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Zbl 1142.47033
Fixed point theorems for generalized contractions in ordered metric spaces.
(English)
[J] J. Math. Anal. Appl. 341, No. 2, 1241-1252 (2008). ISSN 0022-247X

The authors present some fixed point results for self-generalized contractions in ordered metric spaces. These results generalize some recent results of {\it A. C. M. Ran} and {\it M. C. Reurings} [Proc. Am. Math. Soc. 132, No. 5, 1435--1443 (2004; Zbl 1060.47056)] as well as {\it J. J. Nieto} and {\it R. Rodr{\'\i}guez-Lopez} [Order 22, No. 3, 223--239 (2005; Zbl 1095.47013); Acta Math. Sin. Engl. Ser. 23, 2205--2212 (2007; Zbl 1140.47045)], in terms of Picard operators [cf. {\it I. A. Rus}, Sci. Math. Jpn. 58, No. 1, 191--219 (2003; Zbl 1031.47035)]. Moreover, for the case of generalized $\varphi$-contractions, a fixed point theorem is established, as a modification of that of {\it R. P. Agarwal, M. A. El--Gebeily}, and {\it D. O'Regan} [Appl. Anal. 87, No. 1, 109--116 (2008; Zbl 1140.47042)]. Some applications are given to Fredholm and Volterra type integral equations.
[In-Sook Kim (München)]
MSC 2000:
*47H10 Fixed point theorems for nonlinear operators on topol.linear spaces
54H25 Fixed-point theorems in topological spaces
47H07 Positive operators on ordered topological linear spaces
47H09 Mappings defined by "shrinking" properties

Keywords: fixed point; monotone operator; ordered metric space; generalised contraction; Fredholm integral equations; Volterra-type integral equations

Cited in: Zbl 1201.54034

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