Integrated fractional resolvent operator function and fractional abstract Cauchy problem
type: Item
instance of: scholarly article
title: Integrated fractional resolvent operator function and fractional abstract Cauchy problem
zbMATH Open document ID: 1469.34081
DOI: 10.1155/2014/430418
author: Ya-Ning Li
author: Hong-Rui Sun
published in: Abstract and Applied Analysis
publication date: 2019-02-14T00:00:00Z
review text: Summary: We firstly prove that \(\beta\)-times integrated \(\alpha\)-resolvent operator function (\((\alpha, \beta)\)-ROF) satisfies a functional equation which extends that of \(\beta\)-times integrated semigroup and \(\alpha\)-resolvent operator function. Secondly, for the inhomogeneous \(\alpha\)-Cauchy problem \({}^c D_t^\alpha u(t) = A u(t) + f(t)\), \(t \in(0, T)\), \(u(0) = x_0\), \(u'(0) = x_1\), if \(A\) is the generator of an \((\alpha, \beta)\)-ROF, we give the relation between the function \(v(t) = S_{\alpha, \beta}(t) x_0 +(g_1 * S_{\alpha, \beta})(t) x_1 +(g_{\alpha - 1} * S_{\alpha, \beta} * f)(t)\) and mild solution and classical solution of it. Finally, for the problem \({}^c D_t^\alpha v(t) = A v(t) + g_{\beta + 1}(t) x\), \(t > 0\), \(v^{(k)}(0) = 0\), \(k = 0,1, \dots, N - 1\), where \(A\) is a linear closed operator. We show that \(A\) generates an exponentially bounded \((\alpha, \beta)\)-ROF on a Banach space \(X\) if and only if the problem has a unique exponentially bounded classical solution \(v_x\) and \(A v_x \in L_{\text{loc}}^1(\mathbb{R}^+, X) \). Our results extend and generalize some related results in the literature.
Mathematics Subject Classification ID: 34G10
Mathematics Subject Classification ID: 34A08
zbMATH DE Number: 7022377
Wikidata QID: Q59037977
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1155/2014/430418
OpenAlex ID: W2080627587
cites work: \(\alpha\)-times integrated semigroups \((\alpha\in \mathbb{R}^ +)\)
cites work: Q2716079
cites work: Stochastic Calculus for Fractional Brownian Motion I. Theory
cites work: Applications of fractional calculus in physics
cites work: Q5699297
cites work: Fractional dynamics. Applications of fractional calculus to dynamics of particles, fields and media
cites work: A class of fractional evolution equations and optimal controls
cites work: Fractional sewage treatment models with impulses at variable times
cites work: On fractional resolvent operator functions
cites work: \((a,k)\)-regularized \(C\)-resolvent families: regularity and local properties
cites work: On fractional powers of generators of fractional resolvent families
cites work: Fractional abstract Cauchy problems
cites work: Cauchy problems for fractional differential equations with Riemann-Liouville fractional derivatives
cites work: An operator theoretical approach to a class of fractional order differential equations
cites work: Regularized solutions for abstract Volterra equations
cites work: Linear integro-differential equations in Banach spaces
cites work: Fractional evolution equations in Banach spaces
cites work: A novel characteristic of solution operator for the fractional abstract Cauchy problem
cites work: Q4276447
cites work: Q3141902
cites work: Fractional differential equations. An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications
description: scientific article; zbMATH DE number 7022377