Advanced Search

In this paper we prove an existence theorem for the Cauchy problem $$x^{\text{'}}\left(t\right)=f(t,x\left(t\right)),x\left(0\right)=x_0,t\in I_{\alpha }=[0,\alpha ]$$ using the Henstock-Kurzweil-Pettis integral and its properties. The requirements on the function $f$ are not too restrictive: scalar measurability and weak sequential continuity with respect to the second variable. Moreover, we suppose that the function $f$ satisfies some conditions expressed in terms of measures of weak noncompactness.