Three-point fixed point results in b-metric spaces

Authors

Selma Gülyaz-Özyurt

Keywords:

Fixed point, b-metric, (b)-comparison function, triangular α-admissible mapping

Abstract

Motivated by recent three-point generalizations of the Banach contraction principle and by the development of fixed-point theory on b-metric spaces, we establish an α–ψb fixed point theorem for mappings that contract the perimeter of triangles in a complete b-metric space. The contractive condition is expressed in terms of a triangular α-admissible function and a (b)-comparison function ψb, which is adapted to the coefficient s of the underlying b-metric, and it controls a weighted sum of three distances between the images of pairwise distinct points. Under a mild orbital admissibility assumption and either continuity of the mapping or a standard regularity condition, we obtain the existence and uniqueness of a fixed point. Our result unifies and extends several known α–ψ type fixed point theorems in b-metric spaces and, at the same time, complements recent three-point perimeter-type fixed point results due to Petrov and coauthors.

References

[1] A. A. N. Abdou and M. F. S. Alasmari, Fixed point theorems for generalized α–ψ-contractive mappings in extended b-metric spaces, AIMS Math. 6 (2021), no. 6, 5465–5478.

[2] H. Aydi, B. Samet and C. Vetro, Fixed points of contractive mappings in b-metric-like spaces, Abstr. Appl. Anal. 2014 (2014), Article ID 851720, 13 pp.

[3] S. Banach, Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales, Fund. Math. 3 (1922), 133–181.

[4] V. Berinde and M. Păcurar, The early developments in fixed point theory on b-metric spaces: a brief survey and some important related aspects, Carpathian J. Math. 38 (2022), 523–538.

[5] R. K. Bisht and E. Petrov, A three point extension of Chatterjea’s fixed point theorem with at most two fixed points, arXiv preprint, arXiv:2403.07906, 2024.

[6] I. A. Bakhtin, The contraction mapping principle in quasi-metric spaces, Funct. Anal. 30 (1989), 26–37 (in Russian).

[7] S. Czerwik, Contraction mappings in b-metric spaces, Acta Math. Inform. Univ. Ostrav. 1 (1993), 5–11.

[8] M. A. Kutbi, E. Karapinar and H. Aydi, Fixed point results for generalized rational type α-admissible contractive mappings in partially ordered b-metric spaces, J. Funct. Spaces 2022 (2022), Article ID 8189874, 15 pp.

[9] E. Karapinar and A. Fulga, New hybrid contractions on b-metric spaces, Mathematics 7 (2019), no. 7, 578.

[10] E. Karapinar, A short survey on the recent fixed point results on b-metric spaces, Constr. Math. Anal. 1 (2018), 15–44.

[11] A. Mukheimer, α–ψ–φ-contractive mappings in ordered partial b-metric spaces, J. Nonlinear Sci. Appl. 7 (2014), 168–179.

[12] A. F. Roldán-López-de-Hierro, D. Dolián and H. Kafi, A note on the paper “Contraction mappings in b-metric spaces” by Czerwik, Acta Math. Inform. Univ. Ostrav. 25 (2017), no. 2, 223–227.

[13] I. A. Rus, Generalized Contractions and Applications, Cluj University Press, Cluj-Napoca, 2001.

[14] B. Samet, C. Vetro and P. Vetro, Fixed point theorems for α-ψ-contractive type mappings, Nonlinear Anal. 75 (2012), 2154–2165.

[15] E. Petrov, Fixed point theorem for mappings contracting perimeters of triangles, J. Fixed Point Theory Appl. 25 (2023), 1–11.

[16] E. Petrov, Periodic points of mappings contracting total pairwise distance, arXiv preprint, arXiv:2402.02536, 2024.

[17] E. Petrov and R. K. Bisht, Fixed point theorem for generalized Kannan type mappings, Rend. Circ. Mat. Palermo (2) (2024), 1–18.

[18] X. B. Wu and L. N. Zhao, Fixed point theorems for generalized α-ψ type contractive mappings in complete b-metric spaces, J. Math. Comput. Sci. 18 (2018), 49–62.