The article is devoted to the existence and Hyers-Ulam stability of the almost periodic solution to the fractional differential equation with impulse and fractional Brownian motion under nonlocal ...condition. The investigation is mainly based on the semigroups of operators method and Mönch fixed points method, as well as the basic theory of Hyers-Ulam stability.
In this paper, we investigate the existence and exponential stability of mild solutions for a class of impulsive neutral stochastic functional differential equations driven by fBm with noncompact ...semigroup in Hilbert spaces. Sufficient conditions for the existence of mild solutions are obtained using the Hausdorff measure of noncompactness and the Mönch fixed point theorem. Further, we establish a new impulsive-integral inequality to prove the exponential stability of mild solutions in the mean square moment. Finally, an example is presented to illustrate our obtained results.
In this paper, we consider the existence of mild solutions and approximate controllability for Riemann-Liouville fractional stochastic evolution equations with nonlocal conditions of order 1 <
α
< 2. ...As far as we know, there are few articles investigating on this issue. Firstly, the mild solutions to the equations are proved using Laplace transform of the Riemann-Liouville derivative. Moreover, the estimations of resolve operators involving the Riemann-Liouville fractional derivative of order 1 <
α
< 2 are given. Then, the existence results are obtained via the noncompact measurement strategy and the Mönch fixed point theorem. The approximate controllability of this nonlinear Riemann-Liouville fractional nonlocal stochastic systems of order 1 <
α
< 2 is concerned under the assumption that the associated linear system is approximately controllable. Finally, the approximate controllability results are obtained by using Lebesgue dominated convergence theorem.
This paper is concerned with the formula of mild solutions to impulsive fractional evolution equation. For linear fractional impulsive evolution equations 8–25,27,30,31, described mild solution as ...integrals over (tk,tk+1(k=1,2,…,m) and 0, t1. On the other hand, in 26,28,29, their solutions were expressed as integrals over 0, t. However, it is still not clear what are the correct expressions of solutions to the fractional order impulsive evolution equations. In this paper, firstly, we prove that the solutions obtained in 8–25,27,30,31 are not correct; secondly, we present the right form of the solutions to linear fractional impulsive evolution equations with order 0 < α < 1 and 1 < α < 2, respectively; finally, we show that the reason that the solutions to an impulsive ordinary evolution equation are not distinct.
This paper deals with the existence of solution for an impulsive Riemann–Liouville fractional neutral functional stochastic differential equation with infinite delay of order
1
<
β
<
2
and its ...Hyers–Ulam stability. We prove the mild solutions for the equation using basic theorems of fractional differential equation. The existence result of the equation is obtained by Mönch’s fixed point theorem. Finally, we prove the Hyers–Ulam stability of the solution.
In this paper, we investigate the existence and Hyers-Ulam stability for random impulsive stochastic functional differential equations with finite delays. Firstly, we prove the existence of mild ...solutions to the equations by using Krasnoselskii's fixed point. Then, we investigate the Hyers-Ulam stability results under the Lipschitz condition on a bounded and closed interval. Finally, an example is given to illustrate our results.
This paper studies the optimal control problem of random impulsive differential equations. Based on the influence of random impulse generation, we define a more reasonable performance index by ...setting the random function and obtain the HJB equation of random impulse. Using the basic analysis method and stochastic process theory, we prove that the value function satisfies the random impulse HJB equation, and the value function is the viscosity solution of the random impulse HJB. As an application, we present an example of optimal feedback control.
This paper is mainly concerned with the existence of mild solutions for fractional differential equations with nonlocal conditions of order 1<α<2. The results are obtained by the fixed point theorem ...combined with solutions operator theorems.
•This article considers the interplay among four financial factors.•We use integer order and fractional order differential equation systems to model a financial system.•Both mathematical analyses and ...numerical simulations are carried out to illustrate the characteristic of the model.•The model displays a variety of rich dynamic behaviors including chaos over a wide range of system parameters.
In this paper, we use integer order and fractional order differential equation systems to model a financial system. Based on the interaction among several financial factors, a model is constructed. Both mathematical analyses and numerical simulations are carried out to illustrate the characteristic of the model. We find that the system displays a variety of rich dynamic behaviours including chaos over a wide range of system parameters. Our investigation indicates that the interplay among several financial factors lead to chaos under some circumstances. We then design control laws to synchronization two integer order financial systems and two fractional order financial systems. Numerical simulations are presented to verify the effectiveness of the designed control laws.
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