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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 \(A x + B x + C x = x\) 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 Banach spaces or other ordered topological vector spaces
47N20 Applications of operator theory to differential and integral equations
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
35J66 Nonlinear boundary value problems for nonlinear elliptic equations
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