# American Institute of Mathematical Sciences

## Existence and stability results to a class of fractional random implicit differential equations involving a generalized Hilfer fractional derivative

 1 Department of Mathematics, Faculty of Arts and Sciences, Çankaya University, 06790, Ankara, Turkey 2 Department of Mathematics, Sri Ramakrishna Mission Vidyalaya College of Arts and Science, Coimbatore-641020, India 3 Department of Mathematics, University of Malakand, Dir(L), Khyber Pakhtunkhwa, Pakistan

Received  May 2018 Published  March 2019

In this paper, the existence, uniqueness and stability of random implicit fractional differential equations (RIFDs) with nonlocal condition and impulsive effect involving a generalized Hilfer fractional derivative (HFD) are discussed. The arguments are discussed via Krasnoselskii's fixed point theorems, Schaefer's fixed point theorems, Banach contraction principle and Ulam type stability. Some examples are included to ensure the abstract results.

Citation: Fahd Jarad, Sugumaran Harikrishnan, Kamal Shah, Kuppusamy Kanagarajan. Existence and stability results to a class of fractional random implicit differential equations involving a generalized Hilfer fractional derivative. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020040
##### References:
 [1] M. I. Abbas, Ulam stability of fractional impulsive differential equations with Riemann-Liouville integral boundary conditions, J. Contemp. Mathemat. Anal., 50 (2015), 209-219. doi: 10.3103/s1068362315050015. Google Scholar [2] T. Abdeljawad, A Lyapunov type inequality for fractional operators with nonsingular Mittag-Leffler kernel, J of Inequal. Appl., 2017 (2017), Paper No. 130, 11 pp. doi: 10.1186/s13660-017-1400-5. Google Scholar [3] T. Abdeljawad, Fractional operators with exponential kernels and a Lyapunov type inequality, Adv. Differ. Equ., 2017 (2017), Paper No. 313, 11 pp. doi: 10.1186/s13662-017-1285-0. Google Scholar [4] T. Abdeljawad and Q. M. Al-Mdallal, Discrete Mittag-Leffler kernel type fractional difference initial value problems and Gronwall's inequality, J. Comp. Appl. Math., 339 (2018), 218-230. doi: 10.1016/j.cam.2017.10.021. Google Scholar [5] T. Abdeljawad and D. Baleanu, On fractional derivatives with exponential kernel and their discrete versions, Rep. 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Fabrizio, A new definition of fractional derivative without singular kernal, Progr. Fract. Differ. Appl., 1 (2015), 73-85. Google Scholar [11] K. M. Furati, M. D. Kassim and N. E. Tatar, Existence and uniqueness for a problem involving hilfer fractional derivative, Compur. Math. Appl., 64 (2012), 1616-1626. doi: 10.1016/j.camwa.2012.01.009. Google Scholar [12] A. Granas and J. Dugundji, Fixed Point Theory, Springer-Verlag, New York, 2003. doi: 10.1007/978-0-387-21593-8. Google Scholar [13] R. Hilfer, Application Of Fractional Calculus in Physics, World Scientific Publishing Co., Inc., River Edge, NJ, 2000. doi: 10.1142/9789812817747. Google Scholar [14] R. W. Ibrahim, Generalized Ulam-Hyers stability for fractional differential equations, Int. J. Math., 23 (2012), 1250056, 9 pp. doi: 10.1142/S0129167X12500565. Google Scholar [15] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, in: Mathematics Studies, vol.204, Elsevier, 2006. Google Scholar [16] V. Lakshmikantham, D. D. Bainov and P. S. Simeonov, Theory of Impulsive Differential Equations, World scientific, Singapore, 1989. doi: 10.1142/0906. Google Scholar [17] V. Lupulescuand and S. K. Ntouyas, Random fractional differential equations, International Electronic Journal of Pure and Applied Mathematics, 4 (2012), 119-136. Google Scholar [18] I. Podlubny, Fractional Differential Equations: Mathematics in Science and Engineering, vol. 198, Acad. Press, 1999. Google Scholar [19] A. M. Samoilenko and N. A. Perestyuk, Impulsive Differential Equations, World scientific, Singapore, 1995. doi: 10.1142/9789812798664. Google Scholar [20] T. T. Soong, Random Differential Equations in Science and Engineering, Academic Press, New York, 1973. Google Scholar [21] J. Vanterler da, C. Sousa and E. Capelas de Oliveira, On the $\psi$-Hilfer fractional derivative, Commun. Nonlinear Sci. Numer. Simul., 60 (2018), 72-91. doi: 10.1016/j.cnsns.2018.01.005. Google Scholar [22] J. Vanterler da, C. Sousa and E. Capelas de Oliveira, Ulam-Hyers stability of a nonlinear fractional Volterra integro-differential equation, Appl. Math. Lett., 81 (2018), 50-56. doi: 10.1016/j.aml.2018.01.016. Google Scholar [23] H. Vu, Random fractional functional differential equations, Int. J. Nonlinear Anal. and Appl., 7 (2016), 253-267. Google Scholar [24] J. Wang, L. Lv and Y. Zhou, Ulam stability and data dependence for fractional differential equations with Caputo derivative, Electron. J. Qual. Theory Differ. Equ., 63 (2011), 1-10. doi: 10.14232/ejqtde.2011.1.63. Google Scholar [25] Y. Wang and Y. Zhang, Nonlocal initial value problems for differential equations with Hilfer fractional derivative, Appl. Math. Comp., 266 (2015), 850-859. doi: 10.1016/j.amc.2015.05.144. Google Scholar [26] J. Wang, Y. Zhou and M. Fe$\ddot{c}$kanc, Nonlinear impulsive problems for fractional differential equations and Ulam stability, Comp. Math. Appl., 64 (2012), 3389-3405. doi: 10.1016/j.camwa.2012.02.021. Google Scholar [27] H. Ye, J. Gao and Y. Ding, A generalized Gronwall inequality and its application to a fractional differential equation, J. Math. Anal. Appl., 328 (2007), 1075-1081. doi: 10.1016/j.jmaa.2006.05.061. Google Scholar

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##### References:
 [1] M. I. Abbas, Ulam stability of fractional impulsive differential equations with Riemann-Liouville integral boundary conditions, J. Contemp. Mathemat. Anal., 50 (2015), 209-219. doi: 10.3103/s1068362315050015. Google Scholar [2] T. Abdeljawad, A Lyapunov type inequality for fractional operators with nonsingular Mittag-Leffler kernel, J of Inequal. Appl., 2017 (2017), Paper No. 130, 11 pp. doi: 10.1186/s13660-017-1400-5. Google Scholar [3] T. Abdeljawad, Fractional operators with exponential kernels and a Lyapunov type inequality, Adv. Differ. Equ., 2017 (2017), Paper No. 313, 11 pp. doi: 10.1186/s13662-017-1285-0. Google Scholar [4] T. Abdeljawad and Q. M. Al-Mdallal, Discrete Mittag-Leffler kernel type fractional difference initial value problems and Gronwall's inequality, J. Comp. Appl. Math., 339 (2018), 218-230. doi: 10.1016/j.cam.2017.10.021. Google Scholar [5] T. Abdeljawad and D. Baleanu, On fractional derivatives with exponential kernel and their discrete versions, Rep. Math. Phy., 80 (2017), 11-27. doi: 10.1016/S0034-4877(17)30059-9. Google Scholar [6] T. Abdeljawad and D. Baleanu, Discrete fractional differences with nonsingular discrete Mittag-Leffler kernels, Adv. Differ. Equ., 2016 (2016), Paper No. 232, 18 pp. doi: 10.1186/s13662-016-0949-5. Google Scholar [7] A. Atangana and D. Baleanu, New fractional derivative with non-local and non-singular kernel, Thermal Sci., 20 (2016), 757-763. doi: 10.3233/FI-2017-1484. Google Scholar [8] A. Bashir and S. Sivasundaram, Some existence results for fractional integro-differential equations with nonlocal conditions, Commun. Appl. Anal., 12 (2008), 107-112. Google Scholar [9] M. Benchohra and S. Bouriah, Existence and stability results for nonlinear boundary value problem for implicit differential equations of fractional order, Moroccan J. Pure and Appl. Anal., 1 (2015), 22-37. doi: 10.7603/s40956-015-0002-9. Google Scholar [10] M. Caputo and M. Fabrizio, A new definition of fractional derivative without singular kernal, Progr. Fract. Differ. Appl., 1 (2015), 73-85. Google Scholar [11] K. M. Furati, M. D. Kassim and N. E. Tatar, Existence and uniqueness for a problem involving hilfer fractional derivative, Compur. Math. Appl., 64 (2012), 1616-1626. doi: 10.1016/j.camwa.2012.01.009. Google Scholar [12] A. Granas and J. Dugundji, Fixed Point Theory, Springer-Verlag, New York, 2003. doi: 10.1007/978-0-387-21593-8. Google Scholar [13] R. Hilfer, Application Of Fractional Calculus in Physics, World Scientific Publishing Co., Inc., River Edge, NJ, 2000. doi: 10.1142/9789812817747. Google Scholar [14] R. W. Ibrahim, Generalized Ulam-Hyers stability for fractional differential equations, Int. J. Math., 23 (2012), 1250056, 9 pp. doi: 10.1142/S0129167X12500565. Google Scholar [15] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, in: Mathematics Studies, vol.204, Elsevier, 2006. Google Scholar [16] V. Lakshmikantham, D. D. Bainov and P. S. Simeonov, Theory of Impulsive Differential Equations, World scientific, Singapore, 1989. doi: 10.1142/0906. Google Scholar [17] V. Lupulescuand and S. K. Ntouyas, Random fractional differential equations, International Electronic Journal of Pure and Applied Mathematics, 4 (2012), 119-136. Google Scholar [18] I. Podlubny, Fractional Differential Equations: Mathematics in Science and Engineering, vol. 198, Acad. Press, 1999. Google Scholar [19] A. M. Samoilenko and N. A. Perestyuk, Impulsive Differential Equations, World scientific, Singapore, 1995. doi: 10.1142/9789812798664. Google Scholar [20] T. T. Soong, Random Differential Equations in Science and Engineering, Academic Press, New York, 1973. Google Scholar [21] J. Vanterler da, C. Sousa and E. Capelas de Oliveira, On the $\psi$-Hilfer fractional derivative, Commun. Nonlinear Sci. Numer. Simul., 60 (2018), 72-91. doi: 10.1016/j.cnsns.2018.01.005. Google Scholar [22] J. Vanterler da, C. Sousa and E. Capelas de Oliveira, Ulam-Hyers stability of a nonlinear fractional Volterra integro-differential equation, Appl. Math. Lett., 81 (2018), 50-56. doi: 10.1016/j.aml.2018.01.016. Google Scholar [23] H. Vu, Random fractional functional differential equations, Int. J. Nonlinear Anal. and Appl., 7 (2016), 253-267. Google Scholar [24] J. Wang, L. Lv and Y. Zhou, Ulam stability and data dependence for fractional differential equations with Caputo derivative, Electron. J. Qual. Theory Differ. Equ., 63 (2011), 1-10. doi: 10.14232/ejqtde.2011.1.63. Google Scholar [25] Y. Wang and Y. Zhang, Nonlocal initial value problems for differential equations with Hilfer fractional derivative, Appl. Math. Comp., 266 (2015), 850-859. doi: 10.1016/j.amc.2015.05.144. Google Scholar [26] J. Wang, Y. Zhou and M. Fe$\ddot{c}$kanc, Nonlinear impulsive problems for fractional differential equations and Ulam stability, Comp. Math. Appl., 64 (2012), 3389-3405. doi: 10.1016/j.camwa.2012.02.021. Google Scholar [27] H. Ye, J. Gao and Y. Ding, A generalized Gronwall inequality and its application to a fractional differential equation, J. Math. Anal. Appl., 328 (2007), 1075-1081. doi: 10.1016/j.jmaa.2006.05.061. Google Scholar
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