scholarly journals On Stability of Vector Nonlinear Integrodifferential Equations

2016 ◽  
Vol 2016 ◽  
pp. 1-5
Author(s):  
Michael Gil’
Keyword(s):  
Steady State ◽  
Euclidean Space ◽  
Exponential Stability ◽  
Bounded Domain ◽  
Integrodifferential Equations ◽  
Nonlinear Mapping ◽  
Norm Estimate ◽  
Real Euclidean Space

Let Ω be a bounded domain in a real Euclidean space. We consider the equation ∂u(t,x)/∂t=C(x)u(t,x)+∫ΩK(x,s)u(t,s)ds+[F(u)](t,x)  (t>0;  x∈Ω), where C(·) and K(·,·) are matrix-valued functions and F(·) is a nonlinear mapping. Conditions for the exponential stability of the steady state are established. Our approach is based on a norm estimate for operator commutators.

On Certain Functional Identities in EN

1971 ◽  
Vol 23 (2) ◽  
pp. 315-324 ◽  
Author(s):  
A. McD. Mercer
Keyword(s):  
Euclidean Space ◽  
Taylor Series ◽  
Dimensional Space ◽  
The Real ◽  
Higher Dimensional ◽  
Isolated Points ◽  
Functional Identities ◽  
Image Position ◽  
Special Case ◽  
Real Euclidean Space

1. If f is a real-valued function possessing a Taylor series convergent in (a — R, a + R), then it satisfies the following operational identity1.1in which D2 = d2/du2. Furthermore, when g is a solution of y″ + λ2y = 0 in (a – R, a + R), then g is such a function and (1.1) specializes to1.2In this note we generalize these results to the real Euclidean space EN, our conclusions being Theorems 1 and 2 below. Clearly, (1.2) is a special case of (1.1) but in higher-dimensional space it is of interest to allow g, now a solution of1.3to possess singularities at isolated points away from the origin. It is then necessary to consider not only a neighbourhood of the origin but annular regions also.

Inequalities and separation for Schrödinger type operators in L2(Rn)

1978 ◽  
Vol 79 (3-4) ◽  
pp. 257-265 ◽  
Author(s):  
W. N. Everitt ◽  
M. Giertz
Keyword(s):  
Euclidean Space ◽  
Differential Expression ◽  
Integral Inequalities ◽  
Laplacian Operator ◽  
Generalized Derivatives ◽  
Maximal Domain ◽  
Definition Of ◽  
Schrödinger Type Operators ◽  
Real Euclidean Space

SynopsisThe symmetric differential expression M determined by Mf = − Δf;+qf on G, where Δ is the Laplacian operator and G a region of n-dimensional real euclidean space Rn, is said to be separated if qfϵL2(G) for all f ϵ Dt,; here D1 ⊂ L2(G) is the maximal domain of definition of M determined in the sense of generalized derivatives. Conditions are given on the coefficient q to obtain separation and certain associated integral inequalities.

scholarly journals On spatial matchings: The first-in-first-match case

2021 ◽  
Vol 53 (3) ◽  
pp. 757-800
Author(s):  
Mayank Manjrekar
Keyword(s):  
Steady State ◽  
Euclidean Space ◽  
Compact Space ◽  
Arrival Time ◽  
Type Inequality ◽  
Stationary Regime ◽  
Constant Rate ◽  
The One ◽  
Opposite Color

AbstractWe describe a process where two types of particles, marked red and blue, arrive in a domain at a constant rate. When a new particle arrives into the domain, if there are particles of the opposite color present within a distance of 1 from the new particle, then, among these particles, it matches to the one with the earliest arrival time, and both particles are removed. Otherwise, the particle is simply added to the system. Additionally, particles may lose patience and depart at a constant rate. We study the existence of a stationary regime for this process, when the domain is either a compact space or a Euclidean space. In the compact setting, we give a product-form characterization of the stationary distribution, and then prove an FKG-type inequality that establishes certain clustering properties of the particles in the steady state.

Generalization of Schwarz-Pick Lemma to Invariant Volume

1969 ◽  
Vol 21 ◽  
pp. 669-674
Author(s):  
K. T. Hahn ◽  
Josephine Mitchell
Keyword(s):  
Euclidean Space ◽  
Explicit Form ◽  
Bounded Domain ◽  
Kähler Manifold ◽  
Siegel Domain ◽  
Homogeneous Domain ◽  
Kahler Manifold ◽  
Invariant Volume ◽  
Complex Euclidean Space ◽  
Bounded Homogeneous Domain

In this paper we give an extension of (6, Theorem 1), using a similar method of proof, to every homogeneous Siegel domain of second kind which can be mapped biholomorphically into a Kâhler manifold of a certain class (Theorem 1). Then by a well-known result of Vinberg, Gindikin, and Pjateckiï-Sapiro (10) that every bounded homogeneous domain D,contained in a complex euclidean space CN,can be mapped biholomorphically onto an affinely homogeneous Siegel domain of second kind, the theorem follows for D(Theorem 2). (6, Theorem 1) is a generalization of the Ahlfors version of the Schwarz-Pick lemma in C1(1) to invariant volume for a star-like homogeneous bounded domain in CN;see also (4). In § 3 we give the inequality for a special non-symmetric Siegel domain of second kind using an explicit form of TD(z, )due to Lu (7).

EXPONENTIAL STABILITY OF STATIONARY SOLUTIONS TO THE DISCRETE BOLTZMANN EQUATION IN A BOUNDED DOMAIN

1992 ◽  
Vol 02 (02) ◽  
pp. 239-248 ◽  
Author(s):  
SHUICHI KAWASHIMA
Keyword(s):  
Boltzmann Equation ◽  
Exponential Stability ◽  
Bounded Domain ◽  
Large Time ◽  
Stationary Solutions ◽  
Large Time Behavior ◽  
Time Behavior ◽  
Crucial Point ◽  
Boundary Estimates ◽  
Discrete Boltzmann Equation

Large-time behavior of solutions of the discrete Boltzmann equation in a bounded domain is studied. The boundary conditions considered are pure diffuse relection and general reflection. Under suitable assumptions it is proved that a unique solution exists globally in time and converges to the corresponding unique stationary solution exponentially as time goes to infinity. The crucial point of the proof is in the derivation of desired boundary estimates of the solution subordinate to the general reflection.

Existence and exponential stability for some stochastic neutral partial functional integrodifferential equations

2014 ◽  
Vol 22 (2) ◽  
Author(s):  
Mamadou Abdou Diop ◽  
Khalil Ezzinbi ◽  
Modou Lo
Keyword(s):  
Exponential Stability ◽  
Resolvent Operator ◽  
Mild Solutions ◽  
Linear Part ◽  
American Mathematical Society ◽  
Integrodifferential Equations ◽  
Mathematical Society ◽  
Mean Square ◽  
Nonlinear Part ◽  
Stability In Mean Square

Abstract.The aim of this work is to study the existence, uniqueness and exponential stability of mild solutions for some stochastic neutral partial functional integrodifferential equations. We suppose that the linear part has a resolvent operator in the sense given in Grimmer [Transactions of the American Mathematical Society 273 (1982), 333–349]. The nonlinear part is assumed to be continuous and lipschitzian with respect to the second argument. Firstly, we study the existence of mild solutions. Secondly we give some results on the exponential stability in mean square sense. An example is provided to illustrate the results of this work.

The convergence of a sum of independent random variables

1963 ◽  
Vol 59 (2) ◽  
pp. 411-416
Author(s):  
G. De Barra ◽  
N. B. Slater
Keyword(s):  
Distribution Function ◽  
Euclidean Space ◽  
Rate Of Convergence ◽  
Radial Distribution Function ◽  
Random Variables ◽  
Covariance Matrices ◽  
Independent Random Variables ◽  
Second Moments ◽  
Image Position ◽  
Real Euclidean Space

Let Xν, ν= l, 2, …, n be n independent random variables in k-dimensional (real) Euclidean space Rk, which have, for each ν, finite fourth moments β4ii = l,…, k. In the case when the Xν are identically distributed, have zero means, and unit covariance matrices, Esseen(1) has discussed the rate of convergence of the distribution of the sumsIf denotes the projection of on the ith coordinate axis, Esseen proves that ifand ψ(a) denotes the corresponding normal (radial) distribution function of the same first and second moments as μn(a), thenwhere and C is a constant depending only on k. (C, without a subscript, will denote everywhere a constant depending only on k.)

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