tjmcs Turkish Journal of Mathematics and Computer Science 2148-1830 Matematikçiler Derneği 10.47000/tjmcs.1567340 Operator Algebras and Functional Analysis Operatör Cebirleri ve Fonksiyonel Analiz Banach Contractive Mapping Theorem on Partially Ordered Sets Equipped with Vector Metric https://orcid.org/0000-0003-3143-9369 Eminoğlu Şehla ANKARA YILDIRIM BEYAZIT UNIVERSITY 12 30 2025 17 2 586 594 10 14 2024 10 20 2025 Copyright © 2013, Turkish Journal of Mathematics and Computer Science 2013 Turkish Journal of Mathematics and Computer Science

In this paper, left (right) vector metric convergence and left (right) vector metric continuity are definedusing the positive and negative parts of a vector-valued metric. These definitions and the obtained new results are applied to the Banach Contractive Mapping Theorem and examined within the framework of order structures, such as partially ordered sets, Archimedean spaces, and Dedekind complete spaces.

 Left (right) vector metric convergence left (right) vector metric continuity positive (negative) part of vector metric partially ordered set dedekind complete spaces fixed point Aliprantis, C.D., Burkinshaw, O., Positive Operators, Springer, Dordrecht, 2006. Çevik, C., Altun, I., Vector metric spaces and some properties, Topological Methods in Nonlinear Analysis, 34(2)(2009), 375–382. DeMarr, R., Partially ordered linear spaces and locally convex linear topological spaces, Illinois J. Math., 8(1964), 601–606. DeMarr, R., Partially ordered spaces and metric spaces, The American Mathematical Monthly, 72(6)(1965), 628–631. Eminoğlu, Ş ., On non-Archimedean L-Fuzzy vector metric spaces, Journal of New Theory, 45(2023), 46–56. Nieto, J.J., Rodriguez-Lopez, R., Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations, Order, 22(2005), 223–239. Nieto, J.J., Rodriguez-Lopez, R., Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations, Acta. Math. Sin.-English Ser., 23(2007), 2205–2212. Ran, A.C.M., Reurings, M.C.B., A fixed point theorem in partially ordered sets and some applications to matrix equations, Proc. Amer. Math. Soc., 132(2004), 1435–1443. Tarski, A.,A lattice-theoretical fixpoint theorem and its applications, Pacific J. Math., 5(1955), 285–309. Zabrejko, P.P., K-metric and K-normed linear spaces: survey, Universitat de Barcelona Collectanea Mathematica, 48(1997), 825–859.