Positive solutions of systems of perturbed Hammerstein integral equations with arbitrary order dependence

Abstract

Motivated by the study of systems of higher-order boundary value problems with functional boundary conditions, we discuss, by topological methods, the solvability of a fairly general class of systems of perturbed Hammerstein integral equations, where the nonlinearities and the functionals involved depend on some derivatives. We improve and complement earlier results in the literature. We also provide some examples in order to illustrate the applicability of the theoretical results. This article is part of the theme issue ‘Topological degree and fixed point theories in differential and difference equations’.

On the solvability of a parameter-dependent cantilever-type BVP

“…The results are new and complement the ones present in the papers [2,4,5,14,16,17,24,26]. We also complement the results in [10], by obtaining additional qualitative properties (such as monotonicity and localization) of the solution. We illustrate the applicability of our theoretical results in an example.…”

“…The cone (2.2) is a smaller cone than the one of positive functions used in [10], but larger than the one used in [16]. We consider the sets…”

Nonzero positive solutions of elliptic systems with gradient dependence and functional BCs

“…We wish to point out that an advantage of our setting, with respect to the theory developed in [5,6,14,15,17,24], is the possibility to allow also gradient dependence within the functionals occurring in the BCs. This follows the approach used recently in [19,20] within the setting of ODEs.…”

Non-local second-order boundary value problems with derivative-dependent nonlinearity

“…This does not affect fixed points but could influence intermediate calculations. Using that set-up has been done in a number of papers including [17,18] where an integral equation of this type but corresponding to a BVP with nonlinear BCs is studied. It is important for us that v does satisfy the BCs.…”