Abstract
In this paper, we introduce a new metric on the space of fuzzy continuous functions on time scales by using the exponential function, e γ ( t , t 0 ) , where γ > 0 is a constant. Then, we provide some conditions to prove an existence and uniqueness theorem for solutions to nonlinear fuzzy dynamic equations. Furthermore, we present three different examples including a practical example to illustrate the main results.
Highlights
Mathematical modeling of some phenomena becomes more realistic and suitable by non-continuous dynamical equations, so, in this regard, it is necessary to consider both continuous and discrete models for such problems
We are going to incorporate these two theories and describe a new fuzzy theory that is called the theory of fuzzy dynamic equations on time scales, and it is a generalization of the theory of fuzzy differentials and fuzzy difference equations
We aim to study the existence and uniqueness of solutions to fuzzy dynamic equations on time scales with a new metric on fuzzy continuous functions on time scales, which is defined in terms of the exponential functions on time scales
Summary
Mathematical modeling of some phenomena becomes more realistic and suitable by non-continuous dynamical equations, so, in this regard, it is necessary to consider both continuous and discrete models for such problems. In [10], authors introduced and considered the notions of delta derivative and delta integral to fuzzy valued functions on time scales These definitions may accurately describe fuzzy dynamic processes where time may flow continuously and discretely at different stages in the one model; in other words, these concepts are useful in modelling fuzzy start–stop processes. We are going to incorporate these two theories and describe a new fuzzy theory that is called the theory of fuzzy dynamic equations on time scales, and it is a generalization of the theory of fuzzy differentials and fuzzy difference equations To this end, we aim to study the existence and uniqueness of solutions to fuzzy dynamic equations on time scales with a new metric on fuzzy continuous functions on time scales, which is defined in terms of the exponential functions on time scales. In the last section, the existence and uniqueness of the solution to a nonlinear fuzzy dynamic equations on time scales is established
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