ON APPLICATIONS OF MEASURE OF NONCOMPACTNESS IN FRÉCHET SPACES
DOI:
https://doi.org/10.18372/2306-1472.79.13834Keywords:
Fréchet spaces, Fredholm operators, measure of noncompactnessAbstract
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.
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.
