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)
Fixed point theorems for mixed monotone operators with PPF dependence. (English) Zbl 1162.47042
First, the authors prove an existence result for coupled fixed points of mixed monotone operators. The uniqueness of such fixed points is also proved, but just in a region of the domain of the involved operator. In the last section, the authors use this result as well as one of their previous results to prove the existence and uniqueness for the solution of a PVBP with delay, in the minimal class.

MSC:
 47H10 Fixed-point theorems for nonlinear operators on topological linear spaces 34B15 Nonlinear boundary value problems for ODE 47H05 Monotone operators (with respect to duality) and generalizations 47N20 Applications of operator theory to differential and integral equations
Full Text:
References:
 [1] Bernfeld, R. S.; Lakshmikantham, V.; Reddy, Y. M.: Fixed point theorems of operators with PPF dependence in a Banach space, Applicable analysis 6, 271-280 (1977) · Zbl 0375.47027 · doi:10.1080/00036817708839165 [2] Drici, Z.; Mcrae, F. A.; Devi, J. Vasundhara: Fixed point theorems in partially ordered metric spaces for operators with PPF dependence, Nonlinear analysis 67, No. 2, 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 analysis 65, No. 7, 1379-1393 (2006) · Zbl 1106.47047 · doi:10.1016/j.na.2005.10.017 [4] Lakshmikantham, V.; Koksal, S.: Monotone flows and rapid convergence for nonlinear partial differential equations, (2003) [5] Nieto, J. J.; Rodriguez-Lopez, R.: Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations, Order 22, No. 3, 223-239 (2005) · Zbl 1095.47013 · doi:10.1007/s11083-005-9018-5 [6] Ran, A. C. M.; Reurings, M. C. R.: A fixed point theorem in partially ordered sets and some applications to matrix equations, Proceedings of the American mathematical society 132, 1435-1443 (2003) · Zbl 1060.47056 · doi:10.1090/S0002-9939-03-07220-4