Diamond Alpha Differentiability of Interval-Valued Functions and Its Applicability to Interval Differential Equations on Time Scales

Document Type : Research Paper

Authors

1 Department of Mathematics, Faculty of Science, University of Ostrava, Czech Republic

2 Applied Analysis Research Group, Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam

Abstract

Modelling phenomena with interval differential equations (IDEs) is an effective way to consider the uncertainties that are unavoidable when collecting data. Similarly to the theory of ordinary differential equations, IDEs have been parallelly investigated with the interval difference equations from the beginning. These two branches can be regarded as one when unifying continuous and discrete solution domains. A conspicuous advantage when merging these areas is that the proof of several analogous properties in both theories need not be repeated. The paper provides a common and efficient tool for studying IDEs not only with continuous or discrete solution domains but also with more general ones. We propose the diamond-$\alpha$ derivative for interval-valued functions (IVFs) on time scales with respect to the generalized Hukuhara difference. Differently from most of the studies on the derivatives of functions on time scales, using the language of epsilon-delta, the novel concept is naturally studied according to the limit of IVFs on time scales as in classical mathematics. A particular class of IDEs on time scales is then considered with respect to the diamond-$\alpha$ derivative. Numerical problems are elaborated to illustrate the necessity and efficiency of the latter.

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References

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Iranian Journal of Fuzzy Systems
Volume 21, Issue 1
January and February 2024
Pages 143-158

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