Linear and nonlinear abstract equations with parameters (Q988131)

The linear abstract equation \[ -tu^{(2)}(x)+ Au(x)+ t^{1/2} B_1(x)u^{(1)}(x)+ B_2(x) u(x)= f(x) \] with a parameter \(t\) is considered. Here, \(A\) and \(B_1(x)\), \(B_2(x)\) for \(x\in (0,1)\) are linear operators in a Banach space. The nonlocal boundary conditions contain the parameter \(t\) as well. Under some assumptions, the existence of the unique solution in a Sobolev space and a coercive uniform estimation is established. Also, the behavior of the solution for \(t\to 0\) and the smoothness properties of the solution with respect to the parameter \(t\) are investigated and the discreteness of the corresponding differential operator is proved. For the nonlinear problem with right side \(f(x,u, u^{(1)})\), the existence and uniqueness of maximal regular solution is obtained. An application to the equation \[ -t_1 D^2_x u(x,y)- t_2 D^2_y u(x,y)+ du(x,y)+ t^{1/2}_1 D_x u(x,y)+ t^{1/2}_2 D_y u(x,y)= f(x,y) \] on the region \((0,a)\times (0,b)\) is given.