Fixed points of G-monotone mappings in metric and modular spaces
DOI:
https://doi.org/10.12775/TMNA.2024.003Keywords
Monotone mapping, nonexpansive mapping, fixed pointAbstract
Let $C$ be a bounded, closed and convex subset of a reflexive metric space with a digraph $G$ such that $G$-intervals along walks are closed and convex. In the main theorem we show that if $T\colon C\rightarrow C$ is a monotone $G$-nonexpansive mapping and there exists $c\in C$ such that $Tc\in [c,\rightarrow )_{G}$, then $T$ has a fixed point provided for each $a\in C$, $[a,a]_{G}$ has the fixed point property for nonexpansive mappings. In particular, it gives an essential generalization of the Dehaish-Khamsi theorem concerning partial orders in complete uniformly convex hyperbolic metric spaces. Some counterparts of this result for modular spaces, and for commutative families of mappings are given too.References
A.N. Abdou and M.A. Khamsi, Fixed point theorems in modular vector spaces, J. Nonlinear Sci. Appl. 10 (2017), 4046–4057.
M.R. Alfuraidan, Fixed points of monotone nonexpansive mappings with a graph, Fixed Point Theory Appl. 49 (2015), 6 pp.
J. Bang-Jensen and G. Gutin, Digraphs Theory, Algorithms and Applications, Springer, London, 2009.
B.A. Bin Dehaish and M.A. Khamsi, Browder and Göhde fixed point theorem for monotone nonexpansive mappings, Fixed Point Theory Appl. 20 (2016), 1–9.
R. Diestel, Graph Theory, Springer, Berlin, 2017.
T. Domı́nguez-Benavides, M.A. Khamsi and S. Samadi, Asymptotically regular mappings in modular function spaces, Sci. Math. Jpn. 53 (2001), 295–304.
R. Espı́nola and A. Wiśnicki, The Knaster–Tarski theorem versus monotone nonexpansive mappings, Bull. Pol. Acad. Sci. Math. 66 (2018), 1–7.
J. Jachymski, The contraction principle for mappings on a metric space with a graph, Proc. Amer. Math. Soc. 136 (2008), 1359–1373.
M. Kell, Uniformly convex metric spaces, Anal. Geom. Metr. Spaces 2 (2014), 359–380.
M.A. Khamsi, On metric spaces with uniform normal structure, Proc. Amer. Math. Soc. 106 (1989), 723–726.
M.A. Khamsi, Uniform noncompact convexity, fixed point property in modular spaces, Math. Japonica 41 (1994), 1–6.
M.A. Khamsi and A.R. Khan, Inequalities in metric spaces with applications, Nonlinear Anal. 74 (2011), 4036–4045.
M.A. Khamsi and W.M. Kozlowski, Fixed Point Theory in Modular Function Spaces, with a foreword by W.A. Kirk, Birkhauser/Springer, Cham, 2015.
W. Kirk and N. Shahzad, Fixed Point Theory in Distance Spaces, Springer, Cham, 2014.
K. Menger, Untersuchungen über allgemeine Metrik, Math. Ann. 100 (1928), 75–163.
J. Musielak, Orlicz Spaces and Modular Spaces, Springer, Berlin, 1983.
J.J. Nieto and R. Rodrı́guez-López, Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations, Order 3 (2005), 223–239.
D.H. Quan and A. Wiśnicki, Fixed point theorems in Banach spaces endowed with a digraph (to appear).
A.C.M. Ran and M.C.B. Reurings, A fixed point theorem in partially ordered sets and some applications to matrix equations, Proc. Amer. Math. Soc. 5 (2004), 1435–1443.
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