Solutions and Green's Functions for Boundary Value Problems of Second-Order Four-Point Functional Difference Equations
© The Author(s) Yang Shujie and Shi Bao. 2010
Received: 23 April 2010
Accepted: 11 July 2010
Published: 28 July 2010
We consider the Green's functions and the existence of positive solutions for a second-order functional difference equation with four-point boundary conditions.
from ordinary difference equations. Here this condition connects the history with the single . This is suggested by the well-posedness of BVP (1.9), since the function depends on the term (i.e., past values of ).
2. The Green's Function of (1.9)
First we consider the nonexistence of positive solutions of (1.9). We have the following result.
Then (1.9) has no positive solution.
We always assume that
Motivated by Zhao , we have the following conclusions.
We need only to show the uniqueness.
3. Existence of Positive Solutions
In this section, we discuss the BVP (1.9).
Lemma 3.1 (see Guo et al. ).
Then we may transform our existence problem of positive solutions of BVP (3.1) into a fixed point problem of operator (3.13).
We assume that
We have the following main results.
Similar to the above proof, we can show that (1.9) has at least one positive solution. Consequently, (1.9) has at least one positive solution.
4. Eigenvalue Intervals
respectively. We have the following results.
which by combining with (4.21) completes the proof.
The authors would like to thank the editor and the reviewers for their valuable comments and suggestions which helped to significantly improve the paper. This work is supported by Distinguished Expert Science Foundation of Naval Aeronautical and Astronautical University.
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