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

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  $a,b,c \in \mathbb{R}$, $b^2 2$, 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.