Equicontinuous Semigroups Research Articles

C 0 -sequentially equicontinuous semigroups

We present and apply a theory of one-parameter C 0 -semigroups of linear operators in locally convex spaces. Replacing the notion of equicontinuity considered by the literature with the weaker notion of sequential equicontinuity, we prove the basic results of the classical theory of C 0 -equicontinuous semigroups: we show that the semigroup is uniquely identified by its generator and we provide a generation theorem in the spirit of the celebrated Hille–Yosida theorem. Then we particularize the theory in some functional spaces and identify two locally convex topologies that allow us to gather—under a unified framework—various notions of C 0 -semigroups introduced by some authors to deal with Markov transition semigroups. Finally, we apply the results to transition semigroups associated to stochastic differential equations (SDEs).

Strongly continuous and locally equi-continuous semigroups on locally convex spaces

We consider locally equi-continuous strongly continuous semigroups on locally convex spaces \((X,\tau )\) that are also equipped with a ‘suitable’ auxiliary norm. We introduce the set \(\mathcal {N}\) of \(\tau \)-continuous semi-norms that are bounded by the norm. If \((X,\tau )\) has the property that \(\mathcal {N}\) is closed under countable convex combinations, then a number of Banach space results can be generalised in a straightforward way. Importantly, we extend the Hille–Yosida theorem. We relate our results to those on bi-continuous semigroups and show that they can be applied to semigroups on \((C_b(E),\beta )\) and \((\mathcal {B}(\mathfrak {H}),\beta )\) for a Polish space \(E\) and a Hilbert space \(\mathfrak {H}\) and where \(\beta \) is their respective strict topology.

Monotone iterative method for abstract impulsive integro-differential equations with nonlocal conditions in Banach spaces

In this paper we use a monotone iterative technique in the presence of the lower and upper solutions to discuss the existence of mild solutions for a class of semilinear impulsive integro-differential evolution equations of Volterra type with nonlocal conditions in a Banach space E $$\left\{ \begin{gathered} u'(t) + Au(t) = f(t,u(t),Gu(t)) t \in J,t \ne t_k , \hfill \\ \Delta _{\left. u \right|_{t = t_k } } = u\left( {t_k^ + } \right) - u\left( {t_k^ - } \right) = I_k \left( {u\left( {t_k } \right)} \right), k = 1,2, \ldots ,m, \hfill \\ u(0) = g(u) + x_0 , \hfill \end{gathered} \right.$$ where A: D(A) ⊂ E → E is a closed linear operator and −A generates a strongly continuous semigroup T(t) (t ⩾ 0) on E, f ∈ C(J × E × E, E), J = [0, a], 0 < t 1 < t 2 < ... < t m < a, I k ∈ C(E, E), k = 1, 2, ..., m, and g constitutes a nonlocal condition. Under suitable monotonicity conditions and noncompactness measure conditions, we obtain the existence of the extremal mild solutions between the lower and upper solutions assuming that −A generates a compact semigroup, a strongly continuous semigroup or an equicontinuous semigroup. The results improve and extend some relevant results in ordinary differential equations and partial differential equations. Some concrete applications to partial differential equations are considered.

Universal extrapolation spaces for $$\hbox {C}_{0}$$ C 0 -semigroups

The classical theory of Sobolev towers allows for the construction of an infinite ascending chain of extrapolation spaces and an infinite descending chain of interpolation spaces associated with a given $C_0$-semigroup on a Banach space. In this note we first generalize the latter to the case of a strongly continuous and exponentially equicontinuous semigroup on a complete locally convex space. As a new concept - even for $C_0$-semigroups on Banach spaces - we then define a universal extrapolation space as the completion of the inductive limit of the ascending chain. Under mild assumptions we show that the semigroup extends to this space and that it is generated by an automorphism of the latter. Dually, we define a universal interpolation space as the projective limit of the descending chain. We show that the restriction of the initial semigroup to this space is again a semigroup and always has an automorphism as generator.

Positive Solutions for the Initial Value Problems of Fractional Evolution Equation

This paper discusses the existence of positive solutions for the initial value problem of fractional evolution equation with noncompact semigroup , ; in a Banach space , where denotes the Caputo fractional derivative of order , is a closed linear operator, generates an equicontinuous semigroup, and is continuous. In the case where satisfies a weaker measure of noncompactness condition and a weaker boundedness condition, the existence results of positive and saturated mild solutions are obtained. Particularly, an existence result without using measure of noncompactness condition is presented in ordered and weakly sequentially complete Banach spaces. These results are very convenient for application. As an example, we study the partial differential equation of parabolic type of fractional order.

THE SOLUTION OF 3D-PHOTON TRANSPORT PROBLEM IN INTERSTELLAR CLOUD

In this paper we study a problem for photon transport in a host medium (e.g. an interstellar cloud where a localized source is present), that occupies a compact convex region $V$ in $R^{3}$. We find the generalized solution of the photon transport problem by means of the theory of equicontinuous semigroup of bounded linear operators on a sequentially complete locally convex topological vector space.

Adjoint bi-continuous semigroups and semigroups on the space of measures

For a given bi-continuous semigroup (T(t)) t⩾0 on a Banach space X we define its adjoint on an appropriate closed subspace X° of the norm dual X′. Under some abstract conditions this adjoint semigroup is again bi-continuous with respect to the weak topology σ(X°,X). We give the following application: For Ω a Polish space we consider operator semigroups on the space Cb(Ω) of bounded, continuous functions (endowed with the compact-open topology) and on the space M(Ω) of bounded Baire measures (endowed with the weak*-topology). We show that bi-continuous semigroups on M(Ω) are precisely those that are adjoints of bi-continuous semigroups on Cb(Ω). We also prove that the class of bi-continuous semigroups on Cb(ω) with respect to the compact-open topology coincides with the class of equicontinuous semigroups with respect to the strict topology. In general, if is not a Polish space this is not the case.

On a class of continuous fragmentation equations with singular initial conditions

We consider a class of fragmentation equations in which the distribution of daughter particles formed when a parent particle fragments is governed by a homogeneous function. A systematic procedure is presented for constructing a space of distributions in which initial-value problems involving singular initial conditions can be analysed. This procedure makes use of results on sun dual semigroups and equicontinuous semigroups on locally convex spaces. Explicit solutions are obtained for the case when the fragmentation processes are governed by power-law kernels and have monodisperse initial conditions modelled by Dirac delta distributions. Copyright © 2011 John Wiley & Sons, Ltd.

Fragmentation arising from a distributional initial condition

AbstractA standard model for pure fragmentation is subjected to an initial condition of Dirac‐delta type. Results for a corresponding abstract Cauchy problem are derived via the theory of equicontinuous semigroups of operators on locally convex spaces. An explicit solution is obtained for the case of a power‐law kernel. Rigorous justification is thereby provided for results obtained more formally by Ziff and McGrady. Copyright © 2010 John Wiley &amp; Sons, Ltd.

MEAN ERGODIC THEOREMS FOR ALMOST PERIODIC SEMIGROUPS

We show mean ergodic theorems for vector-valued weakly almost periodic functions (in the sense of Eberlein) defined on a semigroup which take values in a locally convex topological vector space. Next, motivated by Fréchet [10], we study the relationship between almost periodicity of semigroups of mappings and their equicontinuity, and also prove mean ergodic theorems for equicontinuous semigroups.

MEAN ERGODIC THEOREMS FOR ALMOST PERIODIC SEMIGROUPS

We show mean ergodic theorems for vector-valued weakly almost periodic functions (in the sense of Eberlein) defined on a semigroup which take values in a locally convex topological vector space. Next, motivated by Fr´echet [10], we study the relationship between almost periodicity of semigroups of mappings and their equicontinuity, and also prove mean ergodic theorems for equicontinuous semigroups.

Invariant measures for equicontinuous semigroups of continuous transformations of a compact Hausdorff space

Equicontinuous semigroups of transformations of a compact Hausdorff space and their sets of all invariant (Borel, regular and probabilistic) measures are studied. Conditions equivalent to the existence of at least one invariant measure are given. The (algebraic and topological) structure of the set of invariant measures is researched.

DIAGONALIZATION OF THE LÉVY LAPLACIAN AND RELATED STABLE PROCESSES

Eigenfunctions of the Lévy Laplacian with an arbitrary real number as an eigenvalue are constructed by means of a coordinate change of white noise distributions. The Lévy Laplacian is diagonalized on the direct integral Hilbert space of such eigenfunctions and the corresponding equi-continuous semigroup is obtained. Moreover, an infinite dimensional stochastic process related to the Lévy Laplacian is constructed from a one-dimensional stable process.

COUNTEREXAMPLES IN ERGODIC THEORY OF EQUICONTINUOUS SEMIGROUPS OF OPERATORS

The paper gives counterexamples in abstract ergodic theory of an equicontinuous semigroup $\mathcal{S}$ of linear operators on a locally convex space $X$. In particular, it is shown that the orbit of an element $x\in X$ may contain a unique fixed point of $\cal{S}$ without $x$ being necessarily ergodic.

COUNTEREXAMPLES IN ERGODIC THEORY OF EQUICONTINUOUS SEMIGROUPS OF OPERATORS

The paper gives counterexamples in abstract ergodic theory of an equicontinuous semigroup $\mathcal{S}$ of linear operators on a locally convex space $X$. In particular, it is shown that the orbit of an element $x\in X$ may contain a unique fixed point of $\cal{S}$ without $x$ being necessarily ergodic.

Fractional powers of generators of equicontinuous semigroups and fractional derivatives

AbstractWe analyze fractional powersHα, α &gt; 0, of the generatorsHof uniformly bounded locally equicontinuous semigroupsS. TheHαare defined as the αth derivative δαof the Dirac measure δ evaluated onS. We demonstrate that theHαare closed operators with the natural properties of fractional powers, for example,HαHβ=Hα+βfor α, β &gt; 0, and (Hα)β=Hαβfor 1 &gt; α &gt; 0 and β &gt; 0. We establish thatHα can be evaluated by the Balakrishnan-Lions-Peetre algorithmwheremis an integer larger than α,Cα, mis a suitable constant, and the limit exists in the appropriate topology if, and only if,x ∈ D(Hα). Finally we prove thatH∈is the fractional derivation ofSin the sensewhere the limit again exists if, and only if,x∈ D(Hα).

Tensor products and almost periodicity

Let E and F be locally convex spaces and G their completed e-tensor product. It is shown that if S and T are weakly almost periodic equicontinuous semigroups of operators on E and F respectively, then, under mild restrictions on E or F, SoT is a weakly almost periodic equicontinuous semigroup of operators on G, and the almost periodic and flight vector subspaces of G are related in a natural way to the corresponding subspaces of E and Fvia the e-tensor product. Furthermore, if Eand Fboth decompose into a direct sum of these subspaces then so does G. 1. Weakly almost periodic semigroups. Let E be a locally convex (Hausdorff) linear topological space with topological dual E', and let L(E) denote the space of continuous linear operators on E. A semigroup of operators on E is a subset S of L(E) containing the identity operator and closed under composition. A vector x E E is said to be weakly (strongly) almost periodic under S if its orbit Sx= {ux: u E S} is relatively compact in the weak (strong) topology of E. The set of all weakly (strongly) almost periodic vectors in E shall be denoted by W(E, S) (A(E, S)). Occasionally we shall suppress the symbols E or S from the notation if they are understood from context. If W(E, S)=E (A(E, S)=E) we say that S is weakly (strongly) almost periodic. It is easily seen that multiplication in a semigroup of operators S on a locally convex space E is separately continuous with respect to the weak or strong operator topologies on L(E), that is to say S is a topological semigroup. Moreover if S is equicontinuous then multiplication is actually jointly continuous in the strong operator topology. The following lemma is at the heart of the theory of weak almost periodicity. A proof can be found in [1]. LEMMA 1.1. Let S be a weakly almost periodic equicontinuous semigroup of operators on a locally convex space E, and let 3 denote the closure Received by the editors March 26, 1973 and, in revised form June 21, 1973. AMS (MOS) subject classiflcations (1970). Primary 47D05, 46M05; Secondary 43A60, 22A15.

Common Fixed-Points for Equicontinuous Semigroups of Mappings

Let S be a semigroup of equicontinuous self maps of X, a compact Hausdorff space. It is shown that if S is left reversible (that is every pair of right ideals of S has nonempty intersection), then there is a compact group G of homeomorphisms of a retract Y of X with the property that S has a common fixed-point in X if and only if G has a common fixed-point in Y. As an application, it is proved that if F is a family of continuous commuting self maps of the closed unit interval I with the property that for each $f \in F$, with one possible exception, the set of all iterates of f is equicontinuous, then I contains a common fixed-point of F.

Common fixed-points for equicontinuous semigroups of mappings

Let S be a semigroup of equicontinuous self maps of X, a compact Hausdorff space. It is shown that if S is left reversible (that is every pair of right ideals of S has nonempty intersection), then there is a compact group G of homeomorphisms of a retract Y of X with the property that S has a common fixed-point in X if and only if G has a common fixed-point in Y. As an application, it is proved that if F is a family of continuous commuting self maps of the closed unit interval I with the property that for each f ∈ F f \in F , with one possible exception, the set of all iterates of f is equicontinuous, then I contains a common fixed-point of F.

Generation of equicontinuous semigroups by hermitian and sectorial operators. II

Generation of equicontinuous semigroups by hermitian and sectorial operators. II