# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
A fixed point theorem for contraction type maps in partially ordered metric spaces and application to ordinary differential equations. (English) Zbl 1197.54054
Let $(X,\le)$ be a partially ordered set and $d$ a complete metric on $X$. The authors present a fixed point theorem for an operator $f:X\to X$ under some suitable conditions in terms of $\le$ and $d$. An application to ordinary differential equations is also given.

##### MSC:
 54H25 Fixed-point and coincidence theorems in topological spaces 54F05 Linearly, generalized, and partial ordered topological spaces 34B15 Nonlinear boundary value problems for ODE
Full Text:
##### References:
 [1] Agarwal, R. P.; El-Gebeily, M. A.; O’regan, D.: Generalized contractions in partially ordered metric spaces, Appl. anal. 87, 109-116 (2008) · Zbl 1140.47042 · doi:10.1080/00036810701556151 [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] Bhaskar, T. Gnana; Lakshmikantham, V.: Fixed point theorems in partially ordered metric spaces and applications, Nonlinear anal. 65, 1379-1393 (2006) · Zbl 1106.47047 · doi:10.1016/j.na.2005.10.017 [4] Harjani, J.; Sadarangani, K.: Fixed point theorems for weakly contractive mappings in partially ordered sets, Nonlinear anal. 71, 3403-3410 (2009) · Zbl 1221.54058 · doi:10.1016/j.na.2009.01.240 [5] Lakshmikantham, V.; Ćirić, L.: Couple fixed point theorems for nonlinear contractions in partially ordered metric spaces, Nonlinear anal. 70, 4341-4349 (2009) · Zbl 1176.54032 · doi:10.1016/j.na.2008.09.020 [6] Nieto, J. J.; Pouso, R. L.; Rodríguez-López, R.: Fixed point theorems in ordered abstract sets, Proc. amer. Math. soc. 135, 2505-2517 (2007) · Zbl 1126.47045 · doi:10.1090/S0002-9939-07-08729-1 [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.; Rodríguez-López, R.: Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations, Acta math. Sin. 23, 2205-2212 (2007) · Zbl 1140.47045 · doi:10.1007/s10114-005-0769-0 [9] O’regan, D.; Petruşel, A.: Fixed point theorems for generalized contractions in ordered metric spaces, J. math. Anal. appl. 341, No. 2, 1241-1252 (2008) · Zbl 1142.47033 · doi:10.1016/j.jmaa.2007.11.026 [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. B.: 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] Geraghty, M.: On contractive mappings, Proc. amer. Math. soc. 40, 604-608 (1973) · Zbl 0245.54027 · doi:10.2307/2039421