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

The linear abstract equation

$-t{u}^{\left(2\right)}\left(x\right)+Au\left(x\right)+{t}^{1/2}{B}_{1}\left(x\right){u}^{\left(1\right)}\left(x\right)+{B}_{2}\left(x\right)u\left(x\right)=f\left(x\right)$

with a parameter $t$ is considered. Here, $A$ and ${B}_{1}\left(x\right)$, ${B}_{2}\left(x\right)$ for $x\in \left(0,1\right)$ 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 $f\left(x,u,{u}^{\left(1\right)}\right)$, the existence and uniqueness of maximal regular solution is obtained.

An application to the equation

$-{t}_{1}{D}_{x}^{2}u\left(x,y\right)-{t}_{2}{D}_{y}^{2}u\left(x,y\right)+du\left(x,y\right)+{t}_{1}^{1/2}{D}_{x}u\left(x,y\right)+{t}_{2}^{1/2}{D}_{y}u\left(x,y\right)=f\left(x,y\right)$

on the region $\left(0,a\right)×\left(0,b\right)$ is given.

MSC:
 34G10 Linear ODE in abstract spaces 35J25 Second order elliptic equations, boundary value problems 35J70 Degenerate elliptic equations 34G20 Nonlinear ODE in abstract spaces 34B10 Nonlocal and multipoint boundary value problems for ODE 47D06 One-parameter semigroups and linear evolution equations