A nonlocal problem with integral conditions
For the equation \(u_{xy} +A(x,y)u_x +B(x,y)u_y +C(x,y)u =f(x,y)\) in the domain \(D=\{(x,y): 0<x<a\), \(0<y<b\}\) we pose the following problem: find a solution of class \(C (\overline D)\cap C^2(D)\) satisfying the conditions \(\int^\alpha_0 u(x,y)dx =\psi(y)\), \(0\leq y\leq b\), \(\int^\beta_0 u(x,y)dx =\varphi(x)\), \(0\leq x\leq a\), where \(\varphi(x)\) and \(\psi(y)\) are given functions, \(0<\alpha\leq a\), and \(0<\beta\leq b\). We prove the following Theorem. If the coefficients satisfy the conditions \(A(x,y)\), \(B(x,y) \in C(\overline D)\cap C^1(D)\), \(C(x,y)\in C (\overline D)\), and \[ \inf_DA_y (x,y)\geq 0,\quad \inf_DB_x(x,y) \geq 0, \] \[ \inf_{x\in [0,\alpha]} A(x,\beta- \delta) \geq 0,\quad \inf_{y\in [0,\beta]} B(\alpha- \delta,y)\geq 0, \quad 0< \delta< \min \{\alpha, \beta\}, \] \[ \sup_D \bigl|A_x(x,y) +B_y(x,y) -C(x,y) \bigr|< {1\over ab}, \] and if the functions \(\varphi(x)\) and \(\psi(y)\) satisfy \(\varphi(x) \in C^1[0, \alpha]\), \(\psi (y)\in C^1 [0, \beta]\), \(\int^\alpha_0 \varphi(x)dx =\int^\beta_0 \psi(y)dy\), \(\alpha \varphi(0) =\beta \psi(0)\), then there exists a unique solution.
scientific article; zbMATH DE number 1028529