We study the long-time behavior of nonlinear stochastic evolution equations in a separable Hilbert space driven by a Q-Wiener process. The linear part of the equation is generated by a strongly continuous semigroup with an exponential dichotomy, which provides fixed rates of decay and growth. The nonlinear drift and diffusion terms are globally Lipschitz and become small as time tends to infinity. Our main result shows that under these conditions, the mean-square Lyapunov exponents of the nonlinear system coincide with those of the linear part. In other words, nonlinear stochastic perturbations that decay in time do not change the main growth or decay rates of solutions in the mean-square sense. This result provides simple and verifiable criteria ensuring that the long-time Lyapunov behavior of the nonlinear stochastic equation is fully determined by the linear semigroup, even in the presence of time-dependent stochastic perturbations.

The paper utilizes the continuous finite element method to solve stiff ordinary differential equations and proves that the linear finite element method and the quadratic finite element method have A-stability in solving autonomous ordinary differential equations, and exponential dichotomy in solving non-autonomous ordinary differential equations. In the numerical experiments of nonlinear autonomous and non-autonomous strongly and moderately stiff ordinary differential equations, a relatively large step size of h=0.1 was adopted over a longer period of time, with the numerical solution accuracy reaching 10−4. The superconvergence order maintained the theoretical order. A new approach is provided for solving stiff ordinary differential equations.

Recurrent Solutions and Mean Exponential Dichotomy for Dynamic Equations on Time Scales

In this work, we study the dynamics of linear non-autonomous differential equations with exponential dichotomy and compact almost automorphic perturbations. First, we prove that if the homogeneous system is exponentially dichotomous and the coefficient matrix is compact almost automorphic, then the associated Green's function is compact bi-almost automorphic and uniformly continuous relative to the principal diagonal of the two-dimensional Euclidean space. Next, we demonstrate the invariance of the compact almost automorphic function space under convolution products with Green's function as the kernel. These results ensures that the unique bounded solution of a linear non-autonomous differential equation, under exponential dichotomy and with compact almost automorphic perturbation, is itself compact almost automorphic. Finally, we investigate the existence and the global exponential stability of a unique positive compact almost automorphic solution for a nonlinear non-autonomous delayed biological model with nonlinear harvesting or immigration terms and mixed delays. For more information and the Latex file see https://ejde.math.txstate.edu/Volumes/2025/91/abstr.html

Abstract We introduce a new conjecture of global asymptotic stability for nonautonomous systems, which is fashioned along the nonuniform exponential dichotomy spectrum and whose restriction to the autonomous case is related to the classical Markus–Yamabe Conjecture: we prove that the conjecture is fulfilled for a family of triangular systems of nonautonomous differential equations satisfying boundedness assumptions. An essential tool to carry out the proof is a necessary and sufficient condition ensuring the property of nonuniform exponential dichotomy for upper block triangular linear differential systems. We also obtain some byproducts having interest on itself, such as the diagonal significance property in terms of the above-mentioned spectrum.

In this paper, combining the theory of discrete exponential dichotomy, Krasnoselskii's fixed point theorem and decomposition technique, we establish two new existent theorems for asymptotically almost periodic solutions of discrete dynamical systems. Our results generalize and improve some previous results, and are implemented for some economical, biological and mathematical models.

ABSTRACTIn this paper, we investigate the robustness of impulsive differential equations in Banach spaces. We establish sufficient conditions to ensure that nonuniform exponential contractions, nonuniform exponential expansions and nonuniform exponential dichotomies of impulsive differential equations persist within a wider range of perturbations. It is worth emphasizing that the new results provide some significant improvements of existing results in the case where the perturbations are not required to be sufficiently small. Moreover, the conditions of our conclusions even for differential equations without impulse are more relaxed. Finally, we illustrate our results with a concrete example.

Entire Solutions of Stochastic Unbounded Delay Evolution Variational Inequalities Driven by Tempered Fractional Noise with an Exponential Dichotomy

In this paper, we present in this work a proof of the exponential dichotomy for a family of dynamically defined matrix-valued Jacobi operators in [Formula: see text], [Formula: see text], where [Formula: see text] is a compact metric space. Namely, we show that for each [Formula: see text], the resolvent set of [Formula: see text] corresponds to the subset of [Formula: see text] for which [Formula: see text], the [Formula: see text]-cocycle induced by the eigenvalue equation [Formula: see text] at [Formula: see text], is uniformly hyperbolic if [Formula: see text] is minimal.

The main aim of this paper is to give characterizations of Datko type for the uniform dichotomy in mean with growth rates concept for reversible stochastic skew-evolution semiflows in Banach spaces. As particular cases, we obtain integral characterizations for uniform exponential dichotomy in mean. The obtained results are generalizations of well-known theorems about uniform \(h\)-dichotomy of variational systems in deterministic case.

Abstract This study focuses on the problem of h-dichotomy for skew-evolution cocycles within Banach spaces. It outlines the necessary conditions and sufficient conditions for this framework, along with those for the notable concepts like exponential dichotomy, polynomial dichotomy and uniform h-dichotomy. These conditions are established through the use of strongly invariant families of projectors.

Abstract The present work is mainly devoted to the investigation of the exponential dichotomy of retarded perturbed boundary control systems with dynamic boundary conditions. Moreover, the well-posedness of these systems is established using the feedback theory of infinite-dimensional regular linear systems. Furthermore, we study the robustness of the exponential dichotomy of their solutions. Finally, the results are applied to a coupled system consisting of an ordinary differential equation and a transport partial differential equation.

In this paper, we establish the strongly topological equivalence on time scales between the quasilinear coupled system and its linear part without the assumption that the whole linear system admits exponential dichotomy. Our results generalize and improve some previous known literature.

We establish a sufficient condition for the existence of a nonuniform exponential dichotomy of an evolution family on the half-line in terms of evolution semigroups. Such endeavour is not at all of a formal type and our proof is new even in the particular case of uniform exponential dichotomy.

Recently, Wu and Xia [Proc. Amer. Math. Soc. 151 (2023), pp. 4389-4403] presented a characterization of nonuniform exponential dichotomy via admissibility for difference equations. They have improved previously known results by removing the use of Lyapunov norms and the assumption of bounded growth of the system. However, they have restricted their attention to the case of finite dimensional and invertible dynamics. In the present work we go one step further and extend their results to the case of possibly noninvertible and infinite dimensional dynamical systems. We emphasize that our method of proof is different and significantly simpler than the one presented in the aforementioned work.

A unified theory for inertial manifolds, saddle point property and exponential dichotomy

In this note we present a new necessary and sufficient condition for having a dichotomy by a continuous linear system with time-varying coefficients. This condition is expressed by Bohl exponents. Based on this condition, we give a formula expressing the spectrum of exponential dichotomy in terms of Bohl exponents. The obtained results are illustrated with a numerical example.

Abstract We consider linear cocycles acting on Banach spaces which satisfy the assumptions of the multiplicative ergodic theorem. A cocycle is nonuniformly hyperbolic if all Lyapunov exponents are non-zero, which is equivalent to the existence of a tempered exponential dichotomy. We provide an equivalent characterization of nonuniform hyperbolicity in terms of a Mather-type admissibility of a pair of weighted function spaces. As an application we give a short proof of the robustness of tempered exponential dichotomies under small linear perturbation.

In this paper, we investigate differential equations with generalized piecewise constant delay, DEGPCD in short, and establish the existence and stability of a unique almost periodic solution that is exponentially stable. Our results are derived by utilizing the properties of the (μ1,μ2)-exponential dichotomy, Cauchy and Green matrices, a Gronwall-type inequality for DEGPCD, and the Banach fixed point theorem. We apply these findings to derive new criteria for the existence, uniqueness, and convergence dynamics of almost periodic solutions in both the linear inhomogeneous and quasilinear DEGPCD systems through the (μ1,μ2)-exponential dichotomy for difference equations. These results are novel and serve to recover, extend, and improve upon recent research.

The problem of exponential dichotomy for systems of difference equations with periodic coefficients is considered. Based on the previously obtained exponential dichotomy criterion, the question of permissible perturbations on the coefficient matrix, under which the exponential dichotomy persists, is investigated.