Oscillation of Solutions of Linear Differential Equations

This paper is devoted to studying the growth problem, the zeros and fixed points distribution of the solutions of linear differential equations f″＋e^-zf′＋Q（z）f=F（z）,whereQ（z）≡h（z）e^cz and c∈R.     

Solutions of Indefinite Hammerstein Equations

Consider the following equation:For positive definite or quasi-positive definite kernel k(x,y),the theory of equa-tion(1)has much been developed.But up to now,little is known about the caseof indefinite kernels.In this paper we get several results of the latter type. Let G R~N have finite measure and f(x,u)satisfy Caratheodory condition,     

Unsteady Solutions of Euler Equations Generated by Steady Solutions

Invariant solutions of partial differential equations are found by solving a reduced system involving one independent variable less. When the solutions are invariant with respect to the so-called projective group, the reduced system is simply the steady version of the original system. This feature enables us to generate unsteady solutions when steady solutions are known. The knowledge of an optimal system of subalgebras of the principal Lie algebra admitted by a system of differential equations provides a method of classifying H-invariant solutions as well as constructing systematically some transformations (essentially different transformations) mapping the given system to a suitable form. Here the transformations allowing to reduce the steady two-dimensional Euler equations of gas dynamics to an equivalent autonomous form are classified by means of the program SymboLie, after that an optimal system of two-dimensional subalgebras of the principal Lie algebra has been calculated. Some steady solutions of two-dimensional Euler equations are determined, and used to build unsteady solutions.     

Properties of Solutions of BSDEs with Integrable Parameters

The main result of this study is to obtain,using the localization method in Briand et al.Levi,Fatou and Lebesgue type theorems for the solutions of certain one-dimensional backward stochastic differentialequation(BSDEs)with integrable parameters with respect to the terminal condition.     

All General Solutions of Post Equations

In a previous paper, we have described all reproductive general solutions of a Post equation, supposing that a general solution is known. In this paper we describe all general solutions of Post equation, supposing that a general solution of this equation is known （Theorem 6）. As a special case we get the previous characterization of reproductive solutions and a similar result for Boolean equations （Theorem 9）.     

Quasi-periodic Solutions of Nonlinear Schrodinger Equations of Higher Dimension

We are concerned with the existence of the quasi-periodic solutions of the nonlinear Schr(?)dinger（NLS）eauationwhere △ is the　d-Laplace and　Mσ is a Fourier multiplier,i.e,     

Existence of Solutions of a Two-Point Boundary Value Problem

A two-point boundary value problem with a non-negative parameter Q arising in the study of surface tension induced flow of a liquid metal or semiconductor is studied. We prove that the problem has at least one solution for Q≥0.This improves a recent result that the problem has at least one solution for 0 ≤Q≤13.21.     

Viscosity Solutions of Monotonic Functional Parabolic PDE

In this paper,by the technique of coupled solutions,the notion of viscosity solution is ex-tended to quasi-monotonic fully nonlinear parabolic equations with delay,which involves many modelsarising from optimal control theory,economy and finance,biology etc.The comparison,existence anduniqueness are proved.And the results are applied to the retarded Bellman equations.     

On Periodic Viable Solutions of Differential Inclusions

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On the Involutive Solutions of Solition Hierarchy

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New Solutions of Heun’s Equation

The Laplace transform and its inverse are used to obtain new solutions of Heuns equation (HE) in the form of readily computable five-fold series. These solutions are a considerable improvement on the extremely complicated solutions of perturbation type deduced by Exton [2].AMS Subject Classification (2000), 33E10, 33E10, 33E20, 33E30, 34AO53     

Boundedness of Solutions for Elliptic Variational Inequalities

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On Some Solutions of A~k=J-i

We present some solutions of the matrix equation A~k=J-I, where A is a(0,l) g-circulant matrix. Ⅰ. lntroboction The problem we consider in this paper is to find the g-circulant solutionsof the matrix equation     

On the Oscillation of Solutions of Second OrderNonlinear Difference Equations

1.IntroductionInthispaperweareconcernedwiththeoscillatorypropertiesofsolutionsofsecondordernonlineardifferenceequationsoftheformwhere△istheoperatordefinedbytheequation△yn=yn 1-ynandItisinterestingtostudysecondordernonlineardifferenceequationsbecausetheyarediscreteanaloguesofdifferentialequations.Inaddition,theydohavephysicalapplicationsasevidencedby[1J.Inequation(l),p={p.},n=l,2,3,..'isagiveninfinitesequenceofrealnumbersandthefunctionfisdefinedonRwhereRisthesetofrealnumbers.Byasolutionof(1…     

Folding of Set-Theoretical Solutions of the Yang–Baxter Equation

We establish a correspondence between the invariant subsets of a non-degenerate symmetric set-theoretical solution of the quantum Yang–Baxter equation and the parabolic subgroups of its structure group, equipped with its canonical Garside structure. Moreover, we introduce the notion of a foldable solution, which extends the one of a decomposable solution.     

Stabilization of Solutions of the Nonlinear Fokker–Planck Equation

One considers the equation $$ \mathrm{div}\left( {{u^{\sigma }}Du} \right)+b(x)Du-{u_t}=f(x)g(u),\quad x\in {{\mathbb{R}}^n},\quad t\in \left( {0,\infty } \right), $$ where $ b:{{\mathbb{R}}^n}\to {{\mathbb{R}}^n} $ and $ f:{{\mathbb{R}}^n}\to [0,\infty ) $ are locally bounded measurable functions, g: (0,∞)?→?(0,∞) is continuous and nondecreasing, One obtains the conditions ensuring that its positive solutions stabilize to zero as t?→?∞.     

The Oscillation of the Solutions of Higher-order Functional Differential Equations

In this paper we study the oscillation of the solutions of higher-order functional differential equations, and have given the new integral conditions when equation (1) is oscillatory and extended some known results.x(n)(t) p(t)x(g(t)) = 0, (1)where n is even number, p(t) ∈ C(R, [0, ∞)), g(t) ∈ C(R; R), g(t) f55 t,.     

Regularity of Solutions of the Cahn-Hilliard Equation with Non-constant Mobility

In this paper, we study the regularity of solutions for two-dimensional Cahn-Hilliard equation with non-constant mobility. Basing on the L^p type estimates and Schauder type estimates, we prove the global existence of classical solutions.     

Multiplicity of Periodic Solutions of Duffing Equations with Jumping Nonlinearities

We provide sufficient conditions for the existence and multiplicity of subharrnonic solutions for Duffing's equations with jumping nonlinearities.