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Existence results for functional semilinear damped integrodifferential equations. (English) Zbl 1180.34088

From the text: We investigate the existence of mild solutions of the Cauchy problems

y ' -Ay=By+Ft , y t , 0 t k (t,s,y s ) d sa.e.tJ=[0,b],
y(t)=φ(t),t[-r,0],

where F:J×C([-r,0],E)×EE is a given function, A is the infinitesimal generator of a family of semigroups {T(t):t0}, B is a bounded linear operator form E into E, k:J×J×C([-r,0],E)E, φC([-r,0],E) and E a real Banach space with norm |·|,

y '' -Ay=By ' +Ft , y t , 0 t k (t,s,y s ) d sa.e.tJ=[0,b],
y 0 =φ,y ' (0)=y 1 ·

Our approach in the both sections is based on the Schaefer’s fixed point theorem and on the Banach contraction principle.

MSC:
34K30Functional-differential equations in abstract spaces
47N20Applications of operator theory to differential and integral equations