Dariusz Bugajewski

BVφ-solutions of nonlinear integral equations

Abstract In this paper we investigate solutions of nonlinear Hammerstein and Volterra–Hammerstein integral equations in the space of functions of bounded φ-variation in the sense of Young. We prove the existence and in some cases the existence and uniqueness of local and global solutions in this class. Real-valued as well as vector-valued functions are under our consideration. The method of our proofs is based on an application of the Banach contraction principle as well as the Leray–Schauder alternative for contractions.

On Some Properties of Hyperconvex Spaces

We are going to answer some open questions in the theory of hyperconvex metric spaces. We prove that in complete -trees hyperconvex hulls are uniquely determined. Next we show that hyperconvexity of subsets of normed spaces implies their convexity if and only if the space under consideration is strictly convex. Moreover, we prove a Krein-Milman type theorem for -trees. Finally, we discuss a general construction of certain complete metric spaces. We analyse its particular cases to investigate hyperconvexity via measures of noncompactness.

Nonautonomous superposition operators in the spaces of functions of bounded variation

The main goal of this paper is to give an answer to the main problem of the theory of nonautonomous superposition operators acting in the space of functions of bounded variation in the sense of Jordan. Namely, we give necessary and sufficient conditions which guarantee that nonautonomous superposition operators map that space into itself and are locally bounded. Moreover, special attention is drawn to nonautonomous superposition operators generated by locally bounded mappings as well as to superposition operators generated by functions with separable variables.

On fixed point theorems of Leray–Schauder type

In this paper we prove a few fixed points theorems of Leray-Schauder type in hyperconvex metric spaces.

On the Darboux problem of neutral type

The aim of this paper is to prove uniqueness theorems for the Darboux problem of neutral type in the space L ∞ and L 1 .

SOME REMARKS ON HYPERCONVEX METRIC SPACES AND THE SMALL BALL PROPERTY

In this paper we deal with hyperconvex metric spaces. We compare a certain class of hyperconvex metric spaces with another one: complete R-trees. An elementary proof of hyperconvexity of “finite” R-trees is also presented. In the second part of the paper we introduce and investigate an index which measures how far a subset of a given metric space is from having the small ball property introduced by Behrends and Kadets.

On Topological Properties of Metrics Defined via Generalized “Linking Construction”

We analyze topological properties of metric spaces obtained by using Szaz’s construction, which we used to call generalized “linking construction.” In particular, we provide necessary and sufficient conditions for completeness of metric spaces obtained in this way. Moreover, we examine the relation between Szaz’s construction and the “linking construction.” A particular attention is drawn to the “floor” metric, the analysis of which provides some interesting observations.