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Keywords:
fractional differential equations; impulse; Caputo fractional order derivative; left-sided mixed Riemann–Liouville integral; Darboux problem; Ulam stability
fractional differential equations; impulse; Caputo fractional order derivative; left-sided mixed Riemann–Liouville integral; Darboux problem; Ulam stability
Summary:
In this paper we investigate the existence of solutions for the initial value problems (IVP for short), for a class of implicit impulsive hyperbolic differential equations by using the lower and upper solutions method combined with Schauder’s fixed point theorem.
References:
[1] Abbas, S., Baleanu, D., Benchohra, M.: Global attractivity for fractional order delay partial integro-differential equations. Adv. Difference Equ. 2012, 62 doi:10.1186/1687-1847-2012-62 (2012), 1–10, online. DOI 10.1186/1687-1847-2012-62 | MR 2958362 | 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 (2009), 597–604. MR 2561676 | Zbl 1219.35345
[3] Abbas, S., Benchohra, M.: Fractional order partial hyperbolic differential equations involving Caputo’s derivative. Stud. Univ. Babeş-Bolyai Math. 57, 4 (2012), 469–479. MR 3034096 | Zbl 1289.26008
[4] Abbas, S., Benchohra, M.: Upper and lower solutions method for Darboux problem for fractional order implicit impulsive partial hyperbolic differential equations. Acta Univ. Palacki. Olomuc., Math. 51, 2 (2012), 5–18. MR 3058869 | Zbl 1302.35393
[5] Abbas, S., Benchohra, M., Cabada, A.: Partial neutral functional integro-differential equations of fractional order with delay. Bound. Value Prob. 2012, 128 (2012), 1–15. MR 3016041 | Zbl 1278.26006
[6] Abbas, S., Benchohra, M., Górniewicz, L.: Existence theory for impulsive partial hyperbolic functional differential equations involving the Caputo fractional derivative. Sci. Math. Jpn. e-2010 (2010), 271–282, online. MR 2666846 | Zbl 1200.26004
[7] Abbas, S., Benchohra, M., Henderson, J.: Asymptotic attractive nonlinear fractional order Riemann-Liouville integral equations in Banach algebras. Nonlinear Studies 20, 1 (2013), 1–10. MR 3058403 | Zbl 1305.45005
[8] Abbas, S., Benchohra, M., N’Guérékata, G. M.: Topics in Fractional Differential Equations. Developments in Mathematics 27, Springer, New York, 2012. MR 2962045 | Zbl 1273.35001
[9] Abbas, S., Benchohra, M., Vityuk, A. N.: On fractional order derivatives and Darboux problem for implicit differential equations. Fract. Calc. Appl. Anal. 15, 2 (2012), 168–182. DOI 10.2478/s13540-012-0012-5 | MR 2897771 | Zbl 1302.35395
[10] Abbas, S., Benchohra, M., Zhou, Y.: Darboux problem for tractional order neutral functional partial hyperbolic differential equations. Int. J. Dynam. Syst. Differ. Equa. 2 (2009), 301–312. MR 2583101
[11] Ahmad, B., Nieto, J. J.: Riemann-Liouville fractional differential equations with fractional boundary conditions. Fixed Point Theory 13 (2012), 329–336. MR 3024321 | Zbl 1315.34006
[12] Baleanu, D., Diethelm, K., Scalas, E., Trujillo, J. J.: Fractional Calculus Models and Numerical Methods. World Scientific Publishing, New York, 2012. MR 2894576 | Zbl 1248.26011
[13] Benchohra, M., Graef, J. R., Hamani, S.: Existence results for boundary value problems of nonlinear fractional differential equations with integral conditions. Appl. Anal. 87, 7 (2008), 851–863. DOI 10.1080/00036810802307579 | MR 2458962
[14] Bota-Boriceanu, M. F., Petrusel, A.: Ulam–Hyers stability for operatorial equations and inclusions. Analele Univ. I. Cuza Iasi 57 (2011), 65–74. MR 2933569
[15] Cabada, A., Staněk, S.: Functional fractional boundary value problems with singular $\phi $-Laplacian. Appl. Math. Comput. 219 (2012), 1383–1390. DOI 10.1016/j.amc.2012.07.062 | MR 2983850 | Zbl 1296.34013
[16] Castro, L. P., Ramos, A.: Hyers–Ulam–Rassias stability for a class of Volterra integral equations. Banach J. Math. Anal. 3 (2009), 36–43. DOI 10.15352/bjma/1240336421 | MR 2461744
[17] Henry, D.: Geometric theory of Semilinear Parabolic Partial Differential Equations. Springer-Verlag, Berlin–New York, 1989.
[18] Hilfer, R., R.: Applications of Fractional Calculus in Physics. World Scientific, Singapore, 2000. MR 1890104 | Zbl 0998.26002
[19] Hyers, D. H.: On the stability of the linear functional equation. Proc. Nat. Acad. Sci. 27 (1941), 222–224. DOI 10.1073/pnas.27.4.222 | MR 0004076 | Zbl 0061.26403
[20] Hyers, D. H., Isac, G., Rassias, Th. M.: Stability of Functional Equations in Several Variables. Birkhäuser, Basel, 1998. MR 1639801 | Zbl 0907.39025
[21] Jung, S.-M.: Hyers–Ulam–Rassias Stability of Functional Equations in Nonlinear Analysis. Springer, New York, 2011. MR 2790773 | Zbl 1221.39038
[22] Jung, S.-M.: A fixed point approach to the stability of a Volterra integral equation. Fixed Point Theory Appl. 2007, Article ID 57064 (2007), 1–9. MR 2318689 | Zbl 1155.45005
[23] Kilbas, A. A., Marzan, S. A.: Nonlinear differential equations with the Caputo fractional derivative in the space of continuously differentiable functions. Differential Equations 41 (2005), 84–89. DOI 10.1007/s10625-005-0137-y | MR 2213269 | Zbl 1160.34301
[24] 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. MR 2218073 | Zbl 1092.45003
[25] Ortigueira, M. D.: Fractional Calculus for Scientists and Engineers. Lecture Notes in Electrical Engineering 84, Springer, Dordrecht, 2011. DOI 10.1007/978-94-007-0747-4 | MR 2768178 | Zbl 1251.26005
[26] Petru, T. P., Bota, M.-F.: Ulam-Hyers stabillity of operational inclusions in complete gauge spaces. Fixed Point Theory 13 (2012), 641–650. MR 3024346
[27] 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. MR 2880400 | Zbl 1246.54049
[28] Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego, 1999. MR 1658022 | Zbl 0924.34008
[29] Rassias, Th. M.: On the stability of linear mappings in Banach spaces. Proc. Amer. Math. Soc. 72 (1978), 297–300. DOI 10.1090/S0002-9939-1978-0507327-1 | MR 0507327
[30] Rus, I. A.: Ulam stability of ordinary differential equations. Studia Univ. Babes-Bolyai, Math. 54, 4 (2009), 125–133. MR 2602351 | Zbl 1224.34165
[31] Rus, I. A.: Remarks on Ulam stability of the operatorial equations. Fixed Point Theory 10 (2009), 305–320. MR 2569004 | Zbl 1204.47071
[32] Staněk, S.: Limit properties of positive solutions of fractional boundary value problems. Appl. Math. Comput. 219 (2012), 2361–2370. DOI 10.1016/j.amc.2012.09.008 | MR 2988118 | Zbl 1308.34104
[33] Tarasov, V. E.: Fractional Dynamics. Applications of Fractional Calculus to Dynamics of Particles, Fields and Media. Springer, Heidelberg, 2010. MR 2796453 | Zbl 1214.81004
[34] Ulam, S. M.: A Collection of Mathematical Problems. Interscience Publishers, New York, 1968. MR 0120127
[35] Vityuk, A. N., Golushkov, A. V.: Existence of solutions of systems of partial differential equations of fractional order. Nonlinear Oscil. 7, 3 (2004), 318–325. DOI 10.1007/s11072-005-0015-9 | MR 2151816
[36] Wang, J., Fečkan, M., Zhou, Y: Ulam’s type stability of impulsive ordinary differential equations. J. Math. Anal. Appl. 395, 1 (2012), 258–264. DOI 10.1016/j.jmaa.2012.05.040 | MR 2943620 | Zbl 1254.34022
[37] Wang, J., Fečkan, M., Zhou, Y: On the new concept of solutions and existence results for impulsive fractional evolution equations. Dyn. Partial Differ. Equ. 8, 4 (2011), 345–361. MR 2901608 | Zbl 1264.34014
[38] Wang, J., Lv, L., Zhou, Y.: Ulam stability and data dependence for fractional differential equations with Caputo derivative. E. J. Qual. Theory Diff. Equ. 63 (2011), 1–10. DOI 10.14232/ejqtde.2011.1.63 | MR 2832769
[39] 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. DOI 10.1016/j.cnsns.2011.09.030 | MR 2877697 | Zbl 1252.35276
[40] Wang, J., Zhou, Y.: Mittag-Leffler-Ulam stabilities of fractional evolution equations. Appl. Math. Lett. 25, 4 (2012), 723–728. DOI 10.1016/j.aml.2011.10.009 | MR 2875807 | Zbl 1246.34012
[41] Wang, J., Zhou, Y., Fečkan, M.: Nonlinear impulsive problems for fractional differential equations and Ulam stability. Comput. Math. Appl. 64 (2012), 3389–3405. DOI 10.1016/j.camwa.2012.02.021 | MR 2989367 | Zbl 1268.34033
[42] Wei, W., Li, X., Li, X.: New stability results for fractional integral equation. Comput. Math. Appl. 64 (2012), 3468–3476. DOI 10.1016/j.camwa.2012.02.057 | MR 2989374 | Zbl 1268.45007
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