CLASSIFICATION AND EXISTENCE OF POSITIVE SOLUTIONS OF SECOND ORDER NONLINEAR DIFFERENTIAL EQUATIONS WITH DELAY DEPENDING ON THE UNKNOWN FUNCTION

A class of second order nonlinear differential equations with delay depenging on the unknown function of the fromin the case where ∫0∞ ds/r(s) < ∞ is studied. Various classifications of their eventually positive solutions are given in terms of their asymptotic magnitudes, and necessary as well as sufficient conditions for the existence of these solutions are also obtained.

MULTIPLICITY AND CONCENTRATION BEHAVIOUR OF POSITIVE SOLUTIONS FOR SCHRDINGER-KIRCHHOFF TYPE EQUATIONS INVOLVING THE p-LAPLACIAN IN R^N

In this article, we study the multiplicity and concentration behavior of positive solutions for the p-Laplacian equation of SchrSdinger-Kirchhoff type -εpM（εp-N∫RN｜△u｜p）△pu＋v（x｜u｜p-2u=f（u）in RN, where △p is the p-Laplacian operator, 1 〈 p 〈 N, M ： R＋ → R＋ and V ： RN →R＋ are continuous functions, ε is a positive parameter, and f is a continuous function with subcritical growth. We assume that V satisfies the local condition introduced by M. del Pino and P. Felmer. By the variational methods, penalization techniques, and Lyusternik- Schnirelmann theory, we prove the existence, multiplicity, and concentration of solutions for the above equation.

Some Equations for the External Space and Their Implications on the Inverse Cone Concept and the Geometry of the Physical Universe

In previous works, they were proposed a photonic model of the Big Bang [1] and several parameters derived from the Hubble-Lemaitre equation [2]. Since these parameters result higher than the classical ones and, otherwise, the General Theory of Relativity does not apply far away the Physical Universe, in this paper, it will be revised the adequacy of such parameters in the external Space and their influence on the relativistic concept of the cone of time. As well, it will be intended to define the Physical Universe geometry accordingly to a thermo-dynamical analysis of the Big Bang.

Asymptotic Behavior and Nonexistence of Positive Solutions of Delay Difference Equations

目的讨论一类带可变系数二阶非线性中立型时滞差分方程正解所具有的性质．方法通过运用差分不等式及几个常用不等式，建立了两个引理．结果与结论获得了有关此类方程正解的渐近性质及不存在正解的新准则，推广并改善了最近文献的某些结果，同时还给出了具体的实例，以说明结果的实用、精确和重要．

Hermite Positive Definite Solution of the Quaternion Matrix Equation Xm + B*XB = C

This paper discusses the necessary and sufficient conditions for the existence of Hermite positive definite solutions of the quaternion matrix equation Xm+ B*XB = C (m > 0) and its iterative solution method. According to the characteristics of the coefficient matrix, a corresponding algebraic equation system is ingeniously constructed, and by discussing the equation system’s solvability, the matrix equation’s existence interval is obtained. Based on the characteristics of the coefficient matrix, some necessary and sufficient conditions for the existence of Hermitian positive definite solutions of the matrix equation are derived. Then, the upper and lower bounds of the positive actual solutions are estimated by using matrix inequalities. Four iteration formats are constructed according to the given conditions and existence intervals, and their convergence is proven. The selection method for the initial matrix is also provided. Finally, using the complexification operator of quaternion matrices, an equivalent iteration on the complex field is established to solve the equation in the Matlab environment. Two numerical examples are used to test the effectiveness and feasibility of the given method. .

PROPER SPLITTINGS FOR RESTRICTED LINEAR EQUATIONS AND THE GENERALIZED INVERSE A_(T,S)~2

This paper presents a proper splitting iterative method for comparing the general restricted linear euqations Ax=b, x ∈T （where, b ∈AT, and T is an arbitrary but fixed subspace of Cm） and the generalized in AT,S For the special case when b ∈AT and dim（T）=dim（AT）, this splitting iterative methverse AT,S hod converges to AT,Sb （the unique solution of the general restricted system Ax=bx ∈T）.

THE EXISTENCE AND LOCAL UNIQUENESS OF MULTI-PEAK POSITIVE SOLUTIONS TO A CLASS OF KIRCHHOFF EQUATION

In the present paper,we consider the nonlocal Kirchhoff problem-(ε^2a+εb∫|■u|^2)Δu+u=Q(x)u^p,u>0 in R^3,,where a,b>0,1<p<5 andε>0 is a parameter.Under some assumptions on Q(x),we show the existence and local uniqueness of positive multi-peak solutions by LyapunovSchmidt reduction method and the local Pohozaev identity method,respectly.

EXISTENCE OF THE MINIMAL POSITIVE SOLUTION OF SOME NONLINEAR ELLIPTIC SYSTEMS WHEN THE NONLINEARITY IS THE SUM OF A SUBLINEAR AND A SUPERLINEAR TERM

It is shown that there exists Λ>0 such that, for every λ∈(0,Λ), the semilinear elliptic system: - Δ u=λu｜u｜ q-1 +u｜u｜ p-1 -v inΩ, - Δ v=δu-γv in Ω, u=v=0 on Ω, where Ω∈R N(N≥2) is a bounded domain with smooth boundary and 0<q<1<p,has a minimal positive solution (u λ,v λ). Moreover: u λ and v λ are strictly increasing with respect to λ.

The Factorization Method to Solve a Class of Inverse Potential Scattering Problems for Schrdinger Equations

This paper is concerned with the inverse scattering problems for Schrdinger equations with compactly supported potentials.For purpose of reconstructing the support of the potential,we derive a factorization of the scattering amplitude operator A and prove that the ranges of （A＊ A） ^1/4 and G which maps more general incident fields than plane waves into the scattering amplitude coincide.As an application we characterize the support of the potential using only the spectral data of the operator A.

GLOBAL ATTRACTIVITY OF NEUTRAL DIFFERENCE EQUATIONS WITH POSITIVE AND NEGATIVE COEFFICIENTS

A neutral difference equation with positive and negative coefficientsΔ(x n-c nx n-k )+p nx n-l -q nx n-r =0, n=0,1,2,...,is considered and a sufficient condition for the global attractivity of the zero solution of this equation is obtained, which improves and extends the all known results in the literature.

FINITE TIME BLOW UP OF THE SOLUTIONS TO BOUSSINESQ EQUATION WITH LINEAR RESTORING FORCE AND ARBITRARY POSITIVE ENERGY

Finite time blow up of the solutions to Boussinesq equation with linear restoring force and combined power nonlinearities is studied. Sufficient conditions on the initial data for nonexistence of global solutions are derived. The results are valid for initial data with arbitrary high positive energy. The proofs are based on the concave method and new sign preserving functionals.

EPSILON-ALGORITHM AND ETA-ALGORITHM OF GENERALIZED INVERSE FUNCTION-VALUED PAD APPROXIMANTS USING FOR SOLUTION OF INTEGRAL EQUATIONS

Two efficient recursive algorithms epsilon_algorithm and eta_algorithm are introduced to compute the generalized inverse function_valued Padé approximants. The approximants were used to accelerate the convergence of the power series with function_valued coefficients and to estimate characteristic value of the integral equations. Famous Wynn identities of the Pad approximants is also established by means of the connection of two algorithms.

A CONTINUATION HOMOTOPY METHOD FOR THE INVERSE PROBLEM OF OPERATOR IDENTIFICATION AND ITS APPLICATION

A new widly convergent method for solving the problem of operator identification is illustrated. Numerical simulations are carried out to test the feasibility and to study the general characteristics of the technique without the real measurement data. This technique is a direct application of the continuation homo-topy method for solving nonlinear systems of equations. It is found that this method does give excellent results in solving the inverse problem of the elliptic differential equations.

STRUCTURE OF NONNEGATIVE NONTRIVIAL AND POSITIVE SOLUTIONS OF SINGULARLY PERTURBED p-LAPLACE EQUATIONS

Structure of nonnegative nontrivial and positive solutions was precisely studied for some singularly perturbed p-Laplace equations. By virtue of sub- and supersolution method, it is shown that there are many nonnegative nontrivial spike-layer solutions and positive intermediate spike-layer solutions. Moreover, the upper and lower bound on the measure of each spike-layer were estimated when the parameter is sufficiently small.

The Generalized Inverse of Singular Integral Operators on an Open Arc and Its Applications

After choosing weight functions suitably, we define a Banach spaceH ω μ (L) and discuss the generalized inverse of singular integral operators on an open arc. Using the generalized inverse, we obtain the solutions for the following singular integral equation

Existence and Multiple of Positive Solution for Nonlinear Fractional Difference Equations with Parameter

Let,. We study the existence and multiple positive solutions of n-th nonlinear discrete fractional boundary value problem of the form By using a fixed-point theorem on cone, the parameter intervals of problem is established.

The General and Centro(Kewsy) Symmetric Solutions to a System of Matrix Equations over an Arbitrary Skew Field

必要、足够的条件在任意的斜域上为一般解决方案，中心对称的解决方案，和线性矩阵方程的一个系统的 centroskewsymmetric 答案的存在被给。系统的如此的答案的代表也被导出。

THE SYMMETRIC AND SYMMETRIC POSITIVE SEMIDEFINITE SOLUTIONS OF LINEAR MATRIX EQUATION——B^TXB = D ON LINEAR MANIFOLDS

This paper discusses the solutions of the linear matrix equation BT X B=Don some linear manifolds.Some necessary and sufficient conditions for the existenceof the solution and the expression of the general solution are given.And also someoptimal approximation solutions are discussed.

Positive solutions of nonlinear elastic beam equations with a fixed end and a movable end

The existence of n positive solutions is studied for a class of fourth-order elastic beam equations where one end is fixed and other end is movable. Here, n is an arbitrary natural number. Our results show that the class of equations may have n positive solutions provided the “heights” of the nonlinear term are appropriate on some bounded sets.

THE NECESSARY AND SUFFICIENT CONDITIONS FOR THE SOLVABILITY OF A CLASS OF THE MATRIX INVERSE PROBLEM

Censider the solutions of the matrix inverse problem, which are symmetric positive semide finite on a subspace. Necessary and sufficient conditions for the solvability, as well as the general solution are obtained. The best approximate solution by the above solution set is given. Thus the open problem in [1] is solved.