\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2007(2007), No. 175, pp. 1--10.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu (login: ftp)} \thanks{\copyright 2007 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2007/175\hfil Boundary-value problems] {Self-adjoint boundary-value problems on time-scales} \author[F. A. Davidson, B. P. Rynne\hfil EJDE-2007/175\hfilneg] {Fordyce A. Davidson, Bryan P. Rynne} % in alphabetical order \address{Fordyce A. Davidson \newline Division of Mathematics, Dundee University, Dundee DD1 4HN, Scotland} \email{fdavidso@maths.dundee.ac.uk} \address{Bryan P. Rynne \newline Department of Mathematics and the Maxwell Institute for Mathematical Sciences, Heriot-Watt University, Edinburgh EH14 4AS, Scotland} \email{bryan@ma.hw.ac.uk} \thanks{Submitted June 6, 2007. Published December 12, 2007.} \subjclass[2000]{34B05, 34L05, 39A05} \keywords{Time-scales; boundary-value problem; self-adjoint linear operators; \hfill\break\indent Sobolev spaces} \begin{abstract} In this paper we consider a second order, Sturm-Liouville-type boundary-value operator of the form $$L u := -[p u^{\nabla}]^{\Delta} + qu,$$ on an arbitrary, bounded time-scale $\mathbb{T}$, for suitable functions $p,q$, together with suitable boundary conditions. We show that, with a suitable choice of domain, this operator can be formulated in the Hilbert space $L^2(\mathbb{T}_\kappa)$, in such a way that the resulting operator is self-adjoint, with compact resolvent (here, self-adjoint' means in the standard functional analytic meaning of this term). Previous discussions of operators of this, and similar, form have described them as self-adjoint', but have not demonstrated self-adjointness in the standard functional analytic sense. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{definition}[theorem]{Definition} \newtheorem{assumption}[theorem]{Assumption} \newtheorem{remark}[theorem]{Remark} \section{Introduction} \label{intro.sec} Over the past decade a large number of papers on second order, Sturm-Liouville-type boundary value problems on bounded time-scales $\mathbb{T}$ have appeared. Most of these deal with a $\Delta\Delta$ formulation of the corresponding differential operator, viz. $$\label{dede_op.eq} L u := -(p u^{\Delta})^{\Delta} + q u^\sigma, \quad u \in D(L),$$ for suitable functions $p,\,q$, on a suitable domain $D(L)$ (the specification of the domain $D(L)$ includes suitable boundary conditions on $u$; in this introductory section we omit details of spaces and domains). Much of the basic theory of such operators is described in, for example, \cite[Chapter 4]{BP1}. Such operators have often been termed self-adjoint'. However, it was shown in \cite{DR1} that expressions of this form do not, in general, yield self-adjoint operators, in the standard functional-analytic meaning of the term self-adjoint'. Indeed, it is shown in \cite{DR1} that a fundamental property of self-adjoint operators can fail for operators of the form \eqref{dede_op.eq}, so that the standard theory of self-adjoint operators cannot readily be applied to such operators. More recently, in an attempt to obtain self-adjointness, differential operators in the following $\nabla\Delta$ form $$\label{nade_op.eq} L u := -(p u^{\nabla})^{\Delta} + qu, \quad u \in D(L)$$ have been considered, see for example, \cite{AGH,GUS2} and the references therein. Such mixed' operators result in a symmetric Green's function, which is taken to indicate that the corresponding operators possess some of the features of self-adjoint operators. However, the operators constructed in these papers map between (different) Banach spaces of continuously differentiable functions on $\mathbb{T}$, whereas, in the standard functional-analytic definition, a self-adjoint operator is defined on a subspace of a Hilbert space $H$, and maps this subspace of $H$ into $H$ itself. This Hilbert space formulation is necessary to obtain many of the desirable properties of such operators. In this paper our goal is to formulate the $\nabla\Delta$ operator in \eqref{nade_op.eq} in the setting of the Hilbert space $L^2(\mathbb{T}_\kappa)$ defined in \cite{RYN}. This formulation is based on the Sobolev-type spaces defined in \cite{RYN} consisting of functions on $\mathbb{T}$ having $L^2$-type generalised derivatives. We then show that the resulting operator $L$, in $L^2(\mathbb{T}_\kappa)$, is an unbounded, self-adjoint operator, with compact resolvent (in the standard functional-analytic sense). The extensive functional-analytic theory of such operators is then available for this operator, although, for brevity, we will not discuss any applications of this general theory to this operator. \begin{remark} \label{Rm1} \rm We consider the $\nabla\Delta$ operator in \eqref{nade_op.eq}, but operators involving $\Delta\nabla$ combinations (see e.g. \cite{AGH, BP2, GUS2}) could be treated similarly, there is no essential difference in these formulations. Using the $\nabla\Delta$ form allows us to apply the results in \cite{RYN} (based on a Lebesgue-type $\Delta$-integral') unaltered. A corresponding Lebesgue-type $\nabla$-integral' could be constructed using the methods in \cite{RYN}, and this would then allow $\Delta\nabla$ operators to be considered in a similar manner. \end{remark} \section{Preliminaries} Papers on time-scales usually go through a set of standard definitions of integration and differentiation on time-scales. For brevity we will omit this and simply refer to \cite[Section~2]{RYN} for this standard material (which is, of course, also discussed in most other time-scales papers). In particular, we will use the Lebesgue-type $\Delta$-integral defined in \cite{RYN}. A similar Lebesgue-type $\nabla$-integral could readily be defined, but will not be required here. However, we will need to use spaces of $\nabla$-differentiable functions, in addition to the spaces of $\Delta$--differentiable functions discussed in \cite{RYN}. To distinguish between these spaces will require some slight modifications to the notation used for various spaces and norms in \cite{RYN}, so we briefly discuss time-scale differentiation, and the notation we will use. Recall that a function $u : \mathbb{T} \to \mathbb{R}$ is \emph{$\nabla$-differentiable} on $\mathbb{T}$ if, at each $t \in \mathbb{T}_\kappa$, there exists $u^\nabla(t)$ such that, for any $\epsilon>0$ there exists $\delta > 0$ such that $$\text{s \in \mathbb{T} and |t-s| < \delta} \implies | u(\rho(t)) - u(s) - u^\nabla(t) (\rho(t)-s) | \le \epsilon | \rho(t)-s| ,$$ see, for example, \cite[Ch. 3]{BP2}; the $\Delta$-derivative is defined similarly, by replacing $\rho(t)$ with $\sigma(t)$ throughout. We let $C^0(\mathbb{T})$ (respectively $C_{\rm rd}^0(\mathbb{T})$, $C_{\rm ld}^0(\mathbb{T})$) denote the set of continuous (respectively rd-continuous, ld-continuous) functions on $\mathbb{T}$; with the norm $$|u|_{\mathbb{T}} := \sup_{t \in \mathbb{T}}|u(t)|, \quad u \in C_{\rm rd}(\mathbb{T}) \cup C_{\rm ld}(\mathbb{T}) ,$$ all these spaces are Banach spaces. We now let $C^1(\mathbb{T},\Delta)$ (respectively $C_{\rm rd}^1(\mathbb{T},\Delta)$) denote the set of functions $u \in C^0(\mathbb{T})$ which are $\Delta$-differentiable and for which $u^\Delta \in C^0(\mathbb{T}^\kappa)$ (respectively $u^\Delta \in C_{\rm rd}^0(\mathbb{T}^\kappa)$); with the norm $$|u|_{\mathbb{T},\Delta} := |u|_{\mathbb{T}} + |u^\Delta|_{\mathbb{T}^\kappa} , \quad u \in C_{\rm rd}^1(\mathbb{T},\Delta),$$ these spaces are Banach spaces. Similarly, we define the Banach spaces $C^1(\mathbb{T},\nabla)$ and $C_{\rm ld}^1(\mathbb{T},\nabla)$ with norm $$|u|_{\mathbb{T},\nabla}:= |u|_{\mathbb{T}} + |u^\nabla|_{\mathbb{T}_\kappa}, \quad u \in C_{\rm ld}^1(\mathbb{T},\nabla).$$ The spaces $C^1(\mathbb{T},\nabla)$ and $C^1(\mathbb{T},\Delta)$ need not be equal. For example, let $\mathbb{T} = [-1,0] \cup [1,2]$ and define the function $u \equiv 0$ on $[-1,0]$, $u(t) = t$ on $[1,2]$. It can be verified that $u \in C^1(\mathbb{T},\nabla)$, but $u \not\in C^1(\mathbb{T},\Delta)$. However, the following result gives a simple relationship between these spaces \begin{lemma}[{\cite[Theorem~6]{GUS2}}] \label{na_de_si.lem} $C^1(\mathbb{T},\nabla) \subset C_{\rm rd}^1(\mathbb{T},\Delta)$. If $u \in C^1(\mathbb{T},\nabla)$ then $u^\Delta = (u^\nabla)^\sigma$. \end{lemma} It will also be necessary to $\nabla$-differentiate indefinite $\Delta$-integrals, for which we will require the following lemma. \begin{lemma}[{\cite[Theorem~2.10]{AG}}]\label{na_deriv_int.lem} If $u \in C^0(\mathbb{T})$, $t_0 \in \mathbb{T}$, and $$U_{t_0}(t) := \int_{t_0}^t u \,\Delta, \quad t \in \mathbb{T},$$ then $U_{t_0} \in C_{\rm ld}^1(\mathbb{T},\nabla)$ and $U_{t_0}^\nabla = u^\rho$ on $\mathbb{T}_\kappa .$ \end{lemma} We will also require the Sobolev-type space of functions with generalised $\Delta$-derivatives defined in \cite{RYN}, which we will denote here by $H^1(\mathbb{T},\Delta)$ with associated norm $$\|u\|_{\mathbb{T},\Delta}:= \|u\|_{\mathbb{T}} + \|u^\Delta\|_{\mathbb{T}}, \quad u \in H^1(\mathbb{T},\Delta),$$ where $$\|u\|^2_{\mathbb{T}}: = \int_{\mathbb{T}} |u|^2 \Delta, \quad u \in L^2(\mathbb{T}).$$ Note that the integral used here is the Lebesgue-type $\Delta$-integral constructed in \cite{RYN}. We also note that \cite[Lemma~3.5]{RYN} shows that $C_{\rm rd}^1(\mathbb{T},\Delta) \subset H^1(\mathbb{T},\Delta)$, so Lemma~\ref{na_de_si.lem} has the following simple corollary, which will be required below. \begin{corollary} \label{na_de_si.cor} $C^1(\mathbb{T},\nabla) \subset H^1(\mathbb{T},\Delta)$. \end{corollary} Finally, in this preliminary section, we recall some basic functional-analytic definitions, see for example \cite[Ch. 13]{RUD}. Let $T : D(T) \subset H \rightarrow H$ be a linear operator in a Hilbert space $H$, with inner product $\langle \cdot\,,\cdot \rangle$. Then $T$ is \emph{symmetric} if $$\langle Tx,y \rangle = \langle x, Ty\rangle , \quad \forall\, x,\,y \in D(T) ,$$ and $T$ is \emph{self-adjoint} if $D(T)$ is dense in $H$ and $$\langle Tx,y \rangle = \langle x,z\rangle , \quad \forall\, x \in D(T) \implies \text{y \in D(T) and  z = Ty.}$$ \section{A boundary value linear operator} \label{form.sec} \subsection{Definition of $L$} \label{defn_L.sec} Let $a=\inf \mathbb{T}$, $b = \sup \mathbb{T}$. We are interested in the class of functions defined on $\mathbb{T}$ which satisfy the boundary conditions $$\label{bc.eq} u^\nabla(\sigma(a)) = \gamma_a u(\sigma(a)), \quad u^\nabla(b) = - \gamma_b u(b) ,$$ with arbitrary constants $\gamma_a \in (-\infty,\infty]$, $\gamma_b \in (-\infty,\infty]$, and we define the following set of functions %which will form the domain of our operator $L$: \begin{gather*} \mathcal{D} := \{ u \in C^1(\mathbb{T},\nabla) : \text{$u^\nabla \in H^1(\mathbb{T}_\kappa,\Delta)$ and $u$ satisfies \eqref{bc.eq}} \}, \\ D(L) := \{ w \in L^2(\mathbb{T}_\kappa) : \text{$w = u|_{\mathbb{T}_\kappa^\kappa}$ for some $u \in \mathcal{D}$} \} \end{gather*} (in the definition of $D(L)$, $w = u|_{\mathbb{T}_\kappa^\kappa}$ denotes the restriction of $u$ to the set $\mathbb{T}_\kappa^\kappa$, and we recall from \cite{RYN} that the point $b$ has $\mu_\mathbb{T}$-measure zero, so in the setting of equivalence classes of $L^2(\mathbb{T}_\kappa)$ functions, the value $u(b)$ is not well-defined). Throughout, we impose the following additional assumption on $\gamma_a$, $\gamma_b$. \begin{assumption} \label{gamma.ass} (i) If $a$ is right-scattered then $\gamma_a < \infty$. (ii) If $b$ is left-scattered then $1+\gamma_b ( b- \rho(b)) \neq0$. \end{assumption} These constructions require some further explanation and remarks. \begin{enumerate} \item[(a)] In the above notation the cases $\gamma_a=\infty$ or $\gamma_b=\infty$ are taken to mean the conditions $u(\sigma(a)) = 0$ or $u(b) = 0$, and in these cases it is the latter form that would be used in the calculations below. Furthermore, if $a$ is right-dense, these cases correspond to the Dirichlet-type conditions $u(a) = 0$ or $u(b) = 0$. It will be seen in Remark~\ref{D_bc_a.rem} below that when $a$ is right-scattered the Dirichlet-type condition at $a$ arises from a different value of $\gamma_a$. \item[(b)] Assumption~\ref{gamma.ass} precludes the boundary conditions $u(\sigma(a))=0$ (when $a$ is right-scattered) or $u(\rho(b))=0$ (when $b$ is left-scattered). Either of these conditions lead to certain pathological properties of the operator $L$ which we wish to avoid. \item[(c)] If $a$ is right-scattered it is natural, in view of the definition of the $\nabla$-derivative, to formulate the first boundary condition in \eqref{bc.eq} in terms of $u^\nabla(\sigma(a))$, but the use of $u(\sigma(a))$, rather than $u(a)$, may seem slightly strange. This formulation is chosen, primarily, to simplify certain formulae arising from various integrations by parts below. The following remark shows that $u(a)$ could be used in \eqref{bc.eq} simply by changing the value of $\gamma_a$. \item[(d)] \label{alt_bc.rem} If $a$ is right-scattered or $b$ is left-scattered, the corresponding boundary conditions in \eqref{bc.eq} can be rewritten in the alternative forms $$\label{alt_BC.eq} \begin{split} u(a) - \bigl(1 + (\sigma(a)-a)\gamma_a \bigr) u(\sigma(a)) & = 0 , \\ u(\rho(b)) - \bigl(1 + (b-\rho(b))\gamma_b \bigr) u(b) & = 0 . \end{split}$$ Hence, by \eqref{alt_BC.eq} and Assumption~\ref{gamma.ass}, if $u \in \mathcal{D}$ then $u(a)$ and $u(b)$ are determined by $u(\sigma(a))$ and $u(\rho(b))$, that is, $u$ is determined entirely by its restriction $w = u|_{\mathbb{T}_\kappa^\kappa} \in D(L)$. Conversely, by using \eqref{alt_BC.eq}, any function $w \in D(L)$ can be extended to $\mathbb{T}$ to yield a function $u \in \mathcal{D}$. Thus, the sets $\mathcal{D}$ and $D(L)$ are (algebraically) isomorphic, and can be naturally identified with each other. In our discussion of $L$ below we will make use of this identification, and we will generally use the symbol $u$ interchangeably for an element of either $\mathcal{D}$ or $D(L)$. \item[(e)] \label{D_bc_a.rem} If $a$ is right-scattered then it follows from \eqref{alt_BC.eq} that we obtain the Dirichlet-type condition $u(a) = 0$ by choosing $\gamma_a$ such that $1 + (\sigma(a)-a)\gamma_a = 0$. \end{enumerate} Having dealt with the boundary conditions, we now define the desired differential operator $L$. Suppose that $p \in H^1(\mathbb{T}_\kappa,\Delta)$, $q \in L^2(\mathbb{T}_\kappa)$, with $$p_{\min} := \min \{ p(t) : t \in \mathbb{T}_\kappa \} > 0 ,$$ and define the linear operator $L : D(L) \subset L^2(\mathbb{T}_\kappa) \to L^2(\mathbb{T}_\kappa)$ by \begin{align*} %D(L) &:= \{ u \in C^1(\mathbb{T}_\kappa,\nabla) : \text{$u^\nabla \in H^1(\mathbb{T}_\kappa,\Delta)$ and $u$ satisfies \eqref{bc.eq}} \}, %\\ L u &:= -[p u^{\nabla}]^{\Delta_g} + qu, \quad u \in D(L), \end{align*} where $\Delta_g$ is the generalised $\Delta-$derivative constructed in \cite{RYN}. The definition $L$ also requires some further explanation and remarks. \begin{enumerate} \item[(a)] The set $\mathcal{D}$ would be a natural domain for the operator $L$. However, to obtain a self-adjoint operator it is necessary that the domain and range of $L$ lie in the same Hilbert space (which we take to be $L^2(\mathbb{T}_\kappa)$). For this reason we introduce the domain $D(L) \subset L^2(\mathbb{T}_\kappa)$, isomorphic to $\mathcal{D}$. \item[(b)] In light of the identification of $\mathcal{D}$ and $D(L)$ described in Remark~\ref{alt_bc.rem} above, we regard the calculation of $Lu \in L^2(\mathbb{T}_\kappa)$ from $u \in D(L)$ as proceeding in the following manner: use \eqref{alt_BC.eq} to extend $u$ from the set $\mathbb{T}_\kappa^\kappa$ to $\mathbb{T}$ (yielding an element of $\mathcal{D}$, which we still write as $u$), and then construct $u^\nabla \in H^1(\mathbb{T}_\kappa,\Delta)$ and $(u^\nabla)^{\Delta_g} \in L^2(\mathbb{T}_\kappa)$ in the usual manner (by the definition of $\mathcal{D}$, these are well-defined for $u \in \mathcal{D}$). \item[(c)] The operator $Lu = -[p u^{\Delta}]^{\Delta_g} + qu^\sigma$, on a similar domain, was considered in \cite{RYN}. However, we will see that the above operator is self-adjoint, (in the functional-analytic sense), whereas the operator in \cite{RYN} is not. Despite this difference, the comments in \cite[Remarks~5.1 and~5.2]{RYN} regarding the definition of $L$ there apply equally well to the above operator. \end{enumerate} \subsection{Properties of $L$} \label{props_L.sec} We now obtain various basic properties of $L$. \begin{lemma} \label{L_symm.lem} The operator $L$ is symmetric with respect to the inner product $\langle \cdot \,, \cdot \rangle_{\mathbb{T}_\kappa}$ on $L^2(\mathbb{T}_\kappa)$, that is, %For any $u,\,v \in D(L)$, $$\label{L_symm.eq} \langle Lu , v \rangle_{\mathbb{T}_\kappa} = \langle u , Lv \rangle_{\mathbb{T}_\kappa} , \quad u,\,v \in D(L) .$$ \end{lemma} \begin{proof} By definition, we can regard $u,\,v$ as belonging to $\mathcal{D}$, that is $u,\,v \in C^1(\mathbb{T},\nabla)$. Thus, by Lemma~\ref{na_de_si.lem}, $u,\,v \in C_{\rm rd}^1(\mathbb{T},\Delta)$, and hence, by \cite[Corollary~4.6~(f)]{RYN}, $$\label{Luv_ip.eq} \begin{split} \langle L u , v \rangle_{\mathbb{T}_\kappa} &= \int_{\sigma(a)}^b (p u^\nabla)^\sigma v^\Delta \,\Delta - [p u^\nabla v ]_{\sigma(a)}^b + \int_{\sigma(a)}^b q u v \, \Delta \\[1 ex]&= \int_{\sigma(a)}^b \bigl( p^\sigma u^\Delta v^\Delta + q u v \bigr)\,\Delta + B(u,v) \end{split}$$ where, by \eqref{bc.eq}, $$B(u,v) = \gamma_a p(\sigma(a)) u(\sigma(a)) v(\sigma(a)) + \gamma_b p(b) u(b) v(b)$$ (if $\gamma_a=\infty$ or $\gamma_b=\infty$ then we omit the corresponding term in this formula). The result now follows from the symmetry in $u$ and $v$ of the right hand side of \eqref{Luv_ip.eq}. \end{proof} \begin{lemma} \label{L_pos_def.lem} There exists a constant $C_L$ such that $$\label{L_pos_def.eq} \langle L u , u \rangle_{\mathbb{T}_\kappa} \ge \tfrac12 p_{\min} \|u\|_{\mathbb{T}_\kappa,\Delta}^2 + C_L \|u\|_{\mathbb{T}_\kappa}^2 , \quad u \in D(L).$$ \end{lemma} \begin{remark} \label{rmk3.4} \rm The constant $C_L$ in Lemma~\ref{L_pos_def.lem} need not be positive. \end{remark} \begin{proof} Suppose that $u \in D(L)$. Then we can regard $u$ as belonging to $\mathcal{D}$, and it follows from \eqref{Luv_ip.eq} and the Cauchy-Schwarz inequality that $$\label{int_parts.eq} \langle L u , u \rangle_{\mathbb{T}_\kappa} \ge p_{\min} \|u^\Delta\|_{\mathbb{T}_\kappa}^2 - C_1 |u|_{\mathbb{T}_\kappa}^2 ,$$ for some constant $C_1 \ge 0$ (independent of $u$), and by \eqref{alt_BC.eq} and Assumption~\ref{gamma.ass}, $$\label{1} |u|_{\mathbb{T}_\kappa}^2 \leq C_2 |u|_{\mathbb{T}^\kappa_\kappa}^2,$$ for some constant $C_2>0$. Also, a straightforward modification of the proof of \cite[Theorem~4.16]{RYN} shows that for any $\epsilon>0$ there exists a constant $C_3(\epsilon)>0$ such that $$\label{4} |w|_{\mathbb{T}_\kappa^\kappa} \leq \epsilon \|w^\Delta\|_{\mathbb{T}_\kappa} + C_3(\epsilon) \|w\|_{\mathbb{T}_\kappa} , \quad w \in H^1(\mathbb{T}_\kappa,\Delta) .$$ By Corollary~\ref{na_de_si.cor} and the definition of $\mathcal{D}$, $u \in H^1(\mathbb{T}_\kappa,\Delta)$, so putting $\epsilon$ sufficiently small and $w = u$ in \eqref{4} and combining this with \eqref{int_parts.eq} and \eqref{1} yields \eqref{L_pos_def.eq}. \end{proof} Invertibility of $L$ will be important below, and it will be seen that invertibility follows from injectivity of $L$, so we now consider this. In general, $L$ need not be injective, but the following result shows that we can obtain injectivity by adding to $L$ a sufficiently large scalar multiple of the identity operator $I : L^2(\mathbb{T}_\kappa) \to L^2(\mathbb{T}_\kappa)$. In many situations, if $L$ itself is not injective then it is possible, with no loss of generality, to replace $L$ with the injective operator $L_c$ given by the following result. \begin{theorem} \label{L_injective.thm} If $c + C_L > 0$ then the operator $L_c := L + c I$ is injective. \end{theorem} \begin{proof} It follows from \eqref{L_pos_def.eq} that $$\langle L_c u , u \rangle_{\mathbb{T}_\kappa} \ge \tfrac12 p_{\min} \|u\|_{\mathbb{T}_\kappa,\Delta}^2 + (c+C_L) \|u\|_{\mathbb{T}_\kappa}^2 >0, \quad 0 \ne u \in D(L),$$ which proves that $L_c$ is injective. \end{proof} The following result gives simple criteria under which $L$ itself is injective. \begin{theorem} \label{L_injective_pos.thm} Suppose that $q \ge 0$ on $\mathbb{T}_\kappa$ and $\gamma_a,\,\gamma_b \ge 0$. Then $L$ is injective under either of the hypotheses$:$ \begin{itemize} \item[(i)] $\gamma_a + \gamma_b > 0;$ \item[(ii)] $\|q\|_{\mathbb{T}_\kappa} > 0.$ \end{itemize} \end{theorem} \begin{proof} We consider hypothesis (i), a similar proof holds for hypothesis (ii). Suppose that $0 \ne u \in D(L)$ and $Lu=0$. It follows from this and \eqref{Luv_ip.eq} that $$0 = \langle Lu,u \rangle_{\mathbb{T}_\kappa} \ge p_{\min} \|u^\Delta\|_{\mathbb{T}_\kappa}^2 + B(u,u),$$ and hence, by \cite[Corollary~4.6]{RYN}, $u \equiv 0$ on $\mathbb{T}_\kappa$. \end{proof} We will also need the following result regarding solutions of the corresponding initial value problem. This result can be proved in a similar manner to that of \cite[Theorem~5.8]{RYN}. \begin{theorem} \label{ivp.thm} For any $h \in L^2(\mathbb{T}_\kappa)$ and $\tau \in \mathbb{T}_\kappa$, $\eta_1,\,\eta_2 \in \mathbb{R}$, the initial value problem $$\label{ivp.eq} \begin{gathered} -(pu^\nabla)^{\Delta_g} + q u = h, \\ u(\tau) = \eta_1, \quad u^\nabla(\tau) = \eta_2, \end{gathered}$$ has a unique solution $u \in C^1(\mathbb{T},\nabla)$, with $u^\nabla \in H^1(\mathbb{T}_\kappa,\Delta)$. \end{theorem} Let $\phi,\,\psi$ be the solutions of \eqref{ivp.eq} given by Theorem~\ref{ivp.thm}, with $h=0$ and the initial' conditions $$\label{ivpbc.eq} \begin{array}{rr@{\ }l} \phi(\sigma(a)) = 1, & \phi^{\nabla}(\sigma(a)) &= \gamma_a,\\ \psi(b) = 1, & \psi^{\nabla}(b) &= -\gamma_b ,\, \end{array}$$ with the obvious modification, here and below, when $\gamma_a=\infty$ or $\gamma_b=\infty$. Also, let $$W := p \bigl(\phi^{\nabla}\psi - \psi^{\nabla}\phi\bigr) \in H^1(\mathbb{T}_\kappa,\Delta)$$ (it follows from the properties of $\phi,\,\psi$ given by Theorem~\ref{ivp.thm}, together with Corollary~\ref{na_de_si.cor} and \cite[Corollary~4.6]{RYN}, that $W \in H^1(\mathbb{T}_\kappa,\Delta)$). \begin{lemma} \label{dn0.lem} $W$ is constant on $\mathbb{T}_\kappa$. The operator $L$ is injective if and only if $W \neq 0$. \end{lemma} \begin{proof} From the definitions of $\phi$, $\psi$, Corollary~\ref{na_de_si.cor} and \cite[Corollary~4.6]{RYN}, \begin{align*} W^{\Delta_g} &= (p\phi^\nabla)^{\Delta_g} \psi + (p\phi^{\nabla})^\sigma \psi^{\Delta} -(p\psi^\nabla)^{\Delta_g} \phi - (p\psi^{\nabla})^\sigma \phi^{\Delta} \\&= q \phi \psi + (p\phi^{\nabla}\psi^{\nabla})^\sigma - q \psi \phi - (p\psi^{\nabla}\phi^{\nabla})^\sigma = 0, \end{align*} so by \cite[Corollary~4.6]{RYN}, $W \equiv \text{const}$. Moreover, by \eqref{ivpbc.eq}, $$W = p(b) \bigl( \phi^\nabla(b) + \gamma_b \phi(b)) ,$$ so $W=0$ if and only if $\phi$ satisfies the boundary conditions \eqref{bc.eq}. Clearly, if $\phi$ satisfies \eqref{bc.eq} then $L$ is not injective, and the converse follows immediately from linearity and the uniqueness of the solution of the initial value problem for $\phi$. \end{proof} We can now begin the construction of the inverse of $L$ (when $L$ is injective). Equivalently, we construct a solution of the boundary value problem $$\label{bvp.eq} Lu = h, \quad h \in L^2(\mathbb{T}_\kappa) , \quad u \in D(L),$$ for any $h \in L^2(\mathbb{T}_\kappa)$. \begin{definition} \label{xst.def} \rm Suppose that $L$ is injective. For $(t,s) \in \mathbb{T} \times \mathbb{T}$ let $$g(t,s) := \begin{cases} W^{-1} \psi(t) \phi(s), \quad \text{if t \geq s,} \\ W^{-1} \phi(t) \psi(s), \quad \text{if t \leq s.} \end{cases}$$ Clearly, $g$ is continuous on $\mathbb{T} \times \mathbb{T}$. For any $h \in L^2(\mathbb{T}_\kappa)$, let $$\label{Gh.eq} Gh(t) := \int_{\sigma(a)}^b g(t,\cdot) h \,\Delta , \quad t \in \mathbb{T}_\kappa^\kappa.$$ \end{definition} \begin{theorem} \label{LGinverse.thm} Suppose that $L$ is injective. Then$:$ \begin{itemize} \item[(i)] for any $h \in L^2(\mathbb{T}_\kappa)$ the function $u = Gh \in D(L),$ and $u$ is the unique solution of \eqref{bvp.eq}$;$ \item [(ii)] the operators $L : D(L) \subset L^2(\mathbb{T}_\kappa) \to L^2(\mathbb{T}_\kappa)$, $G : L^2(\mathbb{T}_\kappa) \to D(L) \subset L^2(\mathbb{T}_\kappa)$, are invertible, linear operators and $L^{-1} = G$, $G^{-1} = L$. The operator $G$ is compact, while if $\dim L^2(\mathbb{T}_\kappa) = \infty$ then $L$ is unbounded. \end{itemize} \end{theorem} \begin{remark} \label{epresults.rem} \rm We call $g$ the \emph{Green's function} and $G$ the \emph{Green's operator} for the operator $L$. \end{remark} \begin{proof} The uniqueness follows immediately from the injectivity of $L$. Now suppose that $h \in C^0(\mathbb{T}_\kappa)$. To simplify the following calculations we will suppose that $p \equiv 1$; the general proof is similar. It is clear that the formula for $u = Gh$ in \eqref{Gh.eq} can be extended to define a function on the whole of $\mathbb{T}$, which we continue to denote by $u$. Now suppose that $a$ is right-dense. Then by direct calculation (using Lemma~\ref{na_deriv_int.lem} above, the product rule for nabla derivatives, see \cite{BP1}, and for generalised derivatives, see Corollary~4.6 in \cite{RYN}, and (3.3) in \cite{RYN}), \begin{equation*} \begin{split} W u(t) &= \psi(t) \int_{\sigma(a)}^t \phi h \,\Delta + \phi(t) \int_t^{b} \psi h \,\Delta , \quad t \in \mathbb{T},\\ W u^{\nabla}(t) &= \psi^\rho(t) \phi^\rho(t) h^\rho(t) + \psi^\nabla(t) \int_{\sigma(a)}^t \phi h \,\Delta\\ &\quad - \phi^\rho(t) \psi^\rho(t) h^\rho(t) +\phi^\nabla(t) \int_t^{b} \psi h \,\Delta \\ &=\psi^\nabla(t) \int_{\sigma(a)}^t \phi h \,\Delta +\phi^\nabla(t) \int_t^{b} \psi h \,\Delta, \quad t \in \mathbb{T}, \\ W (u^\nabla)^{\Delta_g}(t) &=-W h(t) + (\psi^\nabla)^{\Delta_g}(t) \int_{\sigma(a)}^{\sigma(t)} \phi h \,\Delta + (\phi^\nabla)^{\Delta_g}(t) \int_{\sigma(t)}^{b} \psi h \Delta \\ &= -W h(t) + q(t) \biggl\{ \psi(t) \int_{\sigma(a)}^{\sigma(t)} \phi h \,\Delta + \phi(t) \int_{\sigma(t)}^{b} \psi h \,\Delta \biggr\} \\ &= -W h(t) + q(t) \biggl\{ \psi(t) \int_{\sigma(a)}^t \phi h \,\Delta + \phi(t) \int_t^{b} \psi h \,\Delta \biggr\} \\ &= -W h(t) + q(t) W u(t), \quad \text{$\mu_\mathbb{T}$-a.e. $t \in \mathbb{T}$.} \end{split} \end{equation*} It follows directly from these formulae and \eqref{ivpbc.eq} that $u = Gh \in \mathcal{D}$, with $$\label{Gh_bnd.eq} \|Gh\|_{\mathcal{D}} = \|u\|_{\mathcal{D}}:= |u|_{\nabla,\mathbb{T}} + \|u^\nabla\|_{\mathbb{T}_\kappa,\Delta} \le C \|h\|{_{\mathbb{T}_\kappa}}, \quad h \in C^0(\mathbb{T}_\kappa),$$ for some constant $C$. On the other hand, if $a$ is right-scattered the calculation of $Wu(a)$ and $Wu^\nabla(\sigma(a))$ needs to be amended slightly to avoid reference to the (undefined) value $h(a)$, in the following manner. Directly from \eqref{Gh.eq}, \begin{gather*} W u(a) = \phi(a) \int_{\sigma(a)}^b \psi h \,\Delta , \quad W u(\sigma(a)) = \int_{\sigma(a)}^b \psi h \,\Delta , \\ W u^\nabla(\sigma(a)) = W \frac{u(\sigma(a))-u(a)}{\sigma(a)-a} = \phi^\nabla(\sigma(a)) \int_{\sigma(a)}^b \psi h \,\Delta = W \gamma_a u(\sigma(a)). \end{gather*} Thus the boundary condition at $\sigma(a)$ also holds in this case, and it follows from the preceding formulae that if $\sigma(a)$ is right-dense then $\lim_{t\to\sigma(a)^+} u(t) = u(\sigma(a))$, $\lim_{t\to\sigma(a)^+} u^\nabla(t) = u^\nabla(\sigma(a))$, (here, $t\to\sigma(a)^+$ through points in $\mathbb{T}_\kappa$), so that $u \in C^1(\mathbb{T},\nabla)$, that is we again have $u \in \mathcal{D}$. Clearly, the inequality \eqref{Gh_bnd.eq} also holds in this case. Now, since $C^0(\mathbb{T}_\kappa)$ is dense in $L^2(\mathbb{T}_\kappa)$ (see Lemma~3.5 in \cite{RYN}), and $\mathcal{D}$ is a Banach space (with respect to the above norm $\|\cdot\|_{\mathcal{D}}$), the inequality \eqref{Gh_bnd.eq} extends from the set $C^0(\mathbb{T}_\kappa)$ to the whole of $L^2(\mathbb{T}_\kappa)$ by continuity. Hence, in particular, $G$ is bounded as an operator from $L^2(\mathbb{T}_\kappa)$ into $\mathcal{D}$. The assertions about the operators $L$ and $G$ now follow immediately (the compactness of $G$ follows from \eqref{Gh_bnd.eq}, Lemma~\ref{na_de_si.lem} and the compactness of the embedding $C_{\rm rd}^1(\mathbb{T},\Delta) \to C^0(\mathbb{T})$, see \cite[Lemma~2.2]{DR1}). \end{proof} We can now prove that $L$ is self-adjoint (irrespective of injectivity of $L$). \begin{theorem} \label{L_sa.thm} The domain $D(L)$ is dense in $L^2(\mathbb{T}_\kappa)$, and the operator $L$ is self-adjoint with respect to the inner product $\langle \cdot \,, \cdot \rangle_{\mathbb{T}_\kappa}$ on $L^2(\mathbb{T}_\kappa)$. \end{theorem} \begin{proof} It suffices to prove the result for $L_c$, for arbitrary $c \in \mathbb{R}$, so without loss of generality we suppose that $c=0$ and $L$ is injective. If $D(L)$ is not dense in $L^2(\mathbb{T}_\kappa)$ then there exists $0 \ne w \in L^2(\mathbb{T}_\kappa)$ such that $$\langle u , w \rangle_{\mathbb{T}_\kappa} = 0, \quad \forall u \in D(L).$$ Since $R(L) = L^2(\mathbb{T}_\kappa)$, we have $w = Lz$ for some $z \in D(L)$, so by Lemma~\ref{L_symm.lem}, $$0 = \langle u , Lz \rangle_{\mathbb{T}_\kappa} = \langle Lu , z \rangle_{\mathbb{T}_\kappa}, \quad \forall u \in D(L),$$ and hence $z=0$ (again, since $R(L) = L^2(\mathbb{T}_\kappa)$). However, this implies that $w=0$, which contradicts the choice of $w$, and so proves that $D(L)$ is dense in $L^2(\mathbb{T}_\kappa)$. Now suppose that $$\label{sa_test.eq} \langle Lu , v \rangle_{\mathbb{T}_\kappa} = \langle u , w \rangle_{\mathbb{T}_\kappa}, \quad \forall u \in D(L),$$ for some $v ,\,w \in L^2(\mathbb{T}_\kappa)$. We again have $w = Lz$, for some $z \in D(L)$, and so from \eqref{sa_test.eq}, $$\langle Lu , v-z \rangle_{\mathbb{T}_\kappa} = 0, \quad \forall u \in D(L),$$ and hence $v=z$. That is, $v \in D(L)$ and $w = Lv$, which proves that $L$ is self-adjoint. \end{proof} \begin{corollary} \label{G_sa.cor} Suppose that $L$ is injective. Then the operator $G$ is self-adjoint with respect to the inner product $\langle \cdot \,, \cdot \rangle_{\mathbb{T}_\kappa}$ on $L^2(\mathbb{T}_\kappa)$. \end{corollary} \begin{proof} Let $u,\,v \in L^2(\mathbb{T}_\kappa)$ be arbitrary. Then by Theorem~\ref{LGinverse.thm}, $u=Lx$, $v=Ly$, for some $x,\,y \in D(L)$. Hence, by Lemma~\ref{L_symm.lem} and Theorem~\ref{LGinverse.thm}, $$\langle Gu , v \rangle_{\mathbb{T}_\kappa} = \langle x , Ly \rangle_{\mathbb{T}_\kappa} = \langle Lx , y \rangle_{\mathbb{T}_\kappa} = \langle u , Gv \rangle_{\mathbb{T}_\kappa} ,$$ which proves that $G$ is self-adjoint. \end{proof} \begin{thebibliography}{00} \bibitem{AG} {F. M. Atici and G. S. Guseinov}, On Green's functions and positive solutions for boundary-value problmes on time scales, \emph{J. Comput. Appl. 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