Determine all functions $f: \mathbb{R} \mapsto \mathbb{R}$ satisfying

\[f\left(\sqrt{2} \cdot x\right) + f\left(4 + 3 \cdot \sqrt{2} \cdot x \right) = 2 \cdot f\left(\left(2 + \sqrt{2}\right) \cdot x\right)\]

for all $x$.
1. **Substitution and Simplification:**
   Let \( y = x \sqrt{2} \) and \( q = 1 + \sqrt{2} \). The given functional equation can be rewritten as:
   \[
   f\left(\sqrt{2} \cdot x\right) + f\left(4 + 3 \cdot \sqrt{2} \cdot x \right) = 2 \cdot f\left(\left(2 + \sqrt{2}\right) \cdot x\right)
   \]
   Substituting \( y = x \sqrt{2} \), we get:
   \[
   f(y) + f\left(4 + 3y\right) = 2f\left((2 + \sqrt{2}) \cdot \frac{y}{\sqrt{2}}\right)
   \]
   Simplifying the right-hand side:
   \[
   2f\left(\frac{2 + \sqrt{2}}{\sqrt{2}} \cdot y\right) = 2f\left(\left(\sqrt{2} + 1\right) y\right) = 2f(qy)
   \]
   Thus, the equation becomes:
   \[
   f(y) + f(q^2 y) = 2f(q y)
   \]

2. **Sequence Definition:**
   Define a sequence \( a_n \) such that \( a_n = f(q^n y) \). The functional equation now translates to:
   \[
   a_n + a_{n+2} = 2a_{n+1}
   \]
   This is a linear recurrence relation.

3. **Solving the Recurrence Relation:**
   The characteristic equation for the recurrence relation \( a_n + a_{n+2} = 2a_{n+1} \) is:
   \[
   r^2 - 2r + 1 = 0
   \]
   Solving this, we get:
   \[
   (r - 1)^2 = 0 \implies r = 1 \text{ (with multiplicity 2)}
   \]
   Therefore, the general solution to the recurrence relation is:
   \[
   a_n = c_1 + c_2 n
   \]
   where \( c_1 \) and \( c_2 \) are constants.

4. **Expressing \( f \) in Terms of \( x \):**
   Since \( a_n = f(q^n y) \), we have:
   \[
   f(q^n y) = c_1 + c_2 n
   \]
   Let \( x_0 \) be a fixed point such that \( y = x_0 \). Then:
   \[
   f(q^n x_0) = c_1 + c_2 n
   \]
   For \( x = q^n x_0 \), we have:
   \[
   n = \frac{\ln(x/x_0)}{\ln q}
   \]
   Substituting this back, we get:
   \[
   f(x) = c_1 + c_2 \frac{\ln(x/x_0)}{\ln q}
   \]
   Since \( q = 1 + \sqrt{2} \), we have:
   \[
   f(x) = c_1 + c_2 \frac{\ln(x/x_0)}{\ln(1 + \sqrt{2})}
   \]

5. **Conclusion:**
   The general form of the function \( f \) that satisfies the given functional equation is:
   \[
   f(x) = c_1 + c_2 \frac{\ln(x/x_0)}{\ln(1 + \sqrt{2})}
   \]
   where \( c_1 \) and \( c_2 \) are constants, and \( x_0 \) is a fixed point.

The final answer is \( \boxed{ f(x) = c_1 + c_2 \frac{\ln(x/x_0)}{\ln(1 + \sqrt{2})} } \).