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

TL;DR

The paper proves the existence of Caratheodory solutions for certain differential equations with specific conditions on the function and provides examples that challenge previous assumptions, addressing a question posed by Z. Grande.

Contribution

It establishes existence results for Caratheodory solutions under new conditions and presents counterexamples to a question by Z. Grande.

Findings

Existence of solutions under measurable, upper semicontinuous, quasicontinuous, and increasing conditions.

Counterexamples showing the necessity of the increasing condition.

Negative answer to Z. Grande's question regarding solution existence.

Abstract

It is shown that for a function $f:\mathbb R^2\to \mathbb R$ 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 $y(x)=y_0+\int\limits_{x_0}^xf(t,y(t))d\mu(t)$ of the Cauchy problem $y'(x)=f(x,y(x))$ with the initial condition $y(x_0)=y_0$. There are constructed examples which indicate to essentiality of condition of increasing and give the negative answer to a question of Z.~Grande.