Fixed point theorems in partially ordered topological spaces with applications
DOI:
https://doi.org/10.12775/TMNA.2023.013Keywords
Fixed point theorem, partial order, increasing operator, topological space, ordinary differential equations, Hammerstein integral equationsAbstract
In this paper, we establish several new fixed point results in the framework of topological spaces endowed with a partial order. Special attention is paid to the case that the topology is induced by a metric. Our conclusions generalize many well-known results. Several examples and illustrative applications are provided to support the exposed results.References
A. Alahmari, M. Mabrouk, M.A. Taoudi, Fixed point theorems for monotone mappings in ordered Banach spaces under weak topology features J. Math. Appl. 42 (2019), 5–19.
J. Appell, The superposition operator in function spaces – a survey, Expo. Math. 6 (1988), 209–270.
D.C. Biles, Existence of solutions for discontinuous differential equations, Differential Integral Equations 8 (1995), no. 6, 1525–1532.
A. Bressan and W. Shen, On discontinuous differential equations, Differential Inclusions and Optimal Control (J. Andres, L. Górniewicz and P. Nistri, eds.) Julius Schauder Center, Lect. Notes Nonlinear Anal. 2 (1998), 73–87.
C. Carathéodory, Vorlesungen über reelle Funktionen, Teubner, 1918.
J.A. Cid and R. Lopez Pouso, Ordinary differential equations and systems with timedependent discontinuity sets, Proc. Roy. Soc. Edinburgh Sect. A 134 (2004), no. 4, 617–637.
R. Espinola and A. Wiśnicki, The Knaster–Tarski theorem versus monotone nonexpansive mappings, Bull. Pol. Acad. Sci. Math. (2017), DOI: 10.4064/ba8120-1-2018.
R. Figueroa and R. Lopez Pouso, Discontinuous first-order functional boundary value problems, Nonlinear Anal. 69 (2008), 2142–2149.
R. Figueroa and R. Lopez Pouso, Existence of solutions of first-order differential equations via a fixed point theorem for discontinuous operators, Fixed Point Theory Appl. 2015 (2015), 220.
A. Hammerstein, Nichtlineare Integralgleichungen nebst Anwendungen, Acta Math. 54 (1929), 117–176.
E.R. Hassan and W. Rzymowski, Extremal solutions of a discontinuous scalar differential equation, Nonlinear Anal. 37 (1999), 997–1017.
S. Heikkila and V. Lakshmikantham, Monotone Iterative Techniques for Discontinuous Nonlinear Differential Equations, CRC Press, 1994.
S. Hu, Differential equations with discontinuous right-hand sides, J. Math. Anal. Appl. 154 (1991), 377–390.
J. Jachymski, Order-theoretic Aspects of Metric Fixed Point Theory, Handbook of Metric Fixed Point Theory, Kluwer Acad. Publ., Dordrecht, 2001, pp. 613–641.
M. Khazou and M.A. Taoudi, Existence and uniqueness of fixed points for monotone operators in partially ordered Banach spaces and applications, J. Fixed Point Theory Appl. 23 (2021), no. 2, paper no. 12, 26 pp.
K. Lan anf J.R.L. Webb, Positive solutions of semilinear differential equations with singularities, J. Differential Equations 148 (1998), no. 2, 407–421.
G. Peano, Sull’integrabilitá delle equazioni differenzialli di primo ordine, Atti. Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur. 21 (1885), 677–685.
Q. Qiu, Some fixed point theorems of increasing operators and applications, Indian J. Pure Appl. Math. 32 (2001), no. 11, 1679–1687.
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. 132 (2004), no. 5, 1435–1443.
S. Reich and A.J. Zaslavski, Generic well-posedness of the fixed point problem for monotone nonexpansive mappings , Mathematics Almost Everywhere, World Sci. Publ., Hackensack, 2018, pp. 169–179.
W. Rzymowski, Existence of solutions for a class of discontinuous differential equations in Rn , J. Math. Anal. Appl. 233 (1999), 634–643.
B. Schroder, Ordered Sets. An Introduction With Connections From Combinatorics to Topology, second edition, Birkhauser/Springer, 2016.
K.A. Topolski, Upper and lower absolutely continuous functions with applications to discontinuous differential equations, Electron. J. Qual. Theory Differ. Equ. 83 (2017), 1–12.
M. Väth, Topological Analysis: From the Basics to the Triple Degree for Nonlinear Fredholm Inclusions, de Gruyter, Berlin, New York, 2012.
Published
How to Cite
Issue
Section
License
Copyright (c) 2023 Mohamed Aziz Taoudi
This work is licensed under a Creative Commons Attribution-NoDerivatives 4.0 International License.
Stats
Number of views and downloads: 0
Number of citations: 0