Best proximity points in graphical vector metric spaces and its consequences
Submitted: 2025-05-08
|Accepted: 2026-01-02
|Published: 2026-03-31
Copyright (c) 2025 Kamal Fallahi, Halimeh Ardakani, Seyedeh Sara Karimizad
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.
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Keywords:
graphical vector metric spaces, vectorially G-continuous, vectorially G-proximal mapping, X-Dedekind complete Riesz space, best proximity points
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Abstract:
Regarding the vector metric space introduced by Cevik and Altun, we first introduce the concept of a graphical vector metric space and then define several topological notions therein such as $\mathfrak{X}$-Dedekind complete Riesz spaces, vectorially $\mathcal{G}$-continuous mappings, vectorially $\mathcal{G}$-proximal mappings, vector P-property. Then some best proximity point theorems are proven in such spaces using these new concepts. Several consequences and an illustrative example are also provided to support our definitions and results. One important thing that can be distinguished between this work and similar researches is that it can also encompass all of those previously introduced by considering comparable and ε -close elements.
References:
A. A. Armaki, A. Pashapournia, and G. Soleimani Rad, Best proximity points of graphical (ϕ-ψ)-weak contractive mappings and applications to graphical integral-type inequalities, J. Nonlinear Convex Anal. 25 (2024), 1777-1789.
C. D. Aliprantis and K. C. Border, Infinite Dimensional Analysis, Springer-Verlag, Berlin, 1999.https://doi.org/10.1007/978-3-662-03961-8
I. Altun, M. Aslantas and H. Sahin, Best proximity point results for p-proximal contractions, Acta Math. Hungar. 162 (2020), 393-402.https://doi.org/10.1007/s10474-020-01036-3
I. Altun and C. Cevik, Some common fixed point theorems in vector metric spaces, Filomat 25, no. 1 (2011), 105-113.https://doi.org/10.2298/FIL1101105A
C. Cevik and I. Altun, Vector metric space and some properties, Topol. Met. Nonlinear Anal. 34, no. 2 (2009), 375-382.https://doi.org/10.12775/TMNA.2009.048
C. Cevik, On continuity functions between vector metric spaces, Journal of Function Spaces 2014 (2014), 753969.https://doi.org/10.1155/2014/753969
A. A. Eldred and P. Veeramani, Existence and convergence of best proximity points, J. Math. Anal. Appl. 323 (2006), 1001-1006.https://doi.org/10.1016/j.jmaa.2005.10.081
K. Fallahi, H. Ghahramani, and G. Soleimani Rad, Integral type contractions in partially ordered metric spaces and best proximity point, Iran. J. Sci. Technol.Trans. Sci. 44 (2020),177-183.https://doi.org/10.1007/s40995-019-00807-0
K. Fallahi, A. Petruşel, and G. Soleimani Rad, Vectorially G-contraction mappings and best proximity points, Rend. Circ. Mat. Palermo, II. Ser. (2025), 74:106.
K. Fallahi and G. Soleimani Rad, Best proximity points theorem in b-metric spaces endowed with a graph, Fixed Point Theory. 21, no. 2 (2020), 465-474.https://doi.org/10.24193/fpt-ro.2020.2.33
K. Fallahi, G. Soleimani Rad, and A. Fulga, Best proximity points for (ϕ-ψ)-weak contractions and some applications, Filomat 37, no. 6 (2023), 1835-1842.https://doi.org/10.2298/FIL2306835F
J. Jachymski, The contraction principle for mappings on a metric space with a graph, Proc. Amer. Math. Soc. 136 (2008), 1359-1373.https://doi.org/10.1090/S0002-9939-07-09110-1
A. Kostic, E. Karap{i}nar, and V. Rakocević, Best proximity points and fixed points with R-functions in the framework of w-distances, Bull. Australian Math. Soc. 99, no. 3 (2019), 497-507.https://doi.org/10.1017/S0004972718001193
P. Magadevan, S. Karpagam, and E. Karapınar, Existence of fixed point and best proximity point of p-cyclic orbital φ-contraction map, Nonlinear Analysis: Modelling & Control. 27, no. 1 (2022), 91-101.https://doi.org/10.15388/namc.2022.27.25188
J. J. Nieto and R. Rodríguez-López, Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations, Order 22, no. 3 (2005), 223-239.https://doi.org/10.1007/s11083-005-9018-5
A. Petruşel and I. A. Rus, Fixed point theorems in ordered L-spaces, Proc. Amer. Math. Soc. 134 (2006), 411-418.https://doi.org/10.1090/S0002-9939-05-07982-7
H. Rahimi, M. Abbas, and G. Soleimani Rad, Common fixed point results for four mappings on ordered vector metric spaces, Filomat 29, no. 4 (2015), 865-878.https://doi.org/10.2298/FIL1504865R
VS. Raj, A best proximity point theorem for weakly contractive non-self mappings, Nonlinear Anal. 74 (2011), 4804-4808.https://doi.org/10.1016/j.na.2011.04.052
A. C. M. Ran and M. C. B. Reurings, A fixed point theorem in partially ordered sets and some application to matrix equations, Proc. Amer. Math. Soc. 132 (2004), 1435-1443.https://doi.org/10.1090/S0002-9939-03-07220-4
S. Sadig Basha, Best proximity point theorems in the frameworks of fairly and proximally complete spaces, J. Fixed Point Theory Appl. 19, no. 3 (2017), 1939-1951.https://doi.org/10.1007/s11784-016-0324-x
S. Sadiq Basha, Discrete optimization in partially ordered sets, J. Glob. Optim. 54 (2012), 511-517.https://doi.org/10.1007/s10898-011-9774-2
S. Sadiq Basha, N. Shahzad and C. Vetro, Best proximity point theorems for proximal cyclic contractions, J. Fixed Point Theory Appl. 19, no. 4 (2017), 2647-2661.https://doi.org/10.1007/s11784-017-0447-8
H. Sahin, A new approach to homotopy theory via best proximity point, Miskolc Math. Notes. 24, no. 1 (2023), 411-428.https://doi.org/10.18514/MMN.2023.3674
H. Sahin, Best proximity point theory on vetor metric spaces, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 70, no. 1 (2021), 130-142.https://doi.org/10.31801/cfsuasmas.780723
G. Soleimani Rad and I. Altun, Common fixed point results on vector metric spaces, J. Linear Topological Algebra 5, no. 1 (2016), 29-39.
F. Yousefi, H. Rahimi, and G. Soleimani Rad, E-metric spaces and common fixed point theorems, J. Linear Topological Algebra 11, no. 3 (2022), 159-168.
