Solvability of Semilinear Operator Equations with Growing Nonlinearity using Tikhonov regularization

In this paper, we prove approximate solvability of the semilinear operator equation
$Lu+Nu=v$
using Tikhonov regularization and Browder's fixed point theorem under the following assumptions: (i) The corresponding linear operator equation
$Lu=v$
is approximately solvable. (ii) The nonlinear operator
$N:U\rightarrow V$
is compact. (iii)
$\Vert Nu\Vert \leq a+b\Vert u\Vert$
for some positive constants
$a$
and
$b$
with
$b < \frac{2\sqrt{\lambda}}{1+\frac{1}{2\sqrt{\lambda}}}$
where
$\lambda > 0$
is a regularization parameter. Theory is substantiated with an application for computation of regularized control for an approximately controllable semilinear system.