Optimal problems of the best proximity pair by proximal normal structure. (English) Zbl 07850789
Summary: Let \((A_1,A_2,A_3)\) be a triple of nonempty convex subsets of a metric space \(\Omega\). In this paper, we determine optimal problems of the best proximity pair by proximal normal structure between two sets \(A_1\) and \(A_2\) with the help of a third set \(A_3\) and we find some necessary and sufficient conditions for existence this optimal problems. Also, we provide an example to illustrate the convergence behavior of our proposed results.
MSC:
| 41A65 | Abstract approximation theory (approximation in normed linear spaces and other abstract spaces) |
| 41A52 | Uniqueness of best approximation |
| 46N10 | Applications of functional analysis in optimization, convex analysis, mathematical programming, economics |
References:
| [1] | H. Aydi, On common fixed point theorems for (ψ, φ)-generalized f -weakly contractive mappings, Miskolc Mathematical Notes, 14 (1) (2013), 19-30. · Zbl 1299.54082 |
| [2] | H. Aydi, M. Abbas and C. Vetro, Common Fixed points for multi-valued generalized contractions on partial metric spaces, RACSAM, 108 (2014), 483-501. · Zbl 1433.54021 |
| [3] | I. Altun and A. Tasdemir, On best proximity points of in-terpolative proximal contractions, Quaestiones Math. (2020), doi.org/10.2989/16073606.2020.1785576. · Zbl 1477.54046 · doi:10.2989/16073606.2020.1785576 |
| [4] | L. Chen, “Common best proximity points theorems,”Journal of Management Research and Analysis, 39, (3) (2019) 289-294. · Zbl 1449.54056 |
| [5] | A. A. Eldred, W. A. Kirk and P. Veeramani, Proximinal nor-mal structure and relatively nonexpansive mappings, Studia Math., 171(3)(2005) 283-293. · Zbl 1078.47013 |
| [6] | A. A. Eldred and P. Veeramani, Existence and convergence of best proximity points,J. Math. Anal. Appl., 323(2006), 1001-1006. · Zbl 1105.54021 |
| [7] | M. Gabeleh, A characterization of proximal normal structure via proximal diametral sequences,J. Fixed Point Theory Appl. 19(4), (2017) 2909-2925. · Zbl 1493.47063 |
| [8] | M. R. Haddadi and V. Parvaneh, Global Optimal Approximate Solutions of Best Proximity Points, Filomat, 35(5) (2021), 1555-1564. |
| [9] | H. Işık, V. Parvaneh, and M. R. Haddadi, “Strong and pure fixed point properties of mappings on normed spaces,” Mathematical Sci-ences, (2021) 15,305-309. · Zbl 1492.47057 |
| [10] | S.K. Jain, G. Meena, D. Singh, J.K. Maitra, Best proximity point results with their consequences and applications.J. Inequal. Appl. (2022), 2022, 73. · Zbl 1506.54019 |
| [11] | E. Karapinar, R. Agarwal, and H. Aydi, Interpolative Reich-Rus-Ciric contractions on metric spaces,Mathematics 6 (2018), 256, doi.org/10.3390/math6110256. · Zbl 1469.54127 · doi:10.3390/math6110256 |
| [12] | W. A. Kirk and N. Shahzad, Normal structure and orbital fixed point conditions,J. Math. Anal. Appl. 463(2), (2018) 461-476. · Zbl 1392.54032 |
| [13] | P. Kumam and C. Mongkolekeha, “Common best proximity points for proximity commuting mapping with Geraghty”s func-tions,”Carpathian Journal of Mathematics, 31, (3) (2015) 359-364. · Zbl 1389.54096 |
| [14] | D. O’REGAN, Equilibria for abstract economies in Hausdorff topo-logical vector spaces,Constructive Mathematical Analysis, 5(2), 2022, 54-59. · Zbl 1497.91171 |
| [15] | Raj V. Sankar ,P. Veeramani, Best proximity pair theorems for rela-tively nonexpansive mappings,Appl. Gen. Topol., 10(1)(2009)21-28. · Zbl 1213.47062 |
| [16] | S. Sadiq Basha, N. Shahzad, and R. Jeyaraj, “Common best proximity points: global optimization of multi-objective func-tions,”Applied Mathematics Letters, 24, (6)(2011) 883-886 · Zbl 1281.54018 |
| [17] | S. Sadiq Basha, Best proximity points: global optimal approximate solution,J. Global Optim., 49(1) (2011) 15-21. · Zbl 1208.90128 |
| [18] | S. Sadiq Basha, Best proximity points: optimal solutions,J. Optim. Theory Appl. 151 (2011), 210-216. · Zbl 1226.90135 |
| [19] | S. Sadiq Basha, N. Shahzad, K. Sethukumarasamy, Relative Continuity, Proximal Boundedness and Best Proximity Point Theorems,Numerical Functional Analysis and Optimization, 43 (4),(2022), 394-411. · Zbl 07530315 |
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