Best proximity coincidence point results for \((\alpha,D)\)-proximal generalized Geraghty mappings in \(JS\)-metric spaces. (English) Zbl 1477.54162
Summary: We introduce a type of Geraghty contractions in a \(JS\)-metric space \(X\), called \((\alpha,D)\)-proximal generalized Geraghty mappings. By using the triangular-\((\alpha,D)\)-proximal admissible property, we obtain the existence and uniqueness theorem of best proximity coincidence points for these mappings together with some corollaries and illustrative examples. As an application, we give a best proximity coincidence point result in \(X\) endowed with a binary relation.
MSC:
| 54H25 | Fixed-point and coincidence theorems (topological aspects) |
| 54E40 | Special maps on metric spaces |
Keywords:
Geraghty contraction; \(JS\)-metric space; \((\alpha,D)\)-proximal generalized Geraghty mappings; triangular-\((\alpha,D)\)-proximal admissible propertyReferences:
| [1] | Fan, K., Extensions of two fixed point theorems of F.E. Browder, Mathematische Zeitschrift, 112, 3, 234-240 (1969) · Zbl 0185.39503 · doi:10.1007/bf01110225 |
| [2] | Reich, S., Approximate selections, best approximations, fixed points, and invariant sets, Journal of Mathematical Analysis and Applications, 62, 1, 104-113 (1978) · Zbl 0375.47031 · doi:10.1016/0022-247X(78)90222-6 |
| [3] | Kirk, W. A.; Reich, S.; Veeramani, P., Proximinal retracts and best proximity pair theorems, Numerical Functional Analysis and Optimization, 24, 7-8, 851-862 (2003) · Zbl 1054.47040 · doi:10.1081/NFA-120026380 |
| [4] | Eldred, A. A.; Veeramani, P., Existence and convergence of best proximity points, Journal of Mathematical Analysis and Applications, 323, 2, 1001-1006 (2006) · Zbl 1105.54021 · doi:10.1016/j.jmaa.2005.10.081 |
| [5] | Basha, S. S., Best proximity points: global optimal approximate solutions, Journal of Global Optimization, 49, 1, 15-21 (2011) · Zbl 1208.90128 · doi:10.1007/s10898-009-9521-0 |
| [6] | Raj, V. S., A best proximity point theorem for weakly contractive non-self-mappings, Nonlinear Analysis: Theory, Methods & Applications, 74, 14, 4804-4808 (2011) · Zbl 1228.54046 · doi:10.1016/j.na.2011.04.052 |
| [7] | Bilgili, N.; Karapinar, E.; Sadarangani, K., A generalization for the best proximity point of Geraghty-contractions, Journal of Inequalities and Applications, 2013, 1 (2013) · Zbl 1281.54020 · doi:10.1186/1029-242x-2013-286 |
| [8] | Mongkolkeha, C.; Cho, Y. J.; Kumam, P., Best proximity points for Geraghty’s proximal contraction mappings, Fixed Point Theory and Applications, 2013, 1 (2013) · Zbl 1469.54158 · doi:10.1186/1687-1812-2013-180 |
| [9] | Kumam, P.; Mongkolekeha, C., Common best proximity points for proximity commuting mapping with Geraghty’s functions, Carpathian Journal of Mathematics, 31, 359-364 (2015) · Zbl 1389.54096 |
| [10] | Saleem, N.; Ansari, A. H.; Pavlović, M.; Radenović, S., Some new results in the framework of \(S_b\)-metric spaces, Scientific Publications of the State University of Novi Pazar Series A: Applied Mathematics, Informatics and Mechanics, 9, 2, 151-165 (2017) · doi:10.5937/SPSUNP1702151S |
| [11] | Saleem, N.; Vujaković, J.; Baloch, W. U.; Radenović, S., Coincidence point results for multivalued Suzuki type mappings using \(\theta \)-Contraction in \(b\)-metric spaces, Mathematics, 7, 11, article 1017 (2019) · doi:10.3390/math7111017 |
| [12] | Abbas, M.; Saleem, N.; Sohail, K., Optimal coincidence best approximation solution in B-fuzzy metric spaces, Communication in Nonlinear Analysis, 6, 1, 1-12 (2019) |
| [13] | Saleem, N.; Abbas, M.; Ali, B.; Raza, Z., Fixed points of Suzuki-type generalized multivalued \(\left( f , \theta , L\right)\)-almost contractions with applications, Univerzitet u Nišu, 33, 2, 499-518 (2019) · Zbl 1491.54150 · doi:10.2298/FIL1902499S |
| [14] | Gabeleh, M.; Markin, J., Noncyclic Meir-Keeler contractions and best proximity pair theorems, Demonstratio Mathematica, 51, 1, 171-181 (2018) · Zbl 06955035 · doi:10.1515/dema-2018-0017 |
| [15] | Ciríc, L., Some recent results in metriucal fixed point theory (2003), Serbia: University of Belgrade, Beograd, Serbia |
| [16] | Khemphet, A.; Chanthorn, P.; Phudolsitthiphat, N., Common best proximity coincidence point theorem for dominating proximal generalized Geraghty in complete metric spaces, Journal of Function Spaces, 2020 (2020) · Zbl 1516.54032 · doi:10.1155/2020/9620254 |
| [17] | Işık, H.; Aydi, H.; Mlaiki, N.; Radenović, S., Best proximity point results for Geraghty type \(Ƶ\)-Proximal contractions with an application, Axioms, 8, 3, 81 (2019) · Zbl 1432.54060 · doi:10.3390/axioms8030081 |
| [18] | Gabeleh, M.; Künzi, H.-P. A., Equivalence of the existence of best proximity points and best proximity pairs for cyclic and noncyclic nonexpansive mappings, Demonstratio Mathematica, 53, 1, 38-43 (2020) · Zbl 1446.47013 · doi:10.1515/dema-2020-0005 |
| [19] | Geraghty, M. A., On contractive mappings, Proceedings of the American Mathematical Society, 40, 2, 604-608 (1973) · Zbl 0245.54027 · doi:10.1090/S0002-9939-1973-0334176-5 |
| [20] | Ayari, M. I., A best proximity point theorem for \(\alpha \)-proximal Geraghty non-self mappings, Fixed Point Theory and Applications, 2019, 1 (2019) · Zbl 1435.54017 · doi:10.1186/s13663-019-0661-8 |
| [21] | Abbas, M.; Hussain, A.; Kumam, P., A coincidence best proximity point problem in \(G\)-Metric spaces, Abstract and Applied Analysis, 2015 (2015) · Zbl 1470.54024 · doi:10.1155/2015/243753 |
| [22] | de la Sen, M.; Abbas, M.; Saleem, N., On optimal fuzzy best proximity coincidence points of fuzzy order preserving proximal \(\Psi\left( \sigma , \alpha\right)\)-lower-bounding asymptotically contractive mappings in non-archimedean fuzzy metric spaces, SpringerPlus, 5, 1, article 1478 (2016) · doi:10.1186/s40064-016-3116-2 |
| [23] | Hussain, N.; Kutbi, M. A.; Salimi, P., Best proximity point results for modified \(\alpha-\psi \)-proximal rational contractions, Abstract and Applied Analysis, 2013 (2013) · Zbl 1470.54065 · doi:10.1155/2013/927457 |
| [24] | Hussain, N.; Hezarjaribi, M.; Kutbi, M. A.; Salimi, P., Best proximity results for Suzuki and convex type contractions, Fixed Point Theory and Applications, 2016, 1 (2016) · Zbl 1347.54085 · doi:10.1186/s13663-016-0499-2 |
| [25] | Hussain, N.; Khan, A. R.; Iqbal, I., Common best proximity results for multivalued proximal contractions in metric space with applications, Journal of Nonlinear Sciences and Applications, 9, 6, 4814-4828 (2016) · Zbl 1470.54063 · doi:10.22436/jnsa.009.06.117 |
| [26] | Hussain, A.; Iqbal Qamar, M.; Hussain, N., Best proximity point results for Suzuki-Edelstein proximal contractions via auxiliary functions, Filomat, 33, 2, 435-447 (2019) · Zbl 1499.54175 · doi:10.2298/fil1902435h |
| [27] | Al-Sulami, H. H.; Hussain, N.; Ahmad, J., Best proximity results with applications to nonlinear dynamical systems, Mathematics, 7, 10, 900 (2019) · doi:10.3390/math7100900 |
| [28] | Ansari, A. H.; Razani, A.; Hussain, N., Fixed and coincidence points for hybrid rational Geraghty contractive mappings in ordered \(b\)-metric spaces, The International Journal of Nonlinear Analysis and Applications, 8, 315-329 (2017) · Zbl 1428.54013 |
| [29] | Ansari, A. H.; Razani, A.; Hussain, N., New best proximity point results in G-metric space, Journal of Linear and Topological Algebra, 6, 73-89 (2017) · Zbl 1413.54099 |
| [30] | Amini-Harandi, A.; Hussain, N.; Akbar, F., Best proximity point results for generalized contractions in metric spaces, Fixed Point Theory and Applications, 2013, 1 (2013) · Zbl 1297.47061 · doi:10.1186/1687-1812-2013-164 |
| [31] | Amini-Harandi, A.; Fakhar, M.; Hajisharifi, H.; Hussain, N., Some new results on fixed and best proximity points in preordered metric spaces, Fixed Point Theory and Applications, 2013, 1 (2013) · Zbl 1469.54054 · doi:10.1186/1687-1812-2013-263 |
| [32] | Zhang, J.; Su, Y.; Cheng, Q., A note on ‘a best proximity point theorem for Geraghty-contractions’, Fixed Point Theory and Applications, 2013, 1 (2013) · Zbl 1423.54104 · doi:10.1186/1687-1812-2013-99 |
| [33] | Komal, S.; Kumam, P.; Khammahawong, K.; Sitthithakerngkiet, K., Best proximity coincidence point theorems for generalized non-linear contraction mappings, Filomat, 32, 19, 6753-6768 (2018) · Zbl 1491.54105 · doi:10.2298/fil1819753k |
| [34] | Jleli, M.; Samet, B., A generalized metric space and related fixed point theorems, Fixed Point Theory and Applications, 2015, 1 (2015) · Zbl 1312.54024 · doi:10.1186/s13663-015-0312-7 |
| [35] | Samet, B.; Vetro, C.; Vetro, P., Fixed point theorems for \(\alpha-\psi \)-contractive type mappings, Nonlinear Analysis, 75, 4, 2154-2165 (2012) · Zbl 1242.54027 · doi:10.1016/j.na.2011.10.014 |
| [36] | Karapınara, E., \( \alpha-\psi \)-Geraghty contraction type mappings and some related fixed point results, Filomat, 28, 1, 37-48 (2014) · Zbl 1450.54017 · doi:10.2298/fil1401037k |
| [37] | Khemphet, A., The existence theorem for a coincidence point of some admissible contraction mappings in a generalized metric space, Thai Journal of Mathematics: Special Issue, 2020, 223-235 (2020) · Zbl 1474.54180 |
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