\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2013 (2013), No. 153, pp. 1--13.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2013 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2013/153\hfil Psi-exponential dichotomy] {Psi-exponential dichotomy for linear differential equations in a Banach space} \author[A. Georgieva, H. Kiskinov, S. Kostadinov, A. Zahariev \hfil EJDE-2013/153\hfilneg] {Atanaska Georgieva, Hristo Kiskinov,\\ Stepan Kostadinov, Andrey Zahariev} % in alphabetical order \address{ Faculty of Mathematics and Informatics\\ University of Plovdiv\\ 236 Bulgaria Blvd., 4003 Plovdiv, Bulgaria} \email[A. Georgieva]{atanaska@uni-plovdiv.bg} \email[H. Kiskinov]{kiskinov@uni-plovdiv.bg} \email[S. Kostadinov]{stkostadinov@uni-plovdiv.bg} \email[A. Zahariev]{zandrey@uni-plovdiv.bg} \thanks{Submitted March 28, 2013. Published July 2, 2013.} \subjclass[2000]{34G10, 34D09, 34C11} \keywords{Dichotomy for ordinary differential equations; $\psi$-dichotomy; \hfill\break\indent $\psi$-boundedness; $\psi$-stability } \begin{abstract} In this article we extend the concept $\psi$-exponential and $\psi$-ordinary dichotomies for homogeneous linear differential equations in a Banach space. With these two concepts we prove the existence of $\psi$-bounded solutions of the appropriate inhomogeneous equation. A roughness of the $\psi$-dichotomy is also considered. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{definition}[theorem]{Definition} \newtheorem{remark}[theorem]{Remark} \allowdisplaybreaks \section{Introduction} The problem of $\psi$-boundedness and $\psi$-stability of the solutions of differential equations in finite dimensional Euclidean spaces has been studied by many autors; see for example Akinyele \cite{ak1}, Constantin \cite{co2}. In these publications, the function $\psi$ is a scalar continuous function (and increasing, differentiable and bounded in \cite{ak1}, nondecreasing and such that $\psi(t) \geq 1$ on $\mathbb{R}_+$ in \cite{co2}). In Diamandescu \cite{di1,di2,di3,di4,di5} and Boi \cite{bo1,bo2,bo3} the function $\psi$ is a nonnegative continuous diagonal matrix. Inspired by the famous monographs of Coppel \cite{co1}, Daleckii and Krein \cite{da1} and Massera and Schaeffer \cite{ma1}, considered the important notion of exponential and ordinary dichotomy in detail. Diamandescu \cite{di1}-\cite{di5} and Boi \cite{bo1}-\cite{bo3}, introduced and studied the $\psi$-dichotomy for linear differential equations in finite dimensional Euclidean space. Here we introduce the concept of $\psi$-dichotomy for arbitrary Banach spaces instead in finite dimensional Euclidean spaces. Moreover, in our case, $\psi(t)$ is an arbitrary bounded invertible linear operator, instead of the restriction to be a nonnegative diagonal matrix. Conditions for the existence of $\psi$-bounded solutions of the homogeneous and the appropriate inhomogeneous equations are proved. A roughness of the $\psi$-exponential dichotomy is also proved. \section{Preliminaries} Let $X$ be an arbitrary Banach space with norm $|\cdot|$ and identity $I$. Let $LB(X)$ be the space of all linear bounded operators acting in $X$ with the norm $\|\cdot \|$. Let $J=[0,\infty)$. We consider the linear homogenous equation \begin{equation} \label{e1} \frac{{\rm d}x}{{\rm d}t}=A(t)x \end{equation} and the corresponding inhomogeneous equation \begin{equation} \label{e2} \frac{{\rm d}x}{{\rm d}t}=A(t)x+f(t), \end{equation} where $A(.): J \to LB(X)$, $f(.): J \to X$ are strong measurable and Bochner integrable on the finite subintervals of $J$. By a solution of \eqref{e2} (or \eqref{e1}) we will understand a continuous function $x(t)$ that is differentiable (in the sense that it is representable in the form $x(t)=\int_{a}^{t} y(\tau) d\tau$ of a Bochner integral of a strongly measurable function $y$) and satisfies \eqref{e2} (or \eqref{e1}) almost everywhere. By $V(t)$ we will denote the Cauchy operator of \eqref{e1}. Let $RL(X)$ be the subspace of all invertible operators in $LB(X)$ and let $\psi (.) : J \to RL(X)$ be continuous for any $t \in J$ operator-function. \begin{definition} \label{def1}\rm A function $ u(.) : J \to X$ is said to be $\psi$-bounded on $J$ if $\psi(t) u(t)$ is bounded on $J$. A function $ f(.) : J \to X$ is said to be $\psi$-integrally bounded on $J$ if it is measurable and there exists a positive constant $m$ such that$\int_t^{t+1} |\psi(\tau) f(\tau)| d\tau \leq m$ for any $t \in J$. A function $ f(.) : J \to X$ is said to be $\psi$-Bochner integrable on $J$ if it is measurable and $\int_J |\psi(\tau) f(\tau)| d\tau < \infty$. \end{definition} Let $C_\psi(X)$ denote the Banach space of all $\psi$-bounded and continuous functions with values in $X$ with the norm $$ {\||f\||}_{C_\psi}=\sup_{t \in J} |\psi(t)f(t)|. $$ Let $M_\psi(X)$ denote the Banach space of all $\psi$-integrally bounded functions with values in $X$ with the norm $$ {\||f\||}_{M_\psi}=\sup_{t \in J} \int_t^{t+1} |\psi(s)f(s)|{\rm d}s. $$ Let $L_\psi(X)$ denote the Banach space of all $\psi$-Bochner integrable on $J$ functions with values in $X$ with the norm $$ {\||f\||}_{L_\psi}= \int_J |\psi(s)f(s)| {\rm d}s. $$ \begin{definition} \label{def2}\rm The equation \eqref{e1} is said to has a $\psi$-exponential dichotomy on $J$ if there exist a pair mutually complementary projections $P_1$ and $P_2=I-P_1$ and positive constants $N_1, N_2, \nu_1, \nu_2$ such that \begin{gather} \label{e3} \|\psi(t) V(t) P_1 V^{-1}(s) \psi^{-1}(s)\| \leq N_1 e^{-\nu_1 (t-s)} \quad ( 0 \leq s \leq t ) \\ \label{e4} \|\psi(t) V(t) P_2 V^{-1}(s) \psi^{-1}(s)\| \leq N_2 e^{-\nu_2 (s-t)} \quad ( 0 \leq t \leq s ) \end{gather} Equation \eqref{e1} is said to has a $\psi$-ordinary dichotomy on $J$ if \eqref{e3} and \eqref{e4} hold with $\nu_1 = \nu_2 =0$. \end{definition} \begin{remark} \label{rmk1} \rm For $\psi (t) = I$ for all $t \in J$ we obtain the notion of exponential and ordinary dichotomy in \cite{co1, da1, ma1}. \end{remark} \begin{definition} \label{def3} \rm Equation \eqref{e1} is said to have a $\psi$-bounded growth on $J$ if for some fixed $h>0$ there exists a constant $C\geq 1$ such that every solution $x(t)$ of \eqref{e1} satisfies \begin{equation} \label{e5} |\psi(t) x(t)| \leq C |\psi(s) x(s)| \quad ( 0 \leq s \leq t \leq s+h) \end{equation} \end{definition} \section{Main results} \begin{lemma} \label{lem1} Equation \eqref{e1} has a $\psi$-exponential dichotomy on $J$ with positive constants $\nu_1$ and $\nu_2$ if and only if there exist a pair mutually complementary projections $P_1$ and $P_2=I-P_1$ and positive constants $M, \tilde{N}_1, \tilde{N}_2$ such that following inequalities are fulfilled \begin{gather} \label{e6} |\psi(t) V(t) P_1 \xi| \leq \tilde{N}_1 e^{-\nu_1 (t-s)} |\psi(s) V(s) P_1 \xi| \quad (\xi \in X, 0 \leq s \leq t ) \\ \label{e7} |\psi(t) V(t) P_2 \xi| \leq \tilde{N}_2 e^{-\nu_2 (s-t)} |\psi(s) V(s) P_2 \xi| \quad (\xi \in X, 0 \leq t \leq s ) \\ \label{e8} \|\psi(t) V(t) P_1 V^{-1}(t) \psi^{-1}(t)\| \leq M \quad (t \geq 0 ) \end{gather} \end{lemma} \begin{proof} Let \eqref{e1} have a $\psi$-exponential dichotomy on $J$. Then for any $x \in X$ from \eqref{e3} it follows that $$ |\psi(t) V(t) P_1 V^{-1}(s) \psi^{-1}(s) x | \leq N_1 e^{-\nu_1 (t-s)} |x| \quad ( 0 \leq s \leq t ) $$ For $x= \psi(s) V(s) P_1 \xi$ we obtain \eqref{e6}. The proof of\eqref{e7} is analogous. Obviously the inequality \eqref{e8} holds. Now vice versa. Let \eqref{e6}, \eqref{e7} and \eqref{e8} are fulfilled. For any $x \in X$ we can choose $\xi = V^{-1}(s) \psi^{-1}(s) x$ and from \eqref{e6} we obtain \begin{align*} |\psi(t) V(t) P_1 V^{-1}(s) \psi^{-1}(s)x| & \leq \tilde{N}_1 e^{-\nu_1 (t-s)} |\psi(s) V(s) P_1 V^{-1}(s) \psi^{-1}(s) x| \\ & \leq M \tilde{N}_1 e^{-\nu_1 (t-s)} |x| \quad ( 0 \leq s \leq t ) \end{align*} Hence estimate \eqref{e3} holds with $N_1= M \tilde{N}_1$. The proof of \eqref{e4} is analogous. \end{proof} Let us explain in detail the importance of Lemma \ref{lem1}, which obviously can be taken as definition for $\psi$-exponential dichotomy on $J$ instead of Definition \ref{def2}. The pair mutually complementary projections $P_1$ and $P_2=I-P_1$ exists if and only if for some $t_0 \in J$ the space $X$ decomposes into a direct sum of two closed subspaces $X=X_1 + X_2$. Let us introduce the subspaces $X_k(t)=V(t) V^{-1}(t_0) X_k$ $(k=1,2,\; t\in J)$. Then $X_1(t_0)=X_1$ and $X_2(t_0)=X_2$. The projection functions corresponding to the subspaces $X_k(t)$ are $$ P_k(t)=V(t) P_k V^{-1}(t) \quad (k=1,2;\; t\in J). $$ And from the estimates \eqref{e6} and \eqref{e7} it follows, that the complemented subspace $X_1(t_0)$ is exactly the subspace of all initial values $x_1^0 \in X_1(t_0)$ such that the solutions $x_1(t)=V(t) V^{-1}(t_0) x_1^0$ starting at moment $t_0$ from the subspace $X_1(t_0)$ are $\psi$-bounded on $J$. From the existence of the pair mutually complementary projections $P_1$ and $P_2=I-P_1$, it follows also the existence of the projection functions $$ Q_k(t)=\psi(t) V(t) P_k V^{-1}(t) \psi^{-1}(t), \quad (k=1,2; \; t \in J) $$ which induce the decomposition of the spaces $X$ into direct sums of closed subspaces $X=Q_1(t)X + Q_2(t)X=\tilde{X}_1(t) + \tilde{X}_2(t)$ The condition \eqref{e8} for uniform bondedness of the projections $Q_k(t)$ $(k=1,2;\; t\in J)$ is equivalent (see \cite{da1}) to the requirement, that the angular distance between the subspaces $\tilde{X}_1(t)$ and $\tilde{X}_2(t)$ cannot become arbitrary small under a variation of $t$. More precisely there must exist a constant $\gamma >0$ such that \begin{equation} \label{e9} Sn(\tilde{X}_1(t),\tilde{X}_2(t)) \geq \gamma \quad (t \in J) \end{equation} where the angular distance $Sn$ between two subspaces $Y_1$ and $Y_2$ of a space $Y$ is defined as \begin{equation} \label{e10} Sn(Y_1,Y_2)=\inf_{y_k \in Y_k, |y_k|=1, (k=1,2)} |y_1+y_2| \end{equation} The subspaces $\tilde{X}_k(t)$ and projection functions $Q_k(t)$, $(k=1,2;\;t\in J)$ are introduced by us explicitly to fit the concept of the $\psi$-boundedness and $\psi$-dichotomy in an arbitrary Banach space. For $\psi(t)=I$ $(t\in J)$ (i.e. for the exponential dichotomy in \cite{da1, ma1}) $\tilde{X}_k(t) \equiv X_k(t)$ and $Q_k(t)\equiv P_k(t)$ $(k=1,2, t\in J)$. \begin{lemma} \label{lem2} Equation \eqref{e1} has $\psi$-bounded growth on $J$ if and only if there exist positive constants $K\geq 1$ and $\alpha>0$ such that \begin{equation} \label{e11} \|\psi(t) V(t) V^{-1}(s) \psi^{-1}(s)\| \leq K e^{\alpha (t-s)} \quad (0 \leq s \leq t ) \end{equation} \end{lemma} \begin{proof} Let us suppose that \eqref{e1} has $\psi$-bounded growth; i.e. \eqref{e5} holds. Let $t \geq s$ be two arbitrary positive numbers. Setting $n = [\frac{t-s}{h}]$ and $\eta=\frac{t-s}{h}$ we have $n \leq \eta \leq n+1$. Then \begin{align*} |\psi(t) x(t)|& = |\psi(\eta h+s) x(\eta h+s)| \leq C|\psi(nh+s) x(nh+s)| \leq \dots \\ & \leq C^{n+1} |\psi(s) x(s)| \leq C^{\eta+1} |\psi(s) x(s)| \quad ( 0 \leq s \leq t ) \end{align*} We can take $K=C$ and $\alpha=h^{-1} \ln C$. Obviously, $C^{\eta +1} = K e^{\alpha (t-s)}$ and we have the estimate $$ |\psi(t) x(t)| \leq K e^{\alpha (t-s)} |\psi(s) x(s)|. $$ For an arbitrary vector $\xi \in X$ we consider the solution $x(t)$ of \eqref{e1} with $x(0)=V^{-1}(s)\psi^{-1}(s) \xi$. Therefore, $$ |\psi(t) V(t) V^{-1}(s) \psi^{-1}(s) \xi| \leq K e^{\alpha (t-s)} |\xi| $$ is fulfilled for any $\xi \in X$. Hence the estimate \eqref{e11} holds. Vice versa - suppose that \eqref{e11} holds. From $x(t)=V(t)V^{-1}(s) x(s)$ and the estimate \eqref{e11} we obtain $$ |x(t)| \leq K e^{\alpha (t-s)} |x(s)| $$ for some $K\geq 1$ and $\alpha >0$. Then we can take $C=Ke^{\alpha h}$. Obviously $C \geq 1$. Hence \eqref{e1} has $\psi$-bounded growth. \end{proof} \begin{remark} \label{rmk2} \rm The proof shows that the condition for $\psi$-bounded growth (and for bounded growth) of \eqref{e1} is independent from the choice of $h$. \end{remark} \begin{remark} \label{rmk3} \rm In the famous monograph by Coppel \cite[p. 9]{co1}, nessesary and sufficient condition for bounded growth are formulated with $K, \alpha \in \mathbb{R}$, which is an typing error. By Boi \cite[Lemma 2.4]{bo1} necessary and sufficient conditions for $\psi$-bounded growth are formulated with $K, \alpha >0$, which is also wrong. The only correct necessary and sufficient condition for bounded and $\psi$-bounded growth which is independent from the choice of $h$ must be formulated with $K\geq 1, \alpha >0$. \end{remark} \begin{lemma} \label{lem3} If \eqref{e1} has $\psi$-bounded growth on $J$, then \eqref{e8} is a consequence of \eqref{e6} and \eqref{e7}. \end{lemma} \begin{proof} Let suppose that \eqref{e1} has $\psi$-bounded growth. Let $m \geq 0$. Then, using Lemma \ref{lem2} we have the estimate \begin{equation} \label{e12} \|\psi(t+m) V(t+m) V^{-1}(t) \psi^{-1}(t)\| \leq K e^{\alpha m} \end{equation} with $K\geq 1$ and $\alpha >0$. Let us consider, for an arbitrary fixed $t \in J$, a pair unit vectors $y_k(t) \in \tilde{X}_k(t)$ $(k=1,2)$. $$ y_k(t)=\psi(t)V(t) P_k V^{-1}(t) \psi^{-1}(t) \omega_k \quad (\omega_k \in X, |y_k(t)|=1, k=1,2) $$ Let $\xi_k=V^{-1}(t) \psi^{-1}(t) \omega_k$. From \eqref{e6}, \eqref{e7} and \eqref{e12} we obtain \begin{gather} \label{e13} |\psi(t+m) V(t+m) P_1 \xi_1| \leq \tilde{N}_1 e^{-\nu_1 m} |\psi(t) V(t) P_1 \xi_1| = \tilde{N}_1 e^{-\nu_1 m}, \\ \label{e14} |\psi(t+m) V(t+m) P_2 \xi_2| \geq {\tilde{N}_2}^{-1} e^{\nu_2 m} |\psi(t) V(t) P_2 \xi_2| = {\tilde{N}_2}^{-1} e^{\nu_2 m} \end{gather} From \begin{align*} &|\psi(t+m) V(t+m) (P_1 \xi_1+P_2 \xi_2)| = \\ & =|\psi(t+m) V(t+m) V^{-1}(t) \psi^{-1}(t) \psi(t) V(t) (P_1 \xi_1+P_2 \xi_2)| \ \\ & \leq \|\psi(t+m) V(t+m) V^{-1}(t) \psi^{-1}(t)\| \ |\psi(t) V(t) P_1 \xi_1+\psi(t) V(t) P_2 \xi_2| \\ & \leq K e^{\alpha m} |\psi(t) V(t) P_1 \xi_1+\psi(t) V(t) P_2 \xi_2| \\ &= K e^{\alpha m} |y_1(t)+y_2(t)| \end{align*} we conclude that \begin{align*} &|y_1(t)+y_2(t)| \geq \\ & \geq K^{-1} e^{-\alpha m} |\psi(t+m) V(t+m) P_1 \xi_1+ \psi(t+m) V(t+m) P_2 \xi_2| \\ & \geq K^{-1} e^{-\alpha m} ( |\psi(t+m) V(t+m) P_2 \xi_2)| - |\psi(t+m) V(t+m) P_1 \xi_1| ) \\ & \geq K^{-1} e^{-\alpha m} ( {\tilde{N}_2}^{-1} e^{\nu_2 m} - \tilde{N}_1 e^{-\nu_1 m} ) = \gamma_m \end{align*} Making reference to \eqref{e10} it follows $$ Sn(\tilde{X}_1(t),\tilde{X}_2(t)) \geq \gamma_m $$ Taking $m$ large enough the constant $\gamma_m >0$ and we can conclude that the angular distance between the subspaces $\tilde{X}_1(t)$ and $\tilde{X}_2(t)$ is bounded from below. By Daleckii and Krein \cite[Corollary 1.1, Chapter IV]{da1} this is equivalent to the boundedness from above of the corresponding projection function $Q_1(t)$. Hence \eqref{e8} holds and the proof is complete. \end{proof} \begin{theorem} \label{thm1} If the homogeneous equation \eqref{e1} has $\psi$-exponential dichotomy on $J$, then the inhomogeneous equation \eqref{e2} has for every $\psi$-bounded function $f(t) \in C_\psi(X)$ at least one $\psi$-bounded solution $x(t)\in C_\psi(X)$. This solution is \begin{equation} \label{e15} x(t)=\int_0^t V(t)P_1V^{-1}(s)f(s){\rm d}s -\int_t^\infty V(t)P_2V^{-1}(s)f(s){\rm d}s \end{equation} \end{theorem} \begin{proof} Let us consider the function \begin{align*} \tilde{x}(t)&= \int_0^t \psi(t)V(t)P_1V^{-1}(s)f(s){\rm d}s - \int_t^\infty \psi(t)V(t)P_2V^{-1}(s)f(s){\rm d}s \\ &= \int_0^t \psi(t)V(t)P_1V^{-1}(s)\psi^{-1}(s)\psi(s)f(s){\rm d}s \\ &\quad - \int_t^\infty \psi(t)V(t)P_2V^{-1}(s)\psi^{-1}(s)\psi(s)f(s){\rm d}s \end{align*} Because \eqref{e2} has a $\psi$-exponential dichotomy on $J$, from \eqref{e3}, \eqref{e4} and the condition for $\psi$-boundedness of $f(t)$ (i.e. the existence of a constant $c$ such that $|\psi(t)f(t)|\leq c$) we obtain the estimate \begin{align*} |\tilde{x}(t)| &\leq \int_0^t \|\psi(t)V(t)P_1V^{-1}(s)\psi^{-1}(s)\| \ |\psi(s)f(s)|{\rm d}s \\ & \quad + \int_t^\infty \|\psi(t)V(t)P_2V^{-1}(s)\psi^{-1}(s)\| \ |\psi(s)f(s)|{\rm d}s \ \\ & \leq c(\frac{N_1}{\nu_1}+\frac{N_2}{\nu_2}) \end{align*} Hence ${\|| \tilde{x}(t)\||}_{C_\psi} \leq (\frac{N_1}{\nu_1} +\frac{N_2}{\nu_2}) {\||f(t)\||}_{C_\psi}$; i.e. $\tilde{x}(t)$ is bounded on $J$. Let $x(t)= \psi^{-1}(t) \tilde{x}(t)$. Obviously $x(t)$ is $\psi$-bounded on $J$. Then $$ x(t) = \psi(t)^{-1} \Big(\int_0^t \psi(t)V(t)P_1V^{-1}(s)f(s){\rm d}s - \int_t^\infty \psi(t)V(t)P_2V^{-1}(s)f(s){\rm d}s\Big) $$ We have already proved, that the integrals exist. Then \begin{align*} \frac{{\rm d}x}{{\rm d}t} &=A(t)\int_0^t V(t)P_1V^{-1}(s)f(s){\rm d}s+V(t)P_1V^{-1}(t)f(t) \\ &\quad +V(t)P_2V^{-1}(t)f(t)-A(t)\int_t^\infty V(t)P_2V^{-1}(s)f(s){\rm d}s \\ &=A(t)x(t)+V(t)P_1V^{-1}(t)f(t)+V(t)P_2V^{-1}(t)f(t) \\ &=A(t)x(t)+f(t) \end{align*} Hence the function $$ x(t)=\int_0^t V(t)P_1V^{-1}(s)f(s){\rm d}s -\int_t^\infty V(t)P_2V^{-1}(s)f(s){\rm d}s $$ is a $\psi$-bounded solution of the inhomogeneous equation \eqref{e2} on $J$. \end{proof} \begin{remark} \label{rmk4}\rm Let introduce the principal Green function of \eqref{e2} with the projections $P_1$ and $P_2$ from the definition for $\psi$-exponential dichotomy \begin{equation} \label{e16} G(t,s)= \begin{cases} V(t)P_1V^{-1}(s) & (t>s) \\ - V(t)P_2V^{-1}(s) & (t0$ is a constant not depending on $f$. \end{theorem} \begin{proof} Suppose $f(t) \in B_\psi(X)$. By hypothesis, there exists a solution $x(t) \in C_\psi(X)$ of equation \eqref{e2}. Let $P_1$ and $P_2$ be the mutually complementary projections on the subspaces $X_1$ and $X_2$. We denote by $x_1(t)$ the solution of the corresponding homogeneous equation which satisfies the condition $x(0)=P_1 x(0)$. This solution is $\psi$-bounded by definition of the subspace $X_1$. But then the solution $x_2(t)=x(t)-x_1(t)$ of the inhomogeneous equation for which $x_2(0)=x(0)-P_1x(0)=P_2x(0) \in X_2$ is also $\psi$-bounded. The uniqueness follows from the fact that the difference of two such solutions would be bounded by a solution initially in $X_2$ of the homogeneous equation, which is possible only for the zero solution. It remains for us to prove the last assertion of the lemma. We consider the space $C_1$ of all functions $x(t)$ that are solutions of equations of the form $$ x'(t)-A(t)x(t) = f(t) $$ under the conditions $x(0) \in X_2$ and $f(t) \in B_\psi(X)$. It was essentially shown above that the operator $T x(t) =x'(t)-A(t)x(t)$ effects a one-to-one mapping of the linear space $C_1$ onto $B_\psi(X)$ . If in $C_1$ we introduce the norm $$ {\||x\||}_{C_1} = {\||x\||}_{C_\psi} + {\||Tx\||}_{B_\psi} $$ the operator $Tx$ automatically turns out to be continuous. If, in addition, the space $C_1$ turns out to be complete, the inverse operator $T^{-1}$ will also be continuous by Banach's theorem, and the solution $x=T^{-1}f$ of equation \eqref{e2} will then satisfy the estimate $$ {\||x\||}_{C_\psi} \leq {\||x\||}_{C_1} \leq \|T^{-1}\| \ {\||f\||}_{B_\psi} . $$ Thus it remains to prove the completeness of $C_1$. Let $\{x_n(t)\}$ be a Cauchy sequence in it. Such a sequence is also a Cauchy sequence in $C_\psi(X)$ and hence has a limit $x(t)$ in it. In this case clearly $$ x(0)=\lim_{n \to \infty} x_n(0) \in X_2. $$ In exactly the same way it follows that the sequence $\{f_n(t)\} = \{T x_n(t)\}$ has a limit $f(t)$ in $B_\psi(X)$. Therefore for each $t \in J$ \begin{align*} x(t)-x(0)& = \lim_{n \to \infty} \int_0^\infty {x'}_n(\tau) {\rm d}\tau \\ &=\lim_{n \to \infty} \int_0^\infty (f_n(\tau)+ A(\tau) x_n(\tau)) {\rm d}\tau \\ & = \int_0^\infty (f(\tau)+ A(\tau) x(\tau)) {\rm d}\tau \end{align*} which implies that $x(t)$ satisfies the equation $x'(t)-A(t)x(t) = f(t)$. Thus $x(t) \in C_1$ and, as easily seen, ${\||x-x_n\||}_{C_1} \to 0$ for $n \to 0$, i.e. $C_1$ is complete. The theorem is proved. \end{proof} \begin{theorem} \label{thm3} In order for equation \eqref{e1} to has $\psi$-ordinary dichotomy on $J$ it is necessary and sufficient that its $\mathfrak{Y}_\psi$-set be a complemented subspace and that to each function $f(t) \in L_\psi(X)$ there corresponds at least one $\psi$-bounded solution on $J$ of the inhomogeneous equation \eqref{e2}. \end{theorem} \begin{proof} The necessity of the second condition follows from Theorem \ref{thm1} and Remark \ref{rmk6}, because obviously $L_\psi(X) \subset M_\psi(X)$. The necessity of the first was noted in defining the $\mathfrak{Y}$-set. Now the sufficiency. Let $\xi \in X$ be an arbitrary fixed vector and let us consider the function \begin{equation} f(t)= \begin{cases} \psi^{-1}(t) \xi & \text{for } \ s \leq t \leq s+h \\ 0 &\text{otherwise} \end{cases} \end{equation} where $s \geq 0$ and $h>0$. Then $f \in L_\psi(X)$ and ${\||f\||}_{L_\psi}= h |\xi|$. The corresponding solution of \eqref{e2} is $$ x(t)=\int_{J} G(t,\tau) f(\tau) {\rm d}\tau = \int_{s}^{s+h} G(t,\tau) \psi^{-1}(t) \xi {\rm d}\tau. $$ From Theorem \ref{thm2}, it follows the estimate $$ |\psi(t)x(t)| = |\int_{s}^{s+h} \psi(t) G(t,\tau) \psi^{-1}(t) \xi {\rm d}\tau| \leq K_{L_\psi} h |\xi|. $$ It follows that $$ | \psi(t) G(t,\tau) \psi^{-1}(t) \xi | \leq K_{L_\psi} |\xi|. $$ Hence, since $\xi$ is arbitrary, $$ \| \psi(t) G(t,\tau) \psi^{-1}(t) \| \leq K_{L_\psi} . $$ Thus \eqref{e3} and \eqref{e4} hold with $N_1=N_2=K_{L_\psi}$ and $\nu_1=\nu_2=0$. Obviously \eqref{e3} and \eqref{e4} remains valid also in the excepted case $t=s$. \end{proof} \begin{corollary} \label{coro1} In a finite-dimensional phase space the homogeneous equation \eqref{e1} has $\psi$-ordinary dichotomy on $J$ if and only if there corresponds to each function $f(t) \in L_\psi(X)$ at least one $\psi$-bounded solution on $J$ of the inhomogeneous equation \eqref{e2}. \end{corollary} \begin{lemma} \label{lem4} Suppose that \eqref{e2} has a $\psi$-bounded solution for every function $f \in C_\psi$ and let $r=K_{C_\psi}$. Let $x(t)$ be a solution of the corresponding homogeneous equation \eqref{e1} and let $$ x_1(t)=V(t)P_1V^{-1}(t)x(t), \quad x_2(t)=V(t)P_2V^{-1}(t)x(t). $$ If for some fixed $s \geq 0$ is fulfilled $|\psi(t) x_1(t)| \leq N |\psi(s) x(s)|$ for $s \leq t \leq s+r$, then $$ |\psi(t) x_1(t)| \leq eN |\psi(s) x(s)| e^{-r^{-1}(t-s)} \quad \text{for } s \leq t < \infty. $$ If for some fixed $s \geq 0$ is fulfilled $|\psi(t) x_2(t)| \leq N |\psi(s) x(s)|$ for $\max\{0,s-r\} \leq t \leq s$, then $$ |\psi(t) x_2(t)| \leq eN |\psi(s) x(s)| e^{-r^{-1}(s-t)} \quad \text{for } 0 \leq t \leq s. $$ \end{lemma} \begin{proof} Let us take $$ f(t)=\chi(t) x(t) {|\psi(t) x(t)|}^{-1} $$ where $x(t)=V(t)\xi$ is a nontrivial solution of the homogeneous equation \eqref{e1} and $\chi(t)$ be an arbitrary real valued function such that $0 \leq \chi(t) \leq 1$ for all $t \geq 0$ and $\chi(t)=0$ for $f \geq t_1$. Then obviously $f \in C_{\psi}(X)$ and ${\||f\||}_{C_\psi} \leq 1$. Hence by the arbitrary nature of $\chi(t)$ applying Theorem \ref{thm2} we have with $r=K_{C_\psi}$, the estimate $$ |\psi(t) \int_{t_0}^{t_1} G(t,\tau) x(\tau) {|\psi(\tau) x(\tau)|}^{-1} {\rm d}\tau | \leq r \quad ( 0 \leq t_0 \leq t_1, \ t\geq 0). $$ Putting $t_1=t$ and respectively $t_0=t$ we obtain \begin{equation} \label{e17} \begin{gathered} |\psi(t)V(t)P_1 \xi| \int_{t_0}^{t} {|\psi(\tau) x(\tau)|}^{-1} {\rm d}\tau \leq r \quad ( 0 \leq t_0 \leq t), \\ |\psi(t)V(t)P_2 \xi| \int_{t}^{t_1} {|\psi(\tau) x(\tau)|}^{-1} {\rm d}\tau \leq r \quad ( t \leq t_1 \leq \infty). \end{gathered} \end{equation} Replacing $\xi$ by $P_1 \xi$, respectively $P_2 \xi$, it follows by integration that \begin{equation} \label{e18} \begin{gathered} \int_{t_0}^{s} {|\psi(\tau)V(\tau)P_1 \xi|}^{-1} {\rm d}\tau \leq e^{-r^{-1} (t-s)} \int_{t_0}^{t} {|\psi(\tau)V(\tau)P_1 \xi|}^{-1} {\rm d}\tau \quad (t_0\leq s\leq t), \\ \int_{s}^{t_1}{|\psi(\tau)V(\tau)P_1 \xi|}^{-1}{\rm d}\tau \leq e^{-r^{-1} (s-t)} \int_{t}^{t_1} {|\psi(\tau)V(\tau)P_1 \xi|}^{-1} {\rm d}\tau \quad (t\leq s\leq t_1). \end{gathered} \end{equation} Replacing $t_0$ by $s$ and $s$ by $s+r$ in the first inequality \eqref{e18} and using the first assumption of the lemma, for $t \geq s+r$, we obtain $$ r N^{-1} {|\psi(s) x(s)|}^{-1} \leq \int_s^{s+r} {|\psi(\tau)x_1(\tau)|}^{-1} {\rm d}\tau \leq e e^{-r^{-1} (t-s)} \int_{s}^{t} {|\psi(\tau)x_1(\tau)|}^{-1} {\rm d}\tau $$ Using the first inequality \eqref{e17}, for $t \geq s+r$, we have $$ |\psi(t) x_1(t)| \leq r {\left( \int_{s}^{t} {|\psi(\tau)x_1(\tau)|}^{-1} {\rm d}\tau \right)}^{-1} \leq eN |\psi(s) x(s)| e^{-r^{-1}(t-s)} $$ Since obviously the same inequality holds for $s \leq t \leq s+r$, the first assertion of the lemma is proved. The proof of the second assertion of the lemma is similar, using the second assumption of it and replacing $s$ by $s-r$ and $t_1$ by $s$ in the second inequality \eqref{e18}. \end{proof} \begin{theorem}\label{thm4} For equation \eqref{e1} to be $\psi$-exponential dichotomous on $J$ it is necessary and sufficient that its $\mathfrak{Y}_\psi$-set be a complemented subspace and that to each function $f(t) \in M_\psi(X)$ there corresponds at least one $\psi$-bounded solution on $J$ of the inhomogeneous equation \eqref{e2}. \end{theorem} \begin{proof} The necessity of the second condition follows from Theorem \ref{thm1} and Remark \ref{rmk6}, while the necessity of the first was noted in defining the $\mathfrak{Y}$-set. Now the sufficiency. Let the $\mathfrak{Y}_\psi$-set of the homogeneous equation \eqref{e1} be a complemented subspace and suppose that to each function $f(t) \in M_\psi(X)$ there corresponds at least one $\psi$-bounded solution on $J$ of the inhomogeneous equation \eqref{e2}. Since $C_\psi(X) \subset M_\psi(X)$ and $L_\psi(X) \subset M_\psi(X)$ the equation \eqref{e2} has a $\psi$-bounded solution on $J$ for every $f \in C_\psi(X)$ and for every $f \in L_\psi(X)$ too. By Theorem \ref{thm3} and its proof \eqref{e3} and \eqref{e4} hold with $N_1=N_2=K_{L_\psi}$ and $\nu_1=\nu_2=0$. Hence the conditions of Lemma \ref{lem4} are fulfilled with $N=K_{L_\psi}$ for every solution $x(t)$ of \eqref{e1} and for every $s \geq 0$. Applying Lemma \ref{lem4} we obtain \eqref{e3} and \eqref{e4} with $N_1=N_2= e K_{L_\psi}$ and $\nu_1=\nu_2={K_{C_\psi}}^{-1}$. The theorem is proved. \end{proof} \begin{corollary} \label{coro2} In a finite-dimensional phase space the homogeneous equation \eqref{e1} is $\psi$-exponential dichotomous on $J$ if and only if there corresponds to each function $f(t) \in M_\psi(X)$ at least one $\psi$-bounded solution on $J$ of the inhomogeneous equation \eqref{e2}. \end{corollary} \begin{theorem} \label{thm5} Suppose that \eqref{e1} has $\psi$-bounded growth. For equation \eqref{e1} to be $\psi$-exponential dichotomous on $J$ it is necessary and sufficient that its $\mathfrak{Y}_\psi$-set be a complemented subspace and that to each function $f(t) \in C_\psi(X)$ there corresponds at least one $\psi$-bounded solution on $J$ of the inhomogeneous equation \eqref{e2}. \end{theorem} \begin{proof} The necessity of the second condition follows from Theorem \ref{thm1}, while the necessity of the first was noted in defining the $\mathfrak{Y}$-set. Now the sufficiency. Let assume that the equation \eqref{e1} has $\psi$-bounded growth. From Lemma \ref{lem2} it follows $$ \|\psi(t) V(t) V^{-1}(s) \psi^{-1}(s)\| \leq K e^{\alpha (t-s)} \quad (0 \leq s \leq t ) $$ where $K \geq 1$ and $\alpha>0$ are constants. Because the initial conditions of Lemma \ref{lem4} are fulfilled, replacing $\xi$ by $V^{-1}(s)\psi^{-1}(s) \xi$ and putting $t_1=\infty$ in the second inequality \eqref{e17} we obtain for $t \leq s$, \begin{align*} |\psi(t)V(t)P_2V^{-1}(s)\psi^{-1}(s) \xi| &\leq r {\Big( \int_t^{\infty} |\psi(\tau) V(\tau) V^{-1}(s) \psi^{-1}(s) \xi| \Big) }^{-1} \\ &\leq r { \Big(K^{-1}{|\xi|}^{-1} \int_t^{\infty} e^{\alpha (s-\tau)} \Big) }^{-1}. \end{align*} Thus $$ \|\psi(t)V(t)P_2V^{-1}(s)\psi^{-1}(s)\| \leq \alpha r K \quad ( t \leq s). $$ Analogously, we obtain $$ \|\psi(t)V(t)P_2V^{-1}(s)\psi^{-1}(s)\| \leq \alpha r K e^{\alpha (t-s)} \quad ( t \geq s) $$ and hence \begin{equation} \label{e19} \|\psi(t)V(t)P_1V^{-1}(s)\psi^{-1}(s)\| \leq (1+\alpha r) K e^{\alpha (t-s)} \quad ( t \geq s). \end{equation} In the same way, from the first inequality \eqref{e17} it follows \begin{equation} \label{e20} \|\psi(t)V(t)P_1V^{-1}(s)\psi^{-1}(s)\| \leq \alpha r K {\big( 1- e^{-\alpha (t-s)}\big) }^{-1} \quad ( t > s). \end{equation} Let $h= \alpha^{-1} \ln \frac{1+2 \alpha r}{1+ \alpha r}$. By using \eqref{e20} for $t-s \geq h$ and \eqref{e19} for $t-s \leq h$ we obtain $$ \|\psi(t)V(t)P_1V^{-1}(s)\psi^{-1}(s)\| \leq (1+2\alpha r) K \quad \text{for all } ( t \geq s). $$ Now we can apply Lemma \ref{lem4} with $N=(1+2\alpha r) K$ and obtain \begin{gather*} \|\psi(t)V(t)P_1V^{-1}(s)\psi^{-1}(s)\| \leq e (1+2\alpha r) K e^{-r^{-1} (t-s)} \quad (0 \leq s \leq t), \\ \|\psi(t)V(t)P_2V^{-1}(s)\psi^{-1}(s)\| \leq e \alpha r K e^{-r^{-1} (s-t)} \quad (0 \leq t \leq s). \end{gather*} Thus \eqref{e1} has a $\psi$-exponential dichotomy. \end{proof} \begin{corollary} \label{coro3} In a finite-dimensional phase space the homogeneous equation \eqref{e1} with $\psi$-bounded growth is $\psi$-exponential dichotomous on $J$ if and only if there corresponds to each function $f(t) \in C_\psi(X)$ at least one $\psi$-bounded solution on $J$ of the inhomogeneous equation \eqref{e2}. \end{corollary} An important property of the $\psi$-exponential dichotomies is their roughness. That is, they are not destroyed by small perturbations of the coefficient operator. Let consider the perturbed equation \begin{equation} \label{e21} \frac{{\rm d}x}{{\rm d}t}=\left(A(t)+B(t)\right)x\,. \end{equation} \begin{theorem} \label{thm6} Suppose that the equation \eqref{e1} has a $\psi$-exponential dichotomy on $J$. If $\delta=\sup_{t \in J} \|\psi(t) B(t) {\psi}^{-1}(t)\|$ is sufficient small, then the perturbed equation \eqref{e21} has also a $\psi$-exponential dichotomy on $J$. \end{theorem} \begin{proof} Let us consider the inhomogeneous equation \begin{equation} \label{e22} \frac{{\rm d}x(t)}{{\rm d}t}=\left(A(t)+B(t)\right)x(t) + f(t), \end{equation} and introduce the map $$ T z(t) = \int_{J} G(t,\tau) \left( B(\tau) z(\tau) + f(\tau) \right) {\rm d}\tau $$ First we shall prove that $T$ maps $C_\psi$ into itself. Using the same technic and notations as in the proofs of Theorem \ref{thm1} and Remark \ref{rmk6}, we obtain the estimate \begin{align*} |\psi(t) T z(t)| & = |\psi(t) \int_{J} G(t,\tau) \left( B(\tau) z(\tau) + f(\tau) \right) {\rm d}\tau| \leq \\ & \leq \int_{J} \|\psi(t)G(t,\tau)\psi^{-1}(\tau)\| \ \|\psi(\tau) B(\tau)\psi^{-1}(\tau)\| \ |\psi(\tau) z(\tau)| {\rm d}\tau \\ &\quad + \int_{J} \|\psi(t)G(t,\tau)\psi^{-1}(\tau)\| \ |\psi(\tau) f(\tau) | {\rm d}\tau \\ & \leq \delta c \Big( \frac{N_1}{\nu_1} + \frac{N_2}{\nu_2} \Big) + \frac{N_2 m}{1- e^{-\nu_2}} + \frac{N_1 m}{1- e^{-\nu_1}} . \end{align*} Hence $Tz \in C_\psi$ and $T : C_\psi \to C_\psi$. Now we will show that the map $T$ is a contraction. Let $z_1, z_2 \in C_\psi$. Then \begin{align*} &{\||Tz_1 -Tz_2\||}_{C_\psi} \\ &\leq |\psi(t) \int_{J} G(t,\tau) B(\tau) \left( z_1(\tau) - z_2(\tau) \right) {\rm d}\tau| \leq \\ & \leq \int_{J} \|\psi(t)G(t,\tau)\psi^{-1}(\tau)\| \ \|\psi(\tau) B(\tau)\psi^{-1}(\tau)\| \ |\psi(\tau) (z_1(\tau)-z_1(\tau))| {\rm d}\tau \\ & \leq \delta \Big( \frac{N_1}{\nu_1} + \frac{N_2}{\nu_2} \Big) {\||z_1-z_2\||}_{C_\psi}. \end{align*} By selecting a sufficient small $\delta$ we can obtain $\delta \big( \frac{N_1}{\nu_1} + \frac{N_2}{\nu_2} \big) < 1$ and the map $T$ will be a contraction. By the fixed point principle of Banach it follows, that the map $T$ has an unique fixed point. Denoting this point by $z$ we have $$ z(t) = \int_{J} G(t,\tau) \big( B(\tau) z(\tau) + f(\tau) \big) {\rm d}\tau. $$ Thus $z(t)$ is a solution of \eqref{e22}. Hence the equation \eqref{e22} has for every $\psi$-integrally bounded function $f(t)$ at least a $\psi$-bounded solution. 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