The principle for getting examples of nonapplicability of Ekeland's theorem is described by the following simple
Example. With $\mathbb I=[0,1]$ consider the Frechet space $E=F=C^\infty(\mathbb I)$ with the norms $\|x\|_k=\sup\{|{\rm D}^ix\,s|:s\in\mathbb I\text{ and }i\le k\,\}$. Taking $k_0=0$ consider the second order polynomian map $x\mapsto x+x^2$ defined for $\|x\|_0<\frac 14$. Then for $Lxv=\frac v{1+2x}$ Ekeland's condition (5) in his Theorem 3 on page 97 in the Ann. Inst. Henri Poincaré paper [AN 28 (2011) 91–105] requires that $\|\frac v{1+2x}\|_k=\|Lxv\|_k\le m'_k\|v\|_{k+d_2}$ for all $k\in\mathbb N$ and all the appropriate $x$ and all $v$ in $E$. Taking $v:s\mapsto 1$ and $x:s\mapsto\frac 18\sin(2n\pi s)$ and $k=1$ we get $\frac 12n\pi=|{\rm D}(\frac 1{1+2x})0|\le m'_1\|v\|_{1+d_2}=m'_1$ which does not hold if we take $\frac 2\pi m'_1<m'_1=n$.
The same idea with only more complicated computations can be applied to more general spaces and maps.