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Positive solutions for eigenvalue problems of fourth-order elastic beam equations. (English) Zbl 1072.34022
The following fourth-order eigenvalue problem $w^{(4)}(t)=\lambda f(t,w(t)), \quad 0<t<1,\;\lambda >0,\qquad w(0)=w(1)=w'(0)=w'(1)=0,\tag{1}$ is studied. The author uses the Krasnoselskii-Guo fixed-point theorem on cone expansion and compression and the properties of the Green function of the corresponding homogeneous BVP to obtain several results on the existence of positive solutions of (1). Some multiplicity results are established, too.

##### MSC:
 34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations 74K10 Rods (beams, columns, shafts, arches, rings, etc.) 34B15 Nonlinear boundary value problems for ordinary differential equations
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##### References:
 [1] Agarwal, R.P.; Chow, Y.M., Iterative methods for a fourth order boundary value problem, J. comput. appl. math., 10, 203-217, (1984) · Zbl 0541.65055 [2] Gupta, C.P., Existence and uniqueness results for bending of an elastic beam equation at resonance, J. math. anal. appl., 135, 208-225, (1988) · Zbl 0655.73001 [3] Jiang, X.; Yao, Q., An existence theorem of positive solutions for elastic beam equation with both fixed end-points, Applied mathematics—J, Chinese univ., 16B, 237-240, (2001) · Zbl 0995.34009 [4] Yao, Q.; Bai, Z., Existence of positive solutions of BVP for $$u\^{}\{(4)\}(t) − λh(t)ƒ(u(t)) = 0$$ (in Chinese), Chinese annals of mathematics, 20A, 575-578, (1999) · Zbl 0948.34502 [5] Yao, Q., On the positive solutions of lidstone boundary value problem, Appl. math. & comput., 137, 477-485, (2003) · Zbl 1093.34515 [6] Krasnosel’skii, M.A., Positive solutions of operator equations, (1964), Noordhoff Groningen
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