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A note on semilinear abstract functional differential and integrodifferential equations with infinite delay. (English) Zbl 1082.34543
By a fixed-point argument, the authors prove the existence and uniqueness of the solution to the following abstract integrodifferential equation \[ u(t)=g(t)+\int_0^tE(t-s)f(s,u(s),u_s)\,ds,\,\,0\leq t\leq T,\quad u_0=\phi . \] As an application, conditions are derived implying the existence and uniqueness of the solution to the abstract semilinear functional-integrodifferential equation with infinite delay \[ u'(t)=A\left[u(t)+\int_0^tF(t-s)u(s)\,ds\right]+f(t,u(t),u_t),\,\,0\leq t\leq T,\quad u_0=\phi . \]

MSC:
34K30 Functional-differential equations in abstract spaces
45J05 Integro-ordinary differential equations
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