scholarly journals Existence results for some fractional differential equations related to A ∈ B(L2[a, b])

Filomat ◽  
2020 ◽  
Vol 34 (4) ◽  
pp. 1067-1074
Author(s):  
Eshkaftaki Bayati
Keyword(s):  
Integral Operator ◽  
Differential Equations ◽  
Integral Equations ◽  
Fractional Differential Equations ◽  
Existence And Uniqueness ◽  
Existence Of Solutions ◽  
Existence Results ◽  
Uniqueness Of Solutions ◽  
Volterra Integral Operator ◽  
Fractional Differential

In this paper we define the (generalized) linear Volterra integral operator on L2[a, b]. Then the problem of existence and uniqueness of solutions of the second kind Volterra integral equations, corresponding to this operator, will be answered. Finally, some applications of this work to the existence of solutions of some fractional differential equations, are given.

scholarly journals System of fractional differential equations with Erdélyi-Kober fractional integral conditions

2015 ◽  
Vol 13 (1) ◽  
Author(s):  
Natthaphong Thongsalee ◽  
Sorasak Laoprasittichok ◽  
Sotiris K. Ntouyas ◽  
Jessada Tariboon
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Existence And Uniqueness ◽  
Fractional Integral ◽  
Existence Of Solutions ◽  
Integral Conditions ◽  
Liouville Type ◽  
Uniqueness Of Solutions ◽  
Fractional Differential ◽  
Banach’S Contraction Principle

AbstractIn this paper we study existence and uniqueness of solutions for a system consisting from fractional differential equations of Riemann-Liouville type subject to nonlocal Erdélyi-Kober fractional integral conditions. The existence and uniqueness of solutions is established by Banach’s contraction principle, while the existence of solutions is derived by using Leray-Schauder’s alternative. Examples illustrating our results are also presented.

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.

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 Complex Transforms for Systems of Fractional Differential Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-15 ◽  
Author(s):  
Rabha W. Ibrahim
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Fractional Differential Equation ◽  
Fractional Differential Equations ◽  
Existence And Uniqueness ◽  
Uniqueness Of Solutions ◽  
Fractional Differential

We provide a complex transform that maps the complex fractional differential equation into a system of fractional differential equations. The homogeneous and nonhomogeneous cases for equivalence equations are discussed and also nonequivalence equations are studied. Moreover, the existence and uniqueness of solutions are established and applications are illustrated.

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