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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 t β u(t)+Au(t)=f(t,u(t))+ 0 t K(t-s)g(s,u(s))ds,t[0,T],

with u(0)=H(u). Here D t β is a fractional time derivative of order β(0,1), -A generates a compact analytic semigroup T(t) on a Banach space X, uC([0,T];X α ) (X α the domain of A α , 0<α<1); f,g:[0,T]×X α X, KC[0,T] and H:C([0,T];X α )X α . In the example,

H(u)= i=1 N 0 π K 0 (x,y)cosu(t i ;y)dy,0xπ·

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.

MSC:
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
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