On nonlocal boundary value problem for the equation of motion of a homogeneous elastic beam with pinned-pinned ends.

*(English)*Zbl 1406.35395Summary: In the current paper, in the domain \(D=\{(t,x): t\in(0,T), x\in(0,L)\}\) we investigate the boundary value problem for the equation of motion of a homogeneous elastic beam
\[
u_{tt}(t,x)+a^{2}u_{xxxx}(t,x)+b u_{xx}(t,x)+c u(t,x)=0,
\]
where \(a,b,c \in \mathbb{R}\), \(b^2<4a^2c\), with nonlocal two-point conditions
\[
u(0,x)-u(T, x)=\phi(x), \quad u_{t}(0, x)-u_{t}(T, x)=\psi(x)
\]
and local boundary conditions \(u(t, 0)=u(t, L)=u_{xx}(t, 0)=u_{xx}(t, L)=0\). Solvability of this problem is connected with the problem of small denominators, whose estimation from below is based on the application of the metric approach. For almost all (with respect to Lebesgue measure) parameters of the problem, we establish conditions for the solvability of the problem in the Sobolev space. In particular, if \(\phi\in\mathbf{H}_{q+\rho+2}\) and \(\psi \in\mathbf{H}_{q+\rho}\), where \(\rho>2\), then for almost all (with respect to Lebesgue measure in \(\mathbb{R}\)) numbers \(a\) exists a unique solution \(u\in\mathbf{C}^{2}([0,T];\mathbf{H}_{q})\) of the problem considered.

##### MSC:

35Q74 | PDEs in connection with mechanics of deformable solids |

35L25 | Higher-order hyperbolic equations |

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

35A02 | Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness |

74B05 | Classical linear elasticity |

##### Keywords:

nonlocal boundary value problem; homogeneous elastic beam; pinned-pinned ends; small denominator; Lebesgue measure; metric approach
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\textit{T. Goy} et al., Carpathian Math. Publ. 10, No. 1, 105--113 (2018; Zbl 1406.35395)

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