Existence and uniqueness of solutions for initial value problems of multi-order fractional differential equations on the half lines

2012 ◽  
Vol 42 (7) ◽  
pp. 735-756 ◽  
Author(s):  
YuJi LIU
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Existence And Uniqueness ◽  
Initial Value Problems ◽  
Initial Value ◽  
Uniqueness Of Solutions ◽  
Fractional Differential

scholarly journals The Existence and Uniqueness of Solutions and Lyapunov-Type Inequality for CFR Fractional Differential Equations

2018 ◽  
Vol 2018 ◽  
pp. 1-7 ◽  
Author(s):  
Xia Wang ◽  
Run Xu
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Existence And Uniqueness ◽  
Type Inequality ◽  
Uniqueness Theorems ◽  
Initial Value ◽  
Uniqueness Of Solutions ◽  
Maximum Value ◽  
Fractional Differential ◽  
Lyapunov Type Inequality

In this paper, we research CFR fractional differential equations with the derivative of order 3<α<4. We prove existence and uniqueness theorems for CFR-type initial value problem. By Green’s function and its corresponding maximum value, we obtain the Lyapunov-type inequality of corresponding equations. As for application, we study the eigenvalue problem in the sense of CFR.

scholarly journals Impulsive Multiorders Riemann-Liouville Fractional Differential Equations

2015 ◽  
Vol 2015 ◽  
pp. 1-9 ◽  
Author(s):  
Weera Yukunthorn ◽  
Sotiris K. Ntouyas ◽  
Jessada Tariboon
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Existence And Uniqueness ◽  
Initial Value Problems ◽  
Weighted Space ◽  
Initial Value ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results ◽  
Banach’S Fixed Point Theorem ◽  
Fractional Differential

Impulsive multiorders fractional differential equations are studied. Existence and uniqueness results are obtained for first- and second-order impulsive initial value problems by using Banach’s fixed point theorem in an appropriate weighted space. Examples illustrating the main results are presented.

Well-posedness of general Caputo-type fractional differential equations

2018 ◽  
Vol 21 (3) ◽  
pp. 819-832 ◽  
Author(s):  
Chung-Sik Sin
Keyword(s):  
Fixed Point ◽  
Differential Equations ◽  
Fractional Differential Equations ◽  
Initial Value Problems ◽  
Fractional Derivatives ◽  
Local Existence ◽  
Well Posedness ◽  
Initial Value ◽  
Uniqueness Of Solutions ◽  
Fractional Differential

Abstract In the present paper, initial value problems of fractional differential equations are considered. The fractional derivatives are defined as the general Caputo-type fractional derivatives proposed by Anatoly Kochubei. By using the Schauder fixed point theorem, the local existence, the global existence and the uniqueness of solutions are obtained under appropriate conditions. In addition, it is proved that the solution depends on the parameters of the equation in a continuous way.

Existence and uniqueness of global solutions of caputo-type fractional differential equations

2016 ◽  
Vol 19 (3) ◽  
Author(s):  
Chung-Sik Sin ◽  
Liancun Zheng
Keyword(s):  
Fixed Point ◽  
Differential Equations ◽  
Fractional Differential Equations ◽  
Existence And Uniqueness ◽  
Initial Value Problems ◽  
Differential Operators ◽  
Sufficient Conditions ◽  
Global Solutions ◽  
Initial Value ◽  
Fractional Differential

AbstractIn this paper we consider initial value problems for fractional differential equations involving Caputo differential operators. By establishing a new property of the Mittag-Leffler function and using the Schauder fixed point theorem, we obtain new sufficient conditions for the existence and uniqueness of global solutions of initial value problems.

scholarly journals Fractional Integral Equations Tell Us How to Impose Initial Values in Fractional Differential Equations

Mathematics ◽  
2020 ◽  
Vol 8 (7) ◽  
pp. 1093
Author(s):  
Daniel Cao Labora
Keyword(s):  
Differential Equations ◽  
Integral Equations ◽  
Fractional Differential Equations ◽  
Existence And Uniqueness ◽  
Fractional Integral ◽  
Initial Value ◽  
Major Question ◽  
Initial Values ◽  
Fractional Differential

One major question in Fractional Calculus is to better understand the role of the initial values in fractional differential equations. In this sense, there is no consensus about what is the reasonable fractional abstraction of the idea of “initial value problem”. This work provides an answer to this question. The techniques that are used involve known results concerning Volterra integral equations, and the spaces of summable fractional differentiability introduced by Samko et al. In a few words, we study the natural consequences in fractional differential equations of the already existing results involving existence and uniqueness for their integral analogues, in terms of the Riemann–Liouville fractional integral. In particular, we show that a fractional differential equation of a certain order with Riemann–Liouville derivatives demands, in principle, less initial values than the ceiling of the order to have a uniquely determined solution, in contrast to a widely extended opinion. We compute explicitly the amount of necessary initial values and the orders of differentiability where these conditions need to be imposed.

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