THE WELL-POSEDNESS OF FRACTIONAL INTEGRO-DIFFERENTIAL EQUATIONS IN COMPLEX BANACH SPACES∗

doi:10.1007/s10473-023-0410-2

数学物理学报(英文版) ›› 2023, Vol. 43 ›› Issue (4): 1603-1617.doi: 10.1007/s10473-023-0410-2

THE WELL-POSEDNESS OF FRACTIONAL INTEGRO-DIFFERENTIAL EQUATIONS IN COMPLEX BANACH SPACES^∗

Shangquan BU¹, Gang CAI^2,†

1. Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China;
2. School of Mathematical Sciences, Chongqing Normal University, Chongqing 401331, China

收稿日期:2022-03-11 修回日期:2022-06-29 发布日期:2023-08-08
通讯作者: †Gang CAI, E-mail: caigang-aaaa@163.com
作者简介:Shangquan BU, E-mail: bushangquan@mail.tsinghua.edu.cn
基金资助:
* NSF of China (12171266, 12171062) and the NSF of Chongqing (CSTB2022NSCQ-JQX0004).

THE WELL-POSEDNESS OF FRACTIONAL INTEGRO-DIFFERENTIAL EQUATIONS IN COMPLEX BANACH SPACES^∗

Shangquan BU¹, Gang CAI^2,†

1. Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China;
2. School of Mathematical Sciences, Chongqing Normal University, Chongqing 401331, China

Received:2022-03-11 Revised:2022-06-29 Published:2023-08-08
Contact: †Gang CAI, E-mail: caigang-aaaa@163.com
About author:Shangquan BU, E-mail: bushangquan@mail.tsinghua.edu.cn
Supported by:
* NSF of China (12171266, 12171062) and the NSF of Chongqing (CSTB2022NSCQ-JQX0004).

摘要/Abstract

摘要： 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 $B_{p,q}^s(\mathbb{T}; X)$.

关键词: Lebesgue-Bochner spaces, fractional integro-differential equations, multiplier, well-posedness

Abstract: 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 $B_{p,q}^s(\mathbb{T}; X)$.

Key words: Lebesgue-Bochner spaces, fractional integro-differential equations, multiplier, well-posedness

Shangquan BU, Gang CAI. THE WELL-POSEDNESS OF FRACTIONAL INTEGRO-DIFFERENTIAL EQUATIONS IN COMPLEX BANACH SPACES^∗[J]. 数学物理学报(英文版), 2023, 43(4): 1603-1617.

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.

参考文献

[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

THE WELL-POSEDNESS OF FRACTIONAL INTEGRO-DIFFERENTIAL EQUATIONS IN COMPLEX BANACH SPACES^∗

THE WELL-POSEDNESS OF FRACTIONAL INTEGRO-DIFFERENTIAL EQUATIONS IN COMPLEX BANACH SPACES^∗

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 15

Metrics

本文评价

推荐阅读 10

[1]	Yuhui Chen, Qinghe Yao, Minling Li, Zheng-an Yao. GLOBAL WELL-POSEDNESS AND OPTIMAL TIME DECAY RATES FOR THE GENERALIZED PHAN-THIEN-TANNER MODEL IN ${\mathbb{R}}^{3}$^*[J]. 数学物理学报(英文版), 2023, 43(3): 1301-1322.
[2]	Mingxuan Zhu, Zaihong Jiang. THE CAUCHY PROBLEM FOR THE CAMASSA-HOLM-NOVIKOV EQUATION^*[J]. 数学物理学报(英文版), 2023, 43(2): 736-750.
[3]	Changxing MIAO, Junyong ZHANG, Jiqiang ZHENG. A NONLINEAR SCHRÖDINGER EQUATION WITH COULOMB POTENTIAL[J]. 数学物理学报(英文版), 2022, 42(6): 2230-2256.
[4]	Weixi LI, Rui XU, Tong YANG. GLOBAL WELL-POSEDNESS OF A PRANDTL MODEL FROM MHD IN GEVREY FUNCTION SPACES[J]. 数学物理学报(英文版), 2022, 42(6): 2343-2366.
[5]	Qingyou He, Jiawei Sun. LARGE TIME BEHAVIOR OF THE 1D ISENTROPIC NAVIER-STOKES-POISSON SYSTEM[J]. 数学物理学报(英文版), 2022, 42(5): 1843-1874.
[6]	Jinlu LI, Zhaoyang YIN, Xiaoping ZHAI. GLOBAL WELL-POSEDNESS FOR THE FULL COMPRESSIBLE NAVIER-STOKES EQUATIONS[J]. 数学物理学报(英文版), 2022, 42(5): 2131-2148.
[7]	Wei WANG, Bin ZHAO. CHARACTERIZATION OF RESIDUATED LATTICES VIA MULTIPLIERS[J]. 数学物理学报(英文版), 2022, 42(5): 1902-1920.
[8]	晋雪婷, 肖跃龙, 于幻. GLOBAL WELL-POSEDNESS OF THE 2D BOUSSINESQ EQUATIONS WITH PARTIAL DISSIPATION[J]. 数学物理学报(英文版), 2022, 42(4): 1293-1309.
[9]	何骞君, 吴新峰, 燕敦验. BOUNDS FOR MULTILINEAR OPERATORS UNDER AN INTEGRAL TYPE CONDITION ON MORREY SPACES[J]. 数学物理学报(英文版), 2022, 42(3): 1191-1208.
[10]	周向宇, 朱朗峰. L² EXTENSIONS WITH SINGULAR METRICS ON KÄHLER MANIFOLDS[J]. 数学物理学报(英文版), 2021, 41(6): 2021-2038.
[11]	刘永建, Stanis law MIGORSKI, Van Thien NGUYEN, 曾生达. EXISTENCE AND CONVERGENCE RESULTS FOR AN ELASTIC FRICTIONAL CONTACT PROBLEM WITH NONMONOTONE SUBDIFFERENTIAL BOUNDARY CONDITIONS[J]. 数学物理学报(英文版), 2021, 41(4): 1151-1168.
[12]	宋乃琪, 刘和平, 赵纪满. BILINEAR SPECTRAL MULTIPLIERS ON HEISENBERG GROUPS[J]. 数学物理学报(英文版), 2021, 41(3): 968-990.
[13]	Pablo GALINDO, Mikael LINDSTRÖM. THE BLOCH SPACE ON THE UNIT BALL OF A HILBERT SPACE: MAXIMALITY AND MULTIPLIERS[J]. 数学物理学报(英文版), 2021, 41(3): 899-906.
[14]	陈停停, 屈爱芳, 王振. EXISTENCE AND UNIQUENESS OF THE GLOBAL L₁ SOLUTION OF THE EULER EQUATIONS FOR CHAPLYGIN GAS[J]. 数学物理学报(英文版), 2021, 41(3): 941-958.
[15]	陈泳, 刘亚, 秦春桃. REDUCIBILITY FOR A CLASS OF ANALYTIC MULTIPLIERS ON SOBOLEV DISK ALGEBRA[J]. 数学物理学报(英文版), 2021, 41(2): 361-370.