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Ulam-Hyers stability for the Darboux problem for partial fractional differential and integro-differential equations via Picard operators. (English) Zbl 1303.26006

Summary: In the present paper we investigate some uniqueness and Ulam’s type stability concepts of fixed point equations due to Rus, for the Darboux problem of partial differential and integro-differential equations involving the Caputo fractional derivative. Our results are obtained by using weakly Picard operators theory.

MSC:

26A33 Fractional derivatives and integrals
35R09 Integro-partial differential equations
47H10 Fixed-point theorems
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[1] Abbas, S., Baleanu, D., Benchohra, M.: Global attractivity for fractional order delay partial integro-differential equations. Adv. Difference Equ. 2012. doi: 10.1186/1687-1847-2012-62 · Zbl 1302.35392
[2] Abbas S., Benchohra M.: Darboux problem for perturbed partial differential equations of fractional order with finite delay. Nonlinear Anal. Hybrid Syst. 3, 597–604 (2009) · Zbl 1219.35345 · doi:10.1016/j.nahs.2009.05.001
[3] Abbas S., Benchohra M.: Fractional order partial hyperbolic differential equations involving Caputo’s derivative. Stud. Univ. Babeş-Bolyai Math. 57(4), 469–479 (2012) · Zbl 1289.26008
[4] Abbas S., Benchohra M., Cabada A.: Partial neutral functional integro-differential equations of fractional order with delay. Bound. Value Prob. 2012, 128 (2012) · Zbl 1278.26006 · doi:10.1186/1687-2770-2012-128
[5] Abbas S., Benchohra M., N’Guérékata G.M.: Topics in Fractional Differential Equations. Springer, New York (2012)
[6] Abbas S., Benchohra M., Vityuk A.N.: On fractional order derivatives and Darboux problem for implicit differential equations. Frac. Calc. Appl. Anal. 15, 168–182 (2012) · Zbl 1302.35395
[7] Abbas S., Benchohra M., Zhou Y.: Darboux problem for tractional order neutral functional partial hyperbolic differential equations. Int. J. Dyn. Syst. Differ. Equ. 2, 301–312 (2009) · Zbl 1186.35232
[8] Bota-Boriceanu M.F., Petrusel A.: Ulam–Hyers stability for operatorial equations and inclusions. Analele Univ. I. Cuza Iasi 57, 65–74 (2011) · Zbl 1265.54158
[9] Castro L.P., Ramos A.: Hyers-Ulam-Rassias stability for a class of Volterra integral equations. Banach J. Math. Anal. 3, 36–43 (2009) · Zbl 1177.45010 · doi:10.15352/bjma/1240336421
[10] Henry D.: Geometric Theory of Semilinear Parabolic Partial Differential Equations. Springer, Berlin (1989)
[11] Hilfer R.: Applications of Fractional Calculus in Physics. World Scientific, Singapore (2000) · Zbl 0998.26002
[12] Hyers D.H.: On the stability of the linear functional equation. Proc. Nat. Acad. Sci. 27, 222–224 (1941) · Zbl 0061.26403 · doi:10.1073/pnas.27.4.222
[13] Hyers D.H., Isac G., Rassias Th.M.: Stability of Functional Equations in Several Variables. Birkhauser, Boston (1998) · Zbl 0907.39025
[14] Jung S.-M.: Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis. Hadronic Press, Palm Harbor (2001) · Zbl 0980.39024
[15] Jung S.-M.: Hyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analysis. Springer, New York (2011) · Zbl 1221.39038
[16] Jung, S.-M.: A fixed point approach to the stability of a Volterra integral equation. Fixed Point Theory Appl. 2007, Article ID 57064. (2007)
[17] Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, vol. 204. Elsevier Science B.V., Amsterdam (2006) · Zbl 1092.45003
[18] Miller K.S., Ross B.: An Introduction to the Fractional Calculus and Differential Equations. Wiley, New York (1993) · Zbl 0789.26002
[19] Petru T.P., Bota M.-F.: Ulam–Hyers stabillity of operational inclusions in complete gauge spaces. Fixed Point Theory 13, 641–650 (2012) · Zbl 1353.54047
[20] Petru T.P., Petrusel A., Yao J.-C.: Ulam–Hyers stability for operatorial equations and inclusions via nonself operators. Taiwanese J. Math. 15, 2169–2193 (2011)
[21] Podlubny I.: Fractional Differential Equations. Academic Press, San Diego (1999) · Zbl 0924.34008
[22] Rassias Th.M.: On the stability of linear mappings in Banach spaces. Proc. Am. Math. Soc. 72, 297–300 (1978) · Zbl 0398.47040 · doi:10.1090/S0002-9939-1978-0507327-1
[23] Rus I.A.: Ulam stability of ordinary differential equations. Studia Univ. Babes-Bolyai, Math. LIV(4), 125–133 (2009) · Zbl 1224.34165
[24] Rus I.A.: Remarks on Ulam stability of the operatorial equations. Fixed Point Theory 10, 305–320 (2009) · Zbl 1204.47071
[25] Rus I.A.: Fixed points, upper and lower fixed points: abstract Gronwall lemmas. Carpathian J. Math. 20(1), 125–134 (2004) · Zbl 1113.54304
[26] Rus I.A.: Picard operators and applications. Scienticae Mathematicae Japonocae 58(1), 191–219 (2003) · Zbl 1031.47035
[27] Rus I.A.: Generalized Contractions and Applications. Cluj University Press, Cluj-Napoca (2001) · Zbl 0968.54029
[28] Rus I.A.: Weakly Picard operators and applications. Seminar on Fixed Point Theory, Cluj-Napoca 2, 41–58 (2001) · Zbl 1035.47044
[29] Tarasov V.E.: Fractional Dynamics. Applications of Fractional Calculus to Dynamics of Particles, Fields and Media. Springer, Heidelberg (2010) · Zbl 1214.81004
[30] Ulam S.M.: A Collection of Mathematical Problems. Interscience Publishers, New York (1968) · Zbl 0184.14802
[31] 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) · Zbl 1092.35500 · doi:10.1007/s11072-005-0015-9
[32] Wang J., Lv L., Zhou Y.: Ulam stability and data dependence for fractional differential equations with Caputo derivative. Electron. J. Qual. Theory Differ. Equ. 63, 1–10 (2011) · Zbl 1340.34034 · doi:10.1155/2011/783726
[33] 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) · Zbl 1252.35276 · doi:10.1016/j.cnsns.2011.09.030
[34] Wei W., Li X., Li X.: New stability results for fractional integral equation. Comput. Math. Appl. 64, 3468–3476 (2012) · Zbl 1268.45007 · doi:10.1016/j.camwa.2012.02.057
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