International Journal of Nonlinear Analysis and Applications

Interpolative type contraction mappings in $G$-metric spaces

Document Type : Research Paper

Authors

1 Department of Mathematics, University of Management and Technology, Lahore, Pakistan

2 Department of Mathematics and Applied Mathematics, Sefako Makgatho Health Sciences University, Ga-Rankuwa, Pretoria 0204, South Africa

3 Department of Mathematics, Faculty of Science, Ataturk University, Erzurum 25240, Turkey

4 Department of Mathematics, University of Chakwal, Pakistan

Abstract

In this paper, we defined $\alpha_{G}-$admissible interpolative type contraction mappings in $G$-metric spaces. We proved some convergence results for such classes of mappings using the properties of $G$-metric space and found the fixed point results for such contractive mappings. To elaborate on the results we provided some examples, which show that our results hold in the setting of $G-$metric spaces.

Keywords

[1] M.A. Alghamdi and E. Karapinar, G−β−ψ-contractive type mappings in G-metric spaces, Fixed Point Theory Appl. 2013 (2013), 123.
[2] M.A. Alghamdi and E. Karapinar, G−β−ψ contractive-type mappings and related fixed point theorems, J. Inequal. Appl. 2013 (2013), 70.
[3] A.H. Ansari, S. Changdok, N. Hussain, Z. Mustafa, and M.M. M. Jaradat, Some common fixed point theorems for weakly α-admissible pairs in G-metric spaces with auxiliary functions, J. Math. Anal. 8 (2017), no. 3, 80–107.
[4] A.H. Ansari, J.M. Kumar, and N. Saleem, Inverse-C-class function on weak semi compatibility and fixed point theorems for expansive mappings in G-metric spaces, Math. Moravica 24 (2020), no. 1, 93–108.
[5] M. Aslantas, H. Sahin, and I. Altun, Best proximity point theorems for cyclic p−contractions with some consequences and applications, Nonlinear Anal.: Model. Control 26 (2021), no. 1, 113–129.
[6] M. Aslantas, H. Sahin, and D. Turkoglu, Some Caristi type fixed point theorems, J. Anal. 29 (2021), no. 1, 89–103.
[7] I. Altun, M. Aslantas, and H. Sahin, Best proximity point results for p−proximal contractions, Acta Math. Hungar. 162 (2020), no. 2, 393–402.
[8] S.H. Bonab, R. Abazari, A. Bagheri Vakilabad, and H. Hosseinzadeh, Generalized metric spaces endowed with vector-valued metrics and matrix equations by tripled fixed point theorems, J. Inequal. Appl. 2020 (2020), 204.
[9] S. H. Bonab, R. Abazari, A. Bagheri, and H. Hosseinzadeh, Coupled fixed point theorems on G−metric spaces via α−series, Glob. Anal. Discrete Math. 6 (2021), no. 1, 1–12.
[10] P. Debnath, N. Konwar, and S. Radenovic, Metric Fixed Point Theory: Applications in Science, Engineering and Behavioural Sciences, Springer International Publishing, 2021.
[11] Y. Errai, E.M. Marhrani, and M. Aamri, Some remarks on fixed point theorems for interpolative Kannan contraction, J. Funct. Spaces 2020 (2020), Article ID 2075920, 7 pages.
[12] Y.U. Gaba, M. Aphane, and H. Aydi, Interpolative Kannan contractions in T0-quasi-metric spaces, J. Math. 2021 (2021), Article ID 6685638, 5 pages.
[13] H. Hosseinzadeh, S.H. Bonab, K.A. Sefidab, Some common fixed point theorems for four mapping in generalized metric spaces, Thai J. Math. 20 (2022), no. 1, 425–437.
[14] N. Hussain, E. Karapinar, P. Salimi, and P. Vetro, Fixed point results for Gm-Mier-Keeler contractive and G−(α, ψ)-Mier-Keeler contractive mappings, Fixed Point Theory Appl. 2013 (2013), 34.
[15] N. Hussain, V. Parvaneh, and F. Golkarmanesh, Coupled and tripled coincidence point results under (F, g)-invariant sets in Gb-metric spaces and G − α-admissible mappings, Math. Sci. 9 (2015), no. 1, 11–26.
[16] N. Hussain, V. Parvaneh, and S.J. Hoseini Ghoncheh, Generalised contractive mappings and weakly α-admissible pairs in G-metric spaces, Sci. World J. 2014 (2014), Article ID 941086, 15 pages.
[17] E. Karapinar, Revisiting the Kannan type contractions via interpolation, Adv. Theory Nonlinear Anal. Appl. 2 (2018), no. 2, 85–87.
[18] E. Karapinar, R. Agarwal and H. Aydi, Interpolative Reich-Rus-Ciric type contractions on partial metric spaces, Mathematics 6 (2018), no. 11, 256.
[19] E. Karapinar, O. Alqahtani, and H. Aydi, On interpolative Hardy-Rogers type contractions, Symmetry 11 (2018), no. 1, 8.
[20] E. Karapinar, P. Kumam, and P. Salimi, On α-ψ-Meir-Keeler contractive mappings, Fixed Point Theory Appl. 2013 (2013), 94.
[21] B. Khomdram, Y. Rohen, Y. Mahendra Singh, and M. Saeed Khan, Fixed point theorems of generalised S-β-ψ-contractive type mappings, Math. Moravica 22 (2018), no. 1, 81–92.
[22] M.A. Kutbi, N. Hussain, J.R. Roshan, and V. Parvaneh, Coupled and Tripled Coincidence Point Results with Application to Fredholm Integral Equations, Abstr. Appl. Anal. 2014 (2014), Article ID 568718, 18 pages.
[23] N. Mlaiki, A. Mukheimer, Y. Rohen, N. Souayah, and T. Abdeljawad, Fixed point theorems for α-ψ-contractive mapping in Sb-metric spaces, J. Math. Anal. 8 (2017), no. 5, 40–46.
[24] Z. Mustafa, A new structure for generalized metric spaces: With applications to fixed point theory, PhD Thesis, the University of Newcastle, Australia, 2005.
[25] Z. Mustafa and B. Sims, A new approach to generalized metric spaces, J. Nonlinear Convex Anal. 7 (2006), no. 2, 289–297.
[26] Z. Mustafa, H. Obiedat, and F. Awawdeh, Some fixed point theorem for mapping on complete G-metric spaces, Fixed Point Theory Appl. 2008 (2008), Article ID 189870, 12 pages.
[27] Z. Mustafa, W. Shatanawi, and M. Bataineh, Existence of fixed point results in G-metric spaces, Int. J. Math. Math. Sci. 2009 (2009), Article ID 283028, 10 pages.
[28] Z. Mustafa and H. Obiedat, A fixed point theorem of Reich in G-metric spaces, CUBO 12 (2010), no. 1, 83–93.
[29] S. Phiangsungnoen, W. Sintunavarat, and P. Kumam, Fuzzy fixed point theorems for fuzzy mappings via β-admissible with applications, Fixed Point Theory Appl. 2014 (2014), 190.
[30] H. Sahin, M. Aslantas, and I. Altun, Feng-Liu type approach to best proximity point results for multivalued mappings, J. Fixed Point Theory Appl. 22 (2020), no. 1, 11.
[31] N. Saleem, M. Abbas, and Z. Raza, Fixed fuzzy point results of generalized Suzuki type F-contraction mappings in ordered metric spaces, Georg. Math. J. 27 (2020), no. 2, 307–320.
[32] N. Saleem, I. Habib, and M.D.L. Sen, Some new results on coincidence points for multivalued Suzuki-type mappings in fairly complete spaces, Computation 8 (2020), no. 1, 17.
[33] N. Saleem, T. Iqbal, B. Iqbal, and S. Radenovic, Coincidence and fixed points of multivalued F-contractions in generalized metric space with application, J. Fixed Point Theory Appl. 22 (2020), no. 4, 1–24.
[34] B. Samet, C. Vetro, and P. Vetro, Fixed point theorems for α−ψ-contractive type mappings, Nonlinear Anal. 75 (2012), 2154–2165.
[35] I. Yildirim, Fixed point results for interpolative type contractions via Mann iteration process, J. Adv. Math. Stud. 15 (2022), no. 1, 35–45.
[36] I. Yildirim, On extended interpolative single and multivalued F−contractions, Turk. J. Math. 46 (2022), 688–698.
[37] M. Zhou, X.-L. Liu, and S. Radenovic, S-γ-ϕ-φ-contractive type mappings in S-metric spaces, J. Nonlinear Sci. Appl. 10 (2017), 1613–1639.
International Journal of Nonlinear Analysis and Applications
Volume 15, Issue 12
December 2024
Pages 453-464
  • Receive Date: 27 February 2023
  • Revise Date: 22 March 2024
  • Accept Date: 23 April 2024