Automorphic Coefficients Research Articles

Pseudo Almost Automorphic Solutions for Stochastic Differential Equations Driven by Lévy Noise and Its Optimal Control

As we know, the periodic functions are symmetric within a cycle time, and it is meaningful to generalize the periodicity into more general cases, such as almost periodicity or almost automorphy. In this work, we introduce the concept of Poisson Sγ2-pseudo almost automorphy (or Poisson generalized Stepanov-like pseudo almost automorphy) for stochastic processes, which are almost-symmetric within a suitable period, and establish some useful properties of such stochastic processes, including the composition theorems. In addition, we apply a Krasnoselskii–Schaefer type fixed point theorem to obtain the existence of pseudo almost automorphic solutions in distribution for some semilinear stochastic differential equations driven by Lévy noise under Sγ2-pseudo almost automorphic coefficients. In addition, then we establish optimal control results on the bounded interval. Finally, an example is provided to illustrate the theoretical results obtained in this paper.

Delayed fuzzy genetic regulatory networks: Novel results

In this manuscript, we studied a class of delayed Fuzzy Genetic Regulatory Networks (FGRNs) with Stepanov-like weighted pseudo almost automorphic coefficients. New criteria for the existence, uniqueness and global exponential stability of its weighted pseudo almost automorphic solution are established. Our approach is based on Banach fixed point theorem and novel analysis techniques. Moreover, a numerical example is given to illustrate the validity of the obtained results.

The dynamic analysis on a class of stochastic impulsive equations with doubly weighted pseudo almost automorphic coefficients on time scales

This paper is devoted to exploring the translation invariance and convolution invariance of doubly weighted pseudo almost automorphic stochastic processes with impulses on time scales. Based on these results, the existence and uniqueness of the doubly weighted pseudo almost automorphic solutions to a class of stochastic nonlinear impulsive equations on time scales are obtained by taking advantage of a new approach, which enrich the dynamics of doubly weighted pseudo almost automorphic stochastic processes. Finally, an example is researched to illustrate our conclusions.

Study of genetic regulatory networks with Stepanov-like pseudo-weighted almost automorphic coefficients

In this article, we investigated a class of genetic regulatory networks (GRNs) with time-varying delays and Stepanov-like weighted pseudo-almost automorphic coefficients. By Banach fixed point theorem, we firstly established the existence results of the weighted pseudo-almost automorphic solutions for the proposed GRNs. Secondly, by using suitable Lyapunov functional and mathematical analysis skills, new criteria are derived for ensuring the global exponential stability of the solutions. Finally, a numerical example and remarks are provided to substantiate the efficiency of obtained results. The derived results of this article are new and complement many earlier works and the ideas of this work can be applied to investigate other similar systems.

Generalization of Weighted Stepanov-Like Pseudo-Almost Automorphic Space

In this paper we introduce and study a new class of functions called weighted Stepanov-like pseudo-almost automorphic functions with variable exponents, which generalizes the class of weighted Stepanov-like pseudo-almost automorphic functions. Basic properties of these new spaces are established. The existence of weighted pseudo-almost automorphic solutions to some first-order differential equations with Sp,q(x)-pseudo-almost automorphic coefficients will also be studied.

Vanishing theorems for mixed Hodge modules and applications

We prove two Kawamata–Viehweg type vanishing theorems with coefficients, one for polarisable variations of Hodge structures (PVHSs) and the other for general mixed Hodge modules. They generalise previous vanishing theorems of Kawamata, Viehweg, Esnault and Viehweg, Illusie, Saito, and Popa. The two theorems are complementary. One applies to all mixed Hodge modules, but only covers (of necessity) the lowest Hodge graded piece. The other applies only to Deligne’s canonical extension of PVHSs across simple normal crossings divisors, but covers all the Hodge graded pieces. As application, we obtain new results on the cohomology of Shimura varieties and an alternative proof of a known result in birational geometry: (i) Using the one for PVHSs, we obtain new vanishing theorems for the cohomology of arbitrary Shimura varieties (including the ones attached to exceptional groups) with automorphic coefficients, extending works of Faltings, Li and Schwermer, and our own with Lan. (ii) The vanishing theorem for mixed Hodge modules contains, as a special case, Kollár’s vanishing theorem for the higher direct images of dualising sheaves and big and nef line bundles.

Almost automorphic synchronization of quaternion-valued high-order Hopfield neural networks with time-varying and distributed delays

AbstractIn this paper, we consider the problem of the almost automorphic synchronization of quaternion-valued high-order Hopfield neural networks (QVHHNNs) with time-varying and distributed delays. Firstly, to avoid the non-commutativity of quaternion multiplication, we decompose QVHHNNs into an equivalent real-valued system. Secondly, we use the Banach fixed point theorem to obtain the existence of almost automorphic solutions of QVHHNNs. Thirdly, by designing a novel state-feedback controller and constructing suitable Lyapunov functions, we obtain that the drive-response structure of QVHHNNs with almost automorphic coefficients can realize the exponential synchronization. Our results are completely new. Finally, a numerical example is given to illustrate the feasibility of our results.

Poisson $S^2$-almost automorphy for stochastic processes and its applications to SPDEs driven by Lévy noise

In this paper, we introduce and study the concepts and properties of Poisson Stepanov-like almost automorphy (or Poisson \begin{document}$S^2$\end{document} -almost automorphy) for stochastic processes. With appropriate conditions, we apply the results obtained to investigate the asymptotic behavior of the soulutions to SPDEs driven by Levy noise under \begin{document}$S^2$\end{document} -almost automorphic coefficients without global Lipschitz conditions. Moreover, the local asymptotic stability of the solutions under local Lipschitz condition is discussed and the attractive domain is also given. Finally, an illustrative example is provided to justify the practical usefulness of the established theoretical results.

Almost automorphic solutions for stochastic differential equations with state-dependent switching

This paper concerns a kind of reccurence: Almost automorphy for stochastic differential equations with state-dependent switching. The existence and uniqueness of almost automorphic solutions in distribution are established provided the coefficients satisfy some suitable conditions. We also give an example to illustrate that a stochastic differential equation with almost automorphic coefficients might have no square-mean almost automorphic solution.

Weighted pseudo almost automorphic solutions to nonautonomous semilinear evolution equations with delay and $S^{p}$-weighted pseudo almost automorphic coefficients

By some new properties of Stepanov-like weighted pseudo almost automorphic functions established by Chang, Zhang and N'Guerekata recently, we shall deal with weighted pseudo almost automorphic solutions to the nonautonomous semilinear evolution equations with a constant delay: $u'(t)=A(t)u(t)+f(t,u(t-h))$, $ t\in\mathbb{R}$ in a Banach space $\mathbb{X}$, where $A(t),t\in\mathbb{R}$ generates an exponentially stable evolution family $\{U(t,s)\}$ and $f \colon\mathbb{R}\times\mathbb{X} \rightarrow \mathbb{X}$ is a $\S^{p}$-weighted pseudo almost automorphic function satisfying some suitable conditions. We obtain our main results by the Leray-Shauder Alternative theorem.

Almost automorphy and various extensions for stochastic processes

We compare different modes of pseudo almost automorphy and variants for stochastic processes: in probability, in quadratic mean, or in distribution in various senses. We show by a counterexample that square-mean (pseudo) almost automorphy is a property which is too strong for stochastic differential equations (SDEs). Finally, we consider two semilinear SDEs, one with almost automorphic coefficients and the second one with pseudo almost automorphic coefficients, and we prove the existence and uniqueness of a mild solution which is almost automorphic in distribution in the first case, and pseudo almost automorphic in distribution in the second case.

Asymptotic almost automorphic solutions of impulsive neural network with almost automorphic coefficients

In this paper existence and asymptotic stability of asymptotically almost automorphic solution of impulsive neural networks with delay is discussed. The results are established by using various fixed point theorems and Lyapunov-like functional. As far as we know, this is the first paper to discuss such kind of solutions for impulsive neural networks. At the end, we give few examples to illustrate our theoretical findings. One can see that the numerical simulation results show asymptotically almost automorphic behaviour of the solution.

Existence results for some damped second-order Volterra integro-differential equations

In this paper we make a subtle use of operator theory techniques and the well-known Schauder fixed-point principle to establish the existence of pseudo-almost automorphic solutions to some second-order damped integro-differential equations with pseudo-almost automorphic coefficients. In order to illustrate our main results, we will study the existence of pseudo-almost automorphic solutions to a structurally damped plate-like boundary value problem.

Stepanov-like weighted asymptotic behavior of solutions to some stochastic differential equations in Hilbert spaces

In this paper, we first introduce the notation and properties of -weighted pseudo almost automorphy for stochastic processes. And then, we apply the results obtained to consider the existence and uniqueness of -weighted pseudo almost automorphic solutions to some stochastic differential equations in a real separable Hilbert space under global Lipschitz conditions. Moreover, we also investigate asymptotic behavior of solutions to a stochastic differential equation under -weighted pseudo almost automorphic coefficients without global Lipschitz conditions. Our main results extend some known ones in the sense of square-mean weighted pseudo almost automorphy or -pseudo almost automorphy for stochastic processes.

Weighted pseudo almost automorphic mild solutions to semilinear integral equations with $S^{p}$-weighted pseudo almost automorphic coefficients

In this paper, we consider the existence of weighted pseudo almost automorphic solutions of the semilinear integral equation $x(t)= \int_{-\infty}^{t}a(t-s)[Ax(s) + f(s,x(s))]ds, t\in\mathbb{R}$ in a Banach space $\mathbb{X}$, where $a\in L^{1}(\mathbb{R}_{+})$, $A$ is the generator of an integral resolvent family of linear bounded operators defined on the Banach space $\mathbb{X}$, and $f : \mathbb{R}\times\mathbb{X} \rightarrow \mathbb{X}$ is a weighted pseudo almost automorphic function. The main results are proved by using integral resolvent families, suitable composition theorems combined with the theory of fixed points.

非自治中立型无穷时滞泛函微分方程的加权伪概自守解

We first introduce the notion of Stepanov-like weighted pseudo almost automorphic functions of class h and the notion of Stepanov-like weighted pseudo almost automorphic functions of class ∞, and then establish some basic results not only on the completeness of the space that consists of the Sp -weighted pseudo almost automorphic functions of class h and class ∞ but also on the compositions of such functions. Finally, we apply the abstract results to prove the existence and uniqueness of weighted pseudo almost automorphic solutions with Sp -weighted pseudo almost automorphic coefficients to a non-autonomous first-order abstract partial neutral functional differential equation with infinite delay.

Stepanov-like square-mean pseudo almost automorphic solutions to partial neutral stochastic functional differential equations

We obtain the existence and uniqueness of square-mean pseudo almost automorphic mild solutions to first-order partial neutral stochastic functional differential equations with Stepanov-like almost automorphic coefficients in a real separable Hilbert space

New composition theorems of Stepanov-like weighted pseudo almost automorphic functions and applications to nonautonomous evolution equations

In this paper, we establish some new composition theorems for Stepanov-like weighted pseudo almost automorphic functions. We then apply the obtained results to investigate the existence and uniqueness of weighted pseudo almost automorphic mild solutions to a class of nonautonomous evolution equations with Sp-weighted pseudo almost automorphic coefficients.

Weighted pseudo almost automorphic mild solutions to semilinear fractional differential equations

This paper is concerned with the existence and uniqueness of a weighted pseudo almost automorphic mild solution to the semilinear fractional equation: Dtαu(t)=Au(t)+Dtα−1f(t,u(t)),t∈R,1<α<2 in complex Banach spaces with Sp-weighted pseudo almost automorphic coefficients, where A is a linear densely defined operator of sectorial type on a complex Banach space X. Moreover, we present an application to a fractional wave equation.

Stepanov-like C^{(n)}-pseudo almost automorphy and applications to some nonautonomous higher-order differential equations

In this paper we introduce and study a new concept called Stepanov-like \(C^{(n)}\)-pseudo almost automorphy, which generalizes in a natural fashion both the notions of \(C^{(n)}\)-pseudo almost periodicity and that of \(C^{(n)}\)-pseudo almost automorphy recently introduced in the literature by the authors. Basic properties of these new functions are investigated. Furthermore, we study and obtain the existence of \(C^{(N+m)}\)-pseudo almost automorphic solutions to some nonautonomous higher-order systems of differential equations with Stepanov-like \(C^{(m)}\)-pseudo almost automorphic coefficients.