Interval Abel integral equation
type: Item
instance of: scholarly article
title: Interval Abel integral equation
zbMATH Open document ID: 1384.45003
author: Vasile Lupulescu
author: Ngo Van Hoa
published in: Soft Computing
publication date: 2018-02-27T00:00:00Z
review text: The authors study the solvability of the interval Abel integral equation of the form: \[ \frac{1}{\Gamma(\alpha)} \int\limits_a^t (t-s)^{\alpha-1} X(s) \text{d}s = Y(t) ,\, t \in [a, b] \tag{1} \] where \(\alpha \in (0, 1)\), \(K\) is the set of any nonempty compact intervals of the real line \(\mathbb{R}\), \(Y(\cdot) : [a, b] \to K\) being a given interval-valued function and \(X(\cdot) \in L^1([a,b],K)\) is an unknown interval-valued function. The aim of the work is to obtain sufficient conditions for the interval-valued function \(Y(\cdot)\) to ensure that the interval Abel integral equation (1) is uniquely solvable. As first step it is proved that if \(Y(\cdot) \in L^1([a,b],K)\) and the equation (1) has a solution \(X(\cdot) \in L^1([a,b],K)\), then \(Y_{1-\alpha}(\cdot)\) is AC on \([a, b]\), \(Y_{1-\alpha}(a)=\theta\) and \(X(t)=Y_{1-\alpha}'(t)\) for a.e. \(t \in [a, b]\). Here by \(Y_{1-\alpha}(\cdot)\) the interval-valued Riemann-Liouville fractional integral of order \(1-\alpha > 0\) is denoted. The main result about solvability of (1) in the work is proved under the following assumption: Let \(Y(\cdot) \in L^1([a,b],K)\) be an interval-valued function such that \(Y_{1-\alpha}(a)=\theta\), \(Y_{1-\alpha}(\cdot)\) is AC and \(w\)-increasing on \([a, b]\). Then the integral equation (1) has a unique solution \(X(\cdot) \in L^1([a,b],K)\). Several appropriate examples illustrating and clarifying the obtained results are given.
reviewed by: Andrey Zahariev
Mathematics Subject Classification ID: 45E10
Mathematics Subject Classification ID: 26A33
zbMATH DE Number: 6844029
zbMATH Keywords: interval Abel integral equation
zbMATH Keywords: interval-valued Riemann-Liouville fractional integral
MaRDI profile type: Publication
full work available at URL: s00500-015-1980-2
OpenAlex ID: W2202056484
cites work: Fuzzy fractional integral equations under compactness type condition
cites work: On the concept of solution for fractional differential equations with uncertainty
cites work: Explicit solutions of fractional differential equations with uncertainty
cites work: On the fractional differential equations with uncertainty
cites work: Q2995040
cites work: Integrals of set-valued functions
cites work: Generalizations of the differentiability of fuzzy-number-valued functions with applications to fuzzy differential equations
cites work: Generalized differentiability of fuzzy-valued functions
cites work: Generalized derivative and \(\pi \)-derivative for set-valued functions
cites work: Calculus for interval-valued functions using generalized Hukuhara derivative and applications
cites work: Abel integral equations. Analysis and applications
cites work: The initial value problem for interval-valued second-order differential equations under generalized H-differentiability
cites work: Fuzzy fractional functional integral and differential equations
cites work: Integrals, conditional expectations, and martingales of multivalued functions
cites work: Q5543500
cites work: Hukuhara differentiability of interval-valued functions and interval differential equations on time scales
cites work: Fractional calculus for interval-valued functions
cites work: Interval differential equations with a second type Hukuhara derivative
cites work: Interval Cauchy problem with a second type Hukuhara derivative
cites work: Random fuzzy differential equations under generalized Lipschitz condition
cites work: On set differential equations in Banach spaces - a second type Hukuhara differentiability approach
cites work: Calculus for interval functions of a real variable
cites work: Existence and uniqueness results for fractional differential equations with uncertainty
cites work: Q3141895
cites work: A generalization of Hukuhara difference and division for interval and fuzzy arithmetic
cites work: Generalized Hukuhara differentiability of interval-valued functions and interval differential equations
DOI: 10.1007/S00500-015-1980-2
description: scientific article; zbMATH DE number 6844029