Existence, Stability and Controllability Results of Coupled Fractional Dynamical System on Time Scales
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Existence, Stability and Controllability Results of Coupled Fractional Dynamical System on Time Scales Muslim Malik1
· Vipin Kumar1
Received: 13 March 2019 / Revised: 8 November 2019 © Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2019
Abstract In this work, we establish existence, uniqueness, Hyer-Ulam (HU) stability and controllability results for a coupled fractional dynamical system on time scales. Some fixed point theorems and nonlinear functional analysis have been used to establish these results. Also, we have given an example for different time scales, to show the applications of these obtained analytical results. Keywords Time scales · Existence · Stability · Controllability · Fractional coupled system Mathematics Subject Classification 34N05 · 34A12 · 93B05 · 34A08
1 Introduction The theory of fractional calculus started with a correspondence between L’Hospital and Leibniz in 1695. Presently, lots of literature are available on theoretical as well as numerical work on this topic [1–4]. It has applications in numerous fields, for example, biophysics and bioengineering, signal and image processing, mechanics, control theory, electrochemistry, biology, physics and electrical engineering [5,6]. A few years back, many scientists and engineers have shown a great interest in fractional theory due to the memory character of fractional derivative, which is the generalization of integer-order derivative and can describe many phenomena of physics, biology and finance that integer-order derivative cannot explain. In addition, stability analysis
Communicated by Norhashidah Hj. Mohd. Ali.
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Muslim Malik [email protected] Vipin Kumar [email protected]
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School of Basic Sciences, Indian Institute of Technology Mandi, Mandi, H. P., India
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M. Malik, V. Kumar
of dynamical system of an integer as well as fractional order is very important in various fields of sciences and engineering. The concept of HU-type stability has been introduced in the nineteenth century, and now it has been gained a lot of attention. As a matter of fact, HU-type stability has been taken up by a number of mathematicians and the study of this area has grown to be one of the central subjects in the mathematical analysis area. For more details on the recent advances on the HU-type stability of the ordinary and fractional differential equations, one can see [7–9] and references therein. On the other hand, one investigates the continuous and discrete cases separately and there are numerous discrete sets which are most utilizable. Thus, this is a very difficult task to analyze separately for all cases. So to avoid this sort of situation, Hilger [10], in 1988, present time scales theory which cumulates discrete and continuous analysis. This hypothesis presents a powerful actualization for applications to the heat conduction system [11], population model [12] and financial matters [13]. Thus, managing issues of differential equations on time scales turn out to be extremely noteworthy and deliber
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