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July  2022, 15(7): 1767-1776. doi: 10.3934/dcdss.2022005

Existence of solutions for fractional instantaneous and non-instantaneous impulsive differential equations with perturbation and Dirichlet boundary value

College of Electronic and Information Engineering, Southwest University, Chongqing 400715, China

* Corresponding author: Chuandong Li

Received  August 2021 Revised  October 2021 Published  July 2022 Early access  January 2022

A class of fractional instantaneous and non-instantaneous impulsive differential equations under Dirichlet boundary value conditions with perturbation is considered here. The existence of classical solutions is presented by using the Weierstrass theorem. An example is given to verify the validity of the obtained results.

Citation: Yinuo Wang, Chuandong Li, Hongjuan Wu, Hao Deng. Existence of solutions for fractional instantaneous and non-instantaneous impulsive differential equations with perturbation and Dirichlet boundary value. Discrete and Continuous Dynamical Systems - S, 2022, 15 (7) : 1767-1776. doi: 10.3934/dcdss.2022005
References:
[1]

M. Badiale and E. Serra, Semilinear Elliptic Eequations for Beginners: Existence Results Via the Variational Approach, Universitext. Springer, London, 2011. doi: 10.1007/978-0-85729-227-8.

[2]

J. Bernal, Shape analysis, Lebesgue integration and Absolute continuity connections, Universitext. Springer, London, 2019.

[3]

L. ChenZ. HeC. Li and H. Umar, On existence and continuation of solutions of the state-dependent impulsive dynamical system with boundary constraints, Adv. Difference Equ., 2019 (2019), 1-15.  doi: 10.1186/s13662-019-2070-z.

[4]

P. ChenX. Zhang and Y. Li, Iterative method for a new class of evolution equations with non-instantaneous impulses, Taiwanese J. Math., 21 (2017), 913-942.  doi: 10.11650/tjm/7912.

[5]

Z. HeC. LiZ. Cao and H. Li, Periodicity and global exponential periodic synchronization of delayed neural networks with discontinuous activations and impulsive perturbations, Neurocomputing (Amsterdam), 431 (2021), 111-127.  doi: 10.1016/j.neucom.2020.09.080.

[6]

Z. HeC. LiL. Chen and Z. Cao, Dynamic behaviors of the FitzHugh-Nagumo neuron model with state-dependent impulsive effects, Neural Networks, 121 (2020), 497-511.  doi: 10.1016/j.neunet.2019.09.031.

[7]

Z. He, C. Li, H. Li and Q. Zhang, Global exponential stability of high-order hopfield neural networks with state-dependent impulses, Phys. A, 542 (2020), 123434, 21 pp. doi: 10.1016/j. physa. 2019.123434.

[8]

E. Hernández and D. O'Regan, On hybrid impulsive and switching neural networks, Proceedings of the American Mathematical Society, 5 (2013), 1641-1649. 

[9]

F. Jiao and Y. Zhou, Existence of solutions for a class of fractional boundary value problems via critical point theory, Comput. Math. Appl., 62 (2011), 1181-1199.  doi: 10.1016/j.camwa.2011.03.086.

[10]

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

[11]

C. LiG. Feng and T. Huang, On hybrid impulsive and switching neural networks, IEEE Transactions on Systems, Man and Cybernetics. Part B, Cybernetics, 6 (2008), 1549-1560. 

[12]

C. LiS. WuG. Feng and X. Liao, Stabilizing effects of impulses in discrete-time delayed neural networks, IEEE Transactions on Neural Networks, 22 (2011), 323-329.  doi: 10.1109/TNN.2010.2100084.

[13]

X. LiD. Peng and J. Cao, Lyapunov stability for impulsive systems via event-triggered impulsive control, IEEE Trans. Automat. Control, 65 (2020), 4908-4913.  doi: 10.1109/TAC.2020.2964558.

[14]

X. LiS. Song and J. Wu, Exponential stability of nonlinear systems with delayed impulses and applications, IEEE Trans. Automat. Control, 64 (2019), 4024-4034.  doi: 10.1109/TAC.2019.2905271.

[15]

X. Li and J. Wu, Sufficient stability conditions of nonlinear differential systems under impulsive control with state-dependent delay, IEEE Trans. Automat. Control, 63 (2018), 306-311.  doi: 10.1109/TAC.2016.2639819.

[16]

A. RontóI. Rach${\rm{\dot u}}$nkováM. Rontó and L. Rach${\rm{\dot u}}$nek, Investigation of solutions of state-dependent multi-impulsive boundary value problems, Georgian Math. J., 24 (2017), 287-312.  doi: 10.1515/gmj-2016-0084.

[17]

J. Shen and W. Wang, Impulsive boundary value problems with nonlinear boundary conditions, Nonlinear Anal., 69 (2008), 4055-4062.  doi: 10.1016/j.na.2007.10.036.

[18]

Y. Tian and M. Zhang, Variational method to differential equations with instantaneous and non-instantaneous impulses, Appl. Math. Lett., 94 (2019), 160-165.  doi: 10.1016/j.aml.2019.02.034.

[19]

J. Wang, Stability of noninstantaneous impulsive evolution equations, Appl. Math. Lett., 73 (2017), 157-162.  doi: 10.1016/j.aml.2017.04.010.

[20]

J. Wang and X. Li, Periodic BVP for integer/fractional order nonlinear differential equations with non-instantaneous impulses, J. Appl. Math. Comput., 46 (2014), 321-334.  doi: 10.1007/s12190-013-0751-4.

[21]

D. Yang and J. Wang, Integral boundary value problems for nonlinear non-instantaneous impulsive differential equations, J. Appl. Math. Comput., 55 (2017), 59-78.  doi: 10.1007/s12190-016-1025-8.

[22]

X. Yu and J. Wang, Periodic boundary value problems for nonlinear impulsive evolution equations on Banach spaces, Commun. Nonlinear Sci. Numer. Simul., 22 (2015), 980-989.  doi: 10.1016/j.cnsns.2014.10.010.

[23]

W. Zhang and W. Liu, Variational approach to fractional Dirichlet problem with instantaneous and non-instantaneous impulses, Appl. Math. Lett., 99 (2020), 105993, 7 pp. doi: 10.1016/j. aml. 2019.07.024.

[24]

X. Zhang, C. Li and Z. He, Cluster synchronization of delayed coupled neural networks: Delay-dependent distributed impulsive control, Neural Networks, 142 (2021), 34–43. doi: 10.1016/j. neunet. 2021.04.026.

[25]

J. Zhou, Y. Deng and Y. Wang, Variational approach to $p$-Laplacian fractional differential equations with instantaneous and non-instantaneous impulses, Appl. Math. Lett., 104 (2020), 106251, 9 pp. doi: 10.1016/j. aml. 2020.106251.

show all references

References:
[1]

M. Badiale and E. Serra, Semilinear Elliptic Eequations for Beginners: Existence Results Via the Variational Approach, Universitext. Springer, London, 2011. doi: 10.1007/978-0-85729-227-8.

[2]

J. Bernal, Shape analysis, Lebesgue integration and Absolute continuity connections, Universitext. Springer, London, 2019.

[3]

L. ChenZ. HeC. Li and H. Umar, On existence and continuation of solutions of the state-dependent impulsive dynamical system with boundary constraints, Adv. Difference Equ., 2019 (2019), 1-15.  doi: 10.1186/s13662-019-2070-z.

[4]

P. ChenX. Zhang and Y. Li, Iterative method for a new class of evolution equations with non-instantaneous impulses, Taiwanese J. Math., 21 (2017), 913-942.  doi: 10.11650/tjm/7912.

[5]

Z. HeC. LiZ. Cao and H. Li, Periodicity and global exponential periodic synchronization of delayed neural networks with discontinuous activations and impulsive perturbations, Neurocomputing (Amsterdam), 431 (2021), 111-127.  doi: 10.1016/j.neucom.2020.09.080.

[6]

Z. HeC. LiL. Chen and Z. Cao, Dynamic behaviors of the FitzHugh-Nagumo neuron model with state-dependent impulsive effects, Neural Networks, 121 (2020), 497-511.  doi: 10.1016/j.neunet.2019.09.031.

[7]

Z. He, C. Li, H. Li and Q. Zhang, Global exponential stability of high-order hopfield neural networks with state-dependent impulses, Phys. A, 542 (2020), 123434, 21 pp. doi: 10.1016/j. physa. 2019.123434.

[8]

E. Hernández and D. O'Regan, On hybrid impulsive and switching neural networks, Proceedings of the American Mathematical Society, 5 (2013), 1641-1649. 

[9]

F. Jiao and Y. Zhou, Existence of solutions for a class of fractional boundary value problems via critical point theory, Comput. Math. Appl., 62 (2011), 1181-1199.  doi: 10.1016/j.camwa.2011.03.086.

[10]

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

[11]

C. LiG. Feng and T. Huang, On hybrid impulsive and switching neural networks, IEEE Transactions on Systems, Man and Cybernetics. Part B, Cybernetics, 6 (2008), 1549-1560. 

[12]

C. LiS. WuG. Feng and X. Liao, Stabilizing effects of impulses in discrete-time delayed neural networks, IEEE Transactions on Neural Networks, 22 (2011), 323-329.  doi: 10.1109/TNN.2010.2100084.

[13]

X. LiD. Peng and J. Cao, Lyapunov stability for impulsive systems via event-triggered impulsive control, IEEE Trans. Automat. Control, 65 (2020), 4908-4913.  doi: 10.1109/TAC.2020.2964558.

[14]

X. LiS. Song and J. Wu, Exponential stability of nonlinear systems with delayed impulses and applications, IEEE Trans. Automat. Control, 64 (2019), 4024-4034.  doi: 10.1109/TAC.2019.2905271.

[15]

X. Li and J. Wu, Sufficient stability conditions of nonlinear differential systems under impulsive control with state-dependent delay, IEEE Trans. Automat. Control, 63 (2018), 306-311.  doi: 10.1109/TAC.2016.2639819.

[16]

A. RontóI. Rach${\rm{\dot u}}$nkováM. Rontó and L. Rach${\rm{\dot u}}$nek, Investigation of solutions of state-dependent multi-impulsive boundary value problems, Georgian Math. J., 24 (2017), 287-312.  doi: 10.1515/gmj-2016-0084.

[17]

J. Shen and W. Wang, Impulsive boundary value problems with nonlinear boundary conditions, Nonlinear Anal., 69 (2008), 4055-4062.  doi: 10.1016/j.na.2007.10.036.

[18]

Y. Tian and M. Zhang, Variational method to differential equations with instantaneous and non-instantaneous impulses, Appl. Math. Lett., 94 (2019), 160-165.  doi: 10.1016/j.aml.2019.02.034.

[19]

J. Wang, Stability of noninstantaneous impulsive evolution equations, Appl. Math. Lett., 73 (2017), 157-162.  doi: 10.1016/j.aml.2017.04.010.

[20]

J. Wang and X. Li, Periodic BVP for integer/fractional order nonlinear differential equations with non-instantaneous impulses, J. Appl. Math. Comput., 46 (2014), 321-334.  doi: 10.1007/s12190-013-0751-4.

[21]

D. Yang and J. Wang, Integral boundary value problems for nonlinear non-instantaneous impulsive differential equations, J. Appl. Math. Comput., 55 (2017), 59-78.  doi: 10.1007/s12190-016-1025-8.

[22]

X. Yu and J. Wang, Periodic boundary value problems for nonlinear impulsive evolution equations on Banach spaces, Commun. Nonlinear Sci. Numer. Simul., 22 (2015), 980-989.  doi: 10.1016/j.cnsns.2014.10.010.

[23]

W. Zhang and W. Liu, Variational approach to fractional Dirichlet problem with instantaneous and non-instantaneous impulses, Appl. Math. Lett., 99 (2020), 105993, 7 pp. doi: 10.1016/j. aml. 2019.07.024.

[24]

X. Zhang, C. Li and Z. He, Cluster synchronization of delayed coupled neural networks: Delay-dependent distributed impulsive control, Neural Networks, 142 (2021), 34–43. doi: 10.1016/j. neunet. 2021.04.026.

[25]

J. Zhou, Y. Deng and Y. Wang, Variational approach to $p$-Laplacian fractional differential equations with instantaneous and non-instantaneous impulses, Appl. Math. Lett., 104 (2020), 106251, 9 pp. doi: 10.1016/j. aml. 2020.106251.

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