Comparison theorems for oscillation of second-order half-linear differential equations

Summary We establish new comparison theorems on the oscillation of solutions of a class of perturbed half-linear differential equations. These improve the work of Elbert and Schneider [6] in which connections are found between half-linear differential equations and linear differential equations. Our comparison theorems are not of Sturm type or Hille--Wintner type which are very famous. We can apply the main results in combination with Sturm's or Hille--Wintner's comparison theorem to a half-linear differential equation of the general form (|x'|^α-1x')' + a(t) |x|^α-1x = 0.     

Complexity of quasivariety lattices for varieties of differential groupoids. II

We continue the study of the lattice of quasivarieties of differential groupoids. We suggest a method for constructing differential groupoids from graphs. We prove that, for every variety of differential groupoids, the cardinality of the lattice of subquasivarieties is either finite or equal to 2 ^ω .     

Limit-Point Criteria For Not Necessarily Symmetric Ordinary Differential Expressions

A limit-point criterion for not necessarily symmetric differential expressions L of arbitrary order is proved by applying a known limit-point criterion for symmetric even-order differential expressions to (L⁺L)^p for some positive integer p where L⁺ denotes the formal adjoint of L. For certain differential expressions the results obtained from the new criterion are stronger than those obtained from the original one and can even be improved by increasing p.     

Eigenvalues of certain differential operators related to automorphic forms

G.Shimura introduced a certain kind of differential operators when he studied the arithmetic properties of Eisenstein series of several variables. The invariable differential operators of the classical domain of types I and II are included in it as its special cases. In this paper the eigenvalues of this kind of differential operators with respect to the eigenfunction (deti( ^tZ−Z))² are calculated. Projects Supported by the Science Fund of the Chinese Academy of Sciences.     

Degree of the divisor of solutions of a differential equation on a projective variety

Using the data schemes from [1] we give a rigorous definition of algebraic differential equations on the complex projective space Pⁿ. For an algebraic subvariety S?Pⁿ, we present an explicit formula for the degree of the divisor of solutions of a differential equation on S and give some examples of applications. We extend the technique and result to the real case.     

Upper and lower bounds for solutions of generalized two-point boundary value problems

This paper is concerned with two-point boundary value problems for systems of differential equations and integro-differential equations. If ?, ψ and Φ, Ψ are functions which satisfy certain differential (integro-differential) inequalities, then the given problem has a solutionu ^* such that ?≦u ^*≦ψ and Φ≦u ^*′≦Ψ.     

On Iterations of the Green Integrals and their Applications to Elliptic Differential Complexes

Convergence of iterations of special Green integrals for overdetermined elliptic linear partial differential operators P of order p ≥ 1 is proved. Using this result we obtain necessary and sufficient conditions for the solvability of the equation Pu = f in the Sobolev space W^p,²(D) and, as a corollary, necessary and sufficient conditions for the vanishing of the first cohomology group of elliptic differential complexes. Also a criterion for the solvability of a P-Neumann problem for elliptic differential operators is proved.     

On the stability of invariant sets of leaves of three-dimensional periodic systems

The small C ¹ perturbations of differential equations are studied. The concepts of a weakly hyperbolic set K and a leaf ? are introduced for a system of ordinary differential equations. The Lipschitz condition is not supposed. It is shown that, if the perturbation is small enough, then there exists a continuous mapping h: ? → ? ^Y , where ? ^Y is a leaf of the perturbed system.     

Multiplicity of the spectrum of a self-adjoint ordinary differential operator of odd order

A bound is obtained for the multiplicity of the spectrum of the self-adjoint operator generated by a singular ordinary differential operator? of odd order in the Hubert space ?² in terms of solutions of the differential equation?[y]=λy.     

Error analysis of finite element approximation of a stefan problem with nonlinear free boundary condition

By applying the Landau-type transformation, we transform a Stefan problem with nonlinear free boundary condition into a system consisting of a parabolic equation and the ordinary differential equations. Fully discrete finite element method is developed to approximate the solution of a system of a parabolic equation and the ordinary differential equations. We derive optimal orders of convergence of fully discrete approximations inL2, H¹ and H² normed spaces.     

On the stability of sheet invariant sets of two-dimensional periodic systems

In the paper small C ¹-perturbations of differential equations are considered. The concepts of a weakly hyperbolic set K and a sheet ? for a system of ordinary differential equation are introduced. Lipschitz property is not assumed to hold. It is shown that if the perturbation is small enough, then there is a continuous mapping h: ? → ? ^Y , where ? ^Y is a sheet of the perturbed system.     

Primary differential nil-algebras do exist

We construct a monomorphism from the differential algebra k{x}/[x ^m ] to a Grassmann algebra endowed with a structure of differential algebra. Using this monomorphism, we prove the primality of k{x}/[x ^m ] and its algebra of differential polynomials, solve one of so-called Ritt problems related to this algebra, and give a new proof of the integrality of ideal [x ^m ].     

Ordinary differential equations invariant under translation in the independent variable and rescaling: The Lagrangian formulation

The Lagrangian formulation of the class of general second-order ordinary differential equations invariant under translation in the independent variable and rescaling is presented. The differential equations arising from this analysis are analysed using the Painlevé test. The well-known differential equation, y^″+yy^′+ky³=0, is a unique member of this class when k=3 since it is linearisable by a point transformation. A wider subset is shown to be linearisable by means of a nonlocal transformation.     

Integration of differential forms on manifolds with locally-finite variations. II

In part I of the paper, we have defined n-dimensional C⁰-manifolds in ℝⁿ(m ≥ n) with locally-finite n-dimensional variations (a generalization of locally-rectifiable curves to dimensionn > 1) and integration of measurable differential n-forms over such manifolds. The main result of part II states that an n-dimensional manifold that is C¹-embedded into ℝ^m has locally-finite variations and the integral of a measurable differential n-form defined in part I can be calculated by the well-known formula. Bibliography: 5 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 333, 2006, pp. 66–85.     

Real interpolation of spaces of differential forms

In this paper, we study interpolation of Hilbert spaces of differential forms using the real method of interpolation. We show that the scale of fractional order Sobolev spaces of differential l-forms in H ^s with exterior derivative in H ^s can be obtained by real interpolation. Our proof heavily relies on the recent discovery of smoothed Poincaré lifting for differential forms [M. Costabel and A. McIntosh, On Bogovskii and regularized Poincare integral operators for de Rham complexes on Lipschitz domains, Math. Z. 265(2): 297–320, 2010]. They enable the construction of universal extension operators for Sobolev spaces of differential forms, which, in turns, pave the way for a Fourier transform based proof of equivalences of K-functionals.     

Lp-Perturbation of Linear Functional Differential Equations

 For a class of retarded linear differential equations with solutions of exponential form, it is shown that L ^p -perturbations leave the solutions of the same form. As a particular example, the asymptotic integration of a class of differential equations is obtained.     

Isomorphic type of a space of smooth functions generated by a finite family of differential operators. II

The space of smooth function on T ₃ generated by one differential expression may fail to be isomorphic to a complemented subspace of C(K). For instance, this happens for the differential expression ∂ ₁ ² -∂ ₂ ² -∂ ₃ ² . Bibliography: 2 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 333, 2006, pp. 62–65.     

Lp-Perturbation of Linear Functional Differential Equations

 For a class of retarded linear differential equations with solutions of exponential form, it is shown that L ^p -perturbations leave the solutions of the same form. As a particular example, the asymptotic integration of a class of differential equations is obtained. (Received 10 August 1998)     

Some properties of the lattice points and their application to differential algebra *

In this paper we prove some charateristic conditions about the dimension of an autoreduced subset E of N^m . As an application to differential algebra we find a counter-example to a conjecture about an upper bound for the order of a system of algebraic differential equations ([8], p. 199).