December  2021, 10(4): 733-748. 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  December 2021 Early access  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 and Control Theory, 2021, 10 (4) : 733-748. 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.

[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. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[20]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[38]

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

[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.

[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.

[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.

[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.

[43]

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

[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.

[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.

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.

[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. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[20]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[38]

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

[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.

[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.

[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.

[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.

[43]

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

[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.

[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.

[1]

Golamreza Zamani Eskandani, Hamid Vaezi. Hyers--Ulam--Rassias stability of derivations in proper Jordan $CQ^{*}$-algebras. Discrete and Continuous Dynamical Systems, 2011, 31 (4) : 1469-1477. doi: 10.3934/dcds.2011.31.1469

[2]

Yi Guan, Michal Fečkan, Jinrong Wang. Periodic solutions and Hyers-Ulam stability of atmospheric Ekman flows. Discrete and Continuous Dynamical Systems, 2021, 41 (3) : 1157-1176. doi: 10.3934/dcds.2020313

[3]

Chao Wang, Zhien Li, Ravi P. Agarwal. Hyers-Ulam-Rassias stability of high-dimensional quaternion impulsive fuzzy dynamic equations on time scales. Discrete and Continuous Dynamical Systems - S, 2022, 15 (2) : 359-386. doi: 10.3934/dcdss.2021041

[4]

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

[5]

Christopher Bose, Rua Murray. The exact rate of approximation in Ulam's method. Discrete and Continuous Dynamical Systems, 2001, 7 (1) : 219-235. doi: 10.3934/dcds.2001.7.219

[6]

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

[7]

Rua Murray. Ulam's method for some non-uniformly expanding maps. Discrete and Continuous Dynamical Systems, 2010, 26 (3) : 1007-1018. doi: 10.3934/dcds.2010.26.1007

[8]

Paweł Góra, Abraham Boyarsky. Stochastic perturbations and Ulam's method for W-shaped maps. Discrete and Continuous Dynamical Systems, 2013, 33 (5) : 1937-1944. doi: 10.3934/dcds.2013.33.1937

[9]

Anna Cima, Armengol Gasull, Víctor Mañosa. Parrondo's dynamic paradox for the stability of non-hyperbolic fixed points. Discrete and Continuous Dynamical Systems, 2018, 38 (2) : 889-904. doi: 10.3934/dcds.2018038

[10]

Gábor Domokos, Domokos Szász. Ulam's scheme revisited: digital modeling of chaotic attractors via micro-perturbations. Discrete and Continuous Dynamical Systems, 2003, 9 (4) : 859-876. doi: 10.3934/dcds.2003.9.859

[11]

Frédéric Mazenc, Michael Malisoff, Patrick D. Leenheer. On the stability of periodic solutions in the perturbed chemostat. Mathematical Biosciences & Engineering, 2007, 4 (2) : 319-338. doi: 10.3934/mbe.2007.4.319

[12]

Chunqing Wu, Patricia J.Y. Wong. Global asymptotical stability of the coexistence fixed point of a Ricker-type competitive model. Discrete and Continuous Dynamical Systems - B, 2015, 20 (9) : 3255-3266. doi: 10.3934/dcdsb.2015.20.3255

[13]

Gary Froyland. On Ulam approximation of the isolated spectrum and eigenfunctions of hyperbolic maps. Discrete and Continuous Dynamical Systems, 2007, 17 (3) : 671-689. doi: 10.3934/dcds.2007.17.671

[14]

Simone Paleari, Tiziano Penati. Equipartition times in a Fermi-Pasta-Ulam system. Conference Publications, 2005, 2005 (Special) : 710-719. doi: 10.3934/proc.2005.2005.710

[15]

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

[16]

Siqi Chen, Yong-Kui Chang, Yanyan Wei. Pseudo $ S $-asymptotically Bloch type periodic solutions to a damped evolution equation. Evolution Equations and Control Theory, 2022, 11 (3) : 621-633. doi: 10.3934/eect.2021017

[17]

Jifeng Chu, Meirong Zhang. Rotation numbers and Lyapunov stability of elliptic periodic solutions. Discrete and Continuous Dynamical Systems, 2008, 21 (4) : 1071-1094. doi: 10.3934/dcds.2008.21.1071

[18]

Maria Carvalho, Alexander Lohse, Alexandre A. P. Rodrigues. Moduli of stability for heteroclinic cycles of periodic solutions. Discrete and Continuous Dynamical Systems, 2019, 39 (11) : 6541-6564. doi: 10.3934/dcds.2019284

[19]

Rafael Ortega. Stability and index of periodic solutions of a nonlinear telegraph equation. Communications on Pure and Applied Analysis, 2005, 4 (4) : 823-837. doi: 10.3934/cpaa.2005.4.823

[20]

Jifeng Chu, Zaitao Liang, Fangfang Liao, Shiping Lu. Existence and stability of periodic solutions for relativistic singular equations. Communications on Pure and Applied Analysis, 2017, 16 (2) : 591-609. doi: 10.3934/cpaa.2017029

2020 Impact Factor: 1.081

Metrics

  • PDF downloads (334)
  • HTML views (565)
  • Cited by (2)

[Back to Top]