Higher order functional boundary value problems without monotone assumptions
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RESEARCH
Open Access
Higher order functional boundary value problems without monotone assumptions João F Fialho1,3* and Feliz Minhós2,3 *
Correspondence: [email protected] 1 School of Mathematics, Physics and Technology, College of the Bahamas, Nassau, Bahamas 3 Research Centre on Mathematics and Applications, University of Évora (CIMA-UE), Rua Romão Ramalho, 59, Évora, 7000-671, Portugal Full list of author information is available at the end of the article
Abstract In this paper, given f : [a, b] × (C([a, b]))n–2 × R2 → R a L1 -Carathéodory function, it is considered the functional higher order equation u(n) (x) = f (x, u, u , . . . , u(n–2) (x), u(n–1) (x)) together with the nonlinear functional boundary conditions, for i = 0, . . . , n – 2 Li (u, u , . . . , u(n–1) , u(i) (a)) = 0, Ln–1 (u, u , . . . , u(n–1) , u(n–2) (b)) = 0. Here, Li , i = 0, . . . , n – 1, are continuous functions. It will be proved an existence and location result in presence of not necessarily ordered lower and upper solutions, without assuming any monotone properties on the boundary conditions and on the nonlinearity f .
1 Introduction In this paper, it is considered the functional higher order boundary value problem, for n ≥ composed by the equation u(n) (x) = f x, u, . . . , u(n–) , u(n–) (x), u(n–) (x)
()
for a.a. x ∈ I := [a, b], where f : I × (C(I))(n–) × R → R is a L -Carathéodory function, and the function boundary conditions Li u, u , . . . , u(n–) , u(i) (a) = , i = , . . . , n – , Ln– u, u , . . . , u(n–) , u(n–) (b) = ,
()
where Li , i = , . . . , n – , are continuous functions without assuming monotone conditions or another type of variation. The functional differential equation () can be seen as a generalization of several types of full differential and integro-differential equations and allow to consider delays, maxima or minima arguments, or another kind of global variation on the unknown function or its derivatives until order (n – ). On the other hand, the functional dependence in () makes possible its application to a huge variety of boundary conditions, such as Lidstone, © 2013 Fialho and Minhós; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Fialho and Minhós Boundary Value Problems 2013, 2013:81 http://www.boundaryvalueproblems.com/content/2013/1/81
separated, multipoint, nonlocal and impulsive conditions, among others. As example, we mention the problems contained in [–]. A detailed list about the potentialities of functional problems and some applications can be found in []. Recently, functional boundary value problems have been studied by several authors following several approaches, as it can be seen, for example, in [–]. In this work, the lower and upper solutions method is applied together with topological degree th
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