On Compactness of Resolvent of a First Order Singular Differential Operator in Bounded Vector-Valued Function Space

2017 ◽  
pp. 295-301
Author(s):  
Myrzagali N. Ospanov
Keyword(s):  
Differential Operator ◽  
Function Space ◽  
Singular Differential Operator ◽  
First Order ◽  
Vector Valued Function ◽  
Vector Valued

scholarly journals The second dual of C0 (S, A)

1994 ◽  
Vol 56 (3) ◽  
pp. 303-313
Author(s):  
Stephen T. L. Choy ◽  
James C. S. Wong
Keyword(s):  
Function Space ◽  
Simple Proof ◽  
Representation Theorem ◽  
Generalized Functions ◽  
Arens Product ◽  
Nikodym Property ◽  
Second Dual ◽  
Vector Valued Function ◽  
Vector Valued

AbstractThe second dual of the vector-valued function space C0(S, A) is characterized in terms of generalized functions in the case where A* and A** have the Radon-Nikodým property. As an application we present a simple proof that C0 (S, A) is Arens regular if and only if A is Arens regular in this case. A representation theorem of the measure μh is given, where , h ∈ L∞ (|μ;|, A**) and μh is defined by the Arens product.

scholarly journals On some perturbations of a stable process and solutions to the Cauchy problem for a class of pseudo-differential equations

2015 ◽  
Vol 7 (1) ◽  
pp. 101-107 ◽  
Author(s):  
M.M. Osypchuk
Keyword(s):  
Cauchy Problem ◽  
Differential Operator ◽  
Euclidean Space ◽  
Fundamental Solution ◽  
Differential Equations ◽  
Stable Process ◽  
Pseudo Differential Operator ◽  
The Cauchy Problem ◽  
Vector Valued Function ◽  
Vector Valued

A fundamental solution for some class of pseudo-differential equations is constructed by the method based on the theory of perturbations. We consider a symmetric $\alpha$-stable process in multidimensional Euclidean space. Its generator $\mathbf{A}$ is a pseudo-differential operator whose symbol is given by $-c|\lambda|^\alpha$, were the constants $\alpha\in(1,2)$ and $c>0$ are fixed. The vector-valued operator $\mathbf{B}$ has the symbol $2ic|\lambda|^{\alpha-2}\lambda$. We construct a fundamental solution of the equation $u_t=(\mathbf{A}+(a(\cdot),\mathbf{B}))u$ with a continuous bounded vector-valued function $a$.

scholarly journals Regularity for commutators of the local multilinear fractional maximal operators

2020 ◽  
Vol 10 (1) ◽  
pp. 849-876
Author(s):  
Xiao Zhang ◽  
Feng Liu
Keyword(s):  
Sobolev Spaces ◽  
Maximal Operators ◽  
First Order ◽  
Vector Valued Function ◽  
Vector Valued

Abstract In this paper we introduce and study the commutators of the local multilinear fractional maximal operators and a vector-valued function b⃗ = (b1, …, bm). Under the condition that each bi belongs to the first order Sobolev spaces, the bounds for the above commutators are established on the first order Sobolev spaces.

Stability for time-dependent differential equations

Acta Numerica ◽  
1998 ◽  
Vol 7 ◽  
pp. 203-285 ◽  
Author(s):  
Heinz-Otto Kreiss ◽  
Jens Lorenz
Keyword(s):  
Differential Operator ◽  
Asymptotic Stability ◽  
Space Variable ◽  
Time Dependent ◽  
Stationary States ◽  
Asymptotically Stable ◽  
Recent Interest ◽  
Vector Valued Function ◽  
Vector Valued ◽  
Lyapunov Technique

In this paper we review results on asymptotic stability of stationary states of PDEs. After scaling, our normal form is ut = Pu + ε f(u, ux,…) + F(x, t), where the (vector-valued) function u(x, t) depends on the space variable x and time t. The differential operator P is linear, F(x, t) is a smooth forcing, which decays to zero for t → ∞, and εf(u, …) is a nonlinear perturbation. We will discuss conditions that ensure u → 0 for t → ∞ when |ε| is sufficiently small. If this holds, we call the problem asymptotically stable.While there are many approaches to show asymptotic stability, we mainly concentrate on the resolvent technique. However, comparisons with the Lyapunov technique will also be given. The emphasis on the resolvent technique is motivated by the recent interest in pseudospectra.

Generic rotation sets

2020 ◽  
pp. 1-13
Author(s):  
SEBASTIÁN PAVEZ-MOLINA
Keyword(s):  
Dynamical System ◽  
Convex Body ◽  
Probability Measures ◽  
Topological Dynamical System ◽  
Continuous Vector ◽  
Rotation Set ◽  
Vector Valued Function ◽  
Vector Valued ◽  
Uniquely Ergodic ◽  
Dense Set

Abstract Let $(X,T)$ be a topological dynamical system. Given a continuous vector-valued function $F \in C(X, \mathbb {R}^{d})$ called a potential, we define its rotation set $R(F)$ as the set of integrals of F with respect to all T-invariant probability measures, which is a convex body of $\mathbb {R}^{d}$ . In this paper we study the geometry of rotation sets. We prove that if T is a non-uniquely ergodic topological dynamical system with a dense set of periodic measures, then the map $R(\cdot )$ is open with respect to the uniform topologies. As a consequence, we obtain that the rotation set of a generic potential is strictly convex and has $C^{1}$ boundary. Furthermore, we prove that the map $R(\cdot )$ is surjective, extending a result of Kucherenko and Wolf.

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