Published in: Fract. Calc. Appl. Anal. Vol. 5, No 4 (2002), pp. 427 - 436 STRICT L p SOLUTIONS FOR FRACTIONAL EVOLUTION EQUATIONS Emilia Bazhlekova Dedicated to the 60th anniversary of Prof. Francesco Mainardi Abstract The abstract linear problem for the fractional differential equation of order α ∈ (0, 1) is studied. Using the resolvent approach, strict L p solvability is proved provided the initial data belongs to appropriate real interpolation spaces. Mathematics Subject Classification: 26A33, 45N05 Key Words and Phrases: fractional derivative, fractional evolution equa- tion, real interpolation space 1. Introduction Let X be a Banach space and let A : D(A) ⊂ X → X be a closed linear densely defined operator in X . By D α t we denote the Riemann-Liouville fractional derivative of order α ∈ (0, 1): D α t f (t)= d dt J 1−α t f (t),t> 0, where J β t is the Riemann-Liouville fractional integral: J β t f (t)= 1 Γ(β )  t 0 (t − s) β−1 f (s) ds, β > 0,t> 0; J 0 t u(t)= u(t).