March  2020, 13(3): 723-739. doi: 10.3934/dcdss.2020040

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

* Corresponding author: Fahd Jarad

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 and Continuous Dynamical Systems - S, 2020, 13 (3) : 723-739. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[8]

A. Bashir and S. Sivasundaram, Some existence results for fractional integro-differential equations with nonlocal conditions, Commun. Appl. Anal., 12 (2008), 107-112. 

[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.

[10]

M. Caputo and M. Fabrizio, A new definition of fractional derivative without singular kernal, Progr. Fract. Differ. Appl., 1 (2015), 73-85. 

[11]

K. M. FuratiM. 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.

[12]

A. Granas and J. Dugundji, Fixed Point Theory, Springer-Verlag, New York, 2003. doi: 10.1007/978-0-387-21593-8.

[13]

R. Hilfer, Application Of Fractional Calculus in Physics, World Scientific Publishing Co., Inc., River Edge, NJ, 2000. doi: 10.1142/9789812817747.

[14]

R. W. Ibrahim, Generalized Ulam-Hyers stability for fractional differential equations, Int. J. Math., 23 (2012), 1250056, 9 pp. doi: 10.1142/S0129167X12500565.

[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.

[16]

V. Lakshmikantham, D. D. Bainov and P. S. Simeonov, Theory of Impulsive Differential Equations, World scientific, Singapore, 1989. doi: 10.1142/0906.

[17]

V. Lupulescuand and S. K. Ntouyas, Random fractional differential equations, International Electronic Journal of Pure and Applied Mathematics, 4 (2012), 119-136. 

[18]

I. Podlubny, Fractional Differential Equations: Mathematics in Science and Engineering, vol. 198, Acad. Press, 1999.

[19]

A. M. Samoilenko and N. A. Perestyuk, Impulsive Differential Equations, World scientific, Singapore, 1995. doi: 10.1142/9789812798664.

[20] T. T. Soong, Random Differential Equations in Science and Engineering, Academic Press, New York, 1973. 
[21]

J. Vanterler daC. 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.

[22]

J. Vanterler daC. 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.

[23]

H. Vu, Random fractional functional differential equations, Int. J. Nonlinear Anal. and Appl., 7 (2016), 253-267. 

[24]

J. WangL. 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.

[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.

[26]

J. WangY. 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.

[27]

H. YeJ. 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.

show all references

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.

[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.

[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.

[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.

[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.

[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.

[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.

[8]

A. Bashir and S. Sivasundaram, Some existence results for fractional integro-differential equations with nonlocal conditions, Commun. Appl. Anal., 12 (2008), 107-112. 

[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.

[10]

M. Caputo and M. Fabrizio, A new definition of fractional derivative without singular kernal, Progr. Fract. Differ. Appl., 1 (2015), 73-85. 

[11]

K. M. FuratiM. 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.

[12]

A. Granas and J. Dugundji, Fixed Point Theory, Springer-Verlag, New York, 2003. doi: 10.1007/978-0-387-21593-8.

[13]

R. Hilfer, Application Of Fractional Calculus in Physics, World Scientific Publishing Co., Inc., River Edge, NJ, 2000. doi: 10.1142/9789812817747.

[14]

R. W. Ibrahim, Generalized Ulam-Hyers stability for fractional differential equations, Int. J. Math., 23 (2012), 1250056, 9 pp. doi: 10.1142/S0129167X12500565.

[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.

[16]

V. Lakshmikantham, D. D. Bainov and P. S. Simeonov, Theory of Impulsive Differential Equations, World scientific, Singapore, 1989. doi: 10.1142/0906.

[17]

V. Lupulescuand and S. K. Ntouyas, Random fractional differential equations, International Electronic Journal of Pure and Applied Mathematics, 4 (2012), 119-136. 

[18]

I. Podlubny, Fractional Differential Equations: Mathematics in Science and Engineering, vol. 198, Acad. Press, 1999.

[19]

A. M. Samoilenko and N. A. Perestyuk, Impulsive Differential Equations, World scientific, Singapore, 1995. doi: 10.1142/9789812798664.

[20] T. T. Soong, Random Differential Equations in Science and Engineering, Academic Press, New York, 1973. 
[21]

J. Vanterler daC. 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.

[22]

J. Vanterler daC. 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.

[23]

H. Vu, Random fractional functional differential equations, Int. J. Nonlinear Anal. and Appl., 7 (2016), 253-267. 

[24]

J. WangL. 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.

[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.

[26]

J. WangY. 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.

[27]

H. YeJ. 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.

[1]

Ramasamy Subashini, Chokkalingam Ravichandran, Kasthurisamy Jothimani, Haci Mehmet Baskonus. Existence results of Hilfer integro-differential equations with fractional order. Discrete and Continuous Dynamical Systems - S, 2020, 13 (3) : 911-923. doi: 10.3934/dcdss.2020053

[2]

Graeme D. Chalmers, Desmond J. Higham. Convergence and stability analysis for implicit simulations of stochastic differential equations with random jump magnitudes. Discrete and Continuous Dynamical Systems - B, 2008, 9 (1) : 47-64. doi: 10.3934/dcdsb.2008.9.47

[3]

Chun Wang, Tian-Zhou Xu. Stability of the nonlinear fractional differential equations with the right-sided Riemann-Liouville fractional derivative. Discrete and Continuous Dynamical Systems - S, 2017, 10 (3) : 505-521. doi: 10.3934/dcdss.2017025

[4]

Ankit Kumar, Kamal Jeet, Ramesh Kumar Vats. Controllability of Hilfer fractional integro-differential equations of Sobolev-type with a nonlocal condition in a Banach space. Evolution Equations and Control Theory, 2022, 11 (2) : 605-619. doi: 10.3934/eect.2021016

[5]

Seda İğret Araz. New class of volterra integro-differential equations with fractal-fractional operators: Existence, uniqueness and numerical scheme. Discrete and Continuous Dynamical Systems - S, 2021, 14 (7) : 2297-2309. doi: 10.3934/dcdss.2021053

[6]

Priscila Santos Ramos, J. Vanterler da C. Sousa, E. Capelas de Oliveira. Existence and uniqueness of mild solutions for quasi-linear fractional integro-differential equations. Evolution Equations and Control Theory, 2022, 11 (1) : 1-24. doi: 10.3934/eect.2020100

[7]

Qi Yao, Linshan Wang, Yangfan Wang. Existence-uniqueness and stability of the mild periodic solutions to a class of delayed stochastic partial differential equations and its applications. Discrete and Continuous Dynamical Systems - B, 2021, 26 (9) : 4727-4743. doi: 10.3934/dcdsb.2020310

[8]

Ichrak Bouacida, Mourad Kerboua, Sami Segni. Controllability results for Sobolev type $ \psi - $Hilfer fractional backward perturbed integro-differential equations in Hilbert space. Evolution Equations and Control Theory, 2022  doi: 10.3934/eect.2022028

[9]

Saïd Abbas, Mouffak Benchohra, John R. Graef. Coupled systems of Hilfer fractional differential inclusions in banach spaces. Communications on Pure and Applied Analysis, 2018, 17 (6) : 2479-2493. doi: 10.3934/cpaa.2018118

[10]

Pallavi Bedi, Anoop Kumar, Thabet Abdeljawad, Aziz Khan. S-asymptotically $ \omega $-periodic mild solutions and stability analysis of Hilfer fractional evolution equations. Evolution Equations and Control Theory, 2021, 10 (4) : 733-748. doi: 10.3934/eect.2020089

[11]

Yousef Alnafisah, Hamdy M. Ahmed. Neutral delay Hilfer fractional integrodifferential equations with fractional brownian motion. Evolution Equations and Control Theory, 2022, 11 (3) : 925-937. doi: 10.3934/eect.2021031

[12]

Matteo Bonforte, Yannick Sire, Juan Luis Vázquez. Existence, uniqueness and asymptotic behaviour for fractional porous medium equations on bounded domains. Discrete and Continuous Dynamical Systems, 2015, 35 (12) : 5725-5767. doi: 10.3934/dcds.2015.35.5725

[13]

Ilknur Koca. Numerical analysis of coupled fractional differential equations with Atangana-Baleanu fractional derivative. Discrete and Continuous Dynamical Systems - S, 2019, 12 (3) : 475-486. doi: 10.3934/dcdss.2019031

[14]

Aaron Hoffman, Benjamin Kennedy. Existence and uniqueness of traveling waves in a class of unidirectional lattice differential equations. Discrete and Continuous Dynamical Systems, 2011, 30 (1) : 137-167. doi: 10.3934/dcds.2011.30.137

[15]

Tonny Paul, A. Anguraj. Existence and uniqueness of nonlinear impulsive integro-differential equations. Discrete and Continuous Dynamical Systems - B, 2006, 6 (5) : 1191-1198. doi: 10.3934/dcdsb.2006.6.1191

[16]

Farid Tari. Two-parameter families of implicit differential equations. Discrete and Continuous Dynamical Systems, 2005, 13 (1) : 139-162. doi: 10.3934/dcds.2005.13.139

[17]

Bernard Dacorogna, Alessandro Ferriero. Regularity and selecting principles for implicit ordinary differential equations. Discrete and Continuous Dynamical Systems - B, 2009, 11 (1) : 87-101. doi: 10.3934/dcdsb.2009.11.87

[18]

Kolade M. Owolabi, Abdon Atangana. High-order solvers for space-fractional differential equations with Riesz derivative. Discrete and Continuous Dynamical Systems - S, 2019, 12 (3) : 567-590. doi: 10.3934/dcdss.2019037

[19]

Huy Tuan Nguyen, Huu Can Nguyen, Renhai Wang, Yong Zhou. Initial value problem for fractional Volterra integro-differential equations with Caputo derivative. Discrete and Continuous Dynamical Systems - B, 2021, 26 (12) : 6483-6510. doi: 10.3934/dcdsb.2021030

[20]

Rim Bourguiba, Rosana Rodríguez-López. Existence results for fractional differential equations in presence of upper and lower solutions. Discrete and Continuous Dynamical Systems - B, 2021, 26 (3) : 1723-1747. doi: 10.3934/dcdsb.2020180

2020 Impact Factor: 2.425

Metrics

  • PDF downloads (672)
  • HTML views (1001)
  • Cited by (9)

[Back to Top]