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Existence of positive solutions for a class of semi-positive nonlinear elastic beam equation boundary value problems. (Chinese. English summary) Zbl 1463.34108

Summary: By using the fixed point theorem of cone mapping, we obtain the existence and multiplicity of positive solutions for the semi-positive nonlinear elastic beam equation’s boundary value problems \[ \begin{cases} y^{(4)} (x) + (k_1+k_2)y'' (x) + k_1 k_2 y (x) = \lambda f(x, y (x)),\, x \in [0,1], \\ y' (0) = y' (1) = y''' (0) = y''' (1) = 0 \end{cases} \] with Neumann boundary conditions, where \(0 < k_1<k_2\leq \frac{\pi^2}{4}\), \(\lambda > 0\) is a parameter and \(f \in C ([0,1] \times [0, \infty), (-\infty, \infty))\), with \(f(x, y) \geq -X\) for some positive constant \(X\).

MSC:

34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
47N20 Applications of operator theory to differential and integral equations
34B09 Boundary eigenvalue problems for ordinary differential equations
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