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On nonlocal boundary value problem for the equation of motion of a homogeneous elastic beam with pinned-pinned ends. (English) Zbl 1406.35395
Summary: 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
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