1412 CHAPTER 42. MAXIMAL MONOTONE OPERATORS, HILBERT SPACE

∫ t

0|x1 (s)− x2 (s)|2H ds≤

∫ t

0

∫ s

0|v1 (r)− v2 (r)|2H drds.

Iterating this inequality, by replacing Λ with Λk, it follows that for all k large enough, Λk isa contraction map on L2 (0,T ;H). Thus there exists a unique fixed point for Λ, [x] . Thus

L [x]+ [x]+∂Φ([x]) ∋ [ f ]+ [x] .

Let x ∈ [x] be such that

x(t) = x0 +∫ t

0x′ (s)ds.

By Lemma 42.9.6,x′+ x+∂2φ (t,x) ∋ f + x

This function, x(·) is the unique solution to 42.9.74 because if x1 is another solution, then[x1] = [x] and since both functions are continuous, they must coincide.