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.

scholarly journals Weighted fractional differential equations with infinite delay in Banach spaces

2016 ◽  
Vol 14 (1) ◽  
pp. 370-383 ◽  
Author(s):  
Qixiang Dong ◽  
Can Liu ◽  
Zhenbin Fan
Keyword(s):  
Differential Equations ◽  
Banach Spaces ◽  
Fractional Differential Equations ◽  
Fractional Derivatives ◽  
Fixed Point Theorems ◽  
Infinite Delay ◽  
Type Inequality ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results ◽  
Fractional Differential

AbstractThis paper is devoted to the study of fractional differential equations with Riemann-Liouville fractional derivatives and infinite delay in Banach spaces. The weighted delay is developed to deal with the case of non-zero initial value, which leads to the unboundedness of the solutions. Existence and uniqueness results are obtained based on the theory of measure of non-compactness, Schaude’s and Banach’s fixed point theorems. As auxiliary results, a fractional Gronwall type inequality is proved, and the comparison property of fractional integral is discussed.

scholarly journals Multi-term fractional differential equations with nonlocal boundary conditions

2018 ◽  
Vol 16 (1) ◽  
pp. 1519-1536
Author(s):  
Bashir Ahmad ◽  
Najla Alghamdi ◽  
Ahmed Alsaedi ◽  
Sotiris K. Ntouyas
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Existence And Uniqueness ◽  
Fixed Point Theorems ◽  
Nonlocal Boundary ◽  
Nonlocal Boundary Conditions ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results ◽  
Fractional Differential ◽  
The Given

AbstractWe introduce and study a new kind of nonlocal boundary value problems of multi-term fractional differential equations. The existence and uniqueness results for the given problem are obtained by applying standard fixed point theorems. We also construct some examples for demonstrating the application of the main results.

scholarly journals A Note on Fractional Differential Equations with Fractional Separated Boundary Conditions

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
Bashir Ahmad ◽  
Sotiris K. Ntouyas
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Fractional Differential Equations ◽  
Existence And Uniqueness ◽  
Fixed Point Theorems ◽  
Existence And Uniqueness Results ◽  
New Class ◽  
Uniqueness Results ◽  
Separated Boundary Conditions ◽  
Fractional Differential

We consider a new class of boundary value problems of nonlinear fractional differential equations with fractional separated boundary conditions. A connection between classical separated and fractional separated boundary conditions is developed. Some new existence and uniqueness results are obtained for this class of problems by using standard fixed point theorems. Some illustrative examples are also discussed.

scholarly journals EXISTENCE AND UNIQUENESS RESULTS FOR A COUPLED SYSTEM OF HADAMARD FRACTIONAL DIFFERENTIAL EQUATIONS WITH MULTI-POINT BOUNDARY CONDITIONS

2020 ◽  
pp. 843
Author(s):  
Mohamed Houas ◽  
Khellaf Ould Melha
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Existence And Uniqueness ◽  
Coupled System ◽  
Point Boundary ◽  
Uniqueness Of Solutions ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results ◽  
Hadamard Fractional Differential Equations ◽  
Fractional Differential

In this paper, we have studied existence and uniqueness of solutions for a coupled system of multi-point boundary value problems for Hadamard fractional differential equations. By applying principle contraction and Shaefer's fixed point theorem new existence results have been obtained.

scholarly journals On the asymptotic behavior of nonoscillatory solutions of certain fractional differential equations with positive and negative terms

2020 ◽  
Vol 40 (2) ◽  
pp. 227-239
Author(s):  
John R. Graef ◽  
Said R. Grace ◽  
Ercan Tunç
Keyword(s):  
Differential Equation ◽  
Asymptotic Behavior ◽  
Differential Equations ◽  
Fractional Derivative ◽  
Fractional Differential Equation ◽  
Fractional Differential Equations ◽  
Caputo Fractional Derivative ◽  
Nonoscillatory Solutions ◽  
Fractional Differential

This paper is concerned with the asymptotic behavior of the nonoscillatory solutions of the forced fractional differential equation with positive and negative terms of the form \[^{C}D_{c}^{\alpha}y(t)+f(t,x(t))=e(t)+k(t)x^{\eta}(t)+h(t,x(t)),\] where \(t\geq c \geq 1\), \(\alpha \in (0,1)\), \(\eta \geq 1\) is the ratio of positive odd integers, and \(^{C}D_{c}^{\alpha}y\) denotes the Caputo fractional derivative of \(y\) of order \(\alpha\). The cases \[y(t)=(a(t)(x^{\prime}(t))^{\eta})^{\prime} \quad \text{and} \quad y(t)=a(t)(x^{\prime}(t))^{\eta}\] are considered. The approach taken here can be applied to other related fractional differential equations. Examples are provided to illustrate the relevance of the results obtained.

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