Existence and Hyers-Ulam stability of solutions to the implicit second-order differential equation via fractional integral boundary conditions

Volume 31, Issue 1, pp 15--29 http://dx.doi.org/10.22436/jmcs.031.01.02

Publication Date: April 04, 2023 Submission Date: July 06, 2022 Revision Date: December 31, 2022 Accteptance Date: February 07, 2023

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)

• 611 Downloads
• 1690 Views

Authors

Sh. M Al-Issa - Faculty of Arts and Sciences, Department of Mathematics, Lebanese International University, Saida, Lebanon. - Faculty of Arts and Sciences, Department of Mathematics, The International University of Beirut, Beirut, Lebanon. I. H. Kaddoura - Faculty of Arts and Sciences, Department of Mathematics, Lebanese International University, Saida, Lebanon. - Faculty of Arts and Sciences, Department of Mathematics, The International University of Beirut, Beirut, Lebanon. N. J. Rifai - Faculty of Arts and Sciences, Department of Mathematics, Lebanese International University, Saida, Lebanon.

Abstract

In this paper, the existence and Ulam-Hyers stability of solutions for implicit second order fractional differential equations are investigated via fractional-orders integral boundary conditions. Our results are based on Krasnoselskii's fixed point Theorem and Banach contraction principle. We provide examples at the end to clarify our acquired outcomes..

Share and Cite

Keywords

• \(\phi\)-Caputo fractional order
• existence results
• Green's function
• boundary value problems
• Ulam-Hyers stability

MSC

•  26A33
•  34A60
•  45G05

References

• [1] S. Abbas, M. Benchohra, On the generalized Ulam-Hyers-Rassias stability for Darboux problem for partial fractional implicit differential equations, Appl. Math. E-Notes, 14 (2014), 20–28
  • View Article
  • Google Scholar


• [2] S. Abbas, M. Benchohra, G. M. N’Gu´er´ekata, Topics in Fractional Differential Equations, Springer, New York (2012)
  • View Article
  • MathSciNet
  • Google Scholar


• [3] O. P. Agrawal, Some generalized fractional calculus operators and their applications in integral equations, Fract. Calc. Appl. Anal., 15 (2012), 700–711
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [4] R. P. Agarwal, S. Arshad, D. O’Regan, V. Lupulescu, Fuzzy fractional integral equations under compactness type condition, Fract. Calc. Appl. Anal., 15 (2012), 572–590
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [5] R. P. Agarwal, M. Benchohra, S. Hamani, Boundary value problems for fractional differential equations, Georg. Math. J., 16 (2009), 401-411
  • View Article
  • MathSciNet
  • Google Scholar


• [6] Sh. M. Al-Issa, A. M. A. El-Sayed, Y. M. Y. Omar, On some characterizations of the solutions of a hybrid nonlinear functional integral inclusion and applications, J. Math. Comput. Sci., 28 (2023), 21–36
  • View Article
  • Google Scholar


• [7] Sh. M. Al-Issa, N. M. Mawed, Results on solvability of nonlinear quadratic integral equations of fractional orders in Banach algebra, J. Nonlinear Sci. Appl., 14 (2021), 181–195
  • View Article
  • MathSciNet
  • Google Scholar


• [8] R. Almeida, A Caputo fractional derivative of a function with respect to another function, Commun. Nonlinear Sci. Numer. Simul., 44 (2017), 460–481
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [9] T. A. Burton, C. Kirk, A fixed point theorem of Krasnoselskii Schaefer type, Math. Nachr., 189 (1998), 23–31
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [10] A. Babakhani, V. Daftardar-Gejji, Existence of positive solutions for multi-term non-autonomous fractional differential equations with polynomial coefficients, Electron. J. Differ. Equ., 2006 (2006), 1–12
  • View Article
  • Google Scholar
  • MATH


• [11] M. Belmekki, M. Benchohra, Existence results for fractional order semilinear functional differential equations, Proc. A. Razmadze Math. Inst., 146 (2008), 9–20
  • View Article
  • Google Scholar


• [12] M. Benchohra, J. R. Graef, S. Hamani, Existence results for boundary value problems with nonlinear fractional differential equations, Appl. Anal., 87 (2008), 851–863
  • View Article
  • Google Scholar


• [13] M. Benchohra, S. Hamani, S. K. Ntouyas, Boundary value problems for differential equations with fractional order, Surv. Math. Appl., 3 (2008), 1–12
  • View Article
  • Google Scholar


• [14] M. Benchohra, J. Henderson, S. K. Ntouyas, A. Ouahab, Existence results for fractional order functional differential equations with infinite delay, J. Math. Anal. Appl., 338 (2008), 1340–1350
  • View Article
  • MathSciNet
  • Google Scholar


• [15] S. Chasreechai, J. Tariboon, Positive solutions to generalized second-order three-point integral boundary-value problems, Electron. J. Differ. Equ., 2011 (2011), 14 pages
  • View Article
  • Google Scholar
  • MATH


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


• [17] A. M. A. El-Sayed, Nonlinear functional differential equations of arbitrary orders, Nonlinear Anal., 33 (1998), 181–186
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [18] A. M. A. El-Sayed, Sh. M Al-Issa, Existence of integrable solutions for integro-differential inclusions of fractional order; coupled system approach, J. Nonlinear Sci. Appl., 13 (2020), 180–186
  • View Article
  • MathSciNet
  • Google Scholar


• [19] A. M. A. El-Sayed, Sh. M. Al-Issa, M. H. Hijazi, Existence results for a functional integro-differential inclusions with Riemann-Stieltjes integral or infinite-point boundary conditions, Surv. Math. Appl., 16 (2021), 301–325
  • View Article
  • Google Scholar


• [20] A. El-Sayed, H. Hashem, Sh. Al-Issa, Analysis of a hybrid integro-differential inclusion, Bound. Value Probl., 2022 (2022), 1–21
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [21] M. Hu, L. Wang, Existence of solutions for a nonlinear fractional differential equation with integral boundary condition, Int. J. Math. Comput. Sci., 5 (2011), 55–58
  • View Article
  • Google Scholar


• [22] S.-M. Jung, K.-S. Lee, Hyers-Ulam stability of first order linear partial differential equations with constant coeficients, Math. Inequal. Appl., 10 (2007), 261–266
  • View Article
  • Google Scholar


• [23] A. A. Kilbas, S. A. Marzan, Nonlinear differential equations with the Caputo fractional derivative in the space of continuously differentiable functions, Differ. Equ., 41 (2005), 84–89
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [24] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier Science B.V., Amsterdam (2006)
  • View Article
  • Google Scholar


• [25] V. Kiryakova, Generalized Fractional Calculus and Applications, J. Wiley & Sons Inc., New York (1994)
  • View Article
  • Google Scholar


• [26] V. Lakshmikantham, S. Leela, J. V. Devi, Theory of Fractional Dynamic Systems, Cambridge Academic Publishers, (2009)
  • View Article
  • Google Scholar


• [27] S. A. Murad, S. Hadid, Existence and uniqueness theorem for fractional differential equation with integral boundary condition, J. Frac. Calc. Appl., 3 (2012), 1–9
  • View Article
  • Google Scholar
  • MATH


• [28] M. Obłoza, Hyers stability of the linear differential equation, Rocznik Nauk.-Dydakt. Prace Mat., 13 (1993), 259–270
  • View Article
  • Google Scholar


• [29] I. Podlubny, A. M. A. EL-Sayed, On two defintions of fractional calculus, Preprint UEF 03-69, Solvak Academy of science-Institute of Experimental phys., (1996),
  • View Article
  • Google Scholar


• [30] S. G. Samko, A. A. Kilbas, O. I. Mariche, Fractional integrals and derivatives, Gordon and Breach Science Publishers, Yverdon (1993)
  • View Article
  • Google Scholar


• [31] J. R. L. Webb, G. Infante, Positive solutions of nonlocal boundary value problems: a unified approach, J. London Math. Soc., 74 (2006), 673–693
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH