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From (2.11), we have (note that is nondecreasing) (2.18).
In view of (2.21), we arrive at Since is demicontinuous, we have Note that (2.24).
For any, since and is nonexpansive, we have (note that is strictly decreasing) (3.2).
Using (1.14) and (3.13), we have (note that is strictly decreasing) (3.14).
We now integrate the last equality two times over [ t, T ] ⊂ [ t 0, T ] and have (note that v ( T ) = v ′ ( T ) = 0 ) v ( t ) = t ∫ t T s − a − 2 ( ∫ s T ξ a + 1 f ( ξ, v d ξ ) d s, t ∈ [ t 0, T ].
Since we must have (note that w(x j ) → 0 for j → ∞ when x∞ = ∞), we conclude that x∞ = ∞ is impossible as well.
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Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com