A simple spectral theory of polynomially bounded solutions and applications to differential equations

In this paper we present a simple spectral theory of polynomially bounded functions on the half line, and then apply it to study the asymptotic behavior of solutions of fractional differential equations of the form D C α u ( t ) = A u ( t ) + f ( t ) , where D C α u ( t ) is the derivative of the function u in Caputo’s sense, A is generally an unbounded closed operator, f is polynomially bounded. Our main result claims that if u is a mild solution of the Cauchy problem such that lim h ↓ 0 sup t ≥ 0 ‖ u ( t + h ) - u ( t ) ‖ / ( 1 + t ) n = 0 , and sup t ≥ 0 ‖ u ( t ) ‖ / ( 1 + t ) n < ∞ , then, lim t → ∞ u ( t ) / ( 1 + t ) n = 0 provided that the spectral set Σ ( A , α ) ∩ i R is countable, where Σ ( A , α ) = C \ ρ ( A , α ) , ρ ( A , α ) is defined to be the set of complex numbers ξ such that λ α - 1 ( λ α - A ) - 1 is analytic in a neighborhood of ξ , and u satisfies some ergodic conditions with zero means. The obtained result extends known results on strong stability of solutions to fractional equations.