Heriot-Watt Mathematics Report Series
HWM06-18, 21 Apr 2006
$L^2$ spaces and boundary value problems on time-scales
B P Rynne
Abstract
\newcommand\Crd{C_{\rm rd}}
In this paper we consider a
second order, Sturm-Liouville-type boundary value operator
of the form
$$
L u := -[p u^{\De}]^{\De} + qu^\si,
$$
on an arbitrary, bounded time-scale $\T$, for suitable functions $p,\,q$,
together with suitable boundary conditions.
Operators 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 introduce a space $L^2(\T)$
of square-integrable functions on $\T$,
and Sobolev-type spaces $H^n(\T)$, $n \ge 1$,
consisting of $L^2$ functions
with $n$th order, generalised, $L^2$-type derivatives.
We prove some basic functional
analytic results for these spaces,
and then formulate the operator $L$ in this setting.
In particular, we allow $p \in H^1(\T)$,
while $q \in L^2(\T)$ - this generalises the usual conditions that
$p \in \Crd^1(\T^\ka)$, $q \in \Crd^0(\T^{\ka^2})$.
We give some immediate applications of the functional analytic results
to $L$, such as `positivity', injectivity, invertibility
and compactness of the inverse.
We also construct a Green's function for $L$.
The analogues of these results on real intervals are well-known,
and are fundamental to the
usual Sturm-Liouville theory on such intervals.
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