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Existence results for fractional functional integrodifferential equations with nonlocal conditions in Banach spaces. (English) Zbl 1194.34149
From the introduction: We consider the existence of mild solutions for the fractional functional integrodifferential equation with nonlocal condition of the form
$D^\beta x(t)= Ax(t)+F(t,x(\sigma_1(t)),\dots, x(\sigma_n(t)), \int^t_0 h(t,s,x(\sigma_{n+1}(s)))\,ds),\quad t\in J,\tag{1.1}$
where $$J = [0,b]$$, $$0<\beta<1$$, $$D^\beta$$ is the standard Riemann-Liouville fractional derivative, the state $$x(\cdot)$$ takes values in a Banach space $$X$$ with the norm $$\|\cdot\|$$ and $$A$$ generates a strongly continuous semigroup $$T(t)$$ in $$X$$. The nonlinear operators $$F : J\times X^{n+1}\to X$$, $$h:J\times J\times X\to X$$, $$g:C(J,X)\to X$$, $$\sigma_i:J\to J$$, $$i=1,\dots,n+1$$, are given functions.

MSC:
 34K37 Functional-differential equations with fractional derivatives 34K30 Functional-differential equations in abstract spaces
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