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.

scholarly journals Some Existence Results for Systems of Impulsive Stochastic Differential Equations

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Sliman Mekki ◽  
Tayeb Blouhi ◽  
Juan J. Nieto ◽  
Abdelghani Ouahab
Keyword(s):  
Fixed Point ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Banach Spaces ◽  
Existence And Uniqueness ◽  
Fixed Point Theorems ◽  
Sufficient Conditions ◽  
Existence Results ◽  
Impulsive Systems ◽  
Uniqueness Of Solutions

Abstract In this paper we study a class of impulsive systems of stochastic differential equations with infinite Brownian motions. Sufficient conditions for the existence and uniqueness of solutions are established by mean of some fixed point theorems in vector Banach spaces. An example is provided to illustrate the theory.

scholarly journals QUALITATIVE STUDY OF NONLINEAR COUPLED PANTOGRAPH DIFFERENTIAL EQUATIONS OF FRACTIONAL ORDER

Fractals ◽  
2020 ◽  
Vol 28 (08) ◽  
pp. 2040045 ◽  
Author(s):  
ISRAR AHAMAD ◽  
KAMAL SHAH ◽  
THABET ABDELJAWAD ◽  
FAHD JARAD
Keyword(s):  
Fixed Point ◽  
Qualitative Study ◽  
Differential Equations ◽  
Nonlinear Analysis ◽  
Fractional Order ◽  
Existence And Uniqueness ◽  
Fixed Point Theorems ◽  
Coupled System ◽  
Main Work ◽  
Nonlinear Coupled System

In this paper, we investigate a nonlinear coupled system of fractional pantograph differential equations (FPDEs). The respective results address some adequate results for existence and uniqueness of solution to the problem under consideration. In light of fixed point theorems like Banach and Krasnoselskii’s, we establish the required results. Considering the tools of nonlinear analysis, we develop some results regarding Ulam–Hyers (UH) stability. We give three pertinent examples to demonstrate our main work.

scholarly journals Fixed-Point Theorems in Complete Gauge Spaces and Applications to Second-Order Nonlinear Initial-Value Problems

2013 ◽  
Vol 2013 ◽  
pp. 1-8
Author(s):  
Meryam Cherichi ◽  
Bessem Samet ◽  
Calogero Vetro
Keyword(s):  
Fixed Point ◽  
Initial Value Problem ◽  
Existence And Uniqueness ◽  
Initial Value Problems ◽  
Fixed Point Theorems ◽  
Second Order ◽  
Uniqueness Of Solution ◽  
Contractive Condition ◽  
Initial Value ◽  
Gauge Space

We establish fixed-point results for mappings and cyclic mappings satisfying a generalized contractive condition in a complete gauge space. Our theorems generalize and extend some fixed-point results in the literature. We apply our obtained results to the study of existence and uniqueness of solution to a second-order nonlinear initial-value problem.

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 Iterative Bernstein splines technique applied to fractional order differential equations

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Zoltan Satmari
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Differential Equations ◽  
Fractional Order ◽  
Initial Value Problems ◽  
Spline Approximation ◽  
Initial Value ◽  
Numerical Examples ◽  
Fractional Order Differential Equations ◽  
Banach’S Fixed Point Theorem

<p style='text-indent:20px;'>In this work we will discuss about an approximation method for initial value problems associated to fractional order differential equations. For this method we will use Bernstein spline approximation in combination with the Banach's Fixed Point Theorem. In order to illustrate our results, some numerical examples will be presented at the end of this article.</p>

scholarly journals Existence and Uniqueness for a System of Caputo-Hadamard Fractional Differential Equations with Multipoint Boundary Conditions

2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
S. Nageswara Rao ◽  
Ahmed Hussein Msmali ◽  
Manoj Singh ◽  
Abdullah Ali H. Ahmadini
Keyword(s):  
Fixed Point ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Fractional Differential Equations ◽  
Existence And Uniqueness ◽  
Fixed Point Theorems ◽  
Contraction Mapping Principle ◽  
Uniqueness Of Solutions ◽  
Hadamard Fractional Differential Equations ◽  
Fractional Differential

In this paper, we study existence and uniqueness of solutions for a system of Caputo-Hadamard fractional differential equations supplemented with multi-point boundary conditions. Our results are based on some classical fixed point theorems such as Banach contraction mapping principle, Leray-Schauder fixed point theorems. At last, we have presented two examples for the illustration of main results.

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.

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