Abbas, Saïd; Benchohra, Mouffak Upper and lower solutions method for impulsive partial hyperbolic differential equations with fractional order. (English) Zbl 1202.35340 Nonlinear Anal., Hybrid Syst. 4, No. 3, 406-413 (2010). Summary: We investigate the existence of solutions for a class of initial value problems for impulsive partial hyperbolic differential equations involving the Caputo fractional derivative by using the lower and upper solutions method combined with Schauder’s fixed point theorem. Cited in 27 Documents MSC: 35R12 Impulsive partial differential equations 35R11 Fractional partial differential equations 26A33 Fractional derivatives and integrals 35A15 Variational methods applied to PDEs 35A01 Existence problems for PDEs: global existence, local existence, non-existence Keywords:impulsive hyperbolic differential equation; fractional order; upper solution; lower solution; left-sided mixed Riemann-Liouville integral; Caputo fractional order derivative; fixed point PDFBibTeX XMLCite \textit{S. Abbas} and \textit{M. Benchohra}, Nonlinear Anal., Hybrid Syst. 4, No. 3, 406--413 (2010; Zbl 1202.35340) Full Text: DOI References: [1] Kilbas, A. A.; Bonilla, B.; Trujillo, J., Nonlinear differential equations of fractional order in a space of integrable functions, Dokl. Ross. Akad. Nauk, 374, 4, 445-449 (2000) · Zbl 1137.34308 [2] Semenchuk, N. P., On one class of differential equations of noninteger order, Differ. Uravn., 10, 1831-1833 (1982) · Zbl 0522.34012 [3] Vityuk, A. N., Existence of solutions of partial differential inclusions of fractional order, Izv. Vyssh. Uchebn. Ser. Mat., 8, 13-19 (1997) · Zbl 0905.35102 [4] Diethelm, K.; Freed, A. D., On the solution of nonlinear fractional order differential equations used in the modeling of viscoplasticity, (Keil, F.; Mackens, W.; Voss, H.; Werther, J., Scientifice Computing in Chemical Engineering II-Computational Fluid Dynamics, Reaction Engineering and Molecular Properties (1999), Springer-Verlag: Springer-Verlag Heidelberg), 217-224 [5] Gaul, L.; Klein, P.; Kempfle, S., Damping description involving fractional operators, Mech. Syst. Signal Process., 5, 81-88 (1991) [6] Glockle, W. G.; Nonnenmacher, T. F., A fractional calculus approach of selfsimilar protein dynamics, Biophys. J., 68, 46-53 (1995) [7] Hilfer, R., Applications of Fractional Calculus in Physics (2000), World Scientific: World Scientific Singapore · Zbl 0998.26002 [8] Mainardi, F., Fractional calculus: Some basic problems in continuum and statistical mechanics, (Carpinteri, A.; Mainardi, F., Fractals and Fractional Calculus in Continuum Mechanics (1997), Springer-Verlag: Springer-Verlag Wien), 291-348 · Zbl 0917.73004 [9] Metzler, F.; Schick, W.; Kilian, H. G.; Nonnenmacher, T. F., Relaxation in filled polymers: A fractional calculus approach, J. Chem. Phys., 103, 7180-7186 (1995) [10] Podlubny, I., Fractional Differential Equation (1999), Academic Press: Academic Press San Diego · Zbl 0893.65051 [11] Kilbas, A. A.; Srivastava, Hari M.; Trujillo, Juan J., (Theory and Applications of Fractional Differential Equations. Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, vol. 204 (2006), Elsevier Science B.V: Elsevier Science B.V Amsterdam) · Zbl 1092.45003 [12] Lakshmikantham, V.; Leela, S.; Vasundhara, J., Theory of Fractional Dynamic Systems (2009), Cambridge Academic Publishers: Cambridge Academic Publishers Cambridge · Zbl 1188.37002 [13] Miller, K. S.; Ross, B., An Introduction to the Fractional Calculus and Differential Equations (1993), John Wiley: John Wiley New York · Zbl 0789.26002 [14] Samko, S. G.; Kilbas, A. A.; Marichev, O. I., Fractional Integrals and Derivatives. Theory and Applications (1993), Gordon and Breach: Gordon and Breach Yverdon · Zbl 0818.26003 [15] Abbas, S.; Benchohra, M., Partial hyperbolic differential equations with finite delay involving the Caputo fractional derivative, Comm. Math. Anal., 7, 2, 62-72 (2009) · Zbl 1178.35371 [16] 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 [17] R.P Agarwal, M. Benchohra, S. Hamani, A survey on existence result for boundary value problems of nonlinear fractional differential equations and inclusions, Acta. Appl. Math. (2008), doi:10.1007/s10440-008-9356-6; R.P Agarwal, M. Benchohra, S. Hamani, A survey on existence result for boundary value problems of nonlinear fractional differential equations and inclusions, Acta. Appl. Math. (2008), doi:10.1007/s10440-008-9356-6 · Zbl 1198.26004 [18] Belarbi, A.; Benchohra, M.; Ouahab, A., Uniqueness results for fractional functional differential equations with infinite delay in Fréchet spaces, Appl. Anal., 85, 1459-1470 (2006) · Zbl 1175.34080 [19] 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, 851-863 (2008) · Zbl 1198.26008 [20] Benchohra, M.; Hamani, S.; Ntouyas, S. K., Boundary value problems for differential equations with fractional order, Surv. Math. Appl., 3, 1-12 (2008) · Zbl 1157.26301 [21] Benchohra, M.; Henderson, J.; Ntouyas, S. K.; Ouahab, A., Existence results for functional differential equations of fractional order, J. Math. Anal. Appl., 338, 1340-1350 (2008) · Zbl 1209.34096 [22] Diethelm, K.; Ford, N. J., Analysis of fractional differential equations, J. Math. Anal. Appl., 265, 229-248 (2002) · Zbl 1014.34003 [23] Kilbas, A. A.; Marzan, S. A., Nonlinear differential equations with the Caputo fractional derivative in the space of continuously differentiable functions, Differ. Equ., 41, 84-89 (2005) · Zbl 1160.34301 [24] Lakshmikantham, V.; Vatsala, A. S., Basic theory of fractional differential equations, Nonlinear Anal., 69, 2677-2682 (2008) · Zbl 1161.34001 [25] Lakshmikantham, V.; Vatsala, A. S., General uniqueness and monotone iterative technique for fractional differential equations, Appl. Math. Lett., 21, 828-834 (2008) · Zbl 1161.34031 [26] Lakshmikantham, V.; Vatsala, A. S., Theory of fractional differential inequalities and applications, Commun. Appl. Anal., 11, 395-402 (2007) · Zbl 1159.34006 [27] Podlubny, I.; Petraš, I.; Vinagre, B. M.; O’Leary, P.; Dorčak, L., Analogue realizations of fractional-order controllers. fractional order calculus and its applications, Nonlinear Dynam., 29, 281-296 (2002) · Zbl 1041.93022 [28] 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 [29] Zhang, S., Positive solutions for boundary-value problems of nonlinear fractional diffrential equations, Electron. J. Differential Equations, 36, 1-12 (2006) [30] Granas, A.; Dugundji, J., Fixed Point Theory (2003), Springer-Verlag: Springer-Verlag New York · Zbl 1025.47002 [31] Benchohra, M.; Henderson, J.; Ntouyas, S. K., Impulsive Differential Equations and Inclusions, vol. 2 (2006), Hindawi Publishing Corporation: Hindawi Publishing Corporation New York · Zbl 1130.34003 [32] Dawidowski, M.; Kubiaczyk, I., An existence theorem for the generalized hyperbolic equation \(z_{x y}'' \in F(x, y, z)\) in Banach space, Ann. Soc. Math. Pol. Ser. I, Comment. Math., 30, 1, 41-49 (1990) · Zbl 0759.35029 [33] Kamont, Z., Hyperbolic Functional Differential Inequalities and Applications (1999), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht · Zbl 0924.35187 [34] Kamont, Z.; Kropielnicka, K., Differential difference inequalities related to hyperbolic functional differential systems and applications, Math. Inequal. Appl., 8, 4, 655-674 (2005) · Zbl 1091.35105 [35] Lakshmikantham, V.; Pandit, S. G., The Method of upper, lower solutions and hyperbolic partial differential equations, J. Math. Anal. Appl., 105, 466-477 (1985) · Zbl 0569.35056 [36] Pandit, S. G., Monotone methods for systems of nonlinear hyperbolic problems in two independent variables, Nonlinear Anal., 30, 235-272 (1997) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.