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Zbl 1180.34088
Benchohra, M.; Gatsori, E.; Ntouyas, S.K.
Existence results for functional semilinear damped integrodifferential equations.
(English)
[J] Libertas Math. 26, 97-108 (2006). ISSN 0278-5307

From the text: We investigate the existence of mild solutions of the Cauchy problems $$y'-Ay= By+F\bigg(t,y_t,\int_0^t k(t,s,y_s)\,ds\bigg) \ \text{ a.e. }t\in J=[0,b],$$ $$y(t)=\varphi(t), \quad t\in[-r,0],$$ where $F:J\times C([-r,0],E)\times E\to E$ is a given function, $A$ is the infinitesimal generator of a family of semigroups $\{T(t): t\ge 0\}$, $B$ is a bounded linear operator form $E$ into $E$, $k:J\times J\times C([-r,0],E)\to E$, $\varphi\in C([-r,0],E)$ and $E$ a real Banach space with norm $|\cdot|$, $$y''-Ay= By'+F\bigg(t,y_t,\int_0^t k(t,s,y_s)\,ds\bigg) \ \text{ a.e. }t\in J=[0,b],$$ $$y_0=\varphi, \quad y'(0)=y_1.$$ Our approach in the both sections is based on the Schaefer's fixed point theorem and on the Banach contraction principle.
MSC 2000:
*34K30 Functional-differential equations in abstract spaces
47N20 Appl. of operator theory to differential and integral equations

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