Abstract :
Let T ⊂ R be a bounded time-scale, with a = inf T, b = supT. We consider the weighted, linear, eigenvalue
problem
− pu (t) + q(t)uσ (t) = λw(t)uσ (t ), t ∈ Tκ2
,
c00u(a) + c01u (a) = 0, c10u ρ(b) +c11u ρ(b) = 0,
for suitable functions p, q and w and λ ∈ R. Problems of this type on time-scales have normally been
considered in a setting involving Banach spaces of continuous functions on T. In this paper we formulate
the problem in Sobolev-type spaces of functions with generalized L2-type derivatives. This approach allows
us to use the functional analytic theory of Hilbert spaces rather than Banach spaces. Moreover, it allows us
to use more general coefficient functions p, q, and weight function w, than usual, viz., p ∈ H1(Tκ ) and
q,w ∈ L2(Tκ ) compared with the usual hypotheses that p ∈ C1
rd(Tκ ), q,w ∈ C0
rd(Tκ2
). Further to these
conditions, we assume that p c >0 on Tκ , C w c >0 on Tκ2 , for some constants C >c>0. These
conditions are similar to the usual assumptions imposed on Sturm–Liouville, ordinary differential equation
problems. We obtain a min–max characterization of the eigenvalues of the above problem, and various
eigenfunction expansions for functions in suitable function spaces. These results extend certain aspects of
the standard theory of self-adjoint operators with compact resolvent to the above problem, even though
the linear operator associated with the left-hand side of the problem is not in fact self-adjoint on general
time-scales.
© 2007 Elsevier Inc. All rights reserved.
