Dichotomy and well conditioning of two-point boundary value problems on time scale dynamical systems

Abstract

In this paper, we establish close relationships between the stability constants on one hand and the global behaviour of fundamental matrices on the other hand to the two-point boundary value problems on time-scale dynamical systems. We introduce the concept of conditioning number $k$ and show that conditioning number is the right criteria in estimating the global error due to small perturbations of two point boundary value problems on time scale dynamical systems. Further, the moderate stability constants imply a dichotomy with moderate $k$ -bound will be developed. Further, the exponential behaviour of solutions of the Green’s matrix will be investigated. We also investigate the conditions under which strong dichotomy exists for two-point boundary value problems when the boundary conditions are separable.

Keywords

Boundary value problems time-scale dynamical systems

1. Introduction

It is a well recognized fact that mathematical models or equations that describe a physical phenomenon are in most cases nonlinear differential or difference equations of first order. Further difference equations appear as a natural description in the study of discretization methods for differential equations. However, in recent years, the investigation of the theory of difference equations is attracting greater attention in many disciplines. In spite of this tendency of inter dependence there is a striking similarity or even duality between the theory of continuous and discrete dynamic studies. Many results on difference equations are lot more richer than the theory of differential equations. For example a simple difference equation resulting from a first order differential equation executes the chaotic behaviour which can only happen for higher order differential equations. Additional assumptions are required in the discrete case. Lakshmikantham et al. developed dynamic system on measure chains, Kluwer Academic publications [18].

1.1 Related work

Bohner and Peterson [1] wrote A text book of Advances in Dynamic Equations on Time Scales. Keller [2, 3] developed Numerical solution of two-point boundary value problem. It is useful to investigate how these concepts differ on time scale dynamical systems. Murty and Fausett [6, 7, 9] developed Fundamental theory on control systems involving Kronecker Product of Matrices. Lentini and Pereyra [10], Scott and Watts [11], Babu et al. [12] established Some fundamental results on controllability, Observability and Realizability of first order matrix Lyapunov systems. Further, Deuflhard [13] and Mattheij [14, 15] investigated the close relationship between methods for solving two-point boundary value problems. Coppel in [17, 18, 19] Dichotomies Stability Theory, Lecture notes in Mathematics, developed exponential and ordinary dichotomies. Many linear differential methods for dichotomy are available in General Linear methods for Ordinary differential equations by A John Wiley and Sons, inc, Publishing.

To overcome this topological deficiency of lacking connectedness, we introduce the following notions:

The operator $\sigma$ from $T\to T$ defined by [1]

$\displaystyle\sigma=\begin{cases}T\to T,\\ t\to\inf\{s\in T:s>t\}\end{cases}$ $\displaystyle=\begin{cases}T\to T,\\ t\to\sup\{s\in T:s<t\}\end{cases}$

are called jump operators. If $\sigma$ is bounded above and is bounded below: then we define

$\displaystyle\sigma(\operatorname{Max}T)=\operatorname{Max}T\text{ and }\rho({% \operatorname{Min}T})=\operatorname{Min}T.$

For a detailed discussion on measure chains, we refer [1]. In the year 1994, Murty and Rao [4] established existence and uniqueness criteria for two point boundary value problems on time-scale dynamical systems. In the year 2014, Murty, Suryanarayana and Gopalarao [5] established qualitative properties of first order linear systems on time-scale dynamical systems.

Now, we consider the two point boundary value problem

$\displaystyle X^{\Delta}(t)=A(t)X(t)+b(t),$ (1)

satisfying the boundary conditions

$\displaystyle{MX}(a)+{NX}(b)=\alpha,$ (2)

where $A\in{[L_{p}[a,b]]}^{n\times n},f\in{[L_{p}[a,b]]}^{n}$ for some $p$ satisfying $1\leqslant p<\infty$ .

A unique solution of the boundary value problems Eq. (1) satisfying Eq. (2) is given by

$\displaystyle X(t)={\varnothing}_{A}(t)D^{-1}\alpha+\int^{b}_{a}{G(t,\sigma(s)% )b(s)\Delta s,}$ (3)

where $D=M{\varnothing}_{A}(a)+N{\varnothing}_{A}(b)$ and $G(t,\sigma(s))$ is the Green’s matrix for the homogeneous boundary value problem and is given by

$\displaystyle G(t,\sigma(s))=$ $\displaystyle\begin{cases}{\varnothing}_{A}(t)D^{-1}M{\varnothing}_{A}(a){% \varnothing}_{A}^{-1}(\sigma(s)),\\ \quad a\leqslant s<t\leqslant b\\ {-\varnothing}_{A}(t)D^{-1}N{\varnothing}_{A}(b){\varnothing}_{A}^{-1}(\sigma(% s)),\\ \quad a\leqslant t<s\leqslant b\end{cases}$ (4)

which can be written conveniently as

$\displaystyle G(t,\sigma(s))=\begin{cases}{\varnothing}_{A}(t)D^{-1}M{% \varnothing}_{A}(a,\sigma(s))\\ \quad a\leqslant s<t\leqslant b\\ {-\varnothing}_{A}(t)D^{-1}N{\varnothing}_{A}(b,\sigma(s))\\ \quad a\leqslant t<s\leqslant b\end{cases}$

Thus in principle a knowledge of any fundamental solution of $X^{\Delta}(t)=A(t)X(t)$ enables us to calculate Green’s matrix and hence the solution $X(t)$ as given in Eq. (3).

Let $\|{u}\|_{p}={\left[\int^{b}_{a}{{|u(s)|}^{p}{\rm d}s}\right]}^{{1}/{p}},∼{}1% \leqslant p<\infty$ and $\|{u}\|_{\infty}=\operatorname{Sup}\{[u(t)]\}$ .

From Eq. (3)

$\displaystyle X(t)={\varnothing}_{A}(t)D^{-1}\alpha+\int^{b}_{a}{G(t,\sigma(s)% )b(s)\Delta s},$

which implies

$\displaystyle\|{X}\|_{\infty}\leqslant\beta|\textit{BX}|+{\alpha}_{q}\|{b}\|_{% p},∼{}\frac{1}{p}+\frac{1}{q}=1$ (5)

where

$\displaystyle\beta=\|{\varnothing}_{A}(t){[M{\varnothing}_{A}(a)+N{\varnothing% }_{A}(b)]}^{-1}\|$ (6) $\displaystyle\textit{BX}=\textit{MX}(a)+\textit{NX}(b),$ $\displaystyle{\alpha}_{q}=\operatorname{Sup}\left\{{\left[\int^{b}_{a}{{|G(t,% \sigma(s))|}^{p}\Delta s}\right]}^{{1}/{q}}\right\}.$ (7)

We consider only the case $p=1.$ Then Eqs (1.1)–(6) reduce to

$\displaystyle\|{X}\|_{\infty}\leqslant\beta|\textit{BX}|+\alpha\|b\|$ (8) $\displaystyle\beta=\|{\varnothing}_{A}(t){(B{\varnothing}_{A})}^{-1}\|$ (9) $\displaystyle\alpha=\operatorname{Sup}\{|G(t,\sigma(s))|\}$ (10)

and

$\displaystyle\textit{BX}=\textit{MX}(a)+\textit{NX}(b)\text{∼{}and∼{}}$ $\displaystyle B{\varnothing}_{A}=M{\varnothing}_{A}(a)+N{\varnothing}_{A}(b).$

In addition we assume that the boundary condition Eq. (2) is scaled so that

$\displaystyle\textit{MCM}^{\rm T}+\textit{NN}^{\rm T}=I$

where $C={\varnothing}_{A}(a,\sigma(a)){{\varnothing}_{A}}^{\rm T}(\sigma(a))$ .

The Green’s matrix now becomes

$\displaystyle G(t,\sigma(s))=\begin{cases}{\varnothing}_{A}(t)D^{-1}M{% \varnothing}_{A}(a){\varnothing}_{A}^{-1}(\sigma(s)),\\ \quad a\leqslant s<t\leqslant b\\ {-\varnothing}_{A}(t)D^{-1}N{\varnothing}_{A}(b){\varnothing}_{A}^{-1}(\sigma(% s)),\\ \quad a\leqslant t<s\leqslant b\end{cases}$ $\displaystyle G(t,\sigma(a))={\varnothing}_{A}(t)D^{-1}M{\varnothing}_{A}(a){% \varnothing}_{A}^{-1}(\sigma(a))$ $\displaystyle\qquad={\varnothing}_{A}(t)D^{-1}M{\varnothing}_{A}(a,\sigma(a))$ $\displaystyle G(t,\sigma(a))G^{\rm T}(t,\sigma(a))={\varnothing}_{A}(t)D^{-1}M% {\varnothing}_{A}(a,\sigma(a))$ $\displaystyle\quad{[{\varnothing}_{A}(t)D^{-1}M{\varnothing}_{A}(a,\sigma(a))]% }^{\rm T}$ $\displaystyle={\varnothing}_{A}(t)D^{-1}M{\varnothing}_{A}(a,\sigma(a)){{% \varnothing}_{A}}^{\rm T}(a,\sigma(a))M^{\rm T}$ $\displaystyle\quad{({\varnothing}_{A}(t)D^{-1})}^{\rm T}$ $\displaystyle={\varnothing}_{A}(t)D^{-1}\textit{MCM}^{\rm T}{({\varnothing}_{A% }(t)D^{-1})}^{\rm T}$

where $C={\varnothing}_{A}(a,\sigma(a)){{\varnothing}_{A}}^{\rm T}(a,\sigma(a))$ .

Similarly

$\displaystyle G(t,\sigma(b))G^{\rm T}(t,\sigma(b))$ $\displaystyle\quad={\varnothing}_{A}(t)D^{-1}\textit{NN}^{\rm T}{({\varnothing% }_{A}(t)D^{-1})}^{\rm T}$ $\displaystyle G(t,\sigma(a))G^{\rm T}(t,\sigma(a))+G(t,\sigma(b))G^{\rm T}(t,% \sigma(b))$ $\displaystyle\quad=|{\varnothing}_{A}(t)D^{-1}\textit{MCM}^{\rm T}{({% \varnothing}_{A}(t)D^{-1})}^{\rm T}$ $\displaystyle\qquad+{\varnothing}_{A}(t)D^{-1}\textit{NN}^{\rm T}{({% \varnothing}_{A}(t)D^{-1})}^{\rm T}|$ $\displaystyle\quad=|{\varnothing}_{A}(t)D^{-1}[\textit{MCM}^{\rm T}+\textit{NN% }^{\rm T}]{({\varnothing}_{A}(t)D^{-1})}^{\rm T}|$ $\displaystyle\quad=|{\varnothing}_{A}(t)D^{-1}{({\varnothing}_{A}(t)D^{-1})}^{% \rm T}|$ $\displaystyle\quad={|{\varnothing}_{A}(t)D^{-1}|}^{2}.$

Thus

$\displaystyle{|{\varnothing}_{A}(t)D^{-1}|}^{2}$ $\displaystyle=|G(t,\sigma(a))G^{\rm T}(t,\sigma(a))+G(t,\sigma(b))G^{\rm T}(t,% \sigma(b))|$ $\displaystyle\leqslant|G(t,\sigma(a))G^{\rm T}(t,\sigma(a))|+|G(t,\sigma(b))G^% {\rm T}(t,\sigma(b))|$ $\displaystyle\Rightarrow{|{\varnothing}_{A}(t)D^{-1}|}^{2}$ $\displaystyle\leqslant{\alpha}^{2}+{\alpha}^{2}\begin{pmatrix}\therefore\alpha% =\operatorname{Sup}\{|G(t,\sigma(s))|\}\\ s,t\in[a,b]\end{pmatrix}$ $\displaystyle\Rightarrow{\beta}^{2}\leqslant 2{\alpha}^{2}$ $\displaystyle\Rightarrow\beta\leqslant\sqrt{2}\alpha.$

Therefore in this case the stability constant ‘ $\alpha$ ’ gives a measure for the sensitivity of Eqs (1) and (2) to changes in the data.

Definition 1. We say that the solution space $X$ of $LX=0$ is dichotomic if there exists of splitting $X=X_{1}\bigoplus X_{2}$ and a constant ‘ $k$ ’ such that

$\displaystyle y\in X_{1}\Rightarrow\frac{|y(t)|}{|y(s)|}\leqslant K\text{ for % }t\geqslant s$ $\displaystyle y\in X_{2}\Rightarrow\frac{|y(t)|}{|y(s)|}\leqslant K\text{ for % }t\leqslant s.$

Equivalently for every fundamental matrix ${\varnothing}_{A}$ there exists a projection matrix $P\in R^{n\times n}$ such that

$\displaystyle X_{1}=\{{\varnothing}_{A}PC/C\in R^{n}\},$ $\displaystyle X_{2}=\{{\varnothing}_{A}(I-P)C/C\in R^{n}\}$ (11)

for which the above holds. For a finite interval such a dichotomy always exists.

Definition 2. We say that the solution space $\textit{LX}=0$ is strongly dichotomic if there exists a constant ‘ $k$ ’ and a projection matrix $P\in R^{n\times n}$ such that for a fixed fundamental solution ${\varnothing}_{A}$ ,

$\displaystyle|{\varnothing}_{A}(t)P{\varnothing}_{A}^{-1}(\sigma(s))|\leqslant k% \text{ for }t\geqslant s,$ $\displaystyle|{\varnothing}_{A}(t)(I-P){\varnothing}_{A}^{-1}(\sigma(s))|% \leqslant k\text{ for }t\leqslant s.$

Definition 3. The solution space of $\textit{LX}=0$ is said to be exponentially dichotomic if there exists a constant $k>0$ , positive constants $\lambda,\xi$ and a projection matrix ‘ $P$ ’ such that

$\displaystyle|{\varnothing}_{A}(t)P{\varnothing}_{A}^{-1}(\sigma(s))|\leqslant k% \exp[\lambda(s-t)]\quad\text{ for }t\geqslant s,$ $\displaystyle|{\varnothing}_{A}(t)(I-P){\varnothing}_{A}^{-1}(\sigma(s))|% \leqslant k\exp[\xi(t-s)]\quad\text{ for }t\leqslant s.$

2. Dichotomy and strong dichotomy

The concepts of dichotomy and strong dichotomy have been used in the analysis of numerical solutions for boundary value problems in continuous case. Keller developed Numerical solution of two-point boundary value problems, SIAM Regional Conference Series in Applied Mathematics 24, Philadelphia, 1976 [3]. It is useful to investigate how these concepts differ on time scale dynamical systems.

Lemma 4. Let $X_{1}$ and $X_{2}$ be defined as in Eq. (1.1) then

$\displaystyle y\in X_{1}\Rightarrow\frac{|y(t)|}{|y(s)|}\leqslant|{\varnothing% }_{A}(t)P{\varnothing}_{A}^{-1}(\sigma(s))|,t\geqslant s,$ $\displaystyle y\in X_{2}\Rightarrow\frac{|y(t)|}{|y(s)|}\leqslant|{\varnothing% }_{A}(t)(I-P){\varnothing}_{A}^{-1}(\sigma(s))|,% t\leqslant s.$

Proof $y\in X_{1}$ then there exists a constant n-vector $C\in R^{n}$ such that $y(t)={\varnothing}_{A}(t)\textit{PC}$ .

Thus for all $t\geqslant s$

$\displaystyle\frac{|y(t)|}{|y(s)|}=\frac{|{\varnothing}_{A}(t)PC|}{|{% \varnothing}_{A}(\sigma(s))PC|}=\frac{|{\varnothing}_{A}(t)P^{2}C|}{|{% \varnothing}_{A}(\sigma(s))PC|}(\because P^{2}=P)=\frac{|{\varnothing}_{A}(t)P% {\varnothing}_{A}^{-1}(\sigma(s)){\varnothing}_{A}(\sigma(s))PC|}{|{% \varnothing}_{A}(\sigma(s))PC|}\Rightarrow\frac{|y(t)|}{|y(s)|}\leqslant|{% \varnothing}_{A}(t)P{\varnothing}_{A}^{-1}(\sigma(s))|\text{ for }t\geqslant s.$

Now, the second inequality follows in a similar way. Thus strong dichotomy implies dichotomy.

Definition 5. The angle $0\leqslant\theta(t)\leqslant{\pi}/{2}$ between $X_{1}$ and $X_{2}$ is defined as

$\displaystyle{\cos\theta(t)}={\max_{|u|=|v|=1}|u^{\rm T}v|},∼{}u\in X_{1},∼{}v% \in X_{2}$

Theorem 6. Let $|{\varnothing}_{A}(t)P{\varnothing}_{A}^{-1}(\sigma(s))|\leqslant k$ for some $k$ , then ${\cot\theta(t)}\leqslant k.$

Proof Let $u\in X_{1}$ and $v\in X_{2}$ with $|u|=|v|=1$ such that ${\cos\theta}=|u^{\rm T}v|$ , If $u$ is orthogonal to $v$ . Result is trivial.

Suppose $u$ is not orthogonal to $v$ , we define $\hat{u}=u,\hat{v}=-(u^{\rm T}v)v$ .

Clearly $\hat{u}$ is orthogonal to $\hat{u}+\hat{v}$ , hence, $\cot\theta(t)=\frac{\hat{u}}{\hat{u}+\hat{v}}$ .

Since $\hat{u}\in X_{1}$ and $\hat{v}\in X_{2}$ we have $\hat{u}={\varnothing}_{A}(t)PC$ for some $C\in R^{n}$ ,

$\displaystyle{\cot\theta}=\frac{|{\varnothing}_{A}(t)PC|}{|{\varnothing}_{A}(t% )PC+{\varnothing}_{A}(t)(I-P)C|}=\frac{|{\varnothing}_{A}(t)PC|}{|{\varnothing% }_{A}(t)C|}=\frac{|{\varnothing}_{A}(t)P^{2}C|}{|{\varnothing}_{A}(t)C|}% \Rightarrow{\cot\theta}=\frac{|{\varnothing}_{A}(t)P{\varnothing}_{A}^{-1}(t){% \varnothing}_{A}(t))C|}{|{\varnothing}_{A}(t)C|}\Rightarrow{\cot\theta(t)}% \leqslant{\max_{\rm d}\frac{|{\varnothing}_{A}(t)P{\varnothing}_{A}^{-1}(t)d|}% {|d|}}\Rightarrow{\cot\theta}\leqslant k.$

From Theorem 6, we can observe that the angle between two sub spaces $X_{1}$ and $X_{2}$ cannot become smaller than ${\cot^{-1}(k)}$ .

3. Conditioning on time-scale dynamical systems in boundary value problems

The condition numbers indicate, by how much any possible error in the boundary conditions may be amplified. Mattheij wrote The conditioning of linear boundary value problems, in SIAM Journal of Numerical Analysis [15]. They also play an important role in estimating the global error due to small perturbations. So, we give a stability analysis of this algorithm and we also show that condition number is an important quantity in estimating the global error. In this section, we show that $\beta$ is the right criterion to indicate possible error amplification of the perturbed boundary condition. We consider the variation $X(t)$ of Eq. (1) with respect to small perturbations in the boundary conditions. Let us consider the perturbation of Eq. (2) in the form

$\displaystyle(M+\delta M)X(a)+(N+\delta N)X(b)=\alpha+\delta\alpha$ (12)

We also assume that the perturbations are such that the characteristic matrix

$\displaystyle{D}_{1}=(D+\delta D)=(M+\delta M){\varnothing}_{A}(a)+(N+\delta N% ){\varnothing}_{A}(b)$

is non-singular.

Let $X^{*}(t)$ be the unique solution of Eq. (1) satisfying Eq. (12). Then we have the following:

Definition 7. The condition number $\beta$ of the boundary value problem Eq. (1) satisfying Eq. (2) is defined as

$\displaystyle\beta={}^{∼{}∼{}∼{}\operatorname{Sup}}_{a\leqslant t\leqslant b}{% \|{\varnothing}_{A}(t)D^{-1}\|}$

$\beta$ is independent of the choice of the fundamental matrix. It can be easily observe that if ${\tilde{\varnothing}}_{A}(t)$ is another fundamental matrix of $LX=0$ , then there exists a non-singular constant matrix ‘ $C$ ’ such that ${\tilde{\varnothing}}_{A}(t)={\varnothing}_{A}(t)C.$

First we note that $\|\delta DD^{-1}\|$ can be estimated in terms of $\beta$ and the perturbations.

Lemma 8. $\|\delta DD^{-1}\|\leqslant(\|\delta M\|+\|\delta N\|)\beta$ , where, $\beta={}^{∼{}∼{}∼{}\operatorname{Sup}}_{a\leqslant t\leqslant b}{\|{% \varnothing}_{A}(t)D^{-1}\|}$ .

Proof

$\displaystyle\|\delta DD^{-1}\|=\|\delta[M{\varnothing}_{A}(a)+N{\varnothing}_% {A}(b)][M{\varnothing}_{A}(a)\quad+N{\varnothing}_{A}(b)]^{-1}\|\leqslant\|[% \delta M{\varnothing}_{A}(a)+\delta N{\varnothing}_{A}(b)]D^{-1}\|\leqslant[\|% \delta M{\varnothing}_{A}(a)\|+\|\delta N{\varnothing}_{A}(b)\|]\|D^{-1}\|% \leqslant[\|\delta M\|+\|\delta N\|]\|{\varnothing}_{A}(t)D^{-1}\|$ $\displaystyle\|\delta DD^{-1}\|\leqslant[\|\delta M\|+\|\delta N\|]\beta.$

Theorem 9. Let $\epsilon>0$ be such that $0<\epsilon<\frac{1}{(1+k)\delta\beta}$ .

$\displaystyle\delta={\max\{\|\delta M\|,\|\delta N\|,\|\delta\alpha\|,\|\delta D% \|\}}$

and

$\displaystyle k=\int^{b}_{a}{\|{\varnothing}_{A}^{-1}(\sigma(s))b(s)\|\Delta s},$

then the solution $X^{*}(t)$ of Eq. (1) satisfying Eq. (12) is such that

$\displaystyle\delta\beta(1-k)(\|{\varnothing}_{A}(a)\|+\|{\varnothing}_{A}(b)\|)$ $\displaystyle\quad\leqslant{\max_{t\in(a,b)}\|X^{*}(t)-X(t)\|}$ $\displaystyle\quad\leqslant\delta\beta(1+k)(\|{\varnothing}_{A}(a)\|+\|{% \varnothing}_{A}(b)\|)$

Proof Any solution $X(t)$ of Eq. (1) satisfying Eq. (2) is given by

$\displaystyle X(t)={\varnothing}_{A}(t)D^{-1}\alpha+\int^{b}_{a}{G(t,\sigma(s)% )b(s)\Delta s}.$

where $G(t,\sigma(s))$ is the Greens’s matrix given by Eq. (1.1).

Then any solution $X^{*}(t)$ of Eq. (1) satisfying Eq. (12) is given by

$\displaystyle X^{*}(t)={\varnothing}_{A}(t){D_{1}}^{-1}(\alpha+\delta\alpha)+% \int^{b}_{a}{G^{*}(t,\sigma(s))b(s)\Delta s},$

where

$\displaystyle G^{*}(t,\sigma(s))=$ $\displaystyle\left\{\begin{array}[]{cc}{\varnothing}_{A}(t){D_{1}}^{-1}M_{1}{% \varnothing}_{A}(a){\varnothing}_{A}^{-1}(\sigma(s)),&t\geqslant s\\ -{\varnothing}_{A}(t){D_{1}}^{-1}M_{1}{\varnothing}_{A}(a){\varnothing}_{A}^{-% 1}(\sigma(s)),&t\leqslant s\end{array}\right\}$

$M_{1}=M+\delta M$ and $N_{1}=N+\delta N$ . Consider

$\displaystyle\|X^{*}(t)-X(t)\|$ $\displaystyle\leqslant\|{\varnothing}_{A}(t)[{D_{1}}^{-1}(\alpha+\delta\alpha)% -D^{-1}\alpha]\|$ $\displaystyle+\int^{t}_{a}\|{\varnothing}_{A}(t)[{D_{1}}^{-1}M_{1}-D^{-1}M]$ $\displaystyle{\varnothing}_{A}(a){\varnothing}_{A}^{-1}(\sigma(s))\|\Delta s$ $\displaystyle+\int^{b}_{t}\|{\varnothing}_{A}(t)[{D_{1}}^{-1}N_{1}-D^{-1}N_{1}]$ $\displaystyle{\varnothing}_{A}(b){\varnothing}_{A}^{-1}(\sigma(s))\|\Delta s.$ (13)

In accordance with the linear approximations we have the following rough estimates.

$\displaystyle{D}_{1}^{-1}(\alpha+\delta\alpha)-D^{-1}\alpha$ $\displaystyle=(D+\delta D)^{-1}(\alpha+\delta\alpha)-D^{-1}\alpha$ $\displaystyle=D^{-1}(I+D^{-1}\delta D)^{-1}(\alpha+\delta\alpha)-D^{-1}\alpha$ $\displaystyle\approx D^{-1}\lfloor(I-D^{-1}\delta D)^{-1}(\alpha+\delta\alpha)% -\alpha\rfloor$ $\displaystyle\approx D^{-1}\lfloor\alpha+\delta\alpha-D^{-1}\delta D(\alpha+% \delta\alpha)-\alpha\rfloor$ $\displaystyle\approx D^{-1}[\delta\alpha].$

Similarly

$\displaystyle D_{1}^{-1}M_{1}-D^{-1}M\approx D^{-1}\delta M\text{ and }$ $\displaystyle D_{1}^{-1}N_{1}-D^{-1}N\approx D^{-1}\delta N.$

Using these estimates in Eq. (3), we get

$\displaystyle\|X^{*}(t)-X(t)\|\leqslant\|{\varnothing}_{A}(t)D^{-1}\delta\alpha\|$ $\displaystyle\quad+\left\|\int^{t}_{a}{{\varnothing}_{A}(t)D^{-1}\delta M{% \varnothing}_{A}(a)}{\varnothing}_{A}^{-1}(\sigma(s))b(s)\Delta s\right\|$ $\displaystyle\quad+\left\|\int^{b}_{t}{{\varnothing}_{A}(t)D^{-1}\delta N{% \varnothing}_{A}(b)}{\varnothing}_{A}^{-1}(\sigma(s))b(s)\Delta s\right\|$ $\displaystyle\leqslant\|{\varnothing}_{A}(t)D^{-1}\delta\alpha\|+\int^{b}_{a}% \|{\varnothing}_{A}(t)D^{-1}[\delta M{\varnothing}_{A}(a)$ $\displaystyle\quad+\delta N{\varnothing}_{A}(b)]{\varnothing}_{A}^{-1}(\sigma(% s))b(s)\Delta s\|$ $\displaystyle\leqslant\|{\varnothing}_{A}(t)D^{-1}\delta\alpha\|+\|{% \varnothing}_{A}(t)D^{-1}[\delta M{\varnothing}_{A}(a)$ $\displaystyle\quad+\delta N{\varnothing}_{A}(b)]\|\times\int^{b}_{a}{\|{% \varnothing}_{A}^{-1}(\sigma(s))b(s)\Delta s\|}$ $\displaystyle\leqslant\|{\varnothing}_{A}(t)D^{-1}\delta\alpha\|+\|{% \varnothing}_{A}(t)D^{-1}[\delta M{\varnothing}_{A}(a)$ $\displaystyle\quad+\delta N{\varnothing}_{A}(b)]\|k$ $\displaystyle\leqslant\|{\varnothing}_{A}(t)D^{-1}\delta\alpha\|[\|[\delta% \alpha[\delta M{\varnothing}_{A}(a)$ $\displaystyle\quad+\delta N{\varnothing}_{A}(b)]]k\|]$ $\displaystyle\leqslant\beta\delta[1+k(\|{\varnothing}_{A}(a)\|+\|{\varnothing}% _{A}(b)\|)]$ $\displaystyle\delta={\max\{\|\delta\alpha\|,\|\alpha M\|\delta N\|,\|\delta D% \|\}}$ $\displaystyle\|X^{*}(t)-X(t)\|\leqslant$ $\displaystyle\beta\delta(1+k)(\|{\varnothing}_{A}(a)\|+\|{\varnothing}_{A}(b)% \|).$

The reverse inequality follows in a similar way, i.e.

$\displaystyle\|X^{*}(t)-X(t)\|\geqslant\|{\varnothing}_{A}(t)D^{-1}\delta\alpha\|$ $\displaystyle-\int^{b}_{a}{\|[G^{*}(t,\sigma(s))-G(t,\sigma(s))]b(s)\|\Delta s}$

In the next section, we discuss bounds on Dichotomy using Theorem 9.

4. Bounds for dichotomy

In this section, we will prove that moderate stability constants imply a dichotomy with moderate ‘ $k$ ’ bound. Now we can scale the fundamental matrix to simplify the notation and algebra significantly. For, the Characteristic matrix ‘ $D$ ’ defined by

$\displaystyle D=M{\varnothing}_{A}(a)+N{\varnothing}_{A}(b)=I.$

This is not a serious restriction, as $D$ is non-singular.

Generally boundary conditions Eq. (2) must represent ‘ $n$ ’ linearly independent constrains on ${\varnothing}_{A}(a)$ and ${\varnothing}_{B}(a)$ . Thus it is necessary that

$\displaystyle\text{Rank}(M,N)=n.$ (14)

However, it frequently happens that $\text{rank\,}B_{a}<n$ or $\text{rank\,}B_{b}<n$ or both. If either holds, we call boundary conditions partially separated. Indeed, the situation in which rank $M=\text{rank\,}N=n$ must be considered rather rare, its most obvious occurrence being periodic boundary conditions.

Suppose that rank $M=p<n$ for some singular $n\times n$ matrix $Q$ such that

$\displaystyle QM=\begin{pmatrix}Z_{a}\\ 0\end{pmatrix}\begin{array}[]{@{}l@{}}\}p\\ \}n-p\end{array},\text{ rank }Z_{a}=p,$

we also introduce the partitions

$\displaystyle Q\alpha=\begin{pmatrix}{\alpha}_{a}\\ {\alpha}_{b}\end{pmatrix}\begin{array}[]{@{}l@{}}\}p\\ \}n-p\end{array}\quad QN=\begin{pmatrix}z_{\rm ab}\\ z_{b}\end{pmatrix}\begin{array}[]{@{}l@{}}\}p\\ \}n-p\end{array}$

where rank $z_{b}=n-p$ .

Then rank conditions follows from Eq. (12). Thus from Eq. (2) we can find

$\displaystyle Q[M{\varnothing}_{A}(a)+N{\varnothing}_{A}(b)]=Q\alpha$

is equivalent to

$\displaystyle\left.\begin{array}[]{r@{}}Z_{a}{\varnothing}_{A}(a)+z_{\rm ab}{% \varnothing}_{A}(b)={\alpha}_{a}\\ {z}_{b}{\varnothing}_{A}(b)={\alpha}_{b}\end{array}\right\}$ (15)

Obviously if rank $n=q<n,$ we can obtain different matrices and vectors

$\displaystyle\left.\begin{array}[]{r@{}}Z_{a}{\varnothing}_{A}(a)={\alpha}_{a}% \\ {z_{\rm ab}{\varnothing}_{A}(a)+z}_{b}{\varnothing}_{A}(b)={\alpha}_{b}\end{% array}\right\}$ (16)

Either of the forms Eq. (15), Eq. (16) consists of partially separated boundary conditions of $z_{\rm ab}=0$ and $z_{\rm ba}=0.$

Then they are separated boundary conditions which are perhaps the most naturally occurring forms in applications.

Theorem 10. Let the boundary conditions be separable in the sense rank $M=P$ , rank $N=n-p$ then there is a projection matrix $P_{1}$ such that

$\displaystyle|{\varnothing}_{A}(t)P{\varnothing}_{A}^{-1}(\sigma(s))|\leqslant% \alpha,∼{}t>s$ $\displaystyle|{\varnothing}_{A}(t)(I-P){\varnothing}_{A}^{-1}(\sigma(s))|% \leqslant\alpha,∼{}t>s$

where $\alpha$ is a stability constant given by Eq. (1.1).

Proof First we will prove that $P_{1}=N{\varnothing}_{A}(b)$ is a projection. Let $R$ be an orthogonal matrix such that the last $n-p$ rows of RM are zero. Since $R$ is orthogonal, it follows that

$\displaystyle R[M{\varnothing}_{A}(a)+N{\varnothing}_{A}(b)]R^{\rm T}=I.$

On equating the last $(n-p)$ rows of above equation, we find that $\hat{P}=RN{\varnothing}_{A}(b)R^{\rm T}$ has the structure

$\displaystyle\hat{P}=\begin{pmatrix}{\hat{P}}_{11}&{\hat{P}}_{12}\\ 0&I_{n-p}\end{pmatrix}.$

Since rank $\hat{P}=n-p$ , we must have ${\hat{P}}_{11}=0$ and hence $\hat{P}={\hat{P}}^{2}$ .

Thus, ${\hat{P}}^{2}=R^{\rm T}{\hat{P}}^{2}R=R^{\rm T}\hat{P}R=\hat{P}$ . Hence the claim.

The proof now follows from Eq. (1.1) on that

$\displaystyle G(t,\sigma(s))=$ $\displaystyle\left\{\begin{array}[]{cc}{\varnothing}_{A}(t)P{\varnothing}_{A}^% {-1}(\sigma(s)),&t>s\\ -{\varnothing}_{A}(t)(I-P){\varnothing}_{A}^{-1}(\sigma(s)),&t<s\end{array}% \right\}.$

Note that when the boundary conditions are separable, a strong dichotomy exists when $k=\alpha$ . It then follows from Lemma 4 that the same result holds with our weaker version of dichotomy.

For more general boundary conditions the situation is somewhat more difficult. The main reason is that these boundary conditions do not provide a natural projection matrix. Therefore we proceed by constructing separable boundary conditions so that the corresponding boundary value problem is well conditioned. Once this is achieved, we can use Theorem 9 to obtain bounds for the dichotomy.

In order to construct separable boundary conditions, we monitor the growth of solutions over the entire interval. Let the singular-value decomposition of ${\varnothing}_{A}(b){\varnothing}_{A}^{-1}(a)$ be given by

$\displaystyle{\varnothing}_{A}(b){\varnothing}_{A}^{-1}(a)=\textit{UDV}^{\rm T}.$

where $U$ and $V$ are orthogonal matrices and D is a positive diagonal matrix with ordered elements. We use the following notation

$\displaystyle D=\operatorname{diag}\left(\frac{1}{d_{1}},\frac{1}{d_{2}}\ldots% \frac{1}{d_{q}},d_{q+1},\ldots,d_{n}\right)$

with $0<d_{j}\leqslant(j=1,2,\ldots n)$

$\displaystyle D_{1}=\operatorname{diag}(d_{1},d_{2,}\ldots d_{q},1,1\ldots 1)$ $\displaystyle D_{1}=\operatorname{diag}(1,1\ldots 1,d_{q+1},d_{q+2,}\ldots d_{% n})\text{ and}$ $\displaystyle P=\begin{pmatrix}0&0\\ 0&I_{n-p}\end{pmatrix}.$ (17)

Now, we can define the separated boundary conditions specified by

$\displaystyle M=PV^{\rm T}\text{ and }N=(I-P)U^{\rm T}$ (18)

Now, we can easily verify with the structure of P that

$\displaystyle\tilde{M}{\widetilde{\varnothing}}_{A}(a)+\tilde{N}{\widetilde{% \varnothing}}_{A}(b)=I$

where

$\displaystyle{\widetilde{\varnothing}}_{A}(t)={\varnothing}_{A}(t){\varnothing% }_{A}^{-1}(a)VD_{1}={\varnothing}_{A}(t){\varnothing}_{A}^{-1}(b)VD_{2}.$

We can therefore associate with $\tilde{M}$ , $\tilde{N}$ the corresponding Green’s matrix

$\displaystyle\tilde{G}(t,\sigma(s))=$ $\displaystyle\left\{\begin{array}[]{cc}{\widetilde{\varnothing}}_{A}(t)\tilde{% M}{{{\widetilde{\varnothing}}_{A}(a)\varnothing}_{A}}^{-1}(\sigma(s)),&t% \geqslant s\\ -{\widetilde{\varnothing}}_{A}(t)\tilde{N}{{{\widetilde{\varnothing}}_{A}(b)% \varnothing}_{A}}^{-1}(\sigma(s)),&t\leqslant s\end{array}\right\}.$

From Eq. (17), we again have $\text{rank\,}M=q$ and rand $N=n-q$ . Thus, if we can establish the stability constants for the problem Eq. (1) satisfying the boundary conditions.

$\displaystyle\tilde{M}X(a)+\tilde{N}X(b)=\alpha.$

Now, we can use Theorem 9 to establish bounds for dichotomy. First we obtain some relationships between the Green’s matrices $G(t,\sigma(s))$ and $\tilde{G}(t,\sigma(s))$ .

Result 11. For any fundamental matrix $Z$

$Z(t)=G(t,\sigma(s))Z(s)-G(t,\sigma(u))Z(u),a\leqslant s<t<u\leqslant b$ .

$Z^{-1}(t)=Z^{-1}(u)G(u,\sigma(t))-Z^{-1}(s)G(s,$ $\sigma(t)),a\leqslant s<t<u\leqslant b$ .

$\tilde{G}(t,\sigma(s))=G(t,\sigma(s))-{\widetilde{\varnothing}}_{A}(t)[\tilde{% N}G(b,\sigma(s))$ $+\tilde{M}G(a,\sigma(s))]$ .

Proof

(i)

$\displaystyle∼{}∼{}{\varnothing}_{A}(t)={\varnothing}_{A}(t)[M{\varnothing}_{A% }(a)+N{\varnothing}_{A}(b)]$ $\displaystyle{\varnothing}_{A}(t)={\varnothing}_{A}(t)M{\varnothing}_{A}(a){% \varnothing}_{A}^{-1}(\sigma(s))+{\varnothing}_{A}(t)N{\varnothing}_{A}(b){% \varnothing}_{A}^{-1}(\sigma(u))$ $\displaystyle{\varnothing}_{A}(t)=G(t,\sigma(s)){\varnothing}_{A}(\sigma(s))-G% (t,\sigma(u)){\varnothing}_{A}(\sigma(u)).$

By substituting $Z(t)={\varnothing}_{A}(t){\varnothing}_{A}^{-1}(a)Z(a)$ we get result (i).

(ii)

$\displaystyle{\varnothing}_{A}^{-1}(\sigma(t))$ $\displaystyle=[M{\varnothing}_{A}(a)+N{\varnothing}_{A}(b)]{\varnothing}_{A}^{% -1}(\sigma(t))$ $\displaystyle={\varnothing}_{A}^{-1}(u){\varnothing}_{A}(u)M{\varnothing}_{A}(% a){\varnothing}_{A}^{-1}(\sigma(s))$ $\displaystyle\quad+{\varnothing}_{A}^{-1}(s){\varnothing}_{A}(s)N{\varnothing}% _{A}(b){\varnothing}_{A}^{-1}(\sigma(u))$ $\displaystyle={\varnothing}_{A}^{-1}(u)G(u,\sigma(t))-{\varnothing}_{A}^{-1}(s% )G(s,\sigma(t))$

Therefore $Z^{-1}(t)=Z^{-1}(u)G(u,\sigma(t))-Z^{-1}(s)$ $G(s,\sigma(t))$ .

(iii) (a) For $t>s$ we have $\tilde{G}(t,\sigma(s))={\widetilde{\varnothing}}_{A}(t)$ $\tilde{M}{\widetilde{\varnothing}}_{A}(a){{\widetilde{\varnothing}}_{A}}^{-1}(% \sigma(s))$ for ${{\widetilde{\varnothing}}_{A}}^{-1}(\sigma(s))$ we substitute the value of (ii) we get

$\displaystyle\tilde{G}(t,\sigma(s))={\widetilde{\varnothing}}_{A}(t)\tilde{M}{% \widetilde{\varnothing}}_{A}(a)\lfloor{{\widetilde{\varnothing}}_{A}}^{-1}(b)G% (b,\sigma(s))$ $\displaystyle\quad-{{\widetilde{\varnothing}}_{A}}^{-1}(a)G(a,\sigma(s))\rfloor$ $\displaystyle={\widetilde{\varnothing}}_{A}(t)[I-\tilde{N}{\widetilde{% \varnothing}}_{A}(b)]{{\widetilde{\varnothing}}_{A}}^{-1}(b)G(b,\sigma(s))$ $\displaystyle\quad-{\widetilde{\varnothing}}_{A}(t)\tilde{M}{\widetilde{% \varnothing}}_{A}(a){{\widetilde{\varnothing}}_{A}}^{-1}(a)G(a,\sigma(s))$ $\displaystyle={\widetilde{\varnothing}}_{A}(t){{\widetilde{\varnothing}}_{A}}^% {-1}(b)G(b,\sigma(s))-{\widetilde{\varnothing}}_{A}(t){\tilde{N}}_{A}{% \widetilde{\varnothing}}_{A}(b)$ $\displaystyle\quad{{\widetilde{\varnothing}}_{A}}^{-1}(b)G(b,\sigma(s))-{% \widetilde{\varnothing}}_{A}(t)\tilde{M}G(a,\sigma(s))$ $\displaystyle={\widetilde{\varnothing}}_{A}(t){{\widetilde{\varnothing}}_{A}}^% {-1}(b)G(b,\sigma(s))$ $\displaystyle\quad-{\widetilde{\varnothing}}_{A}(t)[\tilde{M}G(a,\sigma(s))+% \tilde{N}G(b,\sigma(s))]$ $\displaystyle=G(t,\sigma(s))-{\widetilde{\varnothing}}_{A}(t)[\tilde{M}G(a,% \sigma(s))+\tilde{N}G(b(s))].$

(b) If $t<s$ , we have

$\displaystyle\tilde{G}(t,\sigma(s))$ $\displaystyle=-{\widetilde{\varnothing}}_{A}(t)\tilde{N}{\widetilde{% \varnothing}}_{A}(b){{\widetilde{\varnothing}}_{A}}^{-1}(s)$ $\displaystyle=-{\widetilde{\varnothing}}_{A}(t)\tilde{N}{\widetilde{% \varnothing}}_{A}(b)\lfloor{{\widetilde{\varnothing}}_{A}}^{-1}(u)G(u,\sigma(s))$ $\displaystyle\quad-{{\widetilde{\varnothing}}_{A}}^{-1}(a)G(a,\sigma(s))\rfloor$ $\displaystyle=-{\widetilde{\varnothing}}_{A}(t)\tilde{N}{\widetilde{% \varnothing}}_{A}(b){{\widetilde{\varnothing}}_{A}}^{-1}(u)G(u,\sigma(s))$ $\displaystyle\quad+{\widetilde{\varnothing}}_{A}(t)[I-M{\widetilde{\varnothing% }}_{A}(a)]{{\widetilde{\varnothing}}_{A}}^{-1}(a)G(a,\sigma(s))$ $\displaystyle=-{\widetilde{\varnothing}}_{A}(t)\tilde{N}\tilde{G}(b,\sigma(s))% +{\widetilde{\varnothing}}_{A}(t){\widetilde{\varnothing}}_{A}(a)G(a,\sigma(s))$ $\displaystyle\quad-{\widetilde{\varnothing}}_{A}(t)\tilde{M}G(a,\sigma(s))$ $\displaystyle=-{\widetilde{\varnothing}}_{A}(t)[\tilde{M}G(a,\sigma(s))+\tilde% {N}G(b,\sigma(s))]$ $\displaystyle\quad+G(t,\sigma(s)).$

Theorem 12.(a)

$\displaystyle\frac{|{\widetilde{\varnothing}}_{A}(t)PX|}{|{\widetilde{% \varnothing}}_{A}(\sigma(s))PX|}\leqslant\alpha+2{\alpha}^{2},∼{}t\geqslant s$

(b)

$\displaystyle\frac{|{\widetilde{\varnothing}}_{A}(t)(I-P)X|}{|{\widetilde{% \varnothing}}_{A}(\sigma(s))(I-P)X|}\leqslant\alpha+2{\alpha}^{2},∼{}t\leqslant s$

Proof

(a) From Result 11, (i) we have

$\displaystyle{\varnothing}_{A}(t)PX$ $\displaystyle=[G(t,\sigma(s)){\widetilde{\varnothing}}_{A}(\sigma(s))-G(t,% \sigma(b))]PX$ $\displaystyle=G(t,\sigma(s)){\widetilde{\varnothing}}_{A}(\sigma(s))PX$ $\displaystyle\quad-G(t,\sigma(b)){{\widetilde{\varnothing}}_{A}}^{-1}(\sigma(s% )){\widetilde{\varnothing}}_{A}(\sigma(s))PX$ $\displaystyle=[G(t,\sigma(s))-G(t,\sigma(b))\tilde{G}(b,\sigma(s))]{\widetilde% {\varnothing}}_{A}(\sigma(s))PX$ $\displaystyle\Rightarrow\frac{|{\widetilde{\varnothing}}_{A}(t)PX|}{|{% \widetilde{\varnothing}}_{A}(\sigma(s))PX|}\leqslant\alpha+2{\alpha}^{2}.$

(b)

$\displaystyle{\widetilde{\varnothing}}_{A}(t)(I-P)X=[G(t,\sigma(a))$ $\displaystyle\quad-G(t,\sigma(u)){\widetilde{\varnothing}}_{A}(\sigma(u))](I-P)X$ $\displaystyle=G(t,\sigma(a))(I-P)X$ $\displaystyle\quad-G(t,\sigma(u)){\widetilde{\varnothing}}_{A}(\sigma(u))(I-P)X$ $\displaystyle=G(t,\sigma(a))(I-P)({\widetilde{\varnothing}}_{A}(\sigma(u)){% \widetilde{\varnothing}}_{A}(\sigma(u)))$ $\displaystyle\quad(I-P)X-G(t,\sigma(u)){\widetilde{\varnothing}}_{A}(\sigma(u)% )(I-P)X$ $\displaystyle=[G(t,\sigma(a))\tilde{G}(a,\sigma(u))$ $\displaystyle\quad-G(t,\sigma(u))]{\widetilde{\varnothing}}_{A}(\sigma(u))(I-P)X$ $\displaystyle\Rightarrow\frac{|{\widetilde{\varnothing}}_{A}(t)(I-P)X|}{|{% \widetilde{\varnothing}}_{A}(\sigma(s))(I-P)X|}\leqslant\alpha+2{\alpha}^{2}.$

Hence the theorem.

We can also given an estimate for strong dichotomy. Since

$\displaystyle\tilde{M}{\widetilde{\varnothing}}_{A}(a)=(PV^{\rm T}){% \varnothing}_{A}(a){\varnothing}_{A}^{-1}(a)VD_{1}=PV^{\rm T}VD_{1}=PD_{1}$ $\displaystyle\tilde{M}{\widetilde{\varnothing}}_{A}(a)=P$

and

$\displaystyle\tilde{N}{\widetilde{\varnothing}}_{A}(b)=\lfloor(I-P)U^{\rm T}% \rfloor\lfloor{\varnothing}_{A}(b){\varnothing}_{A}^{-1}(b)UD_{2}\rfloor=(I-P)% V^{\rm T}VD_{1}=(I-P)D_{2}$ $\displaystyle\tilde{N}{\widetilde{\varnothing}}_{A}(b)=(I-P).$

We can associate the Greeen’s matrix for the boundary conditions.

$\displaystyle\tilde{M}X(a)+\tilde{N}X(b)=\alpha$

$\displaystyle\tilde{G}(t,s)=\left\{\begin{array}[]{cc}{\widetilde{\varnothing}% }_{A}(t)P{{\widetilde{\varnothing}}_{A}}^{-1}(\sigma(s)),&t\geqslant s\\ -{\widetilde{\varnothing}}_{A}(t)(I-P){{\widetilde{\varnothing}}_{A}}^{-1}(% \sigma(s)),&t\leqslant s\end{array}\right\}.$

we have the following lemma:

Lemma 13.

$|{\widetilde{\varnothing}}_{A}(b)P{{\widetilde{\varnothing}}_{A}}^{-1}(\sigma(% s))|=|\tilde{G}(b,\sigma(s))|\leqslant 2\alpha$ ,

$|{\widetilde{\varnothing}}_{A}(a)(I-P){{\widetilde{\varnothing}}_{A}}^{-1}(% \sigma(s))|=|\tilde{G}(a,\sigma(s))|\leqslant 2\alpha$ ,

$|{\widetilde{\varnothing}}_{A}(t)|\leqslant 2\alpha k$ , where $k=\max\{{\widetilde{\varnothing}}_{A}(a),$ ${\widetilde{\varnothing}}_{A}(b)\}$ .

Proof From the Result 11, (ii)

(a)

$\displaystyle|{\widetilde{\varnothing}}_{A}(b)P{{\widetilde{\varnothing}}_{A}}% ^{-1}(\sigma(s))|$ $\displaystyle=|{\widetilde{\varnothing}}_{A}(b)P[{{\widetilde{\varnothing}}_{A% }}^{-1}(b)G(b,\sigma(s))$ $\displaystyle\quad-{{\widetilde{\varnothing}}_{A}}^{-1}(a)G(a,\sigma(s))]|$ $\displaystyle=|{\widetilde{\varnothing}}_{A}(b)P{{\widetilde{\varnothing}}_{A}% }^{-1}(b)G(b,\sigma(s))$ $\displaystyle\quad-{{\widetilde{\varnothing}}_{A}}^{-1}(b)P{{\widetilde{% \varnothing}}_{A}}^{-1}(a)G(a,\sigma(s))|$ $\displaystyle\leqslant|G(b,\sigma(s))|$ $\displaystyle+|{{\widetilde{\varnothing}}_{A}}^{-1}(b)P{{\widetilde{% \varnothing}}_{A}}^{-1}(a)G(a,\sigma(s))|$ $\displaystyle\leqslant\alpha+|UD_{2}P{(VD_{1})}^{-1}G(a,\sigma(s))|$ $\displaystyle\leqslant\alpha+|UD_{2}P{D_{1}}^{-1}V^{-1}G(a,\sigma(s))|$ $\displaystyle\leqslant\alpha+|UD_{2}P{D_{1}}^{-1}V^{-1}||G(a,\sigma(s))|$ $\displaystyle\leqslant\alpha+\alpha=2\alpha.$

(b)

$\displaystyle|{\widetilde{\varnothing}}_{A}(a)(I-P){{\widetilde{\varnothing}}_% {A}}^{-1}(\sigma(s))|$ $\displaystyle=|{\widetilde{\varnothing}}_{A}(a)(I-P)[{{\widetilde{\varnothing}% }_{A}}^{-1}(b)G(b,\sigma(s))$ $\displaystyle\quad-{{\widetilde{\varnothing}}_{A}}^{-1}(a)G(a,\sigma(s))]|$ $\displaystyle=|{\widetilde{\varnothing}}_{A}(a)(I-P){{\widetilde{\varnothing}}% _{A}}^{-1}(b)G(b,\sigma(s))$ $\displaystyle\quad-{{\widetilde{\varnothing}}_{A}}^{-1}(a)(I-P){{\widetilde{% \varnothing}}_{A}}^{-1}(a)G(a,\sigma(s))|$ $\displaystyle\leqslant|VD_{1}(I-P){(UD_{2})}^{-1}G(b,\sigma(s))|$ $\displaystyle+|G(a,\sigma(s))|$ $\displaystyle\leqslant|VD_{1}(I-P){D_{2}}^{-1}U^{\rm T}G(b,\sigma(s))|$ $\displaystyle+|G(a,\sigma(s))|$ $\displaystyle\leqslant\alpha+\alpha=2\alpha.$

$\displaystyle{\widetilde{\varnothing}}_{A}(a)=G(t,a){\widetilde{\varnothing}}_% {A}(a)-G(t,b){\widetilde{\varnothing}}_{A}(b)$

$\displaystyle|{\widetilde{\varnothing}}_{A}(t)|\leqslant|G(t,a){\widetilde{% \varnothing}}_{A}(a)|-|G(t,b){\widetilde{\varnothing}}_{A}(b)|\leqslant\alpha|% {\widetilde{\varnothing}}_{A}(a)|+\alpha|{\widetilde{\varnothing}}_{A}(b)|% \leqslant 2\alpha k$

where $k={\max\{|{\widetilde{\varnothing}}_{A}(a)|,|{\widetilde{\varnothing}}_{A}(b)|\}}$ .

Lemma 14.

$|\tilde{G}(t,\sigma(s))|\leqslant\alpha+4{\alpha}^{2}k$ .

$|\tilde{G}(t,\sigma(s))|\leqslant\alpha+4\alpha\delta k$ .

where $\gamma\!=\!\beta(|M|\!+\!|N|)k,k\!=\!\max\lfloor|{\widetilde{\varnothing}}_{A}% (a)|,|{\widetilde{\varnothing}}_{A}(b)|\rfloor$ and $\beta$ being given by

$\displaystyle\beta={\max_{t\in(a,b)}\|{\varnothing}_{A}(t){\|\|D}^{-1}\|}.$

Proof From Result 11, we have

$\displaystyle\tilde{G}(t,\sigma(s))=G(t,\sigma(s))-{\widetilde{\varnothing}}_{% A}(t)\lfloor PV^{\rm T}G(a,\sigma(s))\quad+(I-P)U^{\rm T}G(b,\sigma(s))\rfloor% =G(t,\sigma(s))+{\widetilde{\varnothing}}_{A}(t)[M\tilde{G}(a,\sigma(s))\quad+% N\tilde{G}(b,\sigma(s))]$

Since ${\widetilde{\varnothing}}_{A}(t)\leqslant 2\alpha k,|\tilde{G}(b,\sigma(s))|% \leqslant 2\alpha$ , we obtain (a). Similarly we can obtain (b).

From this lemma we have the following estimates for strong dichotomy.

Theorem 15.

$|\widetilde{\varnothing}(t)P{{\widetilde{\varnothing}}_{A}}^{-1}(\sigma(s))|% \leqslant\alpha+4{\alpha}^{2}k$ , $t\geqslant s$ ;

$|\widetilde{\varnothing}(t)(I-P){{\widetilde{\varnothing}}_{A}}^{-1}(\sigma(s)% )|\leqslant\alpha+4{\alpha}^{2}k$ , $t\leqslant s$ ;

$|\widetilde{\varnothing}(t)P{{\widetilde{\varnothing}}_{A}}^{-1}(\sigma(s))|% \leqslant\alpha+2\alpha\gamma k$ , $t\geqslant s$ ;

$|\widetilde{\varnothing}(t)(I-P){{\widetilde{\varnothing}}_{A}}^{-1}(\sigma(s)% )|\leqslant\alpha+2\alpha\gamma k$ , $t\leqslant s$ .

5. Bounds for exponential dichotomy

In many realistic problems the solution space is exponentially dichotomy. This implies a certain exponential behavior of the Green’s function G. In this section we replaced the condition Eq. (10) by the following

$\displaystyle|G(t,\sigma(s))|\leqslant\alpha e^{\mu(s-t)},t\geqslant s,\mu>0$ $\displaystyle|G(t,\sigma(s))|\leqslant\alpha e^{\gamma(t-s)},t\leqslant s,% \gamma>0$ (19)

Using previous section we can similarly show that Eq. (5) implies an exponentially solution space.

Theorem 16. Let

$\displaystyle\xi(t)=\alpha\lfloor e^{\mu(a-t)}+e^{\gamma(t-b)}\rfloor$ $\displaystyle\eta(t)=\alpha\lfloor e^{\mu(s-b)}+e^{\gamma(a-s)}\rfloor$

Let $P$ be as defined in Section 4, ${\varnothing}_{A}(t)$ be as Lemma 4.1 and $\gamma=\beta(|M|+|N|)$ the following relations hold good.

(a ${}_{1}$ )

$|{\widetilde{\varnothing}}_{A}(t)P{{\widetilde{\varnothing}}_{A}}^{-1}(\sigma(% s))|\leqslant\alpha e^{\mu(s-t)}+\xi(t)\eta(s)$ . $t\geqslant s$ ;

{}_{1}

)

$|{\widetilde{\varnothing}}_{A}(t)(I-P){{\widetilde{\varnothing}}_{A}}^{-1}(% \sigma(s))|\leqslant\alpha e^{\mu(t-s)}+\xi(t)\eta(s)$ . $t\leqslant s$ ;

{}_{2}

)

$|{\widetilde{\varnothing}}_{A}(t)P{{\widetilde{\varnothing}}_{A}}^{-1}(\sigma(% s))|\leqslant\alpha e^{\mu(s-t)}+\mu\xi(s)$ . $t\geqslant s$ ;

{}_{2}

)

$|{\widetilde{\varnothing}}_{A}(t)(I\!-\!P){{\widetilde{\varnothing}}_{A}}^{-1}% (\sigma(s))|\leqslant\alpha e^{\mu(t-s)}+\mu\eta(s)$ . $t\leqslant s$ .

From Theorem 10, it appears that $\mu$ and $\gamma$ more or less play the roles of the numbers $\lambda$ and $\xi$ in Section 1.

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