scholarly journals Existence of local and global solutions to fractional order fuzzy delay differential equation with non-instantaneous impulses

2021 ◽  
Vol 7 (2) ◽  
pp. 2348-2369
Author(s):  
Anil Kumar ◽  
◽  
Muslim Malik ◽  
Mohammad Sajid ◽  
Dumitru Baleanu ◽  
...  
Keyword(s):  
Fractional Order ◽  
Set Theory ◽  
Fuzzy Set Theory ◽  
Sufficient Conditions ◽  
Global Solutions ◽  
Banach Fixed Point Theorem ◽  
Main Concern ◽  
Local And Global Solutions ◽  
Delay Differential System ◽  
Delay Differential

<abstract><p>The main concern of this manuscript is to examine some sufficient conditions under which the fractional order fuzzy delay differential system with the non-instantaneous impulsive condition has a unique solution. We also study the existence of a global solution for the considered system. Fuzzy set theory, Banach fixed point theorem and Non-linear functional analysis are the major tools to demonstrate our results. In last, an example is given to illustrate these analytical results.</p></abstract>

scholarly journals Sufficient and necessary conditions for oscillation of linear fractional-order delay differential equations

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Qiong Meng ◽  
Zhen Jin ◽  
Guirong Liu
Keyword(s):  
Differential Equation ◽  
Fractional Order ◽  
Sufficient Conditions ◽  
Necessary Condition ◽  
Multiple Delays ◽  
All Solutions ◽  
Sufficient And Necessary Conditions ◽  
Delay Differential ◽  
T Tau ◽  
Sufficient And Necessary Condition

AbstractThis paper studies the linear fractional-order delay differential equation $$ {}^{C}D^{\alpha }_{-}x(t)-px(t-\tau )= 0, $$ D − α C x ( t ) − p x ( t − τ ) = 0 , where $0<\alpha =\frac{\text{odd integer}}{\text{odd integer}}<1$ 0 < α = odd integer odd integer < 1 , $p, \tau >0$ p , τ > 0 , ${}^{C}D_{-}^{\alpha }x(t)=-\Gamma ^{-1}(1-\alpha )\int _{t}^{\infty }(s-t)^{- \alpha }x'(s)\,ds$ D − α C x ( t ) = − Γ − 1 ( 1 − α ) ∫ t ∞ ( s − t ) − α x ′ ( s ) d s . We obtain the conclusion that $$ p^{1/\alpha } \tau >\alpha /e $$ p 1 / α τ > α / e is a sufficient and necessary condition of the oscillations for all solutions of Eq. (*). At the same time, some sufficient conditions are obtained for the oscillations of multiple delays linear fractional differential equation. Several examples are given to illustrate our theorems.

scholarly journals A Class of Stochastic Nonlinear Delay System with Jumps

2014 ◽  
Vol 2014 ◽  
pp. 1-11
Author(s):  
Ling Bai ◽  
Kai Zhang ◽  
Wenju Zhao
Keyword(s):  
Sufficient Conditions ◽  
Delay System ◽  
Lévy Noise ◽  
Local Martingale ◽  
Martingale Inequality ◽  
Strong Law ◽  
Delay Differential System ◽  
Delay Differential ◽  
Levy Noise ◽  
Nonlinear Delay

We consider stochastic suppression and stabilization for nonlinear delay differential system. The system is assumed to satisfy local Lipschitz condition and one-side polynomial growth condition. Since the system may explode in a finite time, we stochastically perturb this system by introducing independent Brownian noises and Lévy noise feedbacks. The contributions of this paper are as follows. (a) We show that Brownian noises or Lévy noise may suppress potential explosion of the solution for some appropriate parameters. (b) Using the exponential martingale inequality with jumps, we discuss the fact that the sample Lyapunov exponent is nonpositive. (c) Considering linear Lévy processes, by the strong law of large number for local martingale, sufficient conditions for a.s. exponentially stability are investigated in Theorem 13.

scholarly journals On the Existence and Uniqueness of Solution to Fractional-Order Langevin Equation

2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
Ahmed Salem ◽  
Noorah Mshary
Keyword(s):  
Fractional Order ◽  
Langevin Equation ◽  
Existence And Uniqueness ◽  
Sufficient Conditions ◽  
Contraction Mapping Principle ◽  
Uniqueness Of Solution ◽  
Main Concern ◽  
Banach Contraction ◽  
Fractional Orders ◽  
Banach Contraction Mapping Principle

In this work, we give sufficient conditions to investigate the existence and uniqueness of solution to fractional-order Langevin equation involving two distinct fractional orders with unprecedented conditions (three-point boundary conditions including two nonlocal integrals). The problem is introduced to keep track of the progress made on exploring the existence and uniqueness of solution to the fractional-order Langevin equation. As a result of employing the so-called Krasnoselskii and Leray-Schauder alternative fixed point theorems and Banach contraction mapping principle, some novel results are presented in regarding to our main concern. These results are illustrated through providing three examples for completeness.

scholarly journals The Oscillation of a Class of the Fractional-Order Delay Differential Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Qianli Lu ◽  
Feng Cen
Keyword(s):  
Differential Equations ◽  
Fractional Order ◽  
Delay Differential Equations ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Analysis Method ◽  
Constant Coefficients ◽  
Delay Differential ◽  
Necessary And Sufficient

Several oscillation results are proposed including necessary and sufficient conditions for the oscillation of fractional-order delay differential equations with constant coefficients, the sufficient or necessary and sufficient conditions for the oscillation of fractional-order delay differential equations by analysis method, and the sufficient or necessary and sufficient conditions for the oscillation of delay partial differential equation with three different boundary conditions. For this,α-exponential function which is a kind of functions that play the same role of the classical exponential functions of fractional-order derivatives is used.

scholarly journals Stability of Atangana - Baleanu Fractional Order Differential Equation with Numerical Approximation

2021 ◽  
Vol 2070 (1) ◽  
pp. 012086
Author(s):  
A. George Maria Selvam ◽  
S. Britto Jacob
Keyword(s):  
Differential Equation ◽  
Fractional Order ◽  
Numerical Scheme ◽  
Numerical Approximation ◽  
Order Differential Equation ◽  
Sufficient Conditions ◽  
Banach Fixed Point Theorem ◽  
Stability Conditions ◽  
System Of Equations ◽  
Fractional Order Differential Equation

Abstract The field of Fractional calculus is more useful to understand the real-world phenomena. In this article, a nonlinear fractional order differential equation with Atangana-Baleanu operator is considered for analysis. Sufficient conditions under which a solution exists and uniqueness are presented using Banach fixed-point theorem method. The well-established Adams-Bashforth numerical scheme is used to solve the system of equations. Stability conditions are presented in details. To corroborate the analytical results, an example is given with numerical simulation. Mathematics Subject Classification [2010]: 26A33, 35B35, 65D25, 65L20.

scholarly journals Boundedness of Stochastic Delay Differential Systems with Impulsive Control and Impulsive Disturbance

2015 ◽  
Vol 2015 ◽  
pp. 1-8 ◽  
Author(s):  
Liming Wang ◽  
Baoqing Yang ◽  
Xiaohua Ding ◽  
Kai-Ning Wu
Keyword(s):  
Stochastic Analysis ◽  
Impulsive Control ◽  
Sufficient Conditions ◽  
Differential Systems ◽  
Stochastic Delay ◽  
Analysis Techniques ◽  
Moment Boundedness ◽  
Delay Differential System ◽  
Delay Differential Systems ◽  
Delay Differential

This paper considers thep-moment boundedness of nonlinear impulsive stochastic delay differential systems (ISDDSs). Using the Lyapunov-Razumikhin method and stochastic analysis techniques, we obtain sufficient conditions which guarantee thep-moment boundedness of ISDDSs. Two cases are considered, one is that the stochastic delay differential system (SDDS) may not be bounded, and how an impulsive strategy should be taken to make the SDDS be bounded. The other is that the SDDS is bounded, and an impulsive disturbance appears in this SDDS, then what restrictions on the impulsive disturbance should be adopted to maintain the boundedness of the SDDS. Our results provide sufficient criteria for these two cases. At last, two examples are given to illustrate the correctness of our results.

scholarly journals Positivity of Fundamental Matrix and Exponential Stability of Delay Differential System

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
Alexander Domoshnitsky ◽  
Roman Shklyar ◽  
Mikhail Gitman ◽  
Valery Stolbov
Keyword(s):  
Differential Equations ◽  
Exponential Stability ◽  
Sufficient Conditions ◽  
Delay System ◽  
Necessary Condition ◽  
Cauchy Matrix ◽  
Linear Ordinary Differential Equations ◽  
Delay Differential System ◽  
Delay Differential ◽  
Necessary And Sufficient

The classical Wazewski theorem established that nonpositivity of all nondiagonal elementspij  (i≠j,  i,j=1,…,n)is necessary and sufficient for nonnegativity of the fundamental (Cauchy) matrix and consequently for applicability of the Chaplygin approach of approximate integration for system of linear ordinary differential equationsxi′t+∑j=1n‍pijtxjt=fit,   i=1,…,n.Results on nonnegativity of the Cauchy matrix for system of delay differential equationsxi′t+∑j=1n‍pijtxjhijt=fit,   i=1,…,n,which were based on nonpositivity of all diagonal elements, were presented in the previous works. Then examples, which demonstrated that nonpositivity of nondiagonal coefficientspijis not necessary for systems of delay equations, were found. In this paper first sufficient results about nonnegativity of the Cauchy matrix of the delay system without this assumption are proven. A necessary condition of nonnegativity of the Cauchy matrix is proposed. On the basis of these results on nonnegativity of the Cauchy matrix, necessary and sufficient conditions of the exponential stability of the delay system are obtained.

scholarly journals Global Attractivity of Positive Periodic Solutions of Delay Differential Equations with Feedback Control

2007 ◽  
Vol 2007 ◽  
pp. 1-11 ◽  
Author(s):  
Zhi-Long Jin
Keyword(s):  
Feedback Control ◽  
Periodic Solutions ◽  
Global Attractivity ◽  
Sufficient Conditions ◽  
Upper And Lower Bounds ◽  
Positive Periodic Solutions ◽  
Liapunov Functionals ◽  
Delay Differential System ◽  
Several Delays ◽  
Delay Differential

By constructing suitable Liapunov functionals and estimating uniform upper and lower bounds of solutions, sufficient conditions are obtained for the global attractivity of positive periodic solutions of the delay differential system with feedback controldy/dt=y(t)F(t,y(t−τ1(t)),…,y(t−τn(t)),u(t−δ(t))),du/dt=−η(t)u(t)+a(t)y(t−σ(t)). When these results are applied to the periodic logistic equation with several delays and feedback control, some new results are obtained.

scholarly journals On solving fuzzy delay differential equation using bezier curves

2020 ◽  
Vol 10 (6) ◽  
pp. 6521
Author(s):  
Ali F. Jameel ◽  
Sardar G. Amen ◽  
Azizan Saaban ◽  
Noraziah H. Man
Keyword(s):  
Differential Equation ◽  
Exact Solution ◽  
Differential Equations ◽  
Set Theory ◽  
Fuzzy Set Theory ◽  
Delay Differential Equations ◽  
Bezier Curves ◽  
Bézier Curves ◽  
First Order ◽  
Delay Differential

In this article, we plan to use Bezier curves method to solve linear fuzzy delay differential equations. A Bezier curves method is presented and modified to solve fuzzy delay problems taking the advantages of the fuzzy set theory properties. The approximate solution with different degrees is compared to the exact solution to confirm that the linear fuzzy delay differential equations process is accurate and efficient. Numerical example is explained and analyzed involved first order linear fuzzy delay differential equations to demonstrate these proper features of this proposed problem.

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