scholarly journals On the Existence and Uniqueness of Solution to Fractional-Order Langevin Equation

2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
Ahmed Salem ◽  
Noorah Mshary
Keyword(s):  
Fractional Order ◽  
Langevin Equation ◽  
Existence And Uniqueness ◽  
Sufficient Conditions ◽  
Contraction Mapping Principle ◽  
Uniqueness Of Solution ◽  
Main Concern ◽  
Banach Contraction ◽  
Fractional Orders ◽  
Banach Contraction Mapping Principle

In this work, we give sufficient conditions to investigate the existence and uniqueness of solution to fractional-order Langevin equation involving two distinct fractional orders with unprecedented conditions (three-point boundary conditions including two nonlocal integrals). The problem is introduced to keep track of the progress made on exploring the existence and uniqueness of solution to the fractional-order Langevin equation. As a result of employing the so-called Krasnoselskii and Leray-Schauder alternative fixed point theorems and Banach contraction mapping principle, some novel results are presented in regarding to our main concern. These results are illustrated through providing three examples for completeness.

scholarly journals Multi-Point and Anti-Periodic Conditions for Generalized Langevin Equation with Two Fractional Orders

2019 ◽  
Vol 3 (4) ◽  
pp. 51 ◽  
Author(s):  
Ahmed Salem ◽  
Balqees Alghamdi
Keyword(s):  
Langevin Equation ◽  
Contraction Mapping Principle ◽  
Generalized Langevin Equation ◽  
Uniqueness Of Solutions ◽  
Caputo Fractional Derivatives ◽  
New Class ◽  
Banach Contraction ◽  
Derivatives Of ◽  
Fractional Orders ◽  
Banach Contraction Mapping Principle

With anti-periodic and a new class of multi-point boundary conditions, we investigate, in this paper, the existence and uniqueness of solutions for the Langevin equation that has Caputo fractional derivatives of two different orders. Existence of solutions is obtained by applying Krasnoselskii–Zabreiko’s and the Leray–Schauder fixed point theorems. The Banach contraction mapping principle is used to investigate the uniqueness. Illustrative examples are provided to apply of the fundamental investigations.

scholarly journals Positive Solutions for Class of State Dependent Boundary Value Problems with Fractional Order Differential Operators

2015 ◽  
Vol 2015 ◽  
pp. 1-11 ◽  
Author(s):  
Dongyuan Liu ◽  
Zigen Ouyang ◽  
Huilan Wang
Keyword(s):  
Positive Solutions ◽  
Fractional Order ◽  
Differential Operators ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Contraction Mapping Principle ◽  
State Dependent ◽  
Banach Contraction ◽  
Liouville Fractional Derivative ◽  
Banach Contraction Mapping Principle

We consider the following state dependent boundary-value problemD0+αy(t)-pD0+βg(t,y(σ(t)))+f(t,y(τ(t)))=0,0<t<1;y(0)=0,ηy(σ(1))=y(1),whereDαis the standard Riemann-Liouville fractional derivative of order1<α<2,0<η<1,p≤0,0<β<1,β+1-α≥0the functiongis defined asg(t,u):[0,1]×[0,∞)→[0,∞), andg(0,0)=0the functionfis defined asf(t,u):[0,1]×[0,∞)→[0,∞)σ(t),τ(t)are continuous ontand0≤σ(t),τ(t)≤t. Using Banach contraction mapping principle and Leray-Schauder continuation principle, we obtain some sufficient conditions for the existence and uniqueness of the positive solutions for the above fractional order differential equations, which extend some references.

scholarly journals Existence and uniqueness of solutions for a mixed p-Laplace boundary value problem involving fractional derivatives

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Shuqi Wang ◽  
Zhanbing Bai
Keyword(s):  
Boundary Value Problem ◽  
Existence And Uniqueness ◽  
Fractional Derivatives ◽  
Boundary Value ◽  
Contraction Mapping Principle ◽  
Uniqueness Of Solutions ◽  
Novel Approach ◽  
Banach Contraction ◽  
Left And Right ◽  
Banach Contraction Mapping Principle

AbstractIn this article, the existence and uniqueness of solutions for a multi-point fractional boundary value problem involving two different left and right fractional derivatives with p-Laplace operator is studied. A novel approach is used to acquire the desired results, and the core of the method is Banach contraction mapping principle. Finally, an example is given to verify the results.

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 Existence and Uniqueness of Solution for a Class of Nonlinear Fractional Order Differential Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
Azizollah Babakhani ◽  
Dumitru Baleanu
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Differential Equations ◽  
Fractional Order ◽  
Existence And Uniqueness ◽  
Initial Conditions ◽  
Uniqueness Of Solution ◽  
Fractional Order Differential Equations ◽  
Banach Contraction ◽  
Uniqueness Of The Solution

We discuss the existence and uniqueness of solution to nonlinear fractional order ordinary differential equations(Dα-ρtDβ)x(t)=f(t,x(t),Dγx(t)),t∈(0,1)with boundary conditionsx(0)=x0,  x(1)=x1or satisfying the initial conditionsx(0)=0,  x′(0)=1, whereDαdenotes Caputo fractional derivative,ρis constant,1<α<2,and0<β+γ≤α. Schauder's fixed-point theorem was used to establish the existence of the solution. Banach contraction principle was used to show the uniqueness of the solution under certain conditions onf.

scholarly journals Revisiting of some outstanding metric fixed point theorems via E-contraction

2018 ◽  
Vol 26 (3) ◽  
pp. 73-98
Author(s):  
Andreea Fulga ◽  
Erdal Karapınar
Keyword(s):  
Fixed Point ◽  
Existence And Uniqueness ◽  
Contraction Mapping ◽  
Fixed Point Theorems ◽  
Contractive Mapping ◽  
Contraction Mapping Principle ◽  
Contraction Mappings ◽  
Banach Contraction ◽  
Banach Contraction Mapping Principle ◽  
Banach Contraction Mapping

AbstractIn this paper, we introduce the notion of α-ψ-contractive mapping of type E, to remedy of the weakness of the existing contraction mappings. We investigate the existence and uniqueness of a fixed point of such mappings. We also list some examples to illustrate our results that unify and generalize the several well-known results including the famous Banach contraction mapping principle.

Solvability and stability of nonlinear hybrid ∆-difference equations of fractional-order

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Jehad Alzabut ◽  
A. George Maria Selvam ◽  
Dhakshinamoorthy Vignesh ◽  
Yousef Gholami
Keyword(s):  
Fractional Order ◽  
Difference Equations ◽  
Contraction Mapping Principle ◽  
Uniqueness Of Solutions ◽  
Banach Contraction ◽  
Hybrid Fixed Point Theorem ◽  
Practical Impact ◽  
Banach Contraction Mapping Principle ◽  
Banach Contraction Mapping ◽  
Nonlinear Hybrid

Abstract In this paper, we study a type of nonlinear hybrid Δ-difference equations of fractional-order. The main objective is to establish some stability criteria including the Ulam–Hyers stability, generalized Ulam–Hyers stability together with the Mittag-Leffler–Ulam–Hyers stability for the addressed problem. Prior to the stabilization processes, solvability criteria for the existence and uniqueness of solutions are considered. For this purpose, a hybrid fixed point theorem for triple operators and the Banach contraction mapping principle are applied, respectively. For the sake of illustrating the practical impact of the proposed theoretical criteria, we finish the paper with particular examples.

scholarly journals Existence results to a ψ- Hilfer neutral fractional evolution equation with infinite delay

2021 ◽  
Vol 8 (1) ◽  
pp. 101-124
Author(s):  
Fatemeh Norouzi ◽  
Gaston M. N’guérékata
Keyword(s):  
Mild Solution ◽  
Existence And Uniqueness ◽  
Evolution Equations ◽  
Infinite Delay ◽  
Contraction Mapping Principle ◽  
Alternative Theorem ◽  
Banach Contraction ◽  
Equivalent Equation ◽  
Banach Contraction Mapping Principle ◽  
Banach Contraction Mapping

Abstract In this paper, we prove the existence and uniqueness of a mild solution to the system of ψ- Hilfer neutral fractional evolution equations with infinite delay H 𝔻0 αβ;ψ [x(t) − h(t, xt )] = A x(t) + f (t, x(t), xt ), t ∈ [0, b], b > 0 and x(t) = ϕ(t), t ∈ (−∞, 0]. We first obtain the Volterra integral equivalent equation and propose the mild solution of the system. Then, we prove the existence and uniqueness of solution by using the Banach contraction mapping principle and the Leray-Schauder alternative theorem.

scholarly journals Almost Periodic Solutions for Wilson-Cowan Type Model with Time-Varying Delays

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Shasha Xie ◽  
Zhenkun Huang
Keyword(s):  
Contraction Mapping ◽  
Sufficient Conditions ◽  
Neuronal Population ◽  
Contraction Mapping Principle ◽  
Type Model ◽  
Time Varying ◽  
Almost Periodic ◽  
Banach Contraction ◽  
Banach Contraction Mapping Principle ◽  
Banach Contraction Mapping

Wilson-Cowan model of neuronal population with time-varying delays is considered in this paper. Some sufficient conditions for the existence and delay-based exponential stability of a unique almost periodic solution are established. The approaches are based on constructing Lyapunov functionals and the well-known Banach contraction mapping principle. The results are new, easily checkable, and complement existing periodic ones.

scholarly journals Existence and uniqueness of solutions for system of Hilfer–Hadamard sequential fractional differential equations with two point boundary conditions

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Warissara Saengthong ◽  
Ekkarath Thailert ◽  
Sotiris K. Ntouyas
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Existence And Uniqueness ◽  
Contraction Mapping Principle ◽  
Uniqueness Of Solutions ◽  
Banach Contraction ◽  
Fractional Differential ◽  
Banach Contraction Mapping Principle ◽  
Banach Contraction Mapping ◽  
Sequential Fractional Differential Equations

AbstractIn this paper, we study existence and uniqueness of solutions for a system of Hilfer–Hadamard sequential fractional differential equations via standard fixed point theorems. The existence is proved by using the Leray–Schauder alternative, while the existence and uniqueness by the Banach contraction mapping principle. Illustrative examples are also discussed.

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