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Existence and uniqueness of convex monotone positive solutions for boundary value problems of an elastic beam equation with a parameter. (English) Zbl 1349.34090

Summary: The purpose of this paper is to investigate the existence and uniqueness of convex monotone positive solutions for a boundary value problem of an elastic beam equation with a parameter. The proofs of the main results rely on a fixed point theorem and some properties of eigenvalue problems for a class of general mixed monotone operators. The results can guarantee the existence of a unique convex monotone positive solution and can be applied to construct two iterative sequences for approximating it. Moreover, we present some properties of convex monotone positive solutions for the boundary value problem dependent on the parameter. Finally, an example is given to illustrate the main results.

MSC:

34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
34A45 Theoretical approximation of solutions to ordinary differential equations
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
34B09 Boundary eigenvalue problems for ordinary differential equations
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