Higher Order ${\phi}$ -Laplacian BVP with Generalized Sturm–Liouville Boundary Conditions

Abstract

This work presents an existence and location result for the higher order boundary value problem

$$\begin{array}{l}-\left( \phi \left( u^{(n-1)}(x)\right) \right) ^{\prime}=f(x,u(x),\ldots,u^{(n-1)}(x)),\\u^{(i)}(0)=A_{i},\text{ \ }i=0,\ldots,n-3,\\ g_{1}\left( u^{(n-2)}(0)\right) -g_{2}\left( u^{(n-1)}(0)\right) =B,\\g_{3}\left( u^{(n-2)}(1)\right) +g_{4}\left( u^{(n-1)}(1)\right) =C, \end{array}$$

where ${\phi :\mathbb{R}\rightarrow \mathbb{R}}$ is an increasing and continuous function such that ${\phi (0)=0, n\geq 2}$ is an integer, ${\ f:[0,1]\times \mathbb{R}^{n}\rightarrow \mathbb{R}}$ is a L ¹-Carathéodory function, ${A_{i}, B, C\in \mathbb{R}}$, and ${g_{j}:\mathbb{R}\rightarrow \mathbb{R}}$ are continuous functions such that g ₁, g ₃ are increasing and g ₂, g ₄ are nondecreasing. In view of the assumptions on ${\phi }$ and f, this paper generalizes several problems due to the dependence on the (n − 1)-st derivative not only in the differential equation but also in the boundary conditions.