Atlas: Solutions of Higher Order Boundary Value Problems by Lingju Kong

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Seventh Mississippi State - UAB Conference on Differential Equations & Computational Simulations
November 1-3, 2007
Doubletree Hotel
Birmingham, AL, USA

Organizers
Mississippi State University & University of Alabama - Birmingham

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Solutions of Higher Order Boundary Value Problems
by
Lingju Kong
University of Tennessee at Chattanooga
Coauthors: John R. Graef (University of Tennessee at Chattanooga ) and Qingkai Kong (Northern Illinois University)

The authors consider the boundary value problem

u⁽ⁿ⁾+f(t, u, u^', ..., u^(n-1))=0, 0 < t < 1,

u⁽ⁱ⁾(0)=g_i(u⁽ⁱ⁾(t₁), ..., u⁽ⁱ⁾(t_m)), i=0, ..., n-2,

u^(n-2)(1)=g_n-1(u^(n-2)(t₁), ..., u^(n-2)(t_m)),

where n ≥ 3 and m ≥ 1 are integers, 0 ≤ t_i ≤ 1 for i=1, ..., m, f and g_i, i=0, ..., n-1, are continuous. Based on the existence of higher order lower and upper solutions, we provide sufficient conditions for the existence of a solution of the above problem. Explicit conditions are also found for the existence of a solution of the problem without assuming the existence of lower and upper solutions. The differential equation has nonlinear dependence on all lower order derivatives of the unknown function and the boundary condition covers many multi-point boundary conditions studied in the literature. Schauder's fixed point theorem and appropriate Nagumo conditions are employed in the analysis. Examples are included to illustrate the results.

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Date received: August 28, 2007