Abstract
Abstract In this paper, a third-order ordinary differential equation coupled to three-point boundary conditions is considered. The related Green’s function changes its sign on the square of definition. Despite this, we are able to deduce the existence of positive and increasing functions on the whole interval of definition, which are convex in a given subinterval. The nonlinear considered problem consists on the product of a positive real parameter, a nonnegative function that depends on the spatial variable and a time dependent function, with negative sign on the first part of the interval and positive on the second one. The results hold by means of fixed point theorems on suitable cones.
Highlights
Third-order three-point boundary value problems arise in several areas of applied mathematics and physics
Li [4] studied the same problems with two-point boundary conditions u (0) = u′(0) = u′′(1) = 0
The aim of this paper is to study the existence, nonexistence and multiplicity of solutions of the thirdorder nonlinear differential equation u′′′(t) = −λp(t)f (u (t)), a.e. t ∈ [0, 1] ≡ I, (1)
Summary
Third-order three-point boundary value problems arise in several areas of applied mathematics and physics. It is worth noticing that the properties of the corresponding sign-changing Green’s function make it necessary to construct a different kind of cone, similar to the one recently used in [10] Using this cone we will impose some conditions in order to assure the existence of positive and increasing solutions of the considered problem, which will be convex in a certain subset of its interval of definition. The paper is organized as follows: in Section 2 we study the linear problem and we deduce the exact expression of the corresponding Green’s function and some of its properties as well as some properties of its first- and second-order derivative
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