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Fixed point theorems for generalized contractions in ordered metric spaces. (English) Zbl 1142.47033
The authors present some fixed point results for self-generalized contractions in ordered metric spaces. These results generalize some recent results of A. C. M. Ran and M. C. Reurings [Proc. Am. Math. Soc. 132, No. 5, 1435–1443 (2004; Zbl 1060.47056)] as well as J. J. Nieto and R. Rodríguez-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. 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 R. P. Agarwal, M. A. El–Gebeily, and 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.

MSC:
47H10 Fixed-point theorems
54H25 Fixed-point and coincidence theorems (topological aspects)
47H07 Monotone and positive operators on ordered Banach spaces or other ordered topological vector spaces
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
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