This paper dealt with the infinite controllability of delayed evolution inclusions with $ \alpha $-order fractional derivatives in Fr$ \acute{e} $chet spaces, where $ \alpha\in (1, 2) $. The controllability conclusion was acquired without any compactness for the nonlinear term, the cosine family, and the sine family. The investigation was based on a nonlinear alternative and the cosine family theory. An application of our findings was provided.
Citation: Yue Liang. Study on the controllability of delayed evolution inclusions involving fractional derivatives[J]. AIMS Mathematics, 2024, 9(7): 17984-17996. doi: 10.3934/math.2024876
This paper dealt with the infinite controllability of delayed evolution inclusions with $ \alpha $-order fractional derivatives in Fr$ \acute{e} $chet spaces, where $ \alpha\in (1, 2) $. The controllability conclusion was acquired without any compactness for the nonlinear term, the cosine family, and the sine family. The investigation was based on a nonlinear alternative and the cosine family theory. An application of our findings was provided.
| [1] |
K. X. Li, J. G. Peng, J. X. Jia, Cauchy problems for fractional differential equations with Riemann-Liouville fractional derivatives, J. Funct. Anal., 263 (2012), 476–510. https://doi.org/10.1016/j.jfa.2012.04.011 doi: 10.1016/j.jfa.2012.04.011
|
| [2] |
R. R. Nigmatullin, To the theoretical explanation of the "universal response", Phys. Stat. Sol. B, 123 (1984), 739–745. https://doi.org/10.1002/pssb.2221230241 doi: 10.1002/pssb.2221230241
|
| [3] |
W. Arendt, Vector-valued Laplace transforms and Cauchy problems, Israel J. Math., 59 (1987), 327–352. https://doi.org/10.1007/BF02774144 doi: 10.1007/BF02774144
|
| [4] |
Y. Zhou, J. W. He, New results on controllability of fractional evolution systems with order $\alpha\in (1, 2)$, Evol. Equ. Control Theory, 10 (2021), 491–509. https://doi.org/10.3934/eect.2020077 doi: 10.3934/eect.2020077
|
| [5] |
H. Yang, Existence and approximate controllability of Riemann-Liouville fractional evolution equations of order $1 < \mu < 2$ with weighted time delay, Bull. Sci. Math., 187 (2023), 103303. https://doi.org/10.1016/j.bulsci.2023.103303 doi: 10.1016/j.bulsci.2023.103303
|
| [6] |
H. D. Gou, Y. X. Li, Existence and approximate controllability of Hilfer fractional evolution equations in Banach spaces, J. Appl. Anal. Comput., 11 (2021), 2895–2920. https://doi.org/10.11948/20210053 doi: 10.11948/20210053
|
| [7] |
J. W. He, L. Peng, Approximate controllability for a class of fractional stochastic wave equations, Comput. Math. Appl., 78 (2019), 1463–1476. https://doi.org/10.1016/j.camwa.2019.01.012 doi: 10.1016/j.camwa.2019.01.012
|
| [8] | M. Benchohra, S. Ntouyas, Controllability on infinite time horizon of nonlinear differential equations in Banach spaces with nonlocal conditions, An. Stiint. Univ. Al. I. Cuza. Iasi. Mat., 47 (2001), 277–286. |
| [9] | M. Benchohra, S. Ntouyas, Controllability of neutral functional differential and integrodifferential inclusions in Banach spaces, Italian J. Pure Appl. Math., 14 (2003), 95–112. |
| [10] |
M. Benchohra, A. Ouahab, Controllability results for functional semilinear differential inclusions in Fr$\acute{e}$chet spaces, Nonlinear Anal. Theor., 61 (2005), 405–423. https://doi.org/10.1016/j.na.2004.12.002 doi: 10.1016/j.na.2004.12.002
|
| [11] | M. Frigon, Fixed point results for multivalued contractions on Gauge spces, In: Set valued mappings with applications in nonlinear analysis, 4 Eds., London: Taylor and Francis, 2002. |
| [12] | A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, Amsterdam, Boston: Elsevier, 2006. |
| [13] | Y. Zhou, Fractional evolution equations and inclusions: Analysis and control, Academic Press, 2016. https://doi.org/10.1016/C2015-0-00813-9 |
| [14] |
C. C. Travis, G. F. Webb, Consine families and abstract nonlinear second order differential equations, Acta Math. Hung., 32 (1978), 75–96. https://doi.org/10.1007/BF01902205 doi: 10.1007/BF01902205
|
| [15] | F. Mainardi, Fractional calculus and waves in linear viscoelasticity: An introduction to mathematical models, Imperial College Press, 2010. |
| [16] | I. Podlubny, Fractional differential equations, San Diego: Academic Press, 1999. |
| [17] | J. W. Hanneken, D. M. Vaught, B. N. Narahari Achar, Enumeration of the real zeros of the Mittag-Leffler function $E_{\alpha}(z), 1 < \alpha < 2$, In: Advances in fractional calculus, Dordrecht: Springer, 2007. https://doi.org/10.1007/978-1-4020-6042-7_2 |
Yue Liang. Study on the controllability of delayed evolution inclusions involving fractional derivatives[J]. AIMS Mathematics, 2024, 9(7): 17984-17996. doi: 10.3934/math.2024876
DownLoad: