Existence and Uniqueness of Positive Solutions for Discrete Fourth-Order Lidstone Problem with a Parameter
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Research Article Existence and Uniqueness of Positive Solutions for Discrete Fourth-Order Lidstone Problem with a Parameter Yanbin Sang,1, 2 Zhongli Wei,2, 3 and Wei Dong4 1
Department of Mathematics, North University of China, Taiyuan, Shanxi 030051, China School of Mathematics, Shandong University, Jinan, Shandong 250100, China 3 Department of Mathematics, Shandong Jianzhu University, Jinan, Shandong 250101, China 4 Department of Mathematics, Hebei University of Engineering, Handan, Hebei 056021, China 2
Correspondence should be addressed to Yanbin Sang, [email protected] Received 9 January 2010; Revised 23 March 2010; Accepted 26 March 2010 Academic Editor: A. Pankov Copyright q 2010 Yanbin Sang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 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.
1. Introduction 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 β given by Δ4 yt − 2 − βΔ2 yt − 1 ht f1 yt f2 yt , ya 0 Δ2 ya − 1,
t ∈ a 1, b − 1 Z ,
yb 0 Δ2 yb − 1,
1.1 1.2
where Δ is the usual forward difference operator given by Δyt yt 1 − yt, Δn yt Δn−1 Δyt, c, d Z : {c, c 1, . . . , d − 1, d}, and β > 0 is a real parameter. 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
2
Advances in Difference Equations
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 Minhos ´ 11 and He and Su 12 , which motivated us to consider the BVP 1.1 and 1.2. In 11 , Anderson and Minhos ´ studied the following nonlinear discrete fourth-order equation with explicit parameters β and λ given by Δ4 yt − 2 − βΔ2 yt − 1 λf t, yt ,
t ∈ a 1, b − 1 Z ,
1.3
with Lidstone boundary conditions 1.2, where β > 0 and λ > 0 are real parameters. The authors obtained the following result. Theorem 1.1 see 11 . Assume that the following condition is satisfied
A1 ft, y gtwy, where g : a 1, b − 1 Z → 0, ∞ w
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