scholarly journals Mixed Fractional Heat Equation Driven by Fractional Brownian Sheet and Lévy Process

2017 ◽  
Vol 2017 ◽  
pp. 1-9 ◽  
Author(s):  
Dengfeng Xia ◽  
Litan Yan ◽  
Weiyin Fei
Keyword(s):  
Fixed Point ◽  
Heat Equation ◽  
White Noise ◽  
Existence And Uniqueness ◽  
Fractional Laplacian ◽  
Brownian Sheet ◽  
Fractional Noise ◽  
Fractional Heat Equation ◽  
Fixed Point Principle ◽  
Uniqueness Of The Solution

We consider the stochastic heat equation of the form∂u/∂t=(Δ+Δα)u+(∂f/∂x)(t,x,u)+σ(t,x,u)L˙+W˙H,whereW˙His the fractional noise,L˙is a (pure jump) Lévy space-time white noise,Δis Laplacian, andΔα=-(-Δ)α/2is the fractional Laplacian generator onR, andf,σ:[0,T]×R×R→Rare measurable functions. We introduce the existence and uniqueness of the solution by the fixed point principle under some suitable assumptions.

scholarly journals On a Fractional SPDE Driven by Fractional Noise and a Pure Jump Lévy Noise inℝd

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Xichao Sun ◽  
Zhi Wang ◽  
Jing Cui
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Fixed Point ◽  
Existence And Uniqueness ◽  
Stochastic Partial Differential Equation ◽  
Arbitrary Dimension ◽  
Derivative Operator ◽  
Fractional Noise ◽  
Fixed Point Principle ◽  
Whole Space

We study a stochastic partial differential equation in the whole spacex∈ℝd, with arbitrary dimensiond≥1, driven by fractional noise and a pure jump Lévy space-time white noise. Our equation involves a fractional derivative operator. Under some suitable assumptions, we establish the existence and uniqueness of the global mild solution via fixed point principle.

scholarly journals Solutions for impulsive fractional pantograph differential equation via generalized anti-periodic boundary condition

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Idris Ahmed ◽  
Poom Kumam ◽  
Jamilu Abubakar ◽  
Piyachat Borisut ◽  
Kanokwan Sitthithakerngkiet
Keyword(s):  
Differential Equation ◽  
Boundary Condition ◽  
Fixed Point ◽  
Fractional Differential Equation ◽  
Existence And Uniqueness ◽  
Periodic Boundary Condition ◽  
Periodic Boundary ◽  
Fixed Point Theorems ◽  
Uniqueness Of The Solution ◽  
Fractional Differential

Abstract This study investigates the solutions of an impulsive fractional differential equation incorporated with a pantograph. This work extends and improves some results of the impulsive fractional differential equation. A differential equation of an impulsive fractional pantograph with a more general anti-periodic boundary condition is proposed. By employing the well-known fixed point theorems of Banach and Krasnoselskii, the existence and uniqueness of the solution of the proposed problem are established. Furthermore, two examples are presented to support our theoretical analysis.

scholarly journals Existence and Uniqueness of the Solution for an Integral Equation with Supremum, via w-Distances

Symmetry ◽  
2020 ◽  
Vol 12 (9) ◽  
pp. 1554 ◽  
Author(s):  
Veronica Ilea ◽  
Diana Otrocol
Keyword(s):  
Integral Equation ◽  
Fixed Point ◽  
Existence And Uniqueness ◽  
Stability Result ◽  
Fixed Point Problem ◽  
Comparison Theorems ◽  
Point Problem ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results ◽  
Uniqueness Of The Solution

Following the idea of T. Wongyat and W. Sintunavarat, we obtain some existence and uniqueness results for the solution of an integral equation with supremum. The paper ends with the study of Gronwall-type theorems, comparison theorems and a result regarding a Ulam–Hyers stability result for the corresponding fixed point problem.

scholarly journals Analysis of logistic equation pertaining to a new fractional derivative with non-singular kernel

2017 ◽  
Vol 9 (2) ◽  
pp. 168781401769006 ◽  
Author(s):  
Devendra Kumar ◽  
Jagdev Singh ◽  
Maysaa Al Qurashi ◽  
Dumitru Baleanu
Keyword(s):  
Numerical Simulation ◽  
Fixed Point ◽  
Fractional Derivative ◽  
Fractional Order ◽  
Existence And Uniqueness ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Logistic Equation ◽  
Iterative Technique ◽  
Uniqueness Of The Solution

In this work, we aim to analyze the logistic equation with a new derivative of fractional order termed in Caputo–Fabrizio sense. The logistic equation describes the population growth of species. The existence of the solution is shown with the help of the fixed-point theory. A deep analysis of the existence and uniqueness of the solution is discussed. The numerical simulation is conducted with the help of the iterative technique. Some numerical simulations are also given graphically to observe the effects of the fractional order derivative on the growth of population.

scholarly journals Solutions to BSDEs Driven by Multidimensional Fractional Brownian Motions

2015 ◽  
Vol 2015 ◽  
pp. 1-12
Author(s):  
Jie Miao ◽  
Xu Yang
Keyword(s):  
Fixed Point ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Conditional Expectation ◽  
Existence And Uniqueness ◽  
Backward Stochastic Differential Equations ◽  
Brownian Motions ◽  
Fractional Brownian Motions ◽  
Fixed Point Principle ◽  
Fractional Ito Formula

We study more general backward stochastic differential equations driven by multidimensional fractional Brownian motions. Introducing the concept of the multidimensional fractional (or quasi-) conditional expectation, we study some of its properties. Using the quasi-conditional expectation and multidimensional fractional Itô formula, we obtain the existence and uniqueness of the solutions to BSDEs driven by multidimensional fractional Brownian motions, where a fixed point principle is employed. Finally, solutions to linear fractional backward stochastic differential equations are investigated.

scholarly journals Optimal existence and uniqueness theory for the fractional heat equation

2017 ◽  
Vol 153 ◽  
pp. 142-168 ◽  
Author(s):  
Matteo Bonforte ◽  
Yannick Sire ◽  
Juan Luis Vázquez
Keyword(s):  
Heat Equation ◽  
Existence And Uniqueness ◽  
Fractional Heat Equation ◽  
Uniqueness Theory

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 On Suzuki and Wardowski-Type Contraction Multivalued Mappings in Partial Symmetric Spaces

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Mohammad Asim ◽  
Hassen Aydi ◽  
Mohammad Imdad
Keyword(s):  
Fixed Point ◽  
Existence And Uniqueness ◽  
Symmetric Spaces ◽  
Multivalued Mappings ◽  
Uniqueness Of The Solution ◽  
Type Contraction

The purpose of this paper is to provide some fixed-point results for Suzuki and Wardowski-type contraction multivalued mappings in partial symmetric spaces. We give some examples to support and substantiate the developed notions and obtained results. Also, we use one of our main results to establish the existence and uniqueness of the solution for a system of integral inclusions.

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