Higher order two-point boundary value problems with asymmetric growth

Abstract

In this work it is studied the higher order nonlinear equation$\u^{( n)} (x)=f(x,u(x),u^{'}(x),\ldots ,u^{( n-1)} (x)) $with $n\in \mathbb{N}$ such that $n\geq 2,$ $f:[ a,b]\times \mathbb{R}^{n}\rightarrow \mathbb{R}$ a continuousfunction, and thetwo-point boundary conditions$u^{(i)}(a) =A_{i},\text{ \ \ }A_{i}\in \mathbb{R},\text{ \}i=0,\ldots,n-3$, $u^{( n-1) }(a) =u^{( n-1) }(b)=0.$From one-sided Nagumo-type condition, allowing that $f$ can beunbounded, it is obtained an existence and location result, thatis, besides the existence, given by Leray-Schauder topologicaldegree, some bounds on the solution and its derivatives till order$(n-2)$ are given by well ordered lower and upper solutions.  &nbspAn application to a continuous model of human-spine, via beamtheory, will be presented.