THE EXISTENCE AND UNIQUENESS FOR NON-LIPSCHITZ STOCHASTIC NEUTRAL DELAY EVOLUTION EQUATIONS DRIVEN BY POISSON JUMPS

2009 ◽  
Vol 09 (01) ◽  
pp. 135-152 ◽  
Author(s):  
JIAOWAN LUO ◽  
TAKESHI TANIGUCHI
Keyword(s):  
Existence And Uniqueness ◽  
Initial Function ◽  
Evolution Equations ◽  
Mild Solutions ◽  
Jump Processes ◽  
Poisson Jumps ◽  
Neutral Delay ◽  
Poisson Jump

In this paper, we consider the existence and uniqueness of mild solutions to non-Lipschitz stochastic neutral delay evolution equations driven by Poisson jump processes: [Formula: see text] with an initial function X(s) = φ(s), -r ≤ s ≤ 0, where φ : [-r, 0] → H is a cadlag function with [Formula: see text].

THE EXISTENCE AND ASYMPTOTIC BEHAVIOUR OF MILD SOLUTIONS TO STOCHASTIC EVOLUTION EQUATIONS WITH INFINITE DELAYS DRIVEN BY POISSON JUMPS

2009 ◽  
Vol 09 (02) ◽  
pp. 217-229 ◽  
Author(s):  
TAKESHI TANIGUCHI ◽  
JIAOWAN LUO
Keyword(s):  
Initial Function ◽  
Evolution Equations ◽  
Mild Solutions ◽  
Jump Processes ◽  
Stochastic Evolution ◽  
Stochastic Evolution Equations ◽  
Poisson Jumps ◽  
Infinite Delays ◽  
Poisson Jump ◽  
Exponentially Stable

In this paper we consider a sufficient condition for mild solutions to exist and to be almost surely exponentially stable or exponentially ultimate bounded in mean square for the following stochastic evolution equation with infinite delays driven by Poisson jump processes: [Formula: see text] with an initial function X(s) = φ (s), -∞ < s ≤ 0, where φ : (-∞, 0] → H is a càdlàg function with [Formula: see text].

Existence results of mild solutions for impulsive fractional integro-differential evolution equations with infinite delay

2014 ◽  
Vol 17 (4) ◽  
Author(s):  
Shengli Xie
Keyword(s):  
Differential Equations ◽  
Differential Evolution ◽  
Existence Theorem ◽  
Banach Spaces ◽  
Existence And Uniqueness ◽  
Evolution Equations ◽  
Infinite Delay ◽  
Mild Solutions ◽  
Existence Results ◽  
Order Case

AbstractIn this paper we prove the existence and uniqueness of mild solutions for impulsive fractional integro-differential evolution equations with infinite delay in Banach spaces. We generalize the existence theorem for integer order differential equations to the fractional order case. The results obtained here improve and generalize many known results.

scholarly journals Fractional Stochastic Differential Equations with Hilfer Fractional Derivative: Poisson Jumps and Optimal Control

2017 ◽  
Vol 2017 ◽  
pp. 1-11 ◽  
Author(s):  
Fathalla A. Rihan ◽  
Chinnathambi Rajivganthi ◽  
Palanisamy Muthukumar
Keyword(s):  
Optimal Control ◽  
Fractional Derivative ◽  
Existence And Uniqueness ◽  
Mild Solutions ◽  
Poisson Jumps ◽  
Hilfer Fractional Derivative ◽  
Analysis Techniques ◽  
Mild Conditions ◽  
Fractional Stochastic Differential Equations ◽  
Existence Of Optimal Control

In this work, we consider a class of fractional stochastic differential system with Hilfer fractional derivative and Poisson jumps in Hilbert space. We study the existence and uniqueness of mild solutions of such a class of fractional stochastic system, using successive approximation theory, stochastic analysis techniques, and fractional calculus. Further, we study the existence of optimal control pairs for the system, using general mild conditions of cost functional. Finally, we provide an example to illustrate the obtained results.

scholarly journals The Neutral Stochastic Integrodifferential Equations with Jumps

2016 ◽  
Vol 2016 ◽  
pp. 1-10 ◽  
Author(s):  
Diem Dang Huan
Keyword(s):  
Successive Approximation ◽  
Existence And Uniqueness ◽  
Continuous Dependence ◽  
Mild Solutions ◽  
Integrodifferential Equations ◽  
Poisson Jumps ◽  
Initial Value ◽  
Carathéodory Conditions ◽  
Global And Local ◽  
Continuous Dependence Of Solutions

We study the existence and uniqueness of mild solutions for neutral stochastic integrodifferential equations with Poisson jumps under global and local Carathéodory conditions on the coefficients by means of the successive approximation. Furthermore, we give the continuous dependence of solutions on the initial value. Finally, an example is provided to illustrate the effectiveness of the obtained results.

scholarly journals Existence and Uniqueness of Pseudo Almost Automorphic Mild Solutions to Some Classes of Partial Hyperbolic Evolution Equations

2008 ◽  
Vol 2008 ◽  
pp. 1-13 ◽  
Author(s):  
Zhanrong Hu ◽  
Zhen Jin
Keyword(s):  
Evolution Equation ◽  
Uniqueness Theorem ◽  
Existence And Uniqueness ◽  
Evolution Equations ◽  
Mild Solutions ◽  
Existence And Uniqueness Theorem ◽  
Abstract Result ◽  
Almost Automorphic ◽  
Pseudo Almost Automorphic

We will establish an existence and uniqueness theorem of pseudo almost automorphic mild solutions to the following partial hyperbolic evolution equation(d/dt)[u(t)+f(t,Bu(t))]=Au(t)+g(t,Cu(t)),  t∈ℝ,under some assumptions. To illustrate our abstract result, a concrete example is given.

scholarly journals A UNIFIED EXISTENCE AND UNIQUENESS THEOREM FOR STOCHASTIC EVOLUTION EQUATIONS

2009 ◽  
Vol 81 (1) ◽  
pp. 33-46
Author(s):  
A. JENTZEN ◽  
P. E. KLOEDEN
Keyword(s):  
Diffusion Coefficient ◽  
Uniqueness Theorem ◽  
Existence And Uniqueness ◽  
Evolution Equations ◽  
Mild Solutions ◽  
Trace Class ◽  
Existence And Uniqueness Theorem ◽  
Stochastic Evolution ◽  
Stochastic Evolution Equations ◽  
Class Noise

AbstractAn existence and uniqueness theorem for mild solutions of stochastic evolution equations is presented and proved. The diffusion coefficient is handled in a unified way which allows a unified theorem to be formulated for different cases, in particular, of multiplicative space–time white noise and trace-class noise.

scholarly journals Fractional Measure-dependent Nonlinear Second-order Stochastic Evolution Equations with Poisson Jumps

2018 ◽  
Vol 5 (1) ◽  
pp. 59-75 ◽  
Author(s):  
Mark A. McKibben ◽  
Micah Webster
Keyword(s):  
Evolution Equations ◽  
Mild Solutions ◽  
Stability Result ◽  
Growth Conditions ◽  
Second Order ◽  
Probability Measures ◽  
Stochastic Evolution ◽  
Stochastic Evolution Equations ◽  
Poisson Jumps ◽  
Fractional Measure

Abstract This paper focuses on a nonlinear second-order stochastic evolution equations driven by a fractional Brownian motion (fBm) with Poisson jumps and which is dependent upon a family of probability measures. The global existence of mild solutions is established under various growth conditions, and a related stability result is discussed. An example is presented to illustrate the applicability of the theory.

Existence of mild solutions for a class of Hilfer fractional evolution equations with nonlocal conditions

2017 ◽  
Vol 20 (3) ◽  
Author(s):  
Min Yang ◽  
Qiru Wang
Keyword(s):  
Fixed Point ◽  
Existence And Uniqueness ◽  
Evolution Equations ◽  
Mild Solutions ◽  
Nonlocal Conditions ◽  
Hilfer Fractional Derivative ◽  
Fractional Evolution Equations ◽  
Noncompact Measure ◽  
The Fixed Point Theorem ◽  
New Criteria

AbstractIn this paper, we consider a class of evolution equations with Hilfer fractional derivative. By employing the fixed point theorem and the noncompact measure method, we establish a number of new criteria to guarantee the existence and uniqueness of mild solutions when the associated semigroup is compact or not.

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