\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2008(2008), No. 68, pp. 1--21.\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 2008 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2008/68\hfil Time scale terminal value problems] {Terminal value problems for first and second order nonlinear equations on time scales} \author[R. Hilscher, C. C. Tisdell\hfil EJDE-2008/68\hfilneg] {Roman Hilscher, Christopher C. Tisdell} % in alphabetical order \address{Roman Hilscher \newline Department of Mathematics and Statistics, Faculty of Science, Masaryk University, Jan\'a\v ckovo n\'am. 2a, CZ-60200 Brno, Czech Republic} \email{hilscher@math.muni.cz} \address{Christopher C. Tisdell \newline School of Mathematics and Statistics, The University of New South Wales, Sydney NSW 2052, Australia} \email{cct@unsw.edu.au} \thanks{Submitted February 11, 2008. Published May 1, 2008.} \subjclass[2000]{34C99, 39A10} \keywords{Time scale; terminal value problem; nonlinear equation; \hfill\break\indent Banach fixed point theorem; bounded solution; weighted norm} \begin{abstract} In this paper we examine terminal'' value problems for dynamic equations on time scales -- that is, a dynamic equation whose solutions are asymptotic at infinity. We present a number of new theorems that guarantee the existence and uniqueness of solutions, as well as some comparison-type results. The methods we employ feature dynamic inequalities, weighted norms, and fixed-point theory. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{remark}[theorem]{Remark} \section{Introduction} \label{S:intro} The theory of terminal'' value problems, where the problem consists of a dynamic equation coupled with asymptotic behavior of the solution at $\infty$, forms an interesting and more challenging field of research than the theory of initial value problems. This is due to even the basic results and methods known for initial value problems, such as the perturbation technique, are unavailable for use in the setting of terminal value problems. For example, the existence of a solution to the terminal value problem $x'=f(t,x)$, $x(\infty)=x_0$, need not imply that the terminal value problem $x'=f(t,x)\pm\frac{1}{n}$, $x(\infty)=x_0\pm\frac{1}{n}$ has a solution, see \cite[pg.~1173]{arA.vL81}. In this work, we examine terminal value problems for dynamic equations on time scales'', which is a new and versatile area of mathematics that is more general than the fields of differential equations and difference equations. The area of time scales originates in the work of Hilger in \cite{sH90}. Such investigations reveal the bonds and distinctions between the two areas and also provide a framework with which to more accurately model stop-start processes. Our main interest herein is in the qualitative properties of solutions to terminal value problems on time scales, including the existence, uniqueness, and comparison theorems. The methods that we employ involve dynamic inequalities, weighted norms, and fixed point theory. The motivation for using the weighted (or Bielecki) norms originates in \cite{ccT.aZ08} and the references quoted therein, where this method was used in order to prove the existence and uniqueness results for nonlinear initial value problems on bounded time scales. The existence of bounded solutions to {\em initial\/} value problems for second order dynamic equations and inequalities on unbounded time scales was studied in \cite{rpA.mB.dO02}, while in \cite{rpA.mB.dO01} results of this type are given for certain first order dynamic equations. In the latter three references, as well as in the present paper, the fixed point theory is utilized. Our results extend some of the ideas in \cite{arA.vL81} and, more recently, those of \cite{lE.aP.ccT?}. More specifically, we provide some extensions of the comparison results in \cite{arA.vL81}, which were formulated for terminal value problems involving ordinary differential equations, to the time scale environment. Furthermore, compared to \cite{lE.aP.ccT?} we allow in Section \ref{S:second.order} the nonlinearity $f(t,x^\sigma)$ or $f(t,x^\sigma,x^{\Delta\sigma})$ to be vector valued and we pose no restriction on the sign of its entries. Also, we assume in those results that the leading coefficient $r(t)$ is merely nonzero as opposed to the assumption of its positivity in \cite{lE.aP.ccT?}. In addition, for the case of positive $r(t)$ and nonnegative nonlinearity $f$ we extend in Section \ref{S:second.order.matrix} the ideas in \cite{lE.aP.ccT?} from the scalar case to the matrix/vector case. Some of the main results (e.g., Theorems \ref{T:compare1}, \ref{T:compare2}, \ref{T:2nd.order}, \ref{T:2nd.order.xD}, \ref{T:2nd.order.matrix} and \ref{T:2nd.order.xD.matrix}) appear to be new even for the special case ${\mathbb T}={\mathbb Z}$, that is, for difference equations. For additional papers that contain comparison and existence and uniqueness results for first-order terminal value problems involving ordinary differential equations, we refer the reader to \cite{tgH70,tgH72,rL.pV83,gV76}. For papers dealing with second-order terminal value problems, the reader is referred to \cite{tgH72,weS69,weS70}. The methods used in the range of the aforementioned papers involve differential inequalities and the fixed-point theorems of Banach or Schauder. The setup of the paper is the following. In Section \ref{S:pre} we introduce necessary notation and terminology as well as some preparatory results about the time scale exponential function. In Section \ref{S:first.order} we derive an existence and uniqueness theorem for the terminal value problem of the first order. Then we continue in deriving comparison results for solutions of first order dynamic inequalities. In Section \ref{S:second.order} we consider terminal value problems for second order dynamic equations with scalar leading coefficient, while in Section \ref{S:second.order.matrix} we deal with such equations with matrix leading coefficient and with nonnegative nonlinearity. In Section \ref{S:examples} we present examples illustrating the applicability of the obtained results. Finally, in Section \ref{S:appl} we discuss further applications and extensions, in particular to nabla dynamic terminal value problems. \section{Prerequisites and notation} \label{S:pre} Let $n\in{\mathbb N}$ be a fixed natural number. For a real symmetric $n\times n$ matrix $A$ we write $A>0$ or $A\geq0$ for $A$ being a positive definite or positive semidefinite matrix, respectively. Moreover, if $B$ is a real symmetric $n\times n$ matrix, then we write $A0$ or $B-A\geq0$, respectively. In this paper we will use the vector norm $|\cdot|_\infty$ on $\mathbb{R}^n$ denoted for simplicity by $|x|:=|x|_\infty=\max\{|x_i|,\ i=1,\dots,n\}$. Given a number $00$ there exists $\delta>0$ such that $0<|(t-t_0,x-x_0,v-v_0)|<\delta$ implies $$\label{E:L.CrdCC} \big| F(t,x,v)-F(t_0,x_0,v_0) \big| <\varepsilon.$$ When the point $t_0$ is left-dense and right-scattered at the same time, then replace $t_0$ in \eqref{E:L.CrdCC} by $t_0^-$ (the left-hand limit). \end{definition} In other words, $f\in{\rm C}_{\rm rd}\times{\rm C}\times{\rm C}$ means that $f$ is continuous at any point $(t_0,y_0,v_0)\in[a,\infty)_{\scriptscriptstyle{\mathbb T}}\times\Omega_q\times\mathbb{R}^n$ when $t_0$ is right-dense, and that $f$ is {\em jointly regulated}, that is, $\lim_{n\to\infty}f(t_n,x_n,v_n)$ exists (finite) whenever $t_n\to t_0^-$ or $t_n\to t_0^+$, and $(x_n,v_n)\to(x_0,v_0)$. The following result is a minor modification of \cite[Proposition~1]{rH.vZ04a}. It shows that the above continuity concept is the right one when considering time scale delta integrals involving a ${\rm C}_{\rm rd}\times{\rm C}\times{\rm C}$-continuous function $f$ in the composition with a ${\rm C}_{\rm rd}^1$ function $x$. \begin{proposition} \label{P:CrdCC.Crd} Let $x\in{\rm C}_{\rm rd}^1[a,\infty)_{\scriptscriptstyle{\mathbb T}}$ and assume that $f\in{\rm C}_{\rm rd}\times{\rm C}\times{\rm C}$ on $[a,\infty)_{\scriptscriptstyle{\mathbb T}}\times\Omega_q\times\mathbb{R}^n$ with $00$ on $[a,\infty)_{\scriptscriptstyle{\mathbb T}}$, i.e., $\ominus p(\cdot)$ is an rd-continuous and positively regressive function. By \cite[Theorem~2.44]{mB.aP01}, we get that $u(t)>0$ on $[a,\infty)_{\scriptscriptstyle{\mathbb T}}$. Consequently, $u^\Delta(t)<0$, the function $u(\cdot)$ is decreasing on $[a,\infty)_{\scriptscriptstyle{\mathbb T}}$, the limit $u_0$ exists, and $u_0\in[0,1)$. The fact that actually $u_0>0$ follows from the assumption \eqref{E:p.int.finite}. We refer to the proof of \cite[Lemma~3.1]{lE.aP.ccT?} for the details. For the proof of \eqref{E:u.sup.int}, we have \begin{equation*} \sup_{t\in[a,\infty)_{\scriptscriptstyle{\mathbb T}}} \frac{1}{u(t)} \int_t^\infty [-u^\Delta(s)]\,\Delta s=\sup_{t\in[a,\infty)_{\scriptscriptstyle{\mathbb T}}} \Big( 1-\frac{u_0}{u(t)} \Big)=1-u_0, \end{equation*} because the function $u$ attains its maximum value $u(a)=1$. \end{proof} \section{First order equations} \label{S:first.order} Consider the first order time scale dynamic equation $$\label{E:1st.order} x^\Delta+f(t,x^\sigma)=0, \quad t\in[a,\infty)_{\scriptscriptstyle{\mathbb T}}.$$ For a given positive number $N$ we define the set $$\label{E:XN.def.1st.order} X_N:=\big\{\,x\in{\rm C}[a,\infty)_{\scriptscriptstyle{\mathbb T}},\ \|x\|_0\leq N\big\},$$ Then $X_N$ is a closed subset of the Banach space $({\rm C}[a,\infty)_{\scriptscriptstyle{\mathbb T}},\|\cdot\|_0)$. \begin{remark} \label{R:psi.Banach} \rm Given a function $\psi:[a,\infty)_{\scriptscriptstyle{\mathbb T}}\to[c,d]$, $0 u(t)$ for all $t \in [a,\infty)_{\scriptscriptstyle{\mathbb T}}$. \end{theorem} \begin{proof} Take $v:=x$ in Lemma \ref{complem} and the result follows. \end{proof} Similarly, we have the following result by taking $u:=x$ in Lemma \ref{complem}. \begin{theorem} \label{T:compare2} Assume that $f:[a,\infty)_{\scriptscriptstyle{\mathbb T}}\times\mathbb{R}\to\mathbb{R}$, $f\in{\rm C}_{\rm rd}\times{\rm C}$, satisfying \eqref{monineq} and there exists a function $v:[a,\infty)_{\scriptscriptstyle{\mathbb T}} \to \mathbb{R}$ such that $v(\infty)$ exists and \eqref{uppineq} holds. If $x$ is a solution to \eqref{E:1st.order.v2}, \eqref{E:IC} and $v(\infty) > A$, then $x(t) < v(t)$ for all $t \in [a,\infty)_{\scriptscriptstyle{\mathbb T}}$. \end{theorem} \section{Second order equations with scalar leading coefficient} \label{S:second.order} The methods used in Section \ref{S:first.order} to derive the existence and uniqueness results for the first order equations can be naturally used in order to derive similar results for the second order dynamic equations. For the second order setting there are two cases depending on whether the nonlinearity $f$ involves the $\Delta$-derivative of $x$ or does not. As we shall see, these two cases differ in the assumption on the leading coefficient $r(t)$. Note that as in the previous section the function $f$ can take both positive and negative values. Consider first the equation $$\label{E:2nd.order} \big(r(t)\,x^\Delta\big)^\Delta+f(t,x^\sigma)=0, \quad t\in[a,\infty)_{\scriptscriptstyle{\mathbb T}}.$$ We note that while the functions $f$ and $x$ in \eqref{E:2nd.order} are $n$-vector valued, the function $r$ will be (in this section) assumed to be scalar valued. Furthermore, compared with some recent oscillation and asymptotic results for second order dynamic equations \cite{rpA.mB.dO02,rpA.mB.pjyW99,eA.mB.lE.aP02,mB.shS04,mB.shS04a,lE.sH93,lE.aP.pR02,lE.aP.ccT?,pR06} in which $r(t)>0$ on $[a,\infty)_{\scriptscriptstyle{\mathbb T}}$, in this paper we assume (if not otherwise stated) that $r(t)\neq0$ only. This type of assumption is common in the oscillation theory of difference equations, see e.g. \cite{zD.sP07,oD.rH99}, and have also been adopted in some papers in the time scale setting \cite{oD.sH02,oD.dM07,rH99,rH00}. The results in this section directly generalize \cite[Theorems 4.2 and 4.5]{lE.aP.ccT?} to vector valued nonlinearity $f$ which can take negative values and the leading coefficient $r(t)$ is assumed to be nonzero only. \begin{remark} \label{R:r.1/r} \rm Given a function $r:[a,\infty)_{\scriptscriptstyle{\mathbb T}}\to\mathbb{R}$, $r\in{\rm C}_{\rm rd}$, such that $$\label{E:r.assume.2nd.order} \inf_{t\in[a,b]_{\scriptscriptstyle{\mathbb T}}}\big|r(t)\big|>0, \quad\text{for all } b\in[a,\infty)_{\scriptscriptstyle{\mathbb T}},$$ it follows that $r(t)\neq0$ for all $t\in[a,\infty)_{\scriptscriptstyle{\mathbb T}}$, and the function $\frac{1}{r}$ also belongs to ${\rm C}_{\rm rd}$ on $[a,\infty)_{\scriptscriptstyle{\mathbb T}}$. Hence, the integrals \begin{equation*} \label{E:R.bR.def} R(t,s):=\int_s^t\frac{1}{r(\tau)}\,\Delta\tau, \quad \bar R(t,s):=\int_s^t\frac{1}{|r(\tau)|}\,\Delta\tau, \quad t,s\in[a,\infty)_{\scriptscriptstyle{\mathbb T}}, \end{equation*} are well-defined. Obviously, for a fixed $s\in[a,\infty)_{\scriptscriptstyle{\mathbb T}}$ both $R(\cdot,s)$ and $\bar R(\cdot,s)$ belong to ${\rm C}_{\rm rd}^1$ with ($\Delta$-differentiating with respect to the first argument) $R^\Delta(t,s)=\frac{1}{r(t)}$ and $\bar R^\Delta(t,s)=\frac{1}{|r(t)|}>0$ for all $t\in[a,\infty)_{\scriptscriptstyle{\mathbb T}}$. Consequently, the function $\bar R(t,s)$ is increasing as $t$ increases or, for the same reason, as $s$ decreases. Moreover, we have $$\label{E:R.bR.ineq} \big|R(t,s)\big|\leq\bar R(t,s)\leq\bar R(t,a), \quad t,s\in[a,\infty)_{\scriptscriptstyle{\mathbb T}},\ t\geq s.$$ \end{remark} In connection with these functions we shall frequently use the identities $$\label{E:R.bR.sata} R\big(\sigma(s),t\big)=R\big(\sigma(s),a\big)-R(t,a), \quad \bar R\big(\sigma(s),t\big)=\bar R\big(\sigma(s),a\big)-\bar R(t,a),$$ for $t,s\in[a,\infty)_{\scriptscriptstyle{\mathbb T}}$. Next we present the first main result of this section. \begin{theorem} \label{T:2nd.order} Assume that $f:[a,\infty)_{\scriptscriptstyle{\mathbb T}}\times\Omega_q\to\mathbb{R}^n$ with $00$ for $t>a$ show that $\big|\int_t^\infty\!g(s)\,\Delta s\big|$ and hence $\int_t^\infty\!g(s)\,\Delta s$ are finite for any $t\in(a,\infty)_{\scriptscriptstyle{\mathbb T}}$. Fix any $t_0\in(a,\infty)_{\scriptscriptstyle{\mathbb T}}$. Since $g\in{\rm C}_{\rm rd}$, the integral $\int_a^{t_0}g(s)\,\Delta s$ exists, and then with respect to the previous conclusion we get that $\int_a^\infty\!g(s)\,\Delta s=\big\{\int_a^{t_0}+\int_{t_0}^\infty\big\}g(s)\,\Delta s$ exists finite. The latter then implies that $$\label{E:g.int.lim} \lim_{t\to\infty}\int_t^\infty\!\!\!g(s)\,\Delta s=0.$$ Thus, by using the first expression in \eqref{E:R.bR.sata}, we may write $$\label{E:G.int.def.expanded} G(t)=\int_t^\infty\!\!\!R\big(\sigma(s),a\big)\,g(s)\,\Delta s-R(t,a)\int_t^\infty\!\!\!g(s)\,\Delta s, \quad t\in[a,\infty)_{\scriptscriptstyle{\mathbb T}},$$ in which both improper integrals exist finite. This shows that $G$ is a ${\rm C}_{\rm rd}^1$ function on $[a,\infty)_{\scriptscriptstyle{\mathbb T}}$. Using the time scale product rule when $\Delta$-differentiating the second term in \eqref{E:G.int.def.expanded} we obtain formula \eqref{E:GD.int}. Finally, the first limit in \eqref{E:G.int.lim} follows from the fact that (for example) $G(a)$ is finite, while the second limit in \eqref{E:G.int.lim} is a consequence of formula \eqref{E:GD.int} in combination with the limit \eqref{E:g.int.lim}. \end{proof} We are now ready to derive Theorem \ref{T:2nd.order}. \begin{proof}[Proof of Theorem \ref{T:2nd.order}] We will apply the Banach fixed point theorem in the space $({\rm C}[a,\infty)_{\scriptscriptstyle{\mathbb T}},\|\cdot\|_\psi)$ for a suitably chosen function $\psi$. Define the operator $F:X_N\to{\mathcal F}$ (the space of $n$-vector functions) by \begin{equation*} \label{E:F.def.2nd.order} [Fx](t):=A-\int_t^\infty\!\!\!R\big(\sigma(s),t\big)\,f\big(s,x^\sigma(s)\big)\,\Delta s, \quad t\in[a,\infty)_{\scriptscriptstyle{\mathbb T}}. \end{equation*} Set $g(t):=f\big(t,x^\sigma(t)\big)$ on $[a,\infty)_{\scriptscriptstyle{\mathbb T}}$. Then Proposition \ref{P:CrdC.Crd} yields that $g\in{\rm C}_{\rm rd}$. By assumption \eqref{E:f.integral.2nd.order} and Lemma \ref{L:Gg.int}, we have that $[Fx](t)$ is well-defined for all $t\in[a,\infty)_{\scriptscriptstyle{\mathbb T}}$, $Fx\in{\rm C}_{\rm rd}^1$, and $$\label{E:Fx.Delta.2nd.order} [Fx]^\Delta(t)=\frac{1}{r(t)}\,\int_t^\infty\!\!\!g(s)\,\Delta s, \quad t\in[a,\infty)_{\scriptscriptstyle{\mathbb T}},$$ with the limits $$\label{E:Fx.lim} \lim_{t\to\infty}[Fx](t)=A, \quad \lim_{t\to\infty}r(t)\,[Fx]^\Delta(t)=0.$$ Furthermore, inequality \eqref{E:R.bR.ineq} and assumption \eqref{E:f.integral.2nd.order} yield that $\big|[Fx](t)\big|\leq N$ for all $t\in[a,\infty)_{\scriptscriptstyle{\mathbb T}}$, so that $\|Fx\|_0\leq N$ and $Fx\in X_N$. \par Next, similarly to the proof of Theorem \ref{T:1st.order}, we choose the function $\psi(t)$ to be the time scale exponential function $e_{\ominus p(\cdot)}(t,a)$, where $p(t):=\bar R\big(\sigma(t),a\big)\,k(t)$ for $t\in[a,\infty)_{\scriptscriptstyle{\mathbb T}}$. That is, $\psi^\Delta(t)=-p(t)\,\psi^\sigma(t)$ on $[a,\infty)_{\scriptscriptstyle{\mathbb T}}$. Then, by Lemma \ref{L:exp.def}, we have $0<\psi_0\leq\psi(t)\leq1$ for all $t\in[a,\infty)_{\scriptscriptstyle{\mathbb T}}$ with $\psi_0\in(0,1)$, where $\psi_0:=\lim_{t\to\infty}\psi(t)$. Thus, by Remark \ref{R:psi.Banach}, $({\rm C}[a,\infty)_{\scriptscriptstyle{\mathbb T}},\|\cdot\|_\psi)$ is a Banach space. By using the Lipschitz condition \eqref{E:Lipschitz.1st.order}, we get for any $x,y\in X_N$ \begin{align} \|Fx-Fy\|_\psi &\leq \sup_{t\in[a,\infty)_{\scriptscriptstyle{\mathbb T}}} \frac{1}{\psi(t)} \int_t^\infty\!\!\!\bar R\big(\sigma(s),t\big)\, \big| f\big(s,x^\sigma(s)\big)-f\big(s,y^\sigma(s)\big) \big|\,\Delta s \notag \\ &\leq \sup_{t\in[a,\infty)_{\scriptscriptstyle{\mathbb T}}} \frac{1}{\psi(t)} \int_t^\infty\!\!\! \bar R\big(\sigma(s),t\big)\,k(s)\,\big|x^\sigma(s)-y^\sigma(s)\big|\,\Delta s. \label{E:FxFy.psi.hlp1} \end{align} Now if $t=a$, then for $s\geq t=a$ we have $\bar R\big(\sigma(s),t\big)\,k(s)=\frac{-\psi^\Delta(s)}{\psi^\sigma(s)}$. If $t>a$, then for $s\geq t$ the quantity $\bar R\big(\sigma(s),a\big)>0$ and $0\leq\bar R\big(\sigma(s),t\big)<\bar R\big(\sigma(s),a\big)$. In this case we have $$\label{E:bRss.ta} \bar R\big(\sigma(s),t\big)\,k(s)= \frac{\bar R\big(\sigma(s),t\big)}{\bar R\big(\sigma(s),a\big)}\,\bar R\big(\sigma(s),a\big)\,k(s)\leq \frac{-\psi^\Delta(s)}{\psi^\sigma(s)},$$ and in combination with the previous case we see that inequality \eqref{E:bRss.ta} holds for any $t\in[a,\infty)_{\scriptscriptstyle{\mathbb T}}$. Thus, we get from \eqref{E:FxFy.psi.hlp1} by using \eqref{E:bRss.ta}, the definition of $\|x-y\|_\psi$, and condition \eqref{E:u.sup.int} with $u:=\psi$ and $u_0:=\psi_0$ that \begin{equation*} \|Fx-Fy\|_\psi\leq\|x-y\|_\psi \sup_{t\in[a,\infty)_{\scriptscriptstyle{\mathbb T}}} \frac{1}{\psi(t)} \int_t^\infty\!\!\![-\psi^\Delta(s)]\,\Delta s =(1-\psi_0)\,\|x-y\|_\psi. \end{equation*} Hence, the mapping $F$ is a contraction in $X_N$. By Proposition \ref{P:Banach}, there is a unique function $x\in X_N$ such that $x=Fx$, i.e., $$\label{E:x.int.eq.2nd.order} x(t)=A-\int_t^\infty\!\!\!R\big(\sigma(s),t\big)\,g(s)\,\Delta s, \quad t\in[a,\infty)_{\scriptscriptstyle{\mathbb T}}.$$ From the limits in \eqref{E:Fx.lim} we get that \begin{equation*} x(\infty)=[Fx](\infty)=A \quad\text{and}\quad (rx^\Delta)(\infty)=\lim_{t\to\infty}r(t)\,[Fx]^\Delta(t)=0. \end{equation*} Moreover, equations \eqref{E:Fx.Delta.2nd.order} and \eqref{E:x.int.eq.2nd.order} show that the function $x$ satisfies $$\label{E:xD.int.f} r(t)\,x^\Delta(t)=\int_t^\infty\!\!\!g(s)\,\Delta s, \quad t\in[a,\infty)_{\scriptscriptstyle{\mathbb T}}.$$ While the right-hand side of \eqref{E:xD.int.f} is $\Delta$-differentiable, it follows that \begin{equation*} \big(r(t)\,x^\Delta(t)\big)^\Delta=-g(t), \quad t\in[a,\infty)_{\scriptscriptstyle{\mathbb T}}, \end{equation*} i.e., the function $x$ satisfies the dynamic equation \eqref{E:2nd.order}. The proof is complete. \end{proof} Next we turn our attention to the more general dynamic equation $$\label{E:2nd.order.xD} \big(r(t)\,x^\Delta\big)^\Delta+f(t,x^\sigma,x^{\Delta\sigma})=0, \quad t\in[a,\infty)_{\scriptscriptstyle{\mathbb T}}.$$ As we saw in Theorem \ref{T:2nd.order}, the problem \eqref{E:2nd.order} which does not involve $x^\Delta$ in $f$ can be treated within the set $X_N$ consisting of certain continuous functions $x$. On the other hand, the problem \eqref{E:2nd.order.xD} must be considered in a narrower space, because it is implicitly assumed in the form of this equation that $x^\Delta$ exists throughout the interval $[a,\infty)_{\scriptscriptstyle{\mathbb T}}$. Therefore, we introduce the set $$\label{E:XN.def.2nd.order.xD} X_N^1:=\big\{\,x\in{\rm C}_{\rm rd}^1[a,\infty)_{\scriptscriptstyle{\mathbb T}},\ \|x\|_1<\infty,\ \|x\|_0\leq N\big\}.$$ Then $X_N^1$ is a closed subset of the Banach space $({\rm C}_{\rm rd}^1[a,\infty)_{\scriptscriptstyle{\mathbb T}},\|\cdot\|_1)$. \begin{remark} \label{R:phi.Banach} \rm Given a function $\varphi:[a,\infty)_{\scriptscriptstyle{\mathbb T}}\to[c,d]$, $00 for some number$r_0>0$, the Lipschitz condition $$\label{E:Lipschitz.2nd.order.xD} \big| f(t,x,u)-f(t,y,v) \big| \leq k(t)\,\big[|x-y|+|u-v|\big]$$ for all$t\in[a,\infty)_{\scriptscriptstyle{\mathbb T}},\ x,y\in\Omega,\ u,v\in\mathbb{R}^n$, where$k:[a,\infty)_{\scriptscriptstyle{\mathbb T}}\to(0,\infty)$,$k\in{\rm C}_{\rm rd}$, and condition $$\label{E:k.integral.2nd.order.xD} \int_a^\infty \big[\bar R\big(\sigma(s),a\big)+1\big]\,k(s)\,\Delta s<\infty.$$ Let$A\in\mathbb{R}^n$be a given vector. If there exists a number$N\in\mathbb{R}$,$|A|\leq N0$, the Lipschitz condition \eqref{E:Lipschitz.2nd.order.xD} in which$k:[a,\infty)_{\scriptscriptstyle{\mathbb T}}\to(0,\infty)$,$k\in{\rm C}_{\rm rd}$, and condition \eqref{E:k.integral.2nd.order.xD} holds. If there exists a vector$M\in\mathbb{R}^n$,$|M|0, \quad\text{for all } b\in[a,\infty)_{\scriptscriptstyle{\mathbb T}}, \end{equation*} and the Lipschitz condition \eqref{E:Lipschitz.1st.order}, in which $k:[a,\infty)_{\scriptscriptstyle{\mathbb T}}\to(0,\infty)$, $k\in{\rm C}_{\rm rd}$, and \begin{equation*} \label{E:k.integral.2nd.order.matrix} \int_a^\infty \big\|Q\big(\sigma(s),a\big)\big\|\,k(s)\,\Delta s<\infty, \end{equation*} where the matrix norm $\|\cdot\|$ is compatible with the vector norm $|\cdot|$ and monotone in the sense of condition \eqref{E:matrix.norm.monotone}. If there exists a vector $M\in\mathbb{R}_+^n$, $|M|0$ as $t\to\infty$. But in any of these two cases the second limit in \eqref{E:Q.lim.hlp2} implies that $$\label{E:Q.lim.hlp4} \lim_{t\to\infty} \int_t^\infty\!\!\!g(s)\,\Delta s =0.$$ Hence, by \eqref{E:Tx.Delta.2nd.order} and \eqref{E:Q.lim.hlp4}, \begin{equation*} (Px^\Delta)(\infty)=\lim_{t\to\infty} P(t)\,[Tx]^\Delta(t)=\int_t^\infty\!\!\!g(s)\,\Delta s =0. \end{equation*} Furthermore, from $x=Tx$ we get \begin{align*} x(t) &\geq M-\int_a^\infty\!\!\!Q\big(\sigma(s),a\big)\,g(s)\,\Delta s+ Q(t,a)\,P(t)\,P^{-1}(t)\int_t^\infty\!\!\!g(s)\,\Delta s, \end{align*} so that by assumption \eqref{E:f.integral.2nd.order.matrix} and by using formula \eqref{E:Tx.Delta.2nd.order} in $x^\Delta(t)=[Tx]^\Delta(t)$ we have inequality \eqref{E:xxD.PQ}. The proof is now complete. \end{proof} Next we define the set \begin{aligned} Y_M^1:=\big\{&x\in{\rm C}_{\rm rd}^1[a,\infty)_{\scriptscriptstyle{\mathbb T}}, &\ 0\leq x(t)\leq M,\ x^\Delta(t)\geq 0 \\ &\text{for all } t\in[a,\infty)_{\scriptscriptstyle{\mathbb T}},\ \|x^\Delta\|_0<\infty\big\}. \end{aligned} \label{E:YM1.def.2nd.order} Then $Y_M^1$ is a closed subset of the Banach space $({\rm C}_{\rm rd}^1[a,\infty)_{\scriptscriptstyle{\mathbb T}},\|\cdot\|_1)$. Consider the second order dynamic equation $$\label{E:2nd.order.xD.matrix} \big(P(t)\,x^\Delta\big)^\Delta+f(t,x^\sigma,x^{\Delta\sigma})=0, \quad t\in[a,\infty)_{\scriptscriptstyle{\mathbb T}}.$$ \begin{theorem} \label{T:2nd.order.xD.matrix} Assume that $f:[a,\infty)_{\scriptscriptstyle{\mathbb T}}\times\Omega_q^+\times\mathbb{R}_+^n\to\mathbb{R}_+^n$ with $00 \end{equation*} for some number$r_0>0$, and the Lipschitz condition \eqref{E:Lipschitz.2nd.order.xD}, in which$k:[a,\infty)_{\scriptscriptstyle{\mathbb T}}\to(0,\infty)$,$k\in{\rm C}_{\rm rd}$, and \begin{equation*} \label{E:k.integral.2nd.order.xD.matrix} \int_a^\infty \big[\big\|Q\big(\sigma(s),a\big)\big\|+1\big]\,k(s)\,\Delta s<\infty, \end{equation*} where the matrix norm$\|\cdot\|$is compatible with the vector norm$|\cdot|$and monotone in the sense of condition \eqref{E:matrix.norm.monotone}. If there exists a vector$M\in\mathbb{R}_+^n$,$|M|0. \end{equation*} Then we claim that the assumptions of Theorem \ref{T:2nd.order} are satisfied. \end{example} \begin{proof} First note that $r(t)$ changes its sign but \begin{equation*} \inf_{t\in[0,b]_{{\scriptscriptstyle{\mathbb Z}}}}\big|r(t)\big|= \inf_{t\in[0,b]_{{\scriptscriptstyle{\mathbb Z}}}}\frac{1}{t+1}=\frac{1}{b+1}>0 \quad\text{for every } b\in[0,\infty)_{{\scriptscriptstyle{\mathbb Z}}}, \end{equation*} hence condition \eqref{E:r.assume.2nd.order} holds. It follows that \begin{equation*} R(t,s)=\sum_{i=s}^{t-1}\frac{i+1}{(-1)^i}=\sum_{i=s+1}^t(-1)^{i-1}\,i, \quad \bar R(t,s)=\sum_{i=s+1}^ti= \frac{t(t+1)}{2}-\frac{s(s+1)}{2}. \end{equation*} Moreover, $f(t,x)$ is Lipschitz with $k(t):=1/(t+1)^{3+\beta}$, i.e., condition \eqref{E:Lipschitz.1st.order} holds. With the estimate $(i+1)(i+2)\leq2(i+1)^2$, a simple calculation shows that \begin{equation*} \int_0^\infty\!\! \bar R(s+1,0)\,k(s)\,\Delta s=\sum_{i=0}^\infty \frac{(i+1)\,(i+2)}{2\,(i+1)^{3+\beta}} \leq \sum_{i=0}^\infty \frac{1}{(i+1)^{1+\beta}}<\infty, \end{equation*} by the integral criterion for infinite series. Thus, condition \eqref{E:k.integral.2nd.order} holds. Finally, since the function $x/(1+x^2)$ is bounded on $\mathbb{R}$ (by $\frac{1}{2}$), we get for any sequence $x\in X_N$ that \begin{align*} \int_0^\infty\!\! \bar R(s+1,0)\,\big|f\big(s,x(s+1)\big)\big|\,\Delta s & \leq \frac{1}{2} \sum_{i=0}^\infty \frac{1}{(i+1)^{\beta+1}} = \frac{1}{2} + \frac{1}{2} \int_1^\infty \!\!\!\!\frac{1}{\tau^{\beta+1}}\,{\mathrm d}\tau \\ &\leq \frac{1}{2}+\frac{1}{2\beta}=\frac{\beta+1}{2\beta}. \end{align*} Since the above estimate is independent of $x(\cdot)$, we may choose $N:=|A|+\frac{\beta+1}{2\beta}$, and then inequality \eqref{E:f.integral.2nd.order} holds. Hence, by Theorem \ref{T:2nd.order}, for any $A\in\mathbb{R}$ the terminal value problem \begin{gather*} \Delta\Big( \frac{(-1)^t}{(t+1)^{3+\beta}}\,\Delta x(t) \Big) + \frac{1}{(t+1)^{3+\beta}}\,\frac{x(t+1)}{1+x^2(t+1)}=0, \quad t\in[0,\infty)_{{\scriptscriptstyle{\mathbb Z}}}, \\ \lim_{t\to\infty} x(t)=A, \quad \lim_{t\to\infty} \frac{(-1)^t}{(t+1)^{3+\beta}}\,\Delta x(t)=0, \end{gather*} has a unique solution $x(\cdot)$ on $[0,\infty)_{{\scriptscriptstyle{\mathbb Z}}}$. \end{proof} \begin{example} \label{Ex:second.order.scalar.r} \rm Note that the leading coefficient $r(t)$ from Example \ref{Ex:second.order.scalar} is not allowed in Theorem \ref{T:2nd.order.xD}, since \begin{equation*} \inf_{t\in[0,\infty)_{{\scriptscriptstyle{\mathbb Z}}}}\big|r(t)\big|= \inf_{t\in[0,\infty)_{{\scriptscriptstyle{\mathbb Z}}}}\frac{1}{t+1}=0, \end{equation*} contradicting condition \eqref{E:r.assume.2nd.order.xD}. However, one can consider a leading coefficient such as $r(t)=(-1)^t$ on $[0,\infty)_{{\scriptscriptstyle{\mathbb Z}}}$, which is admissible in Theorem \ref{T:2nd.order.xD}. \end{example} \begin{example} \label{Ex:second.order.scalar.xD} \rm In this example we illustrate the applicability of Theorem \ref{T:2nd.order.xD}. Let $n=1$, $q=\infty$, ${\mathbb T}=\mathbb{R}$, $a=0$, and \begin{equation*} r(t):=1, \quad f(t,x,y):=\frac{1}{(t+1)^{2+\beta}}\, \Big( \cos x+\frac{\sin y}{1+\sin^2y}\Big), \quad\text{with } \beta>0. \end{equation*} Then we claim that the assumptions of Theorem \ref{T:2nd.order.xD} are satisfied. \begin{proof} We have $R(t,s)=\bar R(t,s)=\int_s^t1\,{\mathrm d}\tau=t-s$, and the Lipschitz condition \eqref{E:Lipschitz.2nd.order.xD} is satisfied with the function $k(t):=1/(t+1)^{2+\beta}$. Conditions \eqref{E:k.integral.2nd.order.xD} and \eqref{E:f.integral.2nd.order.xD} are verified similarly as in Example \ref{Ex:second.order.scalar}. Condition \eqref{E:int.f.finite} follows from \eqref{E:int.f.finite.a} in Remark \ref{R:2nd.order.xD}(iii) and from the estimate \begin{equation*} \int_0^\infty \big|f\big(s,x(s),x'(x)\big)\big|\,{\mathrm d}s \leq \frac{3}{2} \int_0^\infty \!\!\!\! \frac{1}{(s+1)^{2+\beta}}\,{\mathrm d}s=\frac{3}{2\,(\beta+1)}<\infty \end{equation*} for every $x\in X_N^1$, since the function $\cos x+\frac{\sin y}{1+\sin^2y}$ is bounded on $\mathbb{R}$ (by $\frac{3}{2}$). Hence, by Theorem \ref{T:2nd.order.xD}, for any $A\in\mathbb{R}$ the terminal value problem \begin{gather*} x'' + \frac{1}{(t+1)^{2+\beta}}\,\Big( \!\cos x+\frac{\sin x'}{1+\sin^2x'}\Big)=0, \quad t\in[0,\infty), \\ \lim_{t\to\infty} x(t)=A, \quad \lim_{t\to\infty} x'(t)=0, \end{gather*} has a unique solution $x(\cdot)$ on $[0,\infty)$. \end{proof} \end{example} The two examples above motivate the following corollaries of Theorems \ref{T:2nd.order} and \ref{T:2nd.order.xD}, in which the existence of the number $N$ is guaranteed from the assumed estimates on the data. \begin{corollary} \label{C:second.order.scalar} Assume that $g:\Omega_q\to\mathbb{R}^n$ with \$0