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On well-posedness of vector-valued fractional differential-difference equations
1. | Departamento de Matemáticas, Instituto Universitario de Matemáticas y Aplicaciones, Universidad de Zaragoza, 50009 Zaragoza, Spain |
2. | Departamento de Matemática y Ciencia de la Computación, Universidad de Santiago de Chile, Las Sophoras 173, Estación Central, Santiago, Chile |
3. | Escuela Técnica Superior de Ingeniería y Sistemas de Teleomunicación, Universidad Politécnica de Madrid, C/Nikola Tesla, s/n 28031 Madrid, Spain |
$ \begin{equation*} (*) \left\{\begin{array}{rll} \Delta^{\alpha} u(n) & = & Au(n+2) + f(n,u(n)), \quad n \in \mathbb{N}_0, \,\, 1< \alpha \leq 2; \\ u(0) & = & u_0;\\ u(1) & = & u_1, \end{array}\right. \end{equation*} $ |
$ A $ |
$ X $ |
$ A, $ |
$ f $ |
References:
[1] |
L. Abadias, M. De León and J. L. Torrea,
Non-local fractional derivatives. Discrete and continuous, J. Math. Anal. Appl., 449 (2017), 734-755.
doi: 10.1016/j.jmaa.2016.12.006. |
[2] |
L. Abadias and C. Lizama,
Almost automorphic mild solutions to fractional partial difference-differential equations, Appl. Anal., 95 (2016), 1347-1369.
doi: 10.1080/00036811.2015.1064521. |
[3] |
L. Abadias, C. Lizama, P. J. Miana and M. P. Velasco,
Cesàro sums and algebra homomorphisms of bounded operators, Israel J. Math., 216 (2016), 471-505.
doi: 10.1007/s11856-016-1417-3. |
[4] |
L. Abadias and P. J. Miana, A subordination principle on Wright functions and regularized resolvent families, J. Funct. Spaces, Art., 2015 (2015), ID 158145, 9 pp.
doi: 10.1155/2015/158145. |
[5] |
T. Abdeljawad and F. M. Atici, On the definitions of nabla fractional operators, Abstr. Appl. Anal., 2012 (2012), Art. ID 406757, 13 pp.
doi: 10.1155/2012/406757. |
[6] |
G. Akrivis, B. Li and C. Lubich,
Combining maximal regularity and energy estimates for the discretizations of quasilinear parabolic equations, Math. Comp., 86 (2017), 1527-1552.
doi: 10.1090/mcom/3228. |
[7] |
W. Arendt, C. Batty, M. Hieber and F. Neubrander, Vector-valued Laplace Transforms and Cauchy Problems, Monographs in Mathematics. vol. 96. Birkhäuser, Basel, 2001.
doi: 10.1007/978-3-0348-5075-9. |
[8] |
F. M. Atici and P. W. Eloe,
A transform method in discrete fractional calculus, Int. J. Difference Equ., 2 (2007), 165-176.
|
[9] |
F. M. Atici and P. W. Eloe,
Discrete fractional calculus with the nabla operator, Electr. J. Qual. Th. Diff. Equ., 3 (2009), 1-12.
doi: 10.14232/ejqtde.2009.4.3. |
[10] |
F. M. Atici and S. Sengül,
Modeling with fractional difference equations, J. Math. Anal. Appl., 369 (2010), 1-9.
doi: 10.1016/j.jmaa.2010.02.009. |
[11] |
F. M. Atici and P. W. Eloe,
Linear systems of fractional nabla difference equations, Rocky Mountain J. Math., 41 (2011), 353-370.
doi: 10.1216/RMJ-2011-41-2-353. |
[12] |
Y. Bai, D. Baleanu and G. C. Wu,
Existence and discrete approximation for optimization problems governed by fractional differential equations, Commun. Nonlinear Sci. Numer. Simul., 59 (2018), 338-348.
doi: 10.1016/j.cnsns.2017.11.009. |
[13] |
D. Baleanu, K. Diethelm, E. Scalas and J. J. Trujillo, Fractional Calculus: Models and Numerical Methods, World Scientific, 2012.
doi: 10.1142/9789814355216. |
[14] |
E. Bazhlekova, Fractional Evolution Equations in Banach Spaces, Ph.D. Thesis, Eindhoven University of Technology, 2001. |
[15] |
J. Cermák, T. Kisela and L. Nechvátal,
Stability and asymptotic properties of a linear fractional difference equation, Adv. Difference Equ., 122 (2012), 1-14.
doi: 10.1186/1687-1847-2012-122. |
[16] |
J. Cermák, T. Kisela and L. Nechvátal,
Stability regions for linear fractional differential systems and their discretizations, Appl. Math. Comput., 219 (2013), 7012-7022.
doi: 10.1016/j.amc.2012.12.019. |
[17] |
E. Cuesta, C. Lubich and C. Palencia,
Convolution quadrature time discretization of fractional diffusion-wave equations, Math. Comp., 75 (2006), 673-696.
doi: 10.1090/S0025-5718-06-01788-1. |
[18] |
E. Cuesta and C. Palencia,
A numerical method for an integro-differential equation with memory in Banach spaces: Qualitative properties, SIAM J. Numer. Anal., 41 (2003), 1232-1241.
doi: 10.1137/S0036142902402481. |
[19] |
E. Cuesta, Asymptotic behaviour of the solutions of fractional integro-differential equations and some time discretizations, Discrete Contin. Dyn. Syst., 2007, Dynamical Systems and Differential Equations. Proceedings of the 6th AIMS International Conference, suppl., 277–285. |
[20] |
I. K. Dassios, D. I. Baleanu and G. I. Kalogeropoulos,
On non-homogeneous singular systems of fractional nabla difference equations, Appl. Math. Comput., 227 (2014), 112-131.
doi: 10.1016/j.amc.2013.10.090. |
[21] |
K. J. Engel and R. Nagel, One-parameter semigroups for linear evolution equations, Graduate Texts in Mathematics, 194. Springer-Verlag, New York, 2000. |
[22] |
T. Kemmochi and N. Saito,
Discrete maximal regularity and the finite element method for parabolic equations, Num. Math., 138 (2018), 905-937.
doi: 10.1007/s00211-017-0929-z. |
[23] |
V. Keyantuo, C. Lizama and M. Warma,
Spectral criteria for solvability of boundary value problems and positivity of solutions of time-fractional differential equations, Abstr. Appl. Anal., 2013 (2013), 1-11.
doi: 10.1155/2013/614328. |
[24] |
C. Lizama,
The Poisson distribution, abstract fractional difference equations, and stability, Proc. Amer. Math. Soc., 145 (2017), 3809-3827.
doi: 10.1090/proc/12895. |
[25] |
C. Lizama,
$l_p$-maximal regularity for fractional difference equations on UMD spaces, Math. Nach., 288 (2015), 2079-2092.
doi: 10.1002/mana.201400326. |
[26] |
C. Lizama and M. P. Velasco,
Weighted bounded solutions for a class of nonlinear fractional equations, Fract. Calc. Appl. Anal., 19 (2016), 1010-1030.
doi: 10.1515/fca-2016-0055. |
[27] |
C. Lizama and M. Murillo-Arcila,
$\ell_p$-maximal regularity for a class of fractional difference equations on $UMD$ spaces: The case $1 < \alpha \leq 2,$, Banach J. Math. Anal., 11 (2017), 188-206.
doi: 10.1215/17358787-3784616. |
[28] |
C. Lizama and M. Murillo-Arcila,
Maximal regularity in $\ell_p$ spaces for discrete time fractional shifted equations, J. Differential Equations, 263 (2017), 3175-3196.
doi: 10.1016/j.jde.2017.04.035. |
[29] |
K. S. Miller and B. Ross, Fractional difference calculus, In: Univalent Functions, Fractional Calculus, and Their Applications (Kóriyama, 1988), Horwood, Chichester, (1989), 139–152. |
[30] |
A. P. Prudnikov, Yu. A. Brychkov and O. I. Marichev, Integrals and Series. Elementary functions, Vol. 1. Gordon and Breach Science Publishers, New York, 1986. |
[31] |
A. M. Sinclair, Continuous Semigroups in Banach Algebras, London Mathematical Society, Lecture Notes Series 63, Cambridge University Press, New York, 1982.
![]() ![]() |
[32] |
C. C. Travis and G. F. Webb,
Compactness, regularity, and uniform continuity properties of strongly continuous cosine families, Houston Journal of Mathematics, 3 (1977), 555-567.
|
[33] |
L. W. Weis,
A generalization of the Vidav-Jorgens perturbation theorem for semigroups and its application to Transport Theory, J. Math. Anal. Appl., 129 (1988), 6-23.
doi: 10.1016/0022-247X(88)90230-2. |
[34] |
G. C. Wu, D. Baleanu and L. L. Huang,
Novel Mittag-Leffler stability of linear fractional delay difference equations with impulse, Appl. Math. Lett., 82 (2018), 71-78.
doi: 10.1016/j.aml.2018.02.004. |
[35] |
G. C. Wu, D. Baleanu and H. P. Xie, Riesz Riemann–Liouville difference on discrete domains, Chaos, 26 (2016), 084308, 5 pp.
doi: 10.1063/1.4958920. |
[36] |
G. C. Wu and D. Baleanu,
Discrete chaos in fractional delayed logistic maps, Nonlinear Dynam., 80 (2016), 1697-1703.
doi: 10.1007/s11071-014-1250-3. |
[37] |
G. C. Wu and D. Baleanu,
Stability analysis of impulsive fractional difference equations, Fract. Calc. Appl. Anal., 21 (2018), 354-375.
doi: 10.1515/fca-2018-0021. |
[38] |
A. Zygmund, Trigonometric Series, 2nd ed. Vols. Ⅰ, Ⅱ, Cambridge University Press, New York, 1959.
![]() ![]() |
show all references
References:
[1] |
L. Abadias, M. De León and J. L. Torrea,
Non-local fractional derivatives. Discrete and continuous, J. Math. Anal. Appl., 449 (2017), 734-755.
doi: 10.1016/j.jmaa.2016.12.006. |
[2] |
L. Abadias and C. Lizama,
Almost automorphic mild solutions to fractional partial difference-differential equations, Appl. Anal., 95 (2016), 1347-1369.
doi: 10.1080/00036811.2015.1064521. |
[3] |
L. Abadias, C. Lizama, P. J. Miana and M. P. Velasco,
Cesàro sums and algebra homomorphisms of bounded operators, Israel J. Math., 216 (2016), 471-505.
doi: 10.1007/s11856-016-1417-3. |
[4] |
L. Abadias and P. J. Miana, A subordination principle on Wright functions and regularized resolvent families, J. Funct. Spaces, Art., 2015 (2015), ID 158145, 9 pp.
doi: 10.1155/2015/158145. |
[5] |
T. Abdeljawad and F. M. Atici, On the definitions of nabla fractional operators, Abstr. Appl. Anal., 2012 (2012), Art. ID 406757, 13 pp.
doi: 10.1155/2012/406757. |
[6] |
G. Akrivis, B. Li and C. Lubich,
Combining maximal regularity and energy estimates for the discretizations of quasilinear parabolic equations, Math. Comp., 86 (2017), 1527-1552.
doi: 10.1090/mcom/3228. |
[7] |
W. Arendt, C. Batty, M. Hieber and F. Neubrander, Vector-valued Laplace Transforms and Cauchy Problems, Monographs in Mathematics. vol. 96. Birkhäuser, Basel, 2001.
doi: 10.1007/978-3-0348-5075-9. |
[8] |
F. M. Atici and P. W. Eloe,
A transform method in discrete fractional calculus, Int. J. Difference Equ., 2 (2007), 165-176.
|
[9] |
F. M. Atici and P. W. Eloe,
Discrete fractional calculus with the nabla operator, Electr. J. Qual. Th. Diff. Equ., 3 (2009), 1-12.
doi: 10.14232/ejqtde.2009.4.3. |
[10] |
F. M. Atici and S. Sengül,
Modeling with fractional difference equations, J. Math. Anal. Appl., 369 (2010), 1-9.
doi: 10.1016/j.jmaa.2010.02.009. |
[11] |
F. M. Atici and P. W. Eloe,
Linear systems of fractional nabla difference equations, Rocky Mountain J. Math., 41 (2011), 353-370.
doi: 10.1216/RMJ-2011-41-2-353. |
[12] |
Y. Bai, D. Baleanu and G. C. Wu,
Existence and discrete approximation for optimization problems governed by fractional differential equations, Commun. Nonlinear Sci. Numer. Simul., 59 (2018), 338-348.
doi: 10.1016/j.cnsns.2017.11.009. |
[13] |
D. Baleanu, K. Diethelm, E. Scalas and J. J. Trujillo, Fractional Calculus: Models and Numerical Methods, World Scientific, 2012.
doi: 10.1142/9789814355216. |
[14] |
E. Bazhlekova, Fractional Evolution Equations in Banach Spaces, Ph.D. Thesis, Eindhoven University of Technology, 2001. |
[15] |
J. Cermák, T. Kisela and L. Nechvátal,
Stability and asymptotic properties of a linear fractional difference equation, Adv. Difference Equ., 122 (2012), 1-14.
doi: 10.1186/1687-1847-2012-122. |
[16] |
J. Cermák, T. Kisela and L. Nechvátal,
Stability regions for linear fractional differential systems and their discretizations, Appl. Math. Comput., 219 (2013), 7012-7022.
doi: 10.1016/j.amc.2012.12.019. |
[17] |
E. Cuesta, C. Lubich and C. Palencia,
Convolution quadrature time discretization of fractional diffusion-wave equations, Math. Comp., 75 (2006), 673-696.
doi: 10.1090/S0025-5718-06-01788-1. |
[18] |
E. Cuesta and C. Palencia,
A numerical method for an integro-differential equation with memory in Banach spaces: Qualitative properties, SIAM J. Numer. Anal., 41 (2003), 1232-1241.
doi: 10.1137/S0036142902402481. |
[19] |
E. Cuesta, Asymptotic behaviour of the solutions of fractional integro-differential equations and some time discretizations, Discrete Contin. Dyn. Syst., 2007, Dynamical Systems and Differential Equations. Proceedings of the 6th AIMS International Conference, suppl., 277–285. |
[20] |
I. K. Dassios, D. I. Baleanu and G. I. Kalogeropoulos,
On non-homogeneous singular systems of fractional nabla difference equations, Appl. Math. Comput., 227 (2014), 112-131.
doi: 10.1016/j.amc.2013.10.090. |
[21] |
K. J. Engel and R. Nagel, One-parameter semigroups for linear evolution equations, Graduate Texts in Mathematics, 194. Springer-Verlag, New York, 2000. |
[22] |
T. Kemmochi and N. Saito,
Discrete maximal regularity and the finite element method for parabolic equations, Num. Math., 138 (2018), 905-937.
doi: 10.1007/s00211-017-0929-z. |
[23] |
V. Keyantuo, C. Lizama and M. Warma,
Spectral criteria for solvability of boundary value problems and positivity of solutions of time-fractional differential equations, Abstr. Appl. Anal., 2013 (2013), 1-11.
doi: 10.1155/2013/614328. |
[24] |
C. Lizama,
The Poisson distribution, abstract fractional difference equations, and stability, Proc. Amer. Math. Soc., 145 (2017), 3809-3827.
doi: 10.1090/proc/12895. |
[25] |
C. Lizama,
$l_p$-maximal regularity for fractional difference equations on UMD spaces, Math. Nach., 288 (2015), 2079-2092.
doi: 10.1002/mana.201400326. |
[26] |
C. Lizama and M. P. Velasco,
Weighted bounded solutions for a class of nonlinear fractional equations, Fract. Calc. Appl. Anal., 19 (2016), 1010-1030.
doi: 10.1515/fca-2016-0055. |
[27] |
C. Lizama and M. Murillo-Arcila,
$\ell_p$-maximal regularity for a class of fractional difference equations on $UMD$ spaces: The case $1 < \alpha \leq 2,$, Banach J. Math. Anal., 11 (2017), 188-206.
doi: 10.1215/17358787-3784616. |
[28] |
C. Lizama and M. Murillo-Arcila,
Maximal regularity in $\ell_p$ spaces for discrete time fractional shifted equations, J. Differential Equations, 263 (2017), 3175-3196.
doi: 10.1016/j.jde.2017.04.035. |
[29] |
K. S. Miller and B. Ross, Fractional difference calculus, In: Univalent Functions, Fractional Calculus, and Their Applications (Kóriyama, 1988), Horwood, Chichester, (1989), 139–152. |
[30] |
A. P. Prudnikov, Yu. A. Brychkov and O. I. Marichev, Integrals and Series. Elementary functions, Vol. 1. Gordon and Breach Science Publishers, New York, 1986. |
[31] |
A. M. Sinclair, Continuous Semigroups in Banach Algebras, London Mathematical Society, Lecture Notes Series 63, Cambridge University Press, New York, 1982.
![]() ![]() |
[32] |
C. C. Travis and G. F. Webb,
Compactness, regularity, and uniform continuity properties of strongly continuous cosine families, Houston Journal of Mathematics, 3 (1977), 555-567.
|
[33] |
L. W. Weis,
A generalization of the Vidav-Jorgens perturbation theorem for semigroups and its application to Transport Theory, J. Math. Anal. Appl., 129 (1988), 6-23.
doi: 10.1016/0022-247X(88)90230-2. |
[34] |
G. C. Wu, D. Baleanu and L. L. Huang,
Novel Mittag-Leffler stability of linear fractional delay difference equations with impulse, Appl. Math. Lett., 82 (2018), 71-78.
doi: 10.1016/j.aml.2018.02.004. |
[35] |
G. C. Wu, D. Baleanu and H. P. Xie, Riesz Riemann–Liouville difference on discrete domains, Chaos, 26 (2016), 084308, 5 pp.
doi: 10.1063/1.4958920. |
[36] |
G. C. Wu and D. Baleanu,
Discrete chaos in fractional delayed logistic maps, Nonlinear Dynam., 80 (2016), 1697-1703.
doi: 10.1007/s11071-014-1250-3. |
[37] |
G. C. Wu and D. Baleanu,
Stability analysis of impulsive fractional difference equations, Fract. Calc. Appl. Anal., 21 (2018), 354-375.
doi: 10.1515/fca-2018-0021. |
[38] |
A. Zygmund, Trigonometric Series, 2nd ed. Vols. Ⅰ, Ⅱ, Cambridge University Press, New York, 1959.
![]() ![]() |
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