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

Previous Article
Local well-posedness and blow-up for the half Ginzburg-Landau-Kuramoto equation with rough coefficients and potential
DCDS Home
This Issue
Next Article
Global well-posedness for the 2D Boussinesq equations with a velocity damping term

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 & Continuous Dynamical Systems - A, 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.

[1]	Hartmut Pecher. Local well-posedness for the nonlinear Dirac equation in two space dimensions. Communications on Pure & Applied Analysis, 2014, 13 (2) : 673-685. doi: 10.3934/cpaa.2014.13.673
[2]	Junxiong Jia, Jigen Peng, Kexue Li. Well-posedness of abstract distributed-order fractional diffusion equations. Communications on Pure & Applied Analysis, 2014, 13 (2) : 605-621. doi: 10.3934/cpaa.2014.13.605
[3]	Can Li, Weihua Deng, Lijing Zhao. Well-posedness and numerical algorithm for the tempered fractional differential equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1989-2015. doi: 10.3934/dcdsb.2019026
[4]	George Avalos, Roberto Triggiani. Semigroup well-posedness in the energy space of a parabolic-hyperbolic coupled Stokes-Lamé PDE system of fluid-structure interaction. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 417-447. doi: 10.3934/dcdss.2009.2.417
[5]	P. Blue, J. Colliander. Global well-posedness in Sobolev space implies global existence for weighted $L^2$ initial data for $L^2$-critical NLS. Communications on Pure & Applied Analysis, 2006, 5 (4) : 691-708. doi: 10.3934/cpaa.2006.5.691
[6]	Yonggeun Cho, Gyeongha Hwang, Soonsik Kwon, Sanghyuk Lee. Well-posedness and ill-posedness for the cubic fractional Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (7) : 2863-2880. doi: 10.3934/dcds.2015.35.2863
[7]	Jerry Bona, Hongqiu Chen. Well-posedness for regularized nonlinear dispersive wave equations. Discrete & Continuous Dynamical Systems - A, 2009, 23 (4) : 1253-1275. doi: 10.3934/dcds.2009.23.1253
[8]	Giuseppe Floridia. Well-posedness for a class of nonlinear degenerate parabolic equations. Conference Publications, 2015, 2015 (special) : 455-463. doi: 10.3934/proc.2015.0455
[9]	Hartmut Pecher. Corrigendum of "Local well-posedness for the nonlinear Dirac equation in two space dimensions". Communications on Pure & Applied Analysis, 2015, 14 (2) : 737-742. doi: 10.3934/cpaa.2015.14.737
[10]	Wei Qu, Siu-Long Lei, Seak-Weng Vong. A note on the stability of a second order finite difference scheme for space fractional diffusion equations. Numerical Algebra, Control & Optimization, 2014, 4 (4) : 317-325. doi: 10.3934/naco.2014.4.317
[11]	Gabriela Marinoschi. Well posedness of a time-difference scheme for a degenerate fast diffusion problem. Discrete & Continuous Dynamical Systems - B, 2010, 13 (2) : 435-454. doi: 10.3934/dcdsb.2010.13.435
[12]	Hiroyuki Hirayama, Mamoru Okamoto. Well-posedness and scattering for fourth order nonlinear Schrödinger type equations at the scaling critical regularity. Communications on Pure & Applied Analysis, 2016, 15 (3) : 831-851. doi: 10.3934/cpaa.2016.15.831
[13]	Changxing Miao, Bo Zhang. Global well-posedness of the Cauchy problem for nonlinear Schrödinger-type equations. Discrete & Continuous Dynamical Systems - A, 2007, 17 (1) : 181-200. doi: 10.3934/dcds.2007.17.181
[14]	Giulio Schimperna, Antonio Segatti, Ulisse Stefanelli. Well-posedness and long-time behavior for a class of doubly nonlinear equations. Discrete & Continuous Dynamical Systems - A, 2007, 18 (1) : 15-38. doi: 10.3934/dcds.2007.18.15
[15]	Chengchun Hao. Well-posedness for one-dimensional derivative nonlinear Schrödinger equations. Communications on Pure & Applied Analysis, 2007, 6 (4) : 997-1021. doi: 10.3934/cpaa.2007.6.997
[16]	Hiroyuki Hirayama. Well-posedness and scattering for a system of quadratic derivative nonlinear Schrödinger equations with low regularity initial data. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1563-1591. doi: 10.3934/cpaa.2014.13.1563
[17]	Yonggeun Cho, Gyeongha Hwang, Tohru Ozawa. Global well-posedness of critical nonlinear Schrödinger equations below $L^2$. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1389-1405. doi: 10.3934/dcds.2013.33.1389
[18]	Takeshi Wada. A remark on local well-posedness for nonlinear Schrödinger equations with power nonlinearity-an alternative approach. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1359-1374. doi: 10.3934/cpaa.2019066
[19]	G. Fonseca, G. Rodríguez-Blanco, W. Sandoval. Well-posedness and ill-posedness results for the regularized Benjamin-Ono equation in weighted Sobolev spaces. Communications on Pure & Applied Analysis, 2015, 14 (4) : 1327-1341. doi: 10.3934/cpaa.2015.14.1327
[20]	Ewa Schmeidel, Karol Gajda, Tomasz Gronek. On the existence of weighted asymptotically constant solutions of Volterra difference equations of nonconvolution type. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2681-2690. doi: 10.3934/dcdsb.2014.19.2681

American Institute of Mathematical Sciences