A coupled system of fractional difference equations with anti-periodic boundary conditions

Jagan Mohan Jonnalagadda

Abstract


In this article, we propose sufficient conditions on existence, uniqueness and Ulam--Hyers stability of solutions for a coupled system of two-point nabla fractional difference boundary value problems associated with anti-periodic boundary conditions, using the approaches of Precup and Urs. We also illustrate the applicability of established results through examples.

Keywords


Nabla fractional difference; boundary value problem; anti-periodic boundary conditions; existence; uniqueness; Ulam--Hyers stability

Full Text:

PDF

References


bibitem{Ab 2} Abdeljawad, Thabet; Atıcı, Ferhan M. On the definitions of nabla fractional operators. Abstr. Appl. Anal. 2012 (2012), Art. ID 406757, 13 pp.

bibitem{Ah} Ahrendt, K.; Castle, L.; Holm, M.; Yochman, K. Laplace transforms for the nabla-difference operator and a fractional variation of parameters formula. Commun. Appl. Anal. 16 (2012), no. 3, 317--347.

bibitem{At} Atıcı, Ferhan M.; Eloe, Paul W. Discrete fractional calculus with the nabla operator. Electron. J. Qual. Theory Differ. Equ. 2009, Special Edition I, no. 3, 12 pp.

bibitem{Ch 1} Chen, C.; Bohner, M.; Jia, B. Ulam--Hyers stability of Caputo fractional difference equations. Math. Meth. Appl. Sci. 2019, 1--10.

bibitem{Ch 2} Chen, F.; Zhou, Y. Existence and Ulam stability of solutions for discrete fractional boundary value problem. Discrete Dynamics in Nature and Society 2013, Art. ID 459161, 7 pp.

bibitem{Du 1} Dutta, B. K.; Arora, L. K. Hyers--Ulam stability for a class of nonlinear fractional differential equations. Rev. Bull. Calcutta Math. Soc. 21 (2013), no. 1, 95--102.

bibitem{Kh 1} Khan, H.; Li, Y.; Chen, W.; Baleanu, D.; Khan, A. Existence theorems and Hyers--Ulam stability for a coupled system of fractional differential equations with $p$-Laplacian operator. Bound. Value Probl. 2017, Paper No. 157, 16.

bibitem{Gh} Gholami, Yousef; Ghanbari, Kazem. Coupled systems of fractional $nabla$-difference boundary value problems. Differ. Equ. Appl. 8 (2016), no. 4, 459--470.

bibitem{Go} Goodrich, Christopher; Peterson, Allan C. Discrete fractional calculus. Springer, Cham, 2015.

bibitem{He} Hein, J.; McCarthy, S.; Gaswick, N.; McKain, B.; Speer, K. Laplace transforms for the nabla difference operator. Panamer. Math. J. 21 (2011), 79--96.

bibitem{Hy} Hyers, D. H. On the stability of the linear functional equation. Proc. Nat. Acad. Sci. U. S. A. 27 (1941), 222--224.

bibitem{Ik 2} Ikram, Areeba; Lyapunov inequalities for nabla Caputo boundary value problems. J. Difference Equ. Appl. 25 (2019), no. 6, 757--775.

bibitem{Ja} Jagan Mohan, J. Hyers--Ulam stability of fractional nabla difference equations. Int. J. Anal. 2016, Art. ID 7265307, 5 pp.

bibitem{Ju F} Jung, S. M. Hyers--Ulam--Rassias stability of functional equations in nonlinear analysis, Springer, New York, 2011.

bibitem{Ke} Kelley, Walter G.; Peterson, Allan C. Difference equations. An introduction with applications. Second edition. Harcourt/Academic Press, San Diego, CA, 2001.

bibitem{Ki} Kilbas, Anatoly A.; Srivastava, Hari M.; Trujillo, Juan J. Theory and applications of fractional differential equations. North-Holland Mathematics Studies, 204. Elsevier Science B.V., Amsterdam, 2006.

bibitem{Po} Podlubny, Igor Fractional differential equations. Mathematics in Science and Engineering, 198. Academic Press, Inc., San Diego, CA, 1999.

bibitem{Pr} Precup, Radu The role of matrices that are convergent to zero in the study of semilinear operator systems. Math. Comput. Modelling 49 (2009), no. 3-4, 703--708.

bibitem{Ra} Rassias, T. M. On the stability of the linear mapping in Banach spaces. Proc. Amer. Math. Soc. 72 (1978). no. 2, 297--300.

bibitem{Ru 2} Rus, I. A. Ulam stability of ordinary differential equations, Studia Univ. ”Babes Bolyai” Mathematica. 54 (2009), 125--133.

bibitem{Ul} Ulam, S. A Collection of Mathematical Problems. Interscience Tracts in Pure and Applied Mathematics, no. 8. New York-London: Interscience Publishers, 1960.

bibitem{Urs} Urs, C. Coupled fixed point theorems and applications to periodic boundary value problems. Miskolc Mathematical Notes, 14 (2013), no. 1, 323--333.




DOI: http://dx.doi.org/10.24193/subbmath.2023.2.13

Refbacks

  • There are currently no refbacks.