# 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 sum operator equation and applications to nonlinear elastic beam equations and Lane-Emden-Fowler equations. (English) Zbl 1207.47064
The paper studies the nonlinear operator equation $Ax+Bx+Cx=x$ on ordered Banach spaces, where $A$ is an $\alpha$-concave operator, $B$ an increasing sub-homogeneous operator, and $C$ a homogeneous operator. By using the properties of cones and a fixed point theorem for increasing general $\beta$-concave operators, some new results on the existence and uniqueness of positive solutions are obtained. Applications are made to two classes of nonlinear problems; they include fourth-order two-point boundary value problems for elastic beam equations and elliptic boundary value problems for Lane-Emden-Fowler equations.
##### MSC:
 47J05 Equations involving nonlinear operators (general) 47H07 Monotone and positive operators on ordered topological linear spaces 47N20 Applications of operator theory to differential and integral equations 74K10 Rods (beams, columns, shafts, arches, rings, etc.) in solid mechanics 34B10 Nonlocal and multipoint boundary value problems for ODE 35J66 Nonlinear boundary value problems for nonlinear elliptic equations