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 $$\big(Φ(k(t)\,x'(t))\big)' + f(t,\mathcal{G}_x(t))\,ρ(t, x'(t)) = 0$$ on a compact interval $[a,b]$. These equations are quite general due to the presence of a strictly increasing homeomorphism $Φ$, the so-called $Φ$-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.