The contraction principle for mappings on a metric space with a graph. (English) Zbl 1139.47040

Let \((X,d)\) be a metric space and \(\Delta\) denote the diagonal of \(X \times X\). Let \(G\) be a directed graph such that the set of its vertices coincides with \(X\) and the set \(E(G)\) of its edges contains all loops, i.e., \(E(G) \supseteq \Delta\). A map \(f: X \to X\) is called \(G\)-contraction if it preserves the edges of \(G\), i.e., \((x,y) \in E(G)\) implies \((fx,fy) \in E(G)\) and \(f\) decreases weights of edges of \(G\), i.e., \((x,y) \in E(G)\) implies \(d(fx,fy) \leq \alpha d(x,y)\) for some \(\alpha \in (0,1)\). The author presents some fixed point results for \(G\)-contractions, being a hybrid of the Banach and Knaster–Tarski theorems and generalizing a number of known assertions. As an application, the convergence of successive approximations for some linear operators on Banach spaces is considered.


47H10 Fixed-point theorems
05C40 Connectivity
54H25 Fixed-point and coincidence theorems (topological aspects)
Full Text: DOI


[1] Nelson Dunford and Jacob T. Schwartz, Linear Operators. I. General Theory, With the assistance of W. G. Bade and R. G. Bartle. Pure and Applied Mathematics, Vol. 7, Interscience Publishers, Inc., New York; Interscience Publishers, Ltd., London, 1958. · Zbl 0084.10402
[2] Michael Edelstein, An extension of Banach’s contraction principle, Proc. Amer. Math. Soc. 12 (1961), 7 – 10. · Zbl 0096.17101
[3] T. Gnana Bhaskar and V. Lakshmikantham, Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Anal. 65 (2006), no. 7, 1379 – 1393. · Zbl 1106.47047
[4] Andrzej Granas and James Dugundji, Fixed point theory, Springer Monographs in Mathematics, Springer-Verlag, New York, 2003. · Zbl 1025.47002
[5] Jacek Jachymski, Order-theoretic aspects of metric fixed point theory, Handbook of metric fixed point theory, Kluwer Acad. Publ., Dordrecht, 2001, pp. 613 – 641. · Zbl 1027.54065
[6] R. Johnsonbaugh, Discrete Mathematics, Prentice-Hall, Inc., New Jersey, 1997. · Zbl 0860.68078
[7] R. P. Kelisky and T. J. Rivlin, Iterates of Bernstein polynomials, Pacific J. Math. 21 (1967), 511 – 520. · Zbl 0177.31302
[8] Andrzej Lasota, From fractals to stochastic differential equations, Chaos — the interplay between stochastic and deterministic behaviour (Karpacz, 1995) Lecture Notes in Phys., vol. 457, Springer, Berlin, 1995, pp. 235 – 255. · Zbl 0835.60058
[9] Juan J. Nieto, Rodrigo L. Pouso, and Rosana Rodríguez-López, Fixed point theorems in ordered abstract spaces, Proc. Amer. Math. Soc. 135 (2007), no. 8, 2505 – 2517. · Zbl 1126.47045
[10] Juan J. Nieto and Rosana Rodríguez-López, Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations, Order 22 (2005), no. 3, 223 – 239 (2006). · Zbl 1095.47013
[11] J. J. Nieto and R. Rodríguez-López, Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations, Acta Math. Sinica, English Ser. (2007), 2205-2212. · Zbl 1140.47045
[12] Adrian Petruşel and Ioan A. Rus, Fixed point theorems in ordered \?-spaces, Proc. Amer. Math. Soc. 134 (2006), no. 2, 411 – 418. · Zbl 1086.47026
[13] André C. M. Ran and Martine 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. · Zbl 1060.47056
[14] Ioan A. Rus, Iterates of Bernstein operators, via contraction principle, J. Math. Anal. Appl. 292 (2004), no. 1, 259 – 261. · Zbl 1056.41004
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