Best proximity points for alternative \(p\)-contractions. (English) Zbl 1537.54049
Summary: Cyclic mappings describe fixed paths for which each point is sequentially transmitted from one set to another. Cyclic mappings satisfying certain cyclic contraction conditions have been used to obtain the best proximity points, which constitute a suitable framework for the mirror reflection model. Alternative contraction mappings introduced by Y. C. Chen [“Best proximity points for alternative maps”, Symmetry 11, No. 6, 750, 12 p. (2019; doi:10.3390/sym11060750] built a new model containing several mirrors in which the light reflected from a mirror does not go to the next mirror sequentially, and its path may diverge to any other mirror. The aim of this paper is to present a new variant of alternative contraction called alternative \(p\)-contraction and study its properties. The best proximity point result for such contractions under the alternative UC property is proved. An example to support the result proved herein is provided.
MSC:
| 54H25 | Fixed-point and coincidence theorems (topological aspects) |
| 54E40 | Special maps on metric spaces |
| 54E50 | Complete metric spaces |
References:
| [1] | Abbas, M.; Anjumb, R.; Iqbal, H., Generalized enriched cyclic contractions with application to generalized iterated function system, Chaos Solitons Fractals, 154 (2022) · Zbl 1506.47086 · doi:10.1016/j.chaos.2021.111591 |
| [2] | Alghamdi, M. A.; Shahzad, N.; Vetro, F., Best proximity points for some classes of proximal contractions, Abstr. Appl. Anal., 2013 (2013) · Zbl 1470.54034 · doi:10.1155/2013/713252 |
| [3] | Altun, I.; Durmaz, G.; Olgun, M., P-Contractive mappings on metric spaces, J. Nonlinear Funct. Anal., 2018, 1-7 (2018) |
| [4] | Aslantas, M., Some best proximity point results via a new family of F-contraction and an application to homotopy theory, J. Fixed Point Theory Appl., 23 (2021) · Zbl 1476.54051 · doi:10.1007/s11784-021-00895-9 |
| [5] | Aslantas, M., Finding a solution to an optimization problem and an application, J. Optim. Theory Appl., 194, 1, 121-141 (2022) · Zbl 1490.54036 · doi:10.1007/s10957-022-02011-4 |
| [6] | Aslantas, M.; Sahin, H.; Altun, I., Best proximity point theorems for cyclic p-contractions with some consequences and applications, Nonlinear Anal., Model. Control, 26, 1, 113-129 (2021) · Zbl 1476.54052 · doi:10.15388/namc.2021.26.21415 |
| [7] | Aydia, H.; Lakzian, H.; Mitrović, Z. D.; Radenović, S., Best proximity points of MT-cyclic contractions with property UC, Numer. Funct. Anal. Optim., 41, 7, 871-882 (2020) · Zbl 1436.54031 · doi:10.1080/01630563.2019.1708390 |
| [8] | Banach, S., Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales, Fundam. Math., 3, 1, 133-181 (1922) · JFM 48.0201.01 · doi:10.4064/fm-3-1-133-181 |
| [9] | Basha, S. S., Extensions of Banach’s contraction principle, Numer. Funct. Anal. Optim., 31, 5, 569-576 (2010) · Zbl 1200.54021 · doi:10.1080/01630563.2010.485713 |
| [10] | Basha, S. S., Best proximity point theorems for some special proximal contractions, Numer. Funct. Anal. Optim., 40, 10, 1182-1193 (2019) · Zbl 07060057 · doi:10.1080/01630563.2019.1598431 |
| [11] | Chen, Y. C., Best proximity points for alternative maps, Symmetry, 11 (2019) · doi:10.3390/sym11060750 |
| [12] | Eldred, A. A.; Veeramani, P., Existence and convergence of best proximity points, J. Math. Anal. Appl., 323, 1001-1006 (2006) · Zbl 1105.54021 · doi:10.1016/j.jmaa.2005.10.081 |
| [13] | Fan, K., Extensions of two fixed point theorems of F.E. Browder, Math. Z., 112, 234-240 (1969) · Zbl 0185.39503 · doi:10.1007/BF01110225 |
| [14] | Gabeleh, M.; Vetro, C., A note on best proximity point theory using proximal contractions, J. Fixed Point Theory Appl., 20, 4 (2018) · Zbl 1398.90202 · doi:10.1007/s11784-018-0624-4 |
| [15] | Harjani, J.; López, B.; Sadarangani, K., Fixed point theorems for cyclic ϕ-contractions in ordered metric spaces, Fixed Point Theory, 14, 2, 359-368 (2013) · Zbl 1292.54023 |
| [16] | Hristov, M.; Ilchev, A.; Nedelcheva, D.; Zlatanov, B., Existence of coupled best proximity points of p-cyclic contractions, Axioms, 10 (2021) · doi:10.3390/axioms10010039 |
| [17] | Hristov, M.; Ilchev, A.; Zlatanov, B., On the best proximity points for p-cyclic summing contractions, Mathematics, 2020, 8 (2020) · doi:10.3390/math8071060 |
| [18] | Karapınar, E., Fixed point theory for cyclic weak ϕ-contraction, Appl. Math. Lett., 24, 6, 822-825 (2011) · Zbl 1256.54073 · doi:10.1016/j.aml.2010.12.016 |
| [19] | Karapınar, E.; Erhan, I., Cyclic contractions and fixed point theorems, Filomat, 26, 4, 777-782 (2012) · Zbl 1289.47106 · doi:10.2298/FIL1204777K |
| [20] | Kim, W. K.; Lee, K. H., Existence of best proximity pairs and equilibrium pairs, J. Math. Anal. Appl., 316, 2, 433-446 (2006) · Zbl 1101.47040 · doi:10.1016/j.jmaa.2005.04.053 |
| [21] | Kirk, W. A.; Srinivasan, P. S.; Veeramani, P., Fixed points for mappings satisfying cyclical contractive conditions, Fixed Point Theory, 4, 1, 79-89 (2003) · Zbl 1052.54032 |
| [22] | Magadevana, P.; Karpagamb, S.; Karapınar, E., Existence of fixed point and best proximity point of p-cyclic orbital φ-contraction map, Nonlinear Anal., Model. Control, 27, 1, 91-101 (2022) · Zbl 1482.54066 · doi:10.15388/namc.2022.27.25188 |
| [23] | Pǎcurar, M., Fixed point theory for cyclic Berinde operators, Fixed Point Theory, 12, 2, 419-428 (2011) · Zbl 1237.54057 |
| [24] | Pǎcurar, M.; Rus, I. A., Fixed point theory for cyclic ϕ-contractions, Nonlinear Anal., Theory Methods Appl., 72, 1181-1187 (2010) · Zbl 1191.54042 · doi:10.1016/j.na.2009.08.002 |
| [25] | Pasupathi, R.; Chand, A. K.B.; Navascués, M. A., Cyclic Meir-Keeler contraction and its fractals, Numer. Funct. Anal. Optim., 42, 9, 1053-1072 (2021) · Zbl 1494.47090 · doi:10.1080/01630563.2021.1937215 |
| [26] | Popescu, O., A new type of contractive mappings in complete metric spaces, Bull. Transilv. Univ. Brasov, Ser. III, Math. Inform. Phys., 1, 50, 479-482 (2008) · Zbl 1299.54109 |
| [27] | Sahin, H., A new type of F-contraction and their best proximity point results with homotopy application, Acta Appl. Math., 179, 1 (2022) · Zbl 1498.54100 · doi:10.1007/s10440-022-00496-9 |
| [28] | Sahin, H.; Aslantas, M.; Altun, I., Best proximity and best periodic points for proximal nonunique contractions, J. Fixed Point Theory Appl., 2021, 23 (2021) · Zbl 1476.54113 · doi:10.1007/s11784-021-00889-7 |
| [29] | Suzuki, T.; Kikkawa, M.; Vetro, C., The existence of best proximity points in metric spaces with the property UC, Nonlinear Anal., 71, 2918-2926 (2009) · Zbl 1178.54029 · doi:10.1016/j.na.2009.01.173 |
| [30] | Vetro, C., Best proximity points: convergence and existence theorems for p-cyclic mappings, Nonlinear Anal., 73, 2283-2291 (2010) · Zbl 1229.54066 · doi:10.1016/j.na.2010.06.008 |
| [31] | Weng, S.; Liu, X.; Chao, Z., Some fixed point theorems for cyclic mapping in a complete b-metric-like space, Results Nonlinear Anal., 3, 4, 207-213 (2020) |
| [32] | Zhelinski, V., Zlatanovon, B.: The UC and \(UC^*\) properties and the existence of best proximity points in metric spaces. ArXiv (2023). arXiv:2303.05850 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
