\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2005(2005), No. 108, pp. 1--20.\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 2005 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2005/108\hfil Identification of degenerate kernels] {Identification of two degenerate time- and space-dependent kernels in a parabolic equation} \author[E. Pais, J. Janno\hfil EJDE-2005/108\hfilneg] {Enno Pais, Jaan Janno} \address{Enno Pais\hfill\break Tallinn University of Technology\\ 12618 Tallinn, Estonia} \email{ennopais@stv.ee} \address{Jaan Janno \hfill\break Tallinn University of Technology\\ 12618 Tallinn, Estonia} \email{janno@ioc.ee} \date{} \thanks{Submitted April 7, 2005. Published October 7, 2005.} \thanks{Partially supported by Grant 6018 from the Estonian Science Foundation} \subjclass[2000]{35R30, 80A23} \keywords{Inverse problem; memory kernel; parabolic equation} \begin{abstract} An inverse problem to determine two degenerate time- and space-dependent kernels in a parabolic integro-differential equation is considered. Observation data involves given values of the solution of the equation in a finite number of points over the time. Existence and uniqueness of a solution to the inverse problem is proved. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{remark}[theorem]{Remark} \allowdisplaybreaks \section{Introduction} Heat flow in materials with memory is governed by parabolic integro-differential equations containing time-dependent (and in the case of non-homogeneity also space-dependent) memory (or relaxation) kernels \cite{c1,l2,n1,p2}. These kernels are often unknown in the practice. To determine them, inverse problems are used. Various problems to identify time-dependent memory kernels in parabolic equations have been studied in a number of papers (see \cite{g1,g2,j2,j3,l1} and references therein). When the kernels are both time- and space-dependent, inverse problems based on restricted Dirichlet-to-Neumann map in general case \cite{j1} and single trace measurements in stratified cases \cite{c2} are in the use. In some context the kernels can be degenerate, i.e. representable as finite sums of products of known space-dependent functions times unknown time-dependent coefficients. This is so when either the medium is piecewise continuous or a problem for a general kernel is replaced by a related problem for an approximated kernel. The unknown coefficients are recovered by a finite number of measurements of certain time-dependent characteristics of the solution of the direct problem. In \cite{j4,j5} inverse problems of such a type were studied. These papers deal with the simplified case when the model contains only the relaxation kernel of heat flux. However, a more precise model of a material with thermal memory involves two relaxation kernels contained in basic constitutive relations: kernels of internal energy and heat flux \cite{g1,g2,j3,n1}. In the present paper we study an inverse problem to determine degenerate nonhomogeneous relaxation kernels of internal energy and heat flux occurring in a parabolic equation governing heat flow in materials with memory. To recover the kernels, we make use of a finite number of measurements of temperature in fixed points over the time. As in \cite{j4,j5}, we apply the fixed-point argument in weighted norms adjusted to the problem in the Laplace domain. Due to the structure of the problem, the kernels of internal energy and heat flux are recovered with different level of regularity. The corresponding problem with flux-type additional information is more complicated, and not covered by this paper. In Section 2 we formulate the direct and inverse problems and in Section 3 apply the Laplace transform them. In Section 4 we rewrite the transformed problems in the fixed-point form. Sections 5 and 6 contain some auxiliary results for the direct problem. Main existence and uniqueness results for the inverse problem are included in Section 7 of the paper. \section{Physical background and formulation of problem.} We consider the heat flow in a rigid nonhomogeneous bar consisting of a material with thermal memory. For a sake of simplicity we assume the rod to be of the unit length. Then, in the linear approximation the system of constitutive relations and the heat balance equation read as \begin{gather} e(x,t)= \beta(x)u(x,t)+\int_0^tn(x,t-\tau)u(x,\tau)\,d\tau\, , \label{e1.1}\\ q(x,t)= -\lambda(x)u_x(x,t)+\int_{0}^{t}m(x,t-\tau)u_x(x,\tau)\,d\tau\,, \label{e1.2}\\ \frac{\partial}{\partial t}e(x,t)+\frac{\partial}{\partial x}q(x,t)=r(x,t), \label{e1.3} \end{gather} respectively, where $x\in (0,1)$ is the space coordinate and $t\in\mathbb{R}$ is the time \cite{g1,g2,j3,n1} Here $u$ is the temperature of the bar, which is assumed to be zero for $t<0$, $e$ is the internal energy, $q$ the heat flux and $r$ is the heat supply. Moreover, $\beta$ and $\lambda$ stand for the heat capacity and the heat conduction coefficient, respectively. The model contains two memory kernels $n$ and $m$, being the relaxation kernels of the integral energy and the heat flux, respectively. From \eqref{e1.1}--\eqref{e1.3}, we obtain the parabolic integro-differential equation \label{e1.4} \begin{aligned} &\beta(x)\frac{\partial}{\partial t}u(x,t)+\frac{\partial}{\partial t}\int_0^t n(x,t-\tau)u(x,\tau)\,d\tau \\ &= \frac{\partial}{\partial x}(\lambda(x)u_x(x,t)) -\frac{\partial}{\partial x}\int_0^t m(x,t-\tau)u_x(x,\tau)\,d\tau + r(x,t),\quad x\in(0,1), \ t>0. \end{aligned} The function $u(x,t)$ is assumed to satisfy the initial conditions $$\label{e1.5} u(x,0) = \varphi(x),\quad x\in(0,1)$$ and the Dirichlet boundary conditions $$\label{e1.6} u(0,t) = f_1(t), \, u(1,t) = f_2(t), \quad t>0$$ with given functions $\varphi$ on $[0,1]$ and $f_j,\, j=1,2$ on $[0,\infty)$. Equation \eqref{e1.4} with the conditions \eqref{e1.5} and \eqref{e1.6} form the direct problem for the temperature $u$. In an inverse problem we seek for the kernels $n$ and $m$. We restrict ourselves to the case of the kernels in the following degenerate forms $$\label{e1.7} n(x,t)=\sum_{j=1}^{N_1}\nu_j(x)n_j(t), \quad m(x,t)=\sum_{k=1}^{N_2}\mu_k(x)m_k(t)\, ,$$ where $\nu_j$, $j=1,\dots,N_1$, $\mu_k$, $k=1,\dots,N_2$ are given $x$-dependent functions and $n_j$, $j=1,\dots,N_1$, $m_k$, $k=1,\dots,N_2$ are unknown time-dependent coefficients. Formulas \eqref{e1.7} hold, for instance, when the medium is piecewise continuous, where $n_j$ and $m_k$ are characteristic functions or smooth approximations of characteristic functions of the subdomains of homogeneity. In general case \eqref{e1.7} can be interpreted as finite-dimensional approximations of the actual kernels. We are going to recover the unknowns $n_j$ and $m_k$ by the measurement of the temperature $u$ in $N=N_1+N_2$ different interior points $x_i\in (0,1)$, $i=1,\dots,N$, i.e., by the additional conditions $$\label{e1.8} u(x_i,t)=h_i(t),\quad i=1,\dots ,N,\; t>0\,,$$ where $h_i$ are given functions. Summing up, the relations \eqref{e1.4}--\eqref{e1.8} form the inverse problem for $n$ and $m$. \section{Application of Laplace transform} Applying the Laplace transform to the equation \eqref{e1.4} with initial condition \eqref{e1.5} and taking in consideration \eqref{e1.7} we obtain \label{e2.1} \begin{aligned} &\beta(x)[p U(x,p)-\varphi(x)]+p\sum_{j=1}^{N_1}N_j(p)\nu_j(x)U(x,p) \\ &=\frac{\partial}{\partial x}(\lambda(x)U_x(x,p)) -\sum_{k=1}^{N_2}M_k(p)\frac{\partial}{\partial x}(\mu_k(x)U_x(x,p)) +R(x,p) \end{aligned} where $U(x,p)= \mathcal{L}u(x,t)=\int_0^\infty e^{-pt}u(x,t)\,dt, \, \mathop{\rm Re} p>\sigma,\; N_j=\mathcal{L}n_j,\; M_k=\mathcal{L}m_k, \; R=\mathcal{L}r.$ Here $\sigma$ is taken so that the images of the Laplace transform exist in a half-plane $\mathop{\rm Re} p>\sigma$. In the sequel we will study the direct and inverse problems in the Laplace domain and show the existence and uniqueness for these problems in a half-plane $\mathop{\rm Re} p>\sigma$ with sufficiently large $\sigma$. The boundary conditions \eqref{e1.6} are transformed to $$\label{e2.2} U(0,p)=F_1(p),\quad U(1,p)=F_2(p),\quad \mathop{\rm Re} p>\sigma$$ where $F_j=\mathcal{L}f_j$, $j=1,2$. The goal of this section is to rewrite the problem for $U$ in a form of a system of integral equations. To this end we represent equation \eqref{e2.1} in the form \label{e2.3} \begin{aligned} &(LU)(x,p)= p\sum_{j=1}^{N_1}N_j(p)\nu_j(x)U(x,p) + \sum_{k=1}^{N_2}M_k(p)\frac{\partial}{\partial x}(\mu_k(x)U_x(x,p))\\ &\quad -R(x,p)-\beta(x)\varphi(x) \end{aligned} introducing the differential operator $% 2.4 (LU)(x,p)=\frac{\partial}{\partial x}(\lambda(x)U_x(x,p))-\beta(x)pU(x,p),\quad x\in(0,1).$ Let us denote by $G(x,y,p)$ the Green function of operator $L$ with homogeneous boundary conditions \begin{gather} \label{e2.5} L_y G(x,y,p)=\delta(x,y),\quad x\in(0,1),y\in(0,1)\, , \\ G(x,0,p)=G(x,1,p)=0\, . \label{e2.6} \end{gather} Then the solution of \eqref{e2.3} is given by \label{e2.7} \begin{aligned} U(x,p)&= \sum_{j=1}^{N_1}N_j(p)\int_0^1 G(x,y,p)\nu_j(y)pU(y,p)\,dy \\ &\quad +\sum_{k=1}^{N_2}M_k(p)\int_0^1 G(x,y,p)\frac{\partial}{\partial y}\big(\mu_k(y)U_y(y,p)\big)\,dy-F(x,p) \end{aligned} where \label{e2.8} \begin{aligned} F(x,p)&=\int_0^1 G(x,y,p)[\beta(y)\varphi(y)+R(y,p)]\,dy \\ &\quad + \lambda(0)G_y(x,0,p)F_1(p)-\lambda(1)G_y(x,1,p)F_2(p)\, . \end{aligned} Integrating the integrals in the second sum of \eqref{e2.7} by parts and observing \eqref{e2.6} we have \label{e2.9} \begin{aligned} U(x,p)&= \sum_{j=1}^{N_1}pN_j(p)\int_0^1 G(x,y,p)\nu_j(y)U(y,p)\,dy \\ &\quad -\sum_{k=1}^{N_2}M_k(p)\int_0^1 G_y(x,y,p)\mu_k(y)U_y(y,p)\,dy - F(x,p). \end{aligned} Further, differentiating \eqref{e2.7} with respect to $x$ we obtain the equation for $U_x(x,p)$ \label{e2.10} \begin{aligned} U_x(x,p)&= \sum_{j=1}^{N_1}pN_j(p)\int_0^1 G_x(x,y,p)\nu_j(y)U(y,p)\,dy \\ &\quad + \sum_{k=1}^{N_2}M_k(p)\int_0^1 G_x(x,y,p)\frac{\partial}{\partial y}(\mu_k(y)U_y(y,p))\,dy - F_x(x,p). \end{aligned} We split the second integral in \eqref{e2.10} into two parts, from $0$ to $x$ and from $x$ to $1$, and integrate them by parts. Taking into consideration the equalities $G_x(x,0,p)=G_x(x,1,p)=0, \, 00$. \end{enumerate} From 3 and the initial condition \eqref{e1.5} we have $pU(x,p)\to \varphi(x)$ and $pU_x(x,p)\to \varphi'(x)$ as $\mathop{\rm Re}p\to\infty$. Using these asymptotic relations, the items 1, 2 above and the assertions \eqref{e4.12} and \eqref{e4.15} of Lemmas \ref{lem1} and \ref{lem2} below we obtain the equalities $$\label{e3.3} -\sum_{k=1}^{N_1}n_k(0)\frac{1}{\beta(x_i)}\nu_k(x_i)\varphi(x_i) =\lim_{\mathop{\rm Re}p\to\infty} p^2[H_i(p)+F(x_i,p)]\, ,$$ for $i=1,\dots,N=N_1+N_2$, from \eqref{e3.2} in the process $\mathop{\rm Re}p\to\infty$, which form a system for initial values $n_k(0)$ of the unknowns $n_k$. \begin{remark} \label{rmk1} \rm Observing \eqref{e2.9} we see that $-F(x,p)$ is the Laplace transform of a function $\tilde{u}(x,t)$, which is the solution of the direct problem \eqref{e1.4} - \eqref{e1.6} with lacking kernels $(n(x,t)= m(x,t)\equiv0)$. Therefore, according to the basic properties of the Laplace transform (see \cite{d1}), the limits in \eqref{e3.3} exist and are finite under conditions $$h_i(0)-\varphi(x_i)=0, \quad i=1,\dots,N$$ and $h_i-\tilde u(x_i,\cdot)$, $i=1,\dots,N$, are twice continuously differentiable. In particular, the relations $h_i(0)-\varphi(x_i)=0$ can be regarded as consistency conditions. Moreover, under these conditions $\lim_{\mathop{\rm Re}p\to\infty} p^2[H_i(p)+F(x_i,p)]=\lim_{t\to 0^+}(h_i'(t) -\frac{\partial}{\partial t}\tilde u(x_i,t))\, , \quad i=1,\dots,N\, .$ \end{remark} System \eqref{e3.3} has a unique solution provided \label{e3.4} \begin{aligned} &\mathop{\rm rank}\Big(\big(\frac{1}{\beta(x_i)}\nu_k(x_i)\varphi(x_i)\big) _{k=1,\dots,N_1} ,\lim_{\mathop{\rm Re}p\to\infty} p^2[H_i(p)+F(x_i,p)]\Big)_{i=1,\dots,N}\\ &=\mathop{\rm rank}\big(\frac{1}{\beta(x_i)}\nu_k(x_i)\varphi(x_i)\big) _{k,\, i =1,\dots,N_1} = N_1\, . \end{aligned} Thus, \eqref{e3.4} is a necessary condition for the inverse problem to have a unique solution with the properties 1--3. The system \eqref{e3.2} suggests that the kernels $n_k$ and $m_l$ can be determined simultaneously with higher smoothness in $n_k$ than in $m_l$. Therefore we define $$\label{e3.5} Q_k(p) = pN_k(p)-n_k(0) = \mathcal L (n_k')$$ and derive a fixed-point system for $Q_k,M_l$. Observing \eqref{e3.3}, \eqref{e3.5} and having Lemma \ref{lem2} in mind we obtain from \eqref{e3.2} the system \label{e3.6} \begin{aligned} &\sum_{k=1}^{N_1}Q_k(p)\frac{1}{\beta(x_i)}\nu_k(x_i)\varphi(x_i) +\sum_{l=1}^{N_2}M_l(p)\frac{1}{\beta(x_i)}(\mu_l(y)\varphi'(y))'\Big|_{y=x_i} \\ &=\sum_{k=1}^{N_1}[Q_k(p)+n_k(0)]\Big[\int_0^1 pG(x_i,y,p)\nu_k(y)[pU(y,p)-\varphi(y)]dy \\ &\quad +\int_0^1 pG(x_i,y,p)\nu_k(y)\varphi(y)\,dy + \frac{1}{\beta(x_i)}\nu_k(x_i)\varphi(x_i)\Big] \\ &\quad -\sum_{l=1}^{N_2}M_l(p)\Big[\int_0^1 pG_y(x_i,y,p)\mu_l(y)[pU_y(y,p)-\varphi'(y)]\,dy \\ &\quad -\int_0^1 pG(x_i,y,p)(\mu_l(y)\varphi'(y))'\,dy - \frac{1}{\beta(x_i)}(\mu_l(y)\varphi'(y))'\Big|_{y=x_i}\Big] \\ &\quad - p^2[H_i(p)+F(x_i,p)] +\lim_{\mathop{\rm Re}q\to\infty} q^2[H_i(q)+F(x_i,q)],\quad i=1,\dots,N. \end{aligned} In view of assertion \eqref{e4.15} of Lemma \ref{lem2} below and the relation $pU\to \varphi$ as $\mathop{\rm Re}p\to\infty$ the first row in \eqref{e3.6} is the principal part of this system. Therefore we introduce the matrix $\Gamma=(\gamma_{ik})_{i,k=1,\dots,N}$ related to this principal part, where $$\label{e3.7} \gamma_{ik}=\begin{cases} \frac{1}{\beta(x_i)}\nu_k(x_i)\varphi(x_i),& k=1,\dots,N_1,\\ \frac{1}{\beta(x_i)}(\mu_{k-N_1}(y)\varphi'(y))'\big|_{y=x_i},& k=N_{1}+1,\dots,N \end{cases}$$ and assume $\det \Gamma \ne 0$. Further, we introduce the unified notation for the unknowns $$\label{e3.8} Z_k = \begin{cases} Q_k,& k=1,\dots,N_1, \\ M_{k-N_1},& k=N_{1}+1,\dots,N, \end{cases}$$ $Z=(Z_1,\dots,Z_N)$ and vanishing with $\mathop{\rm Re}p\to\infty$ functions $$\label{e3.9} B^0[Z](x,p)=pU[Z](x,p)-\varphi(x)\, ,\quad B^1[Z](x,p)= pU_x[Z](x,p)-\varphi'(x)\, ,$$ where $U[Z](x,p)$ is the Laplace transform of the $Z$-dependent solution of the direct problem. Now system \eqref{e3.6} can be written in the fixed-point form $$\label{e3.10} Z=\Gamma^{-1}{\mathcal{F}}(Z)\, ,$$ where ${\mathcal{F}}(Z)=({\mathcal{F}}_1(Z),\dots,{\mathcal{F}}_N(Z))$, \label{e3.11} \begin{aligned} &{\mathcal{F}}_i[Z](p)\\ &=\sum_{k=1}^{N_1}Z_k(p)\Big[\int_0^1 pG(x_i,y,p)\nu_k(y)B^0[Z](y,p)\,dy \\ &\quad + \int_0^1 pG(x_i,y,p)\nu_k(y)\varphi(y)\,dy+ \frac{1}{\beta(x_i)}\nu_k(x_i)\varphi(x_i)\Big] \\ &\quad + \sum_{k=N_{1}+1}^{N}Z_k(p)\Big[-\int_0^1 pG_y(x_i,y,p)\mu_{k-N_1}(y)B^1[Z](y,p)\,dy \\ &\quad + \int_0^1 pG(x_i,y,p)(\mu_{k-N_1}(y)\varphi'(y))'\,dy + \frac{1}{\beta(x_i)}(\mu_{k-N_1}(x)\varphi'(x))'\Big|_{x=x_i}\Big] \\ &\quad + \sum_{k=1}^{N_1}n_k(0)\int_0^1 pG(x_i,y,p)\nu_k(y)B^0[Z](y,p)\,dy + \widehat{\Psi}_i(p),\quad i=1,\dots,N \end{aligned} and \label{e3.12} \begin{aligned} \widehat{\Psi}_i(p) &= \sum_{k=1}^{N_1}n_k(0)\Big[\int_0^1 pG(x_i,y,p)\nu_k(y)\varphi(y)\,dy + \frac{1}{\beta(x_i)}\nu_k(x_i)\varphi(x_i)\Big] \\ &\quad -p^2[H_i(p)+F(x_i,p)] + \lim_{\mathop{\rm Re}q\to\infty}q^2[H_i(q)+F(x_i,q)]. \end{aligned} For future analysis we need a proper fixed-point system for the quantities $B^0[Z]$ and $B^1[Z]$, too. From \eqref{e2.9} in view of the definitions of $Z$ and $B^0[Z]$ we have \label{e3.13} \begin{aligned} &B^0[Z](x,p)\\ &= \sum_{k=1}^{N_1}Z_k(p)\int_0^1 G(x,y,p)\nu_k(y)[B^0[Z](y,p)+\varphi(y)]\,dy \\ &\quad - \sum_{k=N_{1}+1}^{N}Z_k(p)\int_0^1 G_y(x,y,p)\mu_{k-N_1}(y)[B^1[Z](y,p)+\varphi'(y)]\,dy \\ &\quad + \sum_{k=1}^{N_1}n_k(0)\int_0^1 G(x,y,p)\nu_k(y)B^0[Z](y,p)\,dy + \Phi^0(x,p) \end{aligned} with $$\label{e3.14} \Phi^0(x,p) = \sum_{k=1}^{N_1}n_k(0)\int_0^1 G(x,y,p)\nu_k(y)\varphi(y)\,dy - pF(x,p) - \varphi(x).$$ From \eqref{e2.12} we obtain \begin{align*} B^1[Z](x,p)&= \sum_{k=1}^{N_1}Z_k(p)\int_0^1 G_x(x,y,p)\nu_k(y)[B^0[Z](y,p)+\varphi(y)]\,dy \\ &\quad + \sum_{k=N_{1}+1}^{N}Z_k(p)\Big\{\frac{\mu_{k-N_1}(x)}{\lambda(x)} B^1[Z](x,p)\\ &\quad - \int_0^1G_xy(x,y,p)\mu_{k-N_1}(y)[B^1[Z](y,p) +\varphi'(y)]\,dy\Big\} \\ &\quad +\sum_{k=1}^{N_1}n_k(0)\int_0^1 G_x(x,y,p)\nu_k(y)B^0[Z](y,p)\,dy \\ &\quad +\sum_{k=N_{1}+1}^{N}Z_k(p)\frac{\mu_{k-N_1}(x)\varphi'(x)}{\lambda(x)} + \widetilde{\Phi}^1(x,p) \end{align*} with $$% \label{e3.16} \widetilde{\Phi}^1(x,p) = \sum_{k=1}^{N_1}n_k(0)\int_0^1 G_x(x,y,p)\nu_k(y)\varphi(y)\,dy - pF_x(x,p) - \varphi'(x).$$ For the function $B^0[Z]$, which in contrast to $B^1[Z]$ does not contain a space derivative of $U$, we need a certain higher regularity in the time variable. To this end we will assume that the free term $\Phi^0$ can be decomposed as follows $$\label{e3.17} \Phi^0(x,p) = B^{0,0}(x,p) + \widetilde{\Phi}(x,p)\, ,$$ where $|B^{0,0}(x,p)|\leq\frac{\rm Const}{|p|}$ and $|\widetilde{\Phi}(x,p)|\leq\frac{\rm Const}{|p|^\alpha}$ with some $\alpha>1$ for $\mathop{\rm Re}p>\sigma_0$, $x\in[0,1]$. \begin{remark} \label{rmk2} \rm To clarify under what conditions the decomposition \eqref{e3.17} is valid, let us consider separately two addends \begin{gather*} \Phi^0_1(x,p) = \sum_{k=1}^{N_1}n_k(0)\int_0^1 G(x,y,p)\nu_k(y)\varphi(y)\,dy \, ,\\ \Phi^0_2(x,p) = - pF(x,p) - \varphi(x) \end{gather*} in the formula of the function $\Phi^0(x,p)$. Assuming $\nu(x), \varphi(x)\in C^1[0,1]$, the first addend $\Phi^0_1(x,p)$ satisfies \eqref{e3.17} in view of the assertion \eqref{e4.15} of Lemma \ref{lem2}. Indeed, due to \eqref{e4.15} we have $\Phi^0_1(x,p) =-\frac{1}{p}\sum_{k=1}^{N_1}\frac{\nu_k(x) \varphi(x)}{\beta(x)}+\widetilde{\Phi}_1(x,p)\, ,$ where $|\widetilde{\Phi}_1(x,p)|\le {\rm const}/|p|^{3/2}$. Next, as $$\mathcal{L}^{-1}\Phi^0_2(x,p)=\mathcal{L}^{-1}(-pF(x,p)-\varphi(x)) = \frac{\partial}{\partial t}\tilde{u}(x,t),$$ where $\tilde{u}(x,t)$ is the solution of the direct problem \eqref{e1.4} - \eqref{e1.6} without kernels $m$ and $n$ (see Remark \ref{rmk1}), and $\mathcal{L}^{-1}$ denotes the inverse Laplace transform, then by well-known properties of the Laplace transform (cf \cite{d1}) the addend $\Phi^0_2(x,t)$ satisfies the condition \eqref{e3.17} with $\alpha=2$, if $\tilde{u}(x,t)$ is twice continuously differentiable with respect to $t$. \end{remark} We split $B^0[Z]$ into the sum $$\label{e3.18} B^0[Z](x,p) = B^{0,0}(x,p) + B^{0,1}[Z](x,p)\,,$$ where for $B^{0,1}$ we will require that $|B^{0,1}[Z](x,p)|\,\le\,\frac{\rm Const}{|p|^\alpha}$ for $\mathop{\rm Re}p>\sigma_0,\, x\in[0,1]$. From \eqref{e3.13} and \eqref{e2.12} in view of \eqref{e3.17}, \eqref{e3.18} and the definitions of $Z$ and $B^1[Z]$ we deduce the following fixed-point equation for the vector $B[Z]=(B^{0,1}[Z],B^1[Z])$: $$\label{e3.19} B[Z] = A[Z]B[Z] + b[Z]\, ,$$ where $A[Z]=(A^0[Z],A^1[Z])$ is the $Z$-dependent linear operator of $B$ with the components \label{e3.20} \begin{aligned} (A^{0}[Z]B)(x,p) &= \sum_{k=1}^{N_1}(Z_k(p)+n_k(0))\int_0^1 G(x,y,p)\nu_k(y)B^{0,1}(y,p)\,dy\\ &\quad - \sum_{k=N_{1}+1}^{N}Z_k(p)\int_0^1 G_y(x,y,p)\mu_{k-N_1}(y)B^1(y,p)\,dy, \end{aligned} \label{e3.21} \begin{aligned} &(A^{1}[Z]B)(x,p)\\ &= \sum_{k=1}^{N_1}(Z_k(p)+n_k(0))\int_0^1 G_x(x,y,p)\nu_k(y)B^{0,1}(y,p)\,dy \\ &\quad +\sum_{k=N_{1}+1}^{N}Z_k(p)\Big[ \frac{\mu_{k-N_1}(x)}{\lambda(x)}B^1(x,p) -\int_0^1 G_{xy}(x,y,p)\mu_{k-N_1}(y)B^1(y,p)\,dy\Big] \end{aligned} and $b[Z]=(b^0[Z],b^1[Z])$ is the $Z$-dependent $B$-free term with the components \label{e3.22} \begin{aligned} b^0[Z](x,p) &= \sum_{k=1}^{N_1}Z_k(p)\int_0^1 G(x,y,p)\nu_k(y)[B^{0,0}(y,p)+\varphi(y)]\,dy \\ &\quad - \sum_{k=N_{1}+1}^{N}Z_k(p)\int_0^1 G_y(x,y,p)\mu_{k-N_1}(y)\varphi'(y)\,dy + \Phi^{0,1}(x,p), \end{aligned} \label{e3.23} \begin{aligned} &b^1[Z](x,p)\\ &=\sum_{k=1}^{N_1}Z_k(p)\int_0^1 G_x(x,y,p)\nu_k(y)[B^{0,0}(y,p)+\varphi(y)]\,dy \\ &\quad + \sum_{k=N_{1}+1}^{N}Z_k(p)\Bigl[ \frac{\mu_{k-N_1}(x)\varphi'(x)}{\lambda(x)}-\int_0^1 G_{xy}(x,y,p)\mu_{k-N_1}(y)\varphi'(y)\,dy\Bigl]+\Phi^1(x,p) \end{aligned} and $$\label{e3.24} \Phi^{0,1}(x,p) = \sum_{k=1}^{N_1}n_k(0)\int_0^1 G(x,y,p)\nu_k(y)B^{0,0}(y,p)\,dy + \widetilde{\Phi}(x,p),$$ \label{e3.25} \begin{aligned} \Phi^1(x,p) &= \sum_{k=1}^{N_1}n_k(0)\int_0^1 G_x(x,y,p)\nu_k(y)[B^{0,0}(y,p)+\varphi(y)]\,dy \\ &\quad - pF_x(x,p)-\varphi'(x). \end{aligned} \section{Functional spaces and estimation of Green function} To analyse the direct and inverse problems we define the spaces $$\label{e4.1} \mathcal{A}_{\gamma,\sigma}=\{V: V(p) \mbox{ is holomorphic on }\mathop{\rm Re}p>\sigma, \,\|V\|_{\gamma,\sigma}<\infty\},$$ where $%\label{e4.2} \nonumber \|V\|_{\gamma,\sigma}=\sup_{\mathop{\rm Re}p>\sigma}|p|^\gamma|V(p)|$ and $$\label{e4.3} (\mathcal{A}_{\gamma,\sigma})^N = \{V=(V_1,\dots,V_N):V_k(p)\in\mathcal{A}_{\gamma,\sigma}, k=1,\dots,N\}$$ with the norm $$\label{e4.4} \nonumber \|V\|_{\gamma,\sigma} = \sum_{k=1}^N \|V_k\|_{\gamma,\sigma}\,,\quad V\in (\mathcal{A}_{\gamma,\sigma})^N.$$ We note that $\mathcal{A}_{\gamma,\sigma}\subset\mathcal{A}_{\gamma,\sigma'}$, $(\mathcal{A}_{\gamma,\sigma})^N\subset(\mathcal{A}_{\gamma,\sigma'})^N$ and $\|\cdot\|_{\gamma,\sigma'}\leq\|\cdot\|_{\gamma,\sigma}$ if $\sigma'>\sigma$. Let $\alpha$ be a real number such that $$\label{e4.5} 1<\alpha<\frac{3}{2}\, .$$ Moreover, let $c=(c_1,\dots,c_N)$ be a given vector. We will search the solution $Z=(Z_1,\dots,Z_N)$ of \eqref{e3.10} on the space $$\label{e4.6} \mathcal{M}_{c,\sigma} = \{Z:Z=\frac{c}{p}+V(p),\,V\in(\mathcal{A}_{\alpha,\sigma})^N\}\,.$$ Furthermore, we introduce the spaces of $x$- and $p$-dependent functions $$\label{e4.7} \mathcal{B}_{\gamma,\sigma}=\{ F(x,p): F(x,\cdot)\in \mathcal{A}_{\gamma,\sigma}\mbox{ for } x\in [0,1]\,,\; F(\cdot,p)\in C[0,1] \mbox{ for } \mathop{\rm Re}p>\sigma \}$$ with the norms $$\label{e4.8} \nonumber \|F\|_{\gamma,\sigma}=\max_{0\leq x\leq 1}\sup_{\mathop{\rm Re}p>\sigma}|p|^\gamma|F(x,p)|.$$ Let $\alpha'$ be a given number such that $$\label{e4.9} \alpha<\alpha'<\frac{3}{2}\, .$$ We are going to solve the equation \eqref{e3.19} for the pair $B=(B^{0,1},B^1)$ in the space $\mathcal{B}_\sigma = \mathcal{B}_{\alpha',\sigma}\times\mathcal{B}_{1,\sigma}$ with the norm $$\label{e4.10} \|B\|_\sigma=\|B^{0,1}\|_{\alpha',\sigma} + \|B^1\|_{1,\sigma}\, .$$ For estimating the Green function we use the following lemmas proved in \cite{j5}. \begin{lemma} \label{lem1} Let $\lambda,\beta \in C^{2}[0,1]$ and $\lambda,\beta>0$ in $[0,1]$. Then \begin{gather} K_1=\sup_{0\leq x\leq 1,\; \mathop{\rm Re}p>0}|p|\int_0^1 |G(x,y,p)|\,dy<\infty \, , \label{e4.11}\\ K_2=\sup_{0\leq x\leq 1\,\; \mathop{\rm Re}p>0}\sqrt{|p|}\int_0^1 |G_x(x,y,p)|\,dy<\infty\, ,\label{e4.12}\\ K_3=\sup_{0\leq x\leq 1,\; \mathop{\rm Re}p>0}\sqrt{|p|}\int_0^1 |G_y(x,y,p)|\,dy<\infty \, ,\label{e4.13}\\ K_4=\sup_{0\leq x\leq 1,\; \mathop{\rm Re}p>0}\int_0^1 |G_{xy}(x,y,p)|\,dy<\infty\, . \label{e4.14} \end{gather} \end{lemma} \begin{lemma} \label{lem2} Let $\lambda,\beta \in C^{2}[0,1]$, $\lambda,\beta>0$ in $[0,1]$ and $V\in C^{1}[0,1]$. Then $$\label{e4.15} \sup_{\mathop{\rm Re}p>0}\Big|\sqrt{|p|}\Big(\int_0^1 pG(x,y,p)V(y)\,dy + \frac{V(x)}{\beta(x)}\Big)\Big|\leq K_5\|V\|_{C^{1}[0,1]}$$ for all $x\in (0,1)$, where $K_5$ is independent of $x$ in every closed subinterval of $(0,1)$, and $$\label{e4.16} \sup_{\mathop{\rm Re}p>0}\Big|\sqrt{|p|}\Big(\int_0^1 G_{xy}(x,y,p)V(y)\,dy - \frac{V(x)}{\lambda(x)}\Big)\Big|\leq K_6\|V'\|_{C^{1}[0,1]}$$ for all $x\in [0,1]$, where $K_6$ is independent of $x$. \end{lemma} \section{Analysis of direct problem} Let us assume the following hypotheses: $$\label{e5.1} \parbox{0.8\linewidth}{ \lambda,\beta\in C^2[0,1], \lambda,\beta>0; \Phi^0 given by \eqref{e3.14} admits the decomposition \eqref{e3.17}, where B^{0,0} \in \mathcal{B}_{1,\sigma_0} and \widetilde{\Phi}\in\mathcal{B}_{\alpha',\sigma_0} with some \sigma_0\geq 1 and \alpha,\alpha' that satisfy \eqref{e4.5} \eqref{e4.9}; \Phi^1 given by \eqref{e3.25} belongs to \mathcal{B}_{1,\sigma_0}; \nu_k\in C[0,1], k=1,\dots,N_1, \mu_l\in C^{1}[0,1], l=1,\dots,N_2; \varphi\in C^{2}[0,1].}$$ \begin{lemma} \label{lem3} Let the assumptions {\rm \eqref{e5.1}} hold. If $Z=\frac cp +V\in{\mathcal{M}}_{c,\sigma}$ then the vector function $b[Z]=(b^0[Z],b^1[Z])$, given by \eqref{e3.22}, \eqref{e3.23}, belongs to $\mathcal{B}_{\sigma_0}$ and satisfies the estimate $$\label{e5.2} \|b[Z]\|_{\sigma}\leq C_1\big[1 + \frac{1}{\sigma^{\frac{3}{2}-\alpha'}}\big(|c| +\frac{\|V\|_{\alpha,\sigma}}{\sigma^{\alpha-1}}\big)\big]$$ with $\sigma\ge \sigma_0$, where $C_1$ is a constant and $|c|=\sum_{k=1}^N |c_k|.$ Moreover, for every $\sigma\geq\sigma_0$ and $Z^1=\frac{c}{p}+V^1,\, Z^2=\frac{c}{p}+V^2 \in \mathcal{M}_{c,\sigma}$ the difference $b[Z^1]-b[Z^2]$ fulfils the estimate $$\label{e5.3} \|b[Z^1]-b[Z^2]\|_{\sigma}\leq C_2\,\frac{1}{\sigma^{\alpha-\alpha'+\frac{1}{2}}} \|V^1 - V^2\|_{\alpha,\sigma}$$ with a constant $C_2$. \end{lemma} \begin{lemma} \label{lem4} Let the assumptions {\rm \eqref{e5.1}} hold. If $Z=\frac cp +V\in{\mathcal{M}}_{c,\sigma}$ then the linear operator $A[Z]=(A^0[Z],A^1[Z])$, defined by \eqref{e3.20}, \eqref{e3.21}, is bounded in $\mathcal{B}_{\sigma}$ and satisfies the estimate $$\label{e5.4} \|A[Z]\|_{\mathcal{B}_\sigma\to \mathcal{B}_\sigma} \leq C_3\big[\frac{|c|}{\sigma} +\frac{\|V\|_{\alpha,\sigma}}{\sigma^{\alpha}} +\frac{1}{\sigma^{\alpha'-\frac{1}{2}}}\big]$$ for any $\sigma\geq\sigma_0$ with a constant $C_3$. Moreover, taking $Z^1=\frac{c}{p}+V^1,\, Z^2=\frac{c}{p}+V^2 \in \mathcal{M}_{c, \sigma},$ the estimate for difference $$\label{e5.5} \|(A[Z^1]-A[Z^2])\|_{\mathcal{B}_\sigma\to \mathcal{B}_\sigma}\leq C_4\frac{1}{\sigma^{\alpha}} \|V^1 - V^2\|_{\alpha,\sigma}$$ holds for any $\sigma\geq\sigma_0$ with a constant $C_4$. \end{lemma} Proofs of Lemmas \ref{lem3} and \ref{lem4} are presented in the appendix of this paper. Due to Lemmas \ref{lem3}, \ref{lem4} and the contraction principle equation \eqref{e3.19} has a unique solution $B=B[Z]\in \mathcal{B}_\sigma$ provided $Z=\frac cp +V\in{\mathcal{M}}_{c,\sigma}$ and $\sigma\geq\sigma_0$ satisfy the relation $$\label{e5.6} \eta(Z,\sigma)\, :=\, \frac{|c|}{\sigma} +\frac{\|V\|_{\alpha,\sigma}}{\sigma^{\alpha}} +\frac{1}{\sigma^{\alpha'-\frac{1}{2}}}< \frac{1}{C_3}\, .$$ Furthermore, from \eqref{e3.19} we have $\|B[Z]\|_\sigma\le (1-\|A[Z]\|_{\mathcal{B}_\sigma\to \mathcal{B}_\sigma})^{-1}\|b[Z]\|_\sigma$. This in view of \eqref{e5.2}, \eqref{e5.4} and \eqref{e5.6} yields the estimate $$\label{e5.7} \|B[Z]\|_\sigma \leq C_1\big\{1 - C_3\big[\frac{|c|}{\sigma} +\frac{\|V\|_{\alpha,\sigma}}{\sigma^{\alpha}} +\frac{1}{\sigma^{\alpha'-\frac{1}{2}}}\big]\big\}^{-1} \big[1 + \frac{1}{\sigma^{\frac{3}{2}-\alpha'}}\big(|c| +\frac{\|V\|_{\alpha,\sigma}}{\sigma^{\alpha-1}}\big)\big]$$ for the solution of \eqref{e3.19}. Next let us find an estimate for $B[Z^1]-B[Z^2]$. Let $\sigma\geq\sigma_0$ and $Z^1 = \frac{c}{p}+V^1$, $Z^2=\frac{c}{p}+V^2$ be such that \eqref{e5.6} is valid for $V$ replaced by $V^1$ and $V^2$ i.e. $\eta(Z^j,\sigma)<\frac{1}{C_3}$, $j=1,2$. Subtracting equation \eqref{e3.19} for $Z=Z^2$ from the corresponding equation for $Z=Z^1$ we have $B[Z^1]-B[Z^2]=A[Z^2](B[Z^1]-B[Z^2])+(A[Z^1]-A[Z^2])B[Z^1] +b[Z^1]-b[Z^2].$ This implies \begin{align*} &\|B[Z^1]-B[Z^2]\|_\sigma\\ &\le (1-\|A[Z^2]\|_{\mathcal{B}_\sigma\to \mathcal{B}_\sigma})^{-1} [\|A[Z^1]-A[Z^2]\|_{\mathcal{B}_\sigma\to \mathcal{B}_\sigma}\|B[Z^1]\|_\sigma+\|b[Z^1]-b[Z^2]\|_\sigma]. \end{align*} Using in this relation the estimates \eqref{e5.3} - \eqref{e5.7} we obtain \label{e5.8} \begin{aligned} & \|B[Z^1]-B[Z^2]\|_{\sigma}\\ &\leq C_{5}\big[1 -C_3\big(\frac{|c|}{\sigma} +\frac{\|V^2\|_{\alpha,\sigma}}{\sigma^{\alpha}} +\frac{1}{\sigma^{\alpha'-\frac{1}{2}}}\big)\big]^{-1}\\ &\quad \times\big\{\frac{1}{\sigma^{\alpha-\alpha'+\frac{1}{2}}}+ \frac{1}{\sigma^{\alpha}}\big[1 - C_3\big(\frac{|c|}{\sigma} +\frac{\|V^1\|_{\alpha,\sigma}}{\sigma^{\alpha}} +\frac{1}{\sigma^{\alpha'-\frac{1}{2}}}\big)\big]^{-1}\\ &\quad \times\big[1+\frac{1}{\sigma^{\frac{3}{2}-\alpha'}} \big(|c|+\frac{\|V^1\|_{\alpha,\sigma}}{\sigma^{\alpha-1}}\big)\big] \big\}\|V^1 - V^2\|_{\alpha,\sigma} \end{aligned} with a constant $C_5$. Summing up, we have proved the following theorem. \begin{theorem} \label{thm1} Let the assumptions {\rm \eqref{e5.1}} hold. Then there exists a constant $C_3> 0$ depending on the data of equation \eqref{e3.19} such that for any $\sigma\geq\sigma_0$ and $Z=\frac{c}{p}+V\in \mathcal{M}_{c,\sigma}$, that satisfies the inequality \eqref{e5.6}, equation \eqref{e3.19} has a unique solution $B[Z]=(B^{0,1}[Z],B^1[Z])$ in $\mathcal{B}_{\sigma}$. This solution satisfies estimate \eqref{e5.7}. Moreover, for every $\sigma\geq\sigma_0$ and $Z^1=\frac{c}{p}+V^1$, $Z^2=\frac{c}{p}+V^2\in \mathcal{M}_{c,\sigma}$ such that $\eta(Z^j,\sigma)< \frac{1}{C_3}$, $j=1,2$, the difference $B[Z^1]-B[Z^2]$ fulfils estimate \eqref{e5.8}. \end{theorem} \section{Existence and uniqueness for inverse problem} In this section we study the inverse problem in the fixed-point form \eqref{e3.10} in the Laplace domain and thereupon infer a result for the inverse problem \eqref{e1.4} - \eqref{e1.8} in the time domain. Due to the decomposition \eqref{e3.18} the full $Z$-free term of the operator $\mathcal{F}=({\mathcal{F}}_1,\dots,{\mathcal{F}}_N)$ given by \eqref{e3.11} is $\Psi=(\Psi_1,\dots,\Psi_N)$, where $$\label{e6.1} \Psi_i(p) = \widehat{\Psi}_i(p) + \sum_{k=1}^{N_1}n_k(0)\int_0^1 pG(x_i,y,p)\nu_k(y)B^{0,0}(y,p)\,dy.$$\\ and $\widehat\Psi_i$ is defined in \eqref{e3.12} \begin{theorem} \label{thm2} Assume that {\rm \eqref{e5.1}} holds and $$\label{e6.2} \nu_k\in C^{1}[0,1],\, k=1,\dots,N_1, \quad \mu_l\in C^{2}[0,1],\, l=1,\dots,N_2\quad \varphi\in C^{3}[0,1].$$ Moreover, let $\det\Gamma\ne 0$ for $\Gamma$, given by \eqref{e3.7}, and $$\label{e6.3} \Psi=\frac{d}{p} +Y\in\mathcal{M}_{d,\sigma_0}$$ with some $d\in\mathrm{R}^N$. Then there exists $\sigma_1\ge \sigma_0$ such that equation \eqref{e3.10} has a unique solution $Z=\frac cp +V\in \mathcal{M}_{c,\sigma_1}$. Here $c=\Gamma^{-1}d$. \end{theorem} \begin{proof} Setting $c=\Gamma^{-1}d$ and observing \eqref{e3.18}, problem \eqref{e3.10} in $\mathcal{M}_{c,\sigma}$ is equivalent to the following equation for $V$ in $(\mathcal{A}_{\alpha,\sigma})^N$, $$\label{e6.4} V = F(V)\, ,$$ where $F = \Gamma^{-1}F_1$ and \label{e6.5} \begin{aligned} &(F_{1}(V))_i(p)\\ &= \sum_{k=1}^{N_1}(\frac{c_k}{p}+V_k(p))\Big\{\int_0^1 pG(x_i,y,p)\nu_k(y)[B^{0,0}(y,p)\\ &\quad +B^{0,1}[Z](y,p)]dy+ \int_0^1 pG(x_i,y,p)\nu_k(y)\varphi(y)\,dy + \frac{1}{\beta(x_i)}\nu_k(x_i)\varphi(x_i)\Big\}\\ &\quad +\sum_{k=N_{1}+1}^N (\frac{c_k}{p}+V_k(p)) \Big\{-\int_0^1 pG_y(x_i,y,p)\mu_{k-N_1}(y)B^{1}[Z](y,p)\,dy \\ &\quad +\int_0^1 pG_y(x_i,y,p)(\mu_{k-N_1}(y)\varphi'(y))'\,dy +\frac{1}{\beta(x_i)}(\mu_{k-N_1}(x)\varphi'(x))'\Big|_{x=x_i}\Big\} \\ & \quad +\sum_{k=1}^{N_1}n_k(0)\int_0^1 pG(x_i,y,p)\nu_k(y) B^{0,1}[Z](y,p)\,dy + Y_i(p)\,, \;\;i=1,\dots,N. \end{aligned} We will prove the assertion of theorem using the fixed-point argument in the balls \begin{equation*} D_{\alpha,\sigma}(\rho) =\{V\in(\mathcal{A}_{\alpha,\sigma})^N: \|V\|_{\alpha,\sigma}\leq\rho\}. \end{equation*} Multiplying by $|p|^\alpha$ in \eqref{e6.5} and estimating we have \begin{align*} &|p|^{\alpha}|(F_{1}(V))_i(p)|\\ &\leq \sum_{k=1}^{N_1}\Big(\frac{|c_k|}{|p|^{\frac{3}{2}-\alpha}} +\frac{|p|^{\alpha}|V_k(p)|}{\sqrt{|p|}}\Big) \Big\{|p|\int_0^1 |G(x_i,y,p)|dy\\ &\quad \times \frac{\|\nu_k(y)\|_{C[0,1]}}{\sqrt{|p|}} \Big[|p|\max_{0\leq y\leq 1}|B^{0,0}(y,p)| +\frac{1}{|p|^{\alpha'-1}}|p|^{\alpha'}\max_{0\leq y\leq 1}|B^{0,1}[Z](y,p)|\Big] \\ &\quad +\sqrt{|p|}\Big|\int_0^1 pG(x_i,y,p)\nu_k(y)\varphi(y)\,dy + \frac{1}{\beta(x_i)}\nu_k(x_i)\varphi(x_i)\Big|\Big\} \\ &\quad +\sum_{k=N_{1}+1}^N \Big(\frac{|c_k|}{|p|^{\frac{3}{2}-\alpha}}+ \frac{|p|^{\alpha}|V_k(p)|}{|p|^{\frac{1}{2}}}\Big) \Big[\sqrt{|p|}\int_0^1 |G_y(x_i,y,p)|dy\\ &\times \|\mu_{k-N_1}\|_{C[0,1]}|p| \max_{0\leq y\leq 1}|B^{1}[Z](y,p)| \\ &\quad + \sqrt{|p|}\Big|\int_0^1 pG(x_i,y,p)(\mu_{k-N_1}(y)\varphi'(y))'dy +\frac{1}{\beta(x_i)} (\mu_{k-N_1}(x)\varphi'(x))'\Big|_{x=x_i}\Big|\Big] \\ &\quad +\sum_{k=1}^{N_1}|n_k(0)|\,|p|\int_0^1 |G(x,y,p)|dy\|\nu_k\|_{C[0,1]}\frac{1}{|p|^{\alpha'-\alpha}}|p|^{\alpha'} \max_{0\leq x\leq 1}|B^{0,1}[Z](x,p)| \\ &\quad +|p|^{\alpha}|Y_i(p)| \, ,\quad i=1,\dots,N\, . \end{align*} Using the assumptions \eqref{e5.1}, \eqref{e6.2}, \eqref{e6.3}, the assertions \eqref{e4.11}, \eqref{e4.13} and \eqref{e4.15} of Lemmas \ref{lem1}, \ref{lem2} and the definition of the norm $\|\cdot\|_{\gamma,\sigma}$ we obtain \begin{align*} &|p|^{\alpha}|(F_{1}(V))_i(p)| \\ &\leq \sum_{k=1}^{N_1}\Big(\frac{|c_k|}{|p|^{\frac{3}{2}-\alpha}} +\frac{\|V_k\|_{\alpha,\sigma}}{\sqrt{|p|}}\Big) \Big\{K_1\frac{1}{\sqrt{|p|}}\|\nu_k\|_{C[0,1]}\\ &\quad \times\big[\|B^{0,0}\|_{1,\sigma_0}+ \frac{\|B^{0,1}[Z]\|_{\alpha',\sigma}}{|p|^{\alpha'-1}}\big] +K_5\|\nu_k\varphi\|_{C^1[0,1]}\Big\} \\ &\quad +\sum_{k=N_{1}+1}^N(\frac{|c_k|}{|p|^{\frac{3}{2}-\alpha}} +\frac{\|V_k\|_{\alpha,\sigma}}{\sqrt{|p|}}) \Big(K_3\|\mu_{k-N_1}\|_{C[0,1]} \|B^1[Z]\|_{1,\sigma} \\ &\quad +K_5\|(\mu_{k-N_1}\varphi')'\|_{C^1[0,1]}\Big) +\sum_{k=1}^{N_1}|n_k(0)|\,K_1\|\nu_k\|_{C[0,1]} \frac{\|B^{0,1}[Z]\|_{\alpha',\sigma}}{|p|^{\alpha'-\alpha}} \\ &\quad+ \|Y_i(p)\|_{\alpha,\sigma_0}\, ,\quad i=1,\dots N \end{align*} for $\mathop{\rm Re}p>\sigma$, $\sigma\geq\sigma_0$, $x\in [0,1]$. Taking here the supremum over $\mathop{\rm Re}p>\sigma$, $x\in [0,1]$, observing the relation $|p|^\gamma>\sigma^\gamma$ for $\mathop{\rm Re}p>\sigma$, which holds in the cases $\gamma=3/2-\alpha$, $\alpha'-1$, $\alpha'-\alpha$ due to the assumed inequalities \eqref{e4.5} \eqref{e4.9}, and the inequality $1/\sigma^{\alpha'-1}\le 1$, which holds due to $\sigma\ge\sigma_0\ge 1$, we get \label{e6.6} \begin{aligned} & \|F_1(V)\|_{\alpha,\sigma}\\ &\leq C_{6}\big\{ \big(\frac{|c|}{\sigma^{\frac{3}{2}-\alpha}} +\frac{\|V\|_{\alpha,\sigma}}{\sqrt{\sigma}}\big) (\|B[Z]\|_{\sigma}+1)+ \frac{1}{\sigma^{\alpha'-\alpha}} \|B[Z]\|_{\sigma}\big\} +\|Y\|_{\alpha,\sigma_0},\quad \sigma\geq\sigma_0 \end{aligned} with a constant $C_6$ depending on the data of the problem. Further, let us suppose that $V\in D_{\alpha,\sigma}(\rho)$, where $\sigma$ and $\rho$ satisfy $$\label{e6.7} \eta_0(\rho,\sigma)\,:=\, \frac{|c|}{\sigma}+\frac{\rho}{\sigma^{\alpha}} +\frac{1}{\sigma^{\alpha'-\frac{1}{2}}}\, <\frac{1}{C_3}$$ and $\sigma\geq\sigma_0$. Then \eqref{e5.6} holds, hence we can apply estimate \eqref{e5.7} of Theorem \ref{thm1} for $\|B[Z]\|_\sigma$. Plugging \eqref{e5.7} into \eqref{e6.6} and estimating $\|V\|_{\alpha,\sigma}$ by $\rho$ we have %\label{e6.8} \begin{align*} \|F_1(V)\|_{\alpha,\sigma} &\leq C_{6}\Big\{ \big(\frac{|c|}{\sigma^{\frac{3}{2}-\alpha}} +\frac{1}{\sigma^{\alpha'-\alpha}}+\frac{\rho}{\sqrt{\sigma}}\big) C_1[1-\eta_0(\rho,\sigma)C_3]^{-1} \\ &\quad \times \big[1+\frac{1}{\sigma^{\frac{3}{2}-\alpha'}} (|c|+\frac{\rho}{\sigma^{\alpha-1}})\big] +\frac{|c|}{\sigma^{\frac{3}{2}-\alpha}} +\frac{\rho}{\sqrt{\sigma}}\Big\}+\|Y\|_{\alpha,\sigma_0}. \end{align*} From this inequality, due to the equality $F=\Gamma^{-1}F_1$, we see that for every $\rho>\rho_0 :=|\Gamma^{-1}|\|Y\|_{\alpha,\sigma_0}$ there exists $\sigma_2=\sigma_2(\rho)\geq \sigma_0$ such that the inequalities $\eta_0(\rho,\sigma)< 1/C_3$ and $\|FV\|_{\alpha,\sigma}\leq \rho$ hold for any $\sigma\geq \sigma_2(\rho)$. Consequently, $$\label{e6.9} F:D_{\alpha,\sigma}(\rho)\to D_{\alpha,\sigma}(\rho) \quad \mbox{for \rho>\rho_0 and \sigma\geq \sigma_2(\rho)}.$$ Next, we prove that $F$ is a contraction. From \eqref{e6.5} with $Z=\frac cp +V$ and $\widetilde Z=\frac cp +\widetilde V$ we have \label{e6.10} \begin{aligned} &\big(F_{1}(V)-F_1({\widetilde{V}})\big)_i(p)\\ &= \sum_{k=1}^{N_1}(V_k(p)-\widetilde{V_k}(p)) \Big\{\int_0^1 pG(x_i,y,p)\nu_k(y)\big[B^{0,0}(y,p) +B^{0,1}[Z](y,p)\big]dy\\ &\quad +\int_0^1 pG(x_i,y,p)\nu_k(y)\varphi(y)\,dy + \frac{1}{\beta(x_i)}\nu_k(x_i)\varphi(x_i)\Big\} \\ &\quad +\sum_{k=1}^{N_1}(\frac{c_k}{p}+\widetilde{V_k}(p)) \int_0^1 pG(x_i,y,p)\nu_k(y)[B^{0,1}[Z]-B^{0,1}[\widetilde{Z}]](y,p)\,dy \\ &\quad +\sum_{k=N_{1}+1}^N (V_k(p)-\widetilde{V_k}(p)) \Big\{-\int_0^1 pG_y(x_i,y,p)\mu_{k-N_1}(y)B^{1}[Z](y,p)\,dy \\ & \quad +\int_0^1 pG_y(x_i,y,p)(\mu_{k-N_1}(y)\varphi'(y))'\,dy +\frac{1}{\beta(x_i)}(\mu_{k-N_1}(x)\varphi'(x))'\Big|_{x=x_i}\Big\}\\ &\quad -\sum_{k=N_{1}+1}^N (\frac{c_k}{p}+\widetilde{V_k}(p)) \int_0^1 pG_y(x_i,y,p)\mu_{k-N_1}(y)[B^{1}[Z]- B^{1}[\widetilde{Z}]](y,p)\,dy \\ &\quad +\sum_{k=1}^{N_1}n_k(0)\int_0^1 pG(x_i,y,p)\nu_k(y) [B^{0,1}[Z]-B^{0,1}[\widetilde{Z}]](y,p)\,dy\, , \end{aligned} $i=1,\dots,N$. Performing similar operations as above in deriving \eqref{e6.6} we obtain from \eqref{e6.10} the estimate \begin{align*} &\|F_1(V)-F_1(\widetilde{V})\|_{\alpha,\sigma}\\ &\leq C_{7}\Big\{\frac{\|V-\widetilde{V}\|_{\alpha,\sigma}}{\sqrt{\sigma}} (\|B[Z]\|_{\sigma}+1)\\ &\quad + \big(\frac{|c|}{\sigma^{\frac{3}{2}-\alpha}}+ \frac{\|\widetilde{V}\|_{\alpha,\sigma}}{\sqrt{\sigma}}\big) \|B[Z]-B[\widetilde{Z}]\|_\sigma +\frac{1}{\sigma^{\alpha'-\alpha}}\|B[Z]-B[\widetilde{Z}]\|_\sigma\Big\} \end{align*} for $\sigma\geq \sigma_0$ with a constant $C_7$. Supposing that $V, \widetilde{V}\in D_{\alpha,\sigma}(\rho)$ with $\sigma\ge\sigma_0$ and $\rho$ such that \eqref{e6.7} hold by the estimates \eqref{e5.7} and \eqref{e5.8} of Theorem \ref{thm1} we have \begin{align*} &\|F_1(V)-F_1(\widetilde{V})\|_{\alpha,\sigma} \\ & \leq C_{7}\Big\{\frac{1}{\sqrt{\sigma}} \Big(1+\frac{C_1}{1-\eta_0(\rho,\sigma)C_3} \big[1+\frac{1}{\sigma^{\frac{3}{2}-\alpha'}}(|c|+\frac{\rho}{\sigma^{\alpha-1}}) \big]\Big) \\ & \quad+\big(\frac{|c|}{\sigma^{\frac{3}{2}-\alpha}}+\frac{\rho}{\sqrt{\sigma}}+ \frac{1}{\sigma^{\alpha'-\alpha}}\quad) \frac{C_{5}}{1-\eta_0(\rho,\sigma)C_3}\\ &\quad \times \Big[\frac{1}{\sigma^{\alpha-\alpha'+\frac{1}{2}}} +\frac{1}{\sigma^\alpha(1-\eta_0(\rho,\sigma)C_3)} \big(1+\frac{1}{\sigma^{\frac{3}{2}-\alpha'}}(|c|+ \frac{\rho}{\sigma^{\alpha-1}})\big)\Big]\Big\} \|V-\widetilde{V}\|_{\alpha,\sigma}\, . \end{align*} The coefficient of $\|V-\widetilde{V}\|_{\alpha,\sigma}$ on the right-hand side of this estimate approaches zero as $\sigma\to\infty$ for a fixed $\rho>0$. Hence, for every $\rho>0$ there exists $\sigma_3=\sigma_3(\rho)\ge\sigma_0$, such that the inequality $\eta_0(\rho,\sigma)<1/C_3$ holds and $F=\Gamma^{-1}F_1$ is a contraction in the ball $D_{\alpha,\sigma}(\rho)$ for $\rho>0$ and $\sigma\geq \sigma_3(\rho)$. This together with \eqref{e6.9} shows that equation \eqref{e6.4} has a unique solution $V$ in every ball $D_{\alpha,\sigma}(\rho)$, where $\rho>\rho_0$ and {$\sigma\geq \sigma_4(\rho)=\max(\sigma_2(\rho);\sigma_3(\rho))$}. This proves the existence assertion of theorem with $\sigma_1=\sigma_4(2\rho_0)$. It remains to prove that the solution of \eqref{e6.4} is unique in the whole space $(\mathcal{A}_{\alpha,\sigma_1})^N$. Suppose that \eqref{e6.4} has two solutions $V^1$ and $V^2$ in $(\mathcal{A}_{\alpha,\sigma_1})^N$. Let us define $\bar\rho:=\max(2\rho_0;\|V^1\|_{\alpha,\sigma_1}; \|V^2\|_{\alpha,\sigma_1})$ and $\bar\sigma:=\max(\sigma_1;\sigma_4(\bar\rho))$. Then we have $\|V^j\|_{\alpha,\sigma_1}\leq \bar\rho$, $j=1,2.$ Since the norm $\|.\|_{\alpha,\sigma}$ is nonincreasing with respect to $\sigma$ and $\bar\sigma\geq \sigma_1$, from this relation we derive $$\|V_j\|_{\alpha,\bar\sigma}\leq \bar\rho\;\Longrightarrow\; V^j\,\in\, D_{\alpha,\bar\sigma(\bar\rho)}\,,\quad j=1,2.$$ But due to $\bar\rho>\rho_0$ and $\bar\sigma\geq \sigma_4(\bar\rho)$, the uniqueness in the ball $D_{\alpha,\bar\sigma(\bar\rho)}$ has already been shown. Thus, $V^1=V^2$. Theorem \ref{thm2} is proved. \end{proof} Finally, applying the well-known results about the invertibility of the Laplace transform \cite{d1} we deduce the following corollary from Theorem \ref{thm2}. \begin{corollary} \label{coro1} Let conditions \eqref{e3.4} hold yielding the unique initial values $n_j(0)$ for the unknowns $n_j,\, k=1,\dots, N_1$ from system \eqref{e3.3}. Moreover, let the assumptions of Theorem \ref{thm2} be satisfied for the functions $\lambda_k,\, \mu_l,\, \varphi$ and the quantities $\Phi^0,\, \Phi^1,\, \Psi$ given by formulas \eqref{e3.14}, \eqref{e3.25}, \eqref{e6.1} with \eqref{e3.14}, \eqref{e2.8} in terms of the Laplace transforms $R,\, F_1,\, F_2,\, H_i$ of the data of inverse problem \eqref{e1.4} - \eqref{e1.8}. Then inverse problem \eqref{e1.4}--\eqref{e1.8} has the unique solution $(n,m)$ with coefficients $n_j$ and $m_k$ of the form \begin{gather*} n_j(t)=n_j(0)+c_jt+\frac{1}{2\pi i}\int_0^t\int_{\xi-i\infty}^{\xi+ i\infty} e^{\tau p} V_j(p)\,dp\,d\tau\, ,\quad k=1,\dots N_1\, , \\ m_k(t)=c_{k+N_1}+\frac{1}{2\pi i}\int_{\xi-i\infty}^{\xi+ i\infty} e^{tp} V_{k+N_1}(p)\,dp, ,\quad k=1,\dots N_2\, , \end{gather*} where $\xi>\sigma_1 >1$, $c=(c_1,\dots,c_N)\in \mathrm{R}^N$, $c=\Gamma^{-1}d$ with $d$ from \eqref{e6.3}, $V=(V_1,\dots,V_N)\in ({\mathcal{A}}_{\alpha,\sigma_1})^N$, $N=N_1+N_2$. The functions $n_j$ are continuously differentiable and $m_k$ are continuous for $t\ge 0$. Moreover, $n'_j(0)=c_j$, $j=1,\dots,N_1$ and $m_k(0)=c_{k+N_1}$, $k=1,\dots,N_2$. \end{corollary} \section{Appendix} \begin{proof}[Proof of Lemma \ref{lem3}]. Let us start with the estimation of $b^0[Z]$. Substituting $\frac cp +V$ for $Z$ in \eqref{e3.22}, multiplying by $|p|^{\alpha'}$ and estimating we have \label{e7.1} \begin{aligned} &|p|^{\alpha'}|b^0[Z](x,p)|\\ &\leq \sum_{k=1}^{N_1}\big(|c_k| +\frac{|p|^{\alpha}|V_k(p)|}{|p|^{\alpha-1}}\big) |p|\int_0^1 \big|G(x,y,p)\big|dy \\ &\quad \times\|\nu_k\|_{C[0,1]} \Big[\frac{1}{|p|^{3-\alpha'}}\,|p|\max_{0\leq y \leq 1}|B^{0,0}(y,p)| +\frac{\|\varphi\|_{C[0,1]}}{|p|^{2-\alpha'}}\Big] \\ &\quad +\sum_{k=N_{1}+1}^N \big(|c_k|+\frac{|p|^{\alpha}|V_k(p)|} {|p|^{\alpha-1}}\big) \sqrt{|p|}\int_0^1 |G_y(x,y,p)|dy \|\mu_{k-N_1}\|_{C[0,1]}\frac{\|\varphi'\|_{C[0,1]}}{|p|^{\frac{3}{2}-\alpha'}} \\ &\quad +|p|^{\alpha'}|\Phi^{0,1}(x,p)|. \end{aligned} Note that \eqref{e5.1} implies $\Phi^{0,1}\in \mathcal{B}_{\alpha',\sigma_0}$ for the function $\Phi^{0,1}$ defined in \eqref{e3.24}. Using this relation, the assertions \eqref{e4.11}, \eqref{e4.13} of Lemma \ref{lem1} and the definitions of the norms $\|\cdot\|_{\gamma,\sigma}$, $\|\cdot\|_\sigma$ we obtain from \eqref{e7.1} \begin{align*} & |p|^{\alpha'}|b^0[Z](x,p)|\\ &\leq \sum_{k=1}^{N_1}\big(|c_k| +\frac{\|V_k\|_{\alpha,\sigma}}{|p|^{\alpha-1}}\big)K_1 \|\nu_k\|_{C[0,1]} \big[\frac{\|B^{0,0}\|_{1,\sigma}}{|p|^{3-\alpha'}} +\frac{\|\varphi\|_{C[0,1]}}{|p|^{2-\alpha'}}\big]\\ &\quad + \sum_{k=N_{1}+1}^N \big(|c_k| +\frac{\|V_k\|_{\alpha,\sigma}}{|p|^{\alpha-1}}\big) K_3 \|\mu_{k-N_1}\|_{C[0,1]}\frac{\|\varphi'\|_{C[0,1]}}{|p|^{\frac{3}{2}-\alpha'}} +\|\Phi^{0,1}\|_{\alpha',\sigma_0} \end{align*} for $\mathop{\rm Re}p>\sigma$, $\sigma\geq\sigma_0$, $x\in [0,1]$. Taking here the supremum over $\mathop{\rm Re}p>\sigma$, $x\in [0,1]$ and observing the relation $|p|^\gamma>\sigma^\gamma$ for $\mathop{\rm Re}p>\sigma$, which holds in the cases $\gamma=\alpha-1$, $3-\alpha'$, $2-\alpha'$, $3/2-\alpha'$ due to \eqref{e4.5} \eqref{e4.9}, we have \begin{align*} \|b^0[Z]\|_{\alpha',\sigma}& \leq \sum_{k=1}^{N_1} \big(|c_k|+\frac{\|V_k\|_{\alpha,\sigma}}{\sigma^{\alpha-1}}\big) K_1 \|\nu_k\|_{C[0,1]} \big[\frac{\|B^{0,0}\|_{1,\sigma}}{\sigma^{3-\alpha'}} +\frac{\|\varphi\|_{C[0,1]}}{\sigma^{2-\alpha'}}\big] \\ &\quad + \sum_{k=N_{1}+1}^N \big(|c_k|+\frac{\|V_k\|_{\alpha,\sigma}}{\sigma^{\alpha-1}}\big) K_3 \|\mu_{k-N_1}\|_{C[0,1]}\frac{\|\varphi'\|_{C[0,1]}}{\sigma^{\frac{3}{2}-\alpha'}} + \|\Phi^{0,1}\|_{\alpha',\sigma_0} \end{align*} for $\sigma\ge \sigma_0$. Finally, observing that $\sigma^{\gamma'}\ge\sigma^\gamma$ for $\gamma'>\gamma$, because $\sigma\ge\sigma_0\ge 1$ we arrive at the relation $$\label{e7.2} \|b^0[Z]\|_{\alpha',\sigma}\leq \frac{C_8}{\sigma^{\frac{3}{2}-\alpha'}}\big(|c| +\frac{\|V\|_{\alpha,\sigma}}{\sigma^{\alpha-1}}\big)+ \|\Phi^{0,1}\|_{\alpha',\sigma_0}\, ,\quad \sigma\geq\sigma_0$$ with a constant $C_8$ depending on $K_1, K_3, \nu, \mu, B^{0,0}, \varphi$. Next we perform similar transformations with $b^1[Z]$ in \eqref{e3.23} multiplying by $|p|$ instead of $|p|^{\alpha'}$. We have \begin{align*} |p||b^1[Z](x,p)| &\le \sum_{k=1}^{N_1}\big(|c_k| +\frac{|p|^{\alpha}|V_k(p)|}{|p|^{\alpha-1}}\big) \sqrt{|p|}\int_0^1 |G_x(x,y,p)|dy \\ &\quad \times \|\nu_k\|_{C[0,1]} \big[\frac{1}{|p|^\frac{3}{2} }|p|\max_{0\leq y \leq 1}|B^{0,0}(y,p)| +\frac{\|\varphi\|_{C[0,1]}}{\sqrt{p}}\big]\\ &\quad + \sum_{k=N_{1}+1}^{N} \big(|c_k|+ \frac{|p|^{\alpha}|V_k(p)|}{|p|^{\alpha-1}}\quad )\frac{1}{\sqrt{|p|}} \sqrt{|p|}\Big|\frac{\mu_{k-N_1}(x)\varphi'(x)}{\lambda(x)} \\ &\quad -\int_0^1 G_{xy}(x,y,p)\mu_{k-N_1}(y)\varphi'(y)\,dy\Big| +|p||\Phi^1(x,p)|. \end{align*} Using here the assumption \eqref{e5.1}, the assertions \eqref{e4.12} and \eqref{e4.16} of Lemmas \ref{lem1}, \ref{lem2} and taking the supremum over $\mathop{\rm Re}p>\sigma$, $x\in [0,1]$ we obtain \begin{align*} \|b^1[Z]\|_{1,\sigma}&\leq \sum_{k=1}^{N_1}\big(|c_k| +\frac{\|V_k\|_{\alpha,\sigma}}{\sigma^{\alpha-1}}\big)K_2\|\nu_k\|_{C[0,1]} \times\big[\frac{\|B^{0,0}\|_{1,\sigma}}{\sigma^\frac{3}{2}} +\frac{\|\varphi\|_{C[0,1]}}{\sigma^\frac{1}{2}}\big]\\ &\quad + \sum_{k=N_{1}+1}^{N} \big(|c_k| +\frac{\|V_k\|_{\alpha,\sigma}}{\sigma^{\alpha-1}}\big) K_6\|(\mu_{k-N_1}\varphi')'\|_{C[0,1]}\frac{1}{\sqrt{\sigma}} +\|\Phi^1\|_{1,\sigma_0} \end{align*} for $\sigma\ge \sigma_0$. This yields $$\label{e7.3} \|b^1[Z]\|_{1,\sigma}\leq \frac{C_9}{\sqrt{\sigma}}\big(|c| +\frac{\|V\|_{\alpha,\sigma}}{\sigma^{\alpha-1}}\big) +\|\Phi^1\|_{1,\sigma_0} \, ,\quad \sigma\geq\sigma_0$$ with a constant $C_9$. In particular, \eqref{e7.2} and \eqref{e7.3} imply $b[Z]=(b^0[Z],b^1[Z])\in \mathcal{B}_\sigma$ for $\sigma\geq\sigma_0$ and estimate \eqref{e5.2}. To prove \eqref{e5.3} we denote $Z=Z^1-Z^2$. Then the components $b^{0}[Z]$ and $b^1[Z]$ of the vector $b[Z]=b[Z^1]-b[Z^2]$ are expressed by the formulas \eqref{e3.22} with $\Phi^{0,1}=0$ and \eqref{e3.23} with $\Phi^1=0$, respectively. Using the estimates \eqref{e7.2} and \eqref{e7.3} for the components of $b[Z]$ and observing that $Z=\frac cp +V$ with $c=0$ and $V=V^1-V^2$ we deduce \eqref{e5.3}. The proof is complete. \end{proof} \begin{proof}[Proof of Lemma \ref{lem4}] First we show that the linear operator $A[Z]=(A^0[Z],A^1[Z])$, given by \eqref{e3.20}, \eqref{e3.21}, is bounded in $\mathcal{B}_\sigma$ and satisfies estimate \eqref{e5.4}. From \eqref{e3.20} by $Z=\frac cp +V$ we get \begin{align*} |p|^{\alpha'}|(A^0[Z]B)(x,p)| & \leq \sum_{k=1}^{N_1}\big(\frac{|c_k|}{|p|^2} +\frac{|p|^{\alpha}|V_k(p)|}{|p|^{\alpha+1}}+\frac{|n_k(0)|}{|p|}\big) \\ &\quad \times |p|\int_0^1 |G(x,y,p)|dy\cdot \|\nu_k\|_{C[0,1]}\,|p|^{\alpha'} \max_{0\leq y \leq 1}|B^{0,1}(y,p)| \\ &\quad + \sum_{k=N_{1}+1}^N \big(\frac{|c_k|}{|p|^{\frac{5}{2}-\alpha'}}+ \frac{|p|^{\alpha}|V_k(p)|}{|p|^{\frac{3}{2}-\alpha'+\alpha}}\big) \sqrt{|p|}\int_0^1 |G_y(x,y,p)|dy \\ &\quad \times \|\mu_{k-N_1}\|_{C[0,1]}\,|p|\max_{0\leq y\leq 1} |B^{1}(y,p)|\, . \end{align*} Using Lemma \ref{lem1} and taking the supremum over $\mathop{\rm Re}p>\sigma$, $x\in [0,1]$ we deduce \begin{align*} \|A^0[Z]B\|_{\alpha',\sigma} &\leq \sum_{k=1}^{N_1}\big(\frac{|c_k|}{\sigma^2} +\frac{\|V_k\|_{\alpha,\sigma}}{\sigma^{\alpha+1}}+\frac{|n_k(0)|}{\sigma}\big) K_1\|\nu_k\|_{C[0,1]}\|B^{0,1}\|_{\alpha',\sigma} \\ &\quad + \sum_{k=N_{1}+1}^N \big(\frac{|c_k|}{\sigma^{\frac{5}{2}-\alpha'}}+ \frac{\|V_k\|_{\alpha,\sigma}}{\sigma^{\frac{3}{2}-\alpha'+\alpha}}\big) K_3\|\mu_{k-N_1}\|_{C[0,1]}\|B^{1}\|_{1,\sigma} \end{align*} for $\sigma\geq\sigma_0$. This due to $\alpha'>1/2$ and $\sigma_0\ge 1$ implies $$\label{e7.4} \|A^0[Z]B\|_{\alpha',\sigma}\leq C_{10}\big[\frac{|c|}{\sigma^{\frac{5}{2}-\alpha'}}+ \frac{\|V\|_{\alpha,\sigma}}{\sigma^{\frac{3}{2}-\alpha'+\alpha}}+ \frac{|n(0)|}{\sigma}\big]\|B\|_{\sigma} , \quad \sigma\geq\sigma_0\,,$$ where $C_{10}$ is a constant. Further, from \eqref{e3.21} we derive \begin{align*} & |p||(A^1[Z]B)(x,p)|\\ &\leq \sum_{k=1}^{N_1}\big(\frac{|c_k|}{|p|^{\frac{1}{2}+\alpha'}} +\frac{|p|^{\alpha}|V_k(p)|}{|p|^{\alpha+\alpha'-\frac{1}{2}}} +\frac{|n_k(0)|}{|p|^{\alpha'-\frac{1}{2}}}\big) \\ &\quad \times \sqrt{|p|}\int_0^1 |G_x(x,y,p)|dy\cdot \|\nu_k\|_{C[0,1]}|p|^{\alpha'} \max_{0\leq y \leq 1}|B^{0,1}(y,p)| \\ &\quad + \sum_{k=N_{1}+1}^N \big(\frac{|c_k|}{|p|}+ \frac{|p|^{\alpha}|V_k(p)|}{|p|^{\alpha}}\big) \Big[\frac{\|\mu_{k-N_1}\|_{C[0,1]}}{\lambda_0} |p|\max_{0\leq y\leq 1}|B^{1}(y,p)| \\ &\quad +\int_0^1 |G_{xy}(x,y,p)|dy\cdot \|\mu_{k-N_1}\|_{C[0,1]}|p|\max_{0\leq x\leq 1}|B^{1}(x,p)|\Big]\, . \end{align*} Here $\lambda_0 := \min_{0\leq x \leq 1}\lambda(x) > 0$ because $\lambda\in C[0,1]$, $\lambda(x)>0$, by assumption. Using Lemma \ref{lem1}, taking the supremum over $\mathop{\rm Re}p>\sigma$, $x\in [0,1]$ and observing the inequalities $\alpha'>1/2$ and $\sigma_0\ge 1$ we get $$\label{e7.5} \|A^1[Z]B\|_{1,\sigma}\leq C_{11}[\frac{|c|}{\sigma} +\frac{\|V\|_{\alpha,\sigma}}{\sigma^{\alpha}} +\frac{|n(0)|}{\sigma^{\alpha'-\frac{1}{2}}}]\|B\|_{\sigma},\quad \sigma\geq\sigma_0$$ with a constant $C_{11}$. Putting estimates \eqref{e7.4} and \eqref{e7.5} together and taking the inequalities $\alpha'<3/2$, $\sigma\ge \sigma_0$ into account we have $$\label{e7.6} \|A[Z]B\|_{\sigma}\leq C_{12}[\frac{|c|}{\sigma} +\frac{\|V\|_{\alpha,\sigma}}{\sigma^{\alpha}} +\frac{|n(0)|}{\sigma^{\alpha'-\frac{1}{2}}}]\|B\|_{\sigma},\quad \sigma\geq\sigma_0$$ with a constant $C_{12}$. Due to this relation $A[Z]$ is bounded in $\mathcal{B}_\sigma$ and satisfies estimate \eqref{e5.4}. It remains to prove \eqref{e5.5}. Denoting $Z=Z^1-Z^2$ the components $A^{0}[Z]$ and $A^1[Z]$ of the vector $A[Z]=A[Z^1]-A[Z^2]$ are expressed by the formulas \eqref{e3.20} and \eqref{e3.21}, respectively, containing $n_k(0)=0$. Using the estimate \eqref{e7.6} for $A[Z]$ and observing that $Z=\frac cp +V$ with $c=0$ and $V=V^1-V^2$ we deduce \eqref{e5.5}. The lemma is proved. \end{proof} \begin{thebibliography}{99} \bibitem{c1} B. D. Coleman and M. E. Gurtin; \emph{Equipresence and constitutive equation for rigid heat conduction}, Z. Angew. Math. Phys. {\bf 18}, 199--208 (1967). \bibitem{c2} F. Colombo and A. Lorenzi; \emph{Identification of time and space dependent relaxation kernels for materials with memory related to cylindrical domains}, J. Math. Anal. Appl. {\bf 213}, 32--90 (1997). \bibitem{d1} G. Doetsch; \emph{Einf\"uhrung in Theorie und Anwendung der Laplace-Transformation} (Birkh\"auser Verlag, Basel, 1958). \bibitem{g1} G. Gentili; \emph{Dissipativity conditions and inverse problems for heat conduction with linear memory}, Inverse problems {\bf 7}, 77--84 (1991). \bibitem{g2} M. Grasselli; \emph{An identification problem for a linear integro-differential equation occurring in heat flow}, Math. Meth. Appl. Sci. {\bf 15}, 167--186 (1992). \bibitem{j1} J. Janno; \emph{Determination of a time- and space-dependent heat flux relaxation function by means of a restricted Dirichlet-to-Neumann operator}, Math. Meth. Appl. Sci. {\bf 27}, 1241--1260 (2004). \bibitem{j2} J. Janno and L. v. Wolfersdorf; \emph{Identification of weakly singular memory kernels in heat conduction}, Z. Angew. Math. Mech. {\bf 77}, 243--257 (1997). \bibitem{j3} J. Janno and L. v. Wolfersdorf; \emph{Identification of memory kernels in general linear heat flow}, J. Inv. Ill-Posed Problems {\bf 6}, 141--164 (1998). \bibitem{j4} J. Janno and L. v. Wolfersdorf; \emph{An inverse problem for identification of a time- and space-dependent memory kernel of a special kind in heat conduction}, Inverse Problems {\bf 15}, 1455--1467 (1999). \bibitem{j5} J. Janno and L. v. Wolfersdorf; \emph{Identification of a special class of memory kernels in one-dimensional heat flow}, J. Inv. Ill-Posed Problems {\bf 9}, 389--411 (2001). \bibitem{l1} A. Lorenzi and E. Paparoni; \emph{Direct and inverse problems in the theory of materials with memory}, Rend. Sem. Math. Univ. Padova {\bf 87}, 105--138 (1992). \bibitem{l2} A. Lunardi; \emph{On the linear equation with fading memory}, SIAM J. Math. Anal. {\bf 21}, 1213--1224 (1970). \bibitem{n1} J. W. Nunziato; \emph{On heat conduction in materials with memory}, Quart. Appl. Math. {\bf 29}, 187--204 (1971). \bibitem{p1} I. G. Petrowski; \emph{Vorlesungen \"uber partielle Differentialgleichungen} (Teubner, Leipzig, 1955). \bibitem{p2} J. Pr\"u\ss; \emph{Evolutionary Integral Equations and Applications} (Birkh\"auser Verlag, Boston, 1993). \end{thebibliography} \end{document}