To solve the problem, we need to analyze the conditions given:

1. \( 0 < a < b < c \)
2. \( |a^3 - 4a| = |b^3 - 4b| = |c^3 - 4c| \)
3. \( a + b = c \)

Given these conditions, we proceed as follows:

The expression for each value can be rewritten as:
\[ |x^3 - 4x| = k \]
where \( k \) is a constant equal for \( a \), \( b \), and \( c \).

Considering the expression \( |x^3 - 4x| = k \), we know it involves taking the absolute value, which implies:
\[ x^3 - 4x = k \quad \text{or} \quad x^3 - 4x = -k \]

Now let's test a simple rational solution to satisfy all the conditions:

1. Try a few small integer values or simple rational values for \( a \) to find \( k \), and see if the relations among \( a \), \( b \), and \( c \) are satisfied.

For example, if \( a = 1 \):
\[ |1^3 - 4 \times 1| = |-3| = 3 \]
So, \( k = 3 \).

Plug this into the equation:
For \( b \):
\[ |b^3 - 4b| = 3 \]

and for \( c = a + b \):
\[ |c^3 - 4c| = 3 \]

Next, let's solve \( |x^3 - 4x| = 3 \):
- \( x^3 - 4x = 3 \) gives the equation \( x^3 - 4x - 3 = 0 \)
- \( x^3 - 4x = -3 \) gives the equation \( x^3 - 4x + 3 = 0 \)

By solving these cubic equations, we find the possible \( x \) values that satisfy this:
- For \( x^3 - 4x - 3 = 0 \), factoring or using the Rational Root Theorem gives a possible solution \( x = -1 \), which can be rejected since \( a, b, c > 0 \).
- For \( x^3 - 4x + 3 = 0 \), solve for potential \( a, b, c \).

Let's use the rational root test to verify possible roots or numerical approaches, but ideally:

Assuming each is 1 for simplicity:
\[ a = 1, b = 2, c = 3 \]
Verify if this set works:
\( a^3 - 4a = 1 - 4 = -3 \)
\( b^3 - 4b = 8 - 8 = 0 \) (*Not fulfilled for \( b = 2 \)*)

Thus given the symmetrical nature of these equations, practical whole integer roots for these conditions and constraints won't evenly satisfy them unless \( a, b, c \) are parts of roots in polynomial or irrational results. Therefore a deeper numeric computation or plotting tool should be engaged to solve.

Explore further to find values where \( x^3 - 4x = \pm k \) and satisfy these conditions, potentially using numerical or graph solution methods to find valid \( b, a, \) such that \( c = a + b \).