HIERARCHAL INDUCTIVE PROCESS MODELING AND ANALYSIS ...

4.2 Preliminaries
Both Models A and B have complex structures which differ from more theoretical
models studied by mathematicians. Since solving these differential equations directly
is extremely difficult, we decided to take a more indirect approach by looking at
the bounds of function, using the positive lemma and comparison arguments which
follow.
Lemma 1 A Positivity Lemma. Let W (t) be a smooth function over a domain
[0, T ], T ∈ R . If W satisfies W ′ (t) + M(t)W (t) ≥ 0 in (0, T ] and W (0) ≥ 0, where
M(t) is a bounded function in [0, T ], then W (t) ≥ 0 on [0, T ].
Proof: We prove this lemma by contradiction. Assume that the statement W (t) ≥ 0
in [0, T ] were not true, then there would exist a point t 0 ∈ [0, T ] such that W (t 0 ) is
a negative minimum of W on [0, T ]. Since W (0) ≥ 0, then t 0 ∈ (0, T ] which means
that
W ′ (t 0 ) + M(t 0 )W (t 0 ) ≥ 0.
Since W reaches its minimum value at t 0 , then we have W ′ (t 0 ) = 0 if t 0 ≠ T and
W ′ (t 0 ) ≤ 0 if t 0 = T . This ensures that
M(t 0 )W (t 0 ) ≥ 0
which contradicts our assumption about W (t 0 ) < 0 when M(t 0 ) > 0.
For the case of M(t 0 ) ≤ 0, we let V (t) = e −γt W (t) for some constant γ with
γ > −M(t) in (0, T ], then V will satisfy the relation V ′ (t) + (γ + M)V (t) ≥ 0 in
(0, T ] and V (0) ≥ 0, where γ+M(t) > 0 for all t ∈ (0, T ]. From the above arguments
we have V (t) ≥ 0 in [0, T ]. It follows from W (t) = e γt V (t) that W (t) ≥ 0 on [0, T ].
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