Maximal regularity for fractional difference equations of order 2
DOI:
https://doi.org/10.58997/ejde.2024.20Keywords:
Fractional difference equation; maximal regularity; UMD space; R-bounded.Abstract
In this article, we study the \(\ell^p\)-maximal regularity for the fractional difference equation $$ \Delta^{\alpha}u(n)=Tu(n)+f(n), \quad (n\in \mathbb{N}_0). $$ We introduce the notion of \(\alpha\)-resolvent sequence of bounded linear operators defined by the parameters \(T\) and \(\alpha\), which gives an explicit representation of the solution. Using Blunck's operator-valued Fourier multipliers theorems on \(\ell^p(\mathbb{Z}; X)\), we give a characterization of the \(\ell^p\)-maximal regularity for \(1 < p < \infty\) and \(X\) is a UMD space.
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References
T. Abdeljawad, F. M. Atici; On the definitions of nabla fractional operators, Abstr. Appl. Anal., 2012 (2012), 1-13.
R. P. Agarwal, C. Cuevas, C. Lizama; Regularity of Difference Equations on Banach Spaces, Springer-Verlag, 2014.
W. Arendt, S. Bu; The operator-valued Marcinkiewicz multiplier theorem and maximal regularity, Math. Z., 240 (2002), 311-343.
F. M. Atici, P. W. Eloe; A transform method in discrete fractional calculus, Int. J. Difference Equ., 2 (2007), No. 2, 165-176.
F. M. Atici, P. W. Eloe; Initial value problems in discrete fractional calculus, Proc. Amer. Math. Soc., 137 (2009), No. 3, 981-989.
D. Baleanu, K. Diethelm, E. Scalas, J. J. Trujillo; Fractional Calculus: Models and Numerical Methods, World Scientific, London, 2012.
S. Blunck; Maximal regularity of discrete and continuous time evolution equations, Studia Math., 146 (2001), No. 2, 157-176.
D. L. BurkhNolder; Martingale transforms and the geometry of Banach spaces , Lecture Notes in Mathematics, Vol. 860, Springer, Berlin, 1981, 35-50.
R. E. Corman, L. Rao, N. Ashwin-Bharadwaj, J. T. Allison, R. H. Ewoldt; Setting material function design targets for linear viscoelastic materials and structures, J. Mech. Des., 138 (2016), No. 5, paper No. 051402.
R. Denk, M. Hieber, J. Pr!uss; R-boundedness, Fourier multipliers and problems of elliptic and parabolic type, Mem. Amer. Math. Soc., 166 (2003), 788.
C. S. Goodrich, C. Lizama; A transference principle for nonlocal operators using a convolutional approach: Fractional monotonicity and convexity, Israel J. Math., 236 (2020), 533-589.
H. L. Gray, N. F. Zhang; On a new definition of the fractional difference, Math. Comp., 50 (1988), No. 182, 513-529.
T. HytNonen, J. van Neerven, M. Veraar, L. Weis; Analysis in Banach Spaces, Volume I: Martingales and Littlewood-Paley Theory, Springer International Publishing, 2016.
B. Jin, B. Li, Z. Zhou; Discrete maximal regularity of time-stepping schemes for fractional evolution equations, Numer. Math., 138 (2018), No. 1, 101-131.
C. Leal, C. Lizama, M. Murillo-Arcila; Lebesgue regularity for nonlocal time-discrete equations with delays, Fract. Calc. Appl. Anal., 21 (2018), 696-715.
J. Lindenstrauss, L. Tzafriri; Classical Banach Spaces II, Springer, Berlin 1996.
C. Lizama; The Poisson distribution, abstract fractional difference equations, and stability, Proc. Amer. Math. Soc., 145 (2017), No. 9, 3809-3827.
C. Lizama; .p-maximal regularity for fractional difference equations on UMD spaces, Math. Nachr., 288 (2015), 2079-2092.
C. Lizama, M. Murillo-Arcila; lp-maximal regularity for a class of fractional difference equations on UMD spaces: the case 1 < alpha< 2, Banach J. Math. Anal., 11 (2017), 188-206.
C. Lizama, M. Murillo-Arcila; Maximal regularity in .p spaces for discrete time fractional shifted equations, J. Differential Equations., 263 (2017), 3175-3196.
C. Lizama, M. Murillo-Arcila; Well posedness for semidiscrete abstract fractional Cauchy problems with finite delay, J. Comput. Appl. Math., 339 (2018), 356-366.
C. Lizama, M. Murillo-Arcila; Discrete maximal regularity for Volterra equations and nonlocal time-stepping schemes, Discrete Contin. Dyn. Syst., 40 (2020), 509-528.
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