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This present paper is mainly devoted to investigate the property of spectral decomposition of neutral differential equations in infinite dimensional setting, that is the exponential dichotomy. In fact, we prove that the exponential dichotomy of the associated semigroup to such equations does not depend on that of their associated difference equations. Based on the regular linear systems and feedback theory, we introduce a new transformation of neutral-type equations which plays a key role in our investigation. [ABSTRACT FROM AUTHOR]

In this article we study exponential dichotomies for noninvertible linear difference equations in finite dimensions. After giving the definition, we study the extent to which the projection P (k) in a dichotomy is unique. For equations on Z it is unique but for equations on Z + only its range is unique and for Z − only its nullspace. Here we strengthen Kalkbrenner's results and give a complete characterization of all possible projections. Next we study the possibility of extending the dichotomy to a larger interval. We reproduce the results of Pötzsche but also show exactly when the original projection remains unchanged. Next we prove that the roughness theorem, well known for additive perturbations, holds for multiplicative perturbations also. The proof uses ideas of Zhou, Lu and Zhang. Finally, following Ducrot, Magal and Seydi, we mention that the results by Palmer on finite time conditions on dichotomy for the invertible case can be extended to the noninvertible case. [ABSTRACT FROM AUTHOR]

We propose a new approach toward the existence and uniqueness of periodic solutions to linear and semilinear evolution equations. Our approach is based on the connection of the conditional stability of evolution families (i.e., stability only in a subspace of the Banach space containing the initial data) with the choice of the initial data from which emanates the periodic solution. We also give applications to exponentially dichotomic evolution families as well as to nonautonomous damped wave equations. [ABSTRACT FROM AUTHOR]

We show that functional differential equations of mixed type (MFDEs) with infinite range discrete and/or continuous interactions admit exponential dichotomies, building on the Fredholm theory developed by Faye and Scheel for such systems. For the half line, we refine the earlier approach by Hupkes and Verduyn Lunel. For the full line, we construct these splittings by generalizing the finite-range results obtained by Mallet-Paret and Verduyn Lunel. The finite dimensional space that is 'missed' by these splittings can be characterized using the Hale inner product, but the resulting degeneracy issues raise subtle questions that are much harder to resolve than in the finite-range case. Indeed, there is no direct analogue for the standard 'atomicity' condition that is typically used to rule out degeneracies, since it explicitly references the smallest and largest shifts. We construct alternative criteria that exploit finer information on the structure of the MFDE. Our results are optimal when the coefficients are cyclic with respect to appropriate shift semigroups or when the standard positivity conditions typically associated to comparison principles are satisfied. We illustrate these results with explicit examples and counter-examples that involve the Nagumo equation. [ABSTRACT FROM AUTHOR]

For any sufficiently small perturbation of a tempered exponen- tial dichotomy, we obtain appropriate versions of the Grobman--Hartman theorem and of the Sternberg theorems both for finite and infinite regularity, thus providing, respectively, topological and smooth conjugacies. The constructions are heavily based on the existence of normal forms, with the resonances expressed in terms of the connected components of the nonuniform spectrum, which is a tempered version of the Sacker--Sell spectrum. In order to obtain specific tempered bounds for the derivatives up to a certain order, we first construct smooth stable and unstable invariant manifolds for any sufficiently small perturbation of a tempered exponential dichotomy. We also make several preparations of the dynamics so that the linear part has a block form, the nonlinear part has no terms up to a given order, and the stable and unstable manifolds coincide, respectively, with the stable and unstable spaces. The conjugacies are then constructed via the homotopy method. [ABSTRACT FROM AUTHOR]

We present necessary and sufficient conditions in the spirit of Rolewicz under which all Lyapunov exponents of a given linear cocycle are either positive or negative. As a consequence, we formulate new conditions for the existence of the so-called tempered exponential dichotomies. We consider cocycles over both maps and flows. [ABSTRACT FROM AUTHOR]

In this paper, we prove the existence and uniqueness of pseudo affine-periodic solutions for differential systems with finite or infinite delay via exponential dichotomy and some fixed point theorems. These solutions possess certain spatiotemporal structure and they might be periodic, rotating-periodic, or affine-periodic, even quasi-periodic. [ABSTRACT FROM AUTHOR]

We consider a nonautonomous system x ˙ = A (t) x + f (t , x , y) , y ˙ = g (t , y) and give conditions under which there is a transformation of the form H (t , x , y) = (x + h (t , x , y) , y) taking its solutions onto the solutions of the partially linearized system x ˙ = A (t) x , y ˙ = g (t , y). Shi and Xiong [28] proved a special case where g (t , y) was a linear function of y and x ˙ = A (t) x had an exponential dichotomy. Our assumptions on A and f are of the general form considered by Reinfelds and Steinberga [25] , which include many of the generalizations of Palmer's theorem proved by other authors. Inspired by the work of Shi and Xiong, we also prove Hölder continuity of H and its inverse in x and y. Again the proofs are given in the context of Reinfelds and Steinberga but we show what the results reduce to when x ˙ = A (t) x is assumed to have an exponential dichotomy. The paper is concluded with the discrete version of the results. [ABSTRACT FROM AUTHOR]

We prove dynamical dichotomy into scattering and blow-up (in a weak sense) for all radial solutions of the Zakharov system in the energy space of four spatial dimensions that have less energy than the ground state, which is written using the Aubin-Talenti function. The dichotomy is characterized by the critical mass of the wave component of the ground state. The result is similar to that by Kenig and Merle for the energy-critical nonlinear Schrödinger equation (NLS). Unlike NLS, however, the most difficult interaction in the proof stems from the free wave component. In order to control it, the main novel ingredient we develop in this paper is a uniform global Strichartz estimate for the linear Schrödinger equation with a potential of subcritical mass solving a wave equation. This estimate, as well as the proof, may be of independent interest. For the scattering proof, we follow the idea by Dodson and Murphy. [ABSTRACT FROM AUTHOR]

This article is devoted to the study of certain nonautonomous and nonlinear difference equations of higher order. Our main objective is to formulate sufficient conditions under which the class of difference equations we consider exhibits Hyers–Ulam stability. Our methods rely on the relationship between Hyers–Ulam stability and hyperbolicity for nonautonomous systems. [ABSTRACT FROM AUTHOR]

We obtain characterizations of nonuniform dichotomies, defined by general growth rates, based on admissibility conditions. Additionally, we use the obtained characterizations to derive robustness results for the considered dichotomies. As particular cases, we recover several results in the literature concerning nonuniform exponential dichotomies and nonuniform polynomial dichotomies as well as new results for nonuniform dichotomies with logarithmic growth. [ABSTRACT FROM AUTHOR]

Adsorption process is proposed to be determined by such two successively coupling steps as a transport of adsorbate molecules into system by free convection and a formation of collision-complexes by surface complexing, whose kinetic rate equation can be expressed by a product of exponential monotonous with logistic growth function and quite well experimentally validated by three different adsorption systems. [Display omitted] • Non-linearly coupling molecular transport with surface complexing step leads to a new adsorption kinetic equation. • This equation is suitable for three kinds of adsorption system in vacuum basket, dynamic basket and fixed bed. • Its kinetic parameters can be characterized by molecular motions. Inspired by reaction molecular dynamics together with chain reaction theory, physical adsorption process is proposed to be determined by two successively coupling steps, such as a transport of adsorbate molecules into adsorption system in a free convection velocity and a formation of collision-complex between surface and activated adsorbate molecule in a surface complexing velocity. This leads to establish a new kinetic equation by a product of exponential monotonic and logistic growth functions equal to adsorption extent ratio, which can be quite well validated experimentally by both uptake curves in vacuum or dynamic basket and breakthrough curves from fixed bed for adsorption of benzene vapor by micro-porous carbon. Molecular transport step is an ideal flow with a rate constant of the free convection velocity over the system size; whereas, surface complexing step depends not only on a rate constant of the surface complexing velocity over the apparent free path basically around carbon granular size, but also on its initial surface complex density distribution. The activated adsorbate growth probability is less than unity and equal to the ratio of surface complexing over free convection velocity, both of which are almost nearly proportional to the adsorbate concentration relative to adsorption capacity. [ABSTRACT FROM AUTHOR]

We introduce two new tools for studying CAT(0) spaces: curtains , versions of cubical hyperplanes; and the curtain model , a counterpart of the curve graph. These tools shed new light on CAT(0) spaces, allowing us to prove a dichotomy of a rank-rigidity flavour, establish Ivanov-style rigidity theorems for isometries of the curtain model, find isometry-invariant copies of its Gromov boundary in the visual boundary of the underlying CAT(0) space, and characterise rank-one isometries both in terms of their action on the curtain model and in terms of curtains. Finally, we show that the curtain model is universal for WPD actions over all groups acting properly on the CAT(0) space. [ABSTRACT FROM AUTHOR]

Mean hyperbolic systems permit non-hyperbolic behavior at certain moments during the evolution process T (t , s) , owing to non-fixed error hyperbolic degree ε (t , s). Despite the coexistence of compression and expansion behaviors in generalized stable and unstable subspaces, mean hyperbolic systems admit fixed average contraction and expansion rates measured at sufficiently long time. To discuss the relationship between admissibility and mean hyperbolicity for evolution equations, our main goal is to construct admissible function classes and invariant decomposition with generalized stable and unstable subspaces. As applications, the mean hyperbolicity of damped wave equations with variable coefficients and the roughness of mean hyperbolicity are presented. [ABSTRACT FROM AUTHOR]

Abstract In this paper, a delay Lasota–Wazewska system with feedback control on time scales is proposed. Firstly, by using some differential inequalities on time scales, sufficient conditions which ensure the permanence of the system are obtained. Secondly, by means of the fixed point theory and the exponential dichotomy of linear dynamic equations on time scales, some sufficient conditions for the existence of unique almost periodic solution are obtained. Moreover, exponential stability of the almost periodic positive solution is investigated by applying the Gronwall inequality. Finally, numeric simulations are carried out to show the feasibility of the main results.

In this paper, a delayed Nicholsonos blowflies model with a linear harvesting term is investigated. By transforming the model into an equivalent integral equation, and applying a fixed point theorem inc ones, we establish a sufficient condition which ensure the existence of positive almost periodic solutions for the Nicholsonos blowflies model. The results of this paper are completely new and complement those of the previous studies. The approach is new.

Under consideration is some class of systems of nonlinear differential equations. The exponentially dichotomous linear part of the systems is assumed to have periodic coefficients. Using the author's criterion of exponential dichotomy, we establish the conditions of existence of periodic solutions and prove stability of the solutions under small perturbations of coefficients in the linear part and nonlinear terms. [ABSTRACT FROM AUTHOR]

We study the problem of dichotomy and boundedness for impulsive dynamic equations on arbitrary closed subset of real numbers. The spectral decomposition theorem gives all our main results. The obtained results are fundamentally new, even for the classical case. [ABSTRACT FROM AUTHOR]

In this paper, we first give three definitions with the aid of the shift operators δ±, that is, almost periodicity in shifts δ±, almost periodic function in shifts δ± and ∆-almost periodic function in shifts δ±, then by using the theory of calculus on time scales and some mathematical methods, the existence and uniqueness theorem of solution of linear dynamic system on almost periodic time scale in shifts δ± is obtained. Finally, we applying the obtained results to study the existence and exponential stability of the almost periodic solution in shifts δ± of a class of Hopfield neural networks with delays, several examples and numerical simulations are given to illustrate and reinforce the main results. [ABSTRACT FROM AUTHOR]

The leading-order approximation to a Filippov system ƒ about a generic boundary equilibrium x* is a system F that is affine one side of the boundary and constant on the other side. We prove x* is exponentially stable for ƒ if and only if it is exponentially stable for F when the constant component of F is not tangent to the boundary. We then show exponential stability and asymptotic stability are in fact equivalent for F. We also show exponential stability is preserved under small perturbations to the pieces of F. Such results are well known for homogeneous systems. To prove the results here additional techniques are required because the two components of F have different degrees of homogeneity. The primary function of the results is to reduce the problem of the stability of x* from the general Filippov system ƒ to the simpler system F. Yet in general this problem remains difficult. We provide a four-dimensional example of F for which orbits appear to converge to x* in a chaotic fashion. By utilising the presence of both homogeneity and sliding motion the dynamics of F can in this case be reduced to the combination of a one-dimensional return map and a scalar function. [ABSTRACT FROM AUTHOR]

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In this note we formulate sufficient conditions under which a certain class of nonlinear and nonautonomous differential equations of second order is Hyers–Ulam stable. This class consists of equations obtained by perturbing Hill's equation of the form x ′ ′ = (λ 2 (t) - λ ′ (t)) x . [ABSTRACT FROM AUTHOR]

We show the existence of an exponential attractor for non-autono-mous dynamical system with bounded delay. We considered the case of strong dissipativity then prove that the result remains for the weak dissipativity. We conclude then the existence of the global attractor and ensure the boundedness of its fractal dimension. [ABSTRACT FROM AUTHOR]

Bilayers consisting of La0.67Ca0.33MnO3 (LCMO) and SrMnO3 (SMO) have been prepared by pulsed-laser deposition on SrTiO3 (001) substrates. Unconventional magnetic coupling was found after cooling in a small field. The LCMO/SMO bilayers exhibit an exchange bias field of 209 Oe, which vanishes as the temperature rises above 90 K. A small magnetization has been found above the Curie temperature of the pure LCMO thin films. Spin-cluster-like antiferromagnetic (AFM)/ferromagnetic (FM) clusters have been deduced to exist at the interface due to the competing types of magnetic order at the interface. The magnetic relaxation is found to follow a doubleexponential equation and a slow relaxation process is observed due to the strong exchange coupling between AFM/FM clusters and the LCMO layer. We speculate that the short-range high-temperature FM order of the Mn3+ and Mn4+ moments above the Curie temperature at the interface gives rise to the magnetic regions that pin the FM LCMO layer as the temperature decreases. [ABSTRACT FROM AUTHOR]

In this paper, we consider a class of quaternion-valued neural networks with mixed delays on time scales. By using general Lipschitz condition, contraction mapping principle and exponential dichotomy of linear dynamic equations, we prove the existence and uniqueness and the stability of almost periodic solutions of global exponential neural networks. Finally, one example is given to illustrate the efficiency of our theoretical results. [ABSTRACT FROM AUTHOR]

We characterize the existence of an exponential dichotomy for a nonautonomous linear delay equation via the hyperbolicity of an appropriate cocycle. An important advantage of this approach is that the base is compact under mild additional assumptions. Moreover, we give a few applications of the equivalence of the two notions of hyperbolicity. In particular, we consider the robustness and the admissibility of the equation, and we obtain stable and unstable invariant manifolds. [ABSTRACT FROM AUTHOR]

This paper addresses the distributed state estimation problem for a class of discrete nonlinear system over sensor networks subject to unknown correlated measurement noises. Firstly, under the condition of network connectivity, a novel communication protocol is developed to ensure every sensor node can gather the information distributed throughout the network within finite communication time. Then a fully distributed estimator is designed by periodically fusing the local information and neighbor's information according to the covariance intersection fusion strategy. Theoretically, it is proved that the distributed estimator in each sensor node is stable with the exponentially bounded estimation error in mean square. Finally, some numerous simulations are performed to illustrate the practical effectiveness and superiority of the proposed state estimator. [Display omitted] • Design a novel finite time communication protocol, based on which every node could collect all the information distributed throughout the network within finite communication time under network connectivity. • Develop a new distributed state estimator for a class of discrete nonlinear system with unknown correlated measurement noises. • Establish sufficient conditions for guaranteeing the exponential stability of the new estimator in mean square sense. [ABSTRACT FROM AUTHOR]

We investigate the Tukey order in the class of Fσ ideals of subsets of ω. We show that no nontrivial Fσ ideal is Tukey below a Gδ ideal of compact sets. We introduce the notions of flat ideals and gradually flat ideals. We prove a dichotomy theorem for flat ideals isolating gradual flatness as the side of the dichotomy that is structurally good. We give diverse characterizations of gradual flatness among flat ideals using Tukey reductions and games. For example, we show that gradually flat ideals are precisely those flat ideals that are Tukey below the ideal of density zero sets. [ABSTRACT FROM AUTHOR]

This article investigates quaternion-valued fuzzy cellular neural networks with delays. With the help of fixed point theory, exponential dichotomy of linear equations and related inequalities, some new sufficient conditions which guarantee the existence and global exponentially stability of pseudo almost periodic solutions to quaternion-valued fuzzy cellular neural networks with delays are established. Computer simulations are carried out to check the practicability of the obtained main results. So far, no researchers have investigated this topic. The derived results of this article are completely innovative and complement some earlier investigations to some degree. [ABSTRACT FROM AUTHOR]

We consider unsteady triaxial tension–compression of a moving parallelepiped filled with a Newtonian viscous fluid and changing its linear dimensions (with a constant volume) during motion. The statement of the linearized problem is given in terms of three-dimensional perturbations imposed on the main process. To study this problem, we apply the method of integral relations based on the use of variational inequalities for estimating quadratic functionals. These estimates lead to sufficient integral criteria for stability with respect to the energy measure under small perturbations—to criteria for Lyapunov stability, asymptotic stability, and exponential stability. We derive a system of linear inequalities including two characteristic Reynolds numbers under which the initial three-dimensional picture of perturbations is a priori known to be exponentially stable. [ABSTRACT FROM AUTHOR]

We explore cosmological perturbations in a modified Gauss–Bonnet f (G) gravity, using a 1 + 3 covariant formalism. In such a formalism, we define gradient variables to get perturbed linear evolution equations. We transform these linear evolution equations into ordinary differential equations using a spherical harmonic decomposition method. The obtained ordinary differential equations are time-dependent and then transformed into redshift-dependent. After these transformations, we analyze energy-density perturbations for two fluid systems, namely, for a Gauss–Bonnet field-dust system and for a Gauss–Bonnet field-radiation system for three different pedagogical f (G) models: trigonometric, exponential and logarithmic. For the Gauss–Bonnet field-dust system, energy-density perturbations decay with increase in redshift for all the three models. For the Gauss–Bonnet field-radiation system, the energy-density perturbations decay with increase in redshift for all of the three f (G) models for long wavelength modes whereas for short wavelength modes, the energy-density perturbations decay with increasing redshift for the logarithmic and exponential f (G) models and oscillate with decreasing amplitude for the trigonometric f (G) model. [ABSTRACT FROM AUTHOR]

The exponential stability of complex-valued neural networks with time-varying delays and reaction-diffusion terms was studied. Firstly, the addressed systems were separated into their real parts with the complex-valued activation functions assumed to be divided into the real parts and imaginary parts. Secondly, some sufficient conditions for ensuring the exponential stability of the equilibrium states of the systems were established based on the vector Lyapunov function method and the M-matrix theory. The obtained criteria have no free variables and reduced conservatism compared with the existing results. A numerical example proves the correctness of the obtained results. [ABSTRACT FROM AUTHOR]

This paper is concerned with the study on the existence of attractors for a nonlinear porous elastic system subjected to a delay-type damping in the volume fraction equation. The study will be performed, from the point of view of quasi-stability for infinite dimensional dynamical systems and from then on we will have the result of the existence of global and exponential attractors. [ABSTRACT FROM AUTHOR]

We study a linear cocycle over the irrational rotation of the circle . It is supposed that the cocycle is generated by a -map which depends on a small parameter and has the form of the Poincaré map corresponding to a singularly perturbed Hill equation with quasi-periodic potential. Under the assumption that the norm of the matrix is of order , where is a positive function, we examine the property of the cocycle to possess an exponential dichotomy (ED) with respect to the parameter . We show that in the limit the cocycle "typically" exhibits ED only if it is exponentially close to a constant cocycle. Conversely, if the cocycle is not close to a constant one, it does not possess ED, whereas the Lyapunov exponent is "typically" large. [ABSTRACT FROM AUTHOR]

We prove that any nonautonomous semilinear dynamics on a bounded interval with the property that the nonlinear part is Lipschitz with a sufficiently small Lipschitz constant exhibits Hyers–Ulam stability. [ABSTRACT FROM AUTHOR]

In this paper, we determine the long-time dynamical behaviour of a reaction-diffusion system with free boundaries, which models the spreading of an epidemic whose moving front is represented by the free boundaries. The system reduces to the epidemic model of Capasso and Maddalena [5] when the boundary is fixed, and it reduces to the model of Ahn et al. [1] if diffusion of the infective host population is ignored. We prove a spreading-vanishing dichotomy and determine exactly when each of the alternatives occurs. If the reproduction number R0 obtained from the corresponding ODE model is no larger than 1, then the epidemic modelled here will vanish, while if R0 > 1, then the epidemic may vanish or spread depending on its initial size, determined by the dichotomy criteria. Moreover, when spreading happens, we show that the expanding front of the epidemic has an asymptotic spreading speed, which is determined by an associated semi-wave problem. [ABSTRACT FROM AUTHOR]

In this article, we establish the existence of Lipschitz stable invariant manifolds for the semiflows generated by the delay differential equation $x'= L(t)x_t + f(t,x_t,\lambda)$ with impulses at times $\{\tau_i\}_{i=1}^\infty $, assuming that the perturbation $f(t,x_t,\lambda)$ as well as the impulses are small and the corresponding linear delay differential equation admits a nonuniform exponential dichotomy. We also show that the obtained manifolds are Lipschitz in the parameter $\lambda$.

Let U={U (m,n) : m,n ∈ Z+} n≥m≥0 be the q-periodic discrete evolution family of square size matrices of order m having complex scalars as entries generated by L(C^m-valued, q-periodic sequence of square size matrices (An)n∈Z+ where q≥2 is a natural number. Where the Poincare map U(q,0) is the generator of the discrete evolution family U. The main objective of this article to decompose C^m with the help of discrete evolution family. In fact we decompose Cm in two sub spaces X1 and X2 such that X1 is due to the stability of the discrete evolution family and the vectors of X1 will called stable vectors. While X2 is due to the un-stability of discrete evolution family, and their vectors will be called unstable vectors. More precisely we take the dichotomy of the discrete evolution family with the help of projection P on Cm and we discuss different results of the spaces X1 and X2 .

In this paper, we deal with a class of BAM neural networks with time-varying leakage delays. By applying the exponential dichotomy of linear differential equations, fixed point theorems and differential inequality techniques, we obtain some sufficient conditions which guarantee the existence and exponential stability of almost periodic solutions for such class of BAM neural networks. An example is provided to illustrate the effectiveness of the theoretical predictions. The results obtained in this paper are completely new and complement the previously known publications.

We present a theory of existence and uniqueness of μ-pseudo almost periodic solution for a delayed non-autonomous partial functional differential equation in the exponential dichotomic case, where the nonlinear operator F satisfies the ϕ-Lipschitz condition and ϕ belongs to some admissible spaces. Moreover, we prove the existence of an invariant stable manifold around the μ-pseudo almost periodic solution in that case. We give finally an application to illusrate our results. [ABSTRACT FROM AUTHOR]

A class of inertial neural networks (INNs) with reaction-diffusion terms and distributed delays is studied. The existence and uniqueness of the equilibrium point for the considered system is obtained by topological degree theory, and a sufficient condition is given to guarantee global exponential stability of the equilibrium point. Finally, an example is given to show the effectiveness of the results in this paper. [ABSTRACT FROM AUTHOR]

We define a new type of "shatter function" for set systems that satisfies a Sauer–Shelah type dichotomy, but whose polynomial-growth case is governed by Shelah's two-rank instead of VC dimension. We identify the least exponent bounding the rate of growth of the shatter function, the quantity analogous to VC density, with Shelah's $\omega $ -rank. [ABSTRACT FROM AUTHOR]

It was proved that both the normal hyperbolicity and invariant manifold for a uniformly hyperbolic compact invariant manifold and the invariant manifold for a uniformly hyperbolic noncompact invariant manifold are persistent under small perturbation. In this paper, we weaken the uniform normal hyperbolicity to the nonuniform one and prove that both the nonuniform normal hyperbolicity and invariant manifold for a nonuniformly eventually absolutely normally hyperbolic noncompact invariant manifold are persistent under small perturbation in Banach spaces. Unlike the uniform case, we need infinitely many families of norms to uniformize infinitely many exponential dichotomies correspondingly. We use the approximative exponential dichotomy and the pseudo orbit to overcome the difficulty. [ABSTRACT FROM AUTHOR]

It studied the existence of almost periodic solutions for delay singular systems with almost periodic coefficients in Banach spaces by using dichotomy and fixed point theorems, and obtained a theoremon the sufficient conditions for the existence of almost periodic solutions. [ABSTRACT FROM AUTHOR]

In the search of quantum potential models used in the theoretical treatment of diatomic molecules, some of them have been constructed by using standard hyperbolic functions as well as from the so-called q-deformed hyperbolic functions (sc q-dhf). To improve the scope of hyperbolic functions, in this work, a class of generalized q-deformed hyperbolic functions is presented. Specifically, such improvement contains the particular case sc q-dhf besides that the can be expressed in terms of standard hyperbolic functions. As a useful application of the proposal, new q-deformed exactly solvable exponential-type potentials to be used in quantum chemical calculations are given as examples. At this regard, we are considering multi-parameter exponential-type potentials which means that the single choice of each set of q-dependent parameters leads to a different particular case of q-deformed exponential potential totally characterized. That is, with the proposed approach it is not necessary to make use of specialized methods to solve the involved differential equation for each specific case of exponential q-deformed potentials. [ABSTRACT FROM AUTHOR]

This paper is concerned with the wellposedness of global solution and existence of global attractor to the nonlinear Timoshenko system subject to continuous variable time delay in the angular rotation of the beam filament. The waves are assumed to propagate under the same speed in the transversal and angular direction. A single mechanical damping is implemented to counter the destabilizing effect from the time delay term. By imposing appropriate assumptions on the damping term and sub-linear time delay term, we prove the existence of absorbing set and establish the quasi-stability of the gradient system generated from the solution to the system of equation. The quasi-stability property in turn implies the existence of finite dimensional global and exponential attractors that contain the unstable manifold formed from the set of equilibria. [ABSTRACT FROM AUTHOR]