A note on a class of problems for a higher-order fully nonlinear equation under one-sided Nagumo-type condition Original Research Article

The purpose of this work is to establish existence and location results for the higher-order fully nonlinear differential equation View the MathML sourceu(n)(t)=f(t,u(t),u′(t),…,u(n−1)(t)),n≥2, Turn MathJax on with the boundary conditions View the MathML sourceu(i)(a)=Ai,for i=0,…,n−3, Turn MathJax on View the MathML sourceu(n−1)(a)=B,u(n−1)(b)=C Turn MathJax on or View the MathML sourceu(i)(a)=Ai,for i=0,…,n−3, Turn MathJax on View the MathML sourcec1u(n−2)(a)−c2u(n−1)(a)=B,c3u(n−2)(b)+c4u(n−1)(b)=C, Turn MathJax on with Ai,B,C∈RAi,B,C∈R, for i=0,…,n−3i=0,…,n−3, and c1c1, c2c2, c3c3, c4c4 real positive constants. It is assumed that f:[a,b]×Rn−1→Rf:[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 the Leray–Schauder topological degree and lower and upper solutions method.