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Recognizing the topologies of spaces of metrics with the topology of uniform convergence

Koshino Katsuhisa

Bulletin of the Polish Academy of Sciences. Mathematics

2022

Vol. 70, no. 2

165--171

Given a metrizable space X of density κ, we study the topological structure of the space PM(X) of continuous bounded pseudometrics on X, which is endowed with the topology of uniform convergence. We prove that PM(X) is homeomorphic to [0,1)κ(κ−1)/2 if X is finite, to ℓ2(2<κ) if X is infinite and generalized compact, and to ℓ2(2κ) if X is not generalized compact. We also show that for an infinite σ-compact metrizable space X, the space M(X)⊂PM(X) of continuous bounded metrics on X and the space AM(X)⊂M(X) of bounded admissible metrics on X are homeomorphic to ℓ2 if X is compact, and to ℓ∞ if X is not compact.

On the domain of implicit functions in a projective limit setting without additional norm estimates

Magnot Jean-Pierre

Demonstratio Mathematica

2020

Vol. 53, nr 1

112--120

We examine how implicit functions on ILB-Fréchet spaces can be obtained without metric or norm estimates which are classically assumed. We obtain implicit functions defined on a domain D which is not necessarily open, but which contains the unit open ball of a Banach space. The corresponding inverse function theorem is obtained, and we finish with an open question on the adequate (generalized) notion of differentiation, needed for the corresponding version of the Fröbenius theorem.

Global attractivity for Volterra type Hadamard fractional integral equations in Fréchet spaces

Abbas S., Agarwal R. P., Benchohra M., Berhoun F.

Demonstratio Mathematica

2018

Vol. 51, nr 1

131--140

In this paper, we present some results concerning the existence and the attractivity of solutions for some functional integral equations of Hadamard fractional order. We use an extension of the Burton-Kirk fixed point theorem in Fréchet spaces.

Second order evolution equations with nonlocal conditions

Benchohra M., Nieto J. J., Rezoug N.

Demonstratio Mathematica

2017

Vol. 50, nr 1

309--319

In this paper, we shall establish sufficient conditions for the existence of solutions for second order semilinear functional evolutions equation with nonlocal conditions in Fréchet spaces. Our approach is based on the concepts of Hausdorff measure, noncompactness and Tikhonoff’s fixed point theorem. We give an example for illustration.

Regular covers and metrization

Lin S.

Bulletin of the Polish Academy of Sciences. Mathematics

2002

Vol. 50, no 4

427-432

In this paper it is showed that a regular and k-space with a regular k-network is metrizable, which generalized related results of A. Archangielskiĭ, H. W. Martin, M. Sakai, K. Tamano and Y. Yajima.