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Positive solutions to a third order nonlocal boundary value problem with a parameter

Szajnowska Gabriela, Zima Mirosława

Opuscula Mathematica

2024

Vol. 44, no. 2

267--283

We present some sufficient conditions for the existence of positive solutions to a third order differential equation subject to nonlocal boundary conditions. Our approach is based on the Krasnosel’skiĭ-Guo fixed point theorem in cones and the properties of the Green’s function corresponding to the BVP under study. The main results are illustrated by suitable examples.

On one-dimensional diffusion processes with moving membranes

Kopytko Bohdan, Shevchuk Roman

Journal of Applied Mathematics and Computational Mechanics

2022

Vol. 21, nr 3

45--57

Using the method of the classical potential theory, we construct the two-parameter Feller semigroup associated, on the given interval of the real line, with the Markov proces such that it is a result of pasting together, at some point of the interval, two ordinary diffusion processes given in sub-domains of this interval. It is assumed that the position on the line of boundary points of these sub-domains depends on the time variable. In addition, some variants of the general nonlocal boundary condition of Feller-Wentzell’s type are given in these points. The resulting process can serve as a one-dimensional mathematical model of the physical phenomenon of diffusion in media with moving membranes.

Direct and inverse spectral problems for Dirac systems with nonlocal potentials

Dębowska Kamila, Nizhnik Leonid P.

Opuscula Mathematica

2019

Vol. 39, no. 5

645--673

The main purposes of this paper are to study the direct and inverse spectral problems of the one-dimensional Dirac operators with nonlocal potentials. Based on information about the spectrum of the operator, we find the potential and recover the form of the Dirac system. The methods used allow us to reduce the situation to the one-dimensional case. In accordance with the given assumptions and conditions we consider problems in a specific way. We describe the spectrum, the resolvent, the characteristic function etc. Illustrative examples are also given.

On second order nonlocal boundary value problem at resonance

Szymańska-Dębowska K.

Fasciculi Mathematici

2016

Nr 56

143--153

This work is devoted to the existence of solutions for a system of nonlocal resonant boundary value problem x''=f(t,x), x' (0)=0, x' (1)-∫1 0 x(s)dg(s), where f ∶ [0,1] × Rk → Rk is continuous and g ∶ [0,1] → Rk is a function of bounded variation.