Abstract
Existence and uniqueness theorem are the tool which makes it possible for us to conclude that there exists only one solution to a given problem which satisfies a constraint condition. How does it work? Why is it the case? We believe it, but it would be interesting to see the main ideas behind this. To this end, in this paper, we investigate existence, uniqueness, and other properties of solutions of a certain nonlinear fuzzy Volterra integrodifferential equation under strongly generalized differentiability. The main tools employed in the analysis are based on the applications of the Banach fixed point theorem and a certain integral inequality with explicit estimate. Also, some results for characterizing solution by an equivalent system of crisp Volterra integrodifferential equations are presented. In this way, a new direction for the methods of analytic and approximate solutions is proposed.
Highlights
Many important real-world problems of analytical dynamics are described by the nonlinear mathematical models that, as a rule, are presented and modeled by the nonlinear crisp integrodifferential equations (IDEs)
We present an algorithm to solve the new system which consists of two crisp Volterra IDEs (VIDEs)
We know that solving fuzzy IDEs requires appropriate and applicable definitions and theorems to accomplish the mathematical construction
Summary
Many important real-world problems of analytical dynamics are described by the nonlinear mathematical models that, as a rule, are presented and modeled by the nonlinear crisp (ordinary) integrodifferential equations (IDEs). The third approach is based on the Zadeh’s extension principle, where the associated crisp problem is solved and in the solution the initial fuzzy values are substituted instead of the real constants, and, in the final solution, arithmetic operations are considered to be operations on fuzzy numbers [5, 6]. The purpose of this paper is to investigate the characterization theorem together with the existence and unicity of two solutions, one solution for each lateral derivative, to firstorder fuzzy IDEs of Volterra type under the assumption of strongly generalized differentiability of the general form:. The solvability theory of fuzzy VIDEs has been studied by several researchers by using the strongly generalized differentiability, the Hukuhara derivative, or the Zadeh’s extension principle for the fuzzy-valued mappings of a real variable whose values are normal, convex, upper semicontinuous, and compactly supported fuzzy sets in R.
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