Continuation and maximal regularity of an arbitrary (fractional) order evolutionary integro-differential equation

Abstract

The maximal regularity and continuation of the solution (and its fractional-order derivative) of the Cauchy problem of the non-homogeneous fractional order evolution equation D ^α u(t) = Au(t) + f(t), α ∈ (0,1) have been studied in [A.M.A. El-Sayed and Mohamed. A.E. Herzallah (2004). Continuation and maximal regularity of fractional-order evolution equation. Journal of Mathematical Analysis and Applications, 296 (1), 340–350.]. Here we study the maximal regularity, continuation of the solution (and its fractional derivative), and some other properties of the solution of the Cauchy problem of the non-homogeneous arbitrary (fractional) order evolutionary integral equation.

Keywords:

• Evolution equation
• Evolutionary integral equation
• Fractional order derivative
• Maximal regularity
• AMS: 26A30
• 45D05
• 45N05