New approach to the solution of the classical sine-Gordon equation and its generalizations

We obtain new exact solutions U(x, y, z, t) of the three-dimensional sine-Gordon equation. The three-dimensional solutions depend on an arbitrary function F(α) whose argument is a function α(x, y, z, t). The ansatz α is found from an equation linear in (x, y, z, t) whose coefficients are arbitrary functions of α that should satisfy a system of algebraic equations. By this method, we solve the classical and a generalized sine-Gordon equation; the latter additionally contains first derivatives with respect to (x, y, z, t). We separately consider an equation that contains only the first derivative with respect to time. We present approaches to the solution of the sine-Gordon equation with variable amplitude. The considered methods for solving the sine-Gordon equation admit a natural generalization to the case of integration of the same types of equations in a space of arbitrarily many dimensions.     

Numerical Solutions to the Darboux Problem with Functional Dependence

The paper deals with the Darboux problem for the equation D _{xy z} (x,y) = f(x,y,z( _x,y ) where z( _x,y ) is a function defined by . We construct a general class of difference methods for this problem. We prove the existence and uniqueness of solutions to implicit functional difference equations by means of a comparison method; moreover we give an error estimate. The convergence of explicit difference schemes is proved under a general assumption that given functions satisfy nonlinear estimates of the Perron type. Our results are illustrated by a numerical example.     

Linear Differential Relations Between Solutions of the Class of Euler-Poisson-Darboux Equations

All the linear first-order relations of the form

Existence of positive solutions of higher-order quasilinear ordinary differential equations

In this paper the even-order quasilinear ordinary differential equation is considered under the hypotheses that n is even, D(α _i )x = (|x|αi−1 x)′, α _i > 0(i = 1,2,…, n), β > 0, and p(t) is a continuous, nonnegative, and eventually nontrivial function on an infinite interval [a, ∞), a > 0. The existence of positive solutions of (1.1) is discussed, and basic results to the classical equation are extended to the more general equation (1.1). In particular, necessary and sufficient integral conditions for the existence of positive solutions of (1.1) are established in the case α ₁α₂⋅s α _n ≠ β. This research was partially supported by Grant-in-Aid for Scientific Research (No. 15340048), Japan Society for the Promotion of Science. Mathematics Subject Classification (2000) 34C10, 34C11     

A two-parameter boundary problem for second order differential system

Summary This paper is concerned with second order differential systems involving two parameters with boundary conditions specified at three points. In particular, we consider the system y' = k(x, λ, μ)z, z' = -g(x, λ, μ)y, where k and g are real-valued junctions defined on X: a ≤ x ≤ c, L: L₁ < λ < L₂, and M: M₁ < μ < M₂. This system is studied together with the boundary conditions α(λ, μ)y(a) - β(λ, μ)z(a)=0, γ(λ, μ)y(b) - δ(λ, μ)z(b)=0, ε₁(μ)y(b) - φ₁(μ)z(b)=ε₂(μ)y(c) - φ₂(μ)z(c), where α, β, δ, γ, ε_i, φ_i, i=1, 2, are continuous functions of the parameters. This work establishes the existence of eigenvalue pairs for the boundary problem and the oscillatory behavior of the associated solutions. These results complement those previously obtained by the authors and B. D. Sleeman, where boundary conditions of the ? Sturm-Liouville ? type were studied. Entrata in Redazione il 5 dicembre 1977. The research for this paper was supported by a University College Reasearch Grant, University of Alabama in Birmingham.     

Carleman estimates for one-dimensional degenerate heat equations

In this paper, we are interested in controllability properties of parabolic equations degenerating at the boundary of the space domain. We derive new Carleman estimates for the degenerate parabolic equation $$ w_t + \left( {a\left( x \right)w_x } \right)_x = f,\quad \left( {t,x} \right) \in \left( {0,T} \right) \times \left( {0,1} \right), $$ where the function a mainly satisfies $$ a \in \mathcal{C}^0 \left( {\left[ {0,1} \right]} \right) \cap \mathcal{C}^1 \left( {\left( {0,1} \right)} \right),a \gt 0 \hbox{on }\left( {0,1} \right) \hbox{and }\frac{1} {{\sqrt a }} \in L^1 \left( {0,1} \right). $$ We are mainly interested in the situation of a degenerate equation at the boundary i.e. in the case where a(0)=0 and / or a(1)=0. A typical example is a(x)=x^α (1 − x)^β with α, β ∈ [0, 2). As a consequence, we deduce null controllability results for the degenerate one dimensional heat equation $$ u_t - (a(x)u_x )_x = h\chi _w ,\quad (t,x) \in (0,T) \times (0,1),\quad \omega \subset \subset (0,1). $$ The present paper completes and improves previous works [7, 8] where this problem was solved in the case a(x)=x^α with α ∈[0, 2). Dedicated to Giuseppe Da Prato on the occasion of his 70th birthday     

Nonlinear operator equations with a functional perturbation of the argument of neutral type

We construct small solutions x(t) → 0 as t → 0 of nonlinear operator equations F(x(t), x(α(t)),t) = 0 with a functional perturbation α(t) of the argument. By the Newton diagram method, we reduce the problem to quasilinear operator equations with a functional perturbation of the argument. We show that the solutions of such equations can have not only algebraic but also logarithmic branching points and contain free parameters. The number of free parameters and the form of the solution depend on the properties of the Jordan structure of the operator coefficients of the equation.     

Ergodicity of cylinder flows arising from irregularities of distribution

LetT be the mod 1 circle group, α∈T be irrational and 0<β<1. LetE be the closed subgroup ofR generated by β and 1. DefineX=T×E andT:X→X byT(x, t)=(x+α,t+1 _[0,β] (x)−β). Then we have the theorem:T is ergodic if and only if β is rational or 1, α and β are linearly independent over the rationals. This paper was prepared while I was very graciously hosted by the Centro de Investigacion y Estudios Avanzados, Mexico City.     

一阶时滞微分方程解的零点分布

Abstract. The paper gives two estimates of the distance between adjacent zeros of solutions     

Convolution properties of some classes of analytic functions

Let A denote the class of functions which are analytic in |z|<1 and normalized so that f(0)=0 and f′(0)=1, and let R(α, β)⊂A be the class of functions f such thatRe[f′(z)+αzf″(z)]>β,Re α>0, β<1. We determine conditions under which (i) f ∈ R(α₁, β₁), g ∈ R(α₂, β₂) implies that the convolution f×g of f and g is convex; (ii) f ∈ R(0, β₁), g ∈ R(0, β₂) implies that f×g is starlike; (iii) f≠A such that f′(z)[f(z)/z]^μ-1 ≺ 1 + λz, μ>0, 0<λ<1, is starlike, and (iv) f≠A such that f′(z)+αzf″(z) ≺ 1 + λz, α>0, δ>0, is convex or starlike. Bibliography: 16 titles. Published inZapiski Nauchnykh Seminarov POMI, Vol. 226, 1996, pp. 138–154.     

A necessary and sufficient condition for singular nonlinear second-order boundary value problems to have positive solutions

In this paper the following result is obtained: Suppose f(g,u,v) is nonnegative, continuous in (a, 6) ×R⁺ ×R ⁺ ; f may be singular at κ = a(and/or κ = b) and υ = 0; f is nondecreasing on u for each κ,υ,nonincreasing on υ for each κ,u; there exists a constant q ε (0,1) such that

Existence of Nondecreasing and Continuous Solutions of an Integral Equation with Linear Modification of the Argument

We use a technique associated with measures of noncompactness to prove the existence of nondecreasing solutions to an integral equation with linear modification of the argument in the space C[0, 1]. In the last thirty years there has been a great deal of work in the field of differential equations with a modified argument. A special class is represented by the differential equation with affine modification of the argument which can be delay differential equations or differential equations with linear modifications of the argument. In this case we study the following integral equation x（t） = a（t） ＋ （Tx）（t） ∫0^σ（t） u（t, s, x（s）, x（λs））ds 0 〈 λ 〈 1 which can be considered in connection with the following Cauchy problem x＇（t） = u（t, s, x（t）, x（λt））, t ∈ [0, 1], 0 〈 λ 〈 1 x（0） = u0.     

On the boundedness and the stability properties of solutions of certain fourth order differential equations

Summary The paper investigates the equation(1.1) in the two cases:i) p≡0,ii) p(≠0) is either bounded or satisfies |(p(t,x,y,z,u)|⩽(A₀+|y|+|u|+|z| Ψ(t) where A₀ is a constant. For the casei) the asymptotic stability (in the large) of the trivial solution x=0 is investigated and for the caseii) a general estimate and two boundedness results are obtained for solutions of(1.1). The results extend those obtained by Harrow[1] for the same equation(1.1). Entrata in Redazione il 18 novembre 1971.     

A note on the existence and asymptotic behavior of nonoscillatory solutions of fourth order quasilinear differential equations

This paper is concerned with nonoscillatory solutions of the fourth order quasilinear differential equation

Some inequalities concerning the characteristic frequencies of elastic continuum

Summary We consider the boundary value problem αz″(x)+m(x)y(x)=0, αy″(x)+p(x)z(x)=0, xε[0, 1], y(0)=y(1)=z(0)=0, where the functions m(x) and p(x) are assumed integrable and positive everywhere in [0, 1]. As the main result we obtain the inequalities for n=1, 2, ... where δ_n(m, p) stands for the product of the first n eigenvalues α_i(m, p) of the above system and where δ_n(m) abbreviates δ_n(m, m). Entrata in Redazione il 6 febbraio 1976.     

Uniform exponential stability of solutions to abstract Volterra equations

In terms of the spectral properties of the associated operator matrix, we obtain a new criterion to judge the uniform exponential stability of the solutions to abstract Volterra equations. In particular, we study in detail the case where the kernel function a(t) takes the form α e ^−βt (β > 0, α ≠ 0). Moreover, we give examples to illustrate our results.     

Oscillation and Nonoscillation of Higher Order Self-Adjoint Differential Equations

Oscillation and nonoscillation criteria for the higher order self-adjoint differential equation (-1)ⁿ(t^alphay⁽ⁿ⁾)⁽ⁿ⁾+q(t)y = 0 (*) are established. In these criteria, equation (*) is viewed as a perturbation of the conditionally oscillatory equation (-1)ⁿ(t^alphay⁽ⁿ⁾)⁽ⁿ⁾ - µ,t^2n-y = 0, where _n, is the critical constant in conditional oscillation. Some open problems in the theory of conditionally oscillatory, even order, self-adjoint equations are also discussed.     

Oscillation criteria for nonlinear second order equations

Summary A necessary and a sufficient condition are given for oscillation of all solutions of y″+f(t, y)=0. We sequire that a(t)α(y)≤f(t, y) if y>0, and f(t, y)<-b(t)β(y) if y<0, together with continuity and integrability assumptions on a, b, α, and β. Of speciat interest here is the relaxing of conditions a≥0, b≥0 in Machi - Wong [6]. Entrata in Redazione il 9 ottobre 1968. Supported by NASA Research Grant NGR-45-003-038.     

A direct method of construction of two-zone solutions of nonlinear equations

We suggest an approach that allows one to effectively construct two-zone solutions, including real, of some nonlinear equations without applying the technique of algebraic curves. Thestarting point in the construction is a special addition theorem for theta-functions of two variables. The method is illustrated by the Kadomtsev-Petviashvili, KdV, and sine-Gordon equations. Translated fromMatematicheskie Zametki, Vol. 66, No. 3, pp. 401–406, September, 1999.     

Certain oscillatory integrals on unit square and their applications

Let Q2 = [0, 1]2 be the unit square in two dimension Euclidean space R2. We study the Lp boundedness properties of the oscillatory integral operators Tα,β defined on the set S(R3) of Schwartz test functions f by Tα,βf(x,y,z) = Q2 f(x - t,y - s,z - tksj)e-it-β1s-β2t-1-α1s-1-α2dtds, where β1 > α1 0, β2 > α2 0 and (k, j) ∈ R2. As applications, we obtain some Lp boundedness results of rough singular integral operators on the product spaces.