Estimating the spectral radius of a real matrix by discrete Lyapunov equation

For any real matrix A, this paper is concerned with the estimation of the spectral radius of A. The relationship between the weighted norm and the discrete Lyapunov equation of the matrix A is obtained. On the basis of the relationship, an iterative algorithm is presented to obtain the spectral radius of A and to estimate the solution of the corresponding linear discrete system. Several numerical examples are given to show that the iterative algorithm is effective.     

Existence of Random Solutions of a General Class of Stochastic Functional Integral Equations

Abstract

In this paper, we establish the existence of solutions of a more general class of stochastic functional integral equations. The main tools here are the measure of noncompactness and the fixed point theorem of Darbo type. The results of this paper generalize the results of Rao–Tsokos [Rao, A.N.V.; Tsokos, C.P. A class of stochastic functional integral equations. Coll. Math. 1976, 35, 141–146.] and Szynal–Wedrychowicz [Szynal, D.; Wedrychowicz, S. On existence and an asymptotic behaviour of random solutions of a class of stochastic functional integral equations. Coll. Math. 1987, 51, 349–364.].     

Well-posedness and asymptotic stability results for a viscoelastic plate equation with a logarithmic nonlinearity

ABSTRACT

In this paper, we consider a viscoelastic plate equation with a velocity-dependent material density and a logarithmic nonlinearity. Using the Faedo-Galaerkin approximations and the multiplier method, we establish the existence of the solutions of the problem and we prove an explicit and general decay rate result. These results extend and improve many results in the literature.     

Blow-up of solutions to a nonlinear dispersive rod equation

In this paper, firstly we find the best constant for a convolution problem on the unit circle via a variational method. Then we apply the best constant on a nonlinear rod equation to give sufficient conditions on the initial data, which guarantee finite time singularity formation for the corresponding solutions. Mathematics Subject Classification (2000) 30C70, 37L05, 35Q58, 58E35     

A Note on the Solution of a Stochastic Partial Differential Equation

Abstract

A physical model is described which justifies the appearance of a stochastic term in the two-dimensional Navier–Stokes equations. In this model, a linear oppositional control term accrues as well. The resulting stochastic partial differential equation is shown to have a unique stationary solution.     

On the Toughness of a Graph

Abstract

In this paper, general results on the toughness of a graph are considered. Firstly the link between toughness and connectivity is explored and then results linking toughness and the parameters binding number and integrity are given. Further, the toughness of product graphs is discussed including general results for the lexicographic product. The paper concludes with some observations on toughness and hamiltoni-city.     

Estimation for Translation of a Process Driven by Fractional Brownian Motion

Abstract

We investigate the general problem of estimating the translation of a stochastic process governed by a stochastic differential equation driven by a fractional Brownian motion. The special case of the Ornstein-Uhlenbeck process is discussed in particular.     

On the Action of a Groupoid on a Manifold

In this paper,some properties of reduction for symplectic F-spaces are discussed.The properties of stable subgroups are discussed.We find that the symplectic action of a symplectic groupoid on a symplectic manifold can induce a symplectic map between reduced symplectic manifolds.This symplectic action can be characterized by the action of its induced symplectic groupoid on a symplectic manifold.Lastly,we shall discuss Poisson reduction and give a Poisson reduction theorem.     

The number of rooted Eulerian planar maps

In this paper we provide a solution of the functional equation unsolved in the paper, by the second author, "On functional equations arising from map enumerations" that appeared in Discrete Math, 123: 93-109 (1993). It is also the number of combinatorial distinct rooted general eulerian planar maps with the valency of root-vertex, the number of non-root vertices and non-root faces of the maps as three parameters. In particular, a result in the paper, by the same author, "On the number of eulerian planar map...     

Existence of Solutions of Nonlinear Neutral Stochastic Differential Inclusions in a Hilbert Space

Abstract

In this paper, the existence of mild solutions of a class of non-linear neutral stochastic differential inclusions in Hilbert space is studied. The results are obtained by using a new fixed point theorem for a condensing map due to Martelli. For an application of the result, the neutral stochastic reaction-diffusion inclusion is also discussed.     

On stochastic processes generated by quadric operators

This paper is a continuation of our earlier paper (J. Theoret. Prob.3, 51–70). The existence and uniqueness of solutions of equations for quadric stochastic processes will be studied in this paper.     

The Existence and Uniqueness of Solutions of a Class of Evolution Equations

In this paper, a class of evolution equations which have special boundary conditions is discussed. The spectrum properties of an operator which is associated with this class of equations arc provided, and the existence and uniqueness of solutions of this class of equations is proved by the theory of pertu rbation of C_0-semigroups. In the last part of this paper, a condition which ensures the solutions to be non-negative is given. Some results of the time-independent population systems in [1-5] are our examples.     

一类三阶非线性微分方程解的不稳定性*

文献[1]讨论了非线性缓变系统的渐近稳定性,文献[2]讨论了三阶变系数线性微分方程解的不稳定性。本文应用文献[1]、[2]的方法讨论一类三阶非线性微分方程解的不稳定性。     

Fuzzy linear matrix equation

The main aim of this paper is to discuss Fuzzy Linear Matrix Equations (shown as FLME) of the form AXB = C for finding its fuzzy solutions. In this paper, the parametric form of the fuzzy linear system is used. Necessary and sufficient conditions for the existence of the set of fuzzy solutions are derived, and a numerical procedure for calculating the solutions is designed.     

General decay of solutions to a viscoelastic wave equation with linear damping,nonlinear damping and source term

ABSTRACT

This paper is concerned with the decay property of a nonlinear viscoelastic wave equation with linear damping, nonlinear damping and source term. Under weaker assumption on the relaxation function, we establish a general decay result, which extends the result obtained in Messaoudi [Exponential decay of solutions of a nonlinearly damped wave equation. Nodea-Nonlinear Differ Equat Appl. 2005;12:391–399].     

Comparison of Green's functions for a family of boundary value problems for fractional difference equations

ABSTRACT

In this paper, we obtain sign conditions and comparison theorems for Green's functions of a family of boundary value problems for a Riemann-Liouville type delta fractional difference equation. Moreover, we show that as the length of the domain diverges to infinity, each Green's function converges to a uniquely defined Green's function of a singular boundary value problem.     

Determination of the Robin coefficient in a nonlinear boundary condition for a steady‐state problem

In this paper, a constant heat transfer coefficient present in a nonlinear Robin‐type boundary condition associated with an elliptic equation is reconstructed uniquely from a single boundary energy measurement. Two types of such boundary energy measurement are considered, and solvability theorems for the solution of the resulting nonlinear inverse problems are provided. Further, one‐dimensional numerical results are presented and discussed. Copyright © 2008 John Wiley & Sons, Ltd.     

Compactly Supported Distributional Solutions of Nonstationary Nonhomogeneous Refinement Equations

Let A be a matrix with the absolute values of all eigenvalues strictly larger than one, and let Z ₀ be a subset of Z such than n∈Z ₀ implies n + 1 ∈Z ₀. Denote the space of all compactly supported distributions by D′, and the usual convolution between two compactly supported distributions f and g by f*g. For any bounded sequence G _n and H _n , n∈Z ₀, in D′, define the corresponding nonstationary nonhomogeneous refinement equation Φ _n =H _n *Φ _n+1 (A·)+G _n for all n∈Z ₀ where Φ _n , n∈Z ₀, is in a bounded set of D′. The nonstationary nonhomogeneous refinement equation (*) arises in the construction of wavelets on bounded domain, multiwavelets, and of biorthogonal wavelets on nonuniform meshes. In this paper, we study the existence problem of compactly supported distributional solutions Φ _n , n∈Z ₀, of the equation (*). In fact, we reduce the existence problem to finding a bounded solution of the linear equations for all n∈Z ₀ where the matrices S _n and the vectors , n∈Z ₀, can be constructed explicitly from H _n and G _n respectively. The results above are still new even for stationary nonhomogeneous refinement equations. Received December 30, 1999, Accepted June 15, 2000     

A fast matrix–vector multiplication method for solving the radiosity equation

A “fast matrix–vector multiplication method” is proposed for iteratively solving discretizations of the radiosity equation (I — К)u = E. The method is illustrated by applying it to a discretization based on the centroid collocation method. A convergence analysis is given for this discretization, yielding a discretized linear system (I — K _n )u _n = E _n. The main contribution of the paper is the presentation of a fast method for evaluating multiplications K_n v, avoiding the need to evaluate K_n explicitly and using fewer than O(n ²) operations. A detailed numerical example concludes the paper, and it illustrates that there is a large speedup when compared to a direct approach to discretization and solution of the radiosity equation. The paper is restricted to the surface S being unoccluded, a restriction to be removed in a later paper. This revised version was published online in June 2006 with corrections to the Cover Date.     

Solutions to a Class of Differential—fuctional Equations

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