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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.

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