Interval Cauchy problem with a second type Hukuhara derivative
type: Item
instance of: scholarly article
title: Interval Cauchy problem with a second type Hukuhara derivative
zbMATH Open document ID: 1257.34011
author: Marek T. Malinowski
published in: Information Sciences
publication date: 2012-11-15T00:00:00Z
review text: The paper presents some theoretical studies of an interval-valued initial value problem with the second type Hukuhara derivative of the form \[ X'(t)=F(t,X(t)),\quad X(t_0)=X_0, \] where \(F:[\alpha,\beta]\times I \to I\) and \(X_0=[X_0^-,X_0^+]\in I\) are the data of the equation, and the symbol \('\) denotes the second type Hukuhara derivative. Here, \(I\) is the family of all nonempty, compact real intervals. First, the author collects some necessary facts about the Hukuhara difference and about differentiation and integration of interval-valued functions. Further, he presents a real-world application of an interval differential equation with the second type Hukuhara derivative. The continuous dependence of the solution on the initial value and the right-hand side of the equation is shown and the existence of approximate local solutions is stated. By means of this, the author obtains the existence of at least one local solution to the problem and compactness of the set of solutions. Finally, explicit formulas of unique second type Hukuhara differentiable local solutions to linear interval differential equations are derived.
reviewed by: Irena Rachůnková
Mathematics Subject Classification ID: 34A60
Mathematics Subject Classification ID: 34A12
zbMATH DE Number: 6107033
zbMATH Keywords: interval difference equations
zbMATH Keywords: initial value problem
zbMATH Keywords: approximate solutions
zbMATH Keywords: existence
zbMATH Keywords: second type Hukuhara derivative
MaRDI profile type: Publication
full work available at URL: j.ins.2012.05.022
OpenAlex ID: W2078542729
cites work: A calculus for set-valued maps and set-valued evolution equations
cites work: Generalizations of the differentiability of fuzzy-number-valued functions with applications to fuzzy differential equations
cites work: Solutions of fuzzy differential equations based on generalized differentiability
cites work: First order linear fuzzy differential equations under generalized differentiability
cites work: Q5591689
cites work: Q5612008
cites work: On new solutions of fuzzy differential equations
cites work: Generalized derivative and \(\pi \)-derivative for set-valued functions
cites work: Monotone iterative technique for periodic boundary value problems with causal operators
cites work: Set valued functions in Fréchet spaces: continuity, Hukuhara differentiability and applications to set differential equations
cites work: Set Differential Equations and Flow Invariance
cites work: Nonlinear variation of parameters formula for set differential equations in a metric space
cites work: Stability criteria for set differential equations
cites work: The Henstock-Stieltjes integral for fuzzy-number-valued functions
cites work: An approach to pseudo-integration of set-valued functions
cites work: Q5543500
cites work: A note on convergence properties of interval-valued capacity functionals and Choquet integrals
cites work: Variation of constant formula for first order fuzzy differential equations
cites work: Q4115434
cites work: Q4139042
cites work: Q4098070
cites work: Q5484956
cites work: Q4469660
cites work: Q4450573
cites work: Q3504941
cites work: Interval differential equations with a second type Hukuhara derivative
cites work: Random fuzzy differential equations under generalized Lipschitz condition
cites work: Impulsive set differential equations with delay
cites work: Q3575932
cites work: Some results on sheaf-solutions of sheaf set control problems
cites work: Stability criteria for set control differential equations
cites work: Q4467195
cites work: Generalized Hukuhara differentiability of interval-valued functions and interval differential equations
cites work: Stability of set differential equations and applications
cites work: Q5393180
DOI: 10.1016/J.INS.2012.05.022
description: scientific article; zbMATH DE number 6107033