- Research Article
- Open Access
Positive Solutions for Boundary Value Problems of Second-Order Functional Dynamic Equations on Time Scales
© Ilkay Yaslan Karaca. 2009
- Received: 3 February 2009
- Accepted: 26 February 2009
- Published: 16 March 2009
Criteria are established for existence of least one or three positive solutions for boundary value problems of second-order functional dynamic equations on time scales. By this purpose, we use a fixed-point index theorem in cones and Leggett-Williams fixed-point theorem.
- Banach Space
- Ordinary Differential Equation
- Dynamic Equation
- Green Function
- Difference Equation
We will assume that the following conditions are satisfied.
There have been a number of works concerning of at least one and multiple positive solutions for boundary value problems recent years. Some authors have studied the problem for ordinary differential equations, while others have studied the problem for difference equations, while still others have considered the problem for dynamic equations on a time scale [5–10]. However there are fewer research for the existence of positive solutions of the boundary value problems of functional differential, difference, and dynamic equations [1, 2, 11–13].
Our problem is a dynamic analog of the BVPs (1.1) and (1.2). But it is more general than them. Moreover, conditions for the existence of at least one positive solution were studied for the BVPs (1.1) and (1.2). In this paper, we investigate the conditions for the existence of at least one or three positive solutions for the BVP (1.3). The key tools in our approach are the following fixed-point index theorem , and Leggett-Williams fixed-point theorem .
Theorem 1.1 (see ).
Theorem 1.2 (see ).
First, we give the following definitions of solution and positive solution of BVP (1.3).
Lemma 2.2 (see ).
In this section, we investigate the conditions for the existence of at least one positive solution of the BVP (1.3).
If (H1)–(H6) are satisfied, then the BVP (1.3) has at least one positive solution.
If (H1)–(H5) and (H7) are satisfied, then the BVP (1.3) has at least one positive solution.
In this section, using Theorem 1.2 (the Leggett-Williams fixed-point theorem) we prove the existence of at least three positive solutions to the BVP (1.3).
The remaining conditions of Theorem 1.2 will now be shown to be satisfied.
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