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**Existence and uniqueness results for the bending of an elastic beam equation at resonance.**
*(English)*
Zbl 0655.73001

The purpose of this paper is to study the following nonlinear analogue of the boundary-value problem for bending of an elastic beam which is simply supported at both ends and is at resonance:
\[
(1)\quad d^ 4u/dx^ 4- \pi^ 4u+g(x,u)=e(x),\quad 0<x<1,\quad u(0)=u(1)=u''(0)=u''(1)=0,
\]
where g: [0,1]\(\times R\to R\) satisfies Caratheodory’s conditions and e(x)\(\in L^ 1[0,1]\) with \(\int^{1}_{0}e(x)\) sin \(\pi\) x dx\(=0\). It is shown that (1) has at least one solution if g(x,u)u\(\geq 0\) for all x in [0,1] and u in R. It is also proven that (1) has a unique solution if g(x,u) is strictly increasing in u for every x in [0,1] and \(\int^{1}_{0}g(x,0)\sin \pi x dx=0.\)

Another boundary-value problem studied in the same paper is defined by \[ (2)\quad -d^ 4u/dx^ 4+\pi^ 4u+g(x,u)=e(x),\quad 0<x<1,\quad u(0)=u(1)=u''(0)=u''(1)=0. \] It is proven that a solution of (2) exists when \(e(x)\in L^ 1[0,1]\) with \(\int^{1}_{0}e(x)\quad \sin \pi x dx=0\) and g(x,u) satisfies two supplementary conditions. A proof is also given for the uniqueness of the solutions of (2) when \(e(x)\in L^ 1(0,1)\) with \(\int^{1}_{0}e(x)\sin \pi x dx=0\) and g satisfies two specified conditions.

Another boundary-value problem studied in the same paper is defined by \[ (2)\quad -d^ 4u/dx^ 4+\pi^ 4u+g(x,u)=e(x),\quad 0<x<1,\quad u(0)=u(1)=u''(0)=u''(1)=0. \] It is proven that a solution of (2) exists when \(e(x)\in L^ 1[0,1]\) with \(\int^{1}_{0}e(x)\quad \sin \pi x dx=0\) and g(x,u) satisfies two supplementary conditions. A proof is also given for the uniqueness of the solutions of (2) when \(e(x)\in L^ 1(0,1)\) with \(\int^{1}_{0}e(x)\sin \pi x dx=0\) and g satisfies two specified conditions.

Reviewer: W.A.Bassali

### MSC:

74G30 | Uniqueness of solutions of equilibrium problems in solid mechanics |

74H25 | Uniqueness of solutions of dynamical problems in solid mechanics |

74K10 | Rods (beams, columns, shafts, arches, rings, etc.) |

34B15 | Nonlinear boundary value problems for ordinary differential equations |

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\textit{C. P. Gupta}, J. Math. Anal. Appl. 135, No. 1, 208--225 (1988; Zbl 0655.73001)

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### References:

[1] | Gupta, C.P, Existence and uniqueness theorems for the bending of an elastic beam equation, Applicable analysis, 26, 289-304, (1988) · Zbl 0611.34015 |

[2] | {\scJ. Mawhin}, Landesman-Lazer type problems for nonlinear equations, Confer. Sem. Mat. Univ. Bari, No. 147, 177. · Zbl 0436.47050 |

[3] | Mawhin, J, Topological degree methods in nonlinear boundary value problems, () · Zbl 0414.34025 |

[4] | Mawhin, J, Compacité, monotonie et convexité dans l’étude de problèmes aux limites semi-linéaires, () · Zbl 0497.47033 |

[5] | Mawhin, J; Ward, J.R; Willem, M, Necessary and sufficient conditions for the solvability of a nonlinear two point boundary value problem, (), 667-674 · Zbl 0559.34014 |

[6] | Usmani, R.A, A uniqueness theorem for a boundary value problem, (), 329-335 · Zbl 0424.34019 |

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