A study on impulsive fractional hybrid evolution equations using sequence method
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A study on impulsive fractional hybrid evolution equations using sequence method Haide Gou1
· Yongxiang Li1
Received: 21 March 2020 / Revised: 17 June 2020 / Accepted: 26 June 2020 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2020
Abstract In this paper, we introduce a new concept called α-order cosine-resolvent family, by using the theory of fractional calculus, the concepts of measure of noncompactness and Hybrid fixed point theorem, we consider the existence of PC-mild solutions for a class of impulsive fractional hybrid evolution equation in a Banach space . Furthermore, we obtain some sufficient conditions for approximate controllability of our concern problem. At the end, an example is given to illustrate the feasibility of our results. Keywords Impulsive fractional hybrid evolution equation · Controllability · Fractional cosine family · Mild solutions Mathematics Subject Classification 26A33 · 34K30 · 34K35 · 35R11 · 93B05
1 Introduction In recent decades, the fractional differential equations have been applied to various fields successfully, for example, physics, engineering, finance and so on. Consequently, more and more researchers pay much attention to this subject and have obtained substantial achievements, we refer the read to see (Debbouchea and Baleanu 2011; Lian et al. 2017; Chang et al. 2017; Shu and Shi 2016; Yang et al. 2017; Shu et al. 2011; Xu et al. 2020; Chen et al. 2015, 2020a, b; Kilbas et al. 2006) and the references therein. Controllability is one of the fundamental concepts in mathematical control theory and widely used in many fields of science and technology. Controllability of linear and nonlinear
Communicated by José Tenreiro Machado. Supported by the National Natural Science Foundation of China (Grant no. 11661071).
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Haide Gou [email protected] Yongxiang Li [email protected]
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Department of Mathematics, Northwest Normal University, Lanzhou 730070, People’s Republic of China 0123456789().: V,-vol
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H. Gou, Y. Li
systems represented by ordinary differential equations in finite-dimensional space has been extensively studied. Some authors have extend the concept to infinite-dimensional systems in Banach spaces. The approximate controllability of the systems of integer order ( α = 1, 2) has been proved in Zhou (1983); Shukla et al. (2015); Jeong et al. (2016) and the references therein. To show the results on the approximate controllability a relation between the reachable set of a semilinear system and that of the corresponding linear system is proved. Sakthivel et al. (2011) proved the approximate controllability by assuming that the C0 -semigroup T (t) is compact and the nonlinear function is continuous and uniformly bounded. Recently, Kumar and Sukavanam (2011) proved the approximate controllability for a class of semilinear delayed control system of fractional order by assuming that the corresponding linear system is approximately controllable and nonlinear function satisfies the Lipschitz condition. On the other hand, contro
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