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In this work we proivied a new simpler proof of the global diffeomorphism theorem from [9] which we further apply to consider unique solvability of some abstract semilinear equations. Applications to the second order Dirichlet problem driven by the Laplace operator are given.

In this note we consider the classical variational tools like: Ekelenad's Variational Principle, Mountain Pass Lemma and some of their corollaries subject to a parameter. Next, we investigate the behaviour of critical points obtained once a sequence of parameters is allowed to be convergent. Applications for the Dirichlet Boundary Value Problem on the Sierpi\'{n}ski Gasket are given in presence of assumptions which lead to fulfillment of the mountain geometry.

We investigate solvability of a continuous Dirichlet boundary value problem together with its classical discretization using a gobal diffeomorphism theorem., Comment: paper accepted to Journal of Difference Equations and Applications

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(.)-Laplacian on weighted finite graphs. Several auxiliary inequalities for the discrete p(.)-Laplacian on finite graphs are also derived. Positive solutions are considered.

We consider a functional being a difference of two differentiable convex functionals on a closed ball. Existence and multiplicity of critical points is investigated. Some applications are given.

We provide sufficient conditions for a mapping $f:R^{n}\rightarrow R^{n}$ to be a global diffeomorphism in case it is strictly (Hadamard) differentiable. We use classical local invertibility conditions together with the non-smooth critical point theory.

Using direct variational method we consider the existence of non-spurious solutions to the following Dirichlet problem $\ddot{x}\left( t\right) =f\left( t,x\left( t\right) \right) $, $x\left( 0\right) =x\left( 1\right) =0 $ where $f:\left[ 0,1\right] \times \mathbb{R} \rightarrow \mathbb{R}$ is a jointly continuous function convex in $x$ which does not need to satisfy any further growth conditions.

In this note a critical point result for differentiable functionals is exploited in order to prove that a suitable class of one-dimensional fractional problems admits at least one non-trivial solution under an asymptotical behaviour of the nonlinear datum at zero. A concrete example of an application is then presented.

We investigate the existence of multiple periodic solutions to the anisotropic discrete system. We apply the linking method and a new three critical point theorem which we provide.

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., Comment: Mathematica Slovaca, accepted to be published. arXiv admin note: text overlap with arXiv:1304.2501

We provide sufficient conditions for a mapping acting between two Banach spaces to be a diffeomorphism., Comment: submitted to a journal for consideration

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.

\noindent Using the techniques connected with the measure of noncompactness we investigate the neutral difference equation of the following form \begin{equation*} \Delta \left(r_{n}\left(\Delta \left(x_{n}+p_{n}x_{n-k}\right) \right) ^{\gamma}\right) +q_{n}x_{n}^{\alpha}+a_{n}f(x_{n})=0. \end{equation*}% where $x:{\mathbb{N}}_{0}\rightarrow {\mathbb{R}}$, $a,p,q:{\mathbb{N}}%_{0}\rightarrow {\mathbb{R}}$, $r:{\mathbb{N}}_{0}\rightarrow {\mathbb{R}}% \setminus \{0\}$, $f\colon {\mathbb{R}}\rightarrow {\mathbb{R}}$ is a continuous function, and $k$ is a given positive integer, $\gamma \leq 1$ is ratio of odd positive integers, $\alpha $ is a nonnegative constant. %$\sum a_{n}\left(t\right)$ converges uniformly on ${\mathbb{R}}$. %Here $\bN_0\colon =\left\{0,1,2, \dots \right\}$ and $\bN_k \colon = \left\{k, k+1, -k+2, \dots \right\}$ where $k$ is a given positive integer. Sufficient conditions for the existence of a bounded solution are obtained. Also a special type of stability and asymptotic stability are studied. Some earlier results are generalized. We note that the solution which we obtain does not directly correspond to a fixed point of a certain continuous operator since it is partially iterated. The method which we develop allows for considering through techniques connected with the measure of noncompactness also difference equations with memory. {\small \textbf{Keywords} Difference equation, measures of noncompactness, Darbo's fixed point theorem, boundedness, stability} {\small \textbf{AMS Subject classification} 39A10, 39A22, 39A30}, Comment: submitted to Electronic Journal of the Qualitative Theory of Differential Equations

Using the direct method of the calculus of variations we investigate the existence, uniqueness and continuous dependence on parameters for solutions of second order discrete anisotropic equations with Dirichlet boundary conditions., Comment: paper is accepted by Journal of Difference Equations and Applications, no. GDEA-2012-0047

Using mountain pass arguments and the Karsuh-Kuhn-Tucker Theorem, we prove the existence of at least two positive solution of the anisotropic discrete Dirichlet boundary value problem. Our results generalize and improve those of [15]., Comment: preprint submitted for publication

Using critical point theory methods we undertake the existence and multiplicity of solutions for discrete anisotropic two-point boundary value problems., Comment: preprint submitted to Elsevier

Using monotonicity theory we investigate the continuous dependence on parameters for the discrete BVPs which can be written in a form of a nonlinear system., Comment: The final version of this submission has been accepted for publication and it will be published in Journal of Difference Equations and Applications

Using min-max inequality we investigate the existence of solutions and thier 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 Euler action functional., Comment: This preprint has been submitted for publication

We investigate the dependence on parameters for the discrete boundary value problem connected with the Emden-Fowler equation. A variational method is used in order to obtain a general scheme allowing for investigation the dependence on paramaters of discrete boundary value problems., Comment: This preprint has been submitted for publication

We investigate the dependence on parameters for second order difference equations with two point boundary value conditions by using a variational method in case when the corresponding Euler action functional is coercive. Some applications for discrete Emden-Fowler equation are also given., Comment: The final version of this submission has been accepted for publication and it will be published in Journal of Difference Equations and Applications