Oscillation analysis of a forced fractional order‎ ‎sum-difference equations

Volume 37, Issue 2, pp 214--225 https://dx.doi.org/10.22436/jmcs.037.02.06

Publication Date: September 25, 2024 Submission Date: March 17, 2024 Revision Date: May 06, 2024 Accteptance Date: August 07, 2024

Download PDF

Download XML

• Export citations
  • CITATIONS & REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • ARTICLE CITATION
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)

• 97 Downloads
• 317 Views

Authors

J. Alzabut - Department of Mathematics and Sciences, ‎Prince Sultan University, ‎11586 Riyadh, ‎Saudi Arabia. - Department of Industrial Engineering, OSTIM Technical University, Ankara 06374, Türkiye. A‎. George Maria Selvam - Department of Mathematics, ‎Sacred Heart College, ‎Tirupattur-635601, ‎Tamil Nadu, India. R. ‎Janagaraj - Department of Mathematics‎, ‎Faculty of Engineering, ‎Karpagam Academy of Higher Education, ‎Coimbatore-641021, ‎Tamil Nadu, India.

Abstract

‎This paper contributes some new outcomes about the oscillation of a forced fractional order sum-difference equation of the form‎ ‎\[\Delta^\beta y(\iota)+\sum\limits_{\varkappa=a}^{\iota-1}\mathbb{T}(\iota,\varkappa)\varPsi(\varkappa,y(\varkappa))=\varepsilon(\iota),\ 0<\beta<1,\ \iota\in\mathbb{N}_a,\]‎ ‎with $\Delta^{\beta-1}y(0)=y_0\in\mathbb{R}$‎. ‎Here $\mathbb{T},\varPsi‎ , ‎\varepsilon$ are well-defined functions along with continuity and $\Delta^\beta$ and $\Delta^{\beta-1}$ represent the Riemann-Liouville (R-L) fractional order difference and sum operators‎, ‎respectively‎. ‎Suitable examples are delivered to clarify the strength of the theoretical consequences‎.

Share and Cite

Keywords

• Fractional difference equations
• oscillation
• Riemann-Liouville fractional difference
• Caputo fractional difference
• forcing term

MSC

•  26A33
•  39A10
•  39A13
•  39A21

References

• [1] B. Abdalla, K. Abodayeh, T. Abdeljawad, J. Alzabut, New oscillation criteria for forced nonlinear fractional difference equations, Vietnam J. Math., 45 (2017), 609–618
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [2] T. Abdeljawad, On Riemann and Caputo fractional differences, Comput. Math. Appl., 62 (2011), 1602–1611
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [3] S. Ahmed, A. T. Azar, M. Tounsi, I. K. Ibraheem, Adaptive Control Design for Euler–Lagrange Systems Using Fixed- Time Fractional Integral Sliding Mode Scheme, Fractal Fract., 7 (2023), 1–20
  • View Article
  • Google Scholar


• [4] J. O. Alzabut, T. Abdeljawad, Sufficient conditions for the oscillation of nonlinear fractional difference equations, J. Fract. Calc. Appl., 5 (2014), 177–187
  • View Article
  • Google Scholar
  • MATH


• [5] J. Alzabut, T. Abdeljawad, H. Alrabaiah, Oscillation criteria for forced and damped nable fractional difference equations, J. Comput. Anal. Appl., 24 (2018), 1387–1394
  • View Article
  • Google Scholar


• [6] J. Alzabut, V. Muthulakshmi, A. Özbekler, H. Adıgüzel, On the Oscillation of nonlinear fractional difference equations with damping, Mathematics, 7 (2019), 1–14
  • View Article
  • Google Scholar


• [7] F. M. Atici, P. W. Eloe, A transform method in discrete fractional calculus, Int. J. Difference Equ., 2 (2007), 165–176
  • View Article
  • Google Scholar


• [8] F. M. Atıcı, P. W. Eloe, Discrete fractional calculus with the nabla operator, Electron. J. Qual. Theory Differ. Equ., 2009 (2009), 12 pages
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [9] F. M. Atici, P.W. Eloe, Initial value problems in discrete fractional calculus, Proc. Amer. Math. Soc., 137 (2009), 981–989
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [10] F. M. Atıcı, S. ¸Sengül, Modeling with fractional difference equations, J. Math. Anal. Appl., 369 (2010), 1–9
  • View Article
  • MathSciNet
  • Google Scholar


• [11] D. Baleanu, K. Diethelm, E. Scalas, J. J. Trujillo, Fractional Calculus Models and Numerical Methods, World Scientific, New Jersey (2012)
  • View Article
  • Google Scholar


• [12] N. R. O. Bastos, D. F. M. Torres, Combine Delta-Nabla sum operator in discrete fractional calculus, Commun. Frac. Calc., 1 (2010), 41–47
  • View Article
  • Google Scholar


• [13] G. E. Chatzarakis, A. G. M. Selvam, R. Janagaraj, M. Douka, Oscillation theorems for certain forced nonlinear discrete fractional order equations, Commun. Math. Appl., 10 (2019), 763–772
  • View Article
  • Google Scholar


• [14] G. E. Chatzarakis, G. M. Selvam, R. Janagaraj, G. N. Miliaras, Oscillation criteria for a class of nonlinear discrete fractional order equations with damping term, Math. Slovaca, 70 (2020), 1165–1182
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [15] J. B. Díaz, T. J. Osler, Differences of fractional order, Math. Comp., 28 (1974), 185–202
  • View Article
  • MathSciNet
  • Google Scholar


• [16] A. Dzieli ´ nski, D. Sierociuk, Fractional Order Model of Beam Heating Process and Its Experimental Verification, Springer, New York (2010)
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [17] S. N. Elaydi, An introduction to difference equations, Springer-Verlag, New York (1996)
  • View Article
  • MathSciNet


• [18] G. H. Hardy, J. E. Littlewood, G Pólya, Inequalities, Cambridge University Press, Cambridge (1952)
  • View Article
  • Google Scholar


• [19] M. T. Holm, The theory of discrete fractional calculus: Development and application, ProQuest LLC, Ann Arbor, MI (2011)
  • View Article
  • MathSciNet
  • Google Scholar


• [20] Z. A. Khan, F. Jarad , A. Khan, H. Khan, Nonlinear discrete fractional sum inequalities related to the theory of discrete fractional calculus with applications, Adv. Differ. Equ., 2021 (2021),
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [21] B. Kuttner, On differences of fractional order, Proc. London Math. Soc. (3), 7 (1957), 453–466
  • View Article
  • MathSciNet
  • Google Scholar


• [22] W. Li, W. Sheng, P. Zhang, Oscillatory properties of certain nonlinear fractional nabla difference equations, J. Appl. Anal. Comput., 8 (2018), 1910–1918
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [23] I. Podlubny, Fractional differential equations, Academic Press, San Diego, CA (1999)
  • View Article
  • Google Scholar


• [24] A. G. M. Selvam, J. Alzabut, R. Janagaraj, H. Adiguzel, Oscillation analysis for nonlinear discrete fractional order delay and neutral equations with forcing term, Dyn. Syst. Appl., 29 (2020), 327–342
  • View Article
  • Google Scholar


• [25] A. G. M. Selvam, M. Jacintha, R. Janagaraj, Oscillation theorems for a class of forced nonlinear discrete equations of fractional order with damping term, AIP Conf. Proc., 2246 (2020), 1–7
  • View Article
  • Google Scholar


• [26] A. G. M. Selvam, M. Jacintha, R. Janagaraj, Existence of nonoscillatory solutions of nonlinear neutral delay difference equation of fractional order, Adv. Math.: Sci. J., 9 (2020), 4971–4983
  • View Article
  • Google Scholar


• [27] A. G. M. Selvam, R. Janagaraj, Oscillation theorems for damped fractional order difference equations, AIP Conf. Proc., 2095 (2019), 1–7
  • View Article
  • Google Scholar


• [28] A. G. M. Selvam, R. Janagaraj, Oscillation criteria of a class of fractional order damped difference equations, Int. J. Appl. Math., 32 (2019), 433–441
  • View Article
  • Google Scholar


• [29] A. G. M. Selvam, R. Janagaraj, New oscillation criteria for discrete fractional order forced nonlinear equations, J. Phys.: Conf. Ser., 1597 (2020), 1–8
  • View Article
  • Google Scholar