A theorem of the alternatives for the equation Ax + B|x| = b

The following theorem is proved: given square matrices A, D of the same size, D nonnegative, then either the equation Ax + B|x| = b has a unique solution for each B with |B| ≤ D and for each b, or the equation Ax + B₀|x| = 0 has a nontrivial solution for some matrix B₀ of a very special form, |B₀| ≤ D; the two alternatives exclude each other. Some consequences of this result are drawn. In particular, we define a λ to be an absolute eigenvalue of A if |Ax| = λ|x| for some x ≠ 0, and we prove that each square real matrix has an absolute eigenvalue.     

A unique solution to a nonlinear elliptic equation

The purpose of this paper is to prove the existence of a unique, classical solution to the nonlinear elliptic partial differential equation −∇⋅(a(u(x))∇u(x))=f(x) under periodic boundary conditions, where u(x₀)=u₀ at x₀∈Ω, with Ω=TN, the N-dimensional torus, and N=2,3. The function a is assumed to be smooth, and a(u(x))>0 for , where G⊂R is a bounded interval. We prove that if the functions f and a satisfy certain conditions, then a unique classical solution u exists. The range of the solution u is a subset of a specified interval . Applications of this work include stationary heat/diffusion problems with a source/sink, where the value of the solution is known at a spatial location x₀.     

An existence and uniqueness theorem for a second order nonlinear system

In this paper we give an existence and uniqueness theorem for a nonlinear second order homogeneous Dirichlet system.     

A note on the diophantine equation x 2 + b y = c z

Let a, b, c, r be positive integers such that a ² + b ² = c ^r , min(a, b, c, r) > 1, gcd(a, b) = 1, a is even and r is odd. In this paper we prove that if b ≡ 3 (mod 4) and either b or c is an odd prime power, then the equation x ² + b ^y = c ^z has only the positive integer solution (x, y, z) = (a, 2, r) with min(y, z) > 1.     

A filtering method for the interval eigenvalue problem

We consider the general problem of computing intervals that contain the real eigenvalues of interval matrices. Given an outer approximation (superset) of the real eigenvalue set of an interval matrix, we propose a filtering method that iteratively improves the approximation. Even though our method is based on a sufficient regularity condition, it is very efficient in practice and our experimental results suggest that it improves, in general, significantly the initial outer approximation. The proposed method works for general, as well as for symmetric interval matrices.     

The Cauchy problem for ut=Δu+|∇u|

With q a positive real number, the nonlinear partial differential equation in the title of the paper arises in the study of the growth of surfaces. In that context it is known as the generalized deterministic KPZ equation. The paper is concerned with the initial-value problem for the equation under the assumption that the initial-data function is bounded and continuous. Results on the existence, uniqueness, and regularity of solutions are obtained.     

凝聚映象的不动点及算子方程Ax+Bx+Cx=x的可解性

按文[1]中方法得到几个对凝聚映象的不动点定理,还扩充文[2]中对于算子方程Ax B x=x到Ax B x Cx=x可解性的某些结论.主要结果是定理2、定理3与定理5.     

矩阵方程AX=B的转动不变解及其最佳逼近

利用矩阵的广义逆、奇异值分解、张量积和拉直算子,给出了矩阵方程AX=B有转动不变解的充分必要条件及有解时通解的表达式;给出了矩阵方程解集合中与给定矩阵的最佳逼近解的表达式.     

Positive solutions of the nonlinear fourth-order beam equation with three parameters

This paper is concerned with the existence and nonexistence of positive solutions of the nonlinear fourth-order beam equation u(4)(t)+ηu^″(t)−ζu(t)=λf(t,u(t)), 0<t<1, u(0)=u(1)=u^″(0)=u^″(1)=0, where is continuous and ζ, η and λ are parameters. We show that there exists a such that the above boundary value problem (BVP) has at least two, one and no positive solutions for 0<λ<λ^*, λ=λ^* and λ>λ^*, respectively. Furthermore, by using the semiorder method on cones of Banach space, we establish a uniqueness criterion for positive solution of the BVP. In particular such a positive solution uλ(t) of the BVP depends continuously on the parameter λ, i.e., uλ(t) is nondecreasing in λ, limλ→⁰⁺‖uλ(t)‖=0 and limλ→+∞‖uλ(t)‖=+∞ for any t∈[0,1].     

Non-blow-up phenomenon for the Cahn-Hilliard equation with non-constant mobility

In this paper we prove that the finite time blow-up phenomenon might not occur for the Cahn-Hilliard equation with non-constant mobility and cubic nonlinearity, which is quite different from the case of constant mobility. We reveal such a phenomenon under some structure condition on the mobility.     

Submanifolds in pseudo-Euclidean spaces satisfying the condition Δx=Ax+B

In this paper we study pseudo-Riemannian submanifolds in ℝ _n ^+k/t satisfying the condition Δx=Ax+B, whereA is an endomorphism of ℝ _n ^+k/t andB is a constant vector in ℝ _n ^+k/t . We give a characterization theorem whenA is a self-adjoint endomorphism. As for hypersurfaces we are able to obtain a classification theorem for any endomorphismA. Supported by a FPI Grant, DGICYT, 1990. Partially supported by a DGICYT Grant No. PS87-0115-C03-03.     

A mixed boundary value problem for Chaplygin's hodograph equation

In this paper we will prove existence, uniqueness and regularity of a classical solution to a mixed boundary value problem for Chaplygin's hodograph equation, which is degenerate elliptic on a part of the boundary. This problem is derived from the study of detached bow shock ahead of a straight ramp in uniform supersonic flows in the hodograph plane. The proof depends on Perron's method and some techniques from linear elliptic equations.     

Eigenvalue problems for second-order nonlinear dynamic equations on time scales

This paper is concerned with the existence and nonexistence of positive solutions of the second-order nonlinear dynamic equation uΔΔ(t)+λa(t)f(u(σ(t)))=0, t∈[0,1], satisfying either the conjugate boundary conditions u(0)=u(σ(1))=0 or the right focal boundary conditions u(0)=uΔ(σ(1))=0, where a and f are positive. We show that there exists a λ^∗>0 such that the above boundary value problem has at least two, one and no positive solutions for 0<λ<λ^∗, λ=λ^∗ and λ>λ^∗, respectively. Furthermore, by using the semiorder method on cones of the Banach space, we establish an existence and uniqueness criterion for positive solution of the problem. In particular, such a positive solution uλ(t) of the problem depends continuously on the parameter λ, i.e., uλ(t) is nondecreasing in λ, limλ→⁰⁺‖uλ‖=0 and limλ→+∞‖uλ‖=+∞.     

A bifurcation theorem for the p-logistic equation

We consider a p-logistic equation with an equidiffusive reaction. Using variational methods and truncation techniques, we show that there is a critical parameter value λ^∗ > 0 such that for λ > λ^∗ the problem has a unique positive smooth solution, and for λ ∈ (0, λ^∗] the problem has no positive solution.     

Existence, uniqueness and asymptotic behavior of solutions for a singular parabolic equation

In this paper, we are concerned with a singular parabolic equation in a smooth bounded domain Ω⊂RN subject to zero Dirichlet boundary condition and initial condition φ?0. Under the assumptions on μ, φ and f(x,t), some existence and uniqueness results are obtained by applying parabolic regularization method and the sub-supersolutions method. We also discuss the asymptotic behaviors of solutions in the sense of and L^∞(0,T;L²(Ω)) norms as μ→0 or μ→∞. As a byproduct we obtain the existence of solutions for some problems which blow up on the boundary.     

On the Diophantine equation x 2+5 m =y n

In this paper we consider the Diophantine equation x ²+5 ^m =y ⁿ , n>2, m>0. We prove that the equation has no positive integer solutions when 2∤ m, nor when 2∣m under the additional condition (x,y)=1, with the help of Bilu, Hanrot, and Voutier’s deep result in (J. Reine Angew. Math. 539:75–122, 2001). Supported by the 973 Grant of P.R.C and SRFDP 20040284018.     

Timelike Hypersurfaces in the Standard Lorentzian Space Forms Satisfying L k x = Ax + b

In this paper, we study connected orientable timelike hypersurfaces isometrically immersed by ${x : M_1^{n} \rightarrow {\tilde{M}}_1^{n+1}(c)}$ into the de Sitter space ${\mathbb{S}_1^{n+1}}$ (when c = 1) or anti-de Sitter space ${\mathbb{H}_1^{n+1}}$ (when c = ?1) satisfying the condition L _k x = Ax + b, where the second order differential operator L _k is the linearized operator associated with the first normal variation of the (k + 1)-th mean curvature of M for an integer k, 0 ≤ k < n, A is a matrix and b is a vector. We characterize these hypersurfaces when b = 0 or when the k-th mean curvature of M is constant.     

A remark on the thermistor problem with rapidly growing conductivity

A comparison is made between existing results for the stationary thermistor problem. A theorem of existence is given even when the electrical conductivity is a superlinear function of the temperature.     

关于Diophantine方程x3+1=py2

在素数p=3（8t＋4）（8t＋5）＋1和p=3（8t＋3）（8t＋4）＋1的情形下,运用初等数论的方法给出了丢番图方程x3＋1=py2无正整数解的充分条件,并得到无数个6k＋1型的素数p使得方程x3＋1=py2无正整数解.     

Existence and uniqueness of periodic solutions for a kind of Rayleigh equation with finitely many deviating arguments

In this paper, we consider a kind of Rayleigh equation with finitely many deviating arguments of the form