Measures of noncompactness on \(w\)-distance spaces. (English) Zbl 1462.54026

Summary: The aim of this paper is to provide a new framework for the study of measures of noncompactness in generalized metric spaces. Firstly, we introduce the notion of \(w\)-measure of noncompactness on metric spaces with a \(w\)-distance and extend the diameter and Kuratowski functionals to this setting. At the end, we give a characterization of metric completeness via our main results, providing a new answer to the open question mentioned by I. Arandjelovic [Stavovi o presecanju i njihove primene u nelinearnoj analizi (Serbian). Belgrade: University of Belgrade (PhD Thesis) (1999)].


54E50 Complete metric spaces
47H08 Measures of noncompactness and condensing mappings, \(K\)-set contractions, etc.
55M20 Fixed points and coincidences in algebraic topology
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[1] I. D. Arandjelović,Measures of noncompactness on uniform spaces, Mathematica Moravica, 2 (1998), 1-8. · Zbl 1007.54024
[2] I. Arandjelović,Stavovi o presecanju i njihove primene u nelinearnoj analizi(in Serbian), PhD thesis, Faculty of Mathematics, University of Belgrade, 1999.
[3] J. Banaś, M. Mursaleen,Sequence spaces and measures of noncompactness with applications to differential and integral equations, Springer, 2014. · Zbl 1323.47001
[4] G. Darbo,Punti uniti in trasformazioni a codominio non compatto, Rendiconti del Seminario Matematico della Università di Padova, 24 (1955), 84-92. · Zbl 0064.35704
[5] D. Ilić, V. Rakočević,Common fixed points for maps on metric space withw-distance, Applied Mathematics and Computation, 199 (2008), 599-610. · Zbl 1143.54018
[6] O. Kada, T. Suzuki, W. Takahashi,Nonconvex minimization theorems and fixed point theorems in complete metric spaces, Mathematica Japonica, 44 (1996), 381-391. · Zbl 0897.54029
[7] A. Kostić,Best proximity points revisited, Filomat, 33 (16) (2019), 5159-5166. Aleksandar Kostić69 · Zbl 1491.54107
[8] A. Kostić, E. Karapinar, V. Rakočević,Best proximity points and fixed points withRfunctions in the framework ofw-distances, Bulletin of the Australian Mathematical Society, 99 (3) (2019), 497-507. · Zbl 1502.54043
[9] A. Kostić, V. Rakočević, S. Radenović,Best proximity points involving simulation functions withw0-distances, Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas, 113 (2019), 715-727. · Zbl 1489.54162
[10] K. Kuratowski,Sur les espaces complets, Fundamenta Mathematicae, 15 (1930), 301-309.
[11] E. Malkowsky, V. Rakočević,Introduction into the theory of sequence spaces and measures of noncompactness, Zbornik Radova, 17 (2000), 143-234. · Zbl 0996.46006
[12] V. Rakočević,Measures of noncompactness and some applications, Filomat, 12 (2) (1998), 87-120. · Zbl 1009.47047
[13] M. Singha, K. Sarkar,Towards Cantor intersection theorem and Baire category theorem in partial metric spaces, Matematički Vesnik, 69 (2) (2017), 126-132. · Zbl 1474.54091
[14] T. Suzuki, W. Takahashi,Fixed point theorems and characterizations of metric completeness, Topological Methods in Nonlinear Analysis, 8 (1996), 371-382. · Zbl 0902.47050
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