34.7. SOME IMBEDDING THEOREMS 1223

=Cε

k

∑i=1

∫ ti

ti−1

(∫ ti

ti−1

∥∥u′n (τ)∥∥

X dτ

)p

dt

=Cε

k

∑i=1

((ti− ti−1)

1/p∫ ti

ti−1

∥∥u′n (τ)∥∥

X dτ

)p

Since p≥ 1,

≤ Cε

(k

∑i=1

(ti− ti−1)1/p∫ ti

ti−1

∥∥u′n (τ)∥∥

X dτ

)p

≤ Cε (b−a)k

(k

∑i=1

∫ ti

ti−1

∥∥u′n (τ)∥∥

X dτ

)p

=Cε (b−a)

k

(∥∥u′n∥∥

L1([a,b],X)

)p<

η p

10p

if k is chosen large enough. Now consider the first in 34.7.61. By Jensen’s inequality

k

∑i=1

∫ ti

ti−1

ε

∥∥∥∥ 1ti− ti−1

∫ ti

ti−1

(un (t)−un (s))ds∥∥∥∥p

Edt ≤

k

∑i=1

∫ ti

ti−1

ε1

ti− ti−1

∫ ti

ti−1

∥un (t)−un (s)∥pE dsdt

≤ ε2p−1k

∑i=1

1ti− ti−1

∫ ti

ti−1

∫ ti

ti−1

(∥un (t)∥p +∥un (s)∥p)dsdt

= 2ε2p−1k

∑i=1

∫ ti

ti−1

(∥un (t)∥p)dt = ε (2)(2p−1)∥un∥Lp([a,b],E) ≤Mε

Now pick ε sufficiently small that Mε < η p

10p and then k large enough that the second termin 34.7.61 is also less than η p/10p. Then it will follow that

∥ūn−un∥Lp([a,b],W ) <

(2η p

10p

)1/p

= 21/p η

10≤ η

5

Thus k is fixed and un at a step function with k steps having values in E. Now usecompactness of the embedding of E into W to obtain a subsequence such that {un} isCauchy in Lp ([a,b] ;W ) and use this to contradict 34.7.60. The details follow.

Suppose un (t) = ∑ki=1 un

i X[ti−1,ti) (t) . Thus

||un (t)||E =k

∑i=1||un

i ||E X[ti−1,ti) (t)

and so

R≥∫ b

a||un (t)||pE dt =

Tk

k

∑i=1||un

i ||pE