Volume 9, Issue 2

Motivated by the minimal tower problem, an earlier work studied diagonalizations of covers where the covers are related to linear quasiorders ( τ -covers). We deal with two types of combinatorial questions which arise from this study. 1. Two new cardinals introduced in the topological study are expressed in terms of well known cardinals characteristics of the continuum. 2. We study the additivity numbers of the combinatorial notions corresponding to the topological diagonalization notions. This gives new insights on the structure of the eventual dominance ordering on the Baire space, the almost inclusion ordering on the Rothberger space, and the interactions between them.

The existence of global weak solutions for coupled thermoelasticity with the nonlinear contact boundary condition and Barber's heat exchange condition is proved via the Faedo-Galerkin, monotonicity and compactness methods. Some a priori bounds obtained with Gronwalls inequality in connection with the embedding and trace theorems lead to accomplishing a generalization of our previous study [Bień, Math. Methods Appl. Sci. 19: 1265–1277, 1996]. The heat-exchange coefficient associated with Barber's heat exchange condition is dependent only on the normal displacement. This dependence is described by a bounded Lipschitz function. Moreover, this study is some extension of works due to Andrews et al. [Shi, Shillor, Wright, Appl. Math. Optim. 28: 11–48, 1993] and Elliot et al. [Tang, Nonlinear Anal. 23: 883–898, 1994].

Nonlinear vector integral equations are considered. Solution estimates and solvability conditions are derived. Applications to the periodic boundary value problem are also discussed. Under some restrictions our results improve the well-known ones. The main tool in the paper is the recent estimates for the resolvent of Hilbert-Schmidt operators.

The existence of a positive radial solution to the Dirichlet boundary value problem for the second order elliptic equation –Δ u = f ( u , ∫ U u ), where U = B (0, R ) \ B (0, ρ ), with weak assumptions on the nonlinear term f , is proved. The method based on the Krasnosel'skii Fixed Point Theorem enables to find many solutions to the problem. Solutions for the same problem but with U = B (0, R ) and with nonlinear term f depending explicitely on | x | are found as well.

In this paper using the critical point theory of Chang [J. Math. Anal. Appl. 80: 102–129, 1981] for locally Lipschitz functionals we prove an existence theorem for noncoercive Neumann problems with discontinuous nonlinearities. We use the mountain-pass theorem to obtain a nontrivial solution.

In this paper, we give a coincidence theorem, a minimax theorem, a section theorem, an intersection theorem and two existence theorems of solutions for generalized quasi-variational inequalities in hyperconvex metric spaces.

In the paper there are given new applications of dimension reduction method (from [Liczberski, Starkov, Siberian Math. J. 42: 715–730, 2001]) of studing linearly invariant families of holomorphic mappings in . There also included generalizing modifications of this method.

In the paper we present two continuous selection theorems in hyperconvex metric spaces and apply these to study fixed point and coincidence point problems as well as variational inequality problems in hyperconvex metric spaces.

The present paper gives characterizations of radially u.s.c. convex and pseudoconvex functions f : X → ℝ defined on a convex subset X of a real linear space E in terms of first and second-order upper Dini-directional derivatives. Observing that the property f radially u.s.c. does not require a topological structure of E , we draw the possibility to state our results for arbitrary real linear spaces. For convex functions we extend a theorem of Huang, Ng [Math. Oper. Res. 22: 747–753, 1997]. For pseudoconvex functions we generalize results of Diewert, Avriel, Zang [J. Econom. Theory 25: 397–420, 1981] and Crouzeix [Generalized Convexity, Generalized Monotonicity: Recent Results: 237–256, Kluwer Academic Publisher, 1998]. While some known results on pseudoconvex functions are stated in global concepts (e.g. Komlosi [Math. Pro Programming 26: 232–237, 1983]), we succeeded in realizing the task to confine to local concepts only.

We study MB-representations of algebras and ideals when they are relativized to a subset, and when one considers the operations of sum and intersection for families of algebras and ideals. We observe that the algebras , 3 ≤ α ≤ w 1 , on ℝ are MB-representable under GCH. We find a class of topological spaces in which the algebra of clopen sets is MB-representable.

We present new semilocal convergence theorems for Newton methods in a Banach space. Using earlier general conditions we find more precise error estimates on the distances involved using the majorant principle. Moreover we provide a better information on the location of the solution. In the special case of Newton's method under Lipschitz conditions we show that the famous Newton–Kantorovich hypothesis having gone unchallenged for a long time can be weakened under the same hypotheses/computational cost.