42.2. EVOLUTION INCLUSIONS 1375

Let h,k be small positive numbers. Then

yλ (t +h)− yλ (t)+∫ t+h

tAλ yλ (s)ds =

∫ t+h

tf (s)ds (42.2.8)

Next consider the difference operator

Dkg(t)≡ g(t + k)−g(t)k

Do this Dk to both sides of 42.2.8 where k < h. This gives

Dk (yλ (t +h)− yλ (t))+1k

(∫ t+h+k

t+hAλ yλ (s)ds−

∫ t+k

tAλ yλ (s)ds

)

=1k

(∫ t+h+k

t+hf (s)ds−

∫ t+k

tf (s)ds

)(42.2.9)

Now multiply both sides by yλ (t +h+ k)− yλ (t + k) . Consider the first term. To simplifythe ideas consider instead

(Dkg(t) ,g(t + k)) =1k

(|g(t + k)|2− (g(t) ,g(t + k))

)≥ 1

k

(|g(t + k)|2−|g(t)| |g(t +h)|

)≥ 1

k

(12|g(t + k)|2− 1

2|g(t)|2

)(42.2.10)

Then applying this simple observation to 42.2.9,

12

1k

(|yλ (t +h+ k)− yλ (t + k)|2−|yλ (t +h)− yλ (t)|2

)+

(1k

(∫ t+h+k

t+hAλ yλ (s)ds−

∫ t+k

tAλ yλ (s)ds

),yλ (t +h+ k)− yλ (t + k)

)

≤(

1k

(∫ t+h+k

t+hf (s)ds−

∫ t+k

tf (s)ds

),yλ (t +h+ k)− yλ (t + k)

)Taking limsupk→0 of both sides yields

12

D+(|yλ (t +h)− yλ (t)|2

)+(Aλ yλ (t +h)−Aλ yλ (t) ,yλ (t +h)− yλ (t))

≤ ( f (t +h)− f (t) ,yλ (t +h)− yλ (t))

Now recall that Aλ is monotone. Therefore,

D+(|yλ (t +h)− yλ (t)|2

)≤ | f (t +h)− f (t)|2 + |yλ (t +h)− yλ (t)|2