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Linear and nonlinear abstract equations with parameters. (English) Zbl 1215.34067

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(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 t0 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 x 2 u(x,y)-t 2 D y 2 u(x,y)+du(x,y)+t 1 1/2 D x u(x,y)+t 2 1/2 D y u(x,y)=f(x,y)

on the region (0,a)×(0,b) is given.

MSC:
34G10Linear ODE in abstract spaces
35J25Second order elliptic equations, boundary value problems
35J70Degenerate elliptic equations
34G20Nonlinear ODE in abstract spaces
34B10Nonlocal and multipoint boundary value problems for ODE
47D06One-parameter semigroups and linear evolution equations