- Research Article
- Open Access
Impulsive Periodic Boundary Value Problems for Dynamic Equations on Time Scale
© Eric R. Kaufmann. 2009
- Received: 31 March 2009
- Accepted: 20 May 2009
- Published: 29 June 2009
- Banach Space
- Periodic Solution
- Dynamic Equation
- Periodic Boundary Condition
- Nonlinear Problem
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.
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 ).
All periodic time scales are unbounded above and below.
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).
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.
is the solution to the initial value problem . Other properties of the exponential function are given in the following lemma, [20, Theorem 2.36].
We employ Schaeffer's fixed point theorem, see , to prove the existence of a periodic solution.
Theorem 2.2 (Schaeffer's Theorem).
The converse follows trivially and the proof is complete.
Our first result is an existence and uniqueness theorem.
and such that
Then there exists a unique solution to (1.3).
Our next two results utilize Theorem 2.2 to establish the existence of solutions of (1.3).
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.
Then there exists at least one solution of the boundary value problem (1.3).
The following corollary is an immediate consequences of Theorem 3.3
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