EXISTENCE AND UNIQUENESS OF SOLUTIONS OF THE FRACTIONAL INTEGRO-DIFFERENTIAL EQUATIONS IN VECTOR-VALUED FUNCTION SPACE

The aim of this work is to study the existence and uniqueness of solutions of the fractional integro-differential equations d/dt[x(t) - L(x(t))] = A[x(t) - L(x(t))] + G(x(t)) + + 1/Gamma(alpha) integral(t)(-infinity)(t - s)(alpha-1)(integral(s)(-infinity) a(s -xi )x(xi)d xi)ds + f(t), (alpha > 0) with the periodic condition x(0) = x(2 pi), where a is an element of L-1 (R+). Our approach is based on the R-boundedness of linear operators L-p-multipliers and UMD-spaces.