S condition) if any sequence such that is bounded and as has a convergent subsequence in

Lemma 3.4 (Mountain pass theorem [26, Theorem 2.2], [27]).

Let be a real Hilbert space. Suppose satisfies the P-S condition and the following assumptions:

() there exist constants and such that for all where which will be the open ball in with radius and centered at

() and there exists such that .

## 4. Main Results

Now we introduce some assumptions, which are used hereafter:

(*H* 1) the function
is continuous;

(*H* 2)
holds uniformly for

where and .

Remark 4.1.

is the well-known Ambrosetti-Rabinowitz condition from the paper [27].

Lemma 4.2.

Suppose that the conditions ( )–( ) are satisfied, then satisfies the Palais-Smale condition.

Proof.

for all .

for

for some constants which implies that is bounded by the fact that .

Then is bounded in for Therefore, there exists a subsequence (for simplicity denoted again by ) such that converges weakly to in and by Lemma 2.6, converges strongly to in , that is, as for all

In a similar way to Lemma 3.2, one can prove that

which implies that converges weakly to in .

Thus, possesses a convergent subsequence in Then, the P-S condition is now satisfied.

Theorem 4.3.

Suppose that ( )–( ) hold. Then problem (1.1)–(1.4) has at least one nontrivial weak solution on

Proof.

for all and some positive constant

as for Hence, we can choose sufficiently large such that , and Assumption is verified. Theorem 4.3 is now proved.

Example 4.4.

is solvable.

## Declarations

### Acknowledgment

This research is supported by the National Natural Science Foundation of China (no. 10561004).

## Authors’ Affiliations

## References

- Hilger S:
**Analysis on measure chains—a unified approach to continuous and discrete calculus.***Results in Mathematics*1990,**18**(1-2):18-56.MathSciNetView ArticleMATHGoogle Scholar - Agarwal RP, Bohner M, Wong PJY:
**Sturm-Liouville eigenvalue problems on time scales.***Applied Mathematics and Computation*1999,**99**(2-3):153-166. 10.1016/S0096-3003(98)00004-6MathSciNetView ArticleMATHGoogle Scholar - Agarwal R, Bohner M, O'Regan D, Peterson A:
**Dynamic equations on time scales: a survey.***Journal of Computational and Applied Mathematics*2002,**141**(1-2):1-26. 10.1016/S0377-0427(01)00432-0MathSciNetView ArticleMATHGoogle Scholar - Amster P, Nápoli PD:
**Variational methods for two resonant problems on time scales.***International Journal of Difference Equations*2007,**2**(1):1-12.MathSciNetMATHGoogle Scholar - Anderson DR:
**Eigenvalue intervals for a two-point boundary value problem on a measure chain.***Journal of Computational and Applied Mathematics*2002,**141**(1-2):57-64. 10.1016/S0377-0427(01)00435-6MathSciNetView ArticleMATHGoogle Scholar - Anderson DR, Avery RI:
**An even-order three-point boundary value problem on time scales.***Journal of Mathematical Analysis and Applications*2004,**291**(2):514-525. 10.1016/j.jmaa.2003.11.013MathSciNetView ArticleMATHGoogle Scholar - Atici FM, Guseinov GSh:
**On Green's functions and positive solutions for boundary value problems on time scales.***Journal of Computational and Applied Mathematics*2002,**141**(1-2):75-99. 10.1016/S0377-0427(01)00437-XMathSciNetView ArticleMATHGoogle Scholar - Cabada A:
**Extremal solutions and Green's functions of higher order periodic boundary value problems in time scales.***Journal of Mathematical Analysis and Applications*2004,**290**(1):35-54. 10.1016/j.jmaa.2003.08.018MathSciNetView ArticleMATHGoogle Scholar - He Z:
**Existence of two solutions of****-point boundary value problem for second order dynamic equations on time scales.***Journal of Mathematical Analysis and Applications*2004,**296**(1):97-109. 10.1016/j.jmaa.2004.03.051MathSciNetView ArticleMATHGoogle Scholar - Jiang L, Zhou Z:
**Existence of weak solutions of two-point boundary value problems for second-order dynamic equations on time scales.***Nonlinear Analysis: Theory, Methods & Applications*2008,**69**(4):1376-1388. 10.1016/j.na.2007.06.034MathSciNetView ArticleMATHGoogle Scholar - Topal SG:
**Second-order periodic boundary value problems on time scales.***Computers & Mathematics with Applications*2004,**48**(3-4):637-648. 10.1016/j.camwa.2002.04.005MathSciNetView ArticleMATHGoogle Scholar - Cabada A:
**Existence results for****-Laplacian boundary value problems on time scales.***Advances in Difference Equations*2006,**2006:**-11.Google Scholar - Davidson FA, Rynne BP:
**The formulation of second-order boundary value problems on time scales.***Advances in Difference Equations*2006,**2006:**-10.Google Scholar - Karna B, Lawrence BA:
**An existence result for a multipoint boundary value problem on a time scale.***Advances in Difference Equations*2006,**2006:**-8.Google Scholar - Agarwal RP, Otero-Espinar V, Perera K, Vivero DR:
**Multiple positive solutions in the sense of distributions of singular BVPs on time scales and an application to Emden-Fowler equations.***Advances in Difference Equations*2008,**2008:**-13.Google Scholar - Benchohra M, Henderson J, Ntouyas SK:
**Eigenvalue problems for systems of nonlinear boundary value problems on time scales.***Advances in Difference Equations*2007,**2007:**-10.Google Scholar - 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.View ArticleGoogle Scholar - Samoĭlenko AM, Perestyuk NA:
*Impulsive Differential Equations, World Scientific Series on Nonlinear Science. Series A: Monographs and Treatises*.*Volume 14*. World Scientific, River Edge, NJ, USA; 1995:x+462.Google Scholar - Benchohra M, Ntouyas SK, Ouahab A:
**Existence results for second order boundary value problem of impulsive dynamic equations on time scales.***Journal of Mathematical Analysis and Applications*2004,**296**(1):65-73. 10.1016/j.jmaa.2004.02.057MathSciNetView ArticleMATHGoogle Scholar - 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.2007.10.022MathSciNetView ArticleMATHGoogle Scholar - Zhang H, Li ZX:
**Periodic solutions of second-order nonautonomous impulsive differential equations.***International Journal of Qualitative Theory of Differential Equations and Applications*2008,**2**(1):112-124.MATHGoogle Scholar - Bohner M, Peterson A:
*Dynamic Equations on Time Scales: An Introdution with Applications*. Birkhäuser, Boston, Mass, USA; 2001:x+358.View ArticleGoogle Scholar - Bohner M, Peterson A:
*Dynamic Equations on Time Scales*. Birkhäuser, Boston, Mass, USA; 2001:x+358.View ArticleMATHGoogle Scholar - Agarwal RP, Otero-Espinar V, Perera K, Vivero DR:
**Basic properties of Sobolev's spaces on time scales.***Advances in Difference Equations*2006,**2006:**-14.Google Scholar - Agarwal R, Bohner M, Peterson A:
**Inequalities on time scales: a survey.***Mathematical Inequalities & Applications*2001,**4**(4):535-557.MathSciNetView ArticleMATHGoogle Scholar - Rabinowitz PH:
*Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Regional Conference Series in Mathematics*.*Volume 65*. CBMS AMS, Providence, RI, USA; 1986:viii+100.View ArticleGoogle Scholar - Ambrosetti A, Rabinowitz PH:
**Dual variational methods in critical point theory and applications.***Journal of Functional Analysis*1973,**14**(4):349-381. 10.1016/0022-1236(73)90051-7MathSciNetView ArticleMATHGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.