scholarly journals Asymptotic Stability of Nonlinear Discrete Fractional Pantograph Equations with Non-Local Initial Conditions

Symmetry ◽  
2021 ◽  
Vol 13 (3) ◽  
pp. 473
Author(s):  
Jehad Alzabut ◽  
A. George Maria Selvam ◽  
Rami Ahmad El-Nabulsi ◽  
D. Vignesh ◽  
Mohammad Esmael Samei
Keyword(s):  
Fixed Point ◽  
Asymptotic Stability ◽  
Fixed Point Theorems ◽  
Initial Conditions ◽  
Current Collector ◽  
Stability Of Solutions ◽  
Electric Bus ◽  
Pantograph Equation ◽  
Non Local

Pantograph, the technological successor of trolley poles, is an overhead current collector of electric bus, electric trains, and trams. In this work, we consider the discrete fractional pantograph equation of the form Δ*β[k](t)=wt+β,k(t+β),k(λ(t+β)), with condition k(0)=p[k] for t∈N1−β, 0<β≤1, λ∈(0,1) and investigate the properties of asymptotic stability of solutions. We will prove the main results by the aid of Krasnoselskii’s and generalized Banach fixed point theorems. Examples involving algorithms and illustrated graphs are presented to demonstrate the validity of our theoretical findings.

scholarly journals Existence and Ulam stability for impulsive generalized Hilfer-type fractional differential equations

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Abdelkrim Salim ◽  
Mouffak Benchohra ◽  
Erdal Karapınar ◽  
Jamal Eddine Lazreg
Keyword(s):  
Fixed Point ◽  
Differential Equations ◽  
Banach Spaces ◽  
Fractional Differential Equations ◽  
Fixed Point Theorems ◽  
Measure Of Noncompactness ◽  
Ulam Stability ◽  
Stability Of Solutions ◽  
Fractional Differential ◽  
Implicit Fractional Differential Equations

Abstract In this manuscript, we examine the existence and the Ulam stability of solutions for a class of boundary value problems for nonlinear implicit fractional differential equations with instantaneous impulses in Banach spaces. The results are based on fixed point theorems of Darbo and Mönch associated with the technique of measure of noncompactness. We provide some examples to indicate the applicability of our results.

scholarly journals Existence of Fractional Impulsive Functional Integro-Differential Equations in Banach Spaces

2019 ◽  
Vol 2 (2) ◽  
pp. 18 ◽  
Author(s):  
Dimplekumar Chalishajar ◽  
Chokkalingam Ravichandran ◽  
Shanmugam Dhanalakshmi ◽  
Rangasamy Murugesu
Keyword(s):  
Fixed Point ◽  
Group Theory ◽  
Differential Equations ◽  
Banach Spaces ◽  
Fractional Order ◽  
Fixed Point Theorems ◽  
Mild Solutions ◽  
Order System ◽  
Semi Group ◽  
Non Local

In this paper, we establish the existence of piece wise (PC)-mild solutions (defined in Section 2) for non local fractional impulsive functional integro-differential equations with finite delay. The proofs are obtained using techniques of fixed point theorems, semi-group theory and generalized Bellman inequality. In this paper, we used the distributed characteristic operators to define a mild solution of the system. We also discussed the controversy related to the solution operator for the fractional order system using weak and strong Caputo derivatives. Examples are given to illustrate the theory.

scholarly journals Asymptotic Stability Results for Nonlinear Fractional Difference Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
Fulai Chen ◽  
Zhigang Liu
Keyword(s):  
Fixed Point ◽  
Asymptotic Stability ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Difference Equations ◽  
Stability Of Solutions ◽  
Fractional Difference ◽  
Krasnoselskii’S Fixed Point Theorem ◽  
Fractional Difference Equations ◽  
Stability Results

We present some results for the asymptotic stability of solutions for nonlinear fractional difference equations involvingRiemann-Liouville-likedifference operator. The results are obtained by using Krasnoselskii's fixed point theorem and discrete Arzela-Ascoli's theorem. Three examples are also provided to illustrate our main results.

scholarly journals Study of the Atangana-Baleanu-Caputo type fractional system with a generalized Mittag-Leffler kernel

2021 ◽  
Vol 7 (2) ◽  
pp. 2001-2018
Author(s):  
Mdi Begum Jeelani ◽  
◽  
Abeer S. Alnahdi ◽  
Mohammed A. Almalahi ◽  
Mohammed S. Abdo ◽  
...  
Keyword(s):  
Fixed Point ◽  
Continuous Dependence ◽  
Fixed Point Theorems ◽  
Sufficient Conditions ◽  
Approximate Solutions ◽  
Fractional Operator ◽  
Stability Of Solutions ◽  
Fractional System ◽  
Mathematical Techniques

<abstract><p>We devote our interest in this work to investigate the sufficient conditions for the existence, uniqueness, and Ulam-Hyers stability of solutions for a new fractional system in the frame of Atangana-Baleanu-Caputo fractional operator with multi-parameters Mittag-Leffler kernels investigated lately by Abdeljawad (Chaos: An Interdisciplinary J. Nonlinear Sci. Vol. 29, no. 2, (2019): 023102). Moreover, the continuous dependence of solution and $ \delta $-approximate solutions are analyzed to such a system. Our approach is based on Banach's and Schaefer's fixed point theorems and some mathematical techniques. In order to illustrate the validity of our results, an example is given.</p></abstract>

scholarly journals Existence and Asymptotic Stability of Solutions of a Functional Integral Equation via a Consequence of Sadovskii’s Theorem

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
Agnieszka Chlebowicz ◽  
Mohamed Abdalla Darwish ◽  
Kishin Sadarangani
Keyword(s):  
Integral Equation ◽  
Fixed Point ◽  
Asymptotic Stability ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Functional Integral ◽  
Stability Of Solutions ◽  
Measures Of Noncompactness

Using the technique of measures of noncompactness and, in particular, a consequence of Sadovskii’s fixed point theorem, we prove a theorem about the existence and asymptotic stability of solutions of a functional integral equation. Moreover, in order to illustrate our results, we include one example and compare our results with those obtained in other papers appearing in the literature.

scholarly journals On the existence and Ulam-Hyers stability of a new class of partial (Φ,χ)-fractional differential equations with impulses

Filomat ◽  
2021 ◽  
Vol 35 (6) ◽  
pp. 1977-1991
Author(s):  
Arjumand Seemab ◽  
ur Mujeeb Rehman ◽  
Michal Feckan ◽  
Jehad Alzabut ◽  
Syed Abbas
Keyword(s):  
Fixed Point ◽  
Differential Equations ◽  
Fractional Differential Equations ◽  
Fractional Integral ◽  
Fixed Point Theorems ◽  
Measure Of Noncompactness ◽  
Stability Of Solutions ◽  
New Class ◽  
Fractional Differential ◽  
Partial Fractional Differential Equations

In this paper, we consider the newly defined partial (?,?)-fractional integral and derivative to study a new class of partial fractional differential equations with impulses. The existence and Ulam-Hyers stability of solutions for the proposed equation are investigated via the means of measure of noncompactness and fixed point theorems. The presented results are quite general in their nature and further complement the existing ones.

scholarly journals Impulsive Fractional Semilinear Integrodifferential Equations with Nonlocal Conditions

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Xue Wang ◽  
Bo Zhu
Keyword(s):  
Fixed Point ◽  
Resolvent Operator ◽  
Fixed Point Theorems ◽  
Initial Conditions ◽  
Mild Solutions ◽  
Semigroup Theory ◽  
Nonlocal Conditions ◽  
Integrodifferential Equations ◽  
Existence Theory ◽  
Nonlocal Initial Conditions

This paper is devoted to a class of impulsive fractional semilinear integrodifferential equations with nonlocal initial conditions. Based on the semigroup theory and some fixed point theorems, the existence theory of PC-mild solutions is established under the condition of compact resolvent operator. Furthermore, the uniqueness of PC-mild solutions is proved in the case of the noncompact resolvent operator.

scholarly journals Existence and $\delta $-Approximate solution of implicit fractional pantograph equations in the frame of Hilfer-Katugampola operator

2021 ◽  
Vol 2 (1) ◽  
pp. 1-17
Author(s):  
Mohammed A. Almalahi ◽  
Satish. K Panchal
Keyword(s):  
Fixed Point ◽  
Approximate Solution ◽  
Existence And Uniqueness ◽  
Fixed Point Theorems ◽  
Finite Interval ◽  
Nonlocal Conditions ◽  
Research Paper ◽  
Major Goal ◽  
Uniqueness Of Solutions ◽  
Pantograph Equation

The major goal of this research paper is to investigate the existence and uniqueness of an implicit fractional pantograph equation in the frame of the Hilfer-Katugampola operator on the finite interval $[a,b]$ with mixed nonlocal conditions. Our analysis of the existence and uniqueness of solutions depends on some fixed point theorems such as Banach and Krasnoselskii. Moreover, we discuss the dependence of solutions on mixed nonlocal conditions by means of $\delta $-approximated solution. As an application, we provide an example to illustrate the validity of our results.

scholarly journals Existence of Fractional Impulsive Functional Integro-Differential Equations in Banach Spaces

2018 ◽  
Author(s):  
Dimplekumar Chalishajar ◽  
C. Ravichandran ◽  
S. Dhanalakshmi ◽  
R. Murugesu
Keyword(s):  
Fixed Point ◽  
Group Theory ◽  
Differential Equations ◽  
Banach Spaces ◽  
Mild Solution ◽  
Fixed Point Theorems ◽  
Mild Solutions ◽  
Semi Group ◽  
Finite Delay ◽  
Non Local

In this paper, we established the existence of PC-mild solutions for non local fractional impulsive functional integro-differential equations with finite delay. The proofs are obtained by using the techniques of fixed point theorems, semi-group theory and generalized Bellman inequality. In this paper, we have used the distributed characteristic operators to define the mild solution of the system. Results obtained here improve and extend some known results.

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