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.

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 $\frac 2\pi m'_1<m'_1\le n$.

The same idea with only more complicated computations can be applied to more general spaces and maps.