CARATHEODORY'S SOLUTION OF THE CAUCHY PROBLEM AND A QUESTION OF Z. GRANDE

It is shown that for a function f : inverted right prependicular a, b inverted right prependicular x R -> 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) + integral(x)(x0) f(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 is constructed an example which indicate to essentiality of condition of increasing and give the negative answer to a question of Z. Grande. (C)2018 Mathematical Institute Slovak Academy of Sciences