Dynamics and Stability Results for Nonlinear Neutral Pantograph Equations via Hilfer-Hadamard Fractional Derivative
DOI:
https://doi.org/10.5890/DNC.2019.03.004Abstract
A study on the existence and uniqueness of a solution for nonlinear neutral pantograph equations with Hilfer-Hadamard fractional derivative is the main aim of this paper. The existence result is obtained by using Schauder fixed point theorem. Also, we discuss Ulam stability for these equations based on Banach contraction principle.References
[1] Hilfer, R., Application of fractional Calculus in Physics, World Scientific, Singapore, 1999,
[2] Kilbas, A.A., Srivastava, H.M., and Trujillo, J.J. (2006), Theory and Applications of Fractional Differential Equations, in: Mathematics Studies, 204, Elsevier,
[3] Podlubny, I. (1999), Fractional Differential Equations, in: Mathematics in Science and Engineering, vol. 198, Acad. Press,
[4] Ahmad, B. and Ntouyas, S.K., (2014), Initial value problems for hybrid Hadamard fractional differential equations, Electronic Journal of Differential Equations, 161, 1-8,
[5] Butzer, P.L. , Kilbas, A.A., and Trujillo, J.J., (2002), Compositios of Hadamard-type fractional integration operators and the semigroup property, Journal of Mathematical Analysis and Applications, 269, 1-27,
[6] Butzer, P.L., Kilbas, A.A., and Trujillo, J.J. (2002), Mellin transform analysis and integration by parts for Hadamardtype fractional integrals, Journal of Mathematical Analysis and Applications, 270, 1-15,
[7] Furati, K.M., Kassim, M.D., and Tatar, N.E., (2012), Existence and uniqueness for a problem involvingHilfer fractional derivative, Computers and Mathematics with Applications, 64, 1616-1626,
[8] Furati, K.M., Kassim, M.D., and Tatar, N.E., (2013), Non-existence of global solutions for a differential equation involving Hilfer fractional derivative, Electronic Journal of Differential Equations, 235, 1-10,
[9] Gu, H., and Trujillo, J.J. (2014), Existence of mild solution for evolution equation with Hilfer fractional derivative, Applied Mathematics and Computation, DOI:10.1016/j.amc.2014.10.083,
[10] Hilfer, R., Luchko, Y., and Tomovski, Z. (2009), Operationalmethod for the solution of fractional differential equations with generalized Riemann-Lioville fractional derivative, Fractional calculus and Applications Analysis, 12, 289-318,
[11] Wang, J.R. and Zhang, Y.R. (2015), Nonlocal initial value problems for differential equations with Hilfer fractional derivative, Applied Mathematics and Computation, 266, 850-859,
[12] Samko, S.G., Kilbas, A.A., and Marichev, O.I. (1987), Fractional Integrals and Derivatives, Theory and Applications, Gordon and Breach, Amsterdam, Engl. Trans. from the Russian,
[13] Andras, S. and Kolumban, J.J. (2013), On the Ulam-Hyers stability of first order differential systems with nonlocal initial conditions, Nonlinear Analysis, 82,1-11,
[14] Ibrahim, R.W. (2012), Generalized Ulam-Hyers stability for fractional differential equations, International Journal of mathematics, 23, DOI:10.1142/S0129167X12500565,
[15] Jung, S.M. (2013), Hyers-Ulam stability of linear differential equations of first order, Applied Mathematics Letters, 17, (2004), 1135-1140,
[16] Muniyappan, P. and Rajan, S. (2015), Hyers-Ulam-Rassias stability of fractional differential equation, International Journal of pure and Applied Mathematics, 102, 631-642,
[17] Rus, I.A. (2010), Ulam stabilities of ordinary differential equations in a Banach space, Carpathian journal of Mathematics, 26, 103-107,
[18] Wang, J., Lv, L., and Zhou, Y. (2011), Ulam stability and data dependence for fractional differential equations with Caputo derivative, Electronic Journal of Qualitative Theory of Differential Equations, 63,1-10,
[19] Wang, J. and Zhou, Y. (2012),New concepts and results in stability of fractional differential equations, Communications in Nonlinear Science and Numerical Simulation, 17, 2530-2538,
[20] Balachandran, K., Kiruthika, S., and Trujillo, J.J., (2013), Existence of solutions of Nonlinear fractional pantograph equations, Acta Mathematica Scientia, 33B, 1-9,
[21] Iserles, A. (1997), Exact and discretized stability of the pantograph equation, Applied NumericalMethods, 24, 295-308,
[22] Liu, M.Z. and Li, D. (2004), Runge-Kutta methods for the multi-pantograph delay equation, Applications and Mathematical Computations, 155, 853-871,
[23] Vivek, D., Kanagarajan, K., and Sivasundaram, S. (2016), Dynamics and stability of pantograph equations via Hilfer fractional derivative, Nonlinear Studies, 23(4), 685-698,
[24] Kassim, M.D. and Tatar, N.E., Well-psedness and stability for a differential problem with Hilfer-Hadamard fractional derivative, Abstract and Applied Analysis, 1, 1-12,
[25] Vivek, D., Kanagarajan, K., and Sivasundaram, S. (2017), Theory and analysis of nonlinear neutral pantograph equations via Hilfer fractional derivative, Nonlinear Studies, 24(3), 699-712,
[26] Abbas, S., Benchohra, M., and Sivasundaram, S. (2016), Dynamics and Ulam stability for Hilfer type fractional differential equations, Nonlinear Studies, 23(4), 627-637,
[27] Wang, J. and Zhang, Y. (2014), Ulam-Hyers-Mittag-Leffler stability of fractional-order delay differential equations, Optimization, 63(8), 1181-1190,
[28] Benchohra, M., Henderson, J., Ntouyas, S.K., and Ouahab, A. (2008), Existence results for fractional order functional differential equations with infinite delay, Journal of Mathematical and Applications, 338, 1340-1350.
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