Czechoslovak Mathematical Journal, Vol. 67, No. 3, pp. 579-608, 2017

On boundary value problems for systems of nonlinear generalized ordinary
differential equations

Malkhaz Ashordia

Received May 3, 2011.   First published July 11, 2017.

Malkhaz Ashordia, A. Razmadze Mathematical Institute of I. Javakhishvili Tbilisi State University, 6 Tamarashvili St., Tbilisi 0177, Georgia, and Sukhumi State University, 12, Politkovskaya St., Tbilisi 0186, Georgia, e-mail: ashord@rmi.ge

Abstract: A general theorem (principle of a priori boundedness) on solvability of the boundary value problem ${\rm d} x={\rm d} A(t)\cdot f(t,x),\quad h(x)=0$ is established, where $f\colon[a,b]\times\mathbb{R}^n\to\mathbb{R}^n$ is a vector-function belonging to the Carathéodory class corresponding to the matrix-function $A\colon[a,b]\to\mathbb{R}^{n\times n}$ with bounded total variation components, and $h\colon\operatorname{BV}_s([a,b],\mathbb{R}^n)\to\mathbb{R}^n$ is a continuous operator. Basing on the mentioned principle of a priori boundedness, effective criteria are obtained for the solvability of the system under the condition $x(t_1(x))=\mathcal{B}(x)\cdot x(t_2(x))+c_0,$ where $t_i\colon\operatorname{BV}_s([a,b],\mathbb{R}^n)\to[a,b]$ $(i=1,2)$ and $\mathcal{B}\colon\operatorname{BV}_s([a,b],\mathbb{R}^n)\to\mathbb{R}^n$ are continuous operators, and $c_0\in\mathbb{R}^n$.

Keywords: system of nonlinear generalized ordinary differential equations; Kurzweil-Stieltjes integral; general boundary value problem; solvability; principle of a priori boundedness

Classification (MSC 2010): 34K10

DOI: 10.21136/CMJ.2017.0144-11

Full text available as PDF.


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