Positive solutions for a class of boundary value problems on infinite intervals

This paper studies the existence of nonegative solutions to the two point boundary value , 0<t< with appropriate boundary conditions.     

Multiple solutions for three-point boundary value problem with nonlinear terms depending on the first order derivative

In this paper, we consider the following second-order three-point boundary value problem

Existence theorems for nonlinear functional differential equations of neutral type

Conditions are found upon satisfaction of which the differential equation

On the rapidly decreasing solutions for some systems of differential equations with unbounded coefficients

In this paper we continue the study of solvability of the non-homogeneous system of linear differential equations

Computing Eigenvalues of Lommel-Type Equations by the Sampling Method

Using the semiclassical approximation for where and as , we shall show that theeigenvalues are the zeros of a certain function in a Paley–Wienerspace, which allows us to use the Whittaker–Shannon–Kotelnikovsampling theorem to approximate the eigenvalues. We show how thedistribution of the eigenvalues depends on the asymptotics of thecoefficients and as . After abrief discussion on the truncation error, numerical examples are provided.     

Validity of the multiple scale method for very long intervals

For a class of nonlinear oscillation problems containing a small parameter, it is known that a two-scale method using timest and t gives results valid to any desired order for time (1/). We ask when results can be obtained which are valid for (1/²) or for allt > 0. We show that there is an obstruction to introducing a third time scale ² t, and give an example in which this obstruction does not vanish, so that a third scale cannot be introduced, even though the solution exists for all time. The obstruction does vanish if the first order averaged equation vanishes, in which case the three-scale solution actually involves onlyt and ² t and is valid for time (1/²). The obstruction also vanishes if a certain contracting or dissipative condition is met, but in this case the two-scale solution is already valid for all time and the third scale is not needed. These results correspond to known results for the method of averaging, but are here proved for the multiple scale method without use of averaging.     

The Laplacian on $$ C(\overline \Omega) $$ with generalized Wentzell boundary conditions

In this note we prove that the Laplacian with generalized Wentzell boundary conditions on an open bounded regular domain in defined by generates an analytic semigroup of angle on for every > 0 and (for the definition of cf. (1.3)).Received: 13 July 2002     

Kneser-type oscillation criteria for self-adjoint two-term differential equations

It is proved that the even-order equationy ⁽²ⁿ⁾ +p(t)y=0 is (n,n) oscillatory at if

The spectral property of dual matrix strings as a consequence of the existence of spectral matrix functions for canonical systems

The term dual string for scalar strings was introduced in [KK1], where some connections between the spectra of a string and its dual were studied. In [KK2] it was shown that if () is a spectral function of a scalar stringS ₁ with nonnegative spectrum (in the sense of [KK2]), then the function

Estimates on the growth of meromorphic solutions of linear differential equations

We give a pointwise estimate of meromorphic solutions of linear differential equations with coefficients meromorphic in a finite disk or in the open plane. Our results improve some earlier estimates of Bank and Laine. In particular we show that the growth of meromorphic solutions with ()>0 can be estimated in terms of initial conditions of the solution at or near the origin and the characteristic functions of the coefficients. Examples show that the estimates are sharp in a certain sense. Our results give an affirmative answer to a question of Milne Anderson. Our method consists of two steps. In Theorem 2.1 we construct a path (₀, , t) consisting of the ray followed by the circle on which the coefficients are all bounded in terms of the sum of their characteristic functions on a larger circle. In Theorem 2.2 we show how such an estimate for the coefficients leads to a corresponding bound for the solution on z = t. Putting these two steps together we obtain our main result, Theorem 2.3.     

Singular perturbations as a selection criterion for periodic minimizing sequences

Summary Minimizers of functionals like subject to periodic (or Dirichlet) boundary conditions are investigated. While for =0 the infimum is not attained it is shown that for sufficiently small > 0, all minimizers are periodic with period ^1/3. Connections with solid-solid phase transformations are indicated.     

An oscillation criterion for functional differential equations with deviating arguments

An oscillation criterion is given for the differential equation

Increasing Self-Described Sequences

We proceed with our study of increasing self-described sequences F, beginning with 1 and defined by a functional equation In [1] we exhibited the simple solution f (t)=Ct, for some (0,1), of the associated functional-differential equation and we proved that provided <2/(2+d()), where we have the asymtotic equivalence F(m)~ Cm.In the present paper we show that this last result is optimal, in the sense that the self-described sequence defined by |F^–1(m)|=F(m)², that is

Piecewise projective homeomorphisms and a noncommutative Steinberg extension

Let CP (R) be the group of compactly supported homeomorphisms ofR= which are piecewise with a parabolic defect at each breakpoint. An acyclic extension

On a conditional Gołab-Schinzel equation

Let We show that for every function satisfying the conditional equation

Evolution systems and perturbations of generators of strongly continuous groups

Suppose A generates a strongly continuous linear group on a Banach space X and B is a linear operator on X. It is shown that an extension of generates a strongly continuous semigroup if and only if the family of operators has an appropriate evolution system. This produces simple sufficient conditions for an extension of to generate a strongly continuous semigroup, including

On the number of limit cycles in perturbations of a two dimensional nonlinear differential system

Summary For the nonlinear system , which has a family { _h } of closed orbits, we consider perturbations of the type , whereP andQ are arbitrary polynomials. The abelian integralsA(h) corresponding to this family { _h } are investigated. By deriving differential equations forA(h) and proving monotonicity for quotients of abelian integrals, we obtain results on the number of zeros of abelian integrals and, hence, on the number of closed orbits _h which persist as limit cycles of the perturbed system (^*). In particular, a uniqueness theorem for limit cycles of (^*) with quadratic polynomialsP, Q is proved. Moreover, whenP, Q are of arbitrary degree, a lower bound for the possible number of limit cycles of (^*) is derived.     

The left-definite Legendre type boundary problem

The left-definite Legendre type boundary problem concerns the study of a fourth-order singular differential expressionM _k [–] in a weighted Sobolev spaceH generated by a Dirichlet inner product. The fourth-order differential equation

C*-Modular Vector States

Let be a C*-algebra and X a Hilbert C* -module. If is a projection, let be the p-sphere of X. For φ a state of with support p in and consider the modular vector state φ_x of given by The spheres provide fibrations

Nonselfadjoint spectral problems for linear pencilsN-λP of ordinary differential operators with λ-linear boundary conditions: Completeness results

We study nonselfadjoint spectral problems for ordinary differential equationsN(y)–P(y)=0 with -linear boundary conditions where the orderp of the differential operatorP is less than the ordern ofN. The present paper addresses the question of the completeness of the eigenfunctions and associated functions in the Sobolev spacesW ₂ ^k (0,1) fork=0,1,...,n. To this end we associate a pencil – of operators acting fromL ₂(0,1) to the larger spaceL ₂x(0,1) ⁿ with the given problem. We establish completeness results for normal problems in certain finite codimensional subspaces ofW ₂ ^k (0,1) which are characterized by means of Jordan chains in 0 of the adjoint of the compact operator = ^–1.Dedicated to Professor Heinz Langer on the occasion of his 60th birthday