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

T. Goy, I. Savka, M. Negrych

Abstract


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 $$u(0,x)-u(T, x)=\varphi(x), \quad u_{t}(0, x)-u_{t}(T, x)=\psi(x)$$ 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 $\varphi\in\mathbf{H}_{q+\rho+2}$ 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.

Keywords


nonlocal boundary value problem; homogeneous elastic beam; pinned-pinned ends; small denominator; Lebesque measure; metric approach

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