Solvability of some third order boundary value problem with asymmetric unbounded nonlinearities

Abstract

The purpose of this work is to establish existence and location results for the higher order fully nonlinear differential equation u⁽ⁿ⁾(t)=f(t,u(t),u′(t),…,u⁽ⁿ⁻¹⁾(t)), n≥2, with the boundary conditions

u^{(i)}(a) = A_{i}, for i=0,⋯,n-3, u⁽ⁿ⁻¹⁾(a) = B, u⁽ⁿ⁻¹⁾(b)=C

or

u^{(i)}(a)=A_{i}, for i=0,⋯,n-3, c₁u⁽ⁿ⁻²⁾(a)-c₂u⁽ⁿ⁻¹⁾(a)=B, c₃u⁽ⁿ⁻²⁾(b)+c₄u⁽ⁿ⁻¹⁾(b)=C,

with A_{i},B,C ∈ R, for i=0,⋯,n-3, and c₁, c₂, c₃, c₄ real positive constants. It is assumed that f:[a,b]×Rⁿ⁻¹→R is a continuous function satisfying one-sided Nagumo-type conditions which allows an asymmetric unbounded behaviour on the nonlinearity. The arguments are based on Leray-Schauder topological degree and lower and upper solutions method.