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The aim of the present paper is to emphasize some growth concepts for the dichotomic behavior of evolution operators in Banach spaces. In fact, we approach the exponential growth, the polynomial growth and the h-growthfor both uniform and nonuniform cases. Connections between concepts are established. Majorization criteria for the uniform behaviors are given.

Whale optimization algorithm (WOA) is a novel-innovative swarm-based meta-heuristic algorithm with excellent performance, but it may still be trapped into local extremum for troublesome problems. To this end, an improved multi-objective whale optimization algorithm (IMOWOA) is proposed to cover the shortages. Firstly, in the search stage of WOA, individual difference is considered to strengthen the exploration ability, and evolution operators are introduced to regenerate the stagnated population to prevent premature convergence. Next, the performance of IMOWOA is compared with MOWOA and other classical optimization algorithms, and a series of multi-objective test functions are used. The results on the convergence and diversity of Pareto front confirm that IMOWOA has better feasibility and competitiveness. Finally, integrated with the least squares support vector regression (LSSVR) model, IMOWOA is applied to the deterministic optimization of vehicle structural crashworthiness. The conclusion testified the efficiency of IMOWOA in the field of vehicle crashworthiness. [ABSTRACT FROM AUTHOR]

We study a nonautonomous semilinear parabolic problem with a dynamical boundary condition in an irregular domain with fractal boundary. Local existence, uniqueness and regularity results for the mild solution are established via a family of evolution operators. A sufficient condition on the initial datum for global existence is given. [ABSTRACT FROM AUTHOR]

The present paper deals with two of the most significant behaviors in the theory of dynamical systems: the uniform exponential dichotomy and the uniform polynomial dichotomy for evolution operators in Banach spaces. Assuming that the evolution operator has uniform exponential growth, respectively uniform polynomial growth, we give some characterizations for the uniform exponential dichotomy, respectively for the uniform polynomial dichotomy. The proof techniques that we use for the polynomial case are new. In addition, connections between the concepts approached are established. [ABSTRACT FROM AUTHOR]

The paper considers three concepts of uniform instability for evolution operators: uniform exponential instability, uniform polynomial instability and uniform h-instability. Some characterizations of these notions and connections between these concepts are given. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the global existence of mild solutions as well as uniform h-stability for non-autonomous evolution equations in a Banach space, where the operators in linear part (possibly unbounded) depend on time t and generate an evolution family. Under some estimates on the perturbation terms, we discuss the global uniform h-stability, the uniform h-stability and the global practical uniform h-stability using Gronwall integral inequalities. Some examples are provided to show the validity of the obtained results. [ABSTRACT FROM AUTHOR]

The present paper deals with two of the most significant behaviors in the theory of dynamical systems: the uniform exponential dichotomy and the uniform polynomial dichotomy for evolution operators in Banach spaces. Assuming that the evolution operator has uniform exponential growth, respectively uniform polynomial growth, we give some characterizations for the uniform exponential dichotomy, respectively for the uniform polynomial dichotomy. The proof techniques that we use for the polynomial case are new. In addition, connections between the concepts approached are established.

In this paper, we provide Lyapunov-based tools to establish input-to-state stability (ISS) and integral input-to-state stability (iISS) for non-autonomous infinite-dimensional systems. We prove that for a class of admissible inputs the existence of an ISS Lyapunov function implies the ISS of a system in Banach spaces. Furthermore, it is shown that uniform global asymptotic stability is equivalent to their integral input-to-state stability for non-autonomous generalised bilinear systems over Banach spaces. The Lyapunov method is provided to be very useful for both linear and nonlinear tools including partial differential equations (PDEs). In addition, we present a method for construction of iISS Lyapunov function in Hilbert spaces. Finally, two examples are given to verify the effectiveness of the proposed scheme. [ABSTRACT FROM AUTHOR]

We are interested in the evolution operators defined on commutative and non-associative algebras when the characteristic of the scalar field is 2. We distinguish four types: nilpotent, quasi-constant, ultimately periodic, and plenary train operators. They are studied and classified for non-baric and for baric algebras. [ABSTRACT FROM AUTHOR]

The paper considers a general concept of nonuniform (h, k)-dichotomy for evolution operators in Banach spaces. Two characterizations of this concept in terms of some families of norms compatible with the dichotomy projectors are given.

Recently, systems of coupled renewal and retarded functional differential equations have begun to play a central role in complex and realistic models of population dynamics. In view of studying the local asymptotic stability of equilibria and (mainly) periodic solutions, we propose a pseudospectral collocation method to approximate the eigenvalues of the evolution operators of linear coupled equations, providing rigorous error and convergence analyses and numerical tests. The method combines the ideas of the analogous techniques developed separately for renewal equations and for retarded functional differential equations. Coupling them is not trivial, due to the different state spaces of the two classes of equations, as well as to their different regularization properties. [ABSTRACT FROM AUTHOR]

Copyright of Ordnance Industry Automation is the property of Editorial Board for Ordnance Industry Automation and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

The paper considers a general concept of polynomial dichotomy which includes as particular cases some well-known dichotomy concepts. The main objective is to obtain some characterizations of the nonuniform polynomial dichotomy behavior with respect to a family of norms compatible with the projection families. [ABSTRACT FROM AUTHOR]

The paper considers the general concept of uniform exponential splitting as a generalization of uniform exponential dichotomy property for evolution operators in Banach spaces.

A numerical method based on pseudospectral collocation is proposed to approximate the eigenvalues of evolution operators for linear renewal equations, which are retarded functional equations of Volterra type. Rigorous error and convergence analyses are provided, together with numerical tests. The outcome is an efficient and reliable tool which can be used, for instance, to study the local asymptotic stability of equilibria and periodic solutions of nonlinear autonomous renewal equations. Fundamental applications can be found in population dynamics, where renewal equations play a central role. [ABSTRACT FROM AUTHOR]

The paper considers some concepts of (h, k)-dichotomy and (h, k)-trichotomy for noninvertible evolution operators in Ba- nach spaces. A characterization of the (h, k)-trichotomy of an evolution operator in terms of (h, k)-dichotomy for two associated evolution operators is given. As applications of this result, charac- terizations for nonuniform exponential trichotomy and nonuniform polynomial trichotomy are obtained.

We consider a nonlinear integro-differential equation for prion proliferation that includes prion polymerization, polymer splitting, and polymer joining. The equation can be written as a quasilinear Cauchy problem. For bounded reaction rates we prove global existence and uniqueness of classical solutions by means of evolution operator theory. We also prove global existence of weak solutions for unbounded reaction rates by a compactness argument. [ABSTRACT FROM AUTHOR]

A non-autonomous stochastic linear evolution equation in UMD Banach spaces of type 2 is considered. We construct unique strict solutions to the equation and show their maximal regularity. The abstract results are then applied to a stochastic partial differential equation. [ABSTRACT FROM AUTHOR]

We give necessary integral conditions and suffcient ones for the existence of a general concept of μ-dichotomy for evolution operators defined on the half-line which includes as particular cases the well-known concepts of nonuniform exponential dichotomy and nonuniform polynomial dichotomy, and also contains new situations. Additionally, we consider an adapted notion of Lyapunov function and use our results to obtain necessary and suffcient conditions for the existence of nonuniform μ-dichotomies using these Lyapunov functions. [ABSTRACT FROM AUTHOR]

We prove that a family of linear bounded evolution operators (G(t,s))t ≥ s ∈ I can be associated, in the space of vector-valued bounded and continuous functions, to a class of systems of elliptic operators A with unbounded coefficients defined in I × Rd (where I is a right-halfline or I =R) all having the same principal part. We establish some continuity and representation properties of (G(t,s))t = s ∈ I and a sufficient condition for the evolution operator to be compact in Cb(Rd;Rm). We prove also a uniform weighted gradient estimate and some of its more relevant consequence. [ABSTRACT FROM AUTHOR]

We study the existence of mild solution for impulsive evolution abstract differential equations with state-dependent delay. A concrete application to partial delayed differential equations is considered.

In this paper, we study the asymptotic profile of the solution for a σ-evolution equation with a time-dependent structural damping. We introduce a classification of the damping term, which clarifies whether the solution behaves like the solution to an anomalous diffusion problem. We call this damping effective, whereas we say that the damping is noneffective when the solution shows oscillations in its asymptotic profile that cannot be neglected. Our classification shows a completely new interplay between the strength of the damping and the long time behavior of its coefficient. Copyright © 2015 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this paper we derive asymptotic-in-time linear estimates in Hardy spaces $$H^p(\mathbb {R}^{n})$$ for the Cauchy problem for evolution operators with structural dissipation. The obtained estimates are a natural extension of the known $$L^p-L^q$$ estimates, $$1\le p\le q\le \infty $$ , for these models. Different, standard, tools to work in Hardy spaces, are used to derive optimal estimates. [ABSTRACT FROM AUTHOR]

The paper treats the general concept of uniform h-stability, as a generalization of uniform exponential stability for evolution operators in Banach spaces. The main aim is to give necessary and sufficient conditions of Datko-type and Barbashin-type for this property and also criterias for uniform h-stability using Lyapunov functions. As particular cases, we obtain the results for uniform exponential stability. [ABSTRACT FROM AUTHOR]

Copyright of Journal of Central China Normal University is the property of Huazhong Normal University and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Copyright of Journal of Central China Normal University is the property of Huazhong Normal University and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

We prove some uniform and pointwise gradient estimates for the Dirichlet and the Neumann evolution operators $G_{\mathcal {D}}(t,s)$ and $G_{\mathcal {N}}(t,s)$ associated with a class of nonautonomous elliptic operators ( t) with unbounded coefficients defined in I× $$\mathbb{R}_{+}$$ (where I is a right-halfline or I= ℝ). We also prove the existence and the uniqueness of a tight evolution system of measures $\left \{\mu _{t}^{\mathcal {N}}\right \}_{t \in I}$ associated with $G_{\mathcal {N}}(t,s)$, which turns out to be sub-invariant for $G_{\mathcal {D}}(t,s)$, and we study the asymptotic behaviour of the evolution operators $G_{\mathcal {D}}(t,s)$ and $G_{\mathcal {N}}(t,s)$ in the L-spaces related to the system $\left \{\mu _{t}^{\mathcal {N}}\right \}_{t \in I}$. [ABSTRACT FROM AUTHOR]

We compute the Gabor matrix for Schrödinger-type evolution operators. Precisely, we analyze the heat equation, already presented in Cordero et al. (Gabor representations of evolution operators, ), giving the exact expression of the Gabor matrix which leads to better numerical evaluations. Then, using asymptotic integration techniques, we obtain an upper bound for the Gabor matrix in one-dimension for the generalized heat equation, new in the literature. Using Maple software, we show numeric representations of the coefficients' decay. Finally, we show the super-exponential decay of the coefficients of the Gabor matrix for the harmonic repulsor, together with some numerical evaluations. This work is the natural prosecution of the ideas presented in Cordero et al. (Gabor representations of evolution operators, ) and Cordero et al. (Sparsity of Gabor representation of Schrödinger propagators, Appl Comput Harmonic Appl 26:357-370, ). [ABSTRACT FROM AUTHOR]

Under suitable conditions, we obtain some characterizations of supercontractivity, ultraboundedness and ultracontractivity of the evolution operator G(t, s) associated to a class of nonautonomous second order parabolic equations with unbounded coefficients defined in I x ℝd, where I is a right-halfline. For this purpose, we establish an Harnack type estimate for G(t, s) and a family of logarithmic Sobolev inequalities with respect to the unique tight evolution system of measures {µt : t ∈ I} associated to G(t, s). Sufficient conditions for the supercontractivity, ultraboundedness and ultracontractivity to hold are also provided. [ABSTRACT FROM AUTHOR]

We consider a class of nonautonomous second order parabolic equations with unbounded coefficients defined inI× ℝd, whereIis a right-halfline. We prove logarithmic Sobolev and Poincaré inequalities with respect to an associated evolution system of measures {μt:t ∈ I}, and we deduce hypercontractivity and asymptotic behavior results for the evolution operatorG(t,s). [ABSTRACT FROM AUTHOR]

We determine sufficient conditions for the occurrence of a pointwise gradient estimate for the evolution operators associated with nonautonomous second order parabolic operators with (possibly) unbounded coefficients. Moreover, we exhibit a class of operators which satisfy our conditions. [ABSTRACT FROM AUTHOR]

In this paper we discuss stability problems for a class of discrete-time evolution operators generated by linear positive operators acting on certain ordered Banach spaces. Our approach is based upon a new representation result that links a positive operator with the adjoint operator of its restriction to a Hilbert subspace formed by sequences of Hilbert–Schmidt operators. This class includes the evolution operators involved in stability and optimal control problems for linear discrete-time stochastic systems. The inclusion is strict because, following the results of Choi, we have proved that there are positive operators on spaces of linear, bounded and self-adjoint operators which have not the representation that characterize the completely positive operators. As applications, we introduce a new concept of weak-detectability for pairs of positive operators, which we use to derive sufficient conditions for the existence of global and stabilizing solutions for a class of generalized discrete-time Riccati equations. Finally, assuming weak-detectability conditions and using the method of Lyapunov equations we derive a new stability criterion for positive evolution operators. [ABSTRACT FROM AUTHOR]

In this paper we survey some recent results concerned with nonau-tonomous Kolmogorov elliptic operators. Particular attention is paid to the case of the nonautonomous Ornstein-Uhlenbeck operator [ABSTRACT FROM AUTHOR]

The exponential dichotomy for evolution equations was introduced by O. Perron in 1930 and has gained prominence since the appearance of monographs such as J.L. Massera and J.J. Schaffer (1966); J.L. Daleckii and M.G. Krein (1974); W.A. Coppel (1978); C. Chicone and Y. Latuchkin (1999); L. Barreira and C. Valls (2009). In this paper we study the exponential dichotomy concept introduced by Barreira and Valls. As particular cases we consider the concept of uniform exponential dichotomy and nonuniform exponential dichotomy and as a natural generalization we study the trichotomy concepts. [ABSTRACT FROM AUTHOR]

This paper deals with the approximation of the eigenvalues of evolution operators for linear retarded functional differential equations through the reduction to finite dimensional operators by a pseudospectral collocation. Fundamental applications such as determination of asymptotic stability of equilibria and periodic solutions of nonlinear autonomous retarded functional differential equations follow at once. Numerical tests are provided. [ABSTRACT FROM AUTHOR]

We consider the equations of Navier-Stokes modeling viscous fluid flow past a moving or rotating obstacle in $${\mathbb R^d}$$ subject to a prescribed velocity condition at infinity. In contrast to previously known results, where the prescribed velocity vector is assumed to be parallel to the axis of rotation, in this paper we are interested in a general outflow velocity. In order to use L-techniques we introduce a new coordinate system, in which we obtain a non-autonomous partial differential equation with an unbounded drift term. We prove that the linearized problem in $${\mathbb R^d}$$ is solved by an evolution system on $${L^p_{\sigma}(\mathbb R^d)}$$ for 1 < p < ∞. For this we use results about time-dependent Ornstein-Uhlenbeck operators. Finally, we prove, for p ≥ d and initial data $${u_0\in L^p_{\sigma}(\mathbb R^d)}$$, the existence of a unique mild solution to the full Navier-Stokes system. [ABSTRACT FROM AUTHOR]