In the paper „Local versions of Banach principle and new methods of applications of contracive maps” in a shape of a poster we demonstrate generalizations of local versions of the Banach principle for nonlinear contractions adding new methods of applications of contractive maps.
Generalizations of local versions of Banach principle related to the so called -contractions are shown being the main results of this paper. The famous Cauchy problem is considered here for differential equation in Banach space. In the proof of this theorem we use the new method based on some local version of Banach principle (Theorem 2.5)
In this paper it also proved the inverse function theorem both in Banach spaces. The proofs presented here, based on the local versions of Banach principle, are essentially simpler.
We consider some Dirichlet problems for the differential equations with fractional Laplacian. Both existence and stability results are proved by the use of the varational methods.
Differential equations with delayed argument have numerous applications in biology, like e.g. in immunology, whether of epidemiology. These equation has the form
where the set of values of function may be olso multidimensional. We shall present the applications of more general equations i. e. equations of the form
where is defined on function space and is defined by the formula
The examples of application of such equations will be presented. To such equations the classical method of steps cannot be used.
Co-authors: Anna Poskrobko (Uniwersytet w Białymstoku), Jerzy Leszek Zalasiński (Expert FAO)
We consider the first initial boundary value problem for the following nonlinear Sobolev type equation with a rapid growing nonlinearity
Here is a bounded domain with smooth boundary and is the Laplace operator with respect to the spatial variable. The unique solvability in the classical sense for this problem is proved by M. O. Korpusov and A. G. Sveshnikov. Estimates for the time of the blow-up are given.
We are interested in numerical solving such problem. Stability of the method of lines is investigated.
We consider the Cauchy problem for a nonlocal wave equation in one dimension. We study the existence of solutions by means of bicharacteristics. The existence and uniqueness is obtained in topology. The existence theorem is proved in a subset generated by certain continuity conditions for the derivatives.
We show the applications of hyperbolic and parabolic subdiffusive equation with time fractional derivative to describe the transport process in membrane system and to study the subdiffusive impedance in electrochemical system. Based on solutions of the equations we find characteristic power functions which can be used to extract the subdiffusive parameters of the system from experimental data. To illustrate our considerations we find the values of subdiffusion parameters for a few media.
Co-author: Tadeusz Kosztołowicz
Using the techniques connected with the measure of noncompactness we investigate the neutral difference equation of the following form
where , , , is continuous and is a given positive integer, is a ratio of positive integers with odd denominator, and is ratio of odd positive integers; . Sufficient conditions for the existence of a bounded or stable of a special type solution are presented.
Co-authors: Marek Galewski, Robert Jankowski (Politechnika Łódzka)
Robert Jankowski, Ewa Schmeidel (Uniwersytet w Białymstoku)
First order discontinuous ordinary differential equations are considered. By considering noncontinuous sub and supersolutions (upper and lower absolutely continuous functions) we obtain more general results than in standard theory. We present theorems on the existence of extremal solutions for a large class of boundary value problems. We assume neither continuity nor monotonicity of boundary functions.
We consider the eigenvalue problem for the Laplace-Beltrami operator on the unit sphere :
and for – Laplacian Dirichlet problem:
We have proved the existence of the smallest positive eigenvalue of and Friedrichs-Wirtinger-type inequalities corresponding to our problems.
We have also derived some integro-differential inequalities related to the smallest positive eigenvalue of and .
Co-author: Mariusz Bodzioch
We consider the initial value problem for first-order stochastic functional partial differential equation driven by Brownian motion
where is white noise and is a Hale-type operator
We apply the method of lines and prove the stability of the numerical scheme. This result is proved with the help of representation, existence and uniqueness, and the estimation of solution lemmas.
Co-author: Maria Ziemlańska
Let be a bounded domain. We assume that the boundary is a smooth curve everywhere except at the origin and near the point curves are lateral sides of an angle with the measure and the vertex at near the curve
We shall consider a weak quasilinear elliptic equation with the nonlocal boundary condition connecting the values of the unknown function on the curves with its values of on the
- are diffeomorphisms mapping of onto
We investigate the behavior of weak solutions of the above problem in a neighborhood of the boundary corner point .