## Multiplicity results for a fourth-order boundary value problem.(English)Zbl 0836.73032

Summary: This paper deals with multiplicity results for nonlinear elastic equations of the type $$y^{(IV)} - \alpha_1y + \beta_1y'' + g(x,y,y'') = e$$, $$0 < x < 1$$, $$y(0) = y''(0) = y'(1) = y'''(1) = 0$$, where $$e \in L^2 (0,1)$$, $$g : [0,1] \times \mathbb{R} \times \mathbb{R} \to \mathbb{R}$$ is a bounded continuous function, and the pair $$(\alpha_1, \beta_1)$$ satisfies the conditions $$\alpha_1 + (0 + 0.5)^2 \pi^2 \beta_1 = (0 + 0.5)^4 \pi^4$$ and $$\alpha_1 + (k + 0.5)^2 \pi^2 \beta_1 \neq (k + 0.5)^4 \pi^4$$ for all $$k \in N$$.

### MSC:

 74K10 Rods (beams, columns, shafts, arches, rings, etc.) 34B15 Nonlinear boundary value problems for ordinary differential equations

### Keywords:

elastic beam; two-parameter eigenvalue problem
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### References:

 [1] D. G. Gosta and J. V. A. Goncalves, Existence and multiplicity results for a class of elliptic boundary value problems at resonance.J. Math. Anal. Appl.,84 (1981). 328–338. · Zbl 0479.35037 [2] M. A. Delpino and R. F. Manasevich. Existence for a fourth-order boundary value problem under a two-parameter nonresonance condition.Proc. Amer. Math. Soc.,112 (1991), 81–86. [3] C. P. Gupta. Existence and uniqueness theorems for the bending of an elastic beam equation,Applicable Analysis,26 (1988), 289–304. · Zbl 0611.34015 [4] C. P. Gupta, Existence and uniqueness theorems for some fourth-order fully quasilinear boundary value problems.Applicable Analysis,36 (1991), 157–169. · Zbl 0713.34025 [5] Ma Ruyun, Some multiplicity results for an elastic beam equation at resonance.Applied Mathematics and Mechanics (English Ed.),14, 2 (1993), 193–200. · Zbl 0776.73037 [6] J. Mawhin, J. R. Ward and M. Willem, Necessary and sufficient conditions of a nonlinear two-point boundary value problems.Proc. Amer. Math. Soc.,93 (1985), 667–674. · Zbl 0559.34014 [7] R. A. Usmini, A uniqueness theorem for a boundary value problem,Proc. Amer. Math. Soc.,77 (1979), 329–335. [8] Y. Yang, Fourth-order two-point boundary value problems,Proc. Amer. Math. Soc.,77, (1988), 175–180. · Zbl 0671.34016
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