A Lyapunov type inequality for fractional operators with nonsingular Mittag-Leffler kernel

In this article, we extend fractional operators with nonsingular Mittag-Leffler kernels, a study initiated recently by Atangana and Baleanu, from order α ∈ [ 0 , 1 ] to higher arbitrary order and we formulate their correspondent integral operators. We prove existence and uniqueness theorems for the...

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Published inJournal of inequalities and applications Vol. 2017; no. 1; pp. 130 - 11
Main Author Abdeljawad, Thabet
Format Journal Article
LanguageEnglish
Published Cham Springer International Publishing 2017
Springer Nature B.V
SpringerOpen
Subjects
A B C $ABC$ fractional derivative
A B R $ABR$ fractional derivative
Analysis
Applications of Mathematics
boundary value problem
Boundary value problems
Existence theorems
Fractional calculus
higher order
Kernels
Lyapunov inequality
Mathematical analysis
Mathematics
Mathematics and Statistics
Mittag-Leffler kernel
Operators
Uniqueness theorems
Lyapunov inequality
fractional derivative
boundary value problem
Mittag-Leffler kernel
higher order
[Formula: see text] fractional derivative
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Abstract In this article, we extend fractional operators with nonsingular Mittag-Leffler kernels, a study initiated recently by Atangana and Baleanu, from order α ∈ [ 0 , 1 ] to higher arbitrary order and we formulate their correspondent integral operators. We prove existence and uniqueness theorems for the Caputo ( A B C ) and Riemann ( A B R ) type initial value problems by using the Banach contraction theorem. Then we prove a Lyapunov type inequality for the Riemann type fractional boundary value problems of order 2 < α ≤ 3 in the frame of Mittag-Leffler kernels. Illustrative examples are analyzed and an application as regards the Sturm-Liouville eigenvalue problem in the sense of this fractional calculus is given as well.
AbstractList In this article, we extend fractional operators with nonsingular Mittag-Leffler kernels, a study initiated recently by Atangana and Baleanu, from order [Formula: see text] to higher arbitrary order and we formulate their correspondent integral operators. We prove existence and uniqueness theorems for the Caputo ([Formula: see text]) and Riemann ([Formula: see text]) type initial value problems by using the Banach contraction theorem. Then we prove a Lyapunov type inequality for the Riemann type fractional boundary value problems of order [Formula: see text] in the frame of Mittag-Leffler kernels. Illustrative examples are analyzed and an application as regards the Sturm-Liouville eigenvalue problem in the sense of this fractional calculus is given as well.In this article, we extend fractional operators with nonsingular Mittag-Leffler kernels, a study initiated recently by Atangana and Baleanu, from order [Formula: see text] to higher arbitrary order and we formulate their correspondent integral operators. We prove existence and uniqueness theorems for the Caputo ([Formula: see text]) and Riemann ([Formula: see text]) type initial value problems by using the Banach contraction theorem. Then we prove a Lyapunov type inequality for the Riemann type fractional boundary value problems of order [Formula: see text] in the frame of Mittag-Leffler kernels. Illustrative examples are analyzed and an application as regards the Sturm-Liouville eigenvalue problem in the sense of this fractional calculus is given as well.
Abstract In this article, we extend fractional operators with nonsingular Mittag-Leffler kernels, a study initiated recently by Atangana and Baleanu, from order α ∈ [ 0 , 1 ] $\alpha\in[0,1]$ to higher arbitrary order and we formulate their correspondent integral operators. We prove existence and uniqueness theorems for the Caputo ( A B C $ABC$ ) and Riemann ( A B R $ABR$ ) type initial value problems by using the Banach contraction theorem. Then we prove a Lyapunov type inequality for the Riemann type fractional boundary value problems of order 2 < α ≤ 3 $2<\alpha\leq3$ in the frame of Mittag-Leffler kernels. Illustrative examples are analyzed and an application as regards the Sturm-Liouville eigenvalue problem in the sense of this fractional calculus is given as well.
In this article, we extend fractional operators with nonsingular Mittag-Leffler kernels, a study initiated recently by Atangana and Baleanu, from order [Formula: see text] to higher arbitrary order and we formulate their correspondent integral operators. We prove existence and uniqueness theorems for the Caputo ([Formula: see text]) and Riemann ([Formula: see text]) type initial value problems by using the Banach contraction theorem. Then we prove a Lyapunov type inequality for the Riemann type fractional boundary value problems of order [Formula: see text] in the frame of Mittag-Leffler kernels. Illustrative examples are analyzed and an application as regards the Sturm-Liouville eigenvalue problem in the sense of this fractional calculus is given as well.
In this article, we extend fractional operators with nonsingular Mittag-Leffler kernels, a study initiated recently by Atangana and Baleanu, from order α ∈ [ 0 , 1 ] to higher arbitrary order and we formulate their correspondent integral operators. We prove existence and uniqueness theorems for the Caputo ( A B C ) and Riemann ( A B R ) type initial value problems by using the Banach contraction theorem. Then we prove a Lyapunov type inequality for the Riemann type fractional boundary value problems of order 2 < α ≤ 3 in the frame of Mittag-Leffler kernels. Illustrative examples are analyzed and an application as regards the Sturm-Liouville eigenvalue problem in the sense of this fractional calculus is given as well.
In this article, we extend fractional operators with nonsingular Mittag-Leffler kernels, a study initiated recently by Atangana and Baleanu, from order \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\alpha\in[0,1]$\end{document} α ∈ [ 0 , 1 ] to higher arbitrary order and we formulate their correspondent integral operators. We prove existence and uniqueness theorems for the Caputo ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ABC$\end{document} A B C ) and Riemann ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ABR$\end{document} A B R ) type initial value problems by using the Banach contraction theorem. Then we prove a Lyapunov type inequality for the Riemann type fractional boundary value problems of order \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$2<\alpha\leq3$\end{document} 2 < α ≤ 3 in the frame of Mittag-Leffler kernels. Illustrative examples are analyzed and an application as regards the Sturm-Liouville eigenvalue problem in the sense of this fractional calculus is given as well.
(ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image) In this article, we extend fractional operators with nonsingular Mittag-Leffler kernels, a study initiated recently by Atangana and Baleanu, from order ... to higher arbitrary order and we formulate their correspondent integral operators. We prove existence and uniqueness theorems for the Caputo (...) and Riemann (...) type initial value problems by using the Banach contraction theorem. Then we prove a Lyapunov type inequality for the Riemann type fractional boundary value problems of order ... in the frame of Mittag-Leffler kernels. Illustrative examples are analyzed and an application as regards the Sturm-Liouville eigenvalue problem in the sense of this fractional calculus is given as well.
ArticleNumber 130
Author Abdeljawad, Thabet
Author_xml – sequence: 1
  givenname: Thabet
  surname: Abdeljawad
  fullname: Abdeljawad, Thabet
  email: tabdeljawad@psu.edu.sa
  organization: Department of Mathematics and General Sciences, Prince Sultan University
BackLink https://www.ncbi.nlm.nih.gov/pubmed/28680233$$D View this record in MEDLINE/PubMed
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Keywords Lyapunov inequality
fractional derivative
boundary value problem
Mittag-Leffler kernel
higher order
[Formula: see text] fractional derivative
Language English
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AM Lyapunov (1400_CR15) 1907; 2
I Podlubny (1400_CR2) 1999
M Caputo (1400_CR6) 2015; 1
SG Kilbas (1400_CR1) 1993
JA Tenreiro Machado (1400_CR5) 2011; 16
A Chdouh (1400_CR17) 2017; 312
D O’Regan (1400_CR19) 2015; 2015
A Kilbas (1400_CR3) 2006
T Abdeljawad (1400_CR11) 2016; 2016
M Jleli (1400_CR24) 2016; 9
F Jarad (1400_CR13) 2010; 65
RAC Ferreira (1400_CR23) 2015; 5
J Rong (1400_CR20) 2015; 2015
JA Tenreiro Machado (1400_CR4) 2010; 13
M Jleli (1400_CR18) 2015; 2015
1400_CR10
D Baleanu (1400_CR12) 2008; 41
T Abdeljawad (1400_CR9) 2017; 9
M Jleli (1400_CR21) 2017; 2017
M Jleli (1400_CR22) 2017; 2017
T Odzijewicz (1400_CR14) 2012; 2012
A Atangana (1400_CR8) 2016; 20
J Losada (1400_CR7) 2015; 1
RAC Ferreira (1400_CR16) 2013; 6
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Snippet In this article, we extend fractional operators with nonsingular Mittag-Leffler kernels, a study initiated recently by Atangana and Baleanu, from order α ∈ [ 0...
In this article, we extend fractional operators with nonsingular Mittag-Leffler kernels, a study initiated recently by Atangana and Baleanu, from order...
(ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image) In this article, we extend fractional operators with nonsingular Mittag-Leffler...
In this article, we extend fractional operators with nonsingular Mittag-Leffler kernels, a study initiated recently by Atangana and Baleanu, from order...
Abstract In this article, we extend fractional operators with nonsingular Mittag-Leffler kernels, a study initiated recently by Atangana and Baleanu, from...
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SubjectTerms A B C $ABC$ fractional derivative
A B R $ABR$ fractional derivative
Analysis
Applications of Mathematics
boundary value problem
Boundary value problems
Existence theorems
Fractional calculus
higher order
Kernels
Lyapunov inequality
Mathematical analysis
Mathematics
Mathematics and Statistics
Mittag-Leffler kernel
Operators
Uniqueness theorems
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Title A Lyapunov type inequality for fractional operators with nonsingular Mittag-Leffler kernel
URI https://link.springer.com/article/10.1186/s13660-017-1400-5
https://www.ncbi.nlm.nih.gov/pubmed/28680233
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https://www.proquest.com/docview/1916709543
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Volume 2017
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