On solutions of differential equations in Banach spaces

We prove some existence theorems for the Cauchy problemx′(t)=f(t,x(t)),x(0)=x0,t∈I=[0,α],using different types of integrals and its properties. The requirements on the function f are depended on types of solutions, and as possible as, close to necessary conditions. We present a brief survey of classes of solutions and we give some comparison results for them.