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A note on the fractional Cauchy problems with nonlocal initial conditions. (English) Zbl 1251.45008
The authors consider the Cauchy problem $$D^\beta_t u(t)+ Au(t)= f(t,u(t))+ \int^t_0 K(t-s) g(s,u(s))\,ds,\quad t\in [0,T],$$ with $u(0)= H(u)$. Here $D^\beta_t$ is a fractional time derivative of order $\beta\in(0, 1)$, $-A$ generates a compact analytic semigroup $T(t)$ on a Banach space $X$, $u\in C([0,T]; X_\alpha)$ ($X_\alpha$ the domain of $A^\alpha$, $0<\alpha< 1$); $f,g: [0,T]\times X_\alpha\to X$, $K\in C[0,T]$ and $H: C([0,T]; X_\alpha)\to X_\alpha$. In the example, $$H(u)= \sum^N_{i=1} \int^\pi_0 K_0(x,y)\cos u(t_i; y)\,dy,\quad 0\le x\le\pi.$$ The authors first give a technical definition of mild solutions of the Cauchy problem, and then -- under conditions too lengthy to be included here -- prove the existence of a mild solution. The proof is by Schauders fixed point theorem.

45N05Abstract integral equations, integral equations in abstract spaces
45J05Integro-ordinary differential equations
45G10Nonsingular nonlinear integral equations
26A33Fractional derivatives and integrals (real functions)
45D05Volterra integral equations
Full Text: DOI
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