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Existence results for the distributed order fractional hybrid differential equations. (English) Zbl 1261.34010
Summary: We introduce the distributed-order fractional hybrid differential equations involving the Riemann-Liouville differential operator of order \(0 < q < 1\) with respect to a nonnegative density function. Furthermore, an existence theorem for the fractional hybrid differential equations of distributed order is proved via a fixed point theorem in the Banach algebras under mixed Lipschitz and Carathéodory conditions.
MSC:
34A08 Fractional ordinary differential equations
34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems
47N20 Applications of operator theory to differential and integral equations
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[1] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204 of North-Holland Mathematics Studies, Elsevier, Amsterdam, The Netherlands, 2006. · Zbl 1092.45003
[2] I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999. · Zbl 0924.34008
[3] B. C. Dhage, “Nonlinear quadratic first order functional integro-differential equations with periodic boundary conditions,” Dynamic Systems and Applications, vol. 18, no. 2, pp. 303-322, 2009. · Zbl 1190.45004
[4] B. C. Dhage, “Quadratic perturbations of periodic boundary value problems of second order ordinary differential equations,” Differential Equations & Applications, vol. 2, no. 4, pp. 465-486, 2010. · Zbl 1235.34061
[5] B. C. Dhage and B. D. Karande, “First order integro-differential equations in Banach algebras involving Caratheodory and discontinuous nonlinearities,” Electronic Journal of Qualitative Theory of Differential Equations, vol. 21, 16 pages, 2005. · Zbl 1111.34057
[6] B. Dhage and D. O’Regan, “A fixed point theorem in Banach algebras with applications to functional integral equations,” Functional Differential Equations, vol. 7, no. 3-4, pp. 259-267, 2000. · Zbl 1040.45003
[7] B. C. Dhage, S. N. Salunkhe, R. P. Agarwal, and W. Zhang, “A functional differential equation in Banach algebras,” Mathematical Inequalities & Applications, vol. 8, no. 1, pp. 89-99, 2005. · Zbl 1107.34061
[8] P. Omari and F. Zanolin, “Remarks on periodic solutions for first order nonlinear differential systems,” Unione Matematica Italiana. Bollettino. B. Serie VI, vol. 2, no. 1, pp. 207-218, 1983. · Zbl 0507.34033
[9] B. C. Dhage and V. Lakshmikantham, “Basic results on hybrid differential equations,” Nonlinear Analysis. Hybrid Systems, vol. 4, no. 3, pp. 414-424, 2010. · Zbl 1206.34020
[10] B. C. Dhage, “Theoretical approximation methods for hybrid differential equations,” Dynamic Systems and Applications, vol. 20, no. 4, pp. 455-478, 2011. · Zbl 1245.34013
[11] Y. Zhao, S. Sun, Z. Han, and Q. Li, “Theory of fractional hybrid differential equations,” Computers & Mathematics with Applications, vol. 62, no. 3, pp. 1312-1324, 2011. · Zbl 1228.45017
[12] M. Caputo, Elasticita e Dissipazione, Zanichelli, Bologna, Italy, 1969.
[13] M. Caputo, “Mean fractional-order-derivatives differential equations and filters,” Annali dell’Università di Ferrara. Nuova Serie. Sezione VII. Scienze Matematiche, vol. 41, pp. 73-84, 1995. · Zbl 0882.34007
[14] M. Caputo, “Distributed order differential equations modelling dielectric induction and diffusion,” Fractional Calculus & Applied Analysis, vol. 4, no. 4, pp. 421-442, 2001. · Zbl 1042.34028
[15] B. C. Dhage, “On a fixed point theorem in Banach algebras with applications,” Applied Mathematics Letters, vol. 18, no. 3, pp. 273-280, 2005. · Zbl 1092.47045
[16] A. V. Bobylev and C. Cercignani, “The inverse Laplace transform of some analytic functions with an application to the eternal solutions of the Boltzmann equation,” Applied Mathematics Letters, vol. 15, no. 7, pp. 807-813, 2002. · Zbl 1025.44002
[17] W. Rudin, Real and Complex Analysis, McGraw-Hill, New York, NY, USA, 1966. · Zbl 0142.01701
[18] G. B. Folland, Real Analysis: Modern Techniques and Their Applications, Pure and Applied Mathematics, John Wiley & Sons, New York, NY, USA, 2nd edition, 1999. · Zbl 0924.28001
[19] B. Davis, Integral Transforms and Their Applications, Springer, New York, NY, USA, 3rd edition, 2001.
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