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

Abstract

It is shown that for a function f : [a, b] × 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) = y0 + ∫xx0 f(t, y(t))dμ(t) of the Cauchy problem y′(x) = f(x, y(x)) with the initial condition y(x0) = y0. There is constructed an example which indicate to essentiality of condition of increasing and give the negative answer to a question of Z. Grande.