Existence and Uniqueness of Positive Solutions for Discrete Fourth-Order Lidstone Problem with a Parameter
© Yanbin Sang et al. 2010
Received: 9 January 2010
Accepted: 26 March 2010
Published: 30 March 2010
This work presents sufficient conditions for the existence and uniqueness of positive solutions for a discrete fourth-order beam equation under Lidstone boundary conditions with a parameter; the iterative sequences yielding approximate solutions are also given. The main tool used is monotone iterative technique.
In this paper, we are interested in the existence, uniqueness, and iteration of positive solutions for the following nonlinear discrete fourth-order beam equation under Lidstone boundary conditions with explicit parameter given by
In recent years, the theory of nonlinear difference equations has been widely applied to many fields such as economics, neural network, ecology, and cybernetics, for details, see [1–7] and references therein. Especially, there was much attention focused on the existence and multiplicity of positive solutions of fourth-order problem, for example, [8–10], and in particular the discrete problem with Lidstone boundary conditions [11–17]. However, very little work has been done on the uniqueness and iteration of positive solutions of discrete fourth-order equation under Lidstone boundary conditions. We would like to mention some results of Anderson and Minhós  and He and Su , which motivated us to consider the BVP (1.1) and (1.2).
In , Anderson and Minhós studied the following nonlinear discrete fourth-order equation with explicit parameters and given by
Theorem 1.1 (see ).
Assume that the following condition is satisfied
Very recently, in , He and Su investigated the existence, multiplicity, and nonexistence of nontrivial solutions to the following discrete nonlinear fourth-order boundary value problem
Their main result is the following theorem.
Theorem 1.2 (see ).
The aim of this work is to relax the assumptions and on the nonlinear term, without demanding the existence of upper and lower solutions, we present conditions for the BVP (1.1) and (1.2) to have a unique solution and then study the convergence of the iterative sequence. The ideas come from Zhai et al. [18, 19] and Liang .
Throughout this paper, we need the following hypotheses:
2. Two Lemmas
To prove the main results in this paper, we will employ two lemmas. These lemmas are based on the linear discrete fourth-order equation
with Lidstone boundary conditions (1.2).
Lemma 2.1 (see ).
Lemma 2.2 (see ).
3. Main Results
First, we show that the BVP (1.1) and (1.2) has a solution.
This completes the proof of the theorem.
In the following, we consider the following two cases.
In Theorem 3.1, the more general conditions are imposed on the nonlinear term than Theorem 1.1. In particular, in Theorem 3.3, contains the variable ; therefore, the more comprehensive functions can be incorporated.
4. An Example
The conclusion then follows from Theorem 3.1.
The authors were supported financially by the National Natural Science Foundation of China (10971046), the Natural Science Foundation of Shandong Province (ZR2009AM004), and the Youth Science Foundation of Shanxi Province (2009021001-2).
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