- Research Article
- Open Access
Existence of Weak Solutions for Second-Order Boundary Value Problem of Impulsive Dynamic Equations on Time Scales
© H. Duan and H. Fang. 2009
- Received: 9 April 2009
- Accepted: 28 June 2009
- Published: 20 July 2009
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.
- Weak Solution
- Cauchy Sequence
- Mountain Pass
- Critical Point Theory
- Impulsive Differential Equation
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 . 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.  and Samoĭlenko and Perestyuk . Recently, Benchohra and Ntouyas  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 , Nieto and O'Regan , and Zhang and Li , 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.
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].
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
Lemma 2.5 (see [24, Theorem 2.11]).
Lemma 2.6 (see [24, Corollary 3.8]).
Lemma 2.7 (Hölder inequality [25, Theorem 3.1]).
For more basic properties of Sobolev's spaces on time scales, one may refer to Agarwal et al. .
In this section, we will establish the corresponding variational framework for problem (1.1)–(1.4).
First, we give some lemmas which are useful in the proof of theorems.
The proof is complete.
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
Now we introduce some assumptions, which are used hereafter:
is the well-known Ambrosetti-Rabinowitz condition from the paper .
This research is supported by the National Natural Science Foundation of China (no. 10561004).
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