- 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
Throughout this paper we let be any time scale (nonempty closed subset of ) and be a subset of such that and for is not right scattered and left dense at the same time.
We will assume that the following conditions are satisfied.
(H2) is continuous with respect to and for , where denotes the set of nonnegative real numbers.
if then ; for , where denotes the set of all positively regressive and rd-continuous functions.
(H5) and are defined on and , respectively, where
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 ).
If for , then
If for , then
Theorem 1.2 (see ).
and for all
First, we give the following definitions of solution and positive solution of BVP (1.3).
is nonnegative on .
is -differentiable, is -differentiable on and is continuous.
Furthermore, a solution of (1.3) is called a positive solution if for
Lemma 2.2 (see ).
where and are given in (2.7) and (2.8), respectively. It is obvious from (2.6), (H1) and (H4), that holds.
where is as in (2.12).
It is easy to derive that is a positive solution of BVP (1.3) if and only if is a nontrivial fixed point of , where be defined as before.
is completely continuous.
for all , then there exist such that , for and , for .
Thus, by Theorem 1.1, we conclude that for . The proof is therefore complete.
In this section, we investigate the conditions for the existence of at least one positive solution of the BVP (1.3).
In the next theorem, we will also assume that the following condition on .
If (H1)–(H6) are satisfied, then the BVP (1.3) has at least one positive solution.
Fix and let for . Then, satisfies (2.27). Define by
Hence, has a fixed point in .
Let . Since for and .
If (H1)–(H5) and (H7) are satisfied, then the BVP (1.3) has at least one positive solution.
Therefore, has a fixed point in . The proof is completed.
Using the following (H8) or (H9) instead of (H6) or (H7), the conclusions of Theorems 3.1 and 3.2 are true. For ,
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).
Suppose there exists constants such that
there exists a constant such that for and ,
Thus Consequently, the assumption (D3) holds, then there exist a number such that and
The remaining conditions of Theorem 1.2 will now be shown to be satisfied.
By (D1) and argument above, we can get that Hence, condition (ii) of Theorem 1.2 is satisfied.
As a result, yields
Thus, all conditions of Theorem 1.2 are satisfied. It implies that the TPBVP (1.3) has at least three positive solutions with
Since , . It is clear that (H1)–(H5) and (H8) are satisfied. Thus, by Corollary 3.3, the BVP (5.1) has at least one positive solution.
Clearly, is continuous and increasing . We can also see that . By (2.12), (4.1), and (4.2), we get , and .
from , so that (b) of (D3) is met. Summing up, there exist constants and satisfying
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