scholarly journals Stochastic Volterra integro-differential equations driven by a fractional Brownian motion with delayed impulses

Filomat ◽  
2017 ◽  
Vol 31 (19) ◽  
pp. 5965-5978 ◽  
Author(s):  
Xia Zhou ◽  
Xinzhi Liu ◽  
Shouming Zhong
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Mild Solutions ◽  
Sufficient Conditions ◽  
Fractional Power ◽  
Semi Group ◽  
Volterra Integrodifferential Equation ◽  
The Fixed Point Theorem ◽  
Delayed Impulses ◽  
Fractional Power Of Operators

In this paper, the problem of existence of mild solutions for a stochastic Volterra integrodifferential equation with delayed impulses and driven by a fractional Brownian motion (Hurst parameter H ? (1/2,1)) is investigated. Here, we assume that the delayed impulses are linear and impulsive transients depend on not only their current but also historical states of the system. Utilizing the fixed point theorem combine with fractional power of operators and the semi-group theory, sufficient conditions that guarantee the existence and uniqueness of mild solutions for such equation are obtained. Finally, an example is presented to demonstrate the effectiveness of the proposed results.

scholarly journals Existence and exponential stability in the pth moment for impulsive neutral stochastic integro-differential equations driven by mixed fractional Brownian motion

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Xia Zhou ◽  
Dongpeng Zhou ◽  
Shouming Zhong
Keyword(s):  
Brownian Motion ◽  
Differential Equations ◽  
Exponential Stability ◽  
Fractional Brownian Motion ◽  
Mild Solutions ◽  
Semigroup Theory ◽  
Sufficient Conditions ◽  
Fractional Power ◽  
Banach Fixed Point Theorem ◽  
Mixed Fractional Brownian Motion

Abstract This paper consider the existence, uniqueness and exponential stability in the pth moment of mild solution for impulsive neutral stochastic integro-differential equations driven simultaneously by fractional Brownian motion and by standard Brownian motion. Based on semigroup theory, the sufficient conditions to ensure the existence and uniqueness of mild solutions are obtained in terms of fractional power of operators and Banach fixed point theorem. Moreover, the pth moment exponential stability conditions of the equation are obtained by means of an impulsive integral inequality. Finally, an example is presented to illustrate the effectiveness of the obtained results.

scholarly journals Neutral delay Hilfer fractional integrodifferential equations with fractional brownian motion

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Yousef Alnafisah ◽  
Hamdy M. Ahmed
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Existence And Uniqueness ◽  
Mild Solutions ◽  
Semigroup Theory ◽  
Sufficient Conditions ◽  
Integrodifferential Equations ◽  
Neutral Delay ◽  
Fractional Differential ◽  
The Fixed Point Theorem

<p style='text-indent:20px;'>In this paper, we study the existence and uniqueness of mild solutions for neutral delay Hilfer fractional integrodifferential equations with fractional Brownian motion. Sufficient conditions for controllability of neutral delay Hilfer fractional differential equations with fractional Brownian motion are established. The required results are obtained based on the fixed point theorem combined with the semigroup theory, fractional calculus and stochastic analysis. Finally, an example is given to illustrate the obtained results.</p>

Stochastic partial functional integrodifferential equations driven by a sub-fractional Brownian motion, existence and asymptotic behavior

2019 ◽  
Vol 27 (2) ◽  
pp. 107-122
Author(s):  
Fulbert Kuessi Allognissode ◽  
Mamadou Abdoul Diop ◽  
Khalil Ezzinbi ◽  
Carlos Ogouyandjou
Keyword(s):  
Asymptotic Behavior ◽  
Brownian Motion ◽  
Differential Equations ◽  
Fractional Brownian Motion ◽  
Existence And Uniqueness ◽  
Resolvent Operator ◽  
Mild Solutions ◽  
Sufficient Conditions ◽  
Integrodifferential Equations ◽  
Hurst Parameter

Abstract This paper deals with the existence and uniqueness of mild solutions to stochastic partial functional integro-differential equations driven by a sub-fractional Brownian motion {S_{Q}^{H}(t)} , with Hurst parameter {H\in(\frac{1}{2},1)} . By the theory of resolvent operator developed by R. Grimmer (1982) to establish the existence of mild solutions, we give sufficient conditions ensuring the existence, uniqueness and the asymptotic behavior of the mild solutions. An example is provided to illustrate the theory.

Global attracting sets of neutral stochastic functional integro-differential equations driven by a fractional Brownian motion

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
A. Bakka ◽  
S. Hajji ◽  
D. Kiouach
Keyword(s):  
Hilbert Space ◽  
Fixed Point ◽  
Brownian Motion ◽  
Differential Equations ◽  
Fractional Brownian Motion ◽  
Sufficient Conditions ◽  
Integrodifferential Equations ◽  
Hurst Parameter ◽  
Fixed Point Principle ◽  
Stochastic Functional

Abstract By means of the Banach fixed point principle, we establish some sufficient conditions ensuring the existence of the global attracting sets of neutral stochastic functional integrodifferential equations with finite delay driven by a fractional Brownian motion (fBm) with Hurst parameter H ∈ ( 1 2 , 1 ) {H\in(\frac{1}{2},1)} in a Hilbert space.

scholarly journals Existence Results for Second-Order Impulsive Neutral Functional Differential Equations with Nonlocal Conditions

2009 ◽  
Vol 2009 ◽  
pp. 1-11 ◽  
Author(s):  
Meili Li ◽  
Chunhai Kou
Keyword(s):  
Differential Equations ◽  
Mild Solutions ◽  
Functional Differential Equations ◽  
Nonlocal Conditions ◽  
Fractional Power ◽  
Existence Results ◽  
Second Order ◽  
Neutral Functional Differential Equations ◽  
Functional Differential ◽  
Fractional Power Of Operators

The existence of mild solutions for second-order impulsive semilinear neutral functional differential equations with nonlocal conditions in Banach spaces is investigated. The results are obtained by using fractional power of operators and Sadovskii's fixed point theorem.

scholarly journals Measure-Dependent Stochastic Nonlinear Beam Equations Driven by Fractional Brownian Motion

2013 ◽  
Vol 2013 ◽  
pp. 1-16
Author(s):  
Mark A. McKibben
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Fixed Point Theory ◽  
Mild Solutions ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Point Theory ◽  
Transverse Motion ◽  
Probability Measures ◽  
Uniqueness Results

We study a class of nonlinear stochastic partial differential equations arising in the mathematical modeling of the transverse motion of an extensible beam in the plane. Nonlinear forcing terms of functional-type and those dependent upon a family of probability measures are incorporated into the initial-boundary value problem (IBVP), and noise is incorporated into the mathematical description of the phenomenon via a fractional Brownian motion process. The IBVP is subsequently reformulated as an abstract second-order stochastic evolution equation driven by a fractional Brownian motion (fBm) dependent upon a family of probability measures in a real separable Hilbert space and is studied using the tools of cosine function theory, stochastic analysis, and fixed-point theory. Global existence and uniqueness results for mild solutions, continuous dependence estimates, and various approximation results are established and applied in the context of the model.

scholarly journals Existence and Stability of Square-Mean S-Asymptotically Periodic Solutions to a Fractional Stochastic Diffusion Equation with Fractional Brownian Motion

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-15
Author(s):  
Jia Mu ◽  
Jiecuo Nan ◽  
Yong Zhou
Keyword(s):  
Brownian Motion ◽  
Diffusion Equation ◽  
Periodic Solutions ◽  
Fractional Brownian Motion ◽  
Stochastic Diffusion ◽  
Stochastic Diffusion Equation ◽  
Asymptotically Periodic ◽  
Existence And Stability ◽  
The Fixed Point Theorem ◽  
Asymptotically Periodic Solutions

In this paper, a generalized Gronwall inequality is demonstrated, playing an important role in the study of fractional differential equations. In addition, with the fixed-point theorem and the properties of Mittag–Leffler functions, some results of the existence as well as asymptotic stability of square-mean S-asymptotically periodic solutions to a fractional stochastic diffusion equation with fractional Brownian motion are obtained. In the end, an example of numerical simulation is given to illustrate the effectiveness of our theory results.

scholarly journals Maxima of stochastic processes driven by fractional Brownian motion

2005 ◽  
Vol 37 (03) ◽  
pp. 743-764 ◽  
Author(s):  
Boris Buchmann ◽  
Claudia Klüppelberg
Keyword(s):  
Brownian Motion ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
State Space ◽  
Fractional Brownian Motion ◽  
Sufficient Conditions ◽  
Domain Of Attraction ◽  
Uhlenbeck Process ◽  
Ornstein Uhlenbeck Process ◽  
Maximum Domain Of Attraction

We study stationary processes given as solutions to stochastic differential equations driven by fractional Brownian motion. This rich class includes the fractional Ornstein-Uhlenbeck process and those processes that can be obtained from it by state space transformations. An explicit formula in terms of Euler's Γ-function describes the asymptotic behaviour of the covariance function of the fractional Ornstein-Uhlenbeck process near zero, which, by an application of Berman's condition, guarantees that this process is in the maximum domain of attraction of the Gumbel distribution. Necessary and sufficient conditions on the state space transforms are stated to classify the maximum domain of attraction of solutions to stochastic differential equations driven by fractional Brownian motion.

scholarly journals Approximate Controllability for Impulsive Riemann-Liouville Fractional Differential Inclusions

2013 ◽  
Vol 2013 ◽  
pp. 1-17 ◽  
Author(s):  
Zhenhai Liu ◽  
Maojun Bin
Keyword(s):  
Control Systems ◽  
Differential Inclusions ◽  
Approximate Controllability ◽  
Mild Solutions ◽  
Sufficient Conditions ◽  
Fractional Power ◽  
Fractional Differential Inclusions ◽  
Multivalued Maps ◽  
Impulsive Differential Inclusions ◽  
Fractional Differential

We study the control systems governed by impulsive Riemann-Liouville fractional differential inclusions and their approximate controllability in Banach space. Firstly, we introduce thePC1-α-mild solutions for the impulsive Riemann-Liouville fractional differential inclusions in Banach spaces. Secondly, by using the fractional power of operators and a fixed point theorem for multivalued maps, we establish sufficient conditions for the approximate controllability for a class of Riemann-Liouville fractional impulsive differential inclusions, which is a generalization and continuation of the recent results on this issue. At the end, we give an example to illustrate the application of the abstract results.

scholarly journals Lévy Processes Linked to the Lower-Incomplete Gamma Function

2021 ◽  
Vol 5 (3) ◽  
pp. 72
Author(s):  
Luisa Beghin ◽  
Costantino Ricciuti
Keyword(s):  
Brownian Motion ◽  
Fractional Derivative ◽  
Gamma Function ◽  
Lévy Processes ◽  
Fractional Power ◽  
Levy Processes ◽  
Incomplete Gamma Function ◽  
Governing Equations ◽  
Infinite Moments ◽  
Fractional Power Of Operators

We start by defining a subordinator by means of the lower-incomplete gamma function. This can be considered as an approximation of the stable subordinator, easier to be handled in view of its finite activity. A tempered version is also considered in order to overcome the drawback of infinite moments. Then, we study Lévy processes that are time-changed by these subordinators with particular attention to the Brownian case. An approximation of the fractional derivative (as well as of the fractional power of operators) arises from the analysis of governing equations. Finally, we show that time-changing the fractional Brownian motion produces a model of anomalous diffusion, which exhibits a sub-diffusive behavior.

Sign in / Sign up

Export Citation Format

Share Document