Higher order boundary value problems with φ-Laplacian and functional boundary conditions

Abstract

We study the existence of solutions of the boundary value problem φ(u^(n−1)(t))′ + f (t, u(t), u′(t), . . . , u^(n−1)(t))= 0, t ∈ (0, 1), g_i (u, u′, . . . , u^(n−1), u^(i)(0))= 0, i = 0, . . . , n − 2, g_n−1 (u, u′, . . . , u^(n−1), u^(n−2)(1))= 0, where n ≥ 2, φ and g_i, i = 0, . . . , n − 1, are continuous, and f is a Carathéodory function. We obtain an existence criterion based on the existence of a pair of coupled lower and upper solutions.Wealso apply our existence theorem to derive some explicit conditions for the existence of a solution of a special case of the above problem. In our problem, both the differential equation and the boundary conditions may have dependence on all lower order derivatives of the unknown function, and many boundary value problems with various boundary conditions, studied extensively in the literature, are special cases of our problem. Consequently, our results improve and cover a number of known results in the literature. Examples are given to illustrate the applicability of our theorems.