Research Article
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Year 2021, Volume: 70 Issue: 1, 130 - 142, 30.06.2021
https://doi.org/10.31801/cfsuasmas.780723

Abstract

References

  • Alghamdi, M. A., Shahzad, N., Vetro F., Best proximity points for some classes of proximal contractions, Abstract and Applied Analysis, 2013 (2013), Article ID: 713252.
  • Ali, M. U., Bejenaru, A., Kamran, T., The order-convergence of the Thakur iterative process for Hardy-Rogers contractions in order-Banach spaces, Journal of Mathematical Analysis, 9(4) (2018), 61-74.
  • Aliprantis, C. D., Border, K. C., Infinite dimensional analysis, Berlin, Springer-Verlag, 1999.
  • Aydi, H., Lakzian, H., Mitrović, Z. D., Radenović, S., Best proximity points of MT-cyclic contractions with property UC, Numerical Functional Analysis and Optimization 41 (2020), 1-12.
  • Banach, S., Sur les opérations dans les ensembles abstraits et leur applications aux équations intégrales, Fund. Math, 3 (1922), 133-181.
  • Basha, S. S., Extensions of Banach's contraction principle, Numer. Funct. Anal. Optim., 31 (5) (2010), 569-576.
  • Basha, S. S., Shahzad, N., Vetro, C., Best proximity point theorems for proximal cyclic contractions, Journal of Fixed Point Theory and Applications, 19 (4) (2017), 2647-2661.
  • Çevik, C., On Continuity Functions Between Vector Metric Spaces, Journal of Function Spaces, (2014), Article ID 753969.
  • Çevik, C., Altun, I., Vector metric spaces and some properties, Topol. Methods Nonlinear Anal., 34 (2009), 375--382.
  • Çevik, C., Altun, I., Sahin, H., Ozeken, C. C., Some fixed point theorems for contractive mapping in ordered vector metric spaces, J. Nonlinear Sci. Appl., 10(4) (2017), 1424--1432.
  • Eldred, A. A., Veeramani, P., Existence and convergence of best proximity points, J. Math. Anal. Appl., 323 (2006), 1001-1006.
  • Jleli, M., Samet, B., Best proximity points for α-ψ-proximal contractive type mappings and application, Bull. Sci. Math., 137 (2013), 977-995.
  • Nashine, H. K., Kumam, P., Vetro, C., Best proximity point theorems for rational proximal contractions, Fixed Point Theory and Applications, 2013 (2013), Article ID: 95.
  • Páles, Zs., Petre, I. R., Iterative fixed point theorems in E-metric spaces, Acta Math. Hung., 140(2013), 134--144.
  • Samet, B., Vetro, C., Vetro, P., Fixed point theorem for α-ψ-contractive type mappings, Nonlinear Anal., 75 (2012), 2154-2165.
  • Sankar Raj, V., A best proximity point theorem for weakly contractive non-self-mappings, Nonlinear Anal., 74 (2011), 4804-4808
  • Rahimi, H., Abbas, M., Rad, G., Common fixed point results for four mappings on ordered vector metric spaces, Filomat, 29(4) (2015), 865-878.

Best proximity point theory on vector metric spaces

Year 2021, Volume: 70 Issue: 1, 130 - 142, 30.06.2021
https://doi.org/10.31801/cfsuasmas.780723

Abstract

In this paper, we first give a new definition of Ω-Dedekind complete Riesz space (E,≤) in the frame of vector metric space (Ω,ρ,E) and we investigate the relation between Dedekind complete Riesz space and our new concept. Moreover, we introduce a new contraction so called α-vector proximal contraction mapping. Then, we prove certain best proximity point theorems for such mappings in vector metric spaces (Ω,ρ,E) where (E,≤) is Ω-Dedekind complete Riesz space. Thus, for the first time, we acquire best proximity point results on vector metric spaces. As a result, we generalize some fixed point results proved in both vector metric spaces and partially ordered vector metric spaces such as main results of V4. Further, we provide nontrivial and comparative examples to show the effectiveness of our main results.

References

  • Alghamdi, M. A., Shahzad, N., Vetro F., Best proximity points for some classes of proximal contractions, Abstract and Applied Analysis, 2013 (2013), Article ID: 713252.
  • Ali, M. U., Bejenaru, A., Kamran, T., The order-convergence of the Thakur iterative process for Hardy-Rogers contractions in order-Banach spaces, Journal of Mathematical Analysis, 9(4) (2018), 61-74.
  • Aliprantis, C. D., Border, K. C., Infinite dimensional analysis, Berlin, Springer-Verlag, 1999.
  • Aydi, H., Lakzian, H., Mitrović, Z. D., Radenović, S., Best proximity points of MT-cyclic contractions with property UC, Numerical Functional Analysis and Optimization 41 (2020), 1-12.
  • Banach, S., Sur les opérations dans les ensembles abstraits et leur applications aux équations intégrales, Fund. Math, 3 (1922), 133-181.
  • Basha, S. S., Extensions of Banach's contraction principle, Numer. Funct. Anal. Optim., 31 (5) (2010), 569-576.
  • Basha, S. S., Shahzad, N., Vetro, C., Best proximity point theorems for proximal cyclic contractions, Journal of Fixed Point Theory and Applications, 19 (4) (2017), 2647-2661.
  • Çevik, C., On Continuity Functions Between Vector Metric Spaces, Journal of Function Spaces, (2014), Article ID 753969.
  • Çevik, C., Altun, I., Vector metric spaces and some properties, Topol. Methods Nonlinear Anal., 34 (2009), 375--382.
  • Çevik, C., Altun, I., Sahin, H., Ozeken, C. C., Some fixed point theorems for contractive mapping in ordered vector metric spaces, J. Nonlinear Sci. Appl., 10(4) (2017), 1424--1432.
  • Eldred, A. A., Veeramani, P., Existence and convergence of best proximity points, J. Math. Anal. Appl., 323 (2006), 1001-1006.
  • Jleli, M., Samet, B., Best proximity points for α-ψ-proximal contractive type mappings and application, Bull. Sci. Math., 137 (2013), 977-995.
  • Nashine, H. K., Kumam, P., Vetro, C., Best proximity point theorems for rational proximal contractions, Fixed Point Theory and Applications, 2013 (2013), Article ID: 95.
  • Páles, Zs., Petre, I. R., Iterative fixed point theorems in E-metric spaces, Acta Math. Hung., 140(2013), 134--144.
  • Samet, B., Vetro, C., Vetro, P., Fixed point theorem for α-ψ-contractive type mappings, Nonlinear Anal., 75 (2012), 2154-2165.
  • Sankar Raj, V., A best proximity point theorem for weakly contractive non-self-mappings, Nonlinear Anal., 74 (2011), 4804-4808
  • Rahimi, H., Abbas, M., Rad, G., Common fixed point results for four mappings on ordered vector metric spaces, Filomat, 29(4) (2015), 865-878.
There are 17 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Research Articles
Authors

Hakan Şahin 0000-0002-4671-7950

Publication Date June 30, 2021
Submission Date August 14, 2020
Acceptance Date November 12, 2020
Published in Issue Year 2021 Volume: 70 Issue: 1

Cite

APA Şahin, H. (2021). Best proximity point theory on vector metric spaces. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 70(1), 130-142. https://doi.org/10.31801/cfsuasmas.780723
AMA Şahin H. Best proximity point theory on vector metric spaces. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. June 2021;70(1):130-142. doi:10.31801/cfsuasmas.780723
Chicago Şahin, Hakan. “Best Proximity Point Theory on Vector Metric Spaces”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 70, no. 1 (June 2021): 130-42. https://doi.org/10.31801/cfsuasmas.780723.
EndNote Şahin H (June 1, 2021) Best proximity point theory on vector metric spaces. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 70 1 130–142.
IEEE H. Şahin, “Best proximity point theory on vector metric spaces”, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., vol. 70, no. 1, pp. 130–142, 2021, doi: 10.31801/cfsuasmas.780723.
ISNAD Şahin, Hakan. “Best Proximity Point Theory on Vector Metric Spaces”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 70/1 (June 2021), 130-142. https://doi.org/10.31801/cfsuasmas.780723.
JAMA Şahin H. Best proximity point theory on vector metric spaces. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2021;70:130–142.
MLA Şahin, Hakan. “Best Proximity Point Theory on Vector Metric Spaces”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, vol. 70, no. 1, 2021, pp. 130-42, doi:10.31801/cfsuasmas.780723.
Vancouver Şahin H. Best proximity point theory on vector metric spaces. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2021;70(1):130-42.

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