Exact solution to the periodic boundary value problem for a first-order linear fuzzy differential equation with impulses

Abstract

Using the expression of the exact solution to a periodic boundary value problem for an impulsive first-order linear differential equation, we consider an extension to the fuzzy case and prove the existence and uniqueness of solution for a first-order linear fuzzy differential equation with impulses subject to boundary value conditions. We obtain the explicit solution by calculating the solutions on each level set and justify that the parametric functions obtained define a proper fuzzy function. Our results prove that the solution of the fuzzy differential equation of interest is determined, under the appropriate conditions, by the same Green's function obtained for the real case. Thus, the results proved extend some theorems given for ordinary differential equations.

On periodic solutions to first order linear fuzzy differential equations under differential inclusions’ approach

“…, for all α ∈ [0, 1], and conditions (16), (25) are equivalent to the sufficient conditions for the T -periodicity of the solution to problem (12) under differential inclusions' approach:…”

“…In references [25,28], the authors consider periodic boundary value problems for impulsive fuzzy differential equations under the Hukuhara differentiability. Recently, sufficient conditions for the existence of periodic solutions to fuzzy differential equations under generalized differentiability concept are studied in [20,29].…”

Numerical Solution of Fuzzy Differential Equations with Z-numbers Using Bernstein Neural Networks

“…By interval-valued method, [15] examines the basis solutions nonlinear FDEs with generalized differentiability. [16] uses periodic boundary and Hukuhara differentiability to the impulsive FDE. [17] suggests some suitable criterion to fuzzify the crisp solutions.…”

Boundary value problems for semilinear fuzzy impulsive differential inclusions based on semigroups in Banach spaces

“…If b = 1, then the problem (FIP) is with periodic boundary. sets of the solution, the following periodic boundary value problem for an impulsive linear fuzzy differential equation was studied in [23]:…”