on page no 479 of partial differential equation (Evans) how the condition (iii) of deformation lemma is satisfied.
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After $(12)$, Evans concluded that for $u\in H$ and $t\in [0,1]$ $$\tag{1}\frac{d}{dt}I[\eta_t(u)]\leq 0,\ $$
$(1)$ is telling us that the function $v(t)=I[\eta_t(u)]$ for $u$ fixed is non-increasing, which implies that $v(t)\leq v(0)$ for all $t\in [0,1]$ or equivalently (remember that $\eta_0(u)=u$) $$I[\eta_t(u)]\leq I[u],\ \forall\ t\in [0,1]$$
Tomás
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\tag{1}\frac{d}{dt}I[\eta_t(u)]\leq 0,\from the above equation we can tell $I$ is only non-increasing function. then how you told v(t)(=\eta_t(u)) is non-increasing – nanthini Jul 23 '13 at 06:21
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Sorry, we have to add a $I$ in the text. I have fixed it – Tomás Jul 24 '13 at 11:27
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ya now cleared thank you – nanthini Jul 26 '13 at 04:59