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

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:

(11)

(12)

(13)

(14)

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

(21)

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

(22)

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:

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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

(24)

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:

(25)

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

(26)

equipped with the norm

(27)

Let be the space of the form

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its norm is induced by the inner product given by

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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

(210)

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

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for

Now we consider the following space:

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its norm is induced by the inner product given by

(33)

That is

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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

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which implies that

(36)

Lemma 3.2.

is a Hilbert space.

Proof.

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

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Set

(38)

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

(39)

Noting that

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we have

(311)

Set

(312)

Then we have

(313)

Thus Noting that

(314)

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

Lemma 3.3.

If then for any ,

(315)

where is given in Lemma 3.1.

Proof.

For any by Lemma 3.1, we have

(316)

which implies that

(317)

The proof is complete.

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

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The first term is now

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Hence, one gets

(320)

for all Then we have

(321)

for all

This suggests that one defines by

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where and

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

(323)

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

(324)

where

(325)

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

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(H 4) there exist constants with such that

(42)

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

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for every By we know that there exist constants such that

(44)

for all . By and Lemma 3.1, we have

(45)

(46)

for all .

Set

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for

It follows from (4.3)–(4.5), and that

(48)

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

(49)

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

For any we have

(410)

which implies that converges weakly to in .

By (3.22) and (3.23), we have

(411)

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

(412)

that is,

(413)

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

(414)

for all and Hence, we have

(415)

for all and By Lemma 3.3, we obtain

(416)

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

(417)

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

(418)

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

(419)

for all , and by and Lemma 3.1, we have

(420)

for all and some positive constant

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

(421)

which implies that

(422)

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

(423)

is solvable.