Stability Results for Implicit Fractional Pantograph Differential Equations via <i>ϕ</i>-Hilfer Fractional Derivative with a Nonlocal Riemann-Liouville Fractional Integral Condition

Stability Results for Implicit Fractional Pantograph Differential Equations via <i>ϕ</i>-Hilfer Fractional Derivative with a Nonlocal Riemann-Liouville Fractional Integral Condition

This paper presents a class of implicit pantograph fractional differential equation with more general Riemann-Liouville fractional integral condition. A certain class of generalized fractional derivative is used to set the problem. The existence and uniqueness of the problem is obtained using Schaef...

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Main Authors: Idris Ahmed, Poom Kumam, Kamal Shah, Piyachat Borisut, Kanokwan Sitthithakerngkiet, Musa Ahmed Demba
Format: Article
Language:English
Published: MDPI AG 2020-01-01
Series:Mathematics
Subjects:
hilfer fractional derivative
ulam stability
pantograph differential equation
nonlocal integral condition
Online Access:https://www.mdpi.com/2227-7390/8/1/94
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spelling doaj-fa90cb3edaa04610b26b46941ee0290b2020-11-25T01:42:27ZengMDPI AGMathematics2227-73902020-01-01819410.3390/math8010094math8010094Stability Results for Implicit Fractional Pantograph Differential Equations via <i>ϕ</i>-Hilfer Fractional Derivative with a Nonlocal Riemann-Liouville Fractional Integral ConditionIdris Ahmed0Poom Kumam1Kamal Shah2Piyachat Borisut3Kanokwan Sitthithakerngkiet4Musa Ahmed Demba5KMUTTFixed Point Research Laboratory, Room SCL 802 Fixed Point Laboratory, Science Laboratory Building, Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thrung Khru, Bangkok 10140, ThailandCenter of Excellence in Theoretical and Computational Science (TaCS-CoE), Science Laboratory Building, King Mongkut’s University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thrung Khru, Bangkok 10140, ThailandDepartment of Mathematics, University of Malakand, Chakadara Dir(L), Khyber Pakhtunkhwa 18800, PakistanKMUTTFixed Point Research Laboratory, Room SCL 802 Fixed Point Laboratory, Science Laboratory Building, Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thrung Khru, Bangkok 10140, ThailandIntelligent and Nonlinear Dynamic Innovations Research Center, Department of Mathematics, Faculty of Applied Science, King Mongkut’s University of Technology North Bangkok (KMUTNB), Wongsawang, Bangsue, Bangkok 10800, ThailandKMUTTFixed Point Research Laboratory, Room SCL 802 Fixed Point Laboratory, Science Laboratory Building, Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thrung Khru, Bangkok 10140, ThailandThis paper presents a class of implicit pantograph fractional differential equation with more general Riemann-Liouville fractional integral condition. A certain class of generalized fractional derivative is used to set the problem. The existence and uniqueness of the problem is obtained using Schaefer&#8217;s and Banach fixed point theorems. In addition, the Ulam-Hyers and generalized Ulam-Hyers stability of the problem are established. Finally, some examples are given to illustrative the results.https://www.mdpi.com/2227-7390/8/1/94hilfer fractional derivativeulam stabilitypantograph differential equationnonlocal integral condition
collection DOAJ
language English
format Article
sources DOAJ
author Idris Ahmed
Poom Kumam
Kamal Shah
Piyachat Borisut
Kanokwan Sitthithakerngkiet
Musa Ahmed Demba
spellingShingle Idris Ahmed
Poom Kumam
Kamal Shah
Piyachat Borisut
Kanokwan Sitthithakerngkiet
Musa Ahmed Demba
Stability Results for Implicit Fractional Pantograph Differential Equations via <i>ϕ</i>-Hilfer Fractional Derivative with a Nonlocal Riemann-Liouville Fractional Integral Condition
Mathematics
hilfer fractional derivative
ulam stability
pantograph differential equation
nonlocal integral condition
author_facet Idris Ahmed
Poom Kumam
Kamal Shah
Piyachat Borisut
Kanokwan Sitthithakerngkiet
Musa Ahmed Demba
author_sort Idris Ahmed
title Stability Results for Implicit Fractional Pantograph Differential Equations via <i>ϕ</i>-Hilfer Fractional Derivative with a Nonlocal Riemann-Liouville Fractional Integral Condition
title_short Stability Results for Implicit Fractional Pantograph Differential Equations via <i>ϕ</i>-Hilfer Fractional Derivative with a Nonlocal Riemann-Liouville Fractional Integral Condition
title_full Stability Results for Implicit Fractional Pantograph Differential Equations via <i>ϕ</i>-Hilfer Fractional Derivative with a Nonlocal Riemann-Liouville Fractional Integral Condition
title_fullStr Stability Results for Implicit Fractional Pantograph Differential Equations via <i>ϕ</i>-Hilfer Fractional Derivative with a Nonlocal Riemann-Liouville Fractional Integral Condition
title_full_unstemmed Stability Results for Implicit Fractional Pantograph Differential Equations via <i>ϕ</i>-Hilfer Fractional Derivative with a Nonlocal Riemann-Liouville Fractional Integral Condition
title_sort stability results for implicit fractional pantograph differential equations via <i>ϕ</i>-hilfer fractional derivative with a nonlocal riemann-liouville fractional integral condition
publisher MDPI AG
series Mathematics
issn 2227-7390
publishDate 2020-01-01
description This paper presents a class of implicit pantograph fractional differential equation with more general Riemann-Liouville fractional integral condition. A certain class of generalized fractional derivative is used to set the problem. The existence and uniqueness of the problem is obtained using Schaefer&#8217;s and Banach fixed point theorems. In addition, the Ulam-Hyers and generalized Ulam-Hyers stability of the problem are established. Finally, some examples are given to illustrative the results.
topic hilfer fractional derivative
ulam stability
pantograph differential equation
nonlocal integral condition
url https://www.mdpi.com/2227-7390/8/1/94
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AT poomkumam stabilityresultsforimplicitfractionalpantographdifferentialequationsviaiphihilferfractionalderivativewithanonlocalriemannliouvillefractionalintegralcondition
AT kamalshah stabilityresultsforimplicitfractionalpantographdifferentialequationsviaiphihilferfractionalderivativewithanonlocalriemannliouvillefractionalintegralcondition
AT piyachatborisut stabilityresultsforimplicitfractionalpantographdifferentialequationsviaiphihilferfractionalderivativewithanonlocalriemannliouvillefractionalintegralcondition
AT kanokwansitthithakerngkiet stabilityresultsforimplicitfractionalpantographdifferentialequationsviaiphihilferfractionalderivativewithanonlocalriemannliouvillefractionalintegralcondition
AT musaahmeddemba stabilityresultsforimplicitfractionalpantographdifferentialequationsviaiphihilferfractionalderivativewithanonlocalriemannliouvillefractionalintegralcondition
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