Note that, in the case of $f(u)=-u+g$, with given $g\in L^2(\Omega)$, i.e., when  is linear and strongly elliptic, the norm ${\left|\!\left|\!\left|\cdot\right|\!\right|\!\right|}_{{\epsilon},\Omega}$ is a natural energy norm on $X$. Frequently, for $D=\Omega$, the subindex ‘$D$’ will be omitted. Furthermore, the associated dual norm of ${\mathsf{F}}_{\epsilon}$ from  is given by $${\left|\!\left|\!\left|{\mathsf{F}}_{\epsilon}(u)\right|\!\right|\!\right|}_{X',{\epsilon}} =  \sup_{\genfrac{}{}{0pt}{}{v\in X}{{\left|\!\left|\!\left|v\right|\!\right|\!\right|}_{{\epsilon}}=1}}\int_{\Omega}\left\{{\epsilon}u'v'-f(u)v\right\}{\,{\mathsf{d}}x}.$$ Throughout this work we shall use the abbreviation $x \preccurlyeq y$ to mean $x \leq c y$, for a constant $c>0$ independent of the mesh size $h$, the local polynomial degree $p$, and of ${\epsilon}>0 $.

*Outline:* In Section \[sc:anewton\] we will consider the Newton method within the context of dynamical systems in general Banach spaces, and present a prediction strategy for controlling the Newton step size parameter. Furthermore, Section \[sc:FEM\] focuses on the application of the Newton methodology to semilinear elliptic problems as well as on the discretization of the problems under consideration by $hp$-finite element methods; in addition, we derive an ${\epsilon}$-robust [*a posteriori*]{} residual analysis. An $hp$-adaptive procedure and a series of numerical experiments illustrating the performance of the proposed idea will be presented in Section \[sc:adaptivity\]. Finally, we summarize our findings in Section \[sc:concl\].

Adaptive Newton Methods in Banach Spaces {#sc:anewton}
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In the 