Elsevier

Applied Mathematics Letters

Volume 18, Issue 2, February 2005, Pages 209-218
Applied Mathematics Letters

Positive solutions of two-point boundary value problems for 2n-order differential equations dependent on all derivatives

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Abstract

In this work, we consider the higher-order differential equation(1)(1)nx(2n)(t)=f(t,x(t),x(t),,x(2n1)(t)),0<t<1, subject to one of the following multi-point boundary value conditions: (2)x(2i)(0)=x(2i)(1)=0for i=0,1,,n1, or (3)x(2i+1)(0)=x(2i)(1)=0for i=0,1,,n1,where f(t,x0,x1,,x2n1) is continuous with f(t,x0,x1,,x2n1)0 for all t[0,1] and (x0,x1,,x2n1)R2n. Sufficient conditions for the existence of at least one positive solution of the BVP (1), (2) and BVP (1), (3) are established, respectively. The emphasis in this work is on f depending on all higher-order derivatives. Examples are given to illustrate the main results.

Keywords

Solvability
Two-point boundary value problem
Higher-order differential equation
Positive solution

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The first author is supported by the Science Foundation of Educational Committee of Hunan Province (02C369) and both authors by the National Natural Science Foundation of PR China (10371006).