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Petruşel, Adrian; Rus, Ioan A.

Fixed point theorems in ordered \(L\)-spaces. (English) Zbl 1086.47026

Proc. Am. Math. Soc. 134, No. 2, 411-418 (2006).
An ordered \(L\)-space \((X,\rightarrow ,\leq )\) consists of a nonempty set \(X\), a convergence structure \(\rightarrow\) on the set of the sequences in \(X\) and an ordering relation \(\leq\) on \(X\), the two being connected by some conditions of compatibility. In this paper, the authors establish several sufficient conditions in order that an operator \(f:X\rightarrow X\) defined on an ordered \(L\)-space to be a Picard operator (that is, to have a unique fixed point \(x^*\) and, for each \(x\in X\), the sequence \((f^n(x))\) of successive approximations to converge to \(x^*\)). Finally, some applications to matrix equations are considered.
Reviewer: Mircea Balaj (Oradea)

Cited in 5 Reviews
Cited in 105 Documents

MSC:

54H25 Fixed-point and coincidence theorems (topological aspects)
54E40 Special maps on metric spaces
54F05 Linearly ordered topological spaces, generalized ordered spaces, and partially ordered spaces
54E70 Probabilistic metric spaces
15A24 Matrix equations and identities

Keywords:

L-space; ordered L-space; fixed point; Picard operator; weakly Picard operator; matrix equation

Citations:

Zbl 1060.47056
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\textit{A. Petruşel} and \textit{I. A. Rus}, Proc. Am. Math. Soc. 134, No. 2, 411--418 (2006; Zbl 1086.47026)
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References:

[1] T. A. Burton, Integral equations, implicit functions, and fixed points, Proc. Amer. Math. Soc. 124 (1996), no. 8, 2383 – 2390. · Zbl 0873.45003
[2] E. De Pascale, G. Marino, and P. Pietramala, The use of the \?-metric spaces in the search for fixed points, Matematiche (Catania) 48 (1993), no. 2, 367 – 376 (1994). · Zbl 0833.47049
[3] Maurice Fréchet, Les espaces abstraits, Les Grands Classiques Gauthier-Villars. [Gauthier-Villars Great Classics], Éditions Jacques Gabay, Sceaux, 1989 (French). Reprint of the 1928 original. · Zbl 0033.28901
[4] O. Hadžić, E. Pap, and V. Radu, Generalized contraction mapping principles in probabilistic metric spaces, Acta Math. Hungar. 101 (2003), no. 1-2, 131 – 148. · Zbl 1050.47052
[5] Olga Hadžić and Endre Pap, Fixed point theory in probabilistic metric spaces, Mathematics and its Applications, vol. 536, Kluwer Academic Publishers, Dordrecht, 2001. · Zbl 0994.47077
[6] William A. Kirk and Brailey Sims , Handbook of metric fixed point theory, Kluwer Academic Publishers, Dordrecht, 2001. · Zbl 0970.54001
[7] James Merryfield and James D. Stein Jr., A generalization of the Banach contraction principle, J. Math. Anal. Appl. 273 (2002), no. 1, 112 – 120. · Zbl 1029.54046
[8] Adrian Petruşel, Multivalued weakly Picard operators and applications, Sci. Math. Jpn. 59 (2004), no. 1, 169 – 202. · Zbl 1066.47058
[9] 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
[10] Ioan A. Rus, Generalized contractions and applications, Cluj University Press, Cluj-Napoca, 2001. · Zbl 0968.54029
[11] Ioan A. Rus, Picard operators and applications, Sci. Math. Jpn. 58 (2003), no. 1, 191 – 219. · Zbl 1031.47035
[12] Ioan A. Rus, Adrian Petruşel, and Gabriela Petruşel, Fixed point theory: 1950 – 2000. Romanian contributions, House of the Book of Science, Cluj-Napoca, 2002. · Zbl 1005.54037
[13] P. P. Zabrejko, \?-metric and \?-normed linear spaces: survey, Collect. Math. 48 (1997), no. 4-6, 825 – 859. Fourth International Conference on Function Spaces (Zielona Góra, 1995). · Zbl 0892.46002
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