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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