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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?

This question was previously asked in
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 

\(|x-x_o|< a ; |t-t_o|< b \)

which is an open interval 

Therefore, the Correct Option is Option (2).

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