Normal periodic solutions for the fractional abstract Cauchy problem

We show that if A is a closed linear operator defined in a Banach space X and there exist $t_{0} \geq 0$ t 0 ≥ 0 and $M>0$ M > 0 such that $\{(im)^{\alpha }\}_{|m|> t_{0}} \subset \rho (A)$ { ( i m ) α } | m | > t 0 ⊂ ρ ( A ) , the resolvent set of A, and $$ \bigl\Vert (im)^{\alpha }\bigl(A+(im)^{\alpha }I \bigr)^{-1} \bigr\Vert \leq M \quad \text{ for all } \vert m \vert > t_{0}, m \in \mathbb{Z}, $$ ∥ ( i m ) α ( A + ( i m ) α I ) − 1 ∥ ≤ M for all | m | > t 0 , m ∈ Z , then, for each $\frac{1}{p}<\alpha \leq \frac{2}{p}$ 1 p < α ≤ 2 p and $1< p < 2$ 1 < p < 2 , the abstract Cauchy problem with periodic boundary conditions $$ \textstyle\begin{cases} _{GL}D^{\alpha }_{t} u(t) + Au(t) = f(t), & t \in (0,2\pi ); \\ u(0)=u(2\pi ), \end{cases} $$ { D t α G L u ( t ) + A u ( t ) = f ( t ) , t ∈ ( 0 , 2 π ) ; u ( 0 ) = u ( 2 π ) , where $_{GL}D^{\alpha }$ D α G L denotes the Grünwald–Letnikov derivative, admits a normal 2π-periodic solution for each $f\in L^{p}_{2\pi }(\mathbb{R}, X)$ f ∈ L 2 π p ( R , X ) that satisfies appropriate conditions. In particular, this happens if A is a sectorial operator with spectral angle $\phi _{A} \in (0, \alpha \pi /2)$ ϕ A ∈ ( 0 , α π / 2 ) and $\int _{0}^{2\pi } f(t)\,dt \in \operatorname{Ran}(A)$ ∫ 0 2 π f ( t ) d t ∈ Ran ( A ) .

An optimal linear filter for estimation of random functions in Hilbert space

Let \(\boldsymbol{f}\) be a square-integrable, zero-mean, random vector with observable realizations in a Hilbert space \(H\), and let \(\boldsymbol{g}\) be an associated square-integrable, zero-mean, random vector with realizations which are not observable in a Hilbert space \(K\). We seek an optimal filter in the form of a closed linear operator \(X\) acting on the observable realizations of a proximate vector \(\boldsymbol{f}_{\epsilon} \approx \boldsymbol{f}\) that provides the best estimate \(\widehat{\boldsymbol{g}}_{\epsilon} = X\! \boldsymbol{f}_{\epsilon}\) of the vector \(\boldsymbol{g}\). We assume the required covariance operators are known. The results are illustrated with a typical example. doi:10.1017/S1446181120000188

Stability analysis for discrete time abstract fractional differential equations

In this paper, we consider a discrete-time fractional model of abstract form involving the Riemann-Liouville-like difference operator. On account of the C 0-semigroups generated by a closed linear operator A and based on a distinguished class of sequences of operators, we show the existence of stable solutions for the nonlinear Cauchy problem by means of fixed point technique and the compact method. Moreover, we also establish the Ulam-Hyers-Rassias stability of the proposed problem. Two examples are presented to explain the main results.

AN OPTIMAL LINEAR FILTER FOR ESTIMATION OF RANDOM FUNCTIONS IN HILBERT SPACE

Let $\boldsymbol{f}$ be a square-integrable, zero-mean, random vector with observable realizations in a Hilbert space H, and let $\boldsymbol{g}$ be an associated square-integrable, zero-mean, random vector with realizations which are not observable in a Hilbert space K. We seek an optimal filter in the form of a closed linear operator X acting on the observable realizations of a proximate vector $\boldsymbol{f}_{\epsilon } \approx \boldsymbol{f}$ that provides the best estimate $\widehat{\boldsymbol{g}}_{\epsilon} = X \boldsymbol{f}_{\epsilon}$ of the vector $\boldsymbol{g}$ . We assume the required covariance operators are known. The results are illustrated with a typical example.

Pseudospectra in a non-Archimedean Banach space and essential pseudospectra in Eω

In this work, we introduce and study the pseudospectra and the essential pseudospectra of linear operators in a non-Archimedean Banach space and in the non-Archimedean Hilbert space E?, respectively. In particular, we characterize these pseudospectra. Furthermore, inspired by T. Diagana and F. Ramaroson [12], we establish a relationship between the essential pseudospectrum of a closed linear operator and the essential pseudospectrum of this closed linear operator perturbed by completely continuous operator in the non-Archimedean Hilbert space E?.

Identifications for General Degenerate Problems of Hyperbolic Type in Hilbert Spaces

In a Hilbert space X, we consider the abstract problem M∗ddt(My(t))=Ly(t)+f(t)z,0≤t≤τ,My(0)=My0, where L is a closed linear operator in X and M∈L(X) is not necessarily invertible, z∈X. Given the additional information Φ[My(t)]=g(t) wuth Φ∈X∗, g∈C1([0,τ];C). We are concerned with the determination of the conditions under which we can identify f∈C([0,τ];C) such that y be a strict solution to the abstract problem, i.e., My∈C1([0,τ];X), Ly∈C([0,τ];X). A similar problem is considered for general second order equations in time. Various examples of these general problems are given.

A simple proof of the closed graph theorem

<p>Assume that <em>A</em> is a closed linear operator defined on all of a Hilbert space <em>H</em>. Then, <em>A</em> is bounded. This classical theorem is proved on the basis of uniform boundedness principle. The proof is easily extended to Banach spaces.</p>

Banach’s Continuous Inverse Theorem and Closed Graph Theorem

Summary In this article we formalize one of the most important theorems of linear operator theory - the Closed Graph Theorem commonly used in a standard text book such as [10] in Chapter 24.3. It states that a surjective closed linear operator between Banach spaces is bounded.