In this article, by utilizing the idea of controlled functions, we present a novel notion of triple controlled quasi rectangular metric like spaces and prove Banach fixed point principal in such spaces. A topology in such spaces and its topological properties have been discussed. The result, presented here is a new contribution to the field of fixed point theory. Examples of this new structure are given.
Citation: Mazhar Mehmood, Abdullah Shoaib, Nabil Mlaiki. Fixed point results on triple controlled quasi rectangular metric like spaces[J]. AIMS Mathematics, 2023, 8(5): 10049-10066. doi: 10.3934/math.2023509
In this article, by utilizing the idea of controlled functions, we present a novel notion of triple controlled quasi rectangular metric like spaces and prove Banach fixed point principal in such spaces. A topology in such spaces and its topological properties have been discussed. The result, presented here is a new contribution to the field of fixed point theory. Examples of this new structure are given.
[1] | S. Banach, Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales, Fund. Math., 3 (1922), 133–181. https://doi.org/10.4064/fm-3-1-133-181 doi: 10.4064/fm-3-1-133-181 |
[2] | W. Shatanawi, On w-compatible mappings and common coupled coincidence point in cone metric spaces, Appl. Math. Lett., 25 (2012), 925–931. https://doi.org/10.1016/j.aml.2011.10.037 doi: 10.1016/j.aml.2011.10.037 |
[3] | W. Shatanawi, R. V. Rajic, S. Radenovic, A. Al-Rawashhdeh, Mizoguchi-Takahashi-type theorems in tvs-cone metric spaces, Fixed Point Theory A., 2012 (2012), 106. |
[4] | A. Al-Rawashdeh, A. Hassen, F. Abdelbasset, S. Sehmim, W. Shatanawi, On common fixed points for $\alpha$-$F$-contractions and applications, J. Nonlinear Sci. Appl., 9 (2016), 3445–3458. https://doi.org/10.22436/jnsa.009.05.128 doi: 10.22436/jnsa.009.05.128 |
[5] | W. Shatanawi, Z. Mustafa, N. Tahat, Some coincidence point theorems for nonlinear contraction in ordered metric spaces, Fixed Point Theory A., 2011 (2011), 68. |
[6] | W. Shatanawi, Some fixed point results for a generalized $\psi$-weak contraction mappings in orbitally metric spaces, Chaos Soliton. Fract., 45 (2012), 520–526. https://doi.org/10.1016/j.chaos.2012.01.015 doi: 10.1016/j.chaos.2012.01.015 |
[7] | I. Altun, H. Sahin, M. Aslantas, A new approach to fractals via best proximity point, Chaos Soliton. Fract., 146 (2021), 110850. https://doi.org/10.1016/j.chaos.2021.110850 doi: 10.1016/j.chaos.2021.110850 |
[8] | H. Sahin, M. Aslantas, I. Altun, Best proximity and best periodic points for proximal nonunique contractions, J. Fix. Point Theory A., 23 (2021), 55. https://doi.org/10.1007/s11784-021-00889-7 doi: 10.1007/s11784-021-00889-7 |
[9] | M. Aslantas, H. Sahin, I. Altun, Ciric type cyclic contractions and their best cyclic periodic points, Carpathian J. Math., 38 (2022), 315–326. https://doi.org/10.37193/CJM.2022.02.04 doi: 10.37193/CJM.2022.02.04 |
[10] | I. Bakhtin, The contraction mapping principle in quasimetric spaces, Funct. Anal., 30 (1989), 26–37. |
[11] | T. Kamran, M. Samreen, Q. UL Ain, A generalization of b-metric space and some fixed point theorems, Mathematics, 5 (2017), 19. https://doi.org/10.3390/math5020019 doi: 10.3390/math5020019 |
[12] | S. Shukla, Some fixed point theorems for ordered contractions in partial b-metric spaces, Gazi Univ. J. Sci., 30 (2017), 345–354. |
[13] | S. Shukla, Partial b-metric spaces and fixed point theorems, Mediterr. J. Math., 11 (2014), 703–711. https://doi.org/10.1007/s00009-013-0327-4 doi: 10.1007/s00009-013-0327-4 |
[14] | N. Mlaiki, H. Aydi, N. Souayah, T. Abdeljawad, Controlled metric type spaces and the related contraction principle, Mathematics, 6 (2018), 194. https://doi.org/10.3390/math6100194 doi: 10.3390/math6100194 |
[15] | T. Abdeljawad, N. Mlaiki, H. Aydi, N. Souayah, Double controlled metric type spaces and some fixed point results, Mathematics, 6 (2018), 320. https://doi.org/10.3390/math6120320 doi: 10.3390/math6120320 |
[16] | N. Mlaiki, Double controlled metric-like spaces, J. Inequal. Appl., 2020 (2020), 1–12. https://doi.org/10.1186/s13660-020-02456-z doi: 10.1186/s13660-020-02456-z |
[17] | W. A. Wilson, On quasi-metric spaces, Am. J. Math., 53 (1931), 675–684. https://doi.org/10.2307/2371174 |
[18] | D. Doitchinov, On completeness in quasi-metric spaces, Topol. Appl., 30 (1988), 127–148. https://doi.org/10.1016/0166-8641(88)90012-0 doi: 10.1016/0166-8641(88)90012-0 |
[19] | K. Abodayeh, W. Shatanawi, D. Turkoglu, Some fixed point theorems in quasi-metric spaces under quasi weak contractions, Glob. J. Pure Appl. Math., 12 (2016), 4771–4780. |
[20] | F. M. Zeyada, G. H. Hassan, M. A. Ahmed, A generalization of a fixed point theorem due to Hitzler and Seda in dislocated quasi-metric spaces, Arab. J. Sci. Eng., 31 (2006), 111. |
[21] | S. Haque, A. K. Souayah, N. Mlaiki, D. Rizk, Double controlled quasi metric like spaces, Symmetry, 14 (2022), 618. https://doi.org/10.3390/sym14030618 doi: 10.3390/sym14030618 |
[22] | A. Branciari, A fixed point theorem of Banach-Caccioppoli type on a class of generalized metric spaces, 2000. https://doi.org/10.5486/PMD.2000.2133 |
[23] | R. George, S. Radenović, K. P. Reshma, S. Shukla, Rectangular $b$-metric space and contraction principles, J. Nonlinear Sci. Appl., 8 (2015), 1005–1013. https://doi.org/10.22436/jnsa.008.06.11 doi: 10.22436/jnsa.008.06.11 |
[24] | N. Mlaiki, M. Hajji, T. Abdeljawad, A new extension of the rectangular-metric spaces, Adv. Math. Phys., 2020. |
[25] | S. G. Matthews, Partial metric topology, Ann. N. Y. Acad. Sci., 728 (1994), 183–197. https://doi.org/10.1111/j.1749-6632.1994.tb44144.x |
[26] | S. Shukla, Partial rectangular metric spaces and fixed point theorems, Sci. World J., 2014 (2014), 756298. https://doi.org/10.1155/2014/756298 doi: 10.1155/2014/756298 |
[27] | S. Haque, F. Azmi, N. Mlaiki, Fredholm type integral equation in controlled rectangular metric-like spaces, Symmetry, 14 (2022), 991. https://doi.org/10.3390/sym14050991 doi: 10.3390/sym14050991 |
[28] | M. Asim, K. S. Nisar, A. Morsy, M. Imdad, Extended rectangular $\xi$-metric spaces and fixed point results, Mathematics, 7 (2019), 1136. https://doi.org/10.3390/math7121136 doi: 10.3390/math7121136 |
[29] | M. Asim, M. Imdad, S. Radenovic, Fixed point results in extended rectangular b-metric spaces with an application, UPB Sci. Bull., Ser. A, 81 (2019), 11–20. |
[30] | M. Asim, M. Imdad, S. Shukla, Fixed point results for Geraghty-weak contractions in ordered partial rectangular b-metric spaces, Afr. Mat., 32 (2021), 811–827. https://doi.org/10.1007/s13370-020-00862-6 doi: 10.1007/s13370-020-00862-6 |
[31] | M. Asim, S. Mujahid, I. Uddin, Meir-Keeler contraction in rectangular m-metric space, Topol. Algebra Appl., 9 (2021), 96–104. https://doi.org/10.1515/taa-2021-0106 doi: 10.1515/taa-2021-0106 |
[32] | M. Asim, Meenu, Fixed point theorem via Meir-Keeler contraction in rectangular Mb-metric spaces, Korean J. Math., 30 (2022), 161–173. |