Nonlinear Integrodifferential Equations of Mixed Type in Banach Spaces

We prove two existence theorems for the integrodifferential equation of mixed type: x′(t) = f (t,x(t), t0k1(t,s)g(s,x(s))ds, ∫ a 0k2(t,s)h(s,x(s))ds), x(0) = x0, where in the first part of this paper f , g, h, x are functions with values in a Banach space E and integrals are taken in the sense of Henstock-Kurzweil (HK). In the second part f , g, h, x are weakly-weakly sequentially continuous functions and integrals are taken in the sense of Henstock-Kurzweil-Pettis (HKP) integral. Additionally, the functions f , g, h, x satisfy some conditions expressed in terms of the measure of noncompactness or the measure of weak noncompactness.