Fuzzy Integral Equations and Strong Fuzzy Henstock Integrals

Abstract

By using the strong fuzzy Henstock integral and its controlled convergence theorem, we generalized the existence theorems of solution for initial problems of fuzzy discontinuous integral equation.

Homotopy Analysis Method for the First Order Fuzzy Volterra-Fredholm Integro-differential Equations

“…Provided that this limit exists in the metric d. If the fuzzy function f (t) is continuous in the metric d, its definite integral exists, and also More details about the properties of the fuzzy integral are given in [2,26,25]. Theorem 2..4 [28] (Banach contraction principle).…”

“…Recently, Hence various other methods for solving them such as using homotopy perturbation method [5], expansion method [6], Laplace transformation method [7], homotopy analysis method [8], differential transform method [9], fixed point theorems [10], variational iteration method [11]. Also, some mathematicians have studied fuzzy integral and integro-differential equation by numerical techniques [12]- [21] [23,26]. As we know the fuzzy integral and integro-differential equations are one of the important parts of the fuzzy analysis theory that play a main role in the numerical analysis.…”

A Study of Some Reliable Methods for Solving Fuzzy Volterra-Fredholm Integral Equations

“…However, if f (t) is continuous, both approaches yield the same value. More details about the properties of the fuzzy integral are given in [15,29,27].…”

“…One of the first applications of the fuzzy integral equation was given by Ma and Wu who investigated the fuzzy Fredholm integral equation of the second kind. Recently, some mathematicians have studied fuzzy integral and integro-differential equation by numerical techniques [6,8,9,12,17,29]. As we know the fuzzy integral and differential equations are one of the important parts of the fuzzy analysis theory that play a main role in the numerical analysis.…”

On the existence of generalized weak solutions to discontinuous fuzzy differential equations

“…However, there are discontinuous systems in which the right-hand side functions f : [a, b] × E n → E n are not integrable in the sense of Kaleva [22] on certain intervals and their solutions are not absolute continuous functions. So, in [31][32][33][34][35], we used the strong fuzzy Henstock integral [17], and dealt with the Cauchy problem of discontinuous fuzzy systems. In this paper, according to the idea of [9] and using the concept of generalized differentiability [3], we will deal with the Cauchy problem of discontinuous fuzzy systems and the right side-hand functions are fuzzy Henstock-Pettis integrable.…”