\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2018 (2018), No. 65, pp. 1--11.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu} \thanks{\copyright 2018 Texas State University.} \vspace{8mm}} \begin{document} \title[\hfilneg EJDE-2018/65\hfil Spectral property of a Frankl type problem] {Spectral properties of a Frankl type problem for parabolic-hyperbolic equations} \author[M. A. Sadybekov, G. Dildabek, M. B . Ivanova \hfil EJDE-2018/65\hfilneg] {Makhmud A. Sadybekov, Gulnar Dildabek, Marina B. Ivanova} \address{Makhmud Sadybekov (corresponding author)\newline Institute of Mathematics and Mathematical Modeling, 125 Pushkin str., 050010 Almaty, Kazakhstan} \email{sadybekov@math.kz} \address{Gulnar Dildabek \newline Institute of Mathematics and Mathematical Modeling, 125 Pushkin str., 050010 Almaty, Kazakhstan. \newline Al-Farabi Kazakh National University, Almaty, Kazakhstan} \email{dildabek.g@gmail.com} \address{Marina B. Ivanova \newline Institute of Mathematics and Mathematical Modeling, 125 Pushkin str., 050010 Almaty, Kazakhstan.\newline South-Kazakhstan State Pharmaceutical Academy, 1 al-Farabi sq., 160019, Shymkent, Kazakhstan} \email{marina-iv@mail.ru} \dedicatory{Communicated by Ludmila S. Pulkina} \thanks{Submitted December 14, 2017. Published March 10, 2018.} \subjclass[2010]{35M10, 35M12} \keywords{Equation of the mixed type; parabolic-hyperbolic equation; \hfill\break\indent Non-local boundary value problem; Frankl type problem; spectral properties; eigenvalues} \begin{abstract} In this article we study spectral properties of non-local boundary-value problem for an equation of parabolic-hyperbolic type. The non-local condition binds the solution values at points on boundaries of the parabolic and hyperbolic parts of the domain with each other. Nonlocal boundary conditions of such type are called Frankl-type conditions. This problem was first formulated by Kal'menov and Sadybekov who proved the unique strong solvability. In this article we investigate one particular case of this problem, for which we show that the problem does not have eigenvalues. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{definition}[theorem]{Definition} \allowdisplaybreaks \section{Introduction}\label{intro} The theory of equations of the mixed type is one of the well-developed sections of the modern theory of partial differential equations. This happens because of the appearance of many applied problems such mathematical modeling lead to the studying various types of equations in domains of changing independent variables. Therefore scientists are interested in problems of mixed type. In 1902 Chaplygin was the first to point out the importance of studying equations of the mixed type in his paper ``On gas jet``. Researching boundary value problems for equations of the mixed type began from works of Tricomi, Gellerstedt in 20th-30th of the last century. A new stage of developing this theory was founded by papers of Lavrent'ev, Bitsadze, Frankl, Babenko, where the practical significance of some essential issues of this theory was indicated alongside with theoretical researches of these issues. For the most part, these works were devoted to the theoretical and applied aspects of equations of the mixed elliptic-hyperbolic type. Researching equations of a parabolic-hyperbolic type has gained a rapid development quite recently. These problems are of particular interest due to their application to various problems of mechanics and physics. For example, such problems arise in studying the movement of weak compressible fluid in a channel surrounded by a porous medium: in the channel the pressure of the fluid satisfies the wave equation, but in the porous medium this pressure is described by a diffusion equation. Essential contribution to the development of the theory of boundary value problems for parabolic-hyperbolic equations was done by Salakhitdinov, Dzhuraev, Nakhushev. They justified the well-posedness of formulated problems by the method of reduction to integral equations. Issues of well-posed solvability of problems were researched on the basis of solution representation in the form of bilinear series in papers of Moiseev, Kapustin, Sabitov. Unlike the theory of local boundary value problems, nonlocal boundary value problems are much less researched. In gas dynamics Frankl (in 1945) for the first time set a boundary value problem for the Chaplygin equation \[ k(y)u_{xx}-u_{yy}= 0, \] where $k(0)=0$, $k'(y)>0$. In this problem as a carrier of nonlocal boundary condition ("jump of sealing") \[ u(0,y)-u(0,-y)=f(y) \] is a part $-a0$ by the segments $AA_0$, $A_0B_0$, $B_0B$, $A=(0,0)$, $A_0=(0,1)$, $B_0=(1,1)$, $B=(1,0)$ and for $y<0$ by the characteristics $AC: x+y=0$ and $BC: x-y=1$ of an equation of the mixed parabolic-hyperbolic type \begin{equation} \label{e1} Lu=\bigg\{ \begin{gathered} u_x-u_{yy},\quad y>0 \\ u_{xx}-u_{yy},\quad y<0 \end{gathered} \bigg\}=f(x,y). \end{equation} This is an equation of the mixed type. The equation refers to the first kind because the line of change of type $y=0$ is not a characteristic of the equation. By $W^l_2(\Omega )=H^l(\Omega )$ we denote the Sobolev space with the norm ${\|\cdot \|}_l$, $W^0_2(\Omega )=L_2(\Omega )$; ${\Omega}_1=\Omega \cap\{y>0\}$, ${\Omega }_2=\Omega \cap\{y<0\}$. Consider the following nonlocal boundary value problem being the generalization of an analogue of the Frankl problem for the parabolic-hyperbolic equation \eqref{e1}. This problem was first formulated by Kal'menov and Sadybekov \cite{10,11} . \smallskip \noindent\textbf{Problem $F$.} Find a solution to \eqref{e1} satisfying classical boundary conditions \begin{equation} \label{e2} u\big|_{AA_0}=0,\quad u_y\big|_{A_0B_0}=0, \end{equation} and the non-local boundary condition \begin{equation} \label{e3} \alpha u({\theta }_0(t))+\beta u({\theta }_1(t))=\gamma u(\theta (t)),\quad 0\le t\le 1, \end{equation} where $\theta(t)=(t,1)$, ${\theta}_0(t)=(t/2,-t/2)$, ${\theta}_1(t)=(\frac{t+1}{2},\frac{t-1}{2})$; $\alpha$, $\beta$ and $\gamma$ are given numbers. \smallskip It is easy to see that $\theta (t)\in A_0B_0$, ${\theta }_0(t)\in AC$, ${\theta }_1(t)\in BC$. Therefore the non-local boundary condition \eqref{e3} binds with each other values of the sought-for solution on the parabolic part of the boundary $A_0B_0$ and on the hyperbolic parts of the boundary of the domain (at the characteristics $AC$ and $BC$). Note that for $\gamma=0$ the boundary conditions in the hyperbolic part of the domain of the form \[ \alpha u({\theta }_0(t))+\beta u({\theta }_1(t))=0 \] are well-known and called boundary conditions with displacement. They were first introduced by Nakhushev for a wave equation (see \cite{15}). The particular case of Problem $F$ for $\alpha+\beta=2\gamma$ was considered in \cite{8} and there the unique strong solvability of the problem is proved. \begin{definition} \label{def1}\rm A function $u(x,y)$ from the class \[ {u\in W=C^1(\overline{\Omega })\cap C^{1,2}_{x,y} (\overline{\Omega }}_1)\cap C^2({\overline{\Omega}}_2), \] satisfying the boundary conditions \eqref{e2}--\eqref{e3} of the problem and turning \eqref{e1} into an identity we will call \emph{a classical solution} to Problem $F$. \end{definition} \begin{definition} \label{def2} \rm A function ${u\in L}_2(\Omega )$ we will call \emph{a strong solution} to Problem $F$ if there exists a sequence of functions $\{u_n\}$, $u_n\in W$ satisfying the boundary conditions \eqref{e2}--\eqref{e3} of the problem such that sequences $u_n$ and ${Lu}_n$ reduce in $L_2(\Omega)$ to the functions $u$ and $f$, respectively. \end{definition} Kal'menov and Sadybekov \cite{10,11} proved the unique strong solvability of the problem. \begin{theorem}[\cite{10,11}] \label{thm2.1} Let $\alpha+\beta\neq 0$. Then (a) For any function ${f\in L}_2(\Omega )$ there exists a unique strong solution $u(x,y)$ to Problem $F$. This solution belongs to the class ${H^1(\Omega )\cap H^{1,2}_{x,y} (\Omega }_1)\cap C(\overline{\Omega })$, and satisfies the inequality \begin{equation} \label{e4} {\|u\|}_1\le C {\|f\|}_0. \end{equation} (b) For any function ${f\in C}^1(\overline{\Omega} )$, $f(A)=0$, there exists a unique classical solution $u(x,y)$ to Problem $F$. This solution is stable in the norm \begin{equation}\label{e5} {\|u\|}_{C(\overline{\Omega} )}\le C {\|f\|}_{C(\overline{\Omega})}. \end{equation} \end{theorem} By $L$ denote a closure in $L_2(\Omega )$ of the differential operator given on functions $\{u_n\}$, $u_n\in W$ satisfying the boundary conditions \eqref{e2}--\eqref{e3}. From item (a) of Theorem \ref{thm2.1} follows that the operator $L$ is invertible and ${L}^{-1}$ is a compact operator. Therefore the spectrum of the operator $L$ can consist of only eigenvalues. Naturally there arises a question on existence of eigenvalues of the operator $L$ and, consequently, of Problem $F$. Unlike the theory of solvability, spectral issues of problems for equations of the mixed type are less studied. The papers by Kal'menov \cite{16,17}, Moiseev \cite{18}, Ponomarev \cite{19} have made a sifnificant contribution to this direction. The main bibliography on these issues is given in the monograph of Moiseev \cite{20}. Note that for $\beta=\gamma=0$ Problem $F$ coincides with the Tricomi problem, and for $\alpha=\gamma=0$ it coincides with the Tricomi problem with data on an opposite characteristics. The strong solvability of particular cases of the problem for $\alpha=\gamma=0$ and for $\beta=\gamma=0$ has been researched in paper by Sadybekov, Toizhanova (Dildabek) \cite{21}. It is shown that for $\beta=\gamma=0$ the problem is Volterra, and for $\alpha=\gamma=0$ the problem has an eigenvalue. This method was used in \cite{22} for proving the Volterra property of some problems with the Bitsadze-Samarskii-type conditions for a mixed parabolic-hyperbolic equation. In the next section we present another particular case of Problem $F$ for $\gamma\neq0$ which does not have eigenvalues. In virtue of compactness of the operator ${L}^{-1}$ it will mean that ${L}^{-1}$ is a Volterra operator. Thus Problem $F$ in this case is Volterra. \section{Absence of eigenvalues}\label{sec:2} Consider a particular case of Problem $F$, when $\beta=0$. In virtue of the condition $\alpha+\beta\neq 0$ from Theorem \ref{thm2.1} one can consider that $\alpha=1$. \smallskip \noindent\textbf{Problem $F_0$.} Find a solution to \eqref{e1} satisfying the boundary conditions \begin{gather} \label{e2b} u\big|_{AA_0}=0,\quad u_y\big|_{A_0B_0}=0, \\ \label{e3b} u({\theta }_0(t))=\gamma u(\theta (t)),\quad 0\le t\le 1, \end{gather} where $\theta (t)=(t,1)$, ${\theta }_0(t)=(t/2,-t/2)$, $\gamma$ are given numbers. For proving the Volterra property of Problem $F_0$ we need to obtain a representation of the inverse operator ${L}^{-1}$. A part of the following theorem can be obtained, as a particular case, from \cite{10,11}. But we prove this result anew because it is important to obtain a form of solution in the integral form. \begin{theorem} \label{thm3.1} For any function ${f\in L}_2(\Omega)$ there exists a unique strong solution $u(x,y)$ to Problem $F_0$. This solution belongs to the class ${H^1(\Omega)\cap H^{1,2}_{x,y}\ (\Omega }_1)\cap C(\overline{\Omega })$, satisfies inequality \eqref{e4} and can be represented in the form \begin{equation}\label{e8} u(x,y)=\int_{\Omega}{K(x,y;x_1,y_1)}f(x_1,y_1)dx_1dy_1, \end{equation} where ${K\in L}_2(\Omega \times \Omega )$. \end{theorem} \begin{proof} The proof will be given in several stages. \subsection{Reducing to an integral equation}\label{sec:2.1} At first let ${f\in C}^1(\overline{\Omega} )$, $f(A)=0$. By the unique solvability of the Cauchy problem for a wave equation, the solution to \eqref{e1} for $y<0$ is represented according to the d'Alembert formula \begin{equation}\label{e5_0} u(x,y)=-\int^{\eta }_{\xi }{d{\xi}_1\int^{\eta }_{{\xi }_1}{f_1({\xi }_1,{\eta }_1)d{\eta }_1}}+\frac{1}{2}[\tau (\xi )+\tau (\eta )]-\frac{1}{2}\int^{\eta }_{\xi }{\nu (s)ds}, \end{equation} where \begin{gather*} \tau (x)=u(x,0),\quad \tau (0)=0,\quad \xi =x+y,\quad \eta =x-y,\\ \nu (x)=\frac{\partial u}{\partial y}(x,0),\quad f_1(\xi ,\eta )= \frac{1}{4}f(\frac{\xi +\eta }{2},\frac{\xi -\eta }{2}). \end{gather*} Hence, taking into account $\tau (0)=0$, by direct calculation, we obtain \begin{equation} \label{e5_1} u({\theta }_0(t))=\frac{1}{2}\tau (t)-\frac{1}{2}\int^t_0{\nu (s)ds} - \int^t_0{d{\xi }_1\int^t_{{\xi}_1}{f_1({\xi }_1,{\eta }_1)d{\eta }_1}}. \end{equation} This is the basic relation for $\tau (t)$ and $\nu(t)$ obtained from the hyperbolic part of the domain. In the parabolic part of the domain we consider a problem with mixed boundary condition: \begin{quote} Find in ${\Omega }_1$ a solution to the heat equation \begin{equation} \label{e8_1} u_x-u_{yy}=f(x,y), \end{equation} satisfying the homogeneous initial-boundary conditions \eqref{e2} and non-homogeneous boundary condition \begin{equation} \label{e9} u(x,0)=\tau (x),\quad 0\le x\le 1. \end{equation} \end{quote} It is evident that the natural condition of the sequence $\tau (0)=0$ is a necessary condition of the solution existence. Further we will assume that this condition holds. Considering that the function $\tau (x)$ is well-known, we calculate $\nu (x)= \frac{\partial u}{\partial y}(x,0)$. This is a mixed initial-boundary value problem for the heat equation. Its Green's function has the form \cite[p. 198]{23}: \begin{equation} \label{10} \begin{aligned} G(x,y,y_1) &==\sum^{+\infty }_{n=-\infty } \frac{(-1)^n}{2\sqrt{\pi x}} \Big[\exp\big\{-\frac{{(y-y_1+2n)}^2}{4x}\big\} \\ &\quad - \exp\big\{-\frac{{(y+y_1+2n)}^2}{4x}\big\}\Big]. \end{aligned} \end{equation} Therefore for the solution to problem \eqref{e8_1}, \eqref{e2}, \eqref{e9} we have the representation \begin{equation} \label{10-1} \begin{aligned} u(x,y)&=\int^x_0{dx_1\int^1_0{G(x-x_1,y,y_1)f(x_1,y_1)dy_1}} \\ &\quad +\int^x_0{G_{y_1}(x-s,y,0)\tau(s)ds}. \end{aligned} \end{equation} Hence for $y\to 1$ we find \begin{equation} \label{e24} \begin{aligned} u(\theta (t)) &= \int^t_0{dx_1\int^1_0{G(t-x_1,1,y_1)f(x_1,y_1)dy_1}} \\ &\quad + \int^t_0 G_{y_1}(t-s,1,0)\tau(s)ds, \end{aligned} \end{equation} and differentiating with respect to $y$ and letting $y\to 0$, we obtain \begin{equation} \label{e24_1} \nu(x)= \frac{\partial}{\partial y}\int^x_0{G_{y_1}(x-s,y,0)\tau(s)ds} \big|_{y=0}+\Phi_1(x), \end{equation} where \[ \Phi_1(x)=\frac{\partial}{\partial y}\int^x_0{dx_1 \int^1_0{G(x-x_1,y,y_1)f(x_1,y_1)dy_1}}\big|_{y=0}. \] Formulas \eqref{e24} and \eqref{e24_1} give the basic relation for $\tau (t)$ and $\nu (t)$ obtained from the parabolic part of the domain. Substituting \eqref{e5_1} and \eqref{e24} into the boundary condition \eqref{e3}, after differentiating, taking into account $\tau(0)=0$, we obtain \begin{equation}\label{13} \tau' (t)-\nu (t)-2\gamma \int^t_0{G_{y_1}(t-s,1,0)\tau'(s)ds}=2F(t), \end{equation} where $F(t)=F_1(t)+F_2(t)$, with \begin{gather*} F_1(t)=\gamma\frac{d}{dt}\int^t_0{dx_1\int^1_0{G(t-x_1,1,y_1)f(x_1,y_1)dy_1}},\\ F_2(t)= \int^t_0{f_1({\xi }_1,t)d\xi_1}. \end{gather*} We transform the first summand in the right-hand part of \eqref{e24_1}. For this purpose, taking into account $\tau (0)=0$ and the explicit form of the Green's function \eqref{10}, by integrating by parts, we transform \[ \int^t_0{G_{y_1}(t-s,y,0)\tau(s)ds}=\int^t_0{G_1(t-s,y){\tau}'(s)ds}, \] where \[ G_1(t-s,y) =-\frac{2}{\sqrt{\pi }}\sum^{-1}_{n=-\infty }(-1)^n {\int^{\frac{y+n}{2\sqrt{(t-s)}}}_{-\infty}{e^{-z^2}}dz} + \frac{2}{\sqrt{\pi }}\sum^{+\infty }_{n=0}(-1)^n{\int^{+\infty}_{\frac{y+n}{2\sqrt{(t-s)}}}{e^{-z^2}}dz}. \] Hence it is easy to obtain that \[ \frac{\partial}{\partial y}\int^t_0{G_{y_1}(t-s,y,0)\tau(s)ds}\big|\ _{y=0}=-\int^t_0 k(t-s)\tau'(s)ds, \] where \[ k(t-s)=\frac{1}{\sqrt{\pi (t-s)}} \sum^{+\infty }_{n=-\infty }(-1)^n{\exp\Big\{-\frac{n^2}{4(t-s)}\Big\}}, \] and formula \eqref{e24_1} takes the form \begin{equation} \label{16} \nu(t)= -\int^t_0 k(t-s)\tau'(s)ds+\Phi_1(t). \end{equation} Substituting the obtained result from \eqref{16} into \eqref{13}, we obtain the integral equation \begin{equation} \label{17} \tau' (t)+\int^t_0 k(t-s)\tau'(s)ds- 2\gamma \int^t_0{G_{y_1}(t-s,1,0)\tau'(s)ds}=2\Phi(t),\end{equation} where \begin{equation} \label{17_1} \Phi(t)=F(t)+\frac{1}{2}\Phi_1(t). \end{equation} \subsection{Constructing a solution to the problem}\label{sec:2.2} It is easy to see that $k(t-s)$ is a kernel with a weak polar peculiarity, and the function $G_{y_1}(t-s,1,0)$ is infinitely continuously differentiable for all $s\leq t\leq 1$. Therefore \eqref{17} is an integral Volterra equation of the second kind \begin{equation} \label{17_2} \tau' (t)-\int^t_0 k_1(t-s)\tau'(s)ds=2\Phi(t),\end{equation} where \begin{equation} \label{17_3} k_1(t-s)=-k(t-s)+2\gamma G_{y_1}(t-s,1,0), \end{equation} \begin{align*} \Phi(t)&=\gamma\frac{d}{dt}\int^t_0{dx_1\int^1_0{G(t-x_1,1,y_1)f(x_1,y_1)dy_1}} +\int^t_0{f_1({\xi }_1,t)d\xi_1} \\ &\quad +\Big(\frac{\partial}{\partial y}\int^t_0{dx_1\int^1_0{G(t-x_1,y,y_1)f(x_1,y_1)dy_1}}\Big)\Big|_{y=0}, \end{align*} which always has a unique solution. It is easy to see that the kernel $K(x-t)$ is a kernel with weak peculiarity. Therefore there exists the unique strong solution to \eqref{17_2} and has the form \begin{equation} \tau'(t)=2\Phi (t)+2\int^t_0{\Gamma(t-s)\Phi (s)ds}, \label{GrindEQ__15} \end{equation} where $\Gamma(t)$ is a resolvent of \eqref{17_2}: \[ \Gamma(t)=\sum^{\infty }_{j=1}{k_j(t)},\quad k_{j+1}(t)=\int^t_0{k_1(t-s)k_j(s)ds,\quad j\in N}. \] And the smoothness of this solution depends on the class to which $\Phi(t)$ belongs. \begin{lemma}[\cite{10,11}] \label{lem3.2} Let ${f\in C}^1(\overline{\Omega} )$, $f(A)=0$, then $\Phi(t)\in C^1[0,1]$ and satisfies estimates \begin{gather} \label{18} \|\Phi(t)\|_{C[0,1]}\leq C \|f\|_{C(\overline{\Omega})}, \\ \label{19} \|\Phi(t)\|_{L_2(0,1)}\leq C \|f\|_0. \end{gather} \end{lemma} A proof of the above lemma is obtained by direct calculations, estimating each of summands \eqref{17_1} \cite{10} and \cite{11}. On the basis of this lemma there always exists a unique solution $\tau' (t)$ to \eqref{17}. This solution (depending on the smoothness of $\Phi(t)$) belongs to the class $\tau' (t)\in C^1[0,1]$ or $\tau' (t)\in L_2(0,1)$ and by \eqref{18} and \eqref{19}, it satisfies \begin{equation} \label{20} \|\tau'(t)\|_{C[0,1]}\leq C \|f\|_{C(\overline{\Omega})}, \end{equation} or \begin{equation} \label{21} \|\tau'(t)\|_{L_2(0,1)}\leq C \|f\|_0. \end{equation} Taking into account $\tau (0)=0$, we find a unique $\tau(t)$. Now the solution to Problem $F$ is reestablished in ${\Omega }_1$ as a solution to the first initial-boundary value problem by formula \eqref{10-1}. We find the value of $\nu(x)$ from \eqref{16}. Therefore in the domain ${\Omega }_2$ the solution to Problem $F$ is uniquely reestablished as the solution to the Cauchy problem by the d'Alembert formula \eqref{e5_0}. From the solution properties of the first initial-boundary value problem for the heat equation it follows that the solution to Problem $F$ belongs to the classes of smoothness indicated in the Theorem and (by inequalities \eqref{20} and \eqref{21}) satisfies estimates \eqref{e4} and \eqref{e5}. Let us show that for ${f\in L}_2(\Omega )$ the found solution is strong. Since $C^1_0(\overline{\Omega })$ is dense in $L_2(\Omega )$, then for any function ${f\in L}_2(\Omega )$ there exists a sequence of functions ${f_n\in C}^1_0(\overline{\Omega })$ such that $\|f_n-f\|\to 0$, $n\to \infty$. By $u_n$ we denote the classical solution to Problem $F$ when the right-hand part is $f_n$. Such solution exists by virtue of the above-mentioned proof of the theorem and $u_n\in W$ for all ${f_n\in C}^1_0(\overline{\Omega })$. By inequality \eqref{e4} we have \[ {\|u_n-u\|}_1\le c{\|f_n-f\|}_0\to 0, \quad n\to \infty. \] Consequently, $\{u_n\}$ is the sequence corresponding to the definition of strong solution. Therefore Problem $F$ is strongly solvable for any right-hand part $f$, and the strong solution belongs to the class ${H^1(\Omega )\cap H^{1,2}_{x,y}\ (\Omega }_1)\cap C(\overline{\Omega })$. The existence and uniqueness of the strong solution of Problem $F$ is proved. Let us obtain now a solution in the form \eqref{e8}. From \eqref{GrindEQ__15}, taking into account $\tau(0)=0$, after simple transformations, we obtain \begin{equation} \label{GrindEQ__16} \tau (x)=\int^x_0{\Gamma_1(x-t)\Phi (t)dt}, \end{equation} where $$ \Gamma_1(x)=2+2\int^x_0\Gamma(t)dt. $$ Substituting the value $\Phi (t)$ into \eqref{GrindEQ__16}, after evident transformations, we come to the form \begin{align*} \tau (x)&= \int^x_0{d\xi_1}\int^x_{\xi_1} \Gamma_1(x-\xi_1) f_1(\xi_1,\eta_1)d\eta_1 \\ &\quad +2\gamma\int^x_0 dx_1\int^1_0G(x-x_1,1,y_1)f(x_1,y_1)dy_1 \\ &\quad +\gamma\int^x_0 dx_1\int^1_0 \Big(\int_{x_1}^x \Gamma_1(x-t) G(t-x_1,1,y_1)dt\Big)f(x_1,y_1)dy_1 \\ &\quad +\frac{1}{2}\int^x_0 dx_1\int^1_0 \Big(\int_{x_1}^x \Gamma_1(x-t) \big(\frac{\partial}{\partial y}G(t-x_1,y,y_1)\big) \big|_{y=0}~dt\Big)f(x_1,y_1)dy_1. \end{align*} Substituting this quantity into \eqref{e9} and into \eqref{16}, we obtain formula \eqref{e8}, where the detailed form of the kernel $K(x,y;x_1,y_1)$ can be written in the explicit form. We will not show this form here due to its bulkiness. From the analysis of the kernel representation it is easy to see that \[ K(x,y;x_1,y_1)\in L_2(\Omega \times \Omega). \] Herewith it is easy to see that the kernel $K(x,y;x_1,y_1)$ can be represented in the form \begin{equation} \label{3.14} \begin{aligned} K(x,y;x_1,y_1) &=\theta(y)\Bigl\{\theta(y_1 )\theta(x-x_1 ) G_{11} (x-x_1,y,y_1 ) \\ &\quad +\theta(-y_1 )\theta(x-\eta_1 ) G_{12} (x-x_1,y,y_1)\Bigr\} \\ &\quad + \theta(-y)\Bigl\{\theta(y_1 )\theta(\xi-x_1 ) G_{21} (x-x_1,y,y_1 )\\ &\quad +\theta(-y_1 )\theta(\eta-\eta_1 ) G_{22}(x-x_1,y,y_1)\Bigr\}, \end{aligned} \end{equation} where $G_{kn} (x-x_1,y,y_1 )\in L_2(\Omega_k\times\Omega_n)$, $k,n=1,2$; $\theta(\cdot)$ is a Heaviside step function. The proof is complete. \end{proof} \subsection{Theorem on absence of eigenvalues} \label{sec:2.3} \begin{theorem} \label{thm3.3} The inverse operator $L^{-1}$ of Problem $F_0$ defined by \eqref{e8} is Volterra (that is, compact and quasinilpotent). \end{theorem} \begin{proof} Since $K(x,y;x_1,y_1)\in L_2(\Omega \times \Omega )$, the operator $L^{-1}$ is a Gilbert-Schmidt operator. Consequently, it is compact. For proving the theorem it is sufficient to show that the operator $L^{-1}$ does not have eigenvalues. We need the following definitions and theorem from \cite{24}. \begin{definition} \label{def3} \rm Let $S\subset\Omega\times\Omega$. The kernel $K(P_1,P_2 )$ is called $S$-kernel, if $K(P_1,P_2 )\in L_2 (\Omega\times\Omega)$ and $K(P_1,P_2 )=0$ for $(P_1,P_2 )\in S$. \end{definition} \begin{definition} \label{def4} \rm The open set $S\subset\Omega\times\Omega$ is called a set of type $V$, if any $S$-kernel does not have eigenvalues. \end{definition} As in \cite{24}, we introduce the notation: $P_1 \overset{S}{\to} P_2$, if $(P_1,P_2 )\in S$, and $P_1 \overset{S}{\leftarrow} P_2$, if $(P_1,P_2 )\notin S$. \begin{theorem}[\cite{24}] \label{thm3.4} For the set $S$ to be a set of type $V$ it is necessary and sufficient that for any $k\geq 1$ from conditions \begin{equation} \label{3.15} P_1 \overset{S}{\to} P_2\overset{S}{\to} P_3 \overset{S}{\to} \ldots \overset{S}{\to} P_k \end{equation} it follows that $P_k \overset{S}{\leftarrow} P_1$. \end{theorem} We use this theorem for our operator $L^{-1}$. Consider the sequence of points $P_i=(x_i,y_i )$, $i=\overline{1,k}$. Let the condition \eqref{3.15} hold for any $k\geq 1$. Then for $ix_j, \quad i