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.


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