Existence of Solutions of a New Class of Impulsive Initial Value Problems of Singular Nonlinear Fractional Differential Systems

2016 ◽  
Vol 17 (7-8) ◽  
pp. 343-353
Author(s):  
Yuji Liu
Keyword(s):  
Fixed Point Theorems ◽  
Existence Of Solutions ◽  
Sufficient Conditions ◽  
Initial Value ◽  
Differential Systems ◽  
New Class ◽  
Fractional Differential Systems ◽  
Impulsive Boundary Value Problems ◽  
Fractional Differential ◽  
Conclusion Section

Abstract:Sufficient conditions are given for the existence of solutions of impulsive boundary value problems for singular nonlinear fractional differential systems. We allow the nonlinearities$$p(t)f(t,y)$$and$$q(t)g(t,x)$$in fractional differential equations to be singular at$$t\!=\!0$$. 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. A conclusion section is given at the end of the paper.

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 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.

scholarly journals Some New Fixed Point Theorems in b-Metric Spaces with Application

Mathematics ◽  
2020 ◽  
Vol 8 (5) ◽  
pp. 725
Author(s):  
Badriah A. S. Alamri ◽  
Ravi P. Agarwal ◽  
Jamshaid Ahmad
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorems ◽  
Metric Spaces ◽  
Existence Of Solutions ◽  
Sufficient Conditions ◽  
Contraction Mappings ◽  
New Class

The aim of this article is to introduce a new class of contraction-like mappings, called the almost multivalued ( Θ , δ b )-contraction mappings in the setting of b-metric spaces to obtain some generalized fixed point theorems. As an application of our main result, we present the sufficient conditions for the existence of solutions of Fredholm integral inclusions. An example is also provided to verify the effectiveness and applicability of our main results.

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.

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.

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