Stability Analysis of a Fractional Order Discrete Anti-Periodic Boundary Value Problem
DOI:
https://doi.org/10.26713/cma.v12i1.1445Keywords:
Existence, Ulam Stability, Boundary Value Problem, Caputo Fractional Difference operatorAbstract
This article aims at investigating stability properties for a class of discrete fractional equations with anti-periodic boundary conditions of fractional order \(\delta=(3,4]\). Utilizing Contraction mapping principle and fixed point theorem due to Brouwer, new criteria for the uniqueness and existence of the solutions are developed and two types of Ulam stability are analysed. The theoretical outcomes are corroborated with examples.
Downloads
References
R. P. Agarwal, A. Cabada, V. Otero-Espinar and S. Dontha, Existence and uniqueness of solutions for anti-periodic difference equations, Arch Inequal Appl. 2(4) (2004), 397-411.
A. Bashir, P. Karthikeyan and K. Buvaneswari, Fractional differential equations with coupled slit-strips type integral boundary conditions, AIMS Mathematics 4(6) (2019), 1596-1609, DOI: 10.3934/math.2019.6.1596.
B. Ahmad and J. J. Nieto, Existence and approximation of solutions for a class of nonlinear impulsive functional differential equations with anti-periodic boundary conditions, Nonlinear Anal. 69(10) (2008), 3291-3298, DOI: 10.1016/j.na.2007.09.018.
A. Alsaedi, S. Sivasundaram and B. Ahmad, On the generalization of second order nonlinear anti-periodic boundary value problems, Nonlinear Stud. 16(4) (2009), 415-420.
A. Seemab, M. Rehman, J. Alzabut and A. Hamdi, On the existence of positive solutions for generalized fractional boundary value problems, Bound. Value Probl. 186(2019) (2019), DOI: 10.1186/s13661-019-01300-8.
F. M. Atici and P. W. Eloe, A transform method in discrete fractional calculus, Int. J. Difference Equ. 2(2) (2007), 165-176, DOI: ecommons.udayton.edu/mth-fac-pub/110.
F. M. Atici and P. W. Eloe, Initial value problems in discrete fractional calculus, Proc. Amer. Math. Soc. 137(3) (2008), 981-989, DOI: 10.1090/S0002-9939-08-09626-3.
F. M. Atici and P. W. Eloe, Linear systems of fractional nabla difference equations, Rocky Mt. J. Math. 41(2) (2011), 353-370, DOI: 10.1216/rmj-2011-41-2-353.
A. Carvalho and C. M. A. Pinto, A delay fractional order model for the co-infection of malaria and HIV/AIDS, Int. J. Dyn. Control. 5(1) (2017), 168-186, DOI:10.1007/s40435-016-0224-3.
A. Cernea, On the existence of solutions for fractional differential inclusions with antiperiodic boundary conditions, J. Appl. Math. Comput. 38(1-2) (2012), 133-143, DOI: 10.1007/s12190-010-0468-6.
H. L. Chen, Antiperiodic wavelets, J. Comput. Math. 1996(14) (1996), 32-39.
F. Chen and Y. Zhou, Existence and Ulam stability of solutions for discrete fractional boundary value problem, Discrete Dyn. Nat. Soc. 2013 (2013), 7, DOI: 10.1155/2013/459161.
J. F. Cheng and Y. M. Chu, On the fractional difference equations of order (2,q), Abstr. Appl. Anal. 2011 (2011), 1-16, DOI: 10.1155/2011/497259.
J. F. Cheng and Y. M. Chu, Fractional difference equations with real variable, Abstr. Appl. Anal., 2012 (2012), 1-24, DOI: 10.1155/2012/918529.
J. B. Diaz and T. J. Osler, Differences of fractional order, Math. Comp. 28(125) (1974), 185-202, DOI: http://dx.doi.org/10.2307/2005825.
T. Dogruer and L. Nusret Tan, Lead and lag controller design in fractional-order control systems, Measurement and Control. 52(7-8) (2019), 1017-1028, DOI: 10.1177/0020294019858094.
L. Erbe, C. S. Goodrich, B. Jia and A. Peterson, Survey of the qualitative properties of rational difference operators: monotonicity, convexity and asymptotic behavior of solutions, Adv. Differ. Equ. 2016(1) (2016), DOI: 10.1186/s13662-016-0760-3.
M. Faieghi, S. Kuntanapreeda, H. Delavari and D. Baleanu, LMI - based stabilization of a class of fractional-order chaotic systems, Nonlinear Dyn. 72(1-2) (2013), 301-309, DOI: 10.1007/s11071-012-0714-6.
A. Georege Maria Selvam and R. Dhineshbabu, Hyers-Ulam stability results for discrete antiperiodic boundary value problem with fractional order (2
A. George Maria Selvam and R. Dhineshbabu, Existence and uniqueness of solutions for a discrete fractional boundary value problem, Int. J. Appl. Math. 33(2) (2020), 283-295, DOI: /10.12732/ijam.v33i2.7.
C. S. Goodrich, On a discrete fractional three-point boundary value problem, J. Differ. Equ. Appl. 18(3) (2012), 397-415, DOI: 10.1080/10236198.2010.503240.
C. S. Goodrich and A.C. Peterson, Discrete Fractional Calculus, Springer International Publishing, Switzerland, (2015), DOI: 10.1007/978-3-319-25562-0.
C. S. Goodrich, Existence and uniqueness of solutions to a fractional difference equation with nonlocal conditions, Comput. Math. Appl. 61(2) (2011), 191-202, DOI: /10.1016/j.camwa.2010.10.041.
B. Kuttner, On Differences of Fractional Order, Proceeding of the London Mathematical Society s3-7(1) (1957), 453-466, DOI: 10.1112/plms/s3-7.1.453.
K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, New York, USA, (1993).
A. K. Najeeb, A. R. Oyoon, S. P. Mondal and Q. Rubbab, Fractional order ecological system for complexities of interacting species with harvesting threshold in imprecise environment, Adv. Differ. Equ. 2019(1) (2019), DOI: 10.1186/s13662-019-2331-x.
Y. Pan, Z. Han, S. Sun and Z. Huang, The existence and uniqueness of solutions to boundary value problems of fractional difference equations, Math. Sci. 6(7) (2012), 1-15, DOI: 10.1186/2251-7456-6-7.
F. B. Pelap, G. B. Tanekou, C. F. Fogang and R. Kengne, Fractional-order stability analysis of earthquake dynamics , Journal of Geophysics and Engineering, 15(4) (2018), 1673-1687, DOI: 10.1088/1742-2140/aabe61.
I. Petras and R. L. Magin, Simulation of drug uptake in a two compartmental fractional model for a biological system, Commun. Nonlinear Sci. Numer. Simul. 16(12) (2011), 4588-4595, DOI: /10.1016/j.cnsns.2011.02.012.
R. Shi, T. Lu and C. Wang, Dynamic analysis of a fractional-order model for hepatitis B virus with Holling II functional response, Hindawi Complexity, 2019, 1-13, DOI: 10.1155/2019/1097201.
T. Sitthiwirattham, J. Tariboon and S. K. Ntouyas, Boundary value problems for fractional difference equations with three-point fractional sum boundary conditions, Adv. Differ. Equ. 2013(296) (2013) DOI:10.1186/1687-1847-2013-296.
I. M. Sokolov, J. Klafter and A. Blumen, Fractional kinetics, Phys. Today., 55(11) (2002), 48-54, DOI: 10.1063/1.1535007.
F. Song, Z. Yu and H. Yang, Modeling and analysis of fractional neutral disturbance waves in arterial vessels, Math. Model. Nat. Phenom. 14(3) (2019), 301, Doi: 10.1051/mmnp/2018072.
F. Wang and Z. Liu, Anti-periodic fractional boundary value problems for nonlinear differential equations of fractional order, Adv. Differ. Equ. 2012 (1) (2012), DOI: 10.1186/1687-1847-2012-116.
F. Zhang, G. Chen, C. Li, and J. Kurths, Chaos synchronization in fractional differential systems, Philos. Trans. R. Soc. A. 371(1990) (2013), 20120155, DOI: 10.1098/rsta.2012.0155.
Downloads
Published
How to Cite
Issue
Section
License
Authors who publish with this journal agree to the following terms:
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a CCAL that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work.
