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

Graef John R. (Department of Mathematics, University of Tennessee at Chattanooga, Chattanooga, USA)
Kong Lingju (Department of Mathematics, University of Tennessee at Chattanooga, Chattanooga, USA)
Minhós Feliz M. (Department of Mathematics, University of Évora, Research Centre on Mathematics and its Applications (CIMA-UE), Évora, Portugal)
Fialho João (Researh Centre on Mathematics and its Applications (CIMA-UE), Évora, Portugal)

The authors consider the nth-order differential equation −(ф(u(n−1)(x)))’= f(x, u(x), ..., u(n−1)(x)), for 2Є(0, 1), where ф: R→ R is an increasing homeomorphism such that ф(0) = 0, n≥2, I:= [0,1], and f : I ×Rn → R is a L1-Carathéodory 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.

Keywords: ф-Laplacian, higher order problems, functional boundary conditions, lower and upper solutions