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Existence and uniqueness of mild solutions to a semilinear integrodifferential equation of fractional order. (English) Zbl 1179.45010

The paper deals with the existence and uniqueness of mild solutions to a semilinear integro-differential equation of fractional order. The problem studied here is:

D q x(t)+Ax(t)=ft,x(t), 0 t e(t,s,x(s))ds,t[0,a],x(0)+g(x)=x 0 ,

where 0<q<1, -A is the infinitesimal generator of a noncompact and analytic semigroup on a Banach space and e,f and g some functions. The initial data is taken from the Banach space D(A α ), with 0<α1 with the norm |x| α =|A α x|. Under various conditions on the functions e,f and g, the above problem admits a unique mild solution. The main techniques employed here are the Banach contraction principle and a fixed point theorem.

45J05Integro-ordinary differential equations
45G10Nonsingular nonlinear integral equations
26A33Fractional derivatives and integrals (real functions)
34A08Fractional differential equations