I have the following definition of a pseudo-gradient vector field:
Let $V$ be a Banachspace, $E\in C^1(V)$, $\tilde V = \{u\in V \mid DE(u)\neq 0\}$.
Then $v: \tilde V \to V$ is called a p.g.v.f. of $E$ on $\tilde V$, if $\forall u\in\tilde V$ holds:
(1) $||v(u)|| < 2\min\{||DE(u)||, 1\}$,
(2) $\langle v(u), DE(u)\rangle > \min\{||DE(u)||, 1\}||DE(u)||$.
From this definition I have no idea what this "should be" and how $v$ can look like and what is different to the classic gradient vector field.
So I tried to get some more information from this two inequalities.
It holds that:
(a) $\min\{||DE(u)||, 1\} < ||v(u)|| < 2*\min\{||DE(u)||, 1\}$
(b) $\cos\varphi = \frac{\langle v(u), DE(u)\rangle}{||v(u)||\cdot||DE(u)||} > \frac{1}{2}$.
So the length of a pseudo-gradient should be in-between the length of the corresponding gradient and twice of this length. The angle between a pseudo-gradient and its corresponding gradient should be round about between -60 to 60 degrees.
But I still can't imagine $v$. Can someone explain me the idea behind this concept?
Looking at your 2 statements:
(a) min{||DE(u)||,1}<||v(u)||<2∗min{||DE(u)||,1}
(b) cosφ=⟨v(u),DE(u)⟩||v(u)||⋅||DE(u)||>1/2.
We can see that a pseudo gradient is field which is somewhat similar to a gradient in 2 senses: (a) The magnitude of the pseudo gradient is similar to that of the true gradient (b) The pseudo gradient will always point in "approximately" the same direction as the true gradient. Significantly, when the true gradient is positive, so is the pseudo gradient, and when the true gradient is negative, so is the pseudo gradient.