Non-trivial solutions of local and non-local Neumann boundary-value problems

Abstract

Abstract. We prove new results on the existence, non-existence, localization and multiplicity of nontrivial solutions for perturbed Hammerstein integral equations. Our approach is topological and relies on the classical fixed point index. Some of the criteria involve a comparison with the spectral radius of some related linear operators. We apply our results to some boundary value problems with local and nonlocal boundary conditions of Neumann type. We illustrate in some examples the methodologies used.

Nonzero radial solutions for a class of elliptic systems with nonlocal BCs on annular domains

“…In order to state our eigenvalue comparison results, we consider, in a similar way as in [27], the following operators on…”

“…The approach that we use is topological, relies on classical fixed point index theory and we make use of ideas from the papers [16,26,27,29,35,36,51,55,58]. In the last Section we present an example that illustrates the applicability of our results.…”

Existence, non-existence and multiplicity results for a third order eigenvalue three-point boundary value problem

“…In Section 4, we consider the fixed point index theory for bounded sets in order to obtain some results regarding existence and multiplicity of solutions. We follow the line of results given in [2,10,11]. In Section 5, we give some conditions under which there is not any solution for the considered problem.…”

On conversion of resonant problem to non-resonant one

“…Usually the appropriate conditions on the linear part and on the nonlinearity are imposed in order the problem to be solvable. One may consult recent articles [1,5,12] and references therein for the respective bibliography.…”

“…It is to be mentioned that the natural idea of "a shift" from a resonant linear part to a non-resonant one was employed in the papers [4,5,13]. "Shift" arguments in a broad sense (replacing the linear part of a given equation multiply by different non-resonant linear parts and proving the existence of multiple solutions to a given boundary value problem) were applied for the study of resonant problems in [12] in context of the quasi-linearization approach [3].…”