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doi: 10.3934/eect.2020089

S-asymptotically $ \omega $-periodic mild solutions and stability analysis of Hilfer fractional evolution equations

1. 

Department of Mathematics and Statistics, Central University of Punjab, Bathinda, 151001, Punjab, India

2. 

Department of Mathematics and General Sciences, Prince Sultan University, 66833, 11586 Riyadh, Saudi Arabia

3. 

Department of Medical Research, China Medical University,40402, Taichung, Taiwan

4. 

Department of Computer Science and Information Engineering, Asia University, Taichung, Taiwan

* Corresponding author

Received  April 2020 Revised  June 2020 Published  August 2020

In this article, we deal with the existence of S-asymptotically $ \omega $-periodic mild solutions of Hilfer fractional evolution equations. We also investigate the Ulam-Hyers and Ulam-Hyers-Rassias stability of similar solutions. These results are established in Banach space with the help of resolvent operator functions and fixed point technique on an unbounded interval. An example is also presented for the illustration of obtained results.

Citation: Pallavi Bedi, Anoop Kumar, Thabet Abdeljawad, Aziz Khan. S-asymptotically $ \omega $-periodic mild solutions and stability analysis of Hilfer fractional evolution equations. Evolution Equations & Control Theory, doi: 10.3934/eect.2020089
References:
[1]

S. AbbasM. Benchohra and A. Petrusel, Ulam stability for Hilfer type fractional differential inclusions via the weakly Picard operators theory, Fract. Calc. Appl. Anal., 20 (2017), 384-398.  doi: 10.1515/fca-2017-0020.  Google Scholar

[2]

R. P. AgarwalS. Hristova and D. O'Regand, Lyapunov functions to Caputo reaction-diffusion fractional neural networks with time-varying delays, J. Math. Comput. SCI-JM., 18 (2018), 328-345.   Google Scholar

[3]

H. M. AhmedaM. M. El-BoraibH. M. El-Owaidyc and A. S. Ghanema, Null controllability of fractional stochastic delay integro-differential equations, J. Math. Comput. SCI-JM., 19 (2019), 143-150.  doi: 10.22436/jmcs.019.03.01.  Google Scholar

[4]

I. Ahmed, P. Kumam, K. Shah, P. Borisut, K. Sitthithakerngkiet and M. A. Demba, Stability results for implicit fractional pantograph differential equations via $\phi $-Hilfer fractional derivative with a nonlocal Riemann-Liouville fractional integral condition, Mathematics., 8 (2020), 94. Google Scholar

[5]

M. AhmadA. Zada and J. Alzabut, Hyers-Ulam stability of a coupled system of fractional differential equations of Hilfer -Hadamard type, Demonstratio Math., 52 (2019), 283-295.  doi: 10.1515/dema-2019-0024.  Google Scholar

[6]

S. AliaM. ArifaD. Lateefb and M. Akramc, Stable monotone iterative solutions to a class of bound-ary value problems of nonlinear fractional order differential equations, J. Nonlinear Sci. Appl., 12 (2019), 376-386.  doi: 10.22436/jnsa.012.06.04.  Google Scholar

[7]

A. Atangana and J. F. Gomez-Aguilar, Numerical approximation of Riemann-Liouville definition of fractional derivative: from Riemann-Liouville to Atangana-Baleanu., Numer. Meth. Part. Diff. Eqs., 34 (2018), 1502-1523.  doi: 10.1002/num.22195.  Google Scholar

[8]

P. Bedi, A. Kumar, T. Abdeljawad and A. Khan, Existence of mild solutions for impulsive neutral Hilfer fractional evolution equations, Adv. Diff. Equ., Paper No. 155, 16 pp. doi: 10.1186/s13662-020-02615-y.  Google Scholar

[9]

A. Coronel-Escamilla, J. F. Gomez-Aguilar, E. Alvarado-Mendez, G. V. Guerrero-Ramirez and R. F. Escobar-Jimenez, Fractional dynamics of charged particles in magnetic fields, Int. J. Mod. Phys. C., 27 (2016), 1650084. doi: 10.1142/S0129183116500844.  Google Scholar

[10]

B. Cuahutenango-BarroM. A. Taneco-Hernández and J. F. Gómez-Aguilar, On the solutions of fractional-time wave equation with memory effect involving operators with regular kernel, Chaos Solitons Fractals, 115 (2018), 283-299.  doi: 10.1016/j.chaos.2018.09.002.  Google Scholar

[11]

C. Cuevas and J. C. de Souza, Existence of S-asymptotically $\omega$-periodic solutions for fractional order functional integro-differential equations with infinite delay, Nonlinear Anal. Theory Methods Appl., 72 (2010), 1683-1689.  doi: 10.1016/j.na.2009.09.007.  Google Scholar

[12]

A. Devi, A. Kumar, T. Abdeljawad and A. Khan, Existence and stability analysis of solutions for fractional Langevin equa- tion with nonlocal integral and anti-periodic type boundary conditions, Fractals, (2020). doi: 10.1142/S0218348X2040006X.  Google Scholar

[13]

J. F. Gómez-AguilarM. Miranda-HernandezM. G. López-LópezV. M. Alvarado-Martínez and D. Baleanu, Modeling and simulation of the fractional space-time diffusion equation, Commun. Nonlinear Sci. Numer. Simul., 30 (2016), 115-127.  doi: 10.1016/j.cnsns.2015.06.014.  Google Scholar

[14]

J. F. Gómez-Aguilar and A. Atangana, Fractional Hunter-Saxton equation involving partial operators with bi-order in Riemann-Liouville and Liouville-Caputo sense, Eur. Phys. J. Plus, 132 (2017), 1-18.  doi: 10.1140/epjp/i2017-11371-6.  Google Scholar

[15]

J. F. Gómez-Aguilar, Space-time fractional diffusion equation using a derivative with nonsingular and regular kernel, Physica A., 465 (2017), 562-572.  doi: 10.1016/j.physa.2016.08.072.  Google Scholar

[16]

S. Harikrishnan, K. Shah, D. Baleanu and K. Kanagarajan, Note on the solution of random differential equations via $ \psi$-Hilfer fractional derivative, Adv. Diff. Equ, 2018 (2018), 224. doi: 10.1186/s13662-018-1678-8.  Google Scholar

[17]

H. R. HenríquezM. Pierri and P. Táboas, On S-asymptotically $\omega$-periodic functions on Banach spaces and applications, J. Math. Anal. Appl., 343 (2008), 1119-1130.  doi: 10.1016/j.jmaa.2008.02.023.  Google Scholar

[18]

H. R. HenríquezM. Pierri and P. Táboas, Existence of S-asymptotically $\omega$-periodic solutions for abstract neutral equations, B. Aust. Math Soc., 78 (2008), 365-382.  doi: 10.1017/S0004972708000713.  Google Scholar

[19]

H. R. Henríquez, Asymptotically periodic solutions of abstract differential equations, Nonlinear Anal. Theory Methods Appl., 80 (2013), 135-149.  doi: 10.1016/j.na.2012.10.010.  Google Scholar

[20]

R, Hilfer, Fractional time evolution, in: Applications of Fractional Calculus in Physics, 2000, 87–130 doi: 10.1142/9789812817747_0002.  Google Scholar

[21]

D. H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. U.S.A., 27 (1941), 222-224.  doi: 10.1073/pnas.27.4.222.  Google Scholar

[22]

F. JaradS. HarikrishnanK. Shah and K. Kanagarajan, Existence and stability results to a class of fractional random implicit differential equations involving a generalized Hilfer fractional derivative, Discrete Cont. Dyn-S., 13 (2020), 723-739.  doi: 10.3934/dcdss.2020040.  Google Scholar

[23]

A. KhanH. KhanJ. F. Gómez-Aguilar and T. Abdeljawad, Existence and Hyers-Ulam stability for a nonlinear singular fractional differential equations with Mittag-Leffler kernel, Chaos Solitons Fractals, 127 (2019), 422-427.  doi: 10.1016/j.chaos.2019.07.026.  Google Scholar

[24]

A. KhanJ. F. Gómez-AguilarT. Abdeljawad and H. Khan, Stability and numerical simulation of a fractional order plant-nectar-pollinator model, Alex. Eng. J., 59 (2020), 49-59.  doi: 10.1016/j.aej.2019.12.007.  Google Scholar

[25]

A. Khan, T. S. Khan, M. I. Syam and H. Khan, Analytical solutions of time-fractional wave equation by double Laplace transform method, Eur. Phys. J. Plus., 134 (2019), 163. doi: 10.1140/epjp/i2019-12499-y.  Google Scholar

[26]

H. Khan, A. Khan, T. Abdeljawad and A. Alkhazzan, Existence results in Banach space for a nonlinear impulsive system, Adv. Diff. Equ., 18 (2019), Paper No. 18, 16 pp. doi: 10.1186/s13662-019-1965-z.  Google Scholar

[27]

H. KhanJ. F. Gómez-AguilarA. Khan and T. S. Khan, Stability analysis for fractional order advection- reaction diffusion system, Physica A., 521 (2019), 737-751.  doi: 10.1016/j.physa.2019.01.102.  Google Scholar

[28]

H. KhanC. Tunc and A. Khan, Stability results and existence theorems for nonlinear delay-fractional differential equations with $\varphi^* _p $-operator, J. Appl. Anal. Comp., 10 (2020), 584-597.   Google Scholar

[29]

O. KhanaS. Aracib and M. Saifa, Fractional calculus formulas for Mathieu-type series and generalized Mittag-Leffler function, J. Math. Comput. SCI-JM., 20 (2020), 122-130.  doi: 10.22436/jmcs.020.02.05.  Google Scholar

[30]

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, 204. Elsevier Science B.V., Amsterdam, 2006.  Google Scholar

[31]

Q. Li and M. Wei, Existence and asymptotic stability of periodic solutions for impulsive delay evolution equations, Adv. Diff. Equ., (2019), 1–19. doi: 10.1186/s13662-019-1994-7.  Google Scholar

[32]

J. Mu, Y. Zhou and L. Peng, Periodic Solutions and Asymptotically Periodic Solutions to Fractional Evolution Equations, Discrete Dyn. Nat. Soc., (2017), Art. ID 1364532, 12 pp. doi: 10.1155/2017/1364532.  Google Scholar

[33]

K. M. Saad and J. F. Gómez-Aguilar, Analysis of reaction-diffusion system via a new fractional derivative with non-singular kernel, Physica A., 509 (2018), 703-716.  doi: 10.1016/j.physa.2018.05.137.  Google Scholar

[34]

R. Saadati, E. Pourhadi and B. Samet, On the $PC $-mild solutions of abstract fractional evolution equations with non-instantaneous impulses via the measure of noncompactness, Bound. Value. Probl., (2019), Paper No. 19, 23 pp. doi: 10.1186/s13661-019-1137-9.  Google Scholar

[35]

N. Sene, Stability analysis of the generalized fractional differential equations with and without exogenous inputs, J. Nonlinear Sci. Appl., 12 (2019), 562-572.  doi: 10.22436/jnsa.012.09.01.  Google Scholar

[36]

K. ShahA. Ali and S. Bushnaq, Hyers-Ulam stability analysis to implicit Cauchy problem of fractional differential equations with impulsive conditions, Math. Method Appl. Sci., 41 (2018), 8329-8343.  doi: 10.1002/mma.5292.  Google Scholar

[37]

M. SherK. Shah and J. Rassias, On qualitative theory of fractional order delay evolution equation via the prior estimate method, Math. Method Appl. Sci., 43 (2020), 6464-6475.  doi: 10.1002/mma.6390.  Google Scholar

[38]

J. Sousa, Existence of mild solutions to Hilfer fractional evolution equations in Banach space, preprint, arXiv: 1812.02213. Google Scholar

[39]

J. V. D. C. Sousa and E. C. 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

[40]

J. V. D. C. Sousa and E. C. de Oliveira, A Gronwall inequality and the Cauchy-type problem by means of $\psi $-Hilfer operator, Differ. Equ. Appl., 11 (2019), 87–106. arXiv: 1709.03634. doi: 10.7153/dea-2019-11-02.  Google Scholar

[41]

J. V. D. C. Sousa and E. C. de Oliveira, Leibniz type rule: $\psi $-Hilfer fractional operator, Communications in Nonlinear Science and Numerical Simulation, 77 (2019), 305-311.  doi: 10.1016/j.cnsns.2019.05.003.  Google Scholar

[42]

J. V. D. C. Sousa and E. C. de Oliveira, On the Ulam-Hyers-Rassias stability for nonlinear fractional differential equations using the $\psi $-Hilfer operator, Journal of Fixed Point Theory and Applications, 20 (2018), 96 21 pp. doi: 10.1007/s11784-018-0587-5.  Google Scholar

[43]

S. M. Ulam, A Collection of Mathematical Problems, Interscience Publishers, New York, 8 1960.  Google Scholar

[44]

Asma, G. ur Rahman and K. Shah, Mathematical Analysis of Implicit Impulsive Switched Coupled Evolution Equations, Results Math., 74 (2019), 142. doi: 10.1007/s00025-019-1066-z.  Google Scholar

[45]

J. WangK. Shah and A. Ali, Existence and Hyers-Ulam stability of fractional nonlinear impulsive switched coupled evolution equations, Math. Method Appl. Sci., 41 (2018), 2392-2402.  doi: 10.1002/mma.4748.  Google Scholar

show all references

References:
[1]

S. AbbasM. Benchohra and A. Petrusel, Ulam stability for Hilfer type fractional differential inclusions via the weakly Picard operators theory, Fract. Calc. Appl. Anal., 20 (2017), 384-398.  doi: 10.1515/fca-2017-0020.  Google Scholar

[2]

R. P. AgarwalS. Hristova and D. O'Regand, Lyapunov functions to Caputo reaction-diffusion fractional neural networks with time-varying delays, J. Math. Comput. SCI-JM., 18 (2018), 328-345.   Google Scholar

[3]

H. M. AhmedaM. M. El-BoraibH. M. El-Owaidyc and A. S. Ghanema, Null controllability of fractional stochastic delay integro-differential equations, J. Math. Comput. SCI-JM., 19 (2019), 143-150.  doi: 10.22436/jmcs.019.03.01.  Google Scholar

[4]

I. Ahmed, P. Kumam, K. Shah, P. Borisut, K. Sitthithakerngkiet and M. A. Demba, Stability results for implicit fractional pantograph differential equations via $\phi $-Hilfer fractional derivative with a nonlocal Riemann-Liouville fractional integral condition, Mathematics., 8 (2020), 94. Google Scholar

[5]

M. AhmadA. Zada and J. Alzabut, Hyers-Ulam stability of a coupled system of fractional differential equations of Hilfer -Hadamard type, Demonstratio Math., 52 (2019), 283-295.  doi: 10.1515/dema-2019-0024.  Google Scholar

[6]

S. AliaM. ArifaD. Lateefb and M. Akramc, Stable monotone iterative solutions to a class of bound-ary value problems of nonlinear fractional order differential equations, J. Nonlinear Sci. Appl., 12 (2019), 376-386.  doi: 10.22436/jnsa.012.06.04.  Google Scholar

[7]

A. Atangana and J. F. Gomez-Aguilar, Numerical approximation of Riemann-Liouville definition of fractional derivative: from Riemann-Liouville to Atangana-Baleanu., Numer. Meth. Part. Diff. Eqs., 34 (2018), 1502-1523.  doi: 10.1002/num.22195.  Google Scholar

[8]

P. Bedi, A. Kumar, T. Abdeljawad and A. Khan, Existence of mild solutions for impulsive neutral Hilfer fractional evolution equations, Adv. Diff. Equ., Paper No. 155, 16 pp. doi: 10.1186/s13662-020-02615-y.  Google Scholar

[9]

A. Coronel-Escamilla, J. F. Gomez-Aguilar, E. Alvarado-Mendez, G. V. Guerrero-Ramirez and R. F. Escobar-Jimenez, Fractional dynamics of charged particles in magnetic fields, Int. J. Mod. Phys. C., 27 (2016), 1650084. doi: 10.1142/S0129183116500844.  Google Scholar

[10]

B. Cuahutenango-BarroM. A. Taneco-Hernández and J. F. Gómez-Aguilar, On the solutions of fractional-time wave equation with memory effect involving operators with regular kernel, Chaos Solitons Fractals, 115 (2018), 283-299.  doi: 10.1016/j.chaos.2018.09.002.  Google Scholar

[11]

C. Cuevas and J. C. de Souza, Existence of S-asymptotically $\omega$-periodic solutions for fractional order functional integro-differential equations with infinite delay, Nonlinear Anal. Theory Methods Appl., 72 (2010), 1683-1689.  doi: 10.1016/j.na.2009.09.007.  Google Scholar

[12]

A. Devi, A. Kumar, T. Abdeljawad and A. Khan, Existence and stability analysis of solutions for fractional Langevin equa- tion with nonlocal integral and anti-periodic type boundary conditions, Fractals, (2020). doi: 10.1142/S0218348X2040006X.  Google Scholar

[13]

J. F. Gómez-AguilarM. Miranda-HernandezM. G. López-LópezV. M. Alvarado-Martínez and D. Baleanu, Modeling and simulation of the fractional space-time diffusion equation, Commun. Nonlinear Sci. Numer. Simul., 30 (2016), 115-127.  doi: 10.1016/j.cnsns.2015.06.014.  Google Scholar

[14]

J. F. Gómez-Aguilar and A. Atangana, Fractional Hunter-Saxton equation involving partial operators with bi-order in Riemann-Liouville and Liouville-Caputo sense, Eur. Phys. J. Plus, 132 (2017), 1-18.  doi: 10.1140/epjp/i2017-11371-6.  Google Scholar

[15]

J. F. Gómez-Aguilar, Space-time fractional diffusion equation using a derivative with nonsingular and regular kernel, Physica A., 465 (2017), 562-572.  doi: 10.1016/j.physa.2016.08.072.  Google Scholar

[16]

S. Harikrishnan, K. Shah, D. Baleanu and K. Kanagarajan, Note on the solution of random differential equations via $ \psi$-Hilfer fractional derivative, Adv. Diff. Equ, 2018 (2018), 224. doi: 10.1186/s13662-018-1678-8.  Google Scholar

[17]

H. R. HenríquezM. Pierri and P. Táboas, On S-asymptotically $\omega$-periodic functions on Banach spaces and applications, J. Math. Anal. Appl., 343 (2008), 1119-1130.  doi: 10.1016/j.jmaa.2008.02.023.  Google Scholar

[18]

H. R. HenríquezM. Pierri and P. Táboas, Existence of S-asymptotically $\omega$-periodic solutions for abstract neutral equations, B. Aust. Math Soc., 78 (2008), 365-382.  doi: 10.1017/S0004972708000713.  Google Scholar

[19]

H. R. Henríquez, Asymptotically periodic solutions of abstract differential equations, Nonlinear Anal. Theory Methods Appl., 80 (2013), 135-149.  doi: 10.1016/j.na.2012.10.010.  Google Scholar

[20]

R, Hilfer, Fractional time evolution, in: Applications of Fractional Calculus in Physics, 2000, 87–130 doi: 10.1142/9789812817747_0002.  Google Scholar

[21]

D. H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. U.S.A., 27 (1941), 222-224.  doi: 10.1073/pnas.27.4.222.  Google Scholar

[22]

F. JaradS. HarikrishnanK. Shah and K. Kanagarajan, Existence and stability results to a class of fractional random implicit differential equations involving a generalized Hilfer fractional derivative, Discrete Cont. Dyn-S., 13 (2020), 723-739.  doi: 10.3934/dcdss.2020040.  Google Scholar

[23]

A. KhanH. KhanJ. F. Gómez-Aguilar and T. Abdeljawad, Existence and Hyers-Ulam stability for a nonlinear singular fractional differential equations with Mittag-Leffler kernel, Chaos Solitons Fractals, 127 (2019), 422-427.  doi: 10.1016/j.chaos.2019.07.026.  Google Scholar

[24]

A. KhanJ. F. Gómez-AguilarT. Abdeljawad and H. Khan, Stability and numerical simulation of a fractional order plant-nectar-pollinator model, Alex. Eng. J., 59 (2020), 49-59.  doi: 10.1016/j.aej.2019.12.007.  Google Scholar

[25]

A. Khan, T. S. Khan, M. I. Syam and H. Khan, Analytical solutions of time-fractional wave equation by double Laplace transform method, Eur. Phys. J. Plus., 134 (2019), 163. doi: 10.1140/epjp/i2019-12499-y.  Google Scholar

[26]

H. Khan, A. Khan, T. Abdeljawad and A. Alkhazzan, Existence results in Banach space for a nonlinear impulsive system, Adv. Diff. Equ., 18 (2019), Paper No. 18, 16 pp. doi: 10.1186/s13662-019-1965-z.  Google Scholar

[27]

H. KhanJ. F. Gómez-AguilarA. Khan and T. S. Khan, Stability analysis for fractional order advection- reaction diffusion system, Physica A., 521 (2019), 737-751.  doi: 10.1016/j.physa.2019.01.102.  Google Scholar

[28]

H. KhanC. Tunc and A. Khan, Stability results and existence theorems for nonlinear delay-fractional differential equations with $\varphi^* _p $-operator, J. Appl. Anal. Comp., 10 (2020), 584-597.   Google Scholar

[29]

O. KhanaS. Aracib and M. Saifa, Fractional calculus formulas for Mathieu-type series and generalized Mittag-Leffler function, J. Math. Comput. SCI-JM., 20 (2020), 122-130.  doi: 10.22436/jmcs.020.02.05.  Google Scholar

[30]

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, 204. Elsevier Science B.V., Amsterdam, 2006.  Google Scholar

[31]

Q. Li and M. Wei, Existence and asymptotic stability of periodic solutions for impulsive delay evolution equations, Adv. Diff. Equ., (2019), 1–19. doi: 10.1186/s13662-019-1994-7.  Google Scholar

[32]

J. Mu, Y. Zhou and L. Peng, Periodic Solutions and Asymptotically Periodic Solutions to Fractional Evolution Equations, Discrete Dyn. Nat. Soc., (2017), Art. ID 1364532, 12 pp. doi: 10.1155/2017/1364532.  Google Scholar

[33]

K. M. Saad and J. F. Gómez-Aguilar, Analysis of reaction-diffusion system via a new fractional derivative with non-singular kernel, Physica A., 509 (2018), 703-716.  doi: 10.1016/j.physa.2018.05.137.  Google Scholar

[34]

R. Saadati, E. Pourhadi and B. Samet, On the $PC $-mild solutions of abstract fractional evolution equations with non-instantaneous impulses via the measure of noncompactness, Bound. Value. Probl., (2019), Paper No. 19, 23 pp. doi: 10.1186/s13661-019-1137-9.  Google Scholar

[35]

N. Sene, Stability analysis of the generalized fractional differential equations with and without exogenous inputs, J. Nonlinear Sci. Appl., 12 (2019), 562-572.  doi: 10.22436/jnsa.012.09.01.  Google Scholar

[36]

K. ShahA. Ali and S. Bushnaq, Hyers-Ulam stability analysis to implicit Cauchy problem of fractional differential equations with impulsive conditions, Math. Method Appl. Sci., 41 (2018), 8329-8343.  doi: 10.1002/mma.5292.  Google Scholar

[37]

M. SherK. Shah and J. Rassias, On qualitative theory of fractional order delay evolution equation via the prior estimate method, Math. Method Appl. Sci., 43 (2020), 6464-6475.  doi: 10.1002/mma.6390.  Google Scholar

[38]

J. Sousa, Existence of mild solutions to Hilfer fractional evolution equations in Banach space, preprint, arXiv: 1812.02213. Google Scholar

[39]

J. V. D. C. Sousa and E. C. 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

[40]

J. V. D. C. Sousa and E. C. de Oliveira, A Gronwall inequality and the Cauchy-type problem by means of $\psi $-Hilfer operator, Differ. Equ. Appl., 11 (2019), 87–106. arXiv: 1709.03634. doi: 10.7153/dea-2019-11-02.  Google Scholar

[41]

J. V. D. C. Sousa and E. C. de Oliveira, Leibniz type rule: $\psi $-Hilfer fractional operator, Communications in Nonlinear Science and Numerical Simulation, 77 (2019), 305-311.  doi: 10.1016/j.cnsns.2019.05.003.  Google Scholar

[42]

J. V. D. C. Sousa and E. C. de Oliveira, On the Ulam-Hyers-Rassias stability for nonlinear fractional differential equations using the $\psi $-Hilfer operator, Journal of Fixed Point Theory and Applications, 20 (2018), 96 21 pp. doi: 10.1007/s11784-018-0587-5.  Google Scholar

[43]

S. M. Ulam, A Collection of Mathematical Problems, Interscience Publishers, New York, 8 1960.  Google Scholar

[44]

Asma, G. ur Rahman and K. Shah, Mathematical Analysis of Implicit Impulsive Switched Coupled Evolution Equations, Results Math., 74 (2019), 142. doi: 10.1007/s00025-019-1066-z.  Google Scholar

[45]

J. WangK. Shah and A. Ali, Existence and Hyers-Ulam stability of fractional nonlinear impulsive switched coupled evolution equations, Math. Method Appl. Sci., 41 (2018), 2392-2402.  doi: 10.1002/mma.4748.  Google Scholar

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