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