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

Malkhaz Ashordia

Czechoslovak Mathematical Journal (2017)

  • Volume: 67, Issue: 3, page 579-608
  • ISSN: 0011-4642

Abstract

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A general theorem (principle of a priori boundedness) on solvability of the boundary value problem d x = d A ( t ) · f ( t , x ) , h ( x ) = 0 is established, where f : [ a , b ] × n n is a vector-function belonging to the Carathéodory class corresponding to the matrix-function A : [ a , b ] n × n with bounded total variation components, and h : BV s ( [ a , b ] , n ) 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 ) ) = ( x ) · x ( t 2 ( x ) ) + c 0 , where t i : BV s ( [ a , b ] , n ) [ a , b ] ( i = 1 , 2 ) and : BV s ( [ a , b ] , n ) n are continuous operators, and c 0 n .

How to cite

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Ashordia, Malkhaz. "On boundary value problems for systems of nonlinear generalized ordinary differential equations." Czechoslovak Mathematical Journal 67.3 (2017): 579-608. <http://eudml.org/doc/294112>.

@article{Ashordia2017,
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 $f\colon [a,b]\times \mathbb \{R\}^n\rightarrow \mathbb \{R\}^n$ is a vector-function belonging to the Carathéodory class corresponding to the matrix-function $A\colon [a,b]\rightarrow \mathbb \{R\}^\{n\times n\}$ with bounded total variation components, and $h\colon \operatorname\{BV\}_s([a,b],\mathbb \{R\}^n)\rightarrow \mathbb \{R\}^n$ 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\})\rightarrow [a,b]$$(i=1,2)$ and $\mathcal \{B\}\colon \operatorname\{BV\}_s([a,b],\mathbb \{R\}^\{n\})\rightarrow \mathbb \{R\}^n$ are continuous operators, and $c_0\in \mathbb \{R\}^n$.},
author = {Ashordia, Malkhaz},
journal = {Czechoslovak Mathematical Journal},
keywords = {system of nonlinear generalized ordinary differential equations; Kurzweil-Stieltjes integral; general boundary value problem; solvability; principle of a priori boundedness},
language = {eng},
number = {3},
pages = {579-608},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On boundary value problems for systems of nonlinear generalized ordinary differential equations},
url = {http://eudml.org/doc/294112},
volume = {67},
year = {2017},
}

TY - JOUR
AU - Ashordia, Malkhaz
TI - On boundary value problems for systems of nonlinear generalized ordinary differential equations
JO - Czechoslovak Mathematical Journal
PY - 2017
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 67
IS - 3
SP - 579
EP - 608
AB - 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 $f\colon [a,b]\times \mathbb {R}^n\rightarrow \mathbb {R}^n$ is a vector-function belonging to the Carathéodory class corresponding to the matrix-function $A\colon [a,b]\rightarrow \mathbb {R}^{n\times n}$ with bounded total variation components, and $h\colon \operatorname{BV}_s([a,b],\mathbb {R}^n)\rightarrow \mathbb {R}^n$ 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})\rightarrow [a,b]$$(i=1,2)$ and $\mathcal {B}\colon \operatorname{BV}_s([a,b],\mathbb {R}^{n})\rightarrow \mathbb {R}^n$ are continuous operators, and $c_0\in \mathbb {R}^n$.
LA - eng
KW - system of nonlinear generalized ordinary differential equations; Kurzweil-Stieltjes integral; general boundary value problem; solvability; principle of a priori boundedness
UR - http://eudml.org/doc/294112
ER -

References

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