A constant enclosure method for validating existence and uniqueness of the solution of an initial value problem for a fractional differential equation

2010 ◽  
Vol 217 (5) ◽  
pp. 2162-2168 ◽  
Author(s):  
M. Amairi ◽  
M. Aoun ◽  
S. Najar ◽  
M.N. Abdelkrim
Keyword(s):  
Differential Equation ◽  
Fractional Differential Equation ◽  
Initial Value Problem ◽  
Existence And Uniqueness ◽  
Initial Value ◽  
Enclosure Method ◽  
Uniqueness Of The Solution ◽  
Fractional Differential

scholarly journals On a Hilfer Type Fractional Differential Equation with Nonlinear Right-Hand Side

2021 ◽  
Vol 103 (3) ◽  
pp. 140-155
Author(s):  
T. K. Yuldashev ◽  
◽  
B. J. Kadirkulov ◽  
A. R. Marakhimov ◽  
◽  
...  
Keyword(s):  
Differential Equation ◽  
Fractional Differential Equation ◽  
Initial Value Problem ◽  
Integral Transformation ◽  
Numerical Realization ◽  
Initial Value ◽  
Volterra Type ◽  
Fractional Integral Equation ◽  
Uniqueness Of The Solution ◽  
Fractional Differential

In this article we consider the questions of one-valued solvability and numerical realization of initial value problem for a nonlinear Hilfer type fractional differential equation with maxima. By the aid of uncomplicated integral transformation based on Dirichlet formula, this initial value problem is reduced to the nonlinear Volterra type fractional integral equation. The theorem of existence and uniqueness of the solution of given initial value problem in the segment under consideration is proved. For numerical realization of solution the generalized Jacobi–Galerkin method is applied. Illustrative examples are provided.

scholarly journals On Solvability of an Initial Value Problem for Hilfer type Fractional Differential Equation with Nonlinear Maxima

2020 ◽  
pp. 48-65
Author(s):  
T.K. Yuldashev ◽  
B.J. Kadirkulov
Keyword(s):  
Differential Equation ◽  
Fractional Differential Equation ◽  
Initial Value Problem ◽  
Integral Transformation ◽  
Initial Value ◽  
Volterra Type ◽  
Fractional Integral Equation ◽  
Uniqueness Of The Solution ◽  
Fractional Differential ◽  
The Stability

In this article we consider the questions of one-valued solvability of initial value problem for a nonlinear Hilfer type fractional differential equation with nonlinear maxima. By the aid of uncomplicated integral transformation based on Dirichlet formula, this initial value problem is reduced to the nonlinear Volterra type fractional integral equation with nonlinear maxima. It is proved the theorem of existence and uniqueness of the solution of given initial value problem in an interval under consideration. It is proved also the stability of the desired solution with respect to given parameter.

scholarly journals Solutions for impulsive fractional pantograph differential equation via generalized anti-periodic boundary condition

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Idris Ahmed ◽  
Poom Kumam ◽  
Jamilu Abubakar ◽  
Piyachat Borisut ◽  
Kanokwan Sitthithakerngkiet
Keyword(s):  
Differential Equation ◽  
Boundary Condition ◽  
Fixed Point ◽  
Fractional Differential Equation ◽  
Existence And Uniqueness ◽  
Periodic Boundary Condition ◽  
Periodic Boundary ◽  
Fixed Point Theorems ◽  
Uniqueness Of The Solution ◽  
Fractional Differential

Abstract This study investigates the solutions of an impulsive fractional differential equation incorporated with a pantograph. This work extends and improves some results of the impulsive fractional differential equation. A differential equation of an impulsive fractional pantograph with a more general anti-periodic boundary condition is proposed. By employing the well-known fixed point theorems of Banach and Krasnoselskii, the existence and uniqueness of the solution of the proposed problem are established. Furthermore, two examples are presented to support our theoretical analysis.

scholarly journals Existence and Uniqueness of Solutions for Initial Value Problem of Nonlinear Fractional Differential Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-14
Author(s):  
Qiuping Li ◽  
Shurong Sun ◽  
Ping Zhao ◽  
Zhenlai Han
Keyword(s):  
Differential Equation ◽  
Initial Value Problem ◽  
Existence And Uniqueness ◽  
Upper And Lower Solutions ◽  
Initial Value ◽  
Uniqueness Of Solutions ◽  
Nonlinear Fractional Differential Equation ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results ◽  
Fractional Differential

We discuss the initial value problem for the nonlinear fractional differential equationL(D)u=f(t,u),  t∈(0,1],  u(0)=0, whereL(D)=Dsn-an-1Dsn-1-⋯-a1Ds1,0<s1<s2<⋯<sn<1, andaj<0,j=1,2,…,n-1,Dsjis the standard Riemann-Liouville fractional derivative andf:[0,1]×ℝ→ℝis a given continuous function. We extend the basic theory of differential equation, the method of upper and lower solutions, and monotone iterative technique to the initial value problem. Some existence and uniqueness results are established.

scholarly journals New Fixed Point Theorems in Orthogonal F -Metric Spaces with Application to Fractional Differential Equation

Symmetry ◽  
2020 ◽  
Vol 12 (5) ◽  
pp. 832
Author(s):  
Tanzeela Kanwal ◽  
Azhar Hussain ◽  
Hamid Baghani ◽  
Manuel de la Sen
Keyword(s):  
Differential Equation ◽  
Fixed Point ◽  
Periodic Point ◽  
Fractional Differential Equation ◽  
Existence And Uniqueness ◽  
Fixed Point Theorems ◽  
Metric Spaces ◽  
Nonlinear Fractional Differential Equation ◽  
Uniqueness Of The Solution ◽  
Fractional Differential

We present the notion of orthogonal F -metric spaces and prove some fixed and periodic point theorems for orthogonal ⊥ Ω -contraction. We give a nontrivial example to prove the validity of our result. Finally, as application, we prove the existence and uniqueness of the solution of a nonlinear fractional differential equation.

scholarly journals The theorem existence and uniqueness of the solution of a fractional differential equation

2013 ◽  
Vol 23 ◽  
pp. 16-18
Author(s):  
Juan Martínez Ortiz ◽  
Leticia Ramírez Hernández
Keyword(s):  
Differential Equation ◽  
Fractional Differential Equation ◽  
Existence And Uniqueness ◽  
Banach Theorem ◽  
Uniqueness Of The Solution ◽  
Fractional Differential

This article aims to demonstrate, using the Picard-Banach theorem, the proof of the theorem-based existence and uniqueness of the solution of an fractional orden differential equation with a fractional Caputo-type derivative.

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