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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 Ax+Bx+Cx=x on ordered Banach spaces, where A is an α-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 β-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.
47J05Equations involving nonlinear operators (general)
47H07Monotone and positive operators on ordered topological linear spaces
47N20Applications of operator theory to differential and integral equations
74K10Rods (beams, columns, shafts, arches, rings, etc.) in solid mechanics
34B10Nonlocal and multipoint boundary value problems for ODE
35J66Nonlinear boundary value problems for nonlinear elliptic equations
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