Solution set for fractional differential equations with Riemann-Liouville derivative

2013 ◽  
Vol 16 (3) ◽  
Author(s):  
Yurilev Chalco-Cano ◽  
Juan Nieto ◽  
Abdelghani Ouahab ◽  
Heriberto Román-Flores
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Solution Set ◽  
Topological Properties ◽  
Initial Value ◽  
Structure Solution ◽  
Liouville Derivative ◽  
Liouville Fractional Derivative ◽  
Fractional Differential ◽  
Contractible Spaces

AbstractWe study an initial value problem for a fractional differential equation using the Riemann-Liouville fractional derivative. We obtain some topological properties of the solution set: It is the intersection of a decreasing sequence of compact nonempty contractible spaces. We extend the classical Kneser’s theorem on the structure solution set for ordinary differential equations.

scholarly journals New Gronwall-Bellman Type Inequalities and Applications in the Analysis for Solutions to Fractional Differential Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-13 ◽  
Author(s):  
Bin Zheng ◽  
Qinghua Feng
Keyword(s):  
Quantitative Analysis ◽  
Differential Equations ◽  
Fractional Derivative ◽  
Fractional Differential Equations ◽  
Continuous Dependence ◽  
Initial Value ◽  
Liouville Fractional Derivative ◽  
Explicit Bounds ◽  
Fractional Differential ◽  
Differential And Integral Equations

Some new Gronwall-Bellman type inequalities are presented in this paper. Based on these inequalities, new explicit bounds for the related unknown functions are derived. The inequalities established can also be used as a handy tool in the research of qualitative as well as quantitative analysis for solutions to some fractional differential equations defined in the sense of the modified Riemann-Liouville fractional derivative. For illustrating the validity of the results established, we present some applications for them, in which the boundedness, uniqueness, and continuous dependence on the initial value for the solutions to some certain fractional differential and integral equations are investigated.

scholarly journals Existence and Ulam stability for nonlinear implicit differential equations with Riemann-Liouville fractional derivative

2019 ◽  
Vol 52 (1) ◽  
pp. 437-450 ◽  
Author(s):  
Mouffak Benchohra ◽  
Soufyane Bouriah ◽  
Juan J. Nieto
Keyword(s):  
Differential Equations ◽  
Fractional Derivative ◽  
Initial Value ◽  
Uniqueness Of Solutions ◽  
Banach Contraction ◽  
Liouville Fractional Derivative ◽  
Fractional Differential ◽  
The Stability ◽  
The Banach Contraction Principle ◽  
Implicit Fractional Differential Equations

AbstractIn this paper, we establish the existence and uniqueness of solutions for a class of initial value problem for nonlinear implicit fractional differential equations with Riemann-Liouville fractional derivative, also, the stability of this class of problem. The arguments are based upon the Banach contraction principle and Schaefer’s fixed point theorem. An example is included to show the applicability of our results.

scholarly journals Asymptotically Linear Solutions for Some Linear Fractional Differential Equations

2010 ◽  
Vol 2010 ◽  
pp. 1-8 ◽  
Author(s):  
Dumitru Baleanu ◽  
Octavian G. Mustafa ◽  
Ravi P. Agarwal
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Fractional Differential Equation ◽  
Fractional Differential Equations ◽  
Asymptotically Linear ◽  
Liouville Derivative ◽  
Fractional Differential ◽  
Linear Fractional Differential Equations ◽  
Functional Coefficient

We establish here that under some simple restrictions on the functional coefficienta(t)the fractional differential equationD0tα[tx′−x+x(0)]+a(t)x=0,  t>0, has a solution expressible asct+d+o(1)fort→+∞, whereD0tαdesignates the Riemann-Liouville derivative of orderα∈(0,1)andc,d∈ℝ.

scholarly journals Explicit Solutions of Initial Value Problems for Fractional Generalized Proportional Differential Equations with and without Impulses

Symmetry ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 996
Author(s):  
Snezhana Hristova ◽  
Mohamed I. Abbas
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Initial Value Problem ◽  
Explicit Solution ◽  
Explicit Solutions ◽  
Initial Value ◽  
Impulsive Conditions ◽  
Fractional Differential ◽  
Set Up ◽  
Two Parameters

The object of investigation in this paper is a scalar linear fractional differential equation with generalized proportional derivative of Riemann–Liouville type (LFDEGD). The main goal is the obtaining an explicit solution of the initial value problem of the studied equation. Note that the locally solvability, being the same as the existence of solutions to the initial value problem, is connected with the symmetry of a transformation of a system of differential equations. At the same time, several criteria for existence of the initial value problem for nonlinear fractional differential equations with generalized proportional derivative are connected with the linear ones. It leads to the necessity of obtaining an explicit solution of LFDEGD. In this paper two cases are studied: the case of no impulses in the differential equation are presented and the case when instantaneous impulses at initially given points are involved. All obtained formulas are based on the application of Mittag–Leffler function with two parameters. In the case of impulses, initially the appropriate impulsive conditions are set up and later the explicit solutions are obtained.

scholarly journals Existence and uniqueness results for a nonlinear singular fractional differential equation of order $ \sigma\in(1, 2) $

2021 ◽  
Vol 6 (12) ◽  
pp. 13041-13056
Author(s):  
Sinan Serkan Bilgici ◽  
◽  
Müfit ŞAN
Keyword(s):  
Differential Equation ◽  
Nonlinear Term ◽  
Local Existence ◽  
Hölder Inequality ◽  
Initial Value ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results ◽  
Liouville Fractional Derivative ◽  
Fractional Differential ◽  
Linear Differential

<abstract><p>The first objective of this article is to discuss the local existence of the solution to an initial value problem involving a non-linear differential equation in the sense of Riemann-Liouville fractional derivative of order $ \sigma\in(1, 2), $ when the nonlinear term has a singularity at zero of its independent argument. Hereafter, by using some tools of Lebesgue spaces such as Hölder inequality, we obtain Nagumo-type, Krasnoselskii-Krein-type and Osgood-type uniqueness theorems for the problem.</p></abstract>

scholarly journals Some New Gronwall-Bellman-Type Inequalities Based on the Modified Riemann-Liouville Fractional Derivative

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Bin Zheng
Keyword(s):  
Differential Equation ◽  
Fractional Derivative ◽  
Fractional Differential Equation ◽  
Continuous Dependence ◽  
Initial Value ◽  
Liouville Fractional Derivative ◽  
Explicit Bounds ◽  
Fractional Differential

By use of the properties of the modified Riemann-Liouville fractional derivative, some new Gronwall-Bellman-type inequalities are researched. First, we derive some new explicit bounds for the unknown functions lying in these inequalities, which are of different forms from some existing bounds in the literature. Then, we apply the results established to research the boundedness, uniqueness, and continuous dependence on the initial value for the solution to a certain fractional differential equation.

Existence of Positive Solutions to the Singular Initial Value Problems of Fractional Differential Equations

2011 ◽  
Vol 403-408 ◽  
pp. 563-569
Author(s):  
Qiu Ping Li ◽  
Shu Rong Sun ◽  
Zhen Lai Han ◽  
Yi Ge Zhao
Keyword(s):  
Differential Equations ◽  
Positive Solutions ◽  
Fractional Differential Equations ◽  
Initial Value Problem ◽  
Sufficient Conditions ◽  
Initial Value ◽  
Nonlinear Alternative ◽  
Existence Of Positive Solutions ◽  
Liouville Fractional Derivative ◽  
Fractional Differential

In this paper, we consider the existence of positive solutions for the initial value problem of nonlinear fractional differential equations where and is the Riemann–Liouville fractional derivative. By using the Nonlinear Alternative of Leray and Schauder theorem, some sufficient conditions for the existence of at least one positive solution for the initial value problem are established.

On fractional differential equations with state-dependent delay via Kuratowski measure of noncompactness

Filomat ◽  
2017 ◽  
Vol 31 (2) ◽  
pp. 451-460 ◽  
Author(s):  
Mohammed Belmekki ◽  
Kheira Mekhalfi
Keyword(s):  
Differential Equations ◽  
Measure Of Noncompactness ◽  
Mild Solutions ◽  
Functional Differential Equations ◽  
Kuratowski Measure Of Noncompactness ◽  
State Dependent ◽  
State Dependent Delay ◽  
Liouville Fractional Derivative ◽  
Fractional Differential ◽  
Functional Differential

This paper is devoted to study the existence of mild solutions for semilinear functional differential equations with state-dependent delay involving the Riemann-Liouville fractional derivative in a Banach space and resolvent operator. The arguments are based upon M?nch?s fixed point theoremand the technique of measure of noncompactness.

scholarly journals Lyapunov Functions and Lipschitz Stability for Riemann–Liouville Non-Instantaneous Impulsive Fractional Differential Equations

Symmetry ◽  
2021 ◽  
Vol 13 (4) ◽  
pp. 730
Author(s):  
Ravi Agarwal ◽  
Snezhana Hristova ◽  
Donal O’Regan
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Lyapunov Functions ◽  
Sufficient Conditions ◽  
Initial Time ◽  
Lipschitz Stability ◽  
Comparison Results ◽  
Liouville Fractional Derivative ◽  
Fractional Differential ◽  
Definition Of

In this paper a system of nonlinear Riemann–Liouville fractional differential equations with non-instantaneous impulses is studied. We consider a Riemann–Liouville fractional derivative with a changeable lower limit at each stop point of the action of the impulses. In this case the solution has a singularity at the initial time and any stop time point of the impulses. This leads to an appropriate definition of both the initial condition and the non-instantaneous impulsive conditions. A generalization of the classical Lipschitz stability is defined and studied for the given system. Two types of derivatives of the applied Lyapunov functions among the Riemann–Liouville fractional differential equations with non-instantaneous impulses are applied. Several sufficient conditions for the defined stability are obtained. Some comparison results are obtained. Several examples illustrate the theoretical results.

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