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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 for nonlinear operators on topological linear spaces 54H25 Fixed-point and coincidence theorems in topological spaces 47H07 Monotone and positive operators on ordered topological linear spaces 47H09 Mappings defined by “shrinking” properties
##### References:
 [1] R.P. Agarwal, M.A. El-Gebeily, D. O’Regan, Generalized contractions in partially ordered metric spaces, Appl. Anal., in press [2] Drici, Z.; Mcrae, F. A.; Devi, J. Vasundhara: Fixed point theorems in partially ordered metric spaces for operators with PPF dependence, Nonlinear anal. 7, 641-647 (2007) · Zbl 1127.47049 · doi:10.1016/j.na.2006.06.022 [3] Fréchet, M.: LES espaces abstraits, (1928) · Zbl 54.0614.02 [4] Hadžić, O.; Pap, E.; Radu, V.: Generalized contraction mapping principles in probabilistic metric spaces, Acta math. Hungar. 101, 131-138 (2003) · Zbl 1050.47052 · doi:10.1023/B:AMHU.0000003897.39440.d8 [5] Hadžić, O.; Pap, E.: Fixed point theory in probabilistic metric spaces, (2001) [6] Jachymski, J.; Jóźwik, I.: Nonlinear contractive conditions: A comparison and related problems, Banach center publ. 77, 123-146 (2007) · Zbl 1149.47044 · doi:http://journals.impan.gov.pl/bc/Cont/bc77-0.html [7] Nieto, J. J.; Rodríguez-López, R.: Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations, Order 22, 223-239 (2005) · Zbl 1095.47013 · doi:10.1007/s11083-005-9018-5 [8] Nieto, J. J.; Pouso, R. L.; Rodríguez-López, R.: Fixed point theorem theorems in ordered abstract sets, Proc. amer. Math. soc. 135, 2505-2517 (2007) · Zbl 1126.47045 · doi:10.1090/S0002-9939-07-08729-1 [9] Nieto, J. J.; Rodríguez-López, R.: Existence and uniqueness of fixed points in partially ordered sets and applications to ordinary differential equations, Acta math. Sin. (Engl. Ser.) 23, 2205-2212 (2007) · Zbl 1140.47045 · doi:10.1007/s10114-005-0769-0 [10] Petruşel, A.; Rus, I. A.: Fixed point theorems in ordered L-spaces, Proc. amer. Math. soc. 134, 411-418 (2006) · Zbl 1086.47026 · doi:10.1090/S0002-9939-05-07982-7 [11] Ran, A. C. M.; Reurings, M. C.: A fixed point theorem in partially ordered sets and some applications to matrix equations, Proc. amer. Math. soc. 132, 1435-1443 (2004) · Zbl 1060.47056 · doi:10.1090/S0002-9939-03-07220-4 [12] Rus, I. A.: Generalized contractions and applications, (2001) [13] Rus, I. A.: Picard operators and applications, Sci. math. Jpn. 58, 191-219 (2003) [14] Rus, I. A.; Petruşel, A.; Petruşel, G.: Fixed point theory 1950 – 2000: romanian contributions, (2002) · Zbl 1005.54037