Abstract
In this paper, we investigate some existence and Ulam’s type stability concepts of fixed point inclusions for a class of partial discontinuous fractional-order differential inclusions with impulses in Banach Algebras. Our results are obtained using weakly Picard operators theory.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Abbas, S., Agarwal, R.P., Benchohra, M.: Impulsive discontinuous partial hyperbolic differential equations of fractional order on Banach algebras. Electron. J. Differ. Equ. 2010(91), 17
Abbas S., Benchohra M.: Partial hyperbolic differential equations with finite delay involving the Caputo fractional derivative. Commun. Math. Anal. 7, 62–72 (2009)
Abbas S., Benchohra M.: Impulsive partial hyperbolic functional differential equations of fractional order with state-dependent delay. Frac. Calc. Appl. Anal. 13(3), 225–244 (2010)
Abbas S., Benchohra M.: Upper and lower solutions method for impulsive partial hyperbolic differential equations with fractional order. Nonlinear Anal. Hybrid Syst. 4, 604–613 (2010)
Abbas S., Benchohra M.: Impulsive partial hyperbolic differential inclusions of fractional order. Demonstratio Math. XLIII 4, 775–797 (2010)
Abbas S., Benchohra M., Gorniewicz L.: Fractional order impulsive partial hyperbolic differential inclusions with variable times. Discuss. Math. Differ. Incl. Control Optim. 31(1), 91–114 (2011)
Abbas, S., Benchohra, M., N’Guérékata, G.M.: Topics in Fractional Differential Equations. Springer, New York (2012)
Abbas S., Benchohra M., N’Guérékata G.M., Slimani B.A.: Darboux problem for fractional order discontinuous hyperbolic partial differential equations in Banach algebras. Complex Var. Elliptic Equ. 57(2–4), 337–350 (2012)
Bota-Boriceanu M.F., Petrusel A.: Ulam–Hyers stability for operatorial equations and inclusions. Anal. Univ. I. Cuza Iasi 57, 65–74 (2011)
Castro L.P., Ramos A.: Hyers–Ulam–Rassias stability for a class of Volterra integral equations. Banach J. Math. Anal. 3, 36–43 (2009)
Dhage B.C.: Existence results for neutral functional differential inclusions in Banach algebras. Nonlinear Anal. 64, 1290–1306 (2006)
Hilfer, R.: Applications of Fractional Calculus in Physics. World Scientific, Singapore (2000)
Hyers D.H.: On the stability of the linear functional equation. Proc. Natl. Acad. Sci. 27, 222–224 (1941)
Hyers, D.H., Isac, G., Rassias, T.H.M.: Stability of Functional Equations in Several Variables. Birkhauser, Boston (1998)
Jung, S.-M.: Hyers–Ulam–Rassias Stability of Functional Equations in Mathematical Analysis. Hadronic Press, Palm Harbor (2001)
Jung, S.-M.: Hyers–Ulam–Rassias Stability of Functional Equations in Nonlinear Analysis. Springer, New York (2011)
Jung, S.-M.: A fixed point approach to the stability of a Volterra integral equation. Fixed Point Theory Appl. 2007, 1 (2007) (article ID 57064)
Hu, S.H., Papageorgiou, N.: Handbook of Multivalued Analysis, Theory I. Kluwer, Dordrecht (1997)
Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, 204. Elsevier Science B.V., Amsterdam (2006)
Lakshmikantham, V., Bainov, D.D., Simeonov, P.S.: Theory of Impulsive Differential Equations. World Scientific, Singapore (1989)
Lasota A., Opial Z.: An application of the KakutaniKy Fan theorem in the theory of ordinary differential equations. Bull. Acad. Pol. Sci. Ser. Sci. Math. Astronom. Phys. 13, 781–786 (1965)
Li, X., Wang, J.: Ulam–Hyers–Rassias stability of semilinear differential equations with impulses. Electron. J. Differ. Equ. 2013(172), 8
Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Differential Equations. Wiley, New York (1993)
Petru T.P., Bota M.-F.: Ulam–Hyers stabillity of operational inclusions in complete gauge spaces. Fixed Point Theory 13, 641–650 (2012)
Petru T.P., Petrusel A., Yao J.-C.: Ulam0-Hyers stability for operatorial equations and inclusions via nonself operators. Taiwan. J. Math. 15, 2169–2193 (2011)
Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)
Rassias T.H.M.: On the stability of linear mappings in Banach spaces. Proc. Am. Math. Soc. 72, 297–300 (1978)
Rus I.A.: Ulam stability of ordinary differential equations. Stud. Univ. Babes-Bolyai, Math. LIV 4, 125–133 (2009)
Rus I.A.: Remarks on Ulam stability of the operatorial equations. Fixed Point Theory 10, 305–320 (2009)
Samoilenko, A.M., Perestyuk, N.A.: Impulsive Differential Equations. World Scientific, Singapore (1995)
Ulam, S.M.: A Collection of Mathematical Problems. Interscience Publishers, New York (1968)
Vityuk A.N., Golushkov A.V.: Existence of solutions of systems of partial differential equations of fractional order. Nonlinear Oscil. 7(3), 318–325 (2004)
Vityuk, A.N., Mykhailenko, A.V.: The Darboux problem for an implicit fractional-order differential equation. J. Math. Sci. 175(4), 391–401 (2011)
Wang J., Feckan M., Zhou Y.: Ulam’s type stability of impulsive ordinary differential equations. J. Math. Anal. Appl. 395, 258–264 (2012)
Wang, J., Lv, L., Zhou, Y.: Ulam stability and data dependence for fractional differential equations with Caputo derivative, Electron. J. Qual. Theory Differ. Equ. 2011(63), 10
Wang J., Lv L., Zhou Y.: New concepts and results in stability of fractional differential equations. Commun. Nonlinear Sci. Numer. Simul. 17, 2530–2538 (2012)
Wang J., Zhou Y., Feckan M.: Nonlinear impulsive problems for fractional differential equations and Ulam stability. Comput. Math. Appl. 64, 3389–3405 (2012)
Wei W., Li X., Li X.: New stability results for fractional integral equation. Comput. Math. Appl. 64, 3468–3476 (2012)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Abbas, S., Benchohra, M. Existence and Ulam Stability for Partial Impulsive Discontinuous Fractional Differential Inclusions in Banach Algebras. Mediterr. J. Math. 12, 1245–1264 (2015). https://doi.org/10.1007/s00009-014-0473-3
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00009-014-0473-3