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A characterization of periodic solutions for time-fractional differential equations in \(UMD\) spaces and applications. (English) Zbl 1221.34012

The authors apply the method of operator-valued Fourier multipliers to obtain a characterization of existence, uniqueness and well-posedness for the fractional differential equation \[ D^{\alpha}u(t)+BD^{\beta}u(t)+Au(t)=f(t),\quad 0\leq t\leq 2\pi \]
for \(0\leq \beta<\alpha\leq 2\) in periodic Lebesgue spaces. Applications to fractional problems with periodic boundary condition, which includes time diffusion and fractional wave equations, as well as an abstract version of the Basset-Boussinesq-Oseen equation are treated.

MSC:

34A08 Fractional ordinary differential equations
42A45 Multipliers in one variable harmonic analysis
34G20 Nonlinear differential equations in abstract spaces
35R11 Fractional partial differential equations
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