A~nonlocal boundary value problem for a Bitsadze-Lykov equation that degenerates in the interior of a domain (Q2713887)

Consider the equation NEWLINE\[NEWLINE y^2u_{xx}-u_{yy}+\alpha u_x=0. \tag{1} NEWLINE\]NEWLINE Denote by \(A\), \(B\), \(C\), and \(C_1\) the points of \((0,0)\), \((1,0)\), \(({1\over 2},-1)\), and \(({1\over 2},1)\), respectively. The characteristics of (1), belonging to the different families and issuing from \(A\) and \(B\), meet at \(C\) and \(C_1\). Denote by \(D_1\) and \(D_2\) the characteristic triangles of \(ABC\) and \(ABC_1\), \(I=AB\), \(D=D_1\cup D_2\cup I\).NEWLINENEWLINENEWLINEIn the case of \(|\alpha|\leq 1\) the author studies the problem of finding a solution \(u(x,y)\) to (1) in \(D\) that belongs to the class \(C(\overline D)\cap C^1(\overline D\backslash I)\cap C^2(D\backslash I)\), satisfies the agreement condition NEWLINE\[NEWLINE u(x,+0)=u(x,-0), \lim_{y\to 0+0}u_y(x,y)= \lim_{y\to 0-0}u_y(x,y),\quad x\in I, NEWLINE\]NEWLINE and whose values on the characteristics of \(AC\), \(AC_1\), \(BC\), and \(BC_1\) are interconnected through some operators of generalized fractional integrodifferentiation. Some existence and uniqueness theorems are proved.NEWLINENEWLINEFor the entire collection see [Zbl 0956.00039].