Teori Titik Tetap untuk Tipe Kannan yang Diperumum dalam Ruang b-Metrik Modular Lengkap

Afifah Hayati; Noor Sofiyati; Dwiani Listya Kartika

doi:10.34312/jjom.v5i2.20571

Teori Titik Tetap untuk Tipe Kannan yang Diperumum dalam Ruang b-Metrik Modular Lengkap

Afifah Hayati, Noor Sofiyati, Dwiani Listya Kartika

Abstract

Some generalizations of Banach's contraction principle, which is a fixed-point theorem for contraction mapping in metric spaces, have developed rapidly in recent years. Some of the things that support the development of generalization are the emergence of mappings that are more general than contraction mappings and the emergence of spaces that are more general than metric spaces. The generalized Kannan type mappings are one of the mappings that are more general than contraction mappings. Furthermore, some of the more general spaces than metric spaces are b-metric spaces and modular b-metric spaces, which bring the concept of b-metric spaces into modular spaces. The fixed-point theorems for generalized Kannan-type mappings on b-metric spaces have been given. Therefore, this research aims to define generalized Kannan-type mappings on modular b-metric spaces and provide fixed point theorems for the generalized Kannan-type mappings on complete modular b-metric spaces. The definition of generalized Kannan type mapping in modular b-metric spaces is given by generalizing generalized Kannan type mappings in b-metric spaces. Then, the proof of fixed-point theorems for that mapping in modular b-metric spaces is carried out analogously to the proof of the fixed-point theorems for that mapping given in b-metric space. In this article, we obtain the definition of Kannan-type mappings and fixed-point theorems for generalized Kannan-type mappings in modular b-metric spaces and some consequences of the fixed-point theorem. In proving the theorem, a property of altering distance functions in b-metric spaces is generalized into modular b-metric spaces.

Keywords

Modular b-Metric Space; Altering Distance Function; Kanan-type Mapping

Full Text:

PDF

References

R. P. Agarwal, M. Meehan, and D. O’Regan, Fixed Point Theory and Applications. Cambridge University Press, 2001, doi: 10.1017/CBO9780511543005.

S. Banach, “Sur les operations dans les ensembles abstraits et leur application aux ´ equations ´ integrales,” ´ Fundamenta Mathematicae, vol. 3, no. 1, pp. 133–181, 1922, [online]. available: http://eudml.org/doc/213289.

J. Morales and E. Rojas, “Some fixed point theorems by altering distance functions,” Palestine Journal of Mathematics, vol. 1, no. 2, pp. 110–116, 2012.

M. Khan, M. Swaleh, and S. Sessa, “Fixed point theorems by altering distances between the points,” Bulletin of the Australian Mathematical Society, vol. 30, no. 1, pp. 1–9, 1984, doi: 10.1017/S0004972700001659.

S. Czerwik, “Contraction mappings in b-metric spaces,” Acta Mathematica et Informatica Universitatis Ostraviensis, vol. 01, no. 1, pp. 5–11, 1993, [online]. available: http://eudml.org/doc/23748.

R. Kannan, “Some results on fixed points–ii,” The American Mathematical Monthly, vol. 76, no. 4, p. 405, 1969, doi: 10.2307/2316437.

J. Gornicki, “Fixed point theorems for kannan type mappings,” Journal of Fixed Point Theory and Applications, vol. 19, no. 3, pp. 2145–2152, 2017, doi: 10.1007/s11784-017-0402-8.

N. Haokip and N. Goswami, “Some fixed point theorems for generalized Kannan type mappings in b-metric spaces,” Proyecciones (Antofagasta), vol. 38, no. 4, pp. 763–782, 2019, doi: 10.22199/issn.0717-6279-2019-04-0050.

J. Musielak and W. Orlicz, “On modular spaces,” Studia Mathematica, vol. 18, no. 1, pp. 49–65, 1959, [online]. available: http://eudml.org/doc/216946.

V. V. Chistyakov, “Modular metric spaces, i: Basic concepts,” Nonlinear Analysis: Theory, Methods & Applications, vol. 72, no. 1, pp. 1–14, 2010, doi: https://doi.org/10.1016/j.na.2009.04.057.

M. E. Ege and C. Alaca, “Some results for modular b-metric spaces and an application to system of linear equations,” Azerbaijan Journal of Mathematics, vol. 8, no. 1, pp. 1–11, 2018.

V. Parvaneh and S. J. Hosseini Ghoncheh, “Fixed points of (y, j)w-contractive mappings in ordered p-metric spaces,” Global Analysis and Discrete Mathematics, vol. 4, no. 1, pp. 15–29, 2019, doi: 10.22128/gadm.2019.290.1019.

A. Gholidahneh, S. Sedghi, O. Ege, Z. D. Mitrovic, and M. de la Sen, “The meir-keeler type contractions in extended modular b-metric spaces with an application,” AIMS Mathematics, vol. 6, no. 2, pp. 1781–1799, 2021, doi: 10.3934/math.2021107.

A. Hayati, “Some coincidence point theorems in modular spaces,” Mathline : Jurnal Matematika dan Pendidikan Matematika, vol. 7, no. 1, pp. 91–109, 2022, doi: 10.31943/mathline.v7i1.260.

R. Ozc¸elik and E. E. Kara, “Coincidence point theorems on ¨ b metric spaces via cfFgsimulation functions,” Communications in Advanced Mathematical Sciences, vol. 2, no. 4, pp. 244–250, 2019, doi: 10.33434/cams.567268.

M. Berzig and M. Bouali, “A coincidence point theorem and its applications to fractional differential equations,” Journal of Fixed Point Theory and Applications, vol. 22, no. 3, p. 56, 2020, doi: 10.1007/s11784-020-00794-5.

A. Hayati, L. Harini, A. Winarni, and N. Muhassanah, “Teori titik tetap untuk pemetaan (y,j)w-kontraksi pada ruang p-metrik modular berorder,” PYTHAGORAS Jurnal Pendidikan Matematika, vol. 17, no. 2, 2022, doi: 10.21831/pythagoras.v17i2.52985.

J. Hernadi, “Metoda pembuktian dalam matematika,” Jurnal Pendidikan Matematika, vol. 2, no. 1, pp. 1–13, 2018.

M. Demma, R. Saadati, and P. Vetro, “Fixed point result on b-metric spaces via picard sequences and b-simulation functions,” Iranian Journal of Mathematical Sciences and Informatics, vol. 11, no. 1, pp. 123–136, 2016.

P. Corazza, “Introduction to metric-preserving functions,” The American Mathematical Monthly, vol. 106, no. 4, pp. 309–323, 1999, doi: 10.2307/2589554.

DOI: https://doi.org/10.34312/jjom.v5i2.20571