- Research Article
- Open Access
Positive Solutions of Singular Initial-Boundary Value Problems to Second-Order Functional Differential Equations
© F. Jin and B. Yan. 2008
Received: 23 August 2007
Accepted: 5 August 2008
Published: 26 August 2008
where and the existence of positive solutions to (1.1) is obtained. When in (1.1), Agarwal and O'Regan in , Lin and Xu in  discussed the existence of positive solutions to (1.1) also. We notice that the nonlinearities in all the above-mentioned references depend on .
The more difficult case is that the term depends on for second-order functional differential equations with delay. When has no singularity at and , there are many results on the following (1.2) (see [7–9] and references therein). Up to now, to our knowledge, there are fewer results on (1.2) when the term is allowed to possess singularity for the term at and , which is of more actual significance.
And one defines some functions which one has to use in this paper.
We now introduce the definition of a solution to IBVP(1.2).
Throughout this paper, we assume the following hypotheses hold.
Lemma 1.5 (see ).
Let be the Banach space and let X be any nonempty, convex, closed, and bounded subset of . If is a continuous mapping of into itself and is relatively compact, then the mapping has at least one fixed point (i.e., there exists an with ) .
Using Lemma 1.5, we present the existence of at least one positive solution to (1.2) when is singular at and (notice the new Definition 1.1). To some extent, our paper complements and generalizes these in [1–6, 8–10].
2. Main Results
Assume that (H 1 )–(H 3 ) hold. Then, the IBVP( 1.2 ) has at least one positive solution.
which contradicts (2.7).
which contradicts (2.7).
which contradicts (2.7).
To prove the existence of positive solutions to IBVP(2.5), we seek to transform (2.5) into an integral equation via the use of Green's function and then find a positive solution by using Lemma 1.5.
that is, the claim is true.
Thus, the solution of IBVP(2.5) is also the one of (1.2). The proof is complete.
Equation (3.1) has at least one positive solution.
So, from Theorem 2.1, IBVP(3.1) has at least one positive solution.
The research was supported by NNSF of China (10571111) and the fund of Shandong Education Committee (J07WH08).
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