TY - JOUR
AU - Goy, T.P.
AU - Negrych, M.
AU - Savka, I.Ya.
PY - 2018/07/03
Y2 - 2024/05/20
TI - On nonlocal boundary value problem for the equation of motion of a homogeneous elastic beam with pinned-pinned ends
JF - Carpathian Mathematical Publications
JA - Carpathian Math. Publ.
VL - 10
IS - 1
SE - Scientific articles
DO - 10.15330/cmp.10.1.105-113
UR - https://journals.pnu.edu.ua/index.php/cmp/article/view/1474
SP - 105-113
AB - <p>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.</p>
ER -