Basis properties in a problem of a nonhomogeneous string with damping at the end. (English) Zbl 1320.35186
Summary: This paper is concerned with the equation of a nonhomogeneous string of length one with one end fixed and the other one damped with a parameter \(h\in \mathbb{C}\). This problem can be rewritten as an abstract Cauchy problem for a densely defined closed operator i\(A_h\) acting on an appropriate energy Hilbert space \(H\). Under assumptions that the density function of the string \(\rho \in W_2^1 [0,1]\) is strictly positive and has \(\rho(1)\neq 2\) (if \(h\in \mathbb{R}\)), we prove that the set of root vectors of \(A_h\) form a basis with parentheses in \(H\). We show that with the additional condition
\[
\int_0^1 \frac{\omega_1^2 (\rho^\prime , \tau)}{\tau^2} d\tau <\infty,
\]
where \(\omega_1\) is the integral modulus of continuity, the root vectors of the operator \(A_h\) form a Riesz basis in \(H\).
MSC:
35L20 | Initial-boundary value problems for second-order hyperbolic equations |
74K05 | Strings |
47B44 | Linear accretive operators, dissipative operators, etc. |