On nonlocal boundary value problem for the equatіon of motіon of a homogeneous elastіc beam wіth pіnned-pіnned ends

Abstract

In the current paper, in the domain D = {(t, x) : t ∈ (0, T), x ∈ (0, L)} we investigate the boundary value problem for the equation of motion of a homogeneous elastic beam utt(t, x)+a2uxxxx(t,x)+buxx(t,x)+cu(t, x) = 0, where a, b, c ∈ R, b2 < 4a2c, with nonlocal two-point conditions u(0, x) − u(T, x) = ϕ(x), ut(0, x) − ut(T, x) = ψ(x) and local boundary conditions u(t, 0) = u(t, L) = uxx(t, 0) = uxx(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 spaces. In particular, if ϕ ∈ Hq+ρ+2 and ψ ∈ Hq+ρ, where ρ > 2, then for almost all (with respect to Lebesgue measure in R) numbers a there exists a unique solution u ∈ C2 ([0, T]; Hq) of the problem.