A non-local problem for a loaded mixed-type equation with a integral operator

We study the existence and uniqueness of the solution of non-local boundary value problem for the loaded elliptic-hyperbolic equation $$ u_{xx} + \mathop{\mathrm{sgn}} (y) u_{yy} + \frac{1 - \mathop{\mathrm{sgn}} (y)}{2} \sum\limits_{k = 1}^n {R_k}(x, u(x, 0)) = 0 $$ with integral operator $$ {R_k}\bigl(x, u(x, 0)\bigr) = \left\{ \begin{array}{lc} {p_k}(x)D_{x\,\,1}^{ - {\alpha _k}}u(x, 0), & q \le x \le 1,\\[2mm] {r_k}(x)D_{ - 1\,x}^{ - {\beta _k}}u(x, 0), & - 1 \le x \le - q, \end{array} \right. $$ where $$ \begin{array}{l} \displaystyle D_{ax}^{ - {\alpha _k}}f(x) = \frac{1}{{\Gamma ({\alpha _k})}} \int _a^x \frac{f(t)}{(x - t)^{1-{\alpha _k} }}dt, \\ \displaystyle D_{xb}^{ - {\beta _k}}f(x) = \frac{1}{{\Gamma ({\beta _k})}} \int _x^b \frac{f(t)}{(t - x)^{1-{\beta _k}}}dt , \end{array} $$ in double-connected domain $\Omega $, bounded with two lines: $$ \sigma _1:~x^2 + y^2 = 1,\quad \sigma _2:~ x^2 + y^2 = q^2 \quad \text{at $y > 0$,}$$ and characteristics: $$ A_j C_1:~ x + ( - 1)^j y = ( - 1)^{j + 1},\quad B_j C_2:~x + ( - 1)^j y = ( - 1)^{j + 1} \cdot q$$ of the considered equation at $y < 0$, where $0 < q < 1$, $j = 1, 2$; $A_1 ( 1; 0),$ $A_2( - 1; 0)$, $B_1(q; 0)$, $B_2( - q; 0)$, $C_1(0; - 1)$, $C_2(0; - q)$, $\beta _k$, $\alpha _k > 0$. Uniqueness of the solution of investigated problem was proved by an extremum principle for the mixed type equations. Thus we need to prove that, the loaded part of the equation is identically equal to zero if considerate problem is homogeneous. Existence of the solution of the problem was proved by a method of the integral equations, thus the theory of the singular integral equations and Fredholm integral equations of the second kind were widely used.

loaded equation, integral operator, elliptic-hyperbolic type equations, double-connected domain, existence and uniqueness of solution, extremum principle, integral equations