Volume 9, Issue 2

New fixed point results for a very general class of maps on hyperconvex spaces are given. Our theory relies on the fact that hyperconvex compact spaces are absolute retracts. Also maps on 𝑘-CAR sets are discussed.

A three-dimensional real nonlinear autonomous system of a concrete type is studied. The Hopf bifurcation is analyzed and the existence of a limit cycle is proved. A positively invariant set, which is globally attractive, is found using a suitable Lyapunov-like function. Corollaries for a cubic system are presented. Also, a two-dimensional nonlinear system is studied as a restricted system. An application in economics to the Kodera's model of inflation is presented. In some sense, the model of inflation is an extension of the dynamic version of the neo-keynesian macroeconomic IS-LM model and the presented results correspond to the results for the IS-LM model.

For an operator 𝑇 acting from an infinite-dimensional Hilbert space 𝐻 to a normed space 𝑌 we define the upper AMD-number and the lower AMD-number as the upper and the lower limit of the net ( δ (𝑇| 𝐸 )) 𝐸∈𝐹𝐷(𝐻) , with respect to the family 𝐹𝐷(𝐻) of all finite-dimensional subspaces of 𝐻. When these numbers are equal, the operator is called AMD-regular. It is shown that if an operator 𝑇 is compact, then and, conversely, this property implies the compactness of 𝑇 provided 𝑌 is of cotype 2, but without this requirement may not imply this. Moreover, it is shown that an operator 𝑇 has the property if and only if it is superstrictly singular. As a consequence, it is established that any superstrictly singular operator from a Hilbert space to a cotype 2 Banach space is compact. For an operator 𝑇, acting between Hilbert spaces, it is shown that and are respectively the maximal and the minimal elements of the essential spectrum of , and that 𝑇 is AMD-regular if and only if the essential spectrum of |𝑇| consists of a single point.

An optimal decomposition formula for the norm in the Orlicz space 𝐿(log 𝐿) α is given. New proofs of some results involving 𝐿(log 𝐿) α spaces are given and the decomposition is applied to apriori estimates for elliptic partial differential equations with the right-hand side in Zygmund classes.

The method of lower and upper solutions combined with the monotone iterative technique is used for ordinary differential equations with nonlinear boundary conditions. Some existence results are formulated for such problems.

Let ϕ be a modulus function, i.e., continuous strictly increasing function on [0, ∞), such that ϕ (0) = 0, ϕ (1) = 1, and ϕ (𝑥+𝑦) ≤ ϕ (𝑥)+ ϕ (𝑦) for all 𝑥, 𝑦 in [0, ∞). It is the object of this paper to characterize, for any Banach space 𝑋, extreme points, exposed points, and smooth points of the unit ball of the metric linear space ℓ ϕ (𝑋), the space of all sequences (𝑥 𝑛 ), 𝑥 𝑛 ∈ 𝑋, 𝑛 = 1, 2, . . . , for which ∑ ϕ (‖𝑥 𝑛 ‖) < ∞. Further, extreme, exposed, and smooth points of the unit ball of the space of bounded linear operators on ℓ 𝑝 , 0 < 𝑝 < 1, are characterized.

We consider convexly independent subsets of a given infinite set of points in the plane (Euclidean space) and evaluate the cardinality of such subsets. It is demonstrated, in particular, that situations are essentially different for countable and uncountable point sets.

The solvability conditions are established for a problem of linear conjugation for analytic functions with boundary values from the Zygmund class 𝐿(ln + 𝐿) α when the conjugation coefficient is piecewise-continuous in the Hölder sense. Solutions of the problem are constructed in explicit form.

Some fixed point theorems based on an asymptotic regularity condition have been obtained, which generalize the previously well-known results.

The properties of two families of s -orthogonal polynomials, which are connected with Chebyshev polynomials of third and fourth kind, are studied. Evaluations of the remainders are given and asymptotic formulae are calculated for the corresponding hyper-Gaussian formulae used for an approximate estimation of integrals.

We prove some results about the existence of fixed points, periodic points and chaotic-like dynamics for a class of planar maps which satisfy a suitable property of “arc expansion” type. We also outline some applications to the nonlinear Hill's equations with indefinite weight.

Doubly-connected and triply-connected domains close to each other in a certain sense are considered. Some questions connected with conformal and quasiconformal mappings of such domains are studied using integral equations.

We consider the mean-variance hedging and utility maximization problems under partial information for diffusion models of the stock price process. The special feature of this paper is that we construct a strong innovation process for the stock price process which allows us to reduce the partial information case to the full information one.

The missing References section in the authors' recent paper (Georgian Math. J. 8(2001), No. 3, 571–614) is reproduced.