scholarly journals Existence Results for Langevin Equation Involving Atangana-Baleanu Fractional Operators

Mathematics ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 408 ◽  
Author(s):  
Dumitru Baleanu ◽  
Rahmat Darzi ◽  
Bahram Agheli
Keyword(s):  
Langevin Equation ◽  
Existence Results ◽  
Banach Contraction Principle ◽  
Uniqueness Of Solution ◽  
Fractional Operators ◽  
Nonlinear Alternative ◽  
Banach Contraction ◽  
New Form ◽  
Nonlinear Langevin Equation ◽  
Derivatives Of

A new form of nonlinear Langevin equation (NLE), featuring two derivatives of non-integer orders, is studied in this research. An existence conclusion due to the nonlinear alternative of Leray-Schauder type (LSN) for the solution is offered first and, following that, the uniqueness of solution using Banach contraction principle (BCP) is demonstrated. Eventually, the derivatives of non-integer orders are elaborated in Atangana-Baleanu sense.

Fuzzy double controlled metric spaces and related results

2021 ◽  
Vol 40 (5) ◽  
pp. 9977-9985
Author(s):  
Naeem Saleem ◽  
Hüseyin Işık ◽  
Salman Furqan ◽  
Choonkil Park
Keyword(s):  
Integral Equation ◽  
Metric Space ◽  
Existence And Uniqueness ◽  
Metric Spaces ◽  
Banach Contraction Principle ◽  
Uniqueness Of Solution ◽  
Contraction Principle ◽  
Triangular Inequality ◽  
Banach Contraction ◽  
The Banach Contraction Principle

In this paper, we introduce the concept of fuzzy double controlled metric space that can be regarded as the generalization of fuzzy b-metric space, extended fuzzy b-metric space and controlled fuzzy metric space. We use two non-comparable functions α and β in the triangular inequality as: M q ( x , z , t α ( x , y ) + s β ( y , z ) ) ≥ M q ( x , y , t ) ∗ M q ( y , z , s ) . We prove Banach contraction principle in fuzzy double controlled metric space and generalize the Banach contraction principle in aforementioned spaces. We give some examples to support our main results. An application to existence and uniqueness of solution for an integral equation is also presented in this work.

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 Existence and Uniqueness of Solutions for BVP of Nonlinear Fractional Differential Equation

2017 ◽  
Vol 2017 ◽  
pp. 1-7
Author(s):  
Cheng-Min Su ◽  
Jian-Ping Sun ◽  
Ya-Hong Zhao
Keyword(s):  
Differential Equation ◽  
Boundary Value Problem ◽  
Fractional Differential Equation ◽  
Existence And Uniqueness ◽  
Banach Contraction Principle ◽  
Uniqueness Of Solutions ◽  
Nonlinear Fractional Differential Equation ◽  
Nonlinear Alternative ◽  
Banach Contraction ◽  
Fractional Differential

In this paper, we study the existence and uniqueness of solutions for the following boundary value problem of nonlinear fractional differential equation: D0+qCut=ft,ut,  t∈0,1, u0=u′′0=0,  D0+σ1Cu1=λI0+σ2u1, where 2<q<3, 0<σ1≤1, σ2>0, and λ≠Γ2+σ2/Γ2-σ1. The main tools used are nonlinear alternative of Leray-Schauder type and Banach contraction principle.

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 Multi-Strip and Multi-Point Boundary Conditions for Fractional Langevin Equation

2020 ◽  
Vol 4 (2) ◽  
pp. 18 ◽  
Author(s):  
Ahmed Salem ◽  
Balqees Alghamdi
Keyword(s):  
Fixed Point ◽  
Langevin Equation ◽  
Fixed Point Theorems ◽  
Existence Results ◽  
Banach Fixed Point Theorem ◽  
Integral Conditions ◽  
Fractional Langevin Equation ◽  
Uniqueness Of The Solution ◽  
Nonlocal Integral Conditions ◽  
Nonlinear Langevin Equation

In the present paper, we discuss a new boundary value problem for the nonlinear Langevin equation involving two distinct fractional derivative orders with multi-point and multi-nonlocal integral conditions. The fixed point theorems for Schauder and Krasnoselskii–Zabreiko are applied to study the existence results. The uniqueness of the solution is given by implementing the Banach fixed point theorem. Some examples showing our basic results are provided.

scholarly journals Nonlinear Langevin Equation of Hadamard-Caputo Type Fractional Derivatives with Nonlocal Fractional Integral Conditions

2014 ◽  
Vol 2014 ◽  
pp. 1-15 ◽  
Author(s):  
Jessada Tariboon ◽  
Sotiris K. Ntouyas ◽  
Chatthai Thaiprayoon
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Langevin Equation ◽  
Fractional Derivatives ◽  
Fractional Integral ◽  
Integral Conditions ◽  
Krasnoselskii’S Fixed Point Theorem ◽  
Nonlinear Alternative ◽  
Nonlinear Langevin Equation

We study existence and uniqueness of solutions for a problem consisting of nonlinear Langevin equation of Hadamard-Caputo type fractional derivatives with nonlocal fractional integral conditions. A variety of fixed point theorems are used, such as Banach’s fixed point theorem, Krasnoselskii’s fixed point theorem, Leray-Schauder’s nonlinear alternative, and Leray-Schauder’s degree theory. Enlightening examples illustrating the obtained results are also presented.

scholarly journals The generalized U–H and U–H stability and existence analysis of a coupled hybrid system of integro-differential IVPs involving φ-Caputo fractional operators

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Abdellatif Boutiara ◽  
Sina Etemad ◽  
Azhar Hussain ◽  
Shahram Rezapour
Keyword(s):  
Hybrid System ◽  
Coupled System ◽  
Uniqueness Result ◽  
Existence Results ◽  
Fractional Operators ◽  
Uniqueness Of Solutions ◽  
Banach Contraction ◽  
Hybrid Fixed Point Theorem ◽  
The Given ◽  
The Banach Contraction Principle

AbstractWe investigate the existence and uniqueness of solutions to a coupled system of the hybrid fractional integro-differential equations involving φ-Caputo fractional operators. To achieve this goal, we make use of a hybrid fixed point theorem for a sum of three operators due to Dhage and also the uniqueness result is obtained by making use of the Banach contraction principle. Moreover, we explore the Ulam–Hyers stability and its generalized version for the given coupled hybrid system. An example is presented to guarantee the validity of our existence results.

scholarly journals Extended Rectangular Mrc-Metric Spaces and Fixed Point Results

Mathematics ◽  
2019 ◽  
Vol 7 (12) ◽  
pp. 1136 ◽  
Author(s):  
Asim ◽  
Nisar ◽  
Morsy ◽  
Imdad
Keyword(s):  
Integral Equation ◽  
Fixed Point ◽  
Fredholm Integral Equation ◽  
Metric Spaces ◽  
Banach Contraction Principle ◽  
Uniqueness Of Solution ◽  
Contraction Principle ◽  
Banach Contraction

In this paper, we enlarge the classes of rectangular Mb-metric spaces and extendedrectangular b-metric spaces by considering the class of extended rectangular Mrc-metric spacesand utilize the same to prove an analogue of Banach contraction principle in such spaces. We adoptan example to highlight the utility of our main result. Finally, we apply our result to examine theexistence and uniqueness of solution for a system of Fredholm integral equation.

scholarly journals Fixed point results in $M_{\nu }$-metric spaces with an application

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Mohammad Asim ◽  
Izhar Uddin ◽  
Mohammad Imdad
Keyword(s):  
Integral Equation ◽  
Fixed Point ◽  
Metric Space ◽  
Fredholm Integral Equation ◽  
Existence And Uniqueness ◽  
Metric Spaces ◽  
Banach Contraction Principle ◽  
Uniqueness Of Solution ◽  
Contraction Principle ◽  
Banach Contraction

Abstract In this paper, we introduce the concept of $M_{\nu}$ M ν -metric as a generalization of M-metric and ν-generalized metric and also prove an analogue of Banach contraction principle in an $M_{\nu}$ M ν -metric space. Also, we adopt an example to highlight the utility of our main result which extends and improves the corresponding relevant results of the existing literature. Finally, we use our main result to examine the existence and uniqueness of solution for a Fredholm integral equation.

scholarly journals Existence Results for Impulsive Fractional q-Difference Equation with Antiperiodic Boundary Conditions

2018 ◽  
Vol 2018 ◽  
pp. 1-9 ◽  
Author(s):  
Mingyue Zuo ◽  
Xinan Hao
Keyword(s):  
Difference Equation ◽  
Contraction Mapping ◽  
Contraction Mapping Principle ◽  
Existence Results ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results ◽  
Nonlinear Alternative ◽  
Banach Contraction ◽  
Banach Contraction Mapping Principle ◽  
Banach Contraction Mapping

In this paper, we investigate the impulsive fractional q-difference equation with antiperiodic conditions. The existence and uniqueness results of solutions are established via the theorem of nonlinear alternative of Leray-Schauder type and the Banach contraction mapping principle. Two examples are given to illustrate our results.

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