On Eigenvalue Intervals for Discrete Second Order Boundary Value Problems

On Eigenvalue Intervals for Discrete Second Order Boundary Value Problems In this paper, we consider discrete second order three-point boundary value problems. By exploring the properties of the associated Green’s function and applying Guo-Krasnosel’skii’s fixed point theorem, we show the existence of eigenvalue intervals. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Mathematicae Applicatae Sinica Springer Journals

On Eigenvalue Intervals for Discrete Second Order Boundary Value Problems

On Eigenvalue Intervals for Discrete Second Order Boundary Value Problems

Acta Mathematicae Applicatae Sinica, English Series Vol. 21, No. 1 (2005) 105–114 On Eigenvalue Intervals for Discrete Second Order Boundary Value Problems 1 1,2 1 Zeng-ji Du , Chun-yan Xue ,Wei-gao Ge Department of Mathematics, Beijing Institute of Technology, Beijing 100081, China (E-mail: duzengji@163.com; gew@bit.edu.cn) Department of Mathematics, Shenyang Normal University, Shenyang 110034, China (E-mail: xuechunyanab@bit.edu.cn) Abstract In this paper, we consider discrete second order three-point boundary value problems. By exploring the properties of the associated Green’s function and applying Guo-Krasnosel’skii’s fixed point theorem, we show the existence of eigenvalue intervals. Keywords eigenvalue intervals; positive solution; difference equations; green’s function; fixed point theorem 2000 MR Subject Classification 39A10 1 Introduction In this paper, we consider the three-point boundary value problem for second-order difference equation of the following form ∆ y(k − 1) + λh(k)f y(k) =0,k ∈{1,··· ,T}, (1) y(0) = 0,y(T +1) = ay(n), where ∆ y(k − 1) = y(k) − y(k − 1), ∆ y(k − 1) = y(k +1) − 2y(k)+ y(k − 1),k ∈{1,··· ,T}, T ≥ 3is a fixed positive integer, n ∈{2,··· ,T − 1}, a> 0 such that an < T +1, λ> 0. Boundary value problems for difference equations have been studied extensively by many authors (see, e.g.[1–3,7,8]). One particular area receiving current attention is the question of obtaining optimal eigenvalue intervals of boundary value problems for ordinary differential equations, as well as for finite difference equations. Many of these work have used the Guo- [1,5] Krasnosel’skii fixed-point theorem to obtain eigenvalue intervals based on positive solutions inside a cone (see, e.g. [1,2,4,5,7,9]). [9,10] Recently, Ma studied the existence...
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Publisher
Springer-Verlag
Copyright
Copyright © 2005 by Springer-Verlag Berlin Heidelberg
Subject
Mathematics; Applications of Mathematics; Math Applications in Computer Science; Theoretical, Mathematical and Computational Physics
ISSN
0168-9673
eISSN
1618-3932
D.O.I.
10.1007/s10255-005-0221-3
Publisher site
See Article on Publisher Site

Abstract

In this paper, we consider discrete second order three-point boundary value problems. By exploring the properties of the associated Green’s function and applying Guo-Krasnosel’skii’s fixed point theorem, we show the existence of eigenvalue intervals.

Journal

Acta Mathematicae Applicatae SinicaSpringer Journals

Published: Jan 1, 2005

References

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