Boundary value problems associated with singular strongly nonlinear equations with functional terms

Abstract

We study boundary value problems associated with singular, strongly nonlinear differential equations with functional terms of type $$(\Phi (k(t)x^{\prime }(t)){)}^{\prime }+f(t,{\mathcal{G}}_x(t))\rho (t,x^{\prime }(t))=0$$ on a compact interval $[a,b]$. These equations are quite general due to the presence of a strictly increasing homeomorphism $\Phi$, the so-called $\Phi$-Laplacian operator, of a nonnegative function $k$, which may vanish on a set of null measure, and moreover of a functional term ${\mathcal{G}}_x$. We look for solutions, in a suitable weak sense, which belong to the Sobolev space $W^{1,1}([a,b])$. Under the assumptions of the existence of a well-ordered pair of upper and lower solutions and of a suitable Nagumo-type growth condition, we prove an existence result by means of fixed point arguments.