Existence and Ulam stability results for Hadamard partial fractional integral inclusions via Picard operators
Dedicated to Professor Ioan A. Rus on the occasion of his 80th anniversary
Keywords:
Hadamard fractional integral inclusion, multivalued weekly Picard operator, fixed point inclusion, Ulam-Hyers stability.Abstract
In this paper, by using the weakly Picard operators theory, we investigate some existence and Ulam type stability results for a class of Hadamard partial fractional integral inclusions.
Mathematics Subject Classification (2010): 26A33, 34G20, 34A40, 45N05, 47H10.
References
Abbas, S., and Benchohra, M., Advanced Functional Evolution Equations and Inclusions, Developments in Mathematics, 39, Springer, New York, 2015.
Abbas, S., Benchohra, M., Henderson, J., Ulam stability for partial fractional integral inclusions via Picard operators, J. Frac. Calc. Appl., 5(2014), 133-144.
Abbas, S., Benchohra, M., N'Guerekata, G.M., Advanced Fractional Differential and Integral Equations, Nova Science Publishers, New York, 2015.
Abbas, S., Benchohra, M., N'Gu_er_ekata, G.M., Topics in Fractional Differential Equations, Developments in Mathematics, 27, Springer, New York, 2012.
Abbas, S., Benchohra, M., Petru_sel, A., Ulam stabilities for the Darboux problem for partial fractional differential inclusions via Picard operators, Electron. J. Qual. Theory Differ. Eq., 2014, no. 51, 13 pp.
Abbas, S., Benchohra, M., Sivasundaram, S., Ulam stability for partial fractional differential inclusions with multiple delay and impulses via Picard operators, J. Nonlinear Stud., 20(2013), no. 4, 623-641.
Abbas, S., Benchohra, M., Trujillo, J.J., Upper and lower solutions method for partial fractional differential inclusions with not instantaneous impulses, Prog. Frac. Diff. Appl., 1(2015), no. 1, 11-22.
Aubin, J.P., Frankowska, H., Set-Valued Analysis, Birkhauser, Basel, 1990.
Bota-Boriceanu, M.F., Petrusel, A., Ulam-Hyers stability for operatorial equations and inclusions, Analele Univ. I. Cuza Iasi, 57(2011), 65-74.
Butzer, P.L., Kilbas, A.A., Trujillo, J.J., Fractional calculus in the mellin setting and Hadamard-type fractional integrals, J. Math. Anal. Appl., 269(2002), 1-27.
Butzer, P.L., Kilbas, A.A., Trujillo, J.J., Mellin transform analysis and integration by parts for hadamard-type fractional integrals, J. Math. Anal. Appl., 270(2002), 1-15.
Castro, L.P., Ramos, A., Hyers-Ulam-Rassias stability for a class of Volterra integral equations, Banach J. Math. Anal., 3(2009), 36-43.
Castaing, C., Valadier, M., Convex Analysis and Measurable Multifunctions, Lecture Notes in Mathematics, 580, Springer-Verlag, Berlin-Heidelberg-New York, 1977.
Covitz, H., Nadler Jr., S.B., Multivalued contraction mappings in generalized metric spaces, Israel J. Math., 8(1970), 5-11.
Hadamard, J., Essai sur l'etude des fonctions donnees par leur developpment de Taylor, J. Pure Appl. Math., 4(8)(1892), 101-186.
Hilfer, R., Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000.
Hyers, D.H., On the stability of the linear functional equation, Proc. Nat. Acad. Sci., 27(1941), 222-224.
Hyers, D.H., Isac, G., Rassias, Th.M., Stability of Functional Equations in Several Variables, Birkhauser, 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(2007), Article ID 57064, 9 pages.
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.
Lazar, V.L., Fixed point theory for multivalued 'contractions, Fixed Point Theory Appl., 2011, 2011:50, 12 pp.
Miller, K.S., Ross, B., An Introduction to the Fractional Calculus and Differential Equations, John Wiley, New York, 1993.
Petru, T.P., Bota, M.F., Ulam-Hyers stabillity of operational inclusions in complete gauge spaces, Fixed Point Theory, 13(2012), 641-650.
Petru, T.P., Petrusel, A., Yao, J.C., Ulam-Hyers stability for operatorial equations and inclusions via nonself operators, Taiwanese J. Math., 15(2011), 2169-2193.
Podlubny, I., Fractional Differential Equations, Academic Press, San Diego, 1999.
Pooseh, S., Almeida, R., Torres, D., Expansion formulas in terms of integer-order derivatives for the Hadamard fractional integral and derivative, Numer. Funct. Anal. Optim., 33(2012), no. 3, 301-319.
Samko, S.G., Kilbas, A.A., Marichev, O.I., Fractional Integrals and Derivatives. Theory and Applications, Gordon and Breach, Yverdon, 1993.
Rassias, Th.M., On the stability of linear mappings in Banach spaces, Proc. Amer. Math. Soc., 72(1978), 297-300.
Rus, I.A., Ulam stability of ordinary differential equations, Studia Univ. Babes-Bolyai Math., 54(2009), no. 4, 125-133.
Rus, I.A., Remarks on Ulam stability of the operatorial equations, Fixed Point Theory, 10(2009), 305-320.
Rus, I.A., Fixed points, upper and lower fixed points: abstract Gronwall lemmas, Carpathian J. Math., 20(2004), 125-134.
Rus, I.A., Picard operators and applications, Sci. Math. Jpn., 58(2003), 191-219.
Rus, I.A., Generalized Contractions and Applications, Cluj University Press, Cluj-Napoca, 2001.
Rus, I.A., Petrusel, A., Stamarian, A., Data dependence of the fixed points set of some multivalued weakly Picard operators, Nonlinear Anal., 52(2003), 1947-1959.
Rybinski, L., On Carathedory type selections, Fund. Math., 125(1985), 187-193.
Tarasov, V.E., Fractional Dynamics. Applications of Fractional Calculus to Dynamics of Particles, Fields and Media, Springer, Heidelberg, 2010.
Ulam, S.M., A Collection of Mathematical Problems, Interscience Publ., New York, 1968.
Vityuk, A.N., Golushkov, A.V., Existence of solutions of systems of partial differential equations of fractional order, Nonlinear Oscil., 7(2004), no. 3, 318-325.
Wang, J., Lv, L., Zhou, Y., Ulam stability and data dependence for fractional differential equations with Caputo derivative, Electron. J. Qual. Theory Di_er. Eq., 2011, no. 63, 10
pp.
Wang, J., Lv, L., Zhou, Y., New concepts and results in stability of fractional differential equations, Commun. Nonlinear Sci. Numer. Simul., 17(2012), 2530-2538.
Wei, W., Li, X., Li, X., New stability results for fractional integral equation, Comput. Math. Appl., 64(2012), 3468-3476.
Wegrzyk, R., Fixed point theorems for multifunctions and their applications to functional equations, Dissertationes Math. (Rozprawy Mat.), 201(1982), 28 pp.
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