Basis properties in a problem of a nonhomogeneous string with damping at the end

Abstract

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∈C. This problem can be rewritten as an abstract Cauchy problem for a densely defined closed operator iA_h acting on an appropriate energy Hilbert space H. Under assumptions that the density function of the string ρ∈W₂¹[0,1] is strictly positive and has ρ(1)≠h² (if h∈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

∫₀¹ω₁²(ρ′,τ)/τ² dτ<∞,

where ω₁ is the integral modulus of continuity, the root vectors of the operator A_h form a Riesz basis in H.