scholarly journals The Periodic Solutions of the Compound Singular Fractional Differential System with Delay

2010 ◽  
Vol 2010 ◽  
pp. 1-9 ◽  
Author(s):  
XuTing Wei ◽  
XuanZhu Lu
Keyword(s):  
Periodic Solution ◽  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Two Dimensional ◽  
Differential Systems ◽  
Fractional Differential Systems ◽  
Fractional Differential System ◽  
Singular Fractional Differential System ◽  
Fractional Differential ◽  
Equation With Delay

The paper gives sufficient conditions on the existence of periodic solution for a class of compound singular fractional differential systems with delay, involving Nishimoto fractional derivative. Furthermore, for the particular functions, the necessary conditions on the existence of periodic solution are also derived. Especially, for two-dimensional compound singular fractional differential equation with delay, the criteria of existence of periodic solution are obtained. Finally, two examples are presented to verify the validity of criteria.

scholarly journals Asymptotical Stability of Nonlinear Fractional Differential System with Caputo Derivative

2011 ◽  
Vol 2011 ◽  
pp. 1-12 ◽  
Author(s):  
Fengrong Zhang ◽  
Changpin Li ◽  
YangQuan Chen
Keyword(s):  
Caputo Derivative ◽  
Sufficient Conditions ◽  
Comparison Method ◽  
Asymptotical Stability ◽  
Sufficient Condition ◽  
Differential Systems ◽  
Fractional Differential Systems ◽  
Fractional Differential System ◽  
Fractional Differential ◽  
The Stability

This paper deals with the stability of nonlinear fractional differential systems equipped with the Caputo derivative. At first, a sufficient condition on asymptotical stability is established by using a Lyapunov-like function. Then, the fractional differential inequalities and comparison method are applied to the analysis of the stability of fractional differential systems. In addition, some other sufficient conditions on stability are also presented.

scholarly journals Positive Solutions for a Class of Nonlinear Singular Fractional Differential Systems with Riemann–Stieltjes Coupled Integral Boundary Value Conditions

Symmetry ◽  
2021 ◽  
Vol 13 (1) ◽  
pp. 107
Author(s):  
Daliang Zhao ◽  
Juan Mao
Keyword(s):  
Positive Solutions ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Main Tool ◽  
Integral Boundary ◽  
Differential Systems ◽  
Fractional Differential Systems ◽  
Existence And Multiplicity ◽  
Fractional Differential ◽  
Multiplicity Of Positive Solutions

In this paper, sufficient conditions ensuring existence and multiplicity of positive solutions for a class of nonlinear singular fractional differential systems are derived with Riemann–Stieltjes coupled integral boundary value conditions in Banach Spaces. Nonlinear functions f(t,u,v) and g(t,u,v) in the considered systems are allowed to be singular at every variable. The boundary conditions here are coupled forms with Riemann–Stieltjes integrals. In order to overcome the difficulties arising from the singularity, a suitable cone is constructed through the properties of Green’s functions associated with the systems. The main tool used in the present paper is the fixed point theorem on cone. Lastly, an example is offered to show the effectiveness of our obtained new results.

GENERATING 3-D SCROLL GRID ATTRACTORS OF FRACTIONAL DIFFERENTIAL SYSTEMS VIA STAIR FUNCTION

2007 ◽  
Vol 17 (11) ◽  
pp. 3965-3983 ◽  
Author(s):  
WEIHUA DENG
Keyword(s):  
Differential System ◽  
Autonomous System ◽  
Function Approach ◽  
Differential Systems ◽  
Fractional Differential Systems ◽  
Fractional Differential System ◽  
Linear Autonomous System ◽  
Fractional Differential ◽  
The One ◽  
Number Of Equilibria

This paper discusses the stair function approach for the generation of scroll grid attractors of fractional differential systems. The one-directional (1-D) n-grid scroll, two-directional (2-D) (n × m)-grid scroll and three-directional (3-D) (n × m × l)-grid scroll attractors are created from a fractional linear autonomous system with a simple stair function controller. Being similar to the scroll grid attractors of classical differential systems, the scrolls of 1-D n-grid scroll, 2-D (n × m)-grid scroll and 3-D (n × m × l)-grid scroll attractors are located around the equilibria of fractional differential system on a line, on a plane or in 3D, respectively and the number of scrolls is equal to the corresponding number of equilibria.

State Feedback Control for Fractional Differential Systems with Riemann-Liouville Derivative

2012 ◽  
Vol 562-564 ◽  
pp. 2053-2056
Author(s):  
Yuan Fang
Keyword(s):  
Feedback Control ◽  
State Feedback ◽  
Sufficient Conditions ◽  
State Feedback Control ◽  
Linear Matrix ◽  
Feedback Controllers ◽  
Differential Systems ◽  
Fractional Differential Systems ◽  
Liouville Derivative ◽  
Fractional Differential

This paper studies state feedback control for fractional differential systems with Riemann-Lιiouville derivative, which matrix A not satisfying the condition ιarg(λ(A))ι>α/2 . Based on the state feedback controllers’ designer, and Linear Matrix Inequality (LMI) apαproach, sufficient conditions for the systems with fraction order α (0<α<1) and α (1≤α<2) obtained respectively.

scholarly journals Stability, Boundedness, and Lagrange Stability of Fractional Differential Equations with Initial Time Difference

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Muhammed Çiçek ◽  
Coşkun Yakar ◽  
Bülent Oğur
Keyword(s):  
Fractional Differential Equations ◽  
Sufficient Conditions ◽  
Time Difference ◽  
Initial Time ◽  
Differential Inequalities ◽  
Differential Systems ◽  
Lagrange Stability ◽  
Fractional Differential Systems ◽  
Comparison Results ◽  
Fractional Differential

Differential inequalities, comparison results, and sufficient conditions on initial time difference stability, boundedness, and Lagrange stability for fractional differential systems have been evaluated.

Solvability of Anti-periodic BVPs for Impulsive Fractional Differential Systems Involving Caputo and Riemann–Liouville Fractional Derivatives

2018 ◽  
Vol 19 (2) ◽  
pp. 125-152 ◽  
Author(s):  
Yuji Liu
Keyword(s):  
Fixed Point ◽  
Fractional Derivatives ◽  
Fixed Point Theorems ◽  
Existence Of Solutions ◽  
Sufficient Conditions ◽  
Initial Value ◽  
Differential Systems ◽  
Fractional Differential Systems ◽  
Fractional Differential ◽  
Conclusion Section

AbstractSufficient conditions are given for the existence of solutions of anti-periodic value problems for impulsive fractional differential systems involving both Caputo and Riemann–Liouville fractional derivatives. We allow the nonlinearities$p(t)f(t,x,y,z,w)$and$q(t)g(t,x,y,z,w)$in fractional differential equations to be singular at$t=0$and$t=1$. Both$f$and$g$may be super-linear and sub-linear. The analysis relies on some well known fixed point theorems. The initial value problem discussed may be seen as a generalization of some ecological models. An example is given to illustrate the efficiency of the main theorems. Many unsuitable lemmas in recent published papers are pointed out in order not to mislead readers. A conclusion section is given at the end of the paper.

scholarly journals Successive approximations for random coupled Hilfer fractional differential systems

2021 ◽  
Author(s):  
Fatima Si Bachir ◽  
Saïd Abbas ◽  
Maamar Benbachir ◽  
Mouffak Benchohra
Keyword(s):  
Global Convergence ◽  
Unique Solution ◽  
Differential System ◽  
Successive Approximations ◽  
Random Solution ◽  
Differential Systems ◽  
Fractional Differential Systems ◽  
Fractional Differential System ◽  
Fractional Differential

AbstractIn this paper, we study the global convergence of successive approximations as well as the uniqueness of the random solution of a coupled random Hilfer fractional differential system. We prove a theorem on the global convergence of successive approximations to the unique solution of our problem. In the last section, we present an illustrative example.

scholarly journals Positive solutions of semipositone singular fractional differential systems with a parameter and integral boundary conditions

2018 ◽  
Vol 16 (1) ◽  
pp. 581-596 ◽  
Author(s):  
Xinan Hao ◽  
Huaqing Wang
Keyword(s):  
Boundary Conditions ◽  
Positive Solutions ◽  
Sufficient Conditions ◽  
Integral Boundary Conditions ◽  
Integral Boundary ◽  
Differential Systems ◽  
Fractional Differential Systems ◽  
Existence Of Positive Solutions ◽  
Singular Fractional Differential Equations ◽  
Fractional Differential

AbstractIn this paper, the existence of positive solutions for systems of semipositone singular fractional differential equations with a parameter and integral boundary conditions is investigated. By using fixed point theorem in cone, sufficient conditions which guarantee the existence of positive solutions are obtained. An example is given to illustrate the results.

scholarly journals Controllability Criteria for Linear Fractional Differential Systems with State Delay and Impulses

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Hai Zhang ◽  
Jinde Cao ◽  
Wei Jiang
Keyword(s):  
Fractional Derivative ◽  
Linear Systems ◽  
Necessary Conditions ◽  
State Delay ◽  
Differential Systems ◽  
State Response ◽  
Fractional Differential Systems ◽  
Sufficient And Necessary Conditions ◽  
Fractional Differential ◽  
Controllability Property

This paper is concerned with the controllability of linear fractional differential systems with delay in state and impulses. The factors of such systems including fractional derivative, impulses, and delay are taken into account synchronously. The expression of state response for such systems is derived, and the sufficient and necessary conditions of controllability criteria are established. Both the proposed criteria and illustrative examples show that the controllability property of the linear systems is dependent neither on the order of fractional derivative, on delay nor on impulses.

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