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Nonlinear integrodifferential equations of mixed type in Banach spaces. (English) Zbl 1147.45009
The authors prove two existence theorems for the integrodifferential equation $x^\prime(t)=f(t,x(t),\int_0^t k_1(t,s)g(s,x(s))\,ds,\int_0^a k_2(t,s)h(s,x(s))ds),\;x(0)=x_0,$ where in the first theorem $$f,g,h,x$$ are Banach space valued functions and the integrals are taken in the sense of Henstock and in the second theorem $$f,g,h,x$$ also map weakly convergent sequences into weakly convergent sequences.

##### MSC:
 45N05 Abstract integral equations, integral equations in abstract spaces 45J05 Integro-ordinary differential equations 45G10 Other nonlinear integral equations
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