A characterization of periodic solutions for time-fractional differential equations in UMD spaces and applications

Abstract

Abstract. We study the fractional differential equationwhere X is a Banach space. Using functional calculus and operator valued Fourier multiplier theorems, we characterize, in U M D spaces, the well posedness of ( * ) in terms of R-boundedness of the setsApplications to the fractional Cauchy problems with periodic boundary condition, which includes the time diffusion and fractional wave equations, as well as a abstract version of the Basset-Boussinesq-Oseen equation, are treated.

Well-Posedness of Fractional Differential Equations on Vector-Valued Function Spaces

“…[1][2][3]8,[10][11][12][13][14][15][16][17][18]20,21]). They are useful in the study of the well-posedness of differential equations on Banach spaces.…”

“…We also notice that when α = 1, the well-posedness of (P 2 ) on F s p,q (T, X) was studied by Bu and Fang, a necessary and sufficient condition for (P 2 ) to be well-posed on F s p,q (T, X) was also given under a stronger assumption (H3) on a [7]. When a = 0, the well-posedness of (P 2 ) in different vector-valued function spaces were studied in [5] by Fourier multiplier method, and in [6] by the method of sum of bisectorial operators (see also [17] for further results about the well-posedness of a similar fractional differential equations).…”

Periodic solutions of fractional differential equations with delay

“…Time fractional differential equations with periodic boundary conditions have recently been treated by Bu [7] and by Keyantuo and Lizama [18]. To the knowledge of the authors, time fractional evolution equations with periodic boundary conditions and delay have not been studied until now.…”

Symmetric-periodic solutions for some types of generalized neutral equations

“…The existence of periodic solutions is often a desired property in dynamical systems, constituting one of the most important research directions in the theory of dynamical systems, with applications ranging from celestial mechanics to biology and finance. Fractional differential equations (FDEs) are the most important generalizations of the field of ODE [17][18][19][20][21]. Recent investigations in physics, engineering, biological sciences and other fields have demonstrated that the dynamics of many systems are described more accurately using FDEs, and that FDE with delay are often more realistic to describe natural phenomena than those without delay.…”

“…Periodic solution fractional differential equations have been studied by many researchers. They studied periodic solutions of the equation (see [17,18])…”