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

الأصل · EN

It is shown that for a function f:R²→ 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+∫ₓ₀ˣf(t,y(t))dμ(t) of the Cauchy problem y'(x)=f(x,y(x)) with the initial condition y(x₀)=y₀. There are constructed examples which indicate to essentiality of condition of increasing and give the negative answer to a question of Z. Grande.