Linear fuzzy Volterra integral equations on time scales
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Linear fuzzy Volterra integral equations on time scales M. Shahidi1 · A. Khastan1 Received: 4 July 2019 / Revised: 6 May 2020 / Accepted: 21 May 2020 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2020
Abstract In this paper, we study linear fuzzy Volterra integral equations on time scales and we show that under some appropriate hypotheses, they are equivalent to the first-order linear fuzzy dynamic equations. Moreover, an existence theorem to fuzzy Volterra integral equations on time scales is presented. Some examples are provided to illustrate our results. Keywords Time scales · Linear fuzzy Volterra integral equations · Fuzzy initial value problems · Generalized differentiability Mathematics Subject Classification 34A07 · 45D05
1 Introduction The calculus of time scales was initiated by Hilger (1990). Recently, the theory of time scales has gained much attention. This is not only because it can unify continuous and discrete calculus, but also because the time scale calculus has tremendous potential for applications such as population dynamics, economics, heat transfer, etc. (Bohner and Peterson 2001; Georgiev 2016). Also, because of its hybrid formalism, in the past few years, this area of mathematics has received considerable attention (Guseinov 2003; Hong and Peng 2016; Mozyrska et al. 2019; Zhao et al. 2019). The key concepts of the study of dynamic equations on time scales and integral equations on time scales are the way of unifying and extending the continuous and discrete mathematical analysis, which allows us to generalize a process to deal with both continuous and discrete cases and any combination (Bohner and Peterson 2001; Georgiev 2016). Dynamic equations play a significant role in applications. Therefore, many authors have developed researches in dynamic equations on time scales and other aspects such as using local fractional calculus to study the local scaling behavior of a function, especially in the
Communicated by José Tenreiro Machado.
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M. Shahidi [email protected] A. Khastan [email protected]
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Department of Mathematics, Institute for Advanced Studies in Basic Sciences, Zanjan, Iran 0123456789().: V,-vol
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case of a fractal or a multifractal function (Bohner and Peterson 2001; Yang et al. 2016, 2017a, 2018, 2017b; Yang and Tenreiro Machado 2019). In Georgiev (2016), several methods are applied for solving Volterra integral equations on time scales. Indeed, Volterra integral equations are converted to initial value problems and Fredholm integral equations are converted to boundary value problems on time scales and vice versa. Moreover, some methods are presented to provide a scheme for finding an approximate solution of the Volterra integral equations as infinite series such as the Adomian decomposition method, the successive iterations method to obtain the solutions in an exact form and a series form (Georgiev 2016). Later, dynamic equations on time scales for multivalued functions were developed by Hong (200
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