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Let f ∶ ℝ2 → ℝ be a locally Lipschitz function. Consider the initial value problem

ẋ = f(t, x), x(t0) = x0

for (t0, x0) ∈ ℝ2. Suppose that J(t0, x0) represents the maximal interval of existence for the initial value problem. Which of the following statements is true?

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CSIR UGC (NET) Mathematical Science: Held On (7 June 2023)
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  1. J(t0, x0) = ℝ.
  2. J(t0, x0) is an open set.
  3. J(t0, x0) is a closed set.
  4. J(t0, x0) could be an empty set.

Answer (Detailed Solution Below)

Option 2 : J(t0, x0) is an open set.
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Detailed Solution

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Explanation:

Let f ∶ ℝ2 → ℝ be a locally Lipschitz function. Consider the initial value problem

ẋ = f(t, x), x(t0) = x0

for (t0, x0) ∈ ℝ2. 

By using Picard's theorem we know that solutions lie in the interval 

which is an open interval 

Therefore, the Correct Option is Option (2).

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