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May  2019, 39(5): 2679-2708. doi: 10.3934/dcds.2019112

On well-posedness of vector-valued fractional differential-difference equations

Luciano Abadías 1, , Carlos Lizama 1,2,3,, , Pedro J. Miana 1,2,3, and  M. Pilar Velasco 1,2,3,

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

* Corresponding author: Carlos Lizama

Received  May 2018 Revised  July 2018 Published  January 2019

Fund Project: C. Lizama has been partially supported by DICYT, Universidad de Santiago de Chile and FONDECYT 1180041. L. Abadias and P. J. Miana have been partially supported by Project MTM2016-77710-P, DGI-FEDER, of the MCYTS, and Project E-64 D.G. Aragón. M. P. Velasco has been partially supported by Project ESP2016-79135-R of the MCYTS.

We develop an operator-theoretical method for the analysis on well posedness of partial differential-difference equations that can be modeled in the form
$ \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*} $
where
$ A $
 is a closed linear operator defined on a Banach space
$ X $
. Our ideas are inspired on the Poisson distribution as a tool to sampling fractional differential operators into fractional differences. Using our abstract approach, we are able to show existence and uniqueness of solutions for the problem (*) on a distinguished class of weighted Lebesgue spaces of sequences, under mild conditions on sequences of strongly continuous families of bounded operators generated by
$ A, $
 and natural restrictions on the nonlinearity
$ f $
. Finally we present some original examples to illustrate our results.
Keywords: Difference equations, fractional and nonlinear PDE, Poisson distribution, weighted Lebesgue space, well posedness.
Mathematics Subject Classification: Primary: 35R11; Secondary: 39A14, 47D06.
Citation: Luciano Abadías, Carlos Lizama, Pedro J. Miana, M. Pilar Velasco. On well-posedness of vector-valued fractional differential-difference equations. Discrete and Continuous Dynamical Systems, 2019, 39 (5) : 2679-2708. doi: 10.3934/dcds.2019112
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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