ASYMPTOTIC BEHAVIOR OF FRACTIONAL-ORDER SEMILINEAR EVOLUTION EQUATIONS

Fractional calculus is a subject of great interest in many areas of mathematics, physics, and sciences, including stochastic processes, mechanics, chemistry, and biology. We will call an operator A on a Banach space X w-sectorial (w is an element of R) of angle if there exists theta is an element of [0, pi/2) such that S-theta := {gimel is an element of C \ {0} : vertical bar arg(gimel)vertical bar < theta + pi/2} subset of rho(A) (the resolvent set of A) and sup{vertical bar gimel - w vertical bar vertical bar vertical bar(gimel -A)(-1)vertical bar vertical bar : gimel is an element of w + S-theta} < infinity. Let A be w-sectorial of angle beta pi/2 with w < 0 and f an X-valued function. Using the theory of regularized families, and Banach's fixed-point theorem, we prove existence and uniqueness of mild solutions for the semilinear fractional-order differential equation D(t)(alpha+1)u(t) + mu D(t)(beta)u(t) = Au(t) + t(-alpha)/Gamma(1 - alpha) u'(0) + mu t(-beta)/Gamma(1 - beta) u(0) + f(t, u(t)), t > 0, 0 < alpha <= beta <= 1, mu > 0, with the property that the solution decomposes, uniquely, into a periodic term (respectively almost periodic, almost automorphic, compact almost automorphic) and a second term that decays to zero. We shall make the convention 1/Gamma(0) = 0. The general result on the asymptotic behavior is obtained by first establishing a sharp estimate on the solution family associated to the linear equation.