Eigenfunction expansions in L2 spaces for boundary value problems on time-scales

[摘要] Let T subset of R be a bounded time-scale, with a = inf T, b = sup T. We consider the weighted, linear, eigen-value problem -(pu(Delta) )(Delta)(t)+q(t)u(sigma)(t)=lambda w(t)u(sigma)(t), t is an element of T-k2, c(00)u(a)+c(01)u(Delta)(a)=0, c(10)u(p(b)) + c(11)u(Delta)(p(b))=0, for suitable functions p, q and w and lambda is an element of R. Problems of this type on time-scales have non-nally 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 L-2-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 is an element of H-1 (T-k) and q, w is an element of L-2(T-k) compared with the usual hypotheses that p is an element of C-rd(1) (T-k), q, w is an element of C-rd(0)(T-k2). Further to these conditions, we assume that p >= c > 0 on T-k, C >= w >= c > 0 on T-K2, 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. (c) 2007 Elsevier Inc. All rights reserved.

[发布日期] 2007-11-15

[关键词] time-scales;boundary value problem;weighted eigenvalue problem