Positive and Nontrivial Solutions for the Urysohn Integral Equation

We establish new criteria for the existence of either positive or nonzero solutions of the Urysohn integral equation. We also discuss the existence of an interval of positive eigenvalues and sufficient conditions for the existence of at least a positive eigenvalue with a nonzero or positive eigenfunction for the Urysohn integral operator. Among others, we employ techniques based on fixed point index theory for compact maps, which are new for this type of equation.     

氢氟烃/烷烃二元混合物的气液相平衡

整理了已公开发表的氢氟烃(HFC)/烷烃(HC)二元混合物气液相平衡实验数据,对实验数据进行了热力学一致性检验。采用PR方程加van der Waals混合规则描述该类混合物的气液相平衡性质,优化得到了11种混合物的二元交互作用系数,经比较表明采用优化结果能够高精度再现气液相平衡实验数据,比k_(ij)取0的计算结果有很大改善.     

A TVD Type Wavelet-Galerkin Method for Hamilton-Jacobi Equations

In this paper,we use Daubechies scaling functions as test functions for the Galerkin method,and discuss Wavelet-Galerkin solutions for the Hamilton-Jacobi equations.It can be proved that the schemesare TVD schemes.Numerical tests indicate that the schemes are suitable for the Hamilton-Jacobi equations.Furthermore,they have high-order accuracy in smooth regions and good resolution of singularities.     

Eigenvalue approach to study the effect of rotation and relaxation time in generalized magneto-thermo-viscoelastic medium in one dimension

The fundamental equations of the problems of generalized thermoelasticity with one relaxation parameter including heat sources in infinite rotating magneto-thermo-viscoelastic media have been derived in the form of a vector matrix differential equation in the Laplace transform domain for a one dimensional problem. These equations have been solved by the eigenvalue approach to determine deformations, stress, and temperature. The results have been compared to those available in the existing literature. The graphs have been drawn to show the effect of rotation in the medium.     

Jensen’s Inequality for Backward Stochastic Differential Equations

Abstract Under the Lipschitz assumption and square integrable assumption on g, the author proves that Jensen’s inequality holds for backward stochastic differential equations with generator g if and only if g is independent of y, g(t, 0) ≡ 0 and g is super homogeneous with respect to z. This result generalizes the known results on Jensen’s inequality for g- expectation in [4, 7–9]. *Project supported by the National Natural Science Foundation of China (No.10325101) and the Science Foundation of China University of Mining and Technology.     

On the Lp–Lq maximal regularity for Stokes equations with Robin boundary condition in a bounded domain

We obtain the L_p–L_q maximal regularity of the Stokes equations with Robin boundary condition in a bounded domain in ?ⁿ (n?2). The Robin condition consists of two conditions: v ? u=0 and αu+β(T(u, p)v – 〈T(u, p)v, v〉v)=h on the boundary of the domain with α, β?0 and α+β=1, where u and p denote a velocity vector and a pressure, T(u, p) the stress tensor for the Stokes flow and v the unit outer normal to the boundary of the domain. It presents the slip condition when β=1 and non‐slip one when α=1, respectively. The slip condition is appropriate for problems that involve free boundaries. Copyright © 2006 John Wiley & Sons, Ltd.     

AN ITERATIVE EQUATION ON THE UNIT CIRCLE

A functional equation of nonlinear iterates is discussed on the circle S1 for its continuous solutions and differentiable solutions. By lifting to R, the existence, uniqueness and stability of those solutions are obtained. Techniques of continuation are used to guarantee the preservation of continuity and differentiability in lifting.     

On the invariant measure for the quasi‐linear Lasota equation

The problem of the existence of the invariant measure is important considering its connections with chaotic behaviour. In the papers (Zesz. Nauk. Uniw. Jagiellońskiego, Pr. Mat. 1982; 23 :117–123; Ann. Pol. Math. 1983; XLI :129–137; J. Differential Equations 2004; 196 :448–465) the existence of invariant and ergodic measures according to the dynamical system generated by the Lasota equation was proved, i.e. the equation describing the dynamics and becoming different of the population of cells. In this paper, the existence of such measure for the quasi‐linear Lasota equation is proved. This measure is the carriage of the measure described by Dawidowicz (Zesz. Nauk. Uniw. Jagiellońskiego, Pr. Mat. 1982; 23 :117–123). Copyright © 2006 John Wiley & Sons, Ltd.     

The statement and numerical solution of an optimization problem in X-ray tomography

A nonclassical problem is considered for the transport equation with coefficients depending on the energy of radiation. The task is to find the discontinuity surfaces for the coefficients of the equation from measurements of the radiation flux leaving the medium. For this tomography problem, an optimization problem is stated and numerically analyzed. The latter consists in determining the radiation energy that ensures the best reconstruction of the unknown medium. A simplified optimization problem is solved analytically.     

BOUNDEDNESS AND CONVERGENCE FOR THE NON-LINARD TYPE DIFFERENTIAL EQUATION

In this article, the author studies the boundedness and convergence for the non-Lienard type differential equation (x|·)=a(y)-f(x) (y|·)=b(y)β(x)-g(x) e(t) where a(y),b(y),f(x),g(x),β(x) are real continuous functions in y∈R or x∈R,β(x)≥0 for all x and e(t) is a real continuous function on R = {t: t≥0} such that the equation has a unique solution for the initial value problem. The necessary and sufficient conditions are obtained and some of the results in the literatures are improved and extended.