Existence result for a third-order ODE with nonlinear boundary conditions in presence of a sign-type Nagumo control

Abstract

In this work we provide an existence and location result for the third order nonlinear differential equation

u′′′(t)=f(t,u(t),u′(t),u′′(t))

where f:[a,b]×R³→R is a continuous function, and two types of boundary conditions

u(a)=A, φ(u′(b),u′′(b))=0, u′′(a)=B,

or

u(a)=A, ψ(u′(a),u′′(a))=0, u′′(b)=C,

with φ, ψ:R²→R continuous functions and monotonous in the second variable and A,B,C∈R. We assume that f satisfy a one-sided Nagumo-type condition which allows an asymmetric unbounded behavior on the nonlinearity. The arguments used concern Leray-Schauder degree theory and lower and upper solutions technique.