Existence results for boundary value problems associated with singular strongly nonlinear equations

TL;DR

This paper establishes existence results for boundary value problems involving strongly nonlinear differential equations with the $\

Contribution

It introduces new existence theorems for boundary value problems with $\

Findings

Existence of solutions under Wintner-Nagumo type conditions.

Applicable to equations with $\

demonstrates the effectiveness of fixed point and upper/lower solutions methods.

Abstract

We consider a strongly nonlinear differential equation of the following general type $$(\Phi(a(t,x(t)) \, x'(t)))'= f(t,x(t),x'(t)), \quad \text{a.e. on $[0,T]$}$$ where $f$ is a Carath\'edory function, $\Phi$ is a strictly increasing homeomorphism (the $\Phi$-Laplacian operator) and the function $a$ is continuous and non-negative. We assume that $a(t,x)$ is bounded from below by a non-negative function $h(t)$, independent of $x$ and such that $1/h \in L^p(0,T)$ for some $p> 1$, and we require a weak growth condition of Wintner-Nagumo type. Under these assumptions, we prove existence results for the Dirichlet problem associated to the above equation, as well as for different boundary conditions. Our approach combines fixed point techniques and the upper/lower solutions method.