In this paper, we introduce two new classes of mixed $ (\mathcal{S, T}) $-$ \alpha $-admissible mappings and interspersed $ (\mathcal{S}, \mathfrak{g}, \mathcal{T}) $-$ \alpha $-admissible mappings and study the sufficient conditions for the existence and uniqueness of a common fixed point of generalized $ (\alpha_s, \xi, \hbar, \tau) $-Geraghty contractive mapping in the framework of partial $ b $-metric spaces. We also provide two examples to show the applicability and validity of our results. Moreover, we present an application to the existence of solutions to an integral equation by means of one of our results.
Citation: Ying Chang, Hongyan Guan. Generalized $ (\alpha_s, \xi, \hbar, \tau) $-Geraghty contractive mappings and common fixed point results in partial $ b $-metric spaces[J]. AIMS Mathematics, 2024, 9(7): 19299-19331. doi: 10.3934/math.2024940
In this paper, we introduce two new classes of mixed $ (\mathcal{S, T}) $-$ \alpha $-admissible mappings and interspersed $ (\mathcal{S}, \mathfrak{g}, \mathcal{T}) $-$ \alpha $-admissible mappings and study the sufficient conditions for the existence and uniqueness of a common fixed point of generalized $ (\alpha_s, \xi, \hbar, \tau) $-Geraghty contractive mapping in the framework of partial $ b $-metric spaces. We also provide two examples to show the applicability and validity of our results. Moreover, we present an application to the existence of solutions to an integral equation by means of one of our results.
[1] | S. Banach, Sur les operations dans les ensembles abstraits et leur application aux equations integrales, Fund. Math., 3 (1922), 51–57. https://doi.org/10.4064/fm-3-1-133-181 doi: 10.4064/fm-3-1-133-181 |
[2] | 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 |
[3] | Z. Mustafa, J. R. Roshan, V. Parvaneh, Z. Kadelburg, Some common fixed point results in ordered partial $b$-metric spaces, J. Inequal. Appl., 2013 (2013), 562. https://doi.org/10.1186/1029-242X-2013-562 doi: 10.1186/1029-242X-2013-562 |
[4] | Z. Ma, M. Nazam, S. U. Khan, X. Li, Fixed point theorems for generalized $\alpha_{s}$-$\psi$-contractions with applications, J. Funct. Space., 2018 (2018), 8368546. |
[5] | A. Mukheimer, $\alpha$-$\psi$-$\phi$-contractive mappings in ordered partial $b$-metric spaces, J. Nonliner Sci. Appl., 7 (2014), 168–179. https://doi.org/10.22436/jnsa.007.03.03 doi: 10.22436/jnsa.007.03.03 |
[6] | J. Vujaković, H. Aydi, S. Radenovic, A. Mukheimer, Some remarks and new results in ordered partial $b$-metric spaces, Mathematics, 7 (2019), 334. |
[7] | A. Latif, J. R. Roshan, V. Parvaneh, N. Hussain, Fixed point results via $\alpha$-admissible mappings and cyclic contractive mappings in partial $b$-metric spaces, J. Inequal. Appl., 2014 (2014), 345. https://doi.org/10.1186/1029-242X-2014-345 doi: 10.1186/1029-242X-2014-345 |
[8] | P. Gautam, L. M. Sánchez Ruiz, S. Verma, G. Gupta, Common fixed point results on generalized weak compatible mapping in quasi-partial $b$-metric space, J. Math-UK., 2021 (2021), 5526801. https://doi.org/10.1155/2021/5526801 doi: 10.1155/2021/5526801 |
[9] | F. La'ldolatabad, N. Saleem, M. Abbas, On the fixed points of multivalued mappings in b-metric spaces and their application to linear systems, UPB Sci. Bull. Ser. A-Appl. Math. Phys., 82 (2020), 121–130. |
[10] | H. Aydi, M. Bota, S. Moradi, A common fixed points for weak $\phi$-contractions on $b$-metric spaces, Fixed Point Theor., 13 (2012), 337–346. https://doi.org/10.1186/1687-1812-2012-88 doi: 10.1186/1687-1812-2012-88 |
[11] | M. Pacurar, A fixed point result for $\phi$-contractions and fixed points on $b$-metric spaces without the boundness assumption, Fasc. Math., 43 (2010), 127–137. |
[12] | H. Aydi, E. Karapinar, A Meir-Keeler common type fixed point theorem on partial metric spaces, Fixed Point Theory A., 2012 (2012), 26. https://doi.org/10.1186/1687-1812-2012-26 doi: 10.1186/1687-1812-2012-26 |
[13] | J. R. Roshan, V. Parvaneh, S. Sedghi, N. Shobkolaei, W. Shatanawi, Common fixed points of almost generalized $(\psi, \varphi)_{s}$-contractive mappings in ordered $b$-metric spaces, Fixed Point Theory A., 2013 (2013), 159. https://doi.org/10.1186/1687-1812-2013-123 doi: 10.1186/1687-1812-2013-123 |
[14] | M. Dinarvand, Fixed points of almost contractive type mappings in partially ordered $b$-metric spaces and applications to quadratic integral equations, Facta Univ.-Ser. Math., 31 (2016), 775–800. |
[15] | E. Ameer, M. Arshad, W. Shatanawi, Common fixed point results for generalized $\alpha_{*}$-$\psi$-contraction multivalued mappings in $b$-metric spaces, J. Fix. Point Theory A., 19 (2017), 3069–3086. https://doi.org/10.1007/s11784-017-0477-2 doi: 10.1007/s11784-017-0477-2 |
[16] | H. Tiwari, Padmavati, Berinde-type generalized $\alpha$-$\beta$-$\psi$ contractive mappings in partial metric spaces and some related fixed points, Asian J. Math., 19 (2023), 69–78. https://doi.org/10.9734/arjom/2023/v19i11754 doi: 10.9734/arjom/2023/v19i11754 |
[17] | M. B. Zada, M. Sarwar, F. Jarad, T. Abdeljawad, Common fixed point theorem via cyclic $(\alpha, \beta)$-$(\psi, \phi)_s$-contraction with applications, Symmetry, 11 (2019), 198. https://doi.org/10.3390/sym11020198 doi: 10.3390/sym11020198 |
[18] | H. Tiwari, P. Sudha, Some results on almost generalized $(\alpha, \beta, \psi, \varphi)$-Geraghty type contractive mappings in partial metric spaces, Ann. Math. Comput. Sci., 13 (2023), 84–96. |
[19] | H. Tiwari, P. Sudha, Some fixed point theoerms on almost generalized $\alpha$-$\beta$-$\psi$-$\varphi$-$\vartheta$ contractive mappings in partial metric spaces, Ann. Math. Comput. Sci., 12 (2023), 10–22. |
[20] | P. Debnath, Set-valued Meir-Keeler, Geraghty and Edelstein type fixed point results in $b$-metric spaces, Rend. Circ. Mat. Palerm. (2), 70 (2021), 1389–1398. https://doi.org/10.1007/s12215-020-00561-y doi: 10.1007/s12215-020-00561-y |
[21] | P. Debnath, Best proximity points of multivalued Geraghty contractions, Miskolc Math. Notes, 24 (2023), 119–127. https://doi.org/10.18514/MMN.2023.3984 doi: 10.18514/MMN.2023.3984 |
[22] | A. D. Turkoglu, V. Ozturk, Common fixed point results for four mappings on partial metric spaces, Abstr. Appl. Anal., 2012 (2012), 190862. https://doi.org/10.1155/2012/190862 doi: 10.1155/2012/190862 |
[23] | R. Bouhafs, A. A. Tallafha, W. Shatanawi, Fixed point theorems in ordered $b$-metric spaces with alternating distance functions, Nonlinear Funct. Anal. Appl., 26 (2021), 581–600. |
[24] | J. Li, H. Guan, Common fixed point results for generalized $(g-\alpha _{s^p}, \psi, \varphi)$ contractive mappings with applications, J. Funct. Space., 2021 (2021), 5020027. |
[25] | J. Maheswari, M. Ravichandran, Common fixed points of almost generalized $(\psi, \varphi)$-quasi rational contraction in ordered metric spaces, J. Phys. Conf. Ser., 1724 (2021), 012054. https://doi.org/10.1088/1742-6596/1724/1/012054 doi: 10.1088/1742-6596/1724/1/012054 |
[26] | G. Jungck, Compatible mappings and common fixed points, Int. J. Math., 9 (1986), 771–779. https://doi.org/10.1155/S0161171286000935 doi: 10.1155/S0161171286000935 |
[27] | H. Huang, G. Deng, S. Radenovi, Fixed point theorems in $b$-metric spaces with applications to differential equations, J. Fix. Point Theory A., 20 (2018), 52. https://doi.org/10.1007/s11784-018-0491-z doi: 10.1007/s11784-018-0491-z |
[28] | A. Horvat-Marc, M. Cufoian, A. Mitre, I. Taşcu, Fixed point theorems in rectangular $b$-metric space endowed with a partial order, Axioms, 12 (2023), 1050. https://doi.org/10.3390/axioms12111050 doi: 10.3390/axioms12111050 |
[29] | B. Mitiku, K. Karusala, S. R. Namana, Some fixed point results of generalized $(\phi, \psi)$-contractive mappings in ordered $b$-metric spaces, BMC Res. Notes, 13 (2023), 537. |
[30] | M. Zare, R. Arab, Some common fixed point results for $(\alpha$-$\psi$-$\varphi)$-contractive mappings in metric spaces, Sohag J. Math., 3 (2016), 23–29. https://doi.org/10.18576/sjm/030104 doi: 10.18576/sjm/030104 |
[31] | M. Younis, N. Fabiano, Z. Fadail, Z. Mitrovic, S. Radenovic, Some new observations on fixed point results in rectangular metric spaces with applications to chemical sciences, Vojnoteh. Glas., 69 (2021), 8–30. https://doi.org/10.5937/vojtehg69-29517 doi: 10.5937/vojtehg69-29517 |