Existence of solutions for higher order $\phi-$Laplacian BVPs on the half-line using a one-sided Nagumo condition with nonordered upper and lower solutions

Abstract

In this paper, we consider the following $(n+1)$st order bvp on the half line with a $\phi-$Laplacian operator $$ \begin{cases} (\phi(u^{(n)}))'(t)=f(t,u(t),\ldots,u^{(n)}(t)),& a.e., \,t\in [ 0,+\infty ),\\& n\in \mathbb{N}\setminus\{0\}, \\ u^{(i)}(0)=A_{i},\, i=0,\ldots,n-2,\\ u^{(n-1)}(0)+au^{(n)}(0)=B,\\ u^{(n)}(+\infty )=C. \end{cases} $$ The existence of solutions is obtained by applying Schaefer's fixed point theorem under a one-sided Nagumo condition with nonordered lower and upper solutions method where $f$ is a $L^{1}$-Carath\'eodory function.

Keywords

• boundary value problem
• one-sided nagumo condition
• lower and upper solutions
• a priori estimates