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An inverse problem for the pseudo-parabolic equation with p-Laplacian
$ (\omega,\mathbb{T}) $-periodic solutions of impulsive evolution equations
| 1. | Department of Mathematical Analysis and Numerical Mathematics, Faculty of Mathematics, Physics and Informatics, Comenius University, Mlynská dolina 842 48, Bratislava, Slovakia |
| 2. | Mathematical Institute, Slovak Academy of Sciences, Štefánikova 49,814 73 Bratislava, Slovakia |
| 3. | Department of Mathematics, Guizhou University, Guiyang, Guizhou 550025, P. R. China |
| 4. | College of Science, Guizhou Institute of Technology, Guiyang, Guizhou 550025, P. R. China |
| 5. | School of Mathematical Sciences, Qufu Normal University, , Qufu, Shandong 273165, P. R. China |
In this paper, we study $ (\omega,\mathbb{T}) $-periodic impulsive evolution equations via the operator semigroups theory in Banach spaces $ X $, where $ \mathbb{T}: X\rightarrow X $ is a linear isomorphism. Existence and uniqueness of $ (\omega,\mathbb{T}) $-periodic solutions results for linear and semilinear problems are obtained by Fredholm alternative theorem and fixed point theorems, which extend the related results for periodic impulsive differential equations.
References:
| [1] |
N. U. Ahmed, K. L. Teo and S. H. Hou,
Nonlinear impulsive systems on infinite dimensional spaces, Nonlinear Analysis: TMA, 54 (2003), 907-925.
doi: 10.1016/S0362-546X(03)00117-2. |
| [2] |
E. Alvarez, A. Gómez and M. Pinto, $(\omega, c)$-periodic functions and mild solutions to abstract fractional integro-differential equations, Electron. J. Qual. Theory Differ. Equ., (2018), 16–24.
doi: 10.14232/ejqtde.2018.1.16. |
| [3] |
M. Agaoglou, M. Fečkan and A. Panagiotidou,
Existence and uniqueness of $(\omega, c)$-periodic solutions of semilinear evolution equations, Int. J. Dynamical Systems and Differential Equations, 10 (2020), 149-166.
doi: 10.1504/IJDSDE.2020.106027. |
| [4] |
E. Alvarez, S. Castillo and M. Pinto, $(\omega, c)-$Pseudo periodic functions, first order Cauchy problem and Lasota-Wazewska model with ergodic and unbounded oscillating production of red cells, Bound. Value Prob., (2019), 106–126.
doi: 10.1186/s13661-019-1217-x. |
| [5] |
E. Alvarez, S. Castillo and M. Pinto,
$(\omega, c)-$asymptotically periodic functions, first-order Cauchy problem, and Lasota-Wazewska model with unbounded oscillating production of red cells, Math. Meth. Appl. Sci., 43 (2020), 305-319.
doi: 10.1002/mma.5880. |
| [6] |
D. Bainov and P. Simeonov, Theory of Impulsive Differential Equations, Series on Advances in Mathematics for Applied Sciences, vol. 28. Singapore, World Scientifc, 1995.
doi: 10.1142/9789812831804. |
| [7] |
D. Bainov and P. Simeonov, Oscillation Theory of Impulsive Differential Equations, Interna-tional Publications, 1998. |
| [8] |
C. Cooke and J. Kroll,
The existence of periodic solutions to certain impulsive differential equations, Comput. Math. Appl., 44 (2002), 667-676.
doi: 10.1016/S0898-1221(02)00181-5. |
| [9] |
X. Chang and Y. Li,
Rotating periodic solutions of second order dissipative dynamical systems, Discret. Contin. Dyn. Syst., 36 (2016), 633-652.
|
| [10] |
M. Fečkan, J. Wang and Y. Zhou,
Existence of periodic solutions for nonlinear evolution equations with non- instantaneous impulses, Nonauton. Dyn. Syst., 1 (2014), 93-101.
|
| [11] |
M. Fečkan, R. Ma and B. Thompson,
Forced symmetric oscillations, Bull. Belg. Math. Soc., 14 (2007), 73-85.
doi: 10.36045/bbms/1172852245. |
| [12] |
Y. Li, F. Cong, Z. Lin and W. liu,
Periodic solutions for evolution equations, Nonlinear Analysis: Theory, Methods and Applications, 36 (1999), 275-293.
doi: 10.1016/S0362-546X(97)00626-3. |
| [13] |
X. Li, B. Martin and C. Wang,
Impulsive differential equations: Periodic solutions and applications, Automatica, 52 (2015), 173-178.
doi: 10.1016/j.automatica.2014.11.009. |
| [14] |
M. Li, J. Wang and M. Fečkan,
$(\omega, c)$-periodic solutions for impulsive differential systems, Communications Mathematical Analysis, 21 (2018), 35-46.
|
| [15] |
K. Liu, J. Wang, D. O'Regan and M. Fečkan,
A new class of $(\omega, c)$-periodic non-instantaneous impulsive differential equations, Mediterr. J. Math., 17 (2020), 155-177.
doi: 10.1007/s00009-020-01574-8. |
| [16] |
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983.
doi: 10.1007/978-1-4612-5561-1. |
| [17] |
A. M. Samoilenko and N. A. Perestyuk, Impulsive Differential Equations, World Scientific, Singapore, 1995.
doi: 10.1142/9789812798664. |
| [18] |
C. Wang,
Almost periodic solutions of impulsive BAM neural networks with variable delays on time scales, Commun. Nonlinear Sci. Numer. Simul., 19 (2014), 2828-2842.
doi: 10.1016/j.cnsns.2013.12.038. |
| [19] |
J. Wang and M. Fečkan,
A general class of impulsive evolution equations, Topol. Meth. Nonlinear Anal., 46 (2015), 915-934.
|
| [20] |
J. Wang, X. Xiang and W. Wei, Linear impulsive periodic system with time-varying generating operators on Banach space, Adv. Differ. Equ., (2007), 26196, 16 pp.
doi: 10.1155/2007/26196. |
show all references
References:
| [1] |
N. U. Ahmed, K. L. Teo and S. H. Hou,
Nonlinear impulsive systems on infinite dimensional spaces, Nonlinear Analysis: TMA, 54 (2003), 907-925.
doi: 10.1016/S0362-546X(03)00117-2. |
| [2] |
E. Alvarez, A. Gómez and M. Pinto, $(\omega, c)$-periodic functions and mild solutions to abstract fractional integro-differential equations, Electron. J. Qual. Theory Differ. Equ., (2018), 16–24.
doi: 10.14232/ejqtde.2018.1.16. |
| [3] |
M. Agaoglou, M. Fečkan and A. Panagiotidou,
Existence and uniqueness of $(\omega, c)$-periodic solutions of semilinear evolution equations, Int. J. Dynamical Systems and Differential Equations, 10 (2020), 149-166.
doi: 10.1504/IJDSDE.2020.106027. |
| [4] |
E. Alvarez, S. Castillo and M. Pinto, $(\omega, c)-$Pseudo periodic functions, first order Cauchy problem and Lasota-Wazewska model with ergodic and unbounded oscillating production of red cells, Bound. Value Prob., (2019), 106–126.
doi: 10.1186/s13661-019-1217-x. |
| [5] |
E. Alvarez, S. Castillo and M. Pinto,
$(\omega, c)-$asymptotically periodic functions, first-order Cauchy problem, and Lasota-Wazewska model with unbounded oscillating production of red cells, Math. Meth. Appl. Sci., 43 (2020), 305-319.
doi: 10.1002/mma.5880. |
| [6] |
D. Bainov and P. Simeonov, Theory of Impulsive Differential Equations, Series on Advances in Mathematics for Applied Sciences, vol. 28. Singapore, World Scientifc, 1995.
doi: 10.1142/9789812831804. |
| [7] |
D. Bainov and P. Simeonov, Oscillation Theory of Impulsive Differential Equations, Interna-tional Publications, 1998. |
| [8] |
C. Cooke and J. Kroll,
The existence of periodic solutions to certain impulsive differential equations, Comput. Math. Appl., 44 (2002), 667-676.
doi: 10.1016/S0898-1221(02)00181-5. |
| [9] |
X. Chang and Y. Li,
Rotating periodic solutions of second order dissipative dynamical systems, Discret. Contin. Dyn. Syst., 36 (2016), 633-652.
|
| [10] |
M. Fečkan, J. Wang and Y. Zhou,
Existence of periodic solutions for nonlinear evolution equations with non- instantaneous impulses, Nonauton. Dyn. Syst., 1 (2014), 93-101.
|
| [11] |
M. Fečkan, R. Ma and B. Thompson,
Forced symmetric oscillations, Bull. Belg. Math. Soc., 14 (2007), 73-85.
doi: 10.36045/bbms/1172852245. |
| [12] |
Y. Li, F. Cong, Z. Lin and W. liu,
Periodic solutions for evolution equations, Nonlinear Analysis: Theory, Methods and Applications, 36 (1999), 275-293.
doi: 10.1016/S0362-546X(97)00626-3. |
| [13] |
X. Li, B. Martin and C. Wang,
Impulsive differential equations: Periodic solutions and applications, Automatica, 52 (2015), 173-178.
doi: 10.1016/j.automatica.2014.11.009. |
| [14] |
M. Li, J. Wang and M. Fečkan,
$(\omega, c)$-periodic solutions for impulsive differential systems, Communications Mathematical Analysis, 21 (2018), 35-46.
|
| [15] |
K. Liu, J. Wang, D. O'Regan and M. Fečkan,
A new class of $(\omega, c)$-periodic non-instantaneous impulsive differential equations, Mediterr. J. Math., 17 (2020), 155-177.
doi: 10.1007/s00009-020-01574-8. |
| [16] |
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983.
doi: 10.1007/978-1-4612-5561-1. |
| [17] |
A. M. Samoilenko and N. A. Perestyuk, Impulsive Differential Equations, World Scientific, Singapore, 1995.
doi: 10.1142/9789812798664. |
| [18] |
C. Wang,
Almost periodic solutions of impulsive BAM neural networks with variable delays on time scales, Commun. Nonlinear Sci. Numer. Simul., 19 (2014), 2828-2842.
doi: 10.1016/j.cnsns.2013.12.038. |
| [19] |
J. Wang and M. Fečkan,
A general class of impulsive evolution equations, Topol. Meth. Nonlinear Anal., 46 (2015), 915-934.
|
| [20] |
J. Wang, X. Xiang and W. Wei, Linear impulsive periodic system with time-varying generating operators on Banach space, Adv. Differ. Equ., (2007), 26196, 16 pp.
doi: 10.1155/2007/26196. |
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