arXiv subjects

Mateusz Krukowski

Publications and source records attributed to Mateusz Krukowski.

23 records · Page 2Linked to original sources

arXiv2016

Short note on Smithson's paper concerning Arzelà-Ascoli theorem for multifunctions

As indicated in the title, this paper was inspired by Smithson's work on Arzelà-Ascoli theorem. Up to this point, the author's interest in the topic has led him to generalize Arzelà-Ascoli theorem in the context of uniform spaces. This approach was primarily influenced by J.L. Kelley's \textit{General topology}. Soon after, the author learnt that in 1971, R.E. Smithson carried over Kelley's approach to the case of point-compact multifunctions (rather than single-valued functions). Similarly, in this paper the author would like to carry over the result, presented in his previous paper \textit{Arzelà-Ascoli theorem in uniform spaces}, to the case of point-compact multifunctions.

math.FA↗

arXiv2016

Darbo-type theorem for quasimeasure of noncompactness

The paper introduces the concept of quasimeasure of noncompactness. Motivated by the Arzelà-Ascoli theorem for $C^b(X,E)$, where $X$ is an Euclidean space and $E$ an arbitrary Banach space, we construct a quasimeasure for this space and study its properties. An analogon for Darbo fixed point theorem is obtained with the additional aid of measure of nonconvexity. The paper ends with possible application in integral equations. We prove that a Hammerstein operator with Carathéodory kernel and nonlinearity of a certain type has a fixed point.

math.FA↗

arXiv2016

Arzelà-Ascoli theorem via Wallman compactification

In the paper, we recall the Wallman compactification of a Tychonoff space $T$ (denoted by $\text{Wall}(T)$) and the contribution made by Gillman and Jerison. Motivated by the Gelfand-Naimark theorem, we investigate the homeomorphism between $BC(T,\mathbb{R})$ and $BC(\text{Wall}(T),\mathbb{R})$. Along the way, we attempt to justify the advantages of Wallman compactification over other manifestations of Stone-Čech compactification. The main result of the paper is a new form of Arzelà-Ascoli theorem, which introduces the concept of equicontinuity along $ω$-ultrafilters.

math.GN↗

arXiv2016

Arzelà-Ascoli theorem in uniform spaces

In the paper, we generalize the Arzelà-Ascoli theorem in the setting of uniform spaces. At first, we recall well-known facts and theorems coming from monographs of Kelley and Willard. The main part of the paper introduces the notion of extension property which, similarly as equicontinuity, equates different topologies on $C(X,Y)$. This property enables us to prove the Arzelà-Ascoli theorem for uniform convergence. The paper culminates with applications, which are motivated by Schwartz's distribution theory. Using the Banach-Alaoglu-Bourbaki theorem, we establish relative compactness of subfamily of $C(\mathbb{R},\mathcal{D}'(\mathbb{R}^n))$.

math.FA↗

arXiv2015

Compactness result and its applications in integral equations

A version of Arzelà-Ascoli theorem for $X$ being $σ$-locally compact Hausdorff space is proved. The result is used in proving compactness of Fredholm, Hammerstein and Urysohn operators. Two fixed point theorems, for Hammerstein and Urysohn operator, are derived on the basis of Schauder theorem.

math.FA↗