Fixed points of Kannan maps in modular metric spaces

Afrah. A. N. Abdou; Afrah. A. N. Abdou

doi:10.3934/math.2020411

AIMS Mathematics

2020, Volume 5, Issue 6: 6395-6403. doi: 10.3934/math.2020411

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Research article

Fixed points of Kannan maps in modular metric spaces

Afrah. A. N. Abdou ^,

Department of Mathematics, Faculty of Sciences, University of Jeddah, Jeddah 21589, Saudi Arabia

Received: 01 July 2020 Accepted: 30 July 2020 Published: 12 August 2020
MSC : 46B20, 47H10, 47E10

The notion of a modular metric on an arbitrary set and the corresponding modular spaces, generalizing classical modulars over linear spaces like Orlicz spaces, were recently introduced. In this paper we study the existence of fixed points for contractive and nonexpansive Kannan maps in the setting of modular metric spaces.These are related to the successive approximations of fixed points (via orbits) which converge to the fixed points in the modular sense, which is weaker than the metric convergence.
- fixed point,
- Kannan contraction mapping,
- Kannan nonexpansive mapping,
- modular metric spaces
Citation: Afrah. A. N. Abdou. Fixed points of Kannan maps in modular metric spaces[J]. AIMS Mathematics, 2020, 5(6): 6395-6403. doi: 10.3934/math.2020411

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Abstract

The notion of a modular metric on an arbitrary set and the corresponding modular spaces, generalizing classical modulars over linear spaces like Orlicz spaces, were recently introduced. In this paper we study the existence of fixed points for contractive and nonexpansive Kannan maps in the setting of modular metric spaces.These are related to the successive approximations of fixed points (via orbits) which converge to the fixed points in the modular sense, which is weaker than the metric convergence.

References

[1]	A. A. Abdou, M. A Khamsi, Fixed point results of pointwise contractions in modular metric spaces, Fixed Point Theory Appl., 2013 (2013), 163.
[2]	A. A. Abdou, M. A Khamsi, On the fixed points of nonexpansive maps in modular metric spaces, Fixed Point Theory Appl., 2013 (2013), 229.
[3]	S. Banach, Sur les opérations dans les ensembles abstraits et leurs applications, Fund. Math., 3 (1922), 133-181. doi: 10.4064/fm-3-1-133-181
[4]	V. V. Chistyakov, Modular metric spaces, I: Basic concepts, Nonlinear Anal., 72 (2010), 1-14.
[5]	V. V. Chistyakov, Modular metric spaces, Ⅱ: Application to superposition operators, Nonlinear Anal., 72 (2010), 15-30.
[6]	R. Kannan, Some results on fixed points-Ⅱ, Am. Math. Mon., 76 (1969), 405-408.
[7]	M. A. Khamsi, W. A. Kirk, An Introduction to metric spaces and fixed point theory, John Wiley, New York, 2001.
[8]	M. A. Khamsi, W. K. Kozlowski, S. Reich, Fixed point theory in modular function spaces, Nonlinear Anal., 14 (1990), 935-953.
[9]	M. A. Khamsi, W. M. Kozlowski, Fixed point theory in modular function spaces, Birkhauser New York, 2015.
[10]	W. M. Kozlowski, Modular function spaces, Series of Monographs and Textbooks in Pure and Applied Mathematics, Dekker, New York/Basel, 122 (1988).
[11]	J. Musielak, Orlicz spaces and modular spaces, Lecture Notes in Mathematics, Springer-Verlag, Berlin/Heidelberg/New York/Tokyo, 1034 (1983).
[12]	H. Nakano, Modulared semi-ordered linear spaces, Maruzen Co., Tokyo, 1950.
[13]	W. Orlicz, Über konjugierte exponentenfolgen, Studia Math., 3 (1931), 200-211. doi: 10.4064/sm-3-1-200-211
[14]	M. Paknazar, M. D. Sen, Best proximitym point results in non-archimedean modular metric space, Mathematics, 5 (2017), 23.
[15]	M. Paknazar, M. D. Sen, Some new approaches to modular and fuzzy metric and related best proximity results, Fuzzy Set. Syst., 390 (2020), 138-159. doi: 10.1016/j.fss.2019.12.012

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