Non ordered lower and upper solutions to fourth order problems with functional boundary conditions

Pages: 209 - 218, Issue Special, September 2011

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Alberto Cabada - University of Santiago de Compostela, Department of Mathematical Analysis, Santiago de Compostela, Galicia, Spain (email)
João Fialho - Centro de Investigação em Matemática e Aplicações da U.E. (CIMA-CE), Rua Romão Ramalho 59, 7000-671 Évora, Portugal (email)
Feliz Minhós - Departamento de Matemática. Universidade de Évora, Centro de Investigação em Matemática e Aplicaçoes da U.E. (CIMA-UE), Rua Romão Ramalho, 59. 7000-671 Évora, Portugal (email)

Abstract: In this paper, given $f : I \times (C(I))^2 \times \mathbb{R}^2 \leftarrow \mathbb{R}$ a $L^1$ Carathéodory function, it is considered the functional fourth order equation $u^(iv) (x) = f(x, u, u', u'' (x), u''' (x))$ together with the nonlinear functional boundary conditions $L_0(u, u', u'', u (a)) = 0 = L_1(u, u', u'', u' (a))$ $L_2(u, u', u'', u'' (a), u''' (a)) = 0 = L_3(u, u', u'', u'' (b}, u''' (b)):$ Here $L_i, i$ = 0; 1; 2; 3, are continuous functions satisfying some adequate monotonicity assumptions. It will be proved an existence and location result in presence of non ordered lower and upper solutions and without monotone assumptions on the right hand side of the equation.

Keywords:  Higher order problems, functional boundary value problems, Nagumo condition, lower and upper solutions
Mathematics Subject Classification:  Primary: 34B15, 34K10; Secondary: 34B10, 34K45

Received: July 2010;      Revised: April 2011;      Published: October 2011.