I'm reading this paper about solving semilinear elliptic pde's through iterated approximations. The line i'm trying to understand is
"Then, since $u_k = Tu_{k-1}$ and since $\{u_k\}$ is a bounded, pointwise convergent sequence, it converges in $W_{2,p}$."
The operator $T$ is defined by defining $Tu$ to be the $v$ such that $(L-\Omega)v = f(x,u)+\Omega u$.
They key information seems to be that assumptions imply $f(x,u_n) + \Omega u_n$ is pointwise convergent, and that the solution operator to $(L-\Omega)v = \phi$ (with boundary conditions $v=0$ on boundary for simplicity) maps $L_p$ continuously into $W_{2,p}$.
Why does the quoted line hold?
The link makes it all clear. The sequence $\{u_k\}$ is uniformly bounded monotone decreasing, and hence poinwise convergent. The operator $T$ is a composition of the nonlinear operation $u\to f(x,u)+\Omega u$ with the inversion of the linear elliptic operator bounded from $L^p$ to $W^{2,p}$. With $\{f(x,u_k)\}$ being bounded uniformly on the range of $\{u_k\}$, the nonlinear operation $u_k\to f(x,u_k)+\Omega u_k$ takes uniformly bounded pointwise convergent $\{u_k\}$ into uniformly bounded pointwise convergent sequence, convergent as well in $L^p$ by the Lebesgue's dominated convergence theorem. The inversion of the linear elliptic operator bounded from $L^p$ to $W^{2,p}$ does the rest, taking the sequence convergent in $L^p$ into the sequence convergent in $W^{2,p}$.