A mean ergodic theorem for resolvent operators

Let {R(t)} _t≥0 be a uniformly bounded strongly continuous resolvent operator for the Volterra equation of convolution typeu=g+k*Au, whereA is a closed and densely defined operator on a Banach spaceX andk is a scalar kernel. We show that whenX is reflexive and that the average given by {R(t)} _t≥0 andk converges on the closed subspace to a bounded projection. This work was partially supported by DICYT 92-33LY and FONDECYT 91-0471     

Invariant measures for the linear stochastic Cauchy problem and R-boundedness of the resolvent

We study the asymptotic behaviour of solutions of the stochastic abstract Cauchy problem $$ \left\{ {\begin{array}{*{20}l} {dU\left( t \right) = AU\left( t \right)dt + BdW_H \left( t \right),\quad t \geqslant 0,} \hfill\ {U\left( 0 \right) = 0,} \hfill\ \end{array}} \right. $$ where A is the generator of a C₀-semigroup on a Banach space E, W_H is a cylindrical Brownian motion over a separable Hilbert space H, and $$ B \in \user1{\mathscr L}\left( {H,E} \right) $$ is a bounded operator. Assuming the existence of a solution U, we prove that a unique invariant measure exists if the resolvent R(λ, A) is R-bounded in the right half-plane {Reλ > 0}, and that conversely the existence of an invariant measure implies the R-boundedness of R(λ, A)B in every half-plane properly contained in {Re λ > 0}. We study various abscissae related to the above problem and show, among other things, that the abscissa of R-boundedness of the resolvent of A coincides with the abscissa corresponding to the existence of invariant measures for all γ -radonifying operators B provided the latter abscissa is finite. For Hilbert spaces E this result reduces to the Gearhart-Herbst-Prüss theorem. Dedicated to Giuseppe Da Prato on the occasion of his 70th birthday     

Multiplicatively and non-symmetric multiplicatively norm-preserving maps

Let A and B be Banach function algebras on compact Hausdorff spaces X and Y and let ‖.‖ _X and ‖.‖ _Y denote the supremum norms on X and Y, respectively. We first establish a result concerning a surjective map T between particular subsets of the uniform closures of A and B, preserving multiplicatively the norm, i.e. ‖Tf Tg‖ _Y = ‖fg‖ _X , for certain elements f and g in the domain. Then we show that if α ∈ ℂ {0} and T: A → B is a surjective, not necessarily linear, map satisfying ‖fg + α‖ _X = ‖Tf Tg + α‖ _Y , f,g ∈ A, then T is injective and there exist a homeomorphism φ: c(B) → c(A) between the Choquet boundaries of B and A, an invertible element η ∈ B with η(Y) ⊆ {1, −1} and a clopen subset K of c(B) such that for each f ∈ A,

Exponential decay for distributed bilinear control systems with damping

Let Ω be a bounded open domain in ℝ ^N ,A an unbounded, selfadjoint, positive and coercive linear operator onL ² (Ω). We consider feedback stabilization for the distributed bilinear control systemy″(t)+Ay(t)+Dy′(t)+u(t)By(t)=0, whereD andB are linear bounded operators fromL ²(Ω) toL ²(Ω). Under suitable assumptions onB andD, a nonlinear feedback ensuring uniform exponential decay of solutions is given. Various applications to vibrating processes are presented.     

Global-in-time Uniform Convergence for Linear Hyperbolic-Parabolic Singular Perturbations

We consider the Cauchy problem εu^″ε ＋ δu′ε ＋ Auε = 0, uε（0） = uo, u′ε（0） = ul, where ε 〉 0, δ 〉 0, H is a Hilbert space, and A is a self-adjoint linear non-negative operator on H with dense domain D（A）. We study the convergence of （uε） to the solution of the limit problem ,δu＇ ＋ Au = 0, u（0） = u0. For initial data （u0, u1） ∈ D（A1/2）× H, we prove global-in-time convergence with respect to strong topologies. Moreover, we estimate the convergence rate in the case where （u0, u1）∈ D（A3/2） ∈ D（A1/2）, and we show that this regularity requirement is sharp for our estimates. We give also an upper bound for ｜u′ε（t）｜ which does not depend on ε.     

On the spectral representation of the elementary operator and spectral mapping theorems

LetX andY denote two complex Banach spaces and letB(Y, X) denote the algebra of all bounded linear operators fromY toX. ForA∈B(X) ⁿ ,B∈B(Y) ⁿ , the elementary operator acting onB(Y, X) is defined by . In this paper we obtain the formulae of the spectrum and the essential spectrum of Δ(A, B) by using spectral mapping theorems. Forn=1, we prove thatS _p (L _A ,R _B )=σ(A)×σ(B) and .     

The Cauchy problem for nonstrictly second order hyperbolic equations with nonregular coefficients

In this paper we consider the Cauchy problem for the equation , where the matrix {a _jk(x)} is non-negative, and the first derivatives of the coefficients have a singularity of orderq≥3 att=T>0; under these assumptions, the Cauchy problem is well-posed in all Gevrey classes of indexs<q/(q−1).     

Removability of isolated singularities for anisotropic elliptic equations with gradient absorption

We study the well-posedness of the third-order degenerate differential equation \(\left( {{P_3}} \right):\alpha {\left( {Mu} \right)^{\prime \prime \prime }}\left( t \right) + {\left( {Mu} \right)^{\prime \prime }}\left( t \right) = \beta Au\left( t \right) + f\left( t \right)\), (t ∈ [0, 2p]) with periodic boundary conditions \(Mu\left( 0 \right) = Mu\left( {2\pi } \right),\;Mu'\left( 0 \right) = Mu'\left( {2\pi } \right),\;Mu''\left( 0 \right) = Mu''\left( {2\pi } \right)\), in periodic Lebesgue–Bochner spaces L^p(T,X), periodic Besov spaces B_p,q^s(T,X) and periodic Triebel–Lizorkin spaces F_p,q^s(T,X), where A, B and M are closed linear operators on a Banach space X satisfying D(A) \( \cap \)D(B) ? D(M) and α, β, γ ∈ R. Using known operator-valued Fourier multiplier theorems, we completely characterize the well-posedness of (P₃) in the above three function spaces.     

Packing measure functions of subordinators sample paths

X(t) (t∉[0,∞)) is a subordinator with its upper index β less than one, g(u) is the index function ofX(t), andX(t), andX[0,1]={xϕR:X(t)=x} for sometϕ[0,1]{. If φ(s)(sϕ(0,1)) is a measure function andh , then

Mixed ratio limit theorems for Markov processes

LetP be a conservative and ergodic Markov operator onL ₁(X, Σ,m). We give a sufficient condition for the existence of a decompositionA _f ↑X such that for 0≦f, g ∈L _∞ (A _j ) and any two probability measuresμ andν weaker thanm , whereλ is theσ-finite invariant measure (which necessarily exists). Processes recurrent in the sense of Harris are shown to have this decomposition, and an analytic proof of the convergence of is deduced for such processes. This paper is a part of the author’s Ph.D. thesis prepared at the Hebrew University of Jerusalem under the direction of Professor S. R. Foguel, to whom the author is grateful for his helpful advice and kind encouragement.     

Some inequalities about mixed volumes

We prove inequalities about the quermassintegralsV _k (K) of a convex bodyK in ℝ ⁿ (here,V _k (K) is the mixed volumeV((K, k), (B _n ,n − k)) whereB _n is the Euclidean unit ball). (i) The inequality

Higher integrability for minimizers of variational integrals with nonstandard growth

We consider a (possibly) vector-valued function u: Ω→R ^N, Ω⊂R ⁿ, minimizing the integral , whereD _iu=∂u/∂x _i, or some more general functional retaining the same behaviour; we prove higher integrability forDu:D ₁u,…,D_n−1u∈L^q, under suitable assumptions ona _i(x).

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Sunto Consideriamo una funzione u: Ω→R ^N, Ω⊂R ⁿ che minimizzi l'integrale , doveD _iu=∂u/∂x_i, o un funzionale con un comportamento simile; sotto opportune ipotesi sua _i(x), dimostriamo la seguente maggiore integrabilità perDu:D ₁u,…,D_n−1uεL^q.

     

On the numerical range of a Volterra operator in Lp

LetA be the linear operator inL _p (0, 1), 1<p<∞,p≠2, defined by ,x∈L _p (0, 1),s∈[0,1]. We show that the real values of numbers in the numerical range ofA have maximum , whereq=p/(p−1). This amounts to an inequality between integrals, for which we determine the case of equality.     

Fourier and Hermite series estimates of regression functions

Summary In the paper we estimate a regressionm(x)=E {Y|X=x} from a sequence of independent observations (X ₁,Y ₁),…, (X _n, Y_n) of a pair (X, Y) of random variables. We examine an estimate of a type , whereN depends onn andϕ _N is Dirichlet kernel and the kernel associated with the hermite series. Assuming, that E|Y|<∞ and |Y|≦γ≦∞, we give condition for to converge tom(x) at almost allx, provided thatX has a density. if the regression hass derivatives, then converges tom(x) as rapidly asO(nC^{−(2s−1)/4s}) in probability andO(n ^{−(2s−1)/4s} logn) almost completely.     

On multivalued evolution equations in Hilbert spaces

The Cauchy problemdu/dt+Au+B(t,u)∋0,u(0)=u ₀ is studied in a separable Hilbert space setting, whenA is a multivalued maximal monotone operator, andB is a multivalued operator which is measurable with respect to the time variable and upper semi-continuous with respect to the space variable. Under some boundedness conditions onB, an existence theorem is proved, with the extra assumption, in the infinite dimensional case thatA is the subdifferential of a proper lower semi-continuous inf-compact convex function. A theorem of dependence upon the initial condition is also given.     

Viability for nonlinear multi-valued reaction–diffusion systems

In this paper we consider a nonlinear evolution reaction–diffusion system governed by multi-valued perturbations of m-dissipative operators, generators of nonlinear semigroups of contractions. Let X and Y be real Banach spaces, ${\mathcal{K}}In this paper we consider a nonlinear evolution reaction–diffusion system governed by multi-valued perturbations of m-dissipative operators, generators of nonlinear semigroups of contractions. Let X and Y be real Banach spaces, K{\mathcal{K}} be a nonempty and locally closed subset in \mathbbR ×X×Y, A:D(A) í X\rightsquigarrow X, B:D(B) í Y\rightsquigarrow Y{\mathbb{R} \times X\times Y,\, A:D(A)\subseteq X\rightsquigarrow X, B:D(B)\subseteq Y\rightsquigarrow Y} two m-dissipative operators, F:K ? X{F:\mathcal{K} \rightarrow X} a continuous function and G:K \rightsquigarrow Y{G:\mathcal{K} \rightsquigarrow Y} a nonempty, convex and closed valued, strongly-weakly upper semi-continuous (u.s.c.) multi-function. We prove a necessary and a sufficient condition in order that for each (t,x,h) ? K{(\tau,\xi,\eta)\in \mathcal{K}}, the next system

{ lc u￠(t) ? Au(t)+F(t,u(t),v(t))    t 3 tv￠(t) ? Bv(t)+G(t,u(t),v(t))    t 3 tu(t)=x,    v(t)=h, \left\{ \begin{array}{lc} u'(t)\in Au(t)+F(t,u(t),v(t))\quad t\geq\tau \\ v'(t)\in Bv(t)+G(t,u(t),v(t))\quad t\geq\tau \\ u(\tau)=\xi,\quad v(\tau)=\eta, \end{array} \right.      17. A Fredholm operator approach to Morita equivalence      Ruy Exel 《K-Theory》1993,7(3):285-308 GivenC*-algebrasA andB and an imprimitivityA-B-bimoduleX, we construct an explicit isomorphismX *:K i (A)K i (B), whereK i denotes the complexK-theory functors fori=0, 1. Our techniques do not require separability nor the existence of countable approximate identities. We thus extend to generalC*-algebras the result of Brown, Green, and Rieffel according to which, strongly Morita equivalentC*-algebras have isomorphicK-groups. The method employed includes a study of Fredholm operators on Hilbert modules.On leave from the University of São Paulo, Brazil.      18. Hypergraphs in which all disjoint pairs have distinct unions      Zoltán Füredi 《Combinatorica》1984,4(2-3):161-168 Let ℓ be a set-system ofr-element subsets on ann-element set,r≧3. It is proved that if |ℓ|>3.5 then ℓ contains four distinct membersA, B, C, D such thatA∪B=C∪D andA∩B=C∩D=0.      19. Stability for solutions to nonlinear nonautonomous evolution equations      Seyedeh Marzieh Ghavidel 《Journal of Evolution Equations》2014,14(3):671-680 In this paper, we find an estimate on d(u(t), K(t)), where u is a mild solution to the nonautonomous Cauchy problem \({\dot{u}(t) + A(t)u(t) \ni 0,\, t \geq s, u(s) = u_0}\) . Here, A(t) is a family of nonlinear multivalued, ω-accretive operators in a Banach space X, with D(A(t)) possibly depending on t, and K(t) a family of closed subsets in X.      20. Linear periodic control systems: Controllability with restrained controls   总被引：1，自引：0，他引：1   Nguyen Khoa Son  Nguyen Van Su 《Applied Mathematics and Optimization》1986,14(1):173-185 Consider the problem of null-controllability for a linear non-autonomous system of the form , whereX andU are Banach spaces,A(t) andB(t) are linear bounded operators for eacht 0, and is a subset containing 0 (but not necessarily as an interior point). Some necessary and sufficient conditions for local and global null-controllability are proved whenA(·) andB(·) are periodic continuous functions. The proofs of the main results are based on discretization and on consideration of the corresponding linear discrete-time systems.

A Fredholm operator approach to Morita equivalence

GivenC*-algebrasA andB and an imprimitivityA-B-bimoduleX, we construct an explicit isomorphismX _*:K _i (A)K _i (B), whereK _i denotes the complexK-theory functors fori=0, 1. Our techniques do not require separability nor the existence of countable approximate identities. We thus extend to generalC*-algebras the result of Brown, Green, and Rieffel according to which, strongly Morita equivalentC*-algebras have isomorphicK-groups. The method employed includes a study of Fredholm operators on Hilbert modules.On leave from the University of São Paulo, Brazil.     

Hypergraphs in which all disjoint pairs have distinct unions

Let ℓ be a set-system ofr-element subsets on ann-element set,r≧3. It is proved that if |ℓ|>3.5 then ℓ contains four distinct membersA, B, C, D such thatA∪B=C∪D andA∩B=C∩D=0.     

Stability for solutions to nonlinear nonautonomous evolution equations

In this paper, we find an estimate on d(u(t), K(t)), where u is a mild solution to the nonautonomous Cauchy problem \({\dot{u}(t) + A(t)u(t) \ni 0,\, t \geq s, u(s) = u_0}\) . Here, A(t) is a family of nonlinear multivalued, ω-accretive operators in a Banach space X, with D(A(t)) possibly depending on t, and K(t) a family of closed subsets in X.     

Linear periodic control systems: Controllability with restrained controls

Consider the problem of null-controllability for a linear non-autonomous system of the form , whereX andU are Banach spaces,A(t) andB(t) are linear bounded operators for eacht 0, and is a subset containing 0 (but not necessarily as an interior point). Some necessary and sufficient conditions for local and global null-controllability are proved whenA(·) andB(·) are periodic continuous functions. The proofs of the main results are based on discretization and on consideration of the corresponding linear discrete-time systems.