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Luxemberg [2] addresses the problem of determining the asymptotic behavior of the function Ø defined by [image omitted]  where [image omitted]  analysis for [image omitted]  Luxemberg proves results under certain conditions on the asymptoic behavior of f and on [image omitted] . using these results, it is shown in this paper, that the condition on [image omitted]  can be replaced by the condition [image omitted] .

In [4] M. SAMIMI gives nonuniqueness theorems for initial value problems x'= f(t,x), x(0) =x0. where f is not defined at t = 0. But the result for the scalar case only holds with the additional assumption that one solution exists. The applicability of the revised theorem is shown. The result for the n-dimensional case is generalized.

It is shown that the conditions for existence of periodic traveling wave solution for the KdV equation in theorem 4 in [1] are vacuously satisfied

An integral inequality involving rearrangements of functions is proved. The result implies convergence in Orlicz spaces of the rearrangements if there is convergence in Orlicz spaces of the original functions.

A. Azzan and E.Kreyszig [1] consider a partial differential equation of the form [image omitted]  and assert L εPEO Using the classification scheme introduced by E. Kreyszing in [2] and [3] we shall show L εPOP.

In the paper [1] we have to change the definition of the class introduced at pag. 209. The new definition is now the following: is the subset of whose elements f satisfy the following assumption:for every λ > 0.If the same holds for bf for any constant λ > 0, and moreover Theorem 3, page 209, is now valid for any λ > 0. Theorem 4 still holds in the same form, and the slight modifications to the proofs of theorems 3 and 4 are now obvious. No other essential changements are required in the applications.

This paper uses a time-weighted cross-sectional area measure in order to establish spatial decay estimates for the time-dependent Stokes slow flow of an incompressible viscous fluid in a semi- infinite cylindrical pipe of smooth cross section. The decay rate predicted in this paper depends only on the constant kinematic viscosity and the smallest positive eigenvalue of the free membrane problem as well as two geometric characteristics of the cross section of the pipe.

The use of bifurcation analysis for the study of the long—term dynamic behaviour of food webs is discussed. Food webs are collections of populations with often very complicated interactions. The non—viable nutrients are consumed by populations which are consumed by other populations at higher trophic levels, except for the top—predator species. In the most simple setting the dynamics of each population is described mathematically by an ordinary differential equation (ODE). In this paper we give an overview of local as well as global bifurcations found for a simple food web model. With chaotic behaviour all points in a next—return map appear to lie close to single curve. A cubic map with two critical points is studied and the results obtained are used to clarify global bifurcations of the ODE system. These global bifurcations are homoclinic bifurcations of a saddle limit cycle and a heteroclinic bifurcation from the equilibrium to the limit cycle. The homoclinic bifurcation of the saddle limit cycle is associated with a boundary crisis where a chaotic attractor is abruptly destroyed by a collision with the saddle limit cycle. The complex geometry of the basin of attraction of positive attractors is related to the existence of heteroclinic orbits. Furthermore, a homoclinic bifurcation for the saddle—focus equilibrium forms a 'skeleton' for the period doubling and tangent bifurcations of a limit cycle. Most results were already discussed in earlier papers; in this paper we emphasize the biological interpretations. For example, the transcritical bifurcations will be associated with invasion of species into an ecosystem.

The spectral decomposition theorem for a class of nonselfadjoint operators in a Hilbert space is obtained in the paper. These operators are the dynamics generators for the systems governed by 1–dim hyperbolic equations with spatially nonhomogeneous coefficients containing first order damping terms and subject to linear nonselfadjoint boundary conditions. These equations and boundary conditions describe, in particular, a spatially nonhomogeneous string subject to a distributed viscous damping and also damped at the boundary points. The main result leading to the spectral decomposition is the fact that the generalized eigenvectors (root vectors) of the above operators form Riesz bases in the corresponding energy spaces. The proofs are based on the transformation operators method. The classical concept of transformation operators is extended to the equation of damped string. Originally, this concept was developed by I. M. Gelfand, B. M. Levitan and V. A. Marchenko for 1–dim Schrödinger equation in connection with the inverse scattering problem. In the classical case, the transformation operator maps the exponential function (stationary wave function of the free particle) into the Jost solution of the perturbed Schrödinger equation. For the equation of a nonhomogeneous damped string, it is natural to introduce two transformation operators (outgoingand incomingtransformation operators). The terminology is motivated by an analog with the Lax—Phillips scattering theory. The transformation operators method is used to reduce the Riesz bases property problem for the generalized eigenvectors to the similar problem for a system of nonharmonic exponentials whose complex frequencies are precisely the eigenvalues of our operators. The latter problem is solved based on the spectral asymptotics and known facts about exponential families. The main result presented in the paper means that the generator of a finite string with damping both in the equation and in the boundary conditions is a Riesz spectral operator. The latter result provides a class of nontrivial examples of non—selfadjoint operators which admit an analog of the spectral decomposition. The result also has significant applications in the control theory of distributed parameter systems.