Let $X$ be a complex Banach space and let $B$ and $C$ be two closed linear operators on $X$ satisfying the condition $D(B)\subset D(C)$, and let $d\in L^1(\mathbb{R}_+)$ and $0 \leq \beta < \alpha\leq 2$. We characterize the well-posedness of the fractional integro-differential equations $D^\alpha u(t) + CD^\beta u(t)$ $= Bu(t) + \int_{-\infty}^t d(t-s)Bu(s){\rm d}s + f(t),\ (0\leq t\leq 2\pi)$ on periodic Lebesgue-Bochner spaces $L^p(\mathbb{T}; X)$ and periodic Besov spaces $B_{p,q}^s(\mathbb{T}; X)$.
Shangquan BU
,
Gang CAI
. THE WELL-POSEDNESS OF FRACTIONAL INTEGRO-DIFFERENTIAL EQUATIONS IN COMPLEX BANACH SPACES∗[J]. Acta mathematica scientia, Series B, 2023
, 43(4)
: 1603
-1617
.
DOI: 10.1007/s10473-023-0410-2
[1] Arendt W, Bu S. Operator-valued multipliers on periodic Besov spaces and applications. Proc Edinb Math Soc, 2004, 47: 15-33
[2] Arendt W, Bu S. The operator-valued Marcinkiewicz multiplier theorem and maximal regularity. Math Z, 2002, 240: 311-343
[3] Bourgain J. A Hausdorff-Young inequality for $B$-convex Banach spaces. Pacific J Math, 1982, 101: 255-262
[4] Bourgain J. Some remarks on Banach spaces in which martingale differences sequences are unconditional. Arkiv Math, 1983, 21: 163-168
[5] Bu S. Well-posedness of equations with fractional derivative. Acta Math Sinica, 2010, 26(7): 1223-1232
[6] Bu S, Cai G. Well-posedness of fractional degenerate differential equations in Banach spaces. Fract Calc Appl Anal, 2019, 22: 379-395
[7] Carracedo M, Sanz Alix M.The Theory of Fractional Powers of Operators. Amsterdam: Elsevier, 2001
[8] Haase M.The Functional Calculus for Sectorial Operators. Basel: Birkháuser, 2006
[9] Favini A, Yagi A.Degenerate Differential Equations in Banach Spaces. New York: Marcel Dekker, 1999
[10] Ledoux M, Talagrand M.Probability in Banach Spaces. Berlin: Springer, 1991
[11] Lizama C, Ponce R. Periodic solutions of degenerate differential equations in vector valued function spaces. Studia Math, 2011, 202(1): 49-63
[12] Keyantuo V, Lizama C. A characterization of periodic solutions for time-frcational differential equations in UMD spaces and applications. Math Nachr, 2011, 284(4): 494-506
[13] Pisier G. Sur les espaces de Banach qui ne contiennent pas uniformément de $\ell_1^n$. C R Acad Sci Paris,1973, 277(1): 991-994
[14] Poblete V, Poblete F, Pozo J. C. Strong solutions of a neutral type equation with finite delay. J Evol Equ, 2019, 19(2): 361-386
[15] Poblete V, Pozo J C. Periodic solutions of an abstract third-order differential equation. Studia Math, 2013, 215: 195-219
[16] Ponce R, Poblete V. Maximal $L^p$-regularity for fractional differential equations on the line. Math Nachr, 2017, 290(13): 2009-2023
[17] Zygmund A. Trigonometric Series, Vol II.Cambridge: Cambridge University Press, 1959
