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 \[ \begin{cases} (\phi(u^{(n)}))'(t) = f(t,u(t),\ldots,u^{(n)}(t)), & \text{a.e.},\, t\in [0,+\infty), \\ n \in \mathbb{N}\setminus\{0\}, \\  \\ u^{(i)}(0) = A_{i}, \, i=0,\ldots,n-2, \\ u^{(n-1)}(0) + au^{(n)}(0) = B, \\ u^{(n)}(+\infty) = C. \end{cases} \] 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 \(L^{1}\)-Carathéodory function.