ON APPLICATIONS OF MEASURE OF NONCOMPACTNESS IN FRÉCHET SPACES

Authors

  • Iryna Klyus National Aviation University
  • Kaveh Eftekharinasab National Aviation University

DOI:

https://doi.org/10.18372/2306-1472.79.13834

Keywords:

Fréchet spaces, Fredholm operators, measure of noncompactness

Abstract

In metric and topological vector spaces the notion of measure of noncompactness is used to associate numerical values to sets so that compact sets get zero measures and other ones obtain positive values that indicate how far they are different from compact sets. This concept was initiated by Kuratowski in early 30s and has been defined and developed in many different ways. The indices of noncompactness can give us sufficient conditions for formulating various fixe point theorems in metric spaces. Another important application of these measurements is in characterization of Fredholm operators in infinite dimensional topological vector spaces. The object of this paper is to provide an appropriate criterion that establishes a connection   between Lipschitz-Fredholm operators in more general context of Fréchet spaces, the Hausdorff and lower measures of noncompactness. Furthermore, by using an arbitrary measure of noncompactness in the sense of Banas and Goebel we obtain a fixed point theorem for Fréchet spaces.

Author Biographies

Iryna Klyus, National Aviation University

PhD in Physics and Mathematics, Associate Professor

Associate Professor of the Department of Higher Mathematics, National Aviation University, Kyiv, Ukraine

Education: Drogobych State Pedagogical Institute, Drogobych, Ukraine, 1993

Research areas: Partial Differential Equations

Kaveh Eftekharinasab, National Aviation University

PhD in Physics and Mathematics

Department of Higher Mathematics, National Aviation University, Kyiv, Ukraine

Research areas: Geometry and Topology

References

Akhmerov R. R., Kamenskii M. I., Potapov A. S., Rodkin A. E., Sadovskii B. N. (1992) Measures of Noncompactness and Condensing Operators. Birkhauser Basel, 260 p.

Banas J., Mursaleen M., Rizv S. (2017) Advances in Nonlinear Analysis via the Concept of Measure of Noncompactness. Springer-Verlag, 485 p.

Eftekharinasab K. (2010) Sard's theorem for mappings between Fréchet manifolds. Ukrainian mathematical Journal, vol. 64, no. 12, pp. 1634–164.

Hamilton R. (1987) The inverse function theorem of Nash and Moser. Bull. Amer. Math. Soc. (N.S.), vol.7, no. 1, pp. 65-222.

Banas J., Goebel K. (1980). Measures of Noncompactness in Banach Spaces. New York, Marcel Dekker, 106 p.

Granas A., Dugundji J. (2003) Fixed point theory. Springer-Verlag, 672 p.

Published

13-08-2019

How to Cite

Klyus, I., & Eftekharinasab, K. (2019). ON APPLICATIONS OF MEASURE OF NONCOMPACTNESS IN FRÉCHET SPACES. Proceedings of National Aviation University, 79(2), 71–75. https://doi.org/10.18372/2306-1472.79.13834

Issue

Section

INFORMATION TECHNOLOGY