Best proximity points for generalized \((\mathcal{F},\mathcal{R})\)-proximal contractions. (English) Zbl 07905354
Summary: We present the notion of generalized \((\mathcal{F},\mathcal{R})\)-proximal non-self contractions and prove best proximity point theorems in complete metric spaces endowed with an arbitrary binary relation. An example is given to vindicate our claims. We also show that the edge preserving structure is a particular case of the binary relation \(\mathcal{R}\). Moreover, an application to variational inequality problem is given in order to demonstrate the efficacy of our results.
MSC:
| 47H10 | Fixed-point theorems |
| 54H25 | Fixed-point and coincidence theorems (topological aspects) |
| 46N40 | Applications of functional analysis in numerical analysis |
| 46T99 | Nonlinear functional analysis |
Keywords:
best proximity point; generalized \((\mathcal{F},\mathcal{R})\)-proximal contractions; binary relationReferences:
| [1] | Alam, A., Imdad, M., Relation-theoretic contraction principle, J. Fixed Point Theory Appl. 17(4), 693-702, (2015). · Zbl 1335.54040 |
| [2] | Alam, A., Imdad, M. Relation-theoretic metrical coincidence theorems, Filomat 31, 4421-4439, (2017). · Zbl 1477.54037 |
| [3] | Alam, A., Imdad, M., Monotone generalized contractions in ordered metric spaces, Bull. Korean Math. Soc. 53(1), 61-81, (2016). · Zbl 1337.54033 |
| [4] | Alam, A., Khan, A.R., Imdad, M., Some coincidence theorems for generalized nonlinear contractions in ordered metric spaces with applications, Fixed Point Theory Appl. 216, (2014). · Zbl 1505.54060 |
| [5] | Alam, A., George, R., Imdad, M., Refinements to Relation-Theoretic Contraction Principle, Axioms 11, 316, (2022). |
| [6] | Altun, I., Aslantas, M., Sahin, H., KW-type nonlinear contractions and their best proximity points, Numer. Funct. Anal. Optim. 24, 1-20, (2021). |
| [7] | Altun, I., Aslantas, M., Sahin, H., Best proximity point results for p-proximal contractions, Acta. Math. Hung. 162, 393-402, (2020). · Zbl 1474.54107 |
| [8] | 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 |
| [9] | Basha, S.S., Veeramani, P., Best proximity pair theorems for multifunctions with open fibres, J. Approx. Theory 103, 119-129, (2000). · Zbl 0965.41020 |
| [10] | Basha, S.S., Extensions of Banach’s contraction principle, Numer. Funct. Anal. Optim. 31, 569-576, (2010). · Zbl 1200.54021 |
| [11] | Basha, S.S., Best proximity points: optimal solutions, J. Optim. Theory Appl. 151, 210-216, (2011). · Zbl 1226.90135 |
| [12] | Beg, I., Bartwal, A., Rawat, S., Dimri, R.C., Best proximity points in noncommutative Banach spaces, Comp. Appl. Math. 41, 41 (2022). · Zbl 1489.54077 |
| [13] | Debnath, P., Konwar, N., Radenovic, S., Metric fixed point theory: Applications in science, engineering and behavioural sciences, 314-316, (2021). · Zbl 1478.54001 |
| [14] | Dhivya, P., Radenović, S., Marudai, M., Bin-Mohsin, B., Coupled fixed point and best proximity point results involving simulation functions, Bol. Soc. Paran. Mat. 22, 1-15, (2004). |
| [15] | Eldered, A.A., Veeramani, P., Existence and convergence for best proximity points, J. Math. Anal. Appl. 33, 1001-1006, (2004). |
| [16] | Golubović, Z., Kadelburg, Z., Radenović, S., Coupled coincidence points of mappings in ordered partial metric spaces, Abstr. Appl. Anal. Article ID 192581, (2012). · Zbl 1241.54033 |
| [17] | Harjani, J., Sadarangani, K., Fixed point theorems for weakly contractive mappings in partially ordered sets, Nonlinear Anal. 71, 3403-3410, (2009). · Zbl 1221.54058 |
| [18] | Jachymski, J., The contraction principle for mappings on a metric space with a graph, Proc. Am. Math. Soc. 136(4), 1359-1373, (2008). · Zbl 1139.47040 |
| [19] | Karapınar, E., Roldán, A., Shahzad, N., Sintunavarat, W., Discussion of coupled and tripled coincidence point theorems for φ-contractive mappings without the mixed g-monotone property, Fixed Point Theory Appl. 2014(1), 1-16, (2014). · Zbl 1469.54132 |
| [20] | Lipschitz, S., Schaum’s Outlines of Theory and Problems of Set Theory and Related Topics, McGraw-Hill, New York, (1964). |
| [21] | Raj, V.S. A best proximity point theorem for weakly contractive non-self mappings, Nonlinear Anal. 74(11), 4804-4808, (2011). · Zbl 1228.54046 |
| [22] | Raj, V.S., Best proximity point theorems for nonself mappings, Fixed Point Theory 14(2), 447-454, (2013). · Zbl 1280.41026 |
| [23] | Rawat, S., Dimri, R.C., Bartwal, A., F-Bipolar metric spaces and fixed point theorems with applications, J. Math. Comput. SCI-JM. 26(2), 184-195, (2022). |
| [24] | Rawat, S., Kukreti, S., Dimri, R.C., Fixed point results for enriched ordered contractions in noncommutative Banach spaces, J. Anal. 30, 1555-1566, (2022). · Zbl 1505.47059 |
| [25] | Sahin, H., Aslantas, M., Altun, I., Feng-Liu type approach to best proximity point results for multivalued mappings, J. Fixed Point Theory Appl. 22(1), 1-13, (2020). · Zbl 1431.54039 |
| [26] | Sultana, A., Vetrivel, V., On the existence of best proximity points for generalized contractions, Appl. Gen. Topol. 15(1), 55-63, (2014). · Zbl 1297.54109 |
| [27] | Samet, B., Turinici, M., Fixed point theorems on a metric space endowed with an arbitrary binary relation and appli-cations, Commun. Math. Anal. 13(2), 82-97, (2012). · Zbl 1259.54024 |
| [28] | Senapati, T., Dey, L.K., Relation-theoretic metrical fixed point results via w-distance with applications, J. Fixed Point Theory Appl. 19, 2945-2961, (2017). · Zbl 1489.54221 |
| [29] | Turinici, M., Abstract comparison principles and multi-variable Gronwall-Bellman inequalities, J. Math. Anal. Appl. 117(1), 100-127, (1986). · Zbl 0613.47037 |
| [30] | Turinici, M., Contractive operators in relational metric spaces, Handbook of Functional Equations, Optimization and Its Appl., Springer, 95, 419-458, (2014). · Zbl 1321.54103 |
| [31] | Turinici, M., Fixed points for monotone iteratively local contractions, Dem. Math. 19(1), 171-180, (1986). · Zbl 0651.54020 |
| [32] | Turinici, M., Nieto-López theorems in ordered metric spaces, Math. Student 81(4), 219-229, (2012). · Zbl 1296.54095 |
| [33] | Turinici, M., Ran-Reuring’s theorems in ordered metric spaces, J. Indian Math. Soc. 78, 207-214, (2011). · Zbl 1251.54057 |
| [34] | Wardowski, D., Fixed points of a new type of contractive mappings in complete metric spaces, Fixed Point Theory Appl. 2012, 94, (2012). · Zbl 1310.54074 |
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