Existence and uniqueness of solutions of initial value problems for nonlinear langevin equation involving two fractional orders

2014 ◽  
Vol 19 (6) ◽  
pp. 1661-1668 ◽  
Author(s):  
Tao Yu ◽  
Ke Deng ◽  
Maokang Luo
Keyword(s):  
Langevin Equation ◽  
Existence And Uniqueness ◽  
Initial Value Problems ◽  
Initial Value ◽  
Uniqueness Of Solutions ◽  
Nonlinear Langevin Equation ◽  
Fractional Orders

scholarly journals On fractional Langevin equation involving two fractional orders in different intervals

2019 ◽  
Vol 24 (6) ◽  
Author(s):  
Hamid Baghani ◽  
J. Nieto
Keyword(s):  
Langevin Equation ◽  
Existence And Uniqueness ◽  
Initial Conditions ◽  
Sufficient Conditions ◽  
Uniqueness Of Solutions ◽  
Numerical Examples ◽  
Fractional Equations ◽  
Fractional Langevin Equation ◽  
Nonlinear Langevin Equation ◽  
Fractional Orders

In this paper, we study a nonlinear Langevin equation involving two fractional orders  α ∈ (0; 1] and β ∈ (1; 2] with initial conditions. By means of an interesting fixed point theorem, we establish sufficient conditions for the existence and uniqueness of solutions for the fractional equations. Some illustrative numerical examples are also discussed. 

scholarly journals Application of some new contractions for existence and uniqueness of differential equations involving Caputo–Fabrizio derivative

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Hojjat Afshari ◽  
Hossein Hosseinpour ◽  
H. R. Marasi
Keyword(s):  
Differential Equations ◽  
Existence And Uniqueness ◽  
Initial Value Problems ◽  
Metric Spaces ◽  
Initial Value ◽  
Uniqueness Of Solutions ◽  
Contraction Mappings

AbstractIn this paper we study fractional initial value problems with Caputo–Fabrizio derivative which involves nonsingular kernel. First we apply α-ℓ-contraction and α-type F-contraction mappings to study the existence and uniqueness of solutions for such problems. Finally, we use some contraction mappings in complete $\mathfrak{F}$ F -metric spaces for this purpose.

scholarly journals Applications of new contraction mappings on existence and uniqueness results for implicit ϕ-Hilfer fractional pantograph differential equations

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Hojjat Afshari ◽  
H. R. Marasi ◽  
Jehad Alzabut
Keyword(s):  
Differential Equations ◽  
Existence And Uniqueness ◽  
Initial Value Problems ◽  
Initial Value ◽  
Uniqueness Of Solutions ◽  
Contraction Mappings ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results ◽  
Positive Functions

AbstractIn this paper, we consider initial value problems for two different classes of implicit ϕ-Hilfer fractional pantograph differential equations. We use different approach that is based on $\alpha -\psi $ α − ψ -contraction mappings to demonstrate the existence and uniqueness of solutions for the proposed problems. The mappings are defined in appropriate cones of positive functions. The presented examples demonstrate the efficiency of the used method and the consistency of the proposed results.

On fractional order derivatives and Darboux problem for implicit differential equations

2012 ◽  
Vol 15 (2) ◽  
Author(s):  
Saïd Abbas ◽  
Mouffak Benchohra ◽  
Aleksandr Vityuk
Keyword(s):  
Fixed Point ◽  
Differential Equations ◽  
Fractional Order ◽  
Existence And Uniqueness ◽  
Initial Value Problems ◽  
Fixed Point Theorems ◽  
Initial Value ◽  
Uniqueness Of Solutions ◽  
Darboux Problem ◽  
Hyperbolic Differential Equations

AbstractIn this paper we prove some relations between the Riemann-Liouville and the Caputo fractional order derivatives, and we investigate the existence and uniqueness of solutions for the initial value problems (IVP for short), for a class of functional hyperbolic differential equations by using some fixed point theorems.

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