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Let D 1 be the set of dendrites with the number of the accumulation points of branch points being finite, D ∈ D 1 and f be a continuous map from D to D. Denote by R ( f ) , Ω ( f ) and ω ( x , f ) the set of recurrent points of f, the set of non-wandering points of f and the set of ω-limit points of x under f, respectively. Write ω ( f ) = ∪ x ∈ D ω ( x , f ) and ω n + 1 ( f ) = ∪ x ∈ ω n ( f ) ω ( x , f ) and Ω n + 1 ( f ) = Ω ( f | Ω n ( f ) ) for any n ∈ N . In this paper, we show that Ω 4 ( f ) = R ( f ) ‾ and the depth of f is at most 4, and ω 4 ( f ) = ω 3 ( f ) . Furthermore, we show that there exist dendrites D 1 , D 2 ∈ D 1 and f 1 ∈ C 0 ( D 1 ) and f 2 ∈ C 0 ( D 2 ) such that Ω 3 ( f 1 ) ≠ R ( f 1 ) ‾ and ω 3 ( f 2 ) ≠ ω 2 ( f 2 ) .

In this paper, applying the theory of (mock) modular forms and Zwegers' results on Appell-Lerch sums, we establish two identities on the eighth-order mock theta function V 0 ( q ) . Using these identities, we prove some congruences for V 0 ( q ) .

In this work we study a dynamic contact problem between a thermoelastic mixture and a deformable obstacle. The classical normal compliance condition is used for modeling the contact. The variational formulation of this problem is written as a nonlinear coupled system of three parabolic variational equations. An existence and uniqueness result is proved using the Faedo-Galerkin method. Then, fully discrete approximations are introduced by using the finite element method and the implicit Euler scheme. A discrete stability property and a priori error estimates are proved, from which the decay of the discrete energy and the linear convergence of the algorithm are deduced. Finally, some numerical simulations are presented to show the convergence and the behavior of the solution.

This paper studies in a quantitatively precise manner the field enhancement due to presence of an emitter of the dipole type near the bow-tie structure of perfectly conducting inclusions in the two-dimensional space. We put special emphasis on field enhancement near vertices of the bow-tie structure, and derive upper and lower bounds of the gradient blow-up there. All three different kinds of symmetries are considered by varying locations and directions of the emitter, and a different estimate is derived for each case., Comment: 3 figures

The main purpose of this article is to generalize a characterization of Lipschitz functions in the context of metric-measure spaces. The results are established in the class of metric-measure spaces which satisfy a strong version of the doubling (Bishop-Gromov regularity) condition. Indeed, we establish a necessary and sufficient condition in order that any measurable function which satisfies an integrability condition to be essentially Lipschitzian.

Given a semigroup S and a uniquely divisible group H , for every h ∈ S we define the right difference operator Δ h on functions f : S ⟶ H as follows: Δ h f ( g ) = f ( g h ) − f ( g ) . Each of the following two conditions on a function f can be considered as a characterization of polynomial mappings on S : 1) Δ h 1 ⋯ Δ h m f = 0 for some m ∈ N and every h 1 , … , h m ∈ S . 2) there is m ∈ N such that Δ h m f = 0 for every h ∈ S ; We prove that if S possesses the property g S = S g for every g ∈ S , then two classes of functions, defined by the above conditions respectively, coincide.

This paper is concerned with the multiplicity of positive solutions for the critical Kirchhoff type problems involving indefinite weight functions (0.1) { − M ( ∫ Ω | ∇ u | 2 d x ) Δ u = Q λ ( x ) | u | q − 2 u + K ( x ) | u | 2 ⁎ − 2 u , x ∈ Ω , u = 0 , x ∈ ∂ Ω , where Ω is a smooth bounded domain in R N ( N ≥ 3 ) , 1 q 2 , M ( s ) = a + b s β with β > 0 , a > 0 and b > 0 , the weight functions Q λ and K are continuous and changing-sign. Using the Nehari manifold, fibering maps and Ljusternik-Schnirelmann category, we prove that at least two positive solutions for (0.1) exist provided that β = 1 and 2 ⁎ ≥ 4 . Furthermore, by the mountain pass theorem and Ekeland's variational principle, it is shown that (0.1) possesses at least three positive solutions whenever β > 2 N − 2 , including the case that β = 1 and 2 ⁎ 4 . Our results generalize some recent results in the literature.

Using matrix norm techniques, we give a unified convergence analysis of a general projection method for computing stationary density functions of Frobenius-Perron operators associated with piecewise C 2 and stretching transformations of the unit interval.

Let $S\in\mathcal{M}_d(\mathbb{C})^+$ be a positive semidefinite $d\times d$ complex matrix and let $\mathbf a=(a_i)_{i\in\mathbb{I}_k}\in \mathbb{R}_{>0}^k$, indexed by $\mathbb{I}_k=\{1,\ldots,k\}$, be a $k$-tuple of positive numbers. Let $\mathbb T_{d}(\mathbf a )$ denote the set of families $\mathcal G=\{g_i\}_{i\in\mathbb{I}_k}\in (\mathbb{C}^d)^k$ such that $\|g_i\|^2=a_i$, for $i\in\mathbb{I}_k$; thus, $\mathbb T_{d}(\mathbf a )$ is the product of spheres in $\mathbb{C}^d$ endowed with the product metric. For a strictly convex unitarily invariant norm $N$ in $\mathcal{M}_d(\mathbb{C})$, we consider the generalized frame operator distance function $\Theta_{( N \, , \, S\, , \, \mathbf a)}$ defined on $\mathbb T_{d}(\mathbf a )$, given by $$ \Theta_{( N \, , \, S\, , \, \mathbf a)}(\mathcal G) =N(S-S_{\mathcal G }) \quad \text{where} \quad S_{\mathcal G}=\sum_{i\in\mathbb{I}_k} g_i\,g_i^*\in\mathcal{M}_d(\mathbb{C})^+\,. $$ In this paper we determine the geometrical and spectral structure of local minimizers $\mathcal G_0\in\mathbb T_{d}(\mathbf a )$ of $\Theta_{( N \, , \, S\, , \, \mathbf a)}$. In particular, we show that local minimizers are global minimizers, and that these families do not depend on the particular choice of $N$., Comment: 24 pages. There exists text overlap with other arxiv manuscripts in the preliminary sections

In the current article, we extend Voiculescu's version of Berger-Shaw Theorem to properly infinite semifinite von Neumann algebras acting on separable Hilbert spaces.

The hypergeometric product formula describes the product of two hypergeometric functions with arguments xt and yt, respectively. The result is a power series in t whose coefficients are hypergeometric polynomials. Brychkov has extended the product formula such that in one argument the variable t can be squared. We obtain a further extension such that in both arguments the variable t can be raised to positive integer powers.

The semilinear space-time fractional Schrodinger equation is considered. First, we give the explicit form for the fundamental solutions by using the Fox H-functions in order to establish some L s decay estimates. After that, we give some space-time estimates for the mild solutions from which the local well-posedness is derived on some proper Banach space.

We consider an asymmetric lottery Blotto game with two agents and n items, where both agents wish to maximize their probability of winning a majority value of all n items. Duffy and Matros [2] showed that if there exists a Nash equilibrium, then the equilibrium is unique, and it is found in an explicit expression. They also provided sufficient conditions for the existence of a Nash equilibrium in the cases of n = 3 and n = 4 . In this paper, we prove that the lottery Blotto game always has a unique Nash equilibrium for any value of n.

Estimates for Kolmogorov and Gelfand n-widths of multiplier operators of multiple Walsh series are obtained. Upper and lower bounds are established for n-widths of general multiplier operators. These results are applied to get upper and lower bounds for n-widths of specific multiplier operators, which generate sets of finitely and infinitely differentiable functions in the dyadic sense. It is shown that these estimates have order sharp in various important cases.

Inverse nodal problems for Dirac operators on a finite interval [ 0 , π ] are studied. We prove that the specification of twin-dense nodal subsets on the interior interval [ ( 1 − e 0 ) π 2 , ( 1 + e 1 ) π 2 ] , 0 e 0 , e 1 ≤ 1 , uniquely determines the operator. We also discuss the case e 1 = 0 .

In this paper, we study a class of Finsler metric defined by a Riemannian metric, a 1-form and its norm: general ( α , β ) -metric. We classify locally dually flat general ( α , β ) -metric when the Riemannian metric α is strongly dually flat and give some new examples.

We give necessary and sufficient conditions for an operator A : X → Y on a Banach space having a shrinking FDD to factor through a Banach space Z such that the Szlenk index of Z is equal to the Szlenk index of A. We also prove that for every ordinal ξ ∈ ( 0 , ω 1 ) ∖ { ω η : η ω 1 a limit ordinal } , there exists a Banach space G ξ having a shrinking basis and Szlenk index ω ξ such that for any separable Banach space X and any operator A : X → Y having Szlenk index less than ω ξ , A factors through a subspace and through a quotient of G ξ , and if X has a shrinking FDD, A factors through G ξ .

Let K be a compact Hausdorff space and let H be a real or complex Hilbert space with dim ( H R ) ≥ 2 . We prove that the space C ( K , H ) , of all H-valued continuous functions on K, equipped with the supremum norm, satisfies the Mazur–Ulam property, that is, if Y is any real Banach space, every surjective isometry Δ from the unit sphere of C ( K , H ) onto the unit sphere of Y admits a unique extension to a surjective real linear isometry from C ( K , H ) onto Y. Our strategy relies on the structure of C ( K ) -module of C ( K , H ) and several results in JB⁎-triple theory. For this purpose we determine the facial structure of the closed unit ball of a real JB⁎-triple and its dual space.

A Banach space X is said to have the fixed point property if for every closed convex bounded subset C ⊂ X and for every T : C → C nonexpansive, there is a fixed point. A Banach space is said to have the dual-fixed point property if its dual space X ⁎ verifies the fixed point property. It is a fact that ( C 0 ( K ) , ‖ ⋅ ‖ ∞ ) fails the dual-fixed point property for every locally compact Hausdorff set K. In this article we study the fulfilment of the dual-fixed point property under equivalent norms. We prove that C 0 ( K ) can be renormed to have the dual-fixed point property if and only if K is a countable set. In this case we prove that this renorming can be achieved as close to the standard norm on C 0 ( K ) as we like for the Banach-Mazur distance. We further obtain the corresponding weak⁎-fixed-point-property stability coefficient regarding the duality σ ( C 0 ( K ) ⁎ , C 0 ( K ) ) . It will be displayed that the value of this stability coefficient strongly depends on the distribution of the accumulation points of K. Finally, we prove that the dual-fixed point property is completely ‘‘unstable” for the Banach-Mazur distance in most of nonreflexive Banach spaces.

The Nevanlinna theory is used to derive the general structure of transcendental meromorphic solutions for a wide class of autonomous nonlinear ordinary differential equations with two dominant monomials. An algebro-geometric method, which enables one to obtain these solutions explicitly, is described. New simply periodic solutions of the Lorenz system are obtained.

We reprove the existence of extremals for the critical Trudinger-Moser inequality on a smooth bounded domain of R 2 via the method of energy estimate, which was recently developed by Malchiodi-Martinazzi [11] , Mancini-Martinazzi [12] and Mancini-Thizy [13] . For this purpose, unlike [12] , [13] , it suffices to calculate local energy of maximizers for subcritical Trudinger-Moser functionals near the blow-up point.

In this paper, we generalize the concepts of isotropic vectors and complex symmetric operators from Hilbert spaces to Banach spaces via their dual spaces. With this extension we show the existence of isotropic vectors on Banach spaces whose dimension is at least two and the relation between the simplicity of an eigenvalue and the non-existence of its isotropic eigenvectors.

In many cases, the movement of species within a region depends on the availability of food and other resources necessary for their survival. Starvation-driven diffusion (SDD) is a non-uniform dispersal strategy that increases the motility of biological organisms in unfavorable environments, i.e., a species moves more frequently in search of food if resources are insufficient [4] . In this study, the proposed model represents the dispersion of an invasive species undergoing SDD, where the free boundary represents the expanding front. We observed that the spreading-vanishing dichotomy, which holds in the linear dispersal model, also holds in the model undergoing SDD. In the case that spreading of species occurs, it is shown that the asymptotic spreading speed of the moving front is uniquely determined in relation to the semi-wave speed. Moreover, our results are compared with the results of the linear dispersal model to investigate the advantages of this strategic dispersal with respect to survival in new environments, i.e., the conditions that allowed the species undergoing SDD to spread and the random diffusers to vanish are monitored.

In this paper, we investigate the asymptotic behavior for measure differential equations and dynamic equations on time scales via generalized ordinary differential equations (generalized ODEs, for short). At first, we establish new results that guarantee the existence of unbounded solutions for generalized ODEs, and after that, using the known correspondence between generalized ODEs and measure differential equations, also between measure differential equations and dynamic equations on time scales, we obtain similar results for these equations. Finally, we present an application to illustrate our main results.

The paper treats a nonlinear eigenvalue problem that describes propagation of a transverse-magnetic wave in a plane dielectric waveguide having perfectly conducted walls at both sides. The dielectric's permittivity is characterised by an arbitrary monotonically increasing self-action nonlinearity. The full set of guided modes is described by eigenvalues of the corresponding (nonlinear) Maxwell operator with appropriate boundary conditions. We give a comprehensive analysis of this problem and develop an original approach to study its solvability and properties of solutions. Several results about existence of the eigenvalues are proved, their distribution and asymptotic are found; zeros of the eigenfunctions and their location are also determined; criterion of periodicity for the eigenfunctions is found, comparison theorem is derived, etc. It is shown that unbounded nonlinearities lead to appearance of nonlinearised solutions. This results in the existence of a novel guided regime that cannot be described within the framework of perturbation of linear guided wave regimes.

We deal with the following nonlinear Schrodinger equation with magnetic field and critical growth: { ( e i ∇ − A ( x ) ) 2 u + V ( x ) u = f ( | u | 2 ) u + | u | 2 ⁎ − 2 u in R N , u ∈ H 1 ( R N , C ) , where e > 0 is a small parameter, N ≥ 3 , 2 ⁎ = 2 N N − 2 is the critical Sobolev exponent, A ∈ C 1 ( R N , R N ) is a magnetic vector potential, V : R N → R is a continuous positive potential having a local minimum and f : R → R is a superlinear continuous function with subcritical growth. Using penalization techniques and variational methods, we investigate the existence and concentration of nontrivial solutions for e > 0 small enough.

This paper is concerned with the regularity, forward-compactness and measurability of pullback random attractors of non-autonomous stochastic lattice systems with multiplicative white noise as well as random coefficients in weighted spaces. The existence, uniqueness and forward-compactness of pullback random attractors in weighted space l σ 2 are proved for the systems when the random coefficient in the front of the discrete Laplacian and the time-dependent functions satisfy certain conditions. In addition, this forward-compact ( l σ 2 , l σ 2 ) -attractor is proved to be a forward-compact bi-spatial ( l σ 2 , l σ 2 ∩ l σ p ) -attractor that is forward-compact and measurable in l σ 2 ∩ l p σ , and attracts all random subset of l σ 2 under the topology of l σ 2 ∩ l σ p . The difficulty of establishing the measurability of the attractors in l σ 2 ∩ l σ p is overcome by proving the identity of the attractors on two different universes.

We characterize the real valued functions f defined on perfect subsets P of R which admit n-times differentiable extensions F : R → R . In this characterization no continuity of F ( n ) is imposed. In particular, it generalizes Jarnik's Extension Theorem, according to which f admits differentiable extension F : R → R if, and only if, f is differentiable. The new characterization is also closely related to the Whitney's Extension Theorem, which characterizes partial maps f admitting n-times differentiable extensions F : R → R with continuous nth derivative F ( n ) . We also provide an elegant description of a linear extension operator T n : C ( P ) → C ( R ) such that T n ( f ) ∈ D n ( R ) for every D n ( R ) -extendable f ∈ C ( P ) and T n ( f ) ∈ C n ( R ) whenever f ∈ C ( P ) is C n ( R ) -extendable.

Let α ( x ) : x , ( a , b , c ) ∧ be a one-step backward extension sequence of Stampfli's subnormal completion, where 0 x ≤ a b c , and let W α ( x ) be the associated weighted shift. In this paper, we prove that W α ( x ) is subnormal if and only if W α ( x ) is polynomially hyponormal, which also is equivalent to that W α ( x ) is completely semi-weakly hyponormal.

In this paper, we analyse a Gardner equation with dual power law nonlinearity of any order which has been widely investigated due to its applications in quantum field theory, solid state, plasma and fluid physics. A group theoretic approach is used to perform a comprehensive and detailed analysis of the equation. We derive the symmetry generators of the equation in terms of its arbitrary parameters and used them to obtain symmetry reductions and exact solutions. Furthermore, the conservation laws of the equation are derived via the Noether approach after increasing the order and by the use of the multiplier method. The importance of these conservation laws in finding exact solutions is proved via double reduction theory. Most of these solutions are new and contain many known solutions as special cases. These solutions include important soliton solutions and nontrivial solutions in terms of special functions which are meromorphic in the entire complex plane. Since the Gardner equation can model a variety of wave phenomena in plasma, solid state and fluid physics, these solutions possess significant features in the non-linear mechanics aspects of the work. They can also be used as a basis for solving other related model problems and assessing numerical and approximate analytical methods for nonlinear equations describing solitons in wave mechanics.

We first introduce a new notation so called “asymptotic contractive process” on the time-dependent entire space, and then prove that such asymptotic contractive process is pullback asymptotically compact. Finally, we show that the existence of time-dependent attractor for the plate equation on H 2 ( R n ) × L 2 ( R n ) by using this new abstract result.

It is shown by construction that for some values of p, there exists a polynomial f ∈ l A p such that f fails to factorize as f = J G , where J is p-inner, and G is analytic and nonvanishing in the open unit disk D . In this situation, any p-inner function J vanishing at the roots of f in D , with multiplicities taken into account, must also vanish at some other point of D . It is shown that any such “extra zeros” of J must lie close to the boundary of the disk.

This paper is concerned with input-to-state stability of SFDSs. By using stochastic analysis techniques, Razumikhin techniques and vector Lyapunov function method, vector Razumikhin-type theorem has been established on input-to-state stability for SFDSs. Novel sufficient criteria on the pth moment exponential input-to-state stability are obtained by the established vector Razumikhin-type theorem. When input is zero, an improved criterion on exponential stability is obtained. Two examples are provided to demonstrate validity of the obtained results.

We consider the Dirichlet Laplacian in a straight planar strip perturbed by a bounded periodic symmetric operator. We prove the classical Bethe-Sommerfeld conjecture for this operator, namely, that this operator has finitely many gaps in its spectrum provided a certain special function written as a series satisfies some lower bound. We show that this is indeed the case if the ratio of the period and the width of strip is less than a certain explicit number, which is approximately equal to 0.10121. We also find explicitly the point in the spectrum, above which there is no internal gaps. We then study the case of a sufficiently small period and we prove that in such case the considered operator has no internal gaps in the spectrum. The conditions ensuring the absence are written as certain explicit inequalities.

In this paper, we construct compact Toeplitz operators and trace class Toeplitz operators with unbounded symbols on Hardy-Sobolev spaces. We also characterize the compactness of finite sums of products of two Toeplitz operators on these spaces. Then, we study the Fredholmness of the Toeplitz operators and calculate the essential norm of them. In addition, we consider the C ⁎ -algebra generated by Toeplitz operators.

A unital JB-algebra A is defined to be of rank zero if the set of invertible elements is dense in A . A non-unital JB-algebra A is said to be of rank zero if its unitization A ⊕ R 1 is so. We show that a unital JB-algebra A is of rank zero if and only if the set of elements with finite spectrum is dense in A if and only if every inner ideal of A admits an approximate identity (not necessarily increasing) consisting of projections. Moreover, we establish that zero rank is inherited by every closed ideal and every quotient algebra.

In this paper, we study the asymptotic behavior of the regular solution to a simplified Ericksen–Leslie model for the compressible nematic liquid crystal flow in a bounded smooth domain in R 2 as the Mach number tends to zero. The evolution system consists of the compressible Navier–Stokes equations coupled with the transported heat flow for the averaged molecular orientation. We suppose that the Navier–Stokes equations are characterized by a Navier's slip boundary condition, while the transported heat flow is subject to Neumann boundary condition. By deriving a differential inequality with certain decay property, the low Mach limit of the solutions is verified for all time, provided that the initial data are well-prepared.

The weak and strong ergodic properties of nonautonomous linear ordinary differential equations are considered. It is shown that if the coefficient matrix function is bounded, essentially nonnegative and uniformly irreducible, then the normalized positive solutions are asymptotically equivalent to the Perron vectors of the strongly positive transition matrix at infinity (weak ergodicity). If, in addition, the coefficient matrix function is uniformly continuous, then the convergence of the normalized positive solutions to the same strongly positive limiting vector (strong ergodicity) is equivalent to the convergence of the Perron vectors of the coefficient matrices.

A topological space X is called resolvable if it contains a dense subset with dense complement. Using only basic principles, we show that whenever the space X has a resolving subset that can be written as an at most countably infinite union of subsets, in such a way that a given vector lattice of (not necessarily bounded) continuous functions on X separates every point outside the resolving subset from each of its constituents, then the order continuous dual of this lattice is trivial. In order to apply this result in specific cases, we show that several spaces have resolving subsets that can be written as at most countably infinite unions of closed nowhere dense subsets. An appeal to the main result then yields, for example, that, under appropriate conditions, vector lattices of continuous functions on separable spaces, metric spaces, and topological vector spaces have trivial order continuous duals if they separate points and closed nowhere dense subsets. Our results in this direction extend known results in the literature. We also show that, under reasonably mild separation conditions, vector lattices of continuous functions on locally connected T 1 Baire spaces without isolated points have trivial order continuous duals. A discussion of the relation between our results and the non-existence of non-zero normal measures is included.

In this paper, we deal with the mean-square stability of nonlinear stochastic pantograph equations. An equivalent form of stochastic delay differential equations with constant delay is introduced by the transformation approach. An optimal stability condition is presented, since it is independent of the proportion and coincides with the necessary condition for linear pantograph equations with real coefficients.

Because of individuals' random walk, such as shopping, travel, work, etc., people have different behaviors and thus have different social contact patterns. Therefore, topology of human social contact networks is time-varying. In this paper, we investigate dynamic characteristics of an SIS network epidemic model with Markovian switching. An epidemic threshold is established for the extinction and permanence of the model, which is related to the steady-state distribution of the Markov chain. An interesting result is that when the epidemic is permanent in one network but extinct in another, under network switching mechanisms, it may be either permanent or extinct depending on the steady-state distribution of the Markov chain. This reveals the important role of the Markov chain in epidemic evolution. This work shows that the epidemic propagation in switched networks is quite different from that of static networks. In addition, based on Lyapunov function method, positive recurrence and ergodicity of stochastic spreading processes are also discussed. Finally, numerical simulations are carried out to illustrate our theoretical results.

We establish inequalities on vertical Littlewood--Paley square functions for heat flows in the weighted $L^2$ space over metric measure spaces satisfying the $\RCD^\ast(0,N)$ condition with $N\in [1,\infty)$ and the maximum volume growth assumption. In the noncompact setting, the later assumption can be removed by showing that the volume of the ball growths at least linearly. The estimates are sharp on the growth of the 2-heat weight and the 2-Muckenhoupt weight considered. The $p$-Muckenhoupt weight and the $p$-heat weight are also compared for all $p\in(1,\infty)$., All comments are welcome!

In this article, we have developed a new method regarding asymptotic behavior and stability for the solutions of first order linear impulsive neutral delay differential equations with constant coefficients and constant delays. The asymptotic behavior of the solutions and the stability of the zero solution were investigated using a real root suitable for the characteristic equation. Furthermore, we obtained similar results for impulsive delay differential equations. Finally, three examples are given for the stability of the zero solution of the impulsive neutral delay differential equation.

While there have been extensive studies regarding the theory of composition operators in standard Bergman spaces, there have not been many results pertaining to large Bergman spaces due to a lack of useful tools. In this paper, we give the characterizations of the compact differences of composition operators in Bergman spaces with the exponential type weight using a newly defined Riemannian distance. Furthermore, we give a sufficient condition for the question when two composition operators lie in the same component., This paper is the corrected version of the published paper

We obtain a Sobolev type embedding result for Besov spaces defined on a doubling measure metric space.

We study the size of sets defined by frequencies of digits by means of doubling measures on the unit interval [ 0 , 1 ] . In our results we prove that the set of Borel normal numbers is not fat for doubling measures on [ 0 , 1 ] , although it is of full Lebesgue measure.

It is an important result of Semrl which states that every 2-local automorphism of the full operator algebra over a separable Hilbert space is necessarily an automorphism. In this paper we strengthen that result quite substantially for *-automorphisms. Indeed, we show that one can compress the defining two equations of 2-local *-automorphisms into one single equation, hence weakening the requirement significantly, but still keeping essentially the conclusion that such maps are necessarily *-automorphisms.

In this paper we study classical solutions to the zero–flux attraction–repulsion chemotaxis–system ( ◇ ) { u t = Δ u − χ ∇ ⋅ ( u ∇ v ) + ξ ∇ ⋅ ( u ∇ w ) in Ω × ( 0 , t ⁎ ) , 0 = Δ v + α u − β v in Ω × ( 0 , t ⁎ ) , 0 = Δ w + γ u − δ w in Ω × ( 0 , t ⁎ ) , where Ω is a smooth and bounded domain of R 2 , t ⁎ is the blow–up time and α , β , γ , δ , χ , ξ are positive real numbers. From the literature it is known that under a proper interplay between the above parameters and suitable assumptions on the initial data u ( x , 0 ) = u 0 ∈ C 0 ( Ω ¯ ) , system (◇) has a unique classical solution which becomes unbounded as t ↗ t ⁎ . The main result of this investigation is to provide an explicit lower bound for t ⁎ estimated in terms of ∫ Ω u 0 2 d x and attained by means of well–established techniques based on ordinary differential inequalities.

This paper discusses how the oscillation of solutions changes by adding an impulsive term to a second-order linear differential equation having only a restoring force term but no damping term. Due to the effect of this impulsive term, the moving speed of a mass point of the equation of motion described by the second-order linear differential equation changes discontinuously. Oscillation theorems are given for this impulsive differential equation. As is well known, when the restoring force term is small, all nontrivial solutions of the equation with no impulsive effects do not oscillate. Even when the restoring force term is small, if the action of the impulsive term compensates, the oscillation criteria obtained are satisfied and all nontrivial solutions oscillate. Several examples are given to confirm this fact and some figures are shown to depict the solution curves of these examples. As shown by the obtained oscillation theorems, there are cases that the impulsive effect promotes the oscillation of solutions, but on the contrary, it may suppress the oscillation of solutions. It is shown by a simulation that there is a case where the impulsive effect suppress the oscillation of solutions. Finally, the obtained results are compared with the previous ones.

We consider two systems arising in a two-dimensional viscoelastic fluid-structure interaction, one of them coupled with a wave equation with interior damping of Kelvin–Voigt type. In the second model, the acoustic vibrations of the viscous fluid which fills the two-dimensional interior cavity are coupled with the mechanical vibrations of a one-dimensional thermoelastic beam. Therefore, the systems are related to the problem of the active control of noise in a cavity. Our main results are that the corresponding semigroups are analytic. In particular we show the exponential stability of the models.