We prove a trace theorem that allows the treatment of Neumann problems with nonlinearities on the boundary in anisotropic spaces with variable exponent. Then we proceed to the study of such a problem that involves general operators of the \overrightarrow{p}(.)-Laplace type. We deduce the existence of solutions and we direct attention to the situation where the solution is unique.

COBISS.SI-ID: 16320601

We propose a general approach to defining a contractive-like multivalued mapping F which avoids any use of the Hausdorff distance between the sets F(x) and F(y). Various fixed point theorems are proved under a two-parameter control of the distance function d_F(x) = {\rm dist}(x,F(x)) between a point x \in X and the value F(x) \subset X. Here, both parameters are numerical functions. The first one \alpha \colon [0,+\infty) \to [1,+\infty) controls the distance between x and some appropriate point y \in F(x) in comparison with d_F(x), whereas the second one \beta \colon [0,+\infty) \to [0,1) estimates d_F(y) with respect to d(x,y). It appears that the well harmonized relations between \alpha and \beta are sufficient for the existence of fixed points of F. Our results generalize several known fixed point theorems.

COBISS.SI-ID: 16224857

The notion of max-plus convex subset of Euclidean space can be naturally extended to other linear spaces. The aim of this paper is to describe the topology of hyperspaces of max-plus convex subsets of Tychonov powers {\mathbb R}^\tau of the real line. We show that the corresponding spaces areabsolute retracts if and only if \tau \le \omega_1.

COBISS.SI-ID: 16310361