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

*Advances in Difference Equations*
**volume 2009**, Article number: 907368 (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.

## 1. Introduction

Consider the following boundary value problem:

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.

One defines the forward jump operator the backward jump operator and the graininess by

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

Definition 2.3.

Assume is a function and let Then one defines to be the number (provided it exists) with the property that given any there is a neighborhood of (i.e., for some ) such that

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

As mentioned in [24], the Lebesgue -measure can be characterized as follows:

where is the Lebesgue measure on and is the (at most countable) set of all right-scattered points of A function which is measurable with respect to is called -measurable, and the Lebesgue integral over is denoted by

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

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

If are absolutely continuous functions on , then is absolutely continuous on and the following equality is valid:

For , the Banach space may be defined in the standard way, namely,

equipped with the norm

Let be the space of the form

its norm is induced by the inner product given by

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]).

Let and be the conjugate number of Then

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).

Let and

for

Now we consider the following space:

its norm is induced by the inner product given by

That is

for any

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

Lemma 3.1.

If then for any , where

Proof.

For any and we have

which implies that

Lemma 3.2.

is a Hilbert space.

Proof.

Let be a Cauchy sequence in By Lemma 3.1, we have

Set

for Then be a Cauchy sequence in for Therefore, there exists a such that converges to in It follows from Lemma 2.6 that converges strongly to in , that is, as for all Hence, we have

Noting that

we have

Set

Then we have

Thus Noting that

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

Lemma 3.3.

If then for any ,

where is given in Lemma 3.1.

Proof.

For any by Lemma 3.1, we have

which implies that

The proof is complete.

For any satisfying (1.1)–(1.4), take and multiply (1.1) by then integrate it between and :

The first term is now

Hence, one gets

for all Then we have

for all

This suggests that one defines by

where and

By a standard argument, one can prove that the functional is continuously differentiable at any 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 .

Then possesses a critical value Moreover, can be characterized as

where

## 4. Main Results

Now we introduce some assumptions, which are used hereafter:

(*H* 1) the function is continuous;

(*H* 2) holds uniformly for

(*H* 3) there exist constants and such that

(*H* 4) there exist constants with such that

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.

Let be the sequence in satisfying that is bounded and as Then there exists a constant such that

for every By we know that there exist constants such that

for all . By and Lemma 3.1, we have

for all .

Set

for

It follows from (4.3)–(4.5), and that

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

Set

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

For any we have

which implies that converges weakly to in .

By (3.22) and (3.23), we have

By the fact that as and the continuity of and on we conclude

that is,

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.

In order to show that has at least one nonzero critical point, it suffices to check the conditions and . It follows from that there is a constant such that

for all and Hence, we have

for all and By Lemma 3.3, we obtain

for all and It follows from (4.5) and (4.16) that

for all and Therefore, by (3.6), one gets

for all where and Then is verified. Next we verify By (4.4), one has

for all , and by and Lemma 3.1, we have

for all and some positive constant

Let and For any by (4.19) and (4.20), one obtains

which implies that

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

Example 4.4.

Let Then the system

is solvable.

## References

- 1.
Hilger S:

**Analysis on measure chains—a unified approach to continuous and discrete calculus.***Results in Mathematics*1990,**18**(1-2):18-56. - 2.
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)00004-6 - 3.
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)00432-0 - 4.
Amster P, Nápoli PD:

**Variational methods for two resonant problems on time scales.***International Journal of Difference Equations*2007,**2**(1):1-12. - 5.
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)00435-6 - 6.
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.2003.11.013 - 7.
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-X - 8.
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.2003.08.018 - 9.
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.2004.03.051 - 10.
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.2007.06.034 - 11.
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.2002.04.005 - 12.
Cabada A:

**Existence results for****-Laplacian boundary value problems on time scales.***Advances in Difference Equations*2006,**2006:**-11. - 13.
Davidson FA, Rynne BP:

**The formulation of second-order boundary value problems on time scales.***Advances in Difference Equations*2006,**2006:**-10. - 14.
Karna B, Lawrence BA:

**An existence result for a multipoint boundary value problem on a time scale.***Advances in Difference Equations*2006,**2006:**-8. - 15.
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. - 16.
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. - 17.
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. - 18.
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. - 19.
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.2004.02.057 - 20.
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.2007.10.022 - 21.
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. - 22.
Bohner M, Peterson A:

*Dynamic Equations on Time Scales: An Introdution with Applications*. Birkhäuser, Boston, Mass, USA; 2001:x+358. - 23.
Bohner M, Peterson A:

*Dynamic Equations on Time Scales*. Birkhäuser, Boston, Mass, USA; 2001:x+358. - 24.
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. - 25.
Agarwal R, Bohner M, Peterson A:

**Inequalities on time scales: a survey.***Mathematical Inequalities & Applications*2001,**4**(4):535-557. - 26.
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. - 27.
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)90051-7

## Acknowledgment

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

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Duan, H., Fang, H. Existence of Weak Solutions for Second-Order Boundary Value Problem of Impulsive Dynamic Equations on Time Scales.
*Adv Differ Equ* **2009, **907368 (2009). https://doi.org/10.1155/2009/907368

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### Keywords

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