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In this paper, we investigate four non-autonomous chemostat models with non-monotonic consumption function, where wall growth and nutrient recycling are also taken into account. In each case, we prove the existence and uniqueness of non-negative global solution that generates a non-autonomous dynamical system. In addition, we also prove the existence of a unique (global) pullback attractor whose internal structure provides detailed information about the long-time behaviour of the state variables, for instance, conditions to ensure the extinction and the persistence of the species. We also display numerical simulations to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the pth moment exponential stability of stochastic differential equations driven by G-Brownian motion (G-SDEs) with respect to a part of the variables by means of the G-Lyapunov functions and recently developed Itô's calculus for SDEs driven by G-Brownian motion, as well as Gronwall's inequalities. We establish sufficient conditions to ensure the quasi sure exponential stability of stochastic differential equations perturbed by G-Brownian motion with respect to a part of the variables. Some illustrative examples to show the usefulness of the stability with respect to a part of the variables notion are also provided. [ABSTRACT FROM AUTHOR]

This paper studied a stochastic epidemic model of the spread of the novel coronavirus (COVID-19). Severe factors impacting the disease transmission are presented by white noise and compensated Poisson noise with possibly infinite characteristic measure. Large time estimates are established based on Kunita's inequality rather than Burkholder-Davis-Gundy inequality for continuous diffusions. The effect of stochasticity is taken into account in the formulation of sufficient conditions for the extinction of COVID-19 and its persistence. Our results prove that environmental fluctuations can be privileged in controlling the pandemic behavior. Based on real parameter values, numerical results are presented to illustrate obtained results concerning the extinction and the persistence in mean of the disease. [ABSTRACT FROM AUTHOR]

In this paper, by using the Gronwall inequality, we show two new results on the Ulam–Hyers and the Ulam–Hyers–Rassias stabilities of neutral stochastic functional differential equations. Two examples illustrating our results are exhibited. [ABSTRACT FROM AUTHOR]

The paper provides conditions for the fractional Laplacian and its spectral representation on stationary Gaussian random fields to be well-defined. In addition, we study existence and uniqueness of the weak solution for a stochastic fractional elliptic equation driven by an additive colored noise over an open bounded set. Both spectral and variational approaches are used to provide a solution. Further, the functional covariance associated with the solution is derived. [ABSTRACT FROM AUTHOR]

Stability of nonlinear delay evolution equation with stochastic perturbations is considered. It is shown that if the level of stochastic perturbations fades on the infinity then an exponentially stable deterministic system remains to be exponentially stable (in mean square). This idea has already been checked for ordinary linear stochastic differential equations and linear stochastic difference equations. Here a similar statement is proven for a nonlinear stochastic evolution equation with quickly enough fading stochastic perturbations. More exactly: if the level of the stochastic perturbation is given by a continuous and square integrable function, then the zero solution of the considered exponentially stable deterministic system remains exponentially mean square stable independently on the maximum magnitude of the stochastic perturbations. Applications of the obtained results to stochastic reaction-diffusion equations and stochastic 2D Navier-Stokes model are shown. Consideration of other types of fading stochastic perturbations (for instance, if stochastic perturbations fade on the infinity not so quickly) is an open problem. [ABSTRACT FROM AUTHOR]

We prove an existence and uniqueness result of mild solution for a system of stochastic semilinear differential equations with fractional Brownian motions and Hurst parameter H < 1∕2. Our approach is based on Perov's fixed point theorem, and we establish the transportation inequalities, with respect to the uniform distance, for the law of the mild solution. [ABSTRACT FROM AUTHOR]

In this paper, practical stability with respect to a part of the variables of nonlinear stochastic differential equations is studied. The analysis of the global practical uniform asymptotic stability, the global practical uniform pth moment exponential stability, as well as the global practical uniform exponential stability with respect to a part of the variables of SDEs are carried out by using the Lyapunov techniques. Some illustrative examples to show the usefulness of the stability with respect to a part of variables notion are also provided. [ABSTRACT FROM AUTHOR]

In this paper we prove some results on the mean square asymptotic stability of a class of neutral stochastic differential systems with variable delays by using a contraction mapping principle. Namely, a necessary and sufficient condition ensuring the asymptotic stability is proved. The assumption does not require neither boundedness or differentiability of the delay functions, nor do they ask for a fixed sign on the coefficient functions. In particular, the results improve some previous ones proved by Guo, Y., Xu, C., & Wu, J. [(2017). Stability analysis of neutral stochastic delay differential equations by a generalisation of Banach's contraction principle. International Journal of Control, 90, 1555–1560]. Finally, an example is exhibited to illustrate the effectiveness of the proposed results. [ABSTRACT FROM AUTHOR]

This paper considers a new model in search theory to find a randomly located target in the 3-dimensional space. An approximation algorithm that facilitates searching procedures for searchers or robots is presented. The expected time to detect the target is also proved. The statistical analysis by calculating the optimal search strategy which minimizes the time to detect the target, assuming trivariate standard normal distribution is provided, and the technique by flowcharts is designed as well. The effectiveness of this strategy is illustrated by introducing an application from real world. [ABSTRACT FROM AUTHOR]

In this paper, we prove some results on the existence of solutions and the mean square asymptotic stability for a class of impulsive neutral stochastic differential systems with variable delays by using a contraction mapping principle. Namely, a sufficient condition ensuring the asymptotic stability is proved. The assumptions do not impose any restrictions neither on boundedness nor on the differentiability of the delay functions. In particular, the results improve some previous ones in the literature. Finally, an example is exhibited to illustrate the effectiveness of the results. [ABSTRACT FROM AUTHOR]

In this article, a stochastic version of a SIR nonautonomous model previously introduced in Kloeden and Kozyakin (2011) is considered. The noise considered is a fractional Brownian motion which satisfies the property of long range memory, which roughly implies that the decay of stochastic dependence with respect to the past is only subexponentially slow, what makes this kind of noise a realistic choice for problems with long memory in the applied sciences. The stochastic model containing a standard Brownian motion has been studied in Caraballo and Colucci (2016). In this paper, we analyze the existence and uniqueness of solutions to our stochastic model as well as their positiveness. [ABSTRACT FROM AUTHOR]

In this paper, we prove the local and global existence and attractivity of mild solutions for stochastic impulsive neutral functional differential equations with infinite delay, driven by fractional Brownian motion. [ABSTRACT FROM PUBLISHER]

The method of Lyapunov functions is one of the most effective ones for the investigation of stability of dynamical systems, in particular, of stochastic differential systems. The main purpose of the paper is the analysis of the stability of stochastic differential equations (SDEs) by using Lyapunov functions when the origin is not necessarily an equilibrium point. The global uniform boundedness and the global practical uniform exponential stability of solutions of SDEs based on Lyapunov techniques are investigated. Furthermore, an example is given to illustrate the applicability of the main result. [ABSTRACT FROM AUTHOR]

In this work, we study the existence, uniqueness, and exponential asymptotic behavior of mild solutions to stochastic integrodifferential delay evolution equations. We assume that the non-delay part generates aC0-semigroup. [ABSTRACT FROM PUBLISHER]

In this article, we study the existence of mild solutions to stochastic impulsive evolution equations with time delays, driven by fractional Brownian motion with the Hurst indexH> 1/2 via a new fixed point analysis approach. [ABSTRACT FROM AUTHOR]

In this article we establish some sufficient conditions ensuring almost sure practical asymptotic stability with a non-exponential decay rate for solutions to stochastic evolution equations based on Lyapunov techniques. [ABSTRACT FROM PUBLISHER]

The discrete analogue of the well-known Favard theorem states that the linear difference equationwith Bohr almost periodic coefficients admits at least one Bohr almost periodic solution if it has a bounded solution. The main assumption in this theorem is the separation among bounded solutions of the homogeneous equationswhereIn this paper, we prove that the linear difference equation (1) with Levitan almost periodic coefficients has a unique Levitan almost periodic solution, if it has at least one bounded solution, and the bounded solutions of the homogeneous equationare homoclinic to zero in the positive direction (i.e.for all relatively compact solutionsof (3)). If the coefficients of (1) are Bohr almost periodic and all relatively compact solutions of all limiting equations (2) tend to zero as, then equation (1) admits a unique almost automorphic solution. We study the problem of existence of Bohr/Levitan almost periodic solutions of equation (1) in the framework of general non-autonomous dynamical systems (cocycles). [ABSTRACT FROM AUTHOR]

The aim of this paper is to describe the structure of global attractors for non-autonomous difference systems of equations with recurrent (in particular, almost periodic) coefficients. We consider a special class of this type of systems (the so-called weak convergent systems). We study this problem in the framework of general non-autonomous dynamical systems (cocycles). We apply the general results obtained in our earlier papers to study the almost periodic (almost automorphic, recurrent) and asymptotically almost periodic (asymptotically almost automorphic, asymptotically recurrent) solutions of difference equations. [ABSTRACT FROM PUBLISHER]

Some results on the pathwise asymptotic stability of solutions to stochastic partial differential equations are proved. Special attention is paid in proving sufficient conditions ensuring almost sure asymptotic stability with a non-exponential decay rate. The situation containing some hereditary characteristics is also treated. The results are illustrated with several examples. [ABSTRACT FROM AUTHOR]

Some results on the existence and uniqueness of solutions for stochastic evolution equations containing some hereditary characteristics are proved. In fact, our theory is developed from a variational point of view and in a general functional setting which permit us to deal with several kinds of delay terms in a unified formulation. [ABSTRACT FROM AUTHOR]

In this paper, we point out the different long-time behavior of stochastic partial differential equations when one considers the stochastic term in the Ito or Stratonovich sense. In particular, we prove that the Stratonovich interpretation may not produce modification in the exponential stability of the deterministic model for a wide range of stochastic perturbations, while Ito's one can give different results. In fact, some stabilization or destabilization effect can be obtained. [ABSTRACT FROM AUTHOR]

The article offers information on the career development of Peter Kloeden who has contributed several important works both in science and mathematics. Hhis doctoral thesis on set valued dynamical systems provided early contributions to chaos theory, showing that chaotic mappings are dense in the space of continuous functions. He is regarded as a mathematical pioneer in the fields of differential equations and is awarded with the W. T. and Idalia Reid prize.