New Fixed Point Results for Generalized Θ-Contraction in Extended 𝐆𝐛-Metric Spaces with an Application

Year 2021, Volume: 34 Issue: 3, 808 - 819, 01.09.2021

Sumaiya Tasneem Z Kalpana G Thabet Abdeljawad

https://doi.org/10.35378/gujs.712142

Abstract

Through this work, we analyze the structure of extended Gb-metric spaces and show a fundamental lemma for sequence convergence within the same metric. We also propose the new notion of generalized geraghty type Θ-berinde contraction mappings and demonstrate several fixed point theorems for these mappingsin the sense of extended Gb-metric spaces. Eventually, the existence result for solutions of a Fredholm integral equation is furnished to show the efficacy of the technique developed. 

Keywords

Extended Gb-metric spaces, Θ-contraction, Fixed point

References

• [1] Abbas, M., Kim, J. K., Nazir, T.: Common fixed point of mappings satisfying almost generalized contractive condition in partially ordered G-metric spaces: J. Comput. Anal. Appl., 19(6), 2015, 928-938. [2] Aghajani, A., Abbas, M., Roshan, J. R.: Common fixed point of generalized weak contractive mappings in partially ordered Gb-metric spaces: Filomat., 28(6), 2014, 10871101. [3] Ahmad, J., Al-Mazrooei, A. E., Altun, I.: Generalized θ-contractive fuzzy mappings: J. Intell. Fuzzy Syst., 35, 2018, 1935-1942. [4] Ameer, E., Arshad, M., Shatanawi, W.: Common fixed point results for generalized α∗ψ-contarction multivalued mappings in b-metric spaces: J. Fixed Point Theory Appl., 19(4), 2017, 3069-3086. [5] Aydi, H., Felhi, A., Sahmim, S.: Related fixed point results for cyclic contractions on G-metric spaces and application: Filomat., 31(3), 2017, 853-869. [6] Aydi, H., Postolache, M., Shatanawi, W.: Coupled fixed point results for (ψ-φ)-weakly contractive mappings in ordered G-metric spaces: Comput. Math. Appl., 63 (1), 2012, 298-309. [7] Bakhtin, I. A.: The contraction mapping principle in almost metric space: Funct. Anal., 30, 1989, 26-37. [8] Berinde, V.: Approximating fixed points of weak φ-contractions using the picard iteration: Fixed Point Theory., 4(2), 2003, 131-147. [9] Chandok, S., Mustafa, Z., Postolache, M.: Coupled common fixed point results for mixed g-monotone maps in partially ordered G-metric spaces: Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys., 75(4), 2013, 13-26. [10] Czerwik, C.: Contraction mapping in b-metric spaces: Acta Math. Inform. Univ. Ostrav., 1, 1993, 5-11. [11] Dinarvand, M.: Some fixed point results for admissible Geraghty contraction type mappings in fuzzy metric spaces: Iran. J. Fuzzy Syst., 14(3), 2017, 161-177. [12] Geraghty, M.: On contractive mappings: Proc. Amer. Nath. Soc., 40, (1973), 604-608. [13] Imdad, M., Alfaqih, W. M., Khan, I. A.: Weak θ-contractions and some fixed point results with applications to fractal theory: Adv. Differ. Equ. 2018, 439, 2018. [14] Jleli, M., Samet, B.: A new generalization of the banach contraction principle: J. Inequal. Appl. 2014, 38, (2014). [15] Liu, X., Zhou, M., Damjanovi´c, B.: Common coupled fixed point theorem for geraghtytype contraction in partially ordered metric spaces: J. Funct. Spaces, vol. 2018, Article ID 9063267, 11 pages, 2018. [16] Mudasir Younis, Deepak Singh, Dhananjay Gopal, Anil Goyal, Mahendra Singh Rathore: On applications of generalized F-contraction to differential equations: Nonlinear funct. anal. appl., 24(1), 2019, 155-174. [17] Mustafa, Z., Jaradat, M., Jaradat, H. M.: A remarks on the paper ”Some fixed point theorems for generalized contractive mappings in complete metric spaces”: J. Math. Anal., 8, 2017, 17-22. [18] Mustafa, Z., Parvaneh, V., Mohammed Jaradat, M.M., Kadelburg, Z.: Extended rectangular b-metric spaces and some fixed point theorems for contractive mappings. Symmetry 2019, 11, 594. [19] Mustafa, Z., Aydi, H., Karapinar, E.: On common fixed points in G-metric spaces using (E.A) property: Comput. Math. Appl., 64 (6), 2012, 1944-1956. [20] Parvaneh, V., Kadelburg, Z.: Extended partial b-metric spaces and some fixed point results: Filomat., 32(8), 2018, 2837-2850. [21] Popovi´c, B., Shoaib, M., Sarwar, M.: Coupled fixed point theorems for generalized (ψ,φ)weak contraction in partially ordered G-metric spaces: J. Comput. Anal. Appl., 23(1), 2017. [22] Shatanawi, W., Bataihah, A., Pitea, A.: Fixed and common fixed point results for cyclic mappings of Ω-distance: J. Nonlinear Sci. Appl., 9(3), 2016, 727-735. [23] Zand, M. R. A., Nezhad, A. D.: A generalization of partial metric spaces: J. Contemp. Appl. Math., 1(1), 2011, 86-93. [24] Thabet Abdeljawad, Ravi P Agarwal, Karapinar, E., Sumati Kumari, P.: Solutions of the nonlinear integral equation and fractional differential equation using the technique of a fixed point with a numerical experiment in extended b-metric space: Symmetry 2019, 11, 686.