On the lower and upper solution method for higher order functional boundary value problems

Author:

Graef John1,Kong Lingju1,Minhós Feliz2,Fialho João3

Affiliation:

1. Department of Mathematics, University of Tennessee at Chattanooga, Chattanooga, USA

2. Department of Mathematics, University of Évora, Research Centre on Mathematics and its Applications (CIMA-UE), Évora, Portugal

3. Researh Centre on Mathematics and its Applications (CIMA-UE), Évora, Portugal

Abstract

The authors consider the nth-order differential equation ?(?(u(n?1)(x)))?= f(x, u(x), ..., u(n?1)(x)), for 2?(0, 1), where ?: R? R is an increasing homeomorphism such that ?(0) = 0, n?2, I:= [0,1], and f : I ?Rn ? R is a L1-Carath?odory function, together with the boundary conditions gi(u, u?, ..., u(n?2), u(i)(1)) = 0, i = 0, ..., n? 3, gn?2 (u, u?, ..., u(n?2), u(n?2)(0), u(n?1)(0)) = 0, gn?1 (u, u?, ..., u(n?2), u(n?2)(1), u(n?1)(1)) = 0, where gi : (C(I))n?1?R ? R, i = 0, ..., n?3, and gn?2, gn?1 : (C(I))n?1?R2 ? R are continuous functions satisfying certain monotonicity assumptions. The main result establishes sufficient conditions for the existence of solutions and some location sets for the solution and its derivatives up to order (n?1). Moreover, it is shown how the monotone properties of the nonlinearity and the boundary functions depend on n and upon the relation between lower and upper solutions and their derivatives.

Publisher

National Library of Serbia

Subject

Applied Mathematics,Discrete Mathematics and Combinatorics,Analysis

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