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August  2022, 27(8): 4455-4471. doi: 10.3934/dcdsb.2021236

## Existence of solutions for first-order Hamiltonian random impulsive differential equations with Dirichlet boundary conditions

 College of Mathematics, Hunan University, Changsha 410012, China

* Corresponding author: Xiao-Bao Shu

Yu Guo and Qianbao Yin contributed equally to this paper

Received  May 2021 Revised  June 2021 Published  August 2022 Early access  September 2021

In this paper, we study the sufficient conditions for the existence of solutions of first-order Hamiltonian random impulsive differential equations under Dirichlet boundary value conditions. By using the variational method, we first obtain the corresponding energy functional. And by using Legendre transformation, we obtain the conjugation of the functional. Then the existence of critical point is obtained by mountain pass lemma. Finally, we assert that the critical point of the energy functional is the mild solution of the first order Hamiltonian random impulsive differential equation. Finally, an example is presented to illustrate the feasibility and effectiveness of our results.

Citation: Yu Guo, Xiao-Bao Shu, Qianbao Yin. Existence of solutions for first-order Hamiltonian random impulsive differential equations with Dirichlet boundary conditions. Discrete and Continuous Dynamical Systems - B, 2022, 27 (8) : 4455-4471. doi: 10.3934/dcdsb.2021236
##### References:
 [1] R. P. Agarwal and D. O'Rgean, A multiplicity result for second order impulsive differential equations via the Leggett Williams fixed point theorem, Appl. Math. Comput., 161 (2005), 433-439.  doi: 10.1016/j.amc.2003.12.096. [2] A. Ambrosetti and G. Mancini, Solutions of minimal period for a class of convex Hamiltonian systems, Math. Ann., 255 (1981), 405-421.  doi: 10.1007/BF01450713. [3] T. E. Carter, Optimal impulsive space trajectories based on linear equations, J. Optim. Theory Appl., 70 (1991), 277-297.  doi: 10.1007/BF00940627. [4] L. Chen and J. Sun, Nonlinear boundary value problem of first order impulsive functional differential equations, J. Math. Anal. Appl., 318 (2006), 726-741.  doi: 10.1016/j.jmaa.2005.08.012. [5] P. Chen and X. Tang, Existence and multiplicity of solutions for second-order impulsive differential equations with Dirichlet problems, Appl. Math. Comput., 218 (2012), 11775-11789.  doi: 10.1016/j.amc.2012.05.027. [6] S. Deng, X.-B. Shu and J. Mao, Existence and exponential stability for impulsive neutral stochastic functional differential equations driven by fBm with noncompact semigroup via Mönch fixed point, J. Math. Anal. Appl., 467 (2018), 398-420.  doi: 10.1016/j.jmaa.2018.07.002. [7] P. R. George, A. K. Nandakumaran and A. Arapostathis, A note on controllability of impulsive systems, J. Math. Anal. Appl., 241 (2000), 276-283.  doi: 10.1006/jmaa.1999.6632. [8] Z.-H. Guan, G. Chen and T. Ueta, On impulsive control of a periodically forced chaotic pendulum system, IEEE Trans. Automat. Control, 45 (2000), 1724-1727.  doi: 10.1109/9.880633. [9] Y. Guo, X.-B. Shu, Y. Li and F. Xu, The existence and Hyers–Ulam stability of solution for an impulsive Riemann-Liouville fractional neutral functional stochastic differential equation with infinite delay of order $1 < \beta < 2$, Bound. Value Probl., (2019), Paper No. 59. doi: 10.1186/s13661-019-1172-6. [10] J. Li, J. J. Nieto and J. Shen, Impulsive periodic boundary value problems of first-order differ- ential equations, J. Math. Anal. Appl., 325 (2007), 226-236.  doi: 10.1016/j.jmaa.2005.04.005. [11] S. Li, L. Shu, X.-B. Shu and F. Xu, Existence and Hyers-Ulam stability of random impulsive stochastic functional differential equations with finite delays, Stochastics, 91 (2019), 857-872.  doi: 10.1080/17442508.2018.1551400. [12] J. Q. Liu and Z. Q. Wang, Remarks on subharmonics with minimal periods of Hamiltonian systems, Nonlinear Anal., 20 (1993), 803-821.  doi: 10.1016/0362-546X(93)90070-9. [13] X. Liu and A. R. Willms, Impulsive controllability of linear dynamical systems with applications to maneuvers of Spacecraft, Math. Problems Engineer., 2 (1996), 277-299.  doi: 10.1155/S1024123X9600035X. [14] J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4757-2061-7. [15] J. J. Nieto, Variational formulation of a damped Dirichlet impulsive problem, Appl. Math. Lett., 23 (2010), 940-942.  doi: 10.1016/j.aml.2010.04.015. [16] J. J. Nietoa and D. O'Regan, Variational approach to impulsive differential equations, Nonlinear Anal. Real World Appl., 10 (2009), 680-690.  doi: 10.1016/j.nonrwa.2007.10.022. [17] P. P. Niu, X.-B. Shu and Y. J. Li, The Existence and Hyers-Ulam stability for second order random impulsive differential equation, Dynamic Systems and Applications, 28 (2019), 673-690. [18] A. F. B. A. Prado, Bi-impulsive control to build a satellite constellation, Nonlinear Dyn. Syst. Theory, 5 (2005), 169-175. [19] 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. [20] X.-B. Shu and Y. Shi, A study on the mild solution of impulsive fractional evolution equations, Appl. Math. Comput., 273 (2016), 465-476.  doi: 10.1016/j.amc.2015.10.020. [21] J. Sun, H. Chen and J. J. Nieto, Infinitely many solutions for second-order Hamiltonian system with impulsive effects, Math. Comput. Modelling, 54 (2011), 544-555.  doi: 10.1016/j.mcm.2011.02.044. [22] J. Sun, H. Chen, J. J. Nieto and M. Otero-Novoa, The multiplicity of solutions for perturbed second-order Hamiltonian systems with impulsive effects, Nonlinear Anal., 72 (2010), 4575-4586.  doi: 10.1016/j.na.2010.02.034. [23] Y. Tian and W. G. Ge, Applications of variational methods to boundary value problem for impulsive differ- ential equations, Proc. Edinb. Math. Soc., 51 (2008), 509-527.  doi: 10.1017/S0013091506001532. [24] S. Wu and Y. Duan, Oscillation stability and boundedness of second-order differential systems with random impulses, Comput. Math. Appl., 49 (2005), 1375-1386.  doi: 10.1016/j.camwa.2004.12.009. [25] S.-J. Wu, X.-L. Guo and S.-Q. Lin, Existence and uniqueness of solutions to random impulsive differential systems, Acta Math. Appl. Sin. Engl. Ser., 22 (2006), 627-632.  doi: 10.1007/s10255-006-0336-1. [26] S.-J. Wu and X.-Z Meng, Boundedness of nonlinear differential systems with impulsive effect on random moments, Acta Math. Appl. Sin. Engl. Ser., 20 (2004), 147-154.  doi: 10.1007/s10255-004-0157-z. [27] X. Xian, D. O'Regan and R. P. Agarwa, Multiplicity results via topological degree for impulsive boundary value problems under non-well-ordered upper and lower solution conditions, Bound. Value Probl., (2008), Art. ID 197205, 21 pp. doi: 10.1155/2008/197205. [28] J. Xie, J. Li and Z. Luo, Periodic and subharmonic solutions for a class of the second-order Hamiltonian systems with impulsive effects, Bound Value Probl., 2015 (2015), Article number 52, 10 pp. doi: 10.1186/s13661-015-0313-9. [29] J. Yu, H. Bin and Z. Guo, Periodic solutions for discrete convex Hamiltonian systems via Clarke duality[EB/OL], Discrete Contin. Dyn. Syst., 15 (2006), 939-950.  doi: 10.3934/dcds.2006.15.939. [30] Z. Zhang and R. Yuan, An application of variational methods to Dirichlet boundary value problem with impulses, Nonlinear Analysis: Real World Applications, 11 (2010), 155-162.  doi: 10.1016/j.nonrwa.2008.10.044. [31] J. Zhou and Y. Li, Existence of solutions for a class of second order Hamiltonian systems with impulsive effects, Nonlinear Anal., 72 (2010), 1594-1603.  doi: 10.1016/j.na.2009.08.041.

show all references

##### References:
 [1] R. P. Agarwal and D. O'Rgean, A multiplicity result for second order impulsive differential equations via the Leggett Williams fixed point theorem, Appl. Math. Comput., 161 (2005), 433-439.  doi: 10.1016/j.amc.2003.12.096. [2] A. Ambrosetti and G. Mancini, Solutions of minimal period for a class of convex Hamiltonian systems, Math. Ann., 255 (1981), 405-421.  doi: 10.1007/BF01450713. [3] T. E. Carter, Optimal impulsive space trajectories based on linear equations, J. Optim. Theory Appl., 70 (1991), 277-297.  doi: 10.1007/BF00940627. [4] L. Chen and J. Sun, Nonlinear boundary value problem of first order impulsive functional differential equations, J. Math. Anal. Appl., 318 (2006), 726-741.  doi: 10.1016/j.jmaa.2005.08.012. [5] P. Chen and X. Tang, Existence and multiplicity of solutions for second-order impulsive differential equations with Dirichlet problems, Appl. Math. Comput., 218 (2012), 11775-11789.  doi: 10.1016/j.amc.2012.05.027. [6] S. Deng, X.-B. Shu and J. Mao, Existence and exponential stability for impulsive neutral stochastic functional differential equations driven by fBm with noncompact semigroup via Mönch fixed point, J. Math. Anal. Appl., 467 (2018), 398-420.  doi: 10.1016/j.jmaa.2018.07.002. [7] P. R. George, A. K. Nandakumaran and A. Arapostathis, A note on controllability of impulsive systems, J. Math. Anal. Appl., 241 (2000), 276-283.  doi: 10.1006/jmaa.1999.6632. [8] Z.-H. Guan, G. Chen and T. Ueta, On impulsive control of a periodically forced chaotic pendulum system, IEEE Trans. Automat. Control, 45 (2000), 1724-1727.  doi: 10.1109/9.880633. [9] Y. Guo, X.-B. Shu, Y. Li and F. Xu, The existence and Hyers–Ulam stability of solution for an impulsive Riemann-Liouville fractional neutral functional stochastic differential equation with infinite delay of order $1 < \beta < 2$, Bound. Value Probl., (2019), Paper No. 59. doi: 10.1186/s13661-019-1172-6. [10] J. Li, J. J. Nieto and J. Shen, Impulsive periodic boundary value problems of first-order differ- ential equations, J. Math. Anal. Appl., 325 (2007), 226-236.  doi: 10.1016/j.jmaa.2005.04.005. [11] S. Li, L. Shu, X.-B. Shu and F. Xu, Existence and Hyers-Ulam stability of random impulsive stochastic functional differential equations with finite delays, Stochastics, 91 (2019), 857-872.  doi: 10.1080/17442508.2018.1551400. [12] J. Q. Liu and Z. Q. Wang, Remarks on subharmonics with minimal periods of Hamiltonian systems, Nonlinear Anal., 20 (1993), 803-821.  doi: 10.1016/0362-546X(93)90070-9. [13] X. Liu and A. R. Willms, Impulsive controllability of linear dynamical systems with applications to maneuvers of Spacecraft, Math. Problems Engineer., 2 (1996), 277-299.  doi: 10.1155/S1024123X9600035X. [14] J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4757-2061-7. [15] J. J. Nieto, Variational formulation of a damped Dirichlet impulsive problem, Appl. Math. Lett., 23 (2010), 940-942.  doi: 10.1016/j.aml.2010.04.015. [16] J. J. Nietoa and D. O'Regan, Variational approach to impulsive differential equations, Nonlinear Anal. Real World Appl., 10 (2009), 680-690.  doi: 10.1016/j.nonrwa.2007.10.022. [17] P. P. Niu, X.-B. Shu and Y. J. Li, The Existence and Hyers-Ulam stability for second order random impulsive differential equation, Dynamic Systems and Applications, 28 (2019), 673-690. [18] A. F. B. A. Prado, Bi-impulsive control to build a satellite constellation, Nonlinear Dyn. Syst. Theory, 5 (2005), 169-175. [19] 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. [20] X.-B. Shu and Y. Shi, A study on the mild solution of impulsive fractional evolution equations, Appl. Math. Comput., 273 (2016), 465-476.  doi: 10.1016/j.amc.2015.10.020. [21] J. Sun, H. Chen and J. J. Nieto, Infinitely many solutions for second-order Hamiltonian system with impulsive effects, Math. Comput. Modelling, 54 (2011), 544-555.  doi: 10.1016/j.mcm.2011.02.044. [22] J. Sun, H. Chen, J. J. Nieto and M. Otero-Novoa, The multiplicity of solutions for perturbed second-order Hamiltonian systems with impulsive effects, Nonlinear Anal., 72 (2010), 4575-4586.  doi: 10.1016/j.na.2010.02.034. [23] Y. Tian and W. G. Ge, Applications of variational methods to boundary value problem for impulsive differ- ential equations, Proc. Edinb. Math. Soc., 51 (2008), 509-527.  doi: 10.1017/S0013091506001532. [24] S. Wu and Y. Duan, Oscillation stability and boundedness of second-order differential systems with random impulses, Comput. Math. Appl., 49 (2005), 1375-1386.  doi: 10.1016/j.camwa.2004.12.009. [25] S.-J. Wu, X.-L. Guo and S.-Q. Lin, Existence and uniqueness of solutions to random impulsive differential systems, Acta Math. Appl. Sin. Engl. Ser., 22 (2006), 627-632.  doi: 10.1007/s10255-006-0336-1. [26] S.-J. Wu and X.-Z Meng, Boundedness of nonlinear differential systems with impulsive effect on random moments, Acta Math. Appl. Sin. Engl. Ser., 20 (2004), 147-154.  doi: 10.1007/s10255-004-0157-z. [27] X. Xian, D. O'Regan and R. P. Agarwa, Multiplicity results via topological degree for impulsive boundary value problems under non-well-ordered upper and lower solution conditions, Bound. Value Probl., (2008), Art. ID 197205, 21 pp. doi: 10.1155/2008/197205. [28] J. Xie, J. Li and Z. Luo, Periodic and subharmonic solutions for a class of the second-order Hamiltonian systems with impulsive effects, Bound Value Probl., 2015 (2015), Article number 52, 10 pp. doi: 10.1186/s13661-015-0313-9. [29] J. Yu, H. Bin and Z. Guo, Periodic solutions for discrete convex Hamiltonian systems via Clarke duality[EB/OL], Discrete Contin. Dyn. Syst., 15 (2006), 939-950.  doi: 10.3934/dcds.2006.15.939. [30] Z. Zhang and R. Yuan, An application of variational methods to Dirichlet boundary value problem with impulses, Nonlinear Analysis: Real World Applications, 11 (2010), 155-162.  doi: 10.1016/j.nonrwa.2008.10.044. [31] J. Zhou and Y. Li, Existence of solutions for a class of second order Hamiltonian systems with impulsive effects, Nonlinear Anal., 72 (2010), 1594-1603.  doi: 10.1016/j.na.2009.08.041.
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