scholarly journals Bounds on the solution of a Cauchy-type problem involving a weighted sequential fractional derivative

2013 ◽  
Vol 16 (1) ◽  
Author(s):  
Khaled Furati
Keyword(s):  
Fractional Derivative ◽  
Fractional Differential Equations ◽  
Type Inequality ◽  
Gauss Hypergeometric Function ◽  
Cauchy Type ◽  
Type Problem ◽  
Sequential Fractional Derivative ◽  
Cauchy Type Problem ◽  
Fractional Differential ◽  
Bounds On The Solution

AbstractIn this paper we establish some bounds for the solution of a Cauchytype problem for a class of fractional differential equations with a weighted sequential fractional derivative. The bounds are based on a Bihari-type inequality and a bound on the Gauss hypergeometric function.

scholarly journals Fractional Differential Equations With Dependence on the Caputo–Katugampola Derivative

2016 ◽  
Vol 11 (6) ◽  
Author(s):  
Ricardo Almeida ◽  
Agnieszka B. Malinowska ◽  
Tatiana Odzijewicz
Keyword(s):  
Fractional Derivatives ◽  
Numerical Procedure ◽  
Cauchy Type ◽  
Type Problem ◽  
Existence And Uniqueness Theorem ◽  
Cauchy Type Problem ◽  
Special Cases ◽  
Decomposition Formula ◽  
New Type ◽  
Fractional Differential

In this paper, we present a new type of fractional operator, the Caputo–Katugampola derivative. The Caputo and the Caputo–Hadamard fractional derivatives are special cases of this new operator. An existence and uniqueness theorem for a fractional Cauchy-type problem, with dependence on the Caputo–Katugampola derivative, is proved. A decomposition formula for the Caputo–Katugampola derivative is obtained. This formula allows us to provide a simple numerical procedure to solve the fractional differential equation (FDE).

scholarly journals DIFFERENTIAL EQUATIONS WITH TEMPERED Ψ-CAPUTO FRACTIONAL DERIVATIVE

2021 ◽  
Vol 26 (4) ◽  
pp. 631-650
Author(s):  
Milan Medveď ◽  
Eva Brestovanská
Keyword(s):  
Differential Equations ◽  
Fractional Derivative ◽  
Fractional Differential Equations ◽  
Caputo Fractional Derivative ◽  
Type Inequality ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results ◽  
Representation Of Solutions ◽  
The Cauchy Problem ◽  
Fractional Differential

In this paper we define a new type of the fractional derivative, which we call tempered Ψ−Caputo fractional derivative. It is a generalization of the tempered Caputo fractional derivative and of the Ψ−Caputo fractional derivative. The Cauchy problem for fractional differential equations with this type of derivative is discussed and some existence and uniqueness results are proved. We present a Henry-Gronwall type inequality for an integral inequality with the tempered Ψ−fractional integral. This inequality is applied in the proof of an existence theorem. A result on a representation of solutions of linear systems of Ψ−Caputo fractional differential equations is proved and in the last section an example is presented.

NONLINEAR DIFFERENTIAL EQUATIONS WITH MARCHAUD‐HADAMARD-TYPE FRACTIONAL DERIVATIVE IN THE WEIGHTED SPACE OF SUMMABLE FUNCTIONS

2007 ◽  
Vol 12 (3) ◽  
pp. 343-356 ◽  
Author(s):  
Anatoly Kilbas ◽  
Anatoly Titioura
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Fractional Derivative ◽  
Nonlinear Differential Equations ◽  
Banach Fixed Point Theorem ◽  
Cauchy Type ◽  
Type Problem ◽  
Cauchy Type Problem ◽  
Linear Differential ◽  
Half Axis

The paper is devoted to the study of the Cauchy‐type problem for the nonlinear differential equation of fractional order 0 < α < 1: containing the Marchaud-Hadamard-type fractional derivative (Dα 0+, μ y)(x), on the half-axis R+ = (0, +oo) in the space Xp,α c,0 (R+) defined for α > 0 by where Xp c, 0 (R+) is the subspace of Xp c (R+) of functions g Xp c (R + ) with compact support on infinity: g(x) = 0 for large enough x > R. The equivalence of this problem and of the nonlinear Volterra integral equation is established. The existence and uniqueness of the solution y(x) of the above Cauchy‐type problem is proved by using the Banach fixed point theorem. Solution in closed form of the above problem for the linear differential equation with f[x, y(x)] = λy(x) + f(x) is constructed. The corresponding assertions for the differential equations with the Marchaud‐Hadamard fractional derivative (Dα 0+ y)(x) are presented. Examples are given.

Application of measure of noncompactness to a Cauchy problem for fractional differential equations in Banach spaces

2013 ◽  
Vol 16 (4) ◽  
Author(s):  
Asadollah Aghajani ◽  
Ehsan Pourhadi ◽  
Juan Trujillo
Keyword(s):  
Differential Equation ◽  
Cauchy Problem ◽  
Fixed Point ◽  
Measure Of Noncompactness ◽  
Existence Of Solutions ◽  
Cauchy Type ◽  
Type Problem ◽  
Nonlinear Fractional Differential Equation ◽  
Cauchy Type Problem ◽  
Fractional Differential

AbstractThis paper is devoted to study the existence of solutions of a Cauchy type problem for a nonlinear fractional differential equation, via the techniques of measure of noncompactness. The investigation is based on a new fixed point result which is a generalization of the well known Darbo’s fixed point theorem. The main result is less restrictive than those given in the literature. Some illustrative examples are given.

Initial-value / Nonlocal Cauchy problems for fractional differential equations involving ψ-Hilfer multivariable operators

2020 ◽  
Vol 23 (4) ◽  
pp. 1090-1124
Author(s):  
Jin Liang ◽  
Yunyi Mu ◽  
Ti-Jun Xiao
Keyword(s):  
Cauchy Problem ◽  
Positive Integer ◽  
Differential Equations ◽  
Fractional Derivative ◽  
Fractional Differential Equations ◽  
Type Inequality ◽  
Continuous Functions ◽  
Initial Value ◽  
Fractional Differential ◽  
Nonlocal Cauchy Problem

AbstractIn this paper, we investigate two types of problems (the initial-value problem and nonlocal Cauchy problem) for fractional differential equations involving ψ-Hilfer derivative in multivariable case (ψ-m-Hilfer derivative). First we propose and discuss ψ-fractional integral, ψ-fractional derivative and ψ-Hilfer type fractional derivative of a multivariable function f : ℝm → ℝ (m is a positive integer). Then, using the properties of the ψ-m-Hilfer fractional derivative with m = 1 (the ψ-Hilfer derivative), we derive an equivalent relationship between solutions to the initial-value (Cauchy) problem and solutions to some integral equations, and also present an existence and uniqueness theorem. Based on the equivalency relationship, we establish new and general existence results for the nonlocal Cauchy problem of fractional differential equations involving ψ-Hilfer multivariable operators in the space of weighted continuous functions. Moreover, we obtain a new Gronwall-type inequality with singular kernel, and derive the dependence of the solution on the order and the initial condition for the fractional Cauchy problem with the help of this Gronwall-type inequality. Finally, some examples are given to illustrate our results. Compared with the recent paper [2] and other previous works, the novelties in this paper are in treating the multivariable case of operators (f : ℝm → ℝ, m is a positive integer).

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