Caratheodory's solution of the Cauchy problem and a question of Z. Grande

Year, pages: 2018, 1367 - 1372

It is shown that for a function $f\:[a,b]\times\mathbb{R}\to \mathbb{R}$ which is measurable with respect to the first variable and upper semicontinuous quasicontinuous and increasing with respect to the second variable there exists a Caratheodory's solution $y(x)=y_0+\int\limits_{x_0}^xf(t,y(t))\dd\mu(t)$ of the Cauchy problem $y'(x)=f(x,y(x))$ with the initial condition $y(x_0)=y_0$. There is constructed an example which indicate to essentiality of condition of increasing and give the negative answer to a question of Z. Grande.

Mykhaylyuk, V., Myronyk, V. 2018. Caratheodory's solution of the Cauchy problem and a question of Z. Grande. In Mathematica Slovaca, vol. 68, no.6, pp. 1367-1372. 0139-9918. DOI: https://doi.org/10.1515/ms-2017-0187

Mykhaylyuk, V., Myronyk, V. (2018). Caratheodory's solution of the Cauchy problem and a question of Z. Grande. Mathematica Slovaca, 68(6), 1367-1372. 0139-9918. DOI: https://doi.org/10.1515/ms-2017-0187