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

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

top Access to full text Full (PDF)

## Abstract

top
A general theorem (principle of a priori boundedness) on solvability of the boundary value problem $\mathrm{d}x=\mathrm{d}A\left(t\right)·f\left(t,x\right),\phantom{\rule{1.0em}{0ex}}h\left(x\right)=0$ is established, where $f:\left[a,b\right]×{ℝ}^{n}\to {ℝ}^{n}$ is a vector-function belonging to the Carathéodory class corresponding to the matrix-function $A:\left[a,b\right]\to {ℝ}^{n×n}$ with bounded total variation components, and $h:{BV}_{s}\left(\left[a,b\right],{ℝ}^{n}\right)\to {ℝ}^{n}$ 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 $x\left({t}_{1}\left(x\right)\right)=ℬ\left(x\right)·x\left({t}_{2}\left(x\right)\right)+{c}_{0},$ where ${t}_{i}:{BV}_{s}\left(\left[a,b\right],{ℝ}^{n}\right)\to \left[a,b\right]$$\left(i=1,2\right)$ and $ℬ:{BV}_{s}\left(\left[a,b\right],{ℝ}^{n}\right)\to {ℝ}^{n}$ are continuous operators, and ${c}_{0}\in {ℝ}^{n}$.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.