An extension of Mawhin's continuation theorem and its application to boundary value problems with a \(p\)-Laplacian (Q1879777)

An abstract continuation theorem of coincidence degree type is stated and proved for some relatively compact perturbations of a class of non-invertible nonlinear operators. Applications are given to three-point boundary value problems of the form \[ (|u'|^{p-2}u')' + f(t,u) = 0, \quad u(0)= 0=G(u(\eta),u(1)) \] with \(\eta \in (0,1).\) For example, a solution exists if there exists \(D >0\) such that \[ f(t,D) < 0 < f(t,-D), \quad G(x,D) < 0 < G(x,-D) \] for all \(t \in [0,1]\) and \(|x| \leq D.\)