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The Dirichlet problem for inhomogeneous mixed-type equation with Lavrent'ev–Bitsadze operator K. B. Sabitovab a Mavlyutov Institute of Mechanics, Ufa Centre of the Russian Academy of Sciences, Ufa b Sterlitamak branch of the Ufa University of Science and Technology
Russian version:
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya, 2024, Volume 88, Issue 4, DOI: https://doi.org/10.4213/im9512 
Bibliographic databases:
    § 1. Statement of the problem, available resultsIn the rectangular domain $D=\{(x,y)\mid 0<x<l,\, -\alpha<y<\beta\}$, where $\alpha$, $\beta$, $l$ are positive numbers, $b$ is a real number, consider the mixed-type equation with Lavrent’ev–Bitsadze operator
$$
\begin{equation}
\mathcal{L}u\equiv u_{xx} +(\operatorname{sgn} y) u_{yy}-b u=F(x,y),
\end{equation}
\tag{1.1}
$$
where
$$
\begin{equation}
F(x,y)= \begin{cases} f_1(x)g_1(y), &y>0, \\ f_2(x)g_2(y), &y<0. \end{cases}
\end{equation}
\tag{1.2}
$$
We also consider the following boundary problem. Dirichlet problem. Find a function $u(x,y)$ on $D$ satisfying the conditions Interest in the Dirichlet problem for mixed-type equations arose after Frankl [1], who showed, for the first time, that the problem of passing through the sonic barrier, by a steady two-dimensional vortex-free flow of an ideal gas in nozzles reduces to the Dirichlet problem for a mixed-type equation in the case when the hypersonic waves adhere to the nozzle walls near the minimal section. Shabat [2] was the first to study the Dirichlet problem for the Lavrent’ev equation in the mixed domain $\Omega$ bounded for $y>0$ and $y<0$ by smooth curves $\Gamma$ and $\gamma$ with end-points at $(0,0)$ and $(1,0)$, respectively, where the curve $\gamma$ lies inside the triangle with sides $x+y=0$, $x-y=1$ and $y=0$. This problem was considered in the class of functions
$$
\begin{equation}
C(\overline{\Omega})\cup C^1(\Omega)\cup C^2(\Omega\setminus \{y=0\}).
\end{equation}
\tag{1.8}
$$
If the curve $\gamma\colon y=-l(x)$, where $l(x)\in C^2[0,1]$, satisfies $l(x)> 0$ for $0<x<1$, $l(0)=l(1)=0$, $|l'(x)|\leqslant q<1$, then, according to Shabat [2], the Dirichlet problem for equation (1.7) has a unique solution in the class of functions (1.8). Bitsadze [3] was the first to show that the Dirichlet problem for equation (1.7) in the domain $\Omega$ in the class of functions (1.8) is well posed irrespective of the size and shape of the hyperbolic part of the domain $\Omega$; in other words, this problem is overdetermined by the correctness of the general mixed problem with step out from characteristics. For the Dirichlet problem for equation (1.7) in $\Omega$, the most important results were obtained by Soldatov [4], [5], who proved well-posedness of this problem in the class of functions $C(\overline{\Omega}\setminus A)$ or $C(\overline{\Omega}\setminus B)$, that is, such functions may have power-law singularities at points $A$ or $B$ under some conditions on the curves $\Gamma$ and $\gamma$. These studies have led to the problem of finding the domains for which Dirichlet problem is well-posed in the class of functions (1.8). As such a domain a rectangle was considered. The first studies of problem (1.3)–(1.6) for equation (1.7) in the rectangular domain $D$ for $l=1$ were carried out by Vakhania [6] and Cannon [7], who showed that the problem has a unique solution under the conditions
$$
\begin{equation*}
\tanh(\pi n \beta)\cot(\pi n \alpha)\neq-1,\qquad n=1, 2, \dots.
\end{equation*}
\notag
$$
Cannon [7], applying the method of separation of variables, constructed a solution of the problem in the domains $D_+$ and $D_-$ as a Fourier series. He also proved an existence theorem under the condition $\varphi(x)$, $\psi(x)\in C^4[0,1]$, $\varphi(0)=\varphi(1)=\varphi''(0) =\varphi''(1)=\psi(0)=\psi(1)=\psi''(0)=\psi''(1)=0$, and for $\alpha=p, p/2, p/3, \dots$, where $p=1,2,3, \dots$, and $\alpha=p/q$, $(p,q)=1$, $np=mq+r$, $n\in \mathbb{N}$, $m$, $r\in {\mathbb{N}}_0=\mathbb{N}\cup \{0\}$, $0\leqslant r<q$, $\min_{0\leqslant p<q}|r/q-3/4|\geqslant\delta_q>0$, $n>N_q=\mathrm{const}>0$. Hačev [8] studied the Dirichlet problem for the generalized Lavrent’ev–Bitsadze equation
$$
\begin{equation*}
\operatorname{sgn} y[a(x)u_{xx}+b(x)u_x+c(x)u]+u_{yy}=0
\end{equation*}
\notag
$$
in the domain $D$ for $l=1$, where $a(x),b(x),c(x)\in C[0,1]$, $a(x)\geqslant a_0=\mathrm{const}>0$, $c(x)\leqslant c_0=\mathrm{const}<0$. He established a criterion for uniqueness of a solution, and constructed a solution as a Fourier series in the domains $D_+$ and $D_-$ in the system of eigenfunctions of the Sturm–Liouville problem. However, the proof of uniform convergence of the constructed series, which is a part of the existence theorem of a solution, contains gaps due to small denominators. In the present paper, we show that well-posedness of the problem (1.3)–(1.6) depends substantially on the ratio of the sides $\widetilde{\alpha}=\alpha/l$ of the rectangle $D_-$ due to the hyperbolic part of the mixed domain $D$. We will also establish a criterion for uniqueness of a solution of the Dirichlet problem. Proposition 1. If problem (1.3)–(1.6) has a solution, then this solution is unique if and only if, for all $k\in \mathbb{N}$, does not vanish, where A solution of problem (1.3)–(1.6) under the condition $\Delta_k(\alpha,\beta)\neq 0$ for all $k\in \mathbb{N}$ is constructed as the sum of the Fourier series: where
$$
\begin{equation}
\begin{gathered} \, u_k(y)=\begin{cases} \dfrac{1}{\Delta_k(\alpha,\beta)}\biggl[\varphi_k\Delta_k(\alpha,y)+\psi_k\sinh\lambda_k(\beta-y) \\ \quad{-}\dfrac{2f_{1k}}{\lambda_k}J_{1k}(y)+\dfrac{f_{2k}}{\lambda_k}g_{2k}(-\alpha) \sinh\lambda_k(\beta-y)\biggr], &y\geqslant 0, \\ \dfrac{1}{\Delta_k(\alpha,\beta)}\biggl[\varphi_k\sin\lambda_k(\alpha+y) +\psi_k\Delta_k(-y,\beta) \\ \quad{-}\dfrac{2f_{1k}}{\lambda_k}g_{1k}(\beta)\sin\lambda_k(y+\alpha)+ +\dfrac{f_{2k}}{\lambda_k}J_{2k}(y)\biggr], &y\leqslant 0, \end{cases} \\ X_k(x)=\sqrt{\frac{2}{l}}\sin\mu_k x, \nonumber \\ f_{ik}=\int_0^lf_i(x)X_k(x)\, dx,\qquad i=1,2, \nonumber \\ g_{1k}(y)=\int_0^yg_1(t)\sinh[\lambda_k(y-t)]\, dt,\qquad g_{2k}(y)=\int_y^0g_2(t)\sin[\lambda_k(y-t)]\, dt, \nonumber \\ \varphi_k=\int_0^l\varphi(x)X_k(x)\, dx,\qquad \psi_k=\int_0^l\psi(x)X_k(x)\, dx, \nonumber \\ \begin{aligned} \, J_{1k}(y) &=g_{1k}(\beta)\Delta_k(\alpha,y)-g_{1k}(y)\Delta_k(\alpha, \beta), \\ J_{2k}(y) &=g_{2k}(-\alpha)\Delta_k(-y,\beta)-g_{2k}(y)\Delta_k(\alpha, \beta). \end{aligned} \nonumber \end{gathered}
\end{equation}
\tag{1.10}
$$
For
$$
\begin{equation*}
\widetilde{\alpha}=\frac{n}{k \widetilde{\lambda}_k}-\frac{\gamma_k}{\pi k \widetilde{\lambda}_k}, \qquad n,k\in\mathbb{N},
\end{equation*}
\notag
$$
where
$$
\begin{equation*}
\widetilde{\lambda}_k=\biggl[1+\biggl( \frac{\sqrt{b}\, l}{\pi k}\biggr)^2 \biggr]^{1/2},\qquad \gamma_k =\arcsin\frac{\sinh\lambda_k\beta}{\sqrt{\cosh^2\lambda_k\beta+\sinh^2 \lambda_k\beta}},
\end{equation*}
\notag
$$
the denominator $\Delta_k(\alpha,\beta)$ in (1.10) vanishes, and hence we are led the small denominator problem, as in Arnol’d [9], [10] and Kozlov [11], but with more involved structure. In this problem, we prove in § 3 estimates for $\Delta_k(\alpha,\beta)$ to be bounded from zero with corresponding asymptotics depending on $\widetilde{\alpha}$ and $b$. From these estimates, under some sufficient conditions for functions $\varphi(x)$, $\psi(x)$, and $F(x,y)$, we prove a solvability theorem for the problem (see § 4). For an example, we present one solvability theorem for equation (1.7). Proposition 2. If $\widetilde{\alpha}\in \mathbb{N}$, $b=0$, $\varphi(x), \psi(x)\in C^3[0,l]$, $\varphi(0)=\psi(0)=\varphi''(0)=\psi''(0)=\varphi(l)=\psi(l)=\varphi''(l)=\psi''(l)=0$, $f_i (x)\in C^2[0,l]$, $f_i(0)=f_i(l)= 0$, $i=1,2$, $g_1(t)\in C[0,\beta]$, $g_2(t)\in C[-\alpha,0]$, then there exists a unique solution of problem (1.3)–(1.6), and this solution is given by series (1.9). In § 5, we will obtain estimates for stability of the solution of the problem with respect to given functions $\varphi(x)$, $\psi(x)$, and $f_i(x)$, $i=1,2$. Proposition 3. Under the conditions of Proposition 2, the solution of problem (1.3)–(1.6) satisfies Note that the right-hand side $F(x,y)$ of equation (1.1) is taken in the form (1.2) for convenience of the proof of solvability theorems for the problem, that is, for justification of the convergence of series (1.9) in the class of functions (1.3), and, later, for dealing with inverse problems for determining the functions $(u, f_1=f_2)$, $(u, f_1, f_2)$, $(u,g_1)$, $(u,g_2)$ and $(u,g_1,g_2)$. § 2. Criterion for uniqueness of a solutionLet $u(x,y)$ be a solution of problem (1.3)–(1.6), and let $F(x,y)\in C(D_+\cup D_-)\cap L(D_+\cup D_-)$. Following [12], we consider the functions
$$
\begin{equation}
u_k(y)=\sqrt{\frac{2}{l}}\, \int_0^l u(x,y)\sin\mu_kx\,dx=\int_0^l u(x,y)X_k(x)\, dx,
\end{equation}
\tag{2.1}
$$
where $X_k(x)=\sqrt{2/l}\sin\mu_kx$, $ \mu_k=\pi k/l$, $k\in \mathbb{N}$, form a complete orthonormal basis for $L_2[0,l]$. Differentiating (2.1) twice with respect to $y$, for $y>0$ and $y<0$, using (1.1), and then integrating by parts twice in the integrals involving the derivative $u_{xx}$, we find that
$$
\begin{equation}
u''_k(y)-\lambda_k^2u_k(y) =f_{1k}g_1(y), \qquad y >0,
\end{equation}
\tag{2.2}
$$
$$
\begin{equation}
u''_k(y)+\lambda_k^2u_k(y) =-f_{2k}g_2(y), \qquad y <0,
\end{equation}
\tag{2.3}
$$
where
$$
\begin{equation}
f_{ik}=\int_0^lf_{i}(x)X_k(x)\, dx,\quad i=1,2,\qquad \lambda_k^2=b+\mu_k^2.
\end{equation}
\tag{2.4}
$$
In what follows, we will assume that $b\geqslant 0$, since for $b<0$, there exists $k_0$ such that $b+\mu_k^2>0$ for all $k>k_0$. A general solution of the differential equation (2.2) is given by
$$
\begin{equation}
u_k(y)=a_k e^{\lambda_k y}+b_k e^{-\lambda_k y}+\frac{f_{1k}}{\lambda_k}g_{1k}(y), \qquad y>0,
\end{equation}
\tag{2.5}
$$
where $a_k$ and $b_k$ are arbitrary constants, A general solution of the differential equation (2.3) is given by
$$
\begin{equation}
u_k(y)=c_k \cos\lambda_k y+d_k \sin\lambda_k y-\frac{f_{2k}}{\lambda_k}g_{2k}(y),\qquad y<0,
\end{equation}
\tag{2.6}
$$
where $c_k$ and $d_k$ are arbitrary constants, In view of (1.3), functions (2.5) and (2.6) should satisfy the conjugation conditions Substituting functions (2.5) and (2.6) into conditions (2.7), we find that With these $a_k$ and $b_k$, formulas (2.5) and (2.6) assume the form
$$
\begin{equation}
u_k(y)=\begin{cases} c_k\cosh\lambda_k y+d_k\sinh\lambda_k y+\dfrac{f_{1k}}{\lambda_k}g_{1k}(y), &y\geqslant 0, \\ c_k\cos\lambda_k y+d_k\sin\lambda_k y-\dfrac{f_{2k}}{\lambda_k}g_{2k}(y), &y\leqslant 0. \end{cases}
\end{equation}
\tag{2.8}
$$
To evaluate $c_k$ and $d_k$, we will use the boundary conditions (1.6) and formula (2.1). We have
$$
\begin{equation}
u_k(\beta) =\int_0^lu(x,\beta)X_k(x)\, dx=\int_0^l\varphi(x)X_k(x)\, dx=\varphi_k,
\end{equation}
\tag{2.9}
$$
$$
\begin{equation}
u_k(-\alpha) =\int_0^lu(x,-\alpha)X_k(x)\, dx=\int_0^l\psi(x)X_k(x)\, dx=\psi_k.
\end{equation}
\tag{2.10}
$$
Having satisfied functions (2.8) with the boundary conditions (2.9) and (2.10), we get the system for the unknowns $c_k$ and $d_k$:
$$
\begin{equation}
\begin{cases} c_k\cosh\lambda_k \beta+d_k\sinh\lambda_k\beta =\varphi_k -\dfrac{f_{1k}}{\lambda_k}g_{1k}(\beta), \\ c_k\cos\lambda_k\alpha-d_k\sin\lambda_k \alpha =\psi_k +\dfrac{f_{2k}}{\lambda_k}g_{2k}(-\alpha). \end{cases}
\end{equation}
\tag{2.11}
$$
If the determinant
$$
\begin{equation}
-\Delta_k(\alpha,\beta)=-(\cosh\lambda_k\beta\sin\lambda_k\alpha +\sinh\lambda_k\beta\cos\lambda_k\alpha)
\end{equation}
\tag{2.12}
$$
of system (2.11) does not vanish, then this system has a unique solution
$$
\begin{equation*}
\begin{aligned} \, c_k &=\frac{1}{\Delta_k(\alpha,\beta)}\biggl[\varphi_k\sin\lambda_k\alpha +\psi_k\sinh\lambda_k\beta \\ &\qquad-\frac{f_{1k}}{\lambda_k}g_{1k}(\beta)\sin\lambda_k\alpha+ \frac{f_{2k}}{\lambda_k}g_{2k}(-\alpha)\sinh\lambda_k\beta\biggr], \\ d_k &=\frac{1}{\Delta_k(\alpha,\beta)}\biggl[\varphi_k\cos\lambda_k\alpha -\psi_k\cosh\lambda_k\beta \\ &\qquad-\frac{f_{1k}}{\lambda_k}g_{1k}(\beta)\cos\lambda_k\alpha- \frac{f_{2k}}{\lambda_k}g_{2k}(-\alpha)\cosh\lambda_k\beta\biggr]. \end{aligned}
\end{equation*}
\notag
$$
We now substitute the above $c_k$ and $d_k$ into (2.8). As a result, we find the functions
$$
\begin{equation}
u_k(y)=\begin{cases} \dfrac{1}{\Delta_k(\alpha,\beta)}\biggl[\varphi_k\Delta_k(\alpha,y) +\psi_k\sinh\lambda_k(\beta-y)-\dfrac{f_{1k}}{\lambda_k}g_{1k}(\beta)\Delta_k(\alpha,y) \\ \quad{+}\dfrac{f_{2k}}{\lambda_k}g_{2k}(-\alpha)\sinh\lambda_k(\beta-y)\biggr] +\dfrac{f_{1k}}{\lambda_k}g_{1k}(y), \qquad y\geqslant 0, \\ \dfrac{1}{\Delta_k(\alpha,\beta)}\biggl[\varphi_k\sin\lambda_k(\alpha+y) +\psi_k\Delta_k(-y,\beta)-\dfrac{f_{1k}}{\lambda_k}g_{1k}(\beta)\sin\lambda_k(y+\alpha) \\ \quad{+}\dfrac{f_{2k}}{\lambda_k}g_{2k}(-\alpha)\Delta_k(-y,\beta)\biggr] -\dfrac{f_{1k}}{\lambda_k}g_{2k}(y), \qquad\quad y\leqslant 0, \end{cases}
\end{equation}
\tag{2.13}
$$
in their final form, where
$$
\begin{equation*}
\begin{aligned} \, \Delta_k(\alpha,y) &=\cosh\lambda_k y \sin\lambda_k\alpha+\sinh\lambda_ky\cos\lambda_k\alpha,\qquad y>0, \\ \Delta_k(-y,\beta) &=\sinh\lambda_k \beta \cos\lambda_k y-\cosh\lambda_k\beta\sin\lambda_k y, \qquad y<0. \end{aligned}
\end{equation*}
\notag
$$
We next consider the difference
$$
\begin{equation*}
J_{1k}(y)=g_{1k}(\beta)\Delta_k(\alpha,y)-g_{1k}(y)\Delta_k(\alpha,\beta)
\end{equation*}
\notag
$$
and transform it to single out the function $g_1(t)$. To this end, using the above formulas for $g_{1k}(y)$ and $\Delta_k(\alpha,y)$, we have
$$
\begin{equation}
\begin{aligned} \, &g_{1k}(\beta)\Delta_k(\alpha,y)-g_{1k}(y)\Delta_k(\alpha,\beta) \nonumber \\ &\qquad=\int_0^yg_1(t)[\sinh\lambda_k(\beta-t)\Delta_k(\alpha,y)- \sinh\lambda_k(y-t)\Delta_k(\alpha,\beta)]\, dt \nonumber \\ &\qquad\qquad+\int_y^{\beta}g_1(t)\sinh\lambda_k(\beta-t)\Delta_k(\alpha,y)\, dt =J_{11}+J_{12}. \end{aligned}
\end{equation}
\tag{2.14}
$$
Let us first evaluate the difference in the integral $J_{11}$. We have
$$
\begin{equation*}
\begin{aligned} \, &\sinh\lambda_k(\beta-t)\Delta_k(\alpha,y)-\sinh\lambda_k(y-t)\Delta_k(\alpha,\beta) \\ &\qquad=\sinh\lambda_k(\beta-t)(\cosh\lambda_k y \sin\lambda_k\alpha +\sinh\lambda_ky\cos\lambda_k\alpha) \\ &\qquad\qquad-\sinh\lambda_k(y-t)(\cosh\lambda_k \beta \sin\lambda_k\alpha +\sinh\lambda_k\beta\cos\lambda_k\alpha) \\ &\qquad=\sin\lambda_k\alpha[\sinh\lambda_k(\beta-t)\cosh\lambda_k y -\sinh\lambda_k(y-t)\cosh\lambda_k \beta] \\ &\qquad\qquad+\cos\lambda_k\alpha[\sinh\lambda_k(\beta-t)\sinh\lambda_k y-\sinh\lambda_k(y-t)\sinh\lambda_k \beta] \\ &\qquad=\sinh\lambda_k(\beta-y)\Delta_k(\alpha,t). \end{aligned}
\end{equation*}
\notag
$$
Now the integral $J_{11}$ and equality (2.14) assume the form
$$
\begin{equation}
\begin{aligned} \, J_{11} &=\sinh\lambda_k(\beta-y)\int_0^yg_1(t)\Delta_k(\alpha,t)\, dt, \end{aligned}
\end{equation}
\tag{2.15}
$$
$$
\begin{equation}
\begin{aligned} \, J_{1k}(y) &=\sinh\lambda_k(\beta-y)\int_0^yg_1(t)\Delta_k(\alpha,t)\, dt \nonumber \\ &\qquad+\Delta_k(\alpha,y)\int_y^{\beta}g_1(t)\sinh\lambda_k(\beta-t)\, dt. \end{aligned}
\end{equation}
\tag{2.16}
$$
Proceeding similarly, we consider the difference
$$
\begin{equation}
\begin{aligned} \, J_{2k}(y) &=g_{2k}(-\alpha)\Delta_k(-y,\beta)-g_{2k}(y)\Delta_k(\alpha,\beta) \nonumber \\ &=\int_{-\alpha}^yg_2(t)\sin[\lambda_k(t+\alpha)]\Delta_k(-y,\beta)\, dt \nonumber \\ &\qquad+\int_y^0g_2(t) \bigl[\sin[\lambda_k(t+\alpha)]\Delta_k(-y,\beta) -\sin[\lambda_k(t-y)]\Delta_k(\alpha,\beta)\bigr]\, dt \nonumber \\ &=J_{21}+J_{22} \end{aligned}
\end{equation}
\tag{2.17}
$$
and find the difference in the integral $J_{22}$. We have
$$
\begin{equation*}
\begin{aligned} \, \!&\sin[\lambda_k(t+\alpha)]\Delta_k(-y,\beta)-\sin[\lambda_k(t-y)]\Delta_k(\alpha,\beta) \\ &\qquad =\sin[\lambda_k(t+\alpha)](\sinh\lambda_k \beta \cos\lambda_k y-\cosh\lambda_k\beta\sin\lambda_k y) \\ &\qquad\qquad-\sin[\lambda_k(t-y)](\cosh\lambda_k\beta\sin\lambda_k\alpha +\sinh\lambda_k\beta\cos\lambda_k\alpha) \\ &\qquad=\sinh\lambda_k\beta\bigl(\sin\lambda_k(t+\alpha)\cos\lambda_k y- \sin\lambda_k(t-y)\cos\lambda_k\alpha\bigr) \\ &\qquad\qquad-\cosh\lambda_k\beta\bigl(\sin\lambda_k(t+\alpha)\sin\lambda_k y+ \sin\lambda_k(t-y)\sin\lambda_k\alpha\bigr) \\ &\qquad=\frac{1}{2}\sinh\lambda_k\beta[\sin\lambda_k(t+\alpha+y)+\sin\lambda_k(t+\alpha-y) \\ &\qquad\qquad-\sin\lambda_k(t+\alpha-y)-\sin\lambda_k(t-\alpha-y)] \\ &\qquad\qquad-\frac{1}{2}\cosh\lambda_k\beta[\cos\lambda_k(t+\alpha-y) \\ &\qquad\qquad-\cos\lambda_k(t+\alpha+y) +\cos\lambda_k(t-\alpha-y)-\cos\lambda_k(t+\alpha-y)] \\ &\qquad=\frac{1}{2}\sinh\lambda_k\beta[\sin\lambda_k(t+\alpha+y)-\sin\lambda_k(t-\alpha-y)] \\ &\qquad\qquad-\frac{1}{2}\cosh\lambda_k\beta[\cos\lambda_k(t-\alpha-y) -\cos\lambda_k(t+\alpha+y)] \\ &\qquad=\sinh\lambda_k\beta \sin\lambda_k(\alpha+y)\cos\lambda_k t+\cosh\lambda_k\beta \sin\lambda_k t\sin\lambda_k(-\alpha-y) \\ &\qquad=\sin\lambda_k(\alpha+y)(\sinh\lambda_k\beta\cos\lambda_k t-\cosh\lambda_k\beta\sin\lambda_k t)=\sin\lambda_k(\alpha+y)\Delta_k(-t,\beta). \end{aligned}
\end{equation*}
\notag
$$
Now the integral $J_{22}$ and expression (2.17) assume the form
$$
\begin{equation}
J_{22}=\sin\lambda_k(\alpha+y)\int_y^0g_2(t)\Delta_k(-t,\beta)\, dt
\end{equation}
\tag{2.18}
$$
and
$$
\begin{equation}
\begin{aligned} \, J_{2k}(y) &=\Delta_k(-y,\beta)\int_{-\alpha}^yg_2(t)\sin[\lambda_k(t+\alpha)]\, dt \nonumber \\ &\qquad+\sin\lambda_k(\alpha+y)\int_y^0g_2(t)\Delta_k(-t,\beta)\, dt. \end{aligned}
\end{equation}
\tag{2.19}
$$
In view of equalities (2.16) and (2.19), we can write (2.13) as
$$
\begin{equation}
u_k(y)=\begin{cases} \dfrac{1}{\Delta_k(\alpha,\beta)}\biggl[\varphi_k\Delta_k(\alpha,y)+\psi_k\sinh\lambda_k(\beta-y) \\ \quad{-}\dfrac{f_{1k}}{\lambda_k}J_{1k}(y)+\dfrac{f_{2k}}{\lambda_k}g_{2k}(-\alpha) \sinh\lambda_k(\beta-y)\biggr], &y\geqslant 0, \\ \dfrac{1}{\Delta_k(\alpha,\beta)}\biggl[\varphi_k\sin\lambda_k(\alpha+y) +\psi_k\Delta_k(-y,\beta) \\ \quad{-}\dfrac{f_{1k}}{\lambda_k}g_{1k}(\beta)\sin\lambda_k(y+\alpha)+ \dfrac{f_{2k}}{\lambda_k}J_{2k}(y)\biggr], &y\leqslant 0. \end{cases}
\end{equation}
\tag{2.20}
$$
Now we can prove unique solvability of problem (1.3)–(1.6). Let $\varphi(x)\,{=}\,\psi(x)\,{\equiv}\, 0$, let $F(x,y)\equiv 0$ in $D_+\cup D_-$, and let conditions (2.12) be met for all $k\in \mathbb{N}$. Then $\varphi_k=\psi_k=f_{1k}=f_{2k}\equiv 0$ for all $k$, and now from (2.20) and (2.1) we have, for all $k\in \mathbb{N}$ and any $y\in [-\alpha,\beta]$, The system $X_k(x)$ is compete in $L_2[0,l]$, and hence $u(x,y)=0$ almost everywhere on $[0,l]$ for each $y\in [-\alpha,\beta]$. By (1.3), the function $u(x,y)$ is continuous on $\overline{D}$, and hence $u(x,y)\equiv 0$ in $\overline{D}$.If condition (2.12) is violated for some $l$, $\alpha$, $\beta$ and $k=p\in\mathbb{N}$, that is, $\Delta_p(\alpha,\beta)\,{=}\, 0$, then the homogeneous problem (1.3)–(1.6) (where $\varphi(x)=\psi(x) \equiv 0$, $F(x,y)\equiv 0$) has non-trivial solutions
$$
\begin{equation}
u_{p}(x,y)=u_p(y)X_p(x)= \begin{cases} C_p\sinh\lambda_{p}(\beta-y)X_p(x), &y\geqslant 0, \\ C_p\Delta_p(-y,\beta)X_p(x), &y\leqslant 0, \end{cases}
\end{equation}
\tag{2.21}
$$
where $C_p \neq 0$ is an arbitrary constant. This leads naturally to the problem of whether $\Delta_k(\alpha,\beta)$ has zeros. To this end, we write $\Delta_k(\alpha,\beta)$ in the form
$$
\begin{equation}
\begin{gathered} \, \Delta_k(\alpha,\beta)=\sqrt{\cosh^2\lambda_k\beta+\sinh^2\lambda_k\beta}\sin(\pi k \widetilde{\alpha}\widetilde{\lambda}_k+\gamma_k), \\ \widetilde{\alpha}=\frac{\alpha}{l},\qquad \widetilde{\lambda}_k=\biggl[1+\biggl(\frac{\sqrt{b}\, l}{\pi k}\biggr)^2\biggr]^{1/2}, \nonumber \end{gathered}
\end{equation}
\tag{2.22}
$$
where
$$
\begin{equation*}
\gamma_k=\arcsin\frac{\sinh\lambda_k\beta}{\sqrt{\cosh^2\lambda_k\beta+\sinh^2\lambda_k\beta}}.
\end{equation*}
\notag
$$
From (2.22) we find
$$
\begin{equation}
\widetilde{\alpha}=\frac{n}{k\widetilde{\lambda}_k}-\frac{\gamma_k}{\pi k \widetilde{\lambda}_k}, \qquad n,k\in \mathbb{N},
\end{equation}
\tag{2.23}
$$
for which conditions (2.12) are violated. Therefore, we have the following criterion for uniqueness of solution of problem (1.3)–(1.6). Theorem 1. If problem (1.3)–(1.6) has a solution, then this solution is unique if and only if conditions (2.12) are satisfied for all $k\in \mathbb{N}$, that is, $\Delta_k(\alpha,\beta)\neq 0$ for all $k\in \mathbb{N}$. For $\widetilde{\alpha}=\alpha/l$ from (2.23) expression (2.22) vanishes, and hence we are led to the small denominator problem, as in [9]–[11], but with more involved structure. Hence, to justify solvability of problem (1.3)–(1.6), we need to find estimates for $\Delta_k(\alpha,\beta)$ to be bounded from zero and find the corresponding asymptotic formulas. § 3. Estimates for small denominatorsLet us consider separately the cases $b=0$ and $b\neq 0$. Let $b=0$. In this case, $\lambda_k=\mu_k$, $\widetilde{\lambda}_k\equiv 1$. Now (2.22) assumes the form
$$
\begin{equation}
\Delta_k(\widetilde{\alpha},\widetilde{\beta})=\sqrt{\cosh^2\pi k\widetilde{\beta}+\sinh^2\pi k\widetilde{\beta}}\, \delta_k(\widetilde{\alpha}),
\end{equation}
\tag{3.1}
$$
where
$$
\begin{equation*}
\begin{gathered} \, \delta_k(\widetilde{\alpha})=\sin(\pi k \widetilde{\alpha}+\gamma_k),\qquad \widetilde{\beta}=\frac{\beta}{l},\qquad \widetilde{\alpha}=\frac{\alpha}{l}, \\ \gamma_k=\arcsin\frac{\sinh\pi k\widetilde{\beta}}{\sqrt{\cosh^2\pi k\widetilde{\beta}+\sinh^2\pi k\widetilde{\beta}}}. \end{gathered}
\end{equation*}
\notag
$$
Lemma 1. If $\widetilde{\alpha}\in \mathbb{N}$, then, for each $\widetilde{\beta}>0$, there exists a constant $C_1=C_1(\widetilde{\beta})>0$ such that, for all $k\in \mathbb{N}$, Proof. Let $\widetilde{\alpha}=p$ be a natural number. Then
$$
\begin{equation*}
\begin{aligned} \, |\Delta_k(\widetilde{\alpha},\widetilde{\beta})| &=\sqrt{\cosh^2\pi k\widetilde{\beta}+\sinh^2\pi k\widetilde{\beta}}\, |{\sin(\pi k p+\gamma_k)}| \\ &=\sqrt{\cosh^2\pi k\widetilde{\beta}+\sinh^2\pi k\widetilde{\beta}}\, |{\sin\gamma_k}|=\sinh\pi k \widetilde{\beta} \\ &=\frac{1}{2}\, e^{\pi k \widetilde{\beta}}(1-e^{-2 \pi k \widetilde{\beta}})\geqslant e^{\pi k \widetilde{\beta}}\frac{1-e^{-2\pi \widetilde{\beta}}}{2}=C_1 e^{\pi k \widetilde{\beta}}, \end{aligned}
\end{equation*}
\notag
$$
which proves estimate (3.2), and, therefore, the lemma. Lemma 2. Let $\widetilde{\alpha}$ be a fractional number, that is, $\widetilde{\alpha}=p/q$, $(p,q)=1$, $p/q\notin\mathbb{N}$, $(q,4)=1$. Then there exist positive constants $\beta_0=\beta_0(\widetilde{\alpha})$, $C_2=C_2(\widetilde{\beta})$ such that, for all $\widetilde{\beta}>\beta_0$ Proof. Let $\widetilde{\alpha}=p/q\notin\mathbb{N}$, $(p,q)=1$. We divide $kp$ by $q$ with $kp=sq+r$ as a remainder, where $s,r\in \mathbb{N}_0$, $0\leqslant r< q$, and $s$, $r$ depend in general on $k$. Then $\delta_k(\widetilde{\alpha})$ can be written as
The case $r=0$ reduces to the above case $\widetilde{\alpha}=p\in \mathbb{N}$. Let $0<r<q$. It is clear that $1\leqslant r \leqslant q-1$, $q\geqslant 2$. Note that $\gamma_k\to \pi/4$ as $k\to +\infty$. The function
$$
\begin{equation*}
y=\arcsin u, \qquad u=\frac{\sinh u}{\sqrt{\sinh^2 u+\cosh^2 u}}
\end{equation*}
\notag
$$
is increasing, and hence and where $\varepsilon_k>0$ and $\varepsilon_k\to 0$ as $k\to +\infty$.
Applying the elementary formula
$$
\begin{equation*}
\arcsin x -\arcsin y=\arcsin\bigl(x\sqrt{1-y^2}-y\sqrt{1-x^2}\bigr),\qquad xy>0,
\end{equation*}
\notag
$$
we obtain the estimate
$$
\begin{equation}
\begin{aligned} \, 0 <\varepsilon_k &=\frac{\pi}{4}-\gamma_k=\arcsin\frac{1}{\sqrt{2}}-\arcsin\frac{\sinh\pi k \widetilde{\beta}}{\sqrt{\cosh^2\pi k\widetilde{\beta}+\sinh^2\pi k\widetilde{\beta}}} \nonumber \\ &=\arcsin\frac{1}{\sqrt{2}} \, \frac{\cosh\pi k\widetilde{\beta} -\sinh\pi k \widetilde{\beta}}{\sqrt{\cosh^2\pi k\widetilde{\beta}+\sinh^2\pi k\widetilde{\beta}}} \nonumber \\ &=\arcsin\frac{e^{-\pi k \widetilde{\beta}}}{\sqrt{2}\sqrt{\cosh^2\pi k\widetilde{\beta}+\sinh^2\pi k\widetilde{\beta}}}<\frac{\pi}{2e^{2\pi k\widetilde{\beta} }}. \end{aligned}
\end{equation}
\tag{3.7}
$$
$$
\begin{equation}
|\delta_k(\widetilde{\alpha})|=\biggl|\sin\biggl[\frac{\pi(4r+q)}{4q}-\varepsilon_k\biggr] \biggr|.
\end{equation}
\tag{3.8}
$$
If $2\leqslant q \leqslant 3$, then
$$
\begin{equation*}
\frac{7\pi }{12}\leqslant \frac{\pi(4r+q)}{4q}\leqslant \frac{11\pi}{12}\quad\text{and}\quad \frac{7\pi}{12}-\varepsilon_k< \frac{\pi(4r+q)}{4q}-\varepsilon_k< \frac{11\pi}{12}.
\end{equation*}
\notag
$$
From (3.7) we have
$$
\begin{equation*}
\frac{7\pi}{12}-\varepsilon_k>\frac{7\pi}{12}-\frac{\pi}{2}\, e^{-2\pi k \widetilde{\beta}}\geqslant \frac{\pi}{2}\biggl(\frac{7}{6}-\frac{1}{e^{2\pi \widetilde{\beta}}}\biggr)>0,
\end{equation*}
\notag
$$
and now from (3.8) it follows that
$$
\begin{equation}
|\delta_k(\widetilde{\alpha})|>\min\biggl\{\sin\frac{11\pi}{12},\, \sin\frac{\pi}{2}\biggl(\frac{7}{6}-\frac{1}{e^{2\pi \widetilde{\beta}}}\biggr)\biggr\}>0.
\end{equation}
\tag{3.9}
$$
Now let $q\geqslant 4$. Then the fraction $(4r+q)/(4q)$ can be greater than or equal to 1. Indeed,
$$
\begin{equation*}
\frac{1}{4}<\frac{4+q}{4q}\leqslant \frac{4r+q}{4q}\leqslant \frac{4(q-1)+q}{4q} =1+\frac{q-4}{4q}<1+\frac{1}{4}.
\end{equation*}
\notag
$$
Hence
$$
\begin{equation*}
\frac{4r+q}{4q}=1\quad\Longleftrightarrow\quad r=\frac{3q}{4}.
\end{equation*}
\notag
$$
By the assumption, $(q,4)=1$, and hence $r$ is a fractional number. But this contradicts the fact that $r$ is a natural number. Therefore, $(4r+q)/(4q)\ne 1$.
We again divide $4r+q$ by $4q$ with $4r+q=s_1 4q +r_1$ as a remainder, where $s_1=0$ or $s_1=1$, $r_1\in \mathbb{N}$, $1\leqslant r_1\leqslant 4q-1$. Note that $r_1\ne 0$, for otherwise we get a contradiction with the condition $(q,4)=1$. Now (3.8) assumes the form
$$
\begin{equation}
|\delta_k(\widetilde{\alpha})|=\biggl|\sin\biggl(\frac{\pi r_1}{4q}-\varepsilon_k\biggr)\biggr|.
\end{equation}
\tag{3.10}
$$
By estimate (3.7),
$$
\begin{equation*}
\frac{\pi}{4q}-\frac{\pi}{2e^{2\pi \widetilde{\beta}}}<\frac{\pi r_1}{4q}-\varepsilon_k<\pi-\frac{\pi}{4q}.
\end{equation*}
\notag
$$
Hence if $\widetilde{\beta}>\beta_0=(1/(2\pi))\ln 2q$, then, for all $k\in\mathbb{N}$, from (3.10) we have
$$
\begin{equation}
|\delta_k(\widetilde{\alpha})|>\min\biggl\{ \sin\frac{\pi}{2}\biggl(\frac{1}{2q}-\frac{1}{e^{2\pi \widetilde{\beta}}}\biggr),\, \sin \pi \biggl(1-\frac{1}{4q}\biggr)\biggr\}.
\end{equation}
\tag{3.11}
$$
Now estimate (3.3) for all $k\,{\in}\, \mathbb{N}$ is secured by the above estimates (3.9) and (3.11). Lemma 2 is proved. Remark 1. If $(q,4)\ne 1$, that is, $q=4l$, $l\in \mathbb{N}$, then, in this case $(4r+q)/(4q)=1$, for $r=3l$. Now from (3.7) and (3.10) we have Using the last equality, we can write (3.1) as
$$
\begin{equation}
\Delta_k(\widetilde{\alpha},\widetilde{\beta} )=\frac{1}{\sqrt{2}}\, e^{-\pi k\widetilde{\beta}}.
\end{equation}
\tag{3.12}
$$
From (3.12) it follows that the denominator $\Delta_k(\widetilde{\alpha},\widetilde{\beta})$ tends to zero exponentially fast as $k\to +\infty$. In this case, there is no solution of the Dirichlet problem in the form of a series. Lemma 3. If $\widetilde{\alpha}>0$ is an irrational algebraic number of degree $m\geqslant 2$, then there exist positive constants $\beta_1$ and $C_3$ such that, for all $\beta>\beta_1$ and $k\in \mathbb{N}$, where $\gamma>0$ is a sufficiently small number. Proof. Let $\widetilde{\alpha}$ be any irrational algebraic number of degree $m\geqslant 2$. By the Roth theorem (see [13], Chap. 29, § 2]), for $\widetilde{\alpha}$ and an arbitrary positive number $\gamma>0$, there exists a positive number $\gamma_0$ depending on $\widetilde{\alpha}$ and $\gamma$ such that, for all integer $p$, $q$ $(q>0)$,
According to [14], § 2.1, for each $k\in \mathbb{N}$, there exists $n\in \mathbb{N}$ such that
$$
\begin{equation}
\biggl|\widetilde{\alpha}-\frac{n}{k}\biggr|<\frac{1}{2k}.
\end{equation}
\tag{3.16}
$$
Let $n\in \mathbb{N}$ satisfy inequality (3.16), or what is the same,
$$
\begin{equation}
\biggl|\pi k \biggl(\widetilde{\alpha}-\frac{n}{k}\biggr)\biggr|<\frac{\pi}{2}.
\end{equation}
\tag{3.17}
$$
Now from inequalities (3.15) and (3.17) we have
$$
\begin{equation}
\frac{\pi \gamma_0}{k^{1+\gamma}}<\pi k\biggl| \widetilde{\alpha}-\frac{n}{k}\biggr|<\frac{\pi}{2}.
\end{equation}
\tag{3.18}
$$
In view of estimates (3.18) and (3.5), there are two cases to consider:
$$
\begin{equation*}
\begin{aligned} \, &1) \quad 0<\biggl|\pi k \biggl(\widetilde{\alpha}-\frac{n}{k}\biggr) +\gamma_k\biggr| <\frac{\pi}{2}, \\ &2) \quad \frac{\pi}{2}\leqslant \biggl|\pi k \biggl(\widetilde{\alpha}-\frac{n}{k} \biggr) +\gamma_k\biggr| <\frac{3\pi}{4}. \end{aligned}
\end{equation*}
\notag
$$
In the first case, using the inequality we have
$$
\begin{equation}
\biggl|\sin\biggl[\pi k \biggl(\widetilde{\alpha}-\frac{n}{k}\biggr) +\gamma_k\biggl]\biggr| >\frac{2}{\pi}\biggl|\pi k \biggl(\widetilde{\alpha}-\frac{n}{k}\biggr)+\gamma_k\biggr|.
\end{equation}
\tag{3.20}
$$
Let us estimate the right-hand side of inequality (3.20). We have
$$
\begin{equation}
\begin{aligned} \, |\pi k \widetilde{\alpha}-\pi n+\gamma_k| &= \biggl|\pi k \widetilde{\alpha}-\pi n +\frac{\pi}{4}-\varepsilon_k\biggr| \nonumber \\ &=\biggl|\pi k \widetilde{\alpha}-\pi \frac{4n-1}{4}-\varepsilon_k\biggr| >\pi k \biggl|\widetilde{\alpha}-\frac{4n-1}{4k}\biggr|-\varepsilon_k. \end{aligned}
\end{equation}
\tag{3.21}
$$
In view of (3.15), the first term on the right of (3.21) is estimated as
$$
\begin{equation}
\pi k \biggl|\widetilde{\alpha}-\frac{4n-1}{4k}\biggr|>\frac{\pi \widetilde{\gamma}_0}{16k^{1+\gamma}}, \qquad \widetilde{\gamma}_0=\mathrm{const} >0.
\end{equation}
\tag{3.22}
$$
Now from estimates (3.20)–(3.22) and (3.7) we have
$$
\begin{equation}
\biggl|\sin\biggl[\pi k\biggl(\widetilde{\alpha}-\frac{n}{k}\biggr)+\gamma_k\biggr]\biggr|>\biggl(\frac{\pi \widetilde{\gamma}_0}{16 k^{1+\gamma}}-\frac{\pi}{2e^{2\pi k\beta}}\biggr)\frac{2}{\pi} =\frac{ \widetilde{\gamma}_0}{8 k^{1+\gamma}}-\frac{1}{e^{2\pi k\beta}}.
\end{equation}
\tag{3.23}
$$
Since $e^{2\pi k \widetilde{\beta}}>(2\pi k \widetilde{\beta})^{1+\gamma}$ for all $k$, we have by (3.23)
$$
\begin{equation}
\biggl|\sin\biggl[\pi k\biggl(\widetilde{\alpha}-\frac{n}{k}\biggr) +\gamma_k\biggr]\biggr|>\frac{1}{k^{1+\gamma}}\biggl[\frac{\widetilde{\gamma}_0}{8} -\frac{1}{(2\pi k\widetilde{\beta})^{1+\gamma}}\biggr] =\frac{\overline{\gamma}_0}{k^{1+\gamma}},
\end{equation}
\tag{3.24}
$$
where $\overline{\gamma}_0>0$ for $\widetilde{\beta}>\beta_0=(1/(2\pi))(8/\widetilde{\gamma}_0)^{1/(1+\gamma)}$.
In case 2), we have
$$
\begin{equation}
\biggl|\sin\biggl[\pi k\biggl(\widetilde{\alpha}-\frac{n}{k}\biggr) +\gamma_k\biggr]\biggr| >\sin\frac{3\pi}{4}=\frac{\sqrt{2}}{2}.
\end{equation}
\tag{3.25}
$$
Now estimate (3.13) follows from (3.1), (3.24) and (3.25).
For $m=2$, a more accurate result follows from the Liouville theorem (see [15], Chap. 2, § 9), that is, for each irrational algebraic number $\widetilde{\alpha}$ of degree $m=2$, there exists a positive number $\delta>0$ such that, for all integer $p$, $q$ $(q>0)$,
$$
\begin{equation}
\biggl|\widetilde{\alpha}-\frac{p}{q}\biggr|>\frac{\delta}{q^2}.
\end{equation}
\tag{3.26}
$$
Now using inequality (3.26) and proceeding as in the case $m>2$, we arrive at estimate (3.14). This proves Lemma 3. Note that the case of Lemmas 1–3 with $l=1$ was considered in [16], § 4. In the present paper, the results of [16] are used in the proof of Lemmas 1–3 with some refinements and supplements. In what follows, we will assume that $b\neq 0$. Lemma 4. Let $b>0$, let $\widetilde{\beta}$ be any positive real number, and let $\widetilde{\alpha}=p$ be a natural number. Then there exist positive constants $C_0$ and $k_0$ such that, for all $k>k_0$, Proof. In the case $b>0$, we can write $\widetilde{\lambda}_k$, which depends on $\sqrt{b} \, l$, as
$$
\begin{equation}
\widetilde{\lambda}_k=\biggl(1+\biggl(\frac{\sqrt{b}\, l}{\pi k}\biggr)^2\biggr)^{1/2} =1+\theta_k,
\end{equation}
\tag{3.28}
$$
provided that
$$
\begin{equation}
\frac{\sqrt{b}\, l}{\pi}<1\quad \text{or}\quad k>\frac{\sqrt{b}\, l}{\pi}=k_1.
\end{equation}
\tag{3.29}
$$
In addition, for $\theta_k$ we have the estimate (see [14], § 2.1) Now $\sin(\pi k\widetilde{\lambda}_k\widetilde{\alpha}-\gamma_k)$ can be written as
Let $\widetilde{\alpha}=p\in \mathbb{N}$. Then
$$
\begin{equation*}
\delta_k(p)=(-1)^p\sin\bigl(p \widetilde{\theta}_k-\gamma_k\bigr).
\end{equation*}
\notag
$$
In view of (3.30), the limit
$$
\begin{equation*}
\lim_{k\to \infty}|\delta_k(p)|=\sin\frac{\pi}{4}=\frac{1}{\sqrt{2}}
\end{equation*}
\notag
$$
exists and is finite. Hence there exists a natural number $k_2$ such that, for all $k>k_2$,
$$
\begin{equation*}
|\delta_k(p)|>\frac{1}{2}\lim_{k\to \infty}|\delta_k(p)|=\frac{1}{2\sqrt{2}}.
\end{equation*}
\notag
$$
From representation (3.1) it follows that
$$
\begin{equation*}
|\Delta_k(p,\widetilde{\beta})|\geqslant \frac{1}{\sqrt{2}}\, e^{\pi k \widetilde{\beta}} |\delta_k(p)|.
\end{equation*}
\notag
$$
This implies estimate (3.27) for $k>k_0=\max\{k_1, k_2\}$. Lemma 4 is proved. Lemma 5. Let $b>0$, let $\widetilde{\beta}$ be a positive number, and $\widetilde{\alpha}=p/q$, $(p,q)= 1$, $p,q \in\mathbb{N}$, $q\neq 4$. Then there exist positive constants $C_0$ and $k_0$ such that estimate (3.27) holds for all $k>k_0$. Proof. Proceeding as in the proof of Lemmas 2 and 4, we write $\delta_k(\widetilde{\alpha})$ as
For $r=0$, estimate (3.27) follows from Lemma 4. If $r>0$, then in view of (3.30) the lower limit
$$
\begin{equation*}
\varliminf_{k\to \infty}|\delta_k(\widetilde{\alpha})|=\varliminf_{k\to \infty} \biggl|\sin\biggl(\frac{\pi r}{q}-\frac{1}{4}\biggr)\biggr|=\widetilde{C}_1>0
\end{equation*}
\notag
$$
exists and is finite. Hence there exists $k_2$ such that, for all $k>k_2$,
$$
\begin{equation*}
|\delta_k(\widetilde{\alpha})|>\frac{1}{2}\, \widetilde{C}_1.
\end{equation*}
\notag
$$
Therefore, estimate (3.27) holds for $k>k_0=\max\{k_1,k_2\}$. Lemma 5 is proved. Lemma 6. Let $b>0$, let $\widetilde{\beta}$ be a positive number, and $\widetilde{\alpha}$ be an irrational algebraic number of degree $2$. Then there exists $\delta>0$ depending on $\widetilde{\alpha}$ such that $\pi^2\delta-8\widetilde{\alpha} b l^2> 0$, and there exist positive constants $C_0$ and $k_0$ such that, for all $k>k_0$, Proof. From the proofs of Lemmas 3 and 4 we have
$$
\begin{equation*}
\delta_k(\widetilde{\alpha})=(-1)^n \sin\biggl[\pi k \biggl(\widetilde{\alpha}-\frac{4n-1}{4k}\biggr) +\widetilde{\alpha}\widetilde{\theta}_k-\varepsilon_k\biggr].
\end{equation*}
\notag
$$
By the condition, $\widetilde{\alpha}$ is an algebraic number of degree $2$. Hence by the Liouville theorem (see [15], Chap. 2, § 9), there exists a positive number $\delta>0$ depending on $\widetilde{\alpha}$ such that, for all $m=4n-1$ and $k$,
$$
\begin{equation}
\biggl|\widetilde{\alpha}-\frac{m}{4k}\biggr|>\frac{\delta}{(4k)^2}.
\end{equation}
\tag{3.32}
$$
Note that, for each $k\in \mathbb{N}$, there exists a natural number $m=4n-1$ such that
$$
\begin{equation}
\biggl|\widetilde{\alpha}-\frac{m}{4k}\biggr|<\frac{1}{4k}.
\end{equation}
\tag{3.33}
$$
Using inequalities (3.32) and (3.33), we have
$$
\begin{equation*}
\frac{\pi \delta}{16 k}\leqslant \pi k \biggl|\widetilde{\alpha}-\frac{m}{4k}\biggr| <\frac{\pi}{4}.
\end{equation*}
\notag
$$
Proceeding as in the proof of Lemma 3, using inequality (3.19), and employing estimates (3.32), (3.30), we have
$$
\begin{equation}
\begin{aligned} \, |\delta_k(\widetilde{\alpha})| &>\frac{2}{\pi} \biggl|\pi k\biggl(\widetilde{\alpha}-\frac{m}{4k}\biggr) +\widetilde{\alpha}\theta_k-\varepsilon_k\biggr| \geqslant \frac{2}{\pi}\biggl|\frac{\pi \delta}{16 k}-\frac{\widetilde{\alpha} b l^2}{2\pi k}-\frac{\pi}{\sqrt{2}}\, e^{-2\pi k\widetilde{\beta}}\biggr| \nonumber \\ &=\frac{2}{\pi k}\biggl|\frac{\pi \delta}{16}-\frac{\widetilde{\alpha} b l^2}{2\pi}-\frac{\pi}{\sqrt{2}}\, k e^{-2\pi k\widetilde{\beta}}\biggr|. \end{aligned}
\end{equation}
\tag{3.34}
$$
By the conditions of the lemma,
$$
\begin{equation*}
\frac{\pi \delta}{8}-\frac{\widetilde{\alpha} b l^2}{\pi}>0.
\end{equation*}
\notag
$$
Hence there exist positive constants $\widetilde{C}_3$ and $k_2$ such that, for all $k>k_2$, it follows from inequality (3.34) that
From this estimate and (3.1) we get (3.31) for $k>k_0=\max\{k_1,\, k_2\}$. This proves the lemma. Note that the constants $C_0$ and $k_0$ in Lemmas 4–6 are different in general. § 4. Solvability of the problemUnder conditions (2.12), the solution of problem (1.3)–(1.6) can be formally written (see (2.20)) as the series in the partial solutions $X_k(x)$.Let us now show that, under certain conditions on the given functions $\varphi(x)$, $\psi(x)$ and $F(x,y)$, series (4.1) converges uniformly on the closed domain $\overline{D}$ and can be one time differentiated termwise with respect to $x$ and $y$ and two times differentiated in the closed domains $\overline{D}_+$ and $\overline{D}_-$. We first apply Lemmas 1–6 to estimate the coefficients $u_k(y)$ and their derivatives up to second order inclusive. Lemma 7. Let the conditions of one of Lemmas 1, 2, 4, and 5 be met. Then, for all $k>k_0$, Here and below, $M_i$ are some positive constants which depend only on $\alpha$, $\beta$, $l$, $b$, and Proof. We first estimate the $J_{1k}(y)$ and $J_{2k}(y)$ (defined by (2.16) and (2.19) respectively). To this end, we transform the product as follows:
$$
\begin{equation*}
\begin{aligned} \, \sinh\lambda_k(\beta-y)& \Delta_k(\alpha,t) =\sinh\lambda_k(\beta-y)(\cosh\lambda_kt \sin\lambda_k \alpha+\sinh\lambda_k t \cos\lambda_k\alpha) \\ &=\sin\lambda_k\alpha\sinh\lambda_k(\beta-y)\cosh\lambda_k t+\cos\lambda_k\alpha\sinh\lambda_k(\beta-y)\sinh\lambda_k t \\ &=\frac{1}{2}\sin\lambda_k \alpha[\sinh\lambda_k(\beta-y+t)+\sinh\lambda_k(\beta-y-t)] \\ &\qquad+\frac{1}{2}\cos\lambda_k \alpha[\cosh\lambda_k(\beta-y+t)+\cosh\lambda_k(\beta-y-t)], \end{aligned}
\end{equation*}
\notag
$$
where $\beta-y\leqslant \beta-(y-t)\leqslant \beta$, $\beta-2y\leqslant \beta-y-t\leqslant \beta-y$ for $0\leqslant t\leqslant y$. Hence
A similar analysis shows that
$$
\begin{equation}
|{\sinh\lambda_k(\beta-t)\Delta_k(\alpha,y)}|\leqslant \sinh\lambda_k\beta+\cosh\lambda_k\beta =e^{\lambda_k \beta}.
\end{equation}
\tag{4.7}
$$
Using estimates (4.6) and (4.7), and equality (2.16), we have the estimate
$$
\begin{equation}
|J_{1k}(y)|\leqslant e^{\lambda_k\beta}\int_0^\beta |g_1(t)|\, dt.
\end{equation}
\tag{4.8}
$$
An appeal to (2.19) shows that
$$
\begin{equation}
|J_{2k}(y)|\leqslant (\sinh\lambda_k\beta+\cosh\lambda_k\beta)\int_{-\alpha}^0|g_2(t)|\, dt =e^{\lambda_k\beta}\int_{-\alpha}^0 |g_2(t)|\, dt.
\end{equation}
\tag{4.9}
$$
Now estimate (4.2) follows from (2.20) in view of (4.8) and (4.9). To verify estimate (4.3), let us find the derivative of functions (2.20). We have
$$
\begin{equation}
u'_k(y)=\begin{cases} \dfrac{\lambda_k}{\Delta_k(\alpha,\beta)}\biggl[\varphi_k\Delta^{(1)}_k(\alpha,y) -\psi_k\cosh\lambda_k(\beta-y) \\ \quad{+}\dfrac{f_{1k}}{\lambda_k}J^{(1)}_{1k}(y) -\dfrac{f_{2k}}{\lambda_k}g_{2k}(-\alpha)\cosh\lambda_k(\beta-y)\biggr], &y\geqslant 0, \\ \dfrac{\lambda_k}{\Delta_k(\alpha,\beta)}\biggl[\varphi_k\cos\lambda_k(\alpha+y) -\psi_k\Delta^{(1)}_k(-y,\beta) \\ \quad{-}\dfrac{f_{1k}}{\lambda_k}g_{1k}(\beta)\cos\lambda_k(y+\alpha) -\dfrac{f_{2k}}{\lambda_k}J^{(1)}_{2k}(y)\bigg], &y\leqslant 0, \end{cases}
\end{equation}
\tag{4.10}
$$
where
$$
\begin{equation*}
\begin{gathered} \, \begin{aligned} \, \Delta^{(1)}_k(\alpha,y) &=\sinh\lambda_k y \sin\lambda_k\alpha+\cosh\lambda_k y\cos\lambda_k\alpha, \\ \Delta^{(1)}_k(-y,\beta) &=\sinh\lambda_k \beta \sin\lambda_k y+\cosh\lambda_k \beta\cos\lambda_k y, \end{aligned} \\ \begin{aligned} \, J^{(1)}_{1k}(y) &=\cosh\lambda_k(\beta-y)\int_0^yg_1(t)\Delta_k(\alpha,t)\, dt \\ &\qquad-\Delta_k^{(1)}(\alpha,y)\int^\beta_y g_1(t)\sinh\lambda_k(\beta-t)\, dt, \\ J^{(1)}_{2k}(y) &=\Delta_k^{(1)}(-y,\beta)\int_{-\alpha}^yg_2(t)\sin[\lambda_k(t+\alpha)]\, dt \\ &\qquad -\cos\lambda_k(y+\alpha)\int^0_y g_2(t)\Delta_k(-t,y)\, dt. \end{aligned} \end{gathered}
\end{equation*}
\notag
$$
Proceeding as in (4.6) and (4.7), we have
$$
\begin{equation}
|{\cosh\lambda_k(\beta-y)\Delta_k(\alpha,t)}|\leqslant e^{\lambda_k\beta},\qquad |{\sinh\lambda_k(\beta-t)\Delta^{(1)}_k(\alpha,y)}|\leqslant e^{\lambda_k\beta}.
\end{equation}
\tag{4.11}
$$
Now estimate (4.3) follows from (4.10) in view of (4.11). Note that functions (2.20) satisfy (2.2) and (2.3). This implies estimates (4.4) and (4.5) in view of (4.2). Lemma 7 is proved. Remark 2. Under the conditions of Lemmas 1 and 2, estimates (4.2)–(4.5) hold for all $k\geqslant 1$. Lemma 8. Let the conditions of one of Lemmas 3 and 6 be met. Then, for all $k>k_0$, The proof is similar to that of Lemma 7, estimates (3.13), (3.14), and (3.31) being useful. By Lemma 7, series (4.1), as well as its first and second derivatives, are majorized, for $(x,y)\in\overline{D}$, by the numerical series
$$
\begin{equation}
M_9\sum_{k=k_0+1}^{\infty}k^2|\varphi(x)|+k^2|\psi(x)|+k|f_{1k}|+k|f_{2k}|.
\end{equation}
\tag{4.16}
$$
Lemma 9. Let $\varphi(x), \psi(x)\in C^{3}[0,l]$, $\varphi(0)$ ${}={}$ $\psi(0)$ ${}={}$ $\varphi''(0)$ ${}={}$ $\psi''(0)$ ${}={}$ $\varphi(l)$ ${}={}$ $\psi(l)$ ${}={}$ $\varphi''(l)$ ${}={}$ $\psi''(l)=0$, $f_i (x)\in C^2[0,l]$, $f_i(0)=f_i(l)=0$, $i=1,2$, $g_1(t)\in C[0,\beta]$, $g_2(t)\in C[-\alpha,0]$. Then Proof. Integrating by parts three times in the integrals in (2.9), (2.10) and two times in (2.4), and using the conditions of the lemma, we arrive at (4.17). Estimates (4.18) are just the Bessel inequalities from the theory of Fourier series. This proves Lemma 9. By equalities (4.17) and estimates (4.18), series (4.16) is majorized by the converging series
$$
\begin{equation*}
M_{10}\sum_{k=k_0+1}^{\infty}\frac{1}{k}\, \bigl(|\varphi^{(3)}_k|+|\psi^{(3)}_k|+|f^{(2)}_{1k}|+|f^{(2)}_{2k}|\bigr).
\end{equation*}
\notag
$$
This establishes the following theorem. Theorem 2. Let $b=0$, let functions $\varphi(x)$, $\psi(x)$, $f_1(x)$, $f_2(x)$, $g_1(t)$, and $g_2(t)$ satisfy the conditions of Lemma 9, and let the conditions of one of Lemmas 1 and 2 be met. Then problem (1.3)–(1.6) has a unique solution, which is given by series (4.1). Let the conditions of Lemma 3 be met. Then estimates (3.13) and (3.14) hold. If $\widetilde{\alpha}$ is an algebraic number of degree $m=2$, then estimate (3.14) holds, and the conditions of Lemma 9 should be augmented with the additional smoothness conditions: $\varphi(x), \psi(x)\in C^4[0,l]$, $f_i(x)\in C^3[0,l]$, $f''_i(0)=f''_i(l)=0$, $i=1,2$. In this case, the sum of series (4.1) satisfies conditions (1.3) and (1.4). If $\widetilde{\alpha}$ is an algebraic number of degree $m>2$, then estimate (3.13) holds, and in this case one should require that $\varphi(x),\psi(x)\in C^{4+h}[0,l]$, $f_i(x)\in C^{3+h}[0,l]$, where $0<\gamma<h\leqslant 1$. Hence, by the theorem in [17], Chap. 11, § 4, the coefficients satisfy
$$
\begin{equation*}
|\varphi_k|\leqslant \frac{M_{11}}{k^{4+h}},\qquad |\psi_k|\leqslant \frac{M_{12}}{k^{4+h}},\qquad |f_{ik}|\leqslant \frac{M_{13(14)}}{k^{3+h}}.
\end{equation*}
\notag
$$
Therefore, by Lemma 8, series (4.1), as well as its series of derivatives up to second order inclusive are majorized by the converging numerical series
$$
\begin{equation*}
M_{15}\sum_{k=k_0+1}^{\infty}\frac{1}{k^{1+h-\gamma}}.
\end{equation*}
\notag
$$
We thus arrive at the following results. Theorem 3. Let $b=0$ and let functions $\varphi(x)$, $\psi(x)$, $f_i(x)$, $g_i(t)$, $i=1, 2$, satisfy the conditions of Lemma 9, let $\varphi(x),\psi(x)\in C^4[0,l]$, $f_i(x)\in C^3[0,l]$, $f''_i(0)=f''_i(l)=0$, and let $\widetilde{\alpha}$ be an algebraic number of degree $m=2$. Then there exists a unique solution of problem (1.3)–(1.6), which is given by series (4.1). Theorem 4. Let $b=0$, let functions $\varphi(x)$, $\psi(x)$, $f_i(x)$, $g_i(t)$ satisfy the conditions of Lemma 9, let $\varphi(x),\psi(x)\in C^{4+h}[0,l]$, $f_i(x)\in C^{3+h}[0,l]$, $0<\gamma<h\leqslant 1$, $f''_i(0)=f''_i(l)=0$, $i=1,2$, and let $\widetilde{\alpha}$ be an algebraic number of degree $m>2$. Then problem (1.3)–(1.6) has a unique solution, which is given by series (4.1). We next employ Lemmas 4–8 to derive an existence theorem for a solution of problem (1.3)–(1.6) for $b>0$. If $\delta_p(\widetilde{\alpha})=0$ for some $k=p=k_1, k_2,\dots, k_m$, where $1\leqslant k_1<k_2<\dots<k_m\leqslant k_0$, $k_i$, $i=1,\dots,m$, $m$ are given natural numbers, and $\widetilde{\alpha}$ are numbers from Lemmas 4–6 then problem (1.3)–(1.6) is solvable if and only if
$$
\begin{equation}
\begin{aligned} \, &\varphi_p \cos\lambda_p \alpha-\psi_p \cosh\lambda_p \beta -\frac{f_{1p}}{\lambda_p}\, g_{1p}(\beta)\cos\lambda_p\alpha \nonumber \\ &\qquad-\frac{f_{2p}}{\lambda_p}\, g_{2p}(-\alpha)\cosh\lambda_p\beta=0,\qquad p=k_1, k_2, \dots, k_m. \end{aligned}
\end{equation}
\tag{4.19}
$$
A sufficient condition for solvability of problem (1.3)–(1.6) is as follows:
$$
\begin{equation*}
\varphi_p=\psi_p=f_{1p}=f_{2p}=0,\qquad p=k_1, k_2,\dots, k_m.
\end{equation*}
\notag
$$
In this case, the solution of problem (1.3)–(1.6) is given by the series
$$
\begin{equation}
\begin{aligned} \, u(x,y) &=\biggl(\sum_{k=1}^{k_1-1}+ \dots+ \sum_{k=k_{m-1}+1}^{k_m-1} +\sum_{k=k_m+1}^{+\infty}\biggr) u_k(y)X_k(x) \nonumber \\ &\qquad+\sum_{p}u_p(y)X_p(x), \end{aligned}
\end{equation}
\tag{4.20}
$$
where
$$
\begin{equation}
u_{p}(y)=\begin{cases} \dfrac{\cosh\lambda_p y}{\cosh\lambda_p\beta} \biggl[\varphi_p-\dfrac{f_{1p}}{\lambda_p}g_{1p}(\beta)\biggr] \\ \quad{-}d_p\dfrac{\sinh\lambda_p(\beta-y)}{\cosh\lambda_p\beta}+\dfrac{f_{1p}}{\lambda_p}\, g_{1p}(y), &y\geqslant 0, \\ \dfrac{\cos\lambda_p y}{\cosh\lambda_p\beta} \biggl[\varphi_p\,{-}\,\dfrac{f_{1p}g_{1p}(\beta)}{\lambda_p}\biggr] \\ \quad{-}d_p\dfrac{\Delta_p(-y,\beta)}{\cosh\lambda_p \beta}-\dfrac{f_{2p}}{\lambda_p}\, g_{2p}(y), &y\leqslant 0. \end{cases}
\end{equation}
\tag{4.21}
$$
In the last sum, $p$ assumes the values $k_1, k_2, \dots k_m$, next, $d_p$ are arbitrary constants, and if, in a finite sum in (4.20), the upper limit is smaller than the lower one, then this sum is defined to be zero. Note that if $\varphi_p=f_{1p}=f_{2p}=0$ in (4.21), then we get $u_p(y)$ from (2.21). This establishes the following results. Theorem 5. Let $b\neq 0$, let functions $\varphi(x)$, $\psi(x)$, $f_i(x)$, $g_i(t)$, $i=1,2$, satisfy the conditions of Lemma 9, and let the conditions of one of Lemmas 4 or 5 be met. Assume that $\delta_k(\widetilde{\alpha})\neq 0$ for $k=1,\dots,k_0$. Then there exists a unique solution of problem (1.3)–(1.6), which is given by series (4.1). Further, if $\delta_k(\widetilde{\alpha})= 0$ for $k=p=k_1, k_2,\dots,k_m\leqslant k_0$, then problem (1.3)–(1.6) is solvable if and only if conditions (4.19) are met; in this case, the solution is given by series (4.20). Furthermore, the sum $u(x,y)$ of the series lies in $C^1(\overline{D})\cap C^2(\overline{D}_+)\cap C^2(\overline{D}_-)$. Theorem 6. Let $b\neq0$, let functions $\varphi(x)$, $\psi(x)$, $f_i(x)$, $g_i(t)$, $i=1,2$, satisfy the conditions of Theorem 3, and let the conditions of Lemma 6 be met. Then the conclusion of Theorem 5 holds. § 5. Stability of the solution of the problemLet us now show that the solution of problem (1.3)–(1.6) is stable with respect to given functions $\varphi(x)$, $\psi(x)$ and $f_i(x)$, $i=1,2$. Theorem 7. Let the conditions of one of Theorems 2–4 be meet. Then the solution of problem (1.3)–(1.6) satisfies Proof. The system $X_k(x)$ is orthonormal in $L_2[0,l]$, and hence from (4.1) and Lemma 7 we have
$$
\begin{equation*}
\begin{aligned} \, \|u(x,y)\|^2_{L_2[0,l]} &=\sum_{k=1}^{\infty}u_k^2(y)\leqslant 4M^2_1\sum_{k=1}^{\infty}(\varphi_k^2+\psi_k^2+f_{1k}^2+f_{2k}^2) \\ &\leqslant 4M^2_1\bigl(\|\varphi\|^2_{L_2}+\|\psi\|^2_{L_2} +\|f_1\|^2_{L_2} +\|f_2\|^2_{L_2}\bigr). \end{aligned}
\end{equation*}
\notag
$$
Now estimate (5.1) for each $y\in [-\alpha,\beta]$ follows.
Let $(x,y)$ be an arbitrary point from $\overline{D}$. Then from (4.1) and Lemma 7 we have
$$
\begin{equation}
\begin{aligned} \, |u(x,y)| &\leqslant M_1\sqrt{\frac{2}{l}}\sum_{k=1}^{\infty}|u_k(y)| \nonumber \\ &\leqslant \widetilde{M}_1 \sum_{k=1}^{\infty} \biggl(|\varphi_k|+|\psi_k|+\frac{1}{k}|f_{1k}| +\frac{1}{k}|f_{2k}|\biggr). \end{aligned}
\end{equation}
\tag{5.3}
$$
By Lemma 9, the coefficients $\varphi_k$ and $\psi_k$ can be written as
$$
\begin{equation}
\varphi_k=\frac{\varphi_k^{(1)}}{\mu_k},\qquad \psi_k=\frac{\psi_k^{(1)}}{\mu_k},
\end{equation}
\tag{5.4}
$$
where
$$
\begin{equation*}
\begin{aligned} \, \varphi_k^{(1)} &=\sqrt{\frac{2}{l}}\int_0^l\varphi'(x)\cos\mu_k x \, dx, \\ \psi_k^{(1)} &=\sqrt{\frac{2}{l}}\int_0^l\psi'(x)\cos\mu_k x\, dx. \end{aligned}
\end{equation*}
\notag
$$
Using (5.4) and employing the Cauchy–Schwarz inequality, we have from (5.3)
$$
\begin{equation*}
\begin{aligned} \, |u(x,y)| &\leqslant \widetilde{M}_2\biggl(\sum_{k=1}^{\infty}\frac{1}{k^2}\biggr)^{1/2} \biggl[\biggl(\sum_{k=1}^{\infty}|\varphi_k^{(1)}|^2\biggr)^{1/2} +\biggl(\sum_{k=1}^{\infty}|\psi_k^{(1)}|^2\biggr)^{1/2} \\ &\qquad+\biggl(\sum_{k=1}^{\infty}|f_{1k}|^2\biggr)^{1/2} +\biggl(\sum_{k=1}^{\infty}|f_{1k}|^2\biggr)^{1/2}\biggr] \\ &\leqslant \widetilde{M}_3 \bigl(\|\varphi'\|_{L_2[0,l]}+\|\psi'\|_{L_2[0,l]} +\|f_1\|_{L_2[0,l]} +\|f_2\|_{L_2[0,l]}\bigr) \\ &\leqslant M_{17}\bigl(\|\varphi'\|_{C[0,l]}+\|\psi'\|_{C[0,l]} +\|f_1\|_{C[0,l]}+\|f_2\|_{C[0,l]}\bigr), \end{aligned}
\end{equation*}
\notag
$$
which implies estimate (5.2). This proves Theorem 7. 
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