Marek Galewski

A note on the multiplicity of solutions to anisotropic discrete BVP’s

The aim of this note is to consider the multiplicity of solutions for nonlinear discrete boundary value problems in a case when the difference operator can correspond to a discrete p (x)-Laplacian and its generalizations. Using critical point theory and recently obtained multiplicity results, [1], we formulate a kind of three critical point theorem. Research concerning existence and multiplicity of discrete boundary value problems is somewhat extensive nowadays, see for example [2–4] which pertains to the field of variable exponents. In [1] the authors, while answering a question by Ricceri concerning the uniqueness of a parameter for which some functional has three critical points, prove the following multiplicity result:

On variational impulsive boundary value problems

Using the variational approach, we investigate the existence of solutions and their dependence on functional parameters for classical solutions to the second order impulsive boundary value Dirichlet problems with L1 right hand side.

On the well posed solutions for nonlinear second order neutral difference equations

Abstract In this work we investigate the existence of solutions, their uniqueness and finally dependence on parameters for solutions of second order neutral nonlinear difference equations. The main tool which we apply is Darbo fixed point theorem.

Non-spurious solutions to discrete boundary value problems through variational methods

Using direct variational method we consider the existence of non-spurious solutions to the following Dirichlet problem , where is a jointly continuous function convex in which does not need to satisfy any further growth conditions.

Three solutions to discrete anisotropic problems with two parameters

In this note we derive a type of a three critical point theorem which we further apply to investigate the multiplicity of solutions to discrete anisotropic problems with two parameters.

On a global implicit function theorem for locally Lipschitz maps via non-smooth critical point theory

Abstract We prove a non-smooth generalization of the global implicit function theorem. More precisely we use the non-smooth local implicit function theorem and the non-smooth critical point theory in order to prove a non-smooth global implicit function theorem for locally Lipschitz functions. A comparison between several global inversion theorems is discussed. Applications to algebraic equations are given.

Existence and multiplicity results for boundary value problems connected with the discrete p ( ź ) - Laplacian on weighted finite graphs

We use the direct variational method, the Ekeland variational principle, the mountain pass geometry and Karush-Kuhn-Tucker theorem in order to investigate existence and multiplicity results for boundary value problems connected with the discrete p ( ź ) - Laplacian on weighted finite graphs. Several auxiliary inequalities for the discrete p ( ź ) - Laplacian on finite graphs are also derived. Positive solutions are considered.

Continuous dependence on parameters for second order discrete BVP’s

Using Fan’s Min-Max Theorem we investigate existence of solutions and their dependence on parameters for some second order discrete boundary value problem. The approach is based on variational methods and solutions are obtained as saddle points to the relevant Euler action functional.

Solvability of Abstract Semilinear Equations by a Global Diffeomorphism Theorem

In this work we develop a method towards unique solvability of abstract semilinear equations. We use a global diffeomorphism theorem for which we provide a simplified proof. Applications to second order partial differential equations are given. Some additional technical tools about the properties of the Niemytskij operator are also given.

A note on a global invertibility of mappings on R n

Abstract We provide sufficient conditions for a mapping f : Rn → Rn to be a global diffeomorphism in case f need not be continuously differentiable. Instead it is assumed to be strictly (Hadamard-like) and Fréchet differentiable. We use classical local invertibility conditions together with the non-smooth critical point theory.