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Ulam stabilities for the Darboux problem for partial fractional differential inclusions

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Języki publikacji
EN
Abstrakty
EN
In this article, we investigate some Ulam’s type stability concepts for the Darboux problem of partial fractional differential inclusions with a nonconvex valued right hand side. Our results are based upon Covitz-Nadler fixed point theorem and fractional version of Gronwall’s inequality.
Wydawca
Rocznik
Strony
826--838
Opis fizyczny
Bibliogr. 37 poz.
Twórcy
autor
  • 2320, Rue De Salaberry, Apt 10 Montréal, QC H3M 1K9, Canada
autor
  • Laboratoire de Mathématiques, Université de Sidi Bel-Abbès, B.P. 89, 22000, Sidi Bel-Abbès, Algérie
  • Department of Mathematics, Faculty of Scienc,e King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
Bibliografia
  • [1] S. Abbas, D. Baleanu, M. Benchohra, Global attractivity for fractional order delay partial integro-differential equations, Adv. Difference Equ. 2012, 19 pages. doi:10.1186/1687-1847-2012-62
  • [2] S. Abbas, M. Benchohra, Darboux problem for perturbed partial differential equations of fractional order with finite delay, Nonlinear Anal. Hybrid Syst. 3 (2009), 597–604.
  • [3] S. Abbas, M. Benchohra, Fractional order partial hyperbolic differential equations involving Caputo’s derivative, Stud. Univ. Babes-Bolyai Math. 57(4) (2012), 469–479.
  • [4] S. Abbas, M. Benchohra, A. Cabada, Partial neutral functional integro-differential equations of fractional order with delay, Bound. Value Prob. 128 (2012), 15 pp.
  • [5] S. Abbas, M. Benchohra, J. Henderson, Asymptotic attractive nonlinear fractional order Riemann–Liouville integral equations in Banach algebras, Nonlinear Stud. 20(1) (2013), 1–10.
  • [6] S. Abbas, M. Benchohra, G. M. N’Guérékata, Topics in Fractional Differential Equations, Springer, New York, 2012.
  • [7] S. Abbas, M. Benchohra, A. N. Vityuk, On fractional order derivatives and Darboux problem for implicit differential equations, Fract. Calc. Appl. Anal. 15 (2012), 168–182.
  • [8] J. P. Aubin, A. Cellina, Differential Inclusions, Springer-Verlag, Berlin-Heidelberg, New York, 1984.
  • [9] M. F. Bota-Boriceanu, A. Petrusel, Ulam–Hyers stability for operatorial equations and inclusions, An. Stiint. Univ. Al. I. Cuza Iasi Mat. 57 (2011), 65–74.
  • [10] L. Gorniewicz, Topological Fixed Point Theory of Multivalued Mappings, Mathematics and its Applications, 495, Kluwer Academic Publishers, Dordrecht, 1999.
  • [11] L. P. Castro, A. Ramos, Hyers–Ulam–Rassias stability for a class of Volterra integral equations, Banach J. Math. Anal. 3 (2009), 36–43.
  • [12] C. Castaing, M. Valadier, Convex Analysis and Measurable Multifunctions, Lecture Notes in Mathematics 580, Springer-Verlag, Berlin-Heidelberg-New York, 1977.
  • [13] H. Covitz, S. B. Nadler Jr., Multivalued contraction mappings in generalized metric spaces, Israel J. Math. 8 (1970), 5–11.
  • [14] D. Henry, Geometric Theory of Semilinear Parabolic Partial Differential Equations, Springer-Verlag, Berlin-New York, 1989.
  • [15] R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000.
  • [16] Sh. Hu, N. Papageorgiou, Handbook of Multivalued Analysis, Volume I: Theory, Kluwer, Dordrecht, Boston, London, 1997.
  • [17] D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. 27 (1941), 222–224.
  • [18] D. H. Hyers, G. Isac, Th. M. Rassias, Stability of Functional Equations in Several Variables, Birkhauser, 1998.
  • [19] S.-M. Jung, Hyers–Ulam–Rassias Stability of Functional Equations in Mathematical Analysis, Hadronic Press, Palm Harbor, 2001.
  • [20] S.-M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analysis, Springer, New York, 2011.
  • [21] S.-M. Jung, A fixed point approach to the stability of a Volterra integral equation, Fixed Point Theory Appl. (2007), Article ID 57064, 9 pages.
  • [22] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204, Elsevier Science B.V., Amsterdam, 2006.
  • [23] M. Kisielewicz, Differential Inclusions and Optimal Control, Kluwer, Dordrecht, The Netherlands, 1991.
  • [24] K. S. Miller, B. Ross, An Introduction to the Fractional Calculus and Differential Equations, John Wiley, New York, 1993.
  • [25] M. D. Ortigueira, Fractional Calculus for Scientists and Engineers, Lecture Notes in Electrical Engineering, 84, Springer, Dordrecht, 2011.
  • [26] T. P. Petru, M.-F. Bota, Ulam–Hyers stabillity of operational inclusions in complete gauge spaces, Fixed Point Theory 13 (2012), 641–650.
  • [27] T. P. Petru, A. Petrusel. J.-C. Yao, Ulam–Hyers stability for operatorial equations and inclusions via nonself operators, Taiwanese J. Math. 15 (2011), 2169–2193.
  • [28] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
  • [29] Th. M. Rassias, On the stability of linear mappings in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297–300.
  • [30] I. A. Rus, Ulam stability of ordinary differential equations, Studia Univ. Babes-Bolyai Math. 54(4) (2009), 125–133.
  • [31] I. A. Rus, Remarks on Ulam stability of the operatorial equations, Fixed Point Theory 10 (2009), 305–320.
  • [32] V. E. Tarasov, Fractional Dynamics, Applications of Fractional Calculus to Dynamics of Particles, Fields and Media, Springer, Heidelberg, 2010.
  • [33] S. M. Ulam, A Collection of Mathematical Problems, Interscience Publishers, New York, 1968.
  • [34] A. N. Vityuk, A. V. Golushkov, Existence of solutions of systems of partial differential equations of fractional order, Nonlinear Oscil. 7(3) (2004), 318–325.
  • [35] J. Wang, L. Lv, Y. Zhou, Ulam stability and data dependence for fractional differential equations with Caputo derivative, Electron. J. Qual. Theory Differ. Equ. 63 (2011), 1–10.
  • [36] J. Wang, L. Lv, Y. Zhou, New concepts and results in stability of fractional differential equations, Commun. Nonlinear Sci. Numer. Simul. 17 (2012), 2530–2538.
  • [37] W. Wei, X. Li, X. Li, New stability results for fractional integral equation, Comput. Math. Appl. 64 (2012), 3468–3476.
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Bibliografia
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