Exercise 5.27

Exercise 27: Let ϕ be a real function defined on a rectangle R in the plane, given by a ≤ x ≤ b , α ≤ y ≤ β . A solution of the initial-value problem

y ′ = ϕ ( x , y ) , y ( a ) = c ( a ≤ c ≤ b )

is, by definition, a differentiable function f on [ a , b ] such that f ( a ) = c , α ≤ f ( x ) ≤ β , and

f ′ ( x ) = ϕ ( x , f ( x ) ) ( a ≤ x ≤ b ) .

Prove that such a problem has at most one solution if there is a constant A such that

| ϕ ( x , y 2 ) − ϕ ( x , y 1 ) | ≤ A | y 2 − y 1 |

whenever ( x , y 1 ) ∈ R and ( x , y 2 ) ∈ R .

Answers

Following the hint, let f 1 , f 2 be two solutions of the initial value problem and let f = f 1 − f 2 . Then f is differentiable on [ a , b ] , f ( a ) = f 1 ( a ) − f 2 ( a ) = 0 , and

| f ′ ( x ) | = | f 1 ′ ( x ) − f 2 ′ ( x ) | = | ϕ ( x , f 1 ( x ) ) − ϕ ( x , f 2 ( x ) ) | ≤ A | f 1 ( x ) − f 2 ( x ) | = A | f ( x ) | .

Applying Exercise 27, we get f ( x ) = 0 for all x ∈ [ a , b ] .

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2023-08-07 00:00
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