Impulsive Periodic Boundary Value Problems for Dynamic Equations on Time Scale

Abstract

Let T be a periodic time scale with period such that , and . Assume each is dense. Using Schaeffer's theorem, we show that the impulsive dynamic equation where , , and is the -derivative on T, has a solution.

1. Introduction

Due to their importance in numerous application, for example, physics, population dynamics, industrial robotics, optimal control, and other areas, many authors are studying dynamic equations with impulse effects; see [1 - 19] and references therein.

The primary motivation for this work are the papers by Kaufmann et al. [9] and Li et al. [12]. In [9], the authors used a fixed point theorem due to Krasnosel'skiĭ to establish the existence theorems for the impulsive dynamic equation:

(1.1)

where and is the -derivative on .

In [12], the authors gave sufficient conditions for the existence of solutions for the impulsive periodic boundary value problem equation:

(1.2)

where , and . This paper extends and generalized the above results to dynamic equations on time scales.

We assume the reader is familiar with the notation and basic results for dynamic equations on time scales. While the books [20, 21] are indispensable resources for those who study dynamic equations on time scales, these manuscripts do not explicitly cover the concept of periodicity. The following definitions are essential in our analysis.

Definition 1.1 (see [8]).

We say that a time scale is periodic if there exist a such that if , then . For , the smallest positive is called the period of the time scale.

Example 1.2.

The following time scales are periodic:

• (1) has period ,

• (2) ,

• (3) has period ,

• (4) , where , has period .

Remark 1.3.

All periodic time scales are unbounded above and below.

Definition 1.4.

Let be a periodic time scale with period . We say that the function  is periodic with period if there exists a natural number such that , for all and is the smallest number such that .

If , we say that is periodic with period if is the smallest positive number such that for all .

Remark 1.5.

If is a periodic time scale with period , then . Consequently, the graininess function satisfies and so, is a periodic function with period .

Let be a periodic time scale with period such that , for , where for some , , and assume that each is dense in for each . We show the existence of solutions for the nonlinear periodic impulsive dynamic equation:

(1.3)

where and . Define and note that the intervals and are defined similarly.

In Section 2 we present some preliminary ideas that will be used in the remainder of the paper. In Section 3 we give sufficient conditions for the existence of at least one solution of the nonlinear problem (1.3).

2. Preliminaries

In this section we present some important concepts found in [20, 21] that will be used throughout the paper. We also define the space in which we seek solutions, state Schaeffer's theorem, and invert the linearized dynamic equation.

A function is said to be regressive provided for all . The set of all regressive rd-continuous functions is denoted by .

Let and for all . The exponential function on , defined by

(2.1)

is the solution to the initial value problem . Other properties of the exponential function are given in the following lemma, [20, Theorem  2.36].

Lemma 2.1.

Let . Then

• (i) and ;

• (ii) ;

• (iii) where, ;

• (iv) ;

• (v) ;

• (vi) .

Define and let . For , let . Define

(2.2)

and

(2.3)

where is the space of all real-valued continuous functions on , and is the space of all continuously delta-differentiable functions on . The set is a Banach space when it is endowed with the supremum norm:

(2.4)

where .

We employ Schaeffer's fixed point theorem, see [22], to prove the existence of a periodic solution.

Theorem 2.2 (Schaeffer's Theorem).

Let be a normed linear space and let the operator be compact. Define

(2.5)

Then either

• (i) the set is unbounded, or

• (ii) the operator has a fixed point in .

The following conditions hold throughout the paper:

• (A) is periodic with period ; for all .

• (F) and for all , .

Furthermore, to ensure that the boundary value problem is not at resonance, we assume that .

Consider the linear boundary value problem:

(2.6)

where . Our first result inverts the operator (2.6).

Lemma 2.3.

The function is a solution of (2.6) if and only if is a solution of

(2.7)

where

(2.8)

Proof.

It is easy to see that if is a solution of (2.6), then for we have

(2.9)

Apply the periodic boundary condition to obtain

(2.10)

Since , we can solve the above equation for . Thus,

(2.11)

Substitute (2.11) into (2.9). Since , we have, for all ,

(2.12)

We can rewrite this equation as follows:

(2.13)

Since , then

(2.14)

That is, satisfies (2.7).

The converse follows trivially and the proof is complete.

3. The Nonlinear Problem

In this section we give sufficient conditions for the existence of periodic solutions of (1.3). To this end, define the operator by

(3.1)

Then is a solution of (1.3) if and only if is a fixed point of . A standard application of the Arzelà-Ascoli theorem yields that is compact.

Our first result is an existence and uniqueness theorem.

Theorem 3.1.

Suppose there exist constants and for which

(3.2)

and

(3.3)

and such that

(3.4)

Then there exists a unique solution to (1.3).

Proof.

We will show that there exists a unique solution of (3.1). By Lemma 2.3 this solution is the unique solution of (1.3).

Let . Then for all

(3.5)

Hence, . By the Contraction Mapping Principal, there exists a unique solution of (3.1) and the proof is complete.

Our next two results utilize Theorem 2.2 to establish the existence of solutions of (1.3).

Theorem 3.2.

Assume there exist functions with

(3.6)

such that

(3.7)

Suppose that . Then there exists at least one solution of (1.3).

Proof.

Define

(3.8)

and let . We show is bounded by a constant that depends only on the constants , and . For all ,

(3.9)

Consequently,

(3.10)

which implies that We have that if , then is bounded by the constant The set is bounded and so by Schaeffer's theorem, the operator has a fixed point. This fixed point is a solution of (1.3) and the proof is complete.

In our next theorem we assume that and are sublinear at infinity with respect to the second variable.

Theorem 3.3.

Assume that

(F_I), uniformly, and

(I), uniformly.

Then there exists at least one solution of the boundary value problem (1.3).

Proof.

Suppose that the set

(3.11)

is unbounded. Then there exists sequences and , with and , such that

(3.12)

Define . Then and

(3.13)

By conditions () and () we have

(3.14)

(3.15)

From (3.13), (3.14), and (3.15), we have that

(3.16)

as , which contradicts for all . Thus the set is bounded. By Theorem 2.2, the operator has a fixed point. This fixed point is a solution of (1.3) and the proof is complete.

The following corollary is an immediate consequences of Theorem 3.3

Corollary 3.4.

Assume that and are bounded. Then there exists at least one solution of (1.3).