A nonlocal problem with integral conditions for a quasilinear hyperbolic equation (Q1809960)

This paper is devoted to a quasilinear equation of hyperbolic type with integral conditions. Indeeds the author considers the equation \[ u_{xy}+ (Au)_x+ (Bu)_y+ Cu= f(x, y, u),\tag{1} \] whose coefficients \(A(x,y)\), \(B(x,y)\) and \(C(x,y)\) are bounded and have bounded derivatives of first-order and the coefficient \(C(x,y)\) has also a bounded mixed derivative in the rectangle \(\{(x,y)\mid 0< x< a, 0< y< b\}\). The author studies existence and uniqueness of a generalized solution of (1) satisfying the conditions \[ \int^a_0 u(x,y) dy= \psi(y),\quad \int^b_0 u(x,y) dy= \varphi(x). \]