# Solve differential system of equations

Consider the following system: $$\begin{cases} x_1 + 3 x_3 = 4a, \\ f(x_1) + 3 f(x_3) = 8 f(a), \\ f'(x_1) = 3 f'(x_3). \end{cases}$$

I want to find all functions (or at least learn some properties that hold for all of them) $$f : [0,1] \to [0,1]$$ that are continuous, differentiable on $$[0,1]$$, monotonically decreasing on $$[0,1]$$ and the aforementioned system has a solution for every $$a \in (0,1)$$. In other words, if $$a$$ is fixed, there should exist $$x_1, x_3 \in [0,1]$$ that satisfy the system.

Or, a bit re-phrased: find $$f$$ such that for all $$a \in (0,1)$$, the aforementioned system has at least one pair $$(x_1, x_3) \in [0,1]^2$$ satisfying the system. And the general question is of course to find all such $$f$$.

UPD: Apparently, the solution is trivial only. I created another question that is hopefully more of interest: Solve differential system with a parameter

• I don't understand the assumptions. $x_1,x_3 \in [0,1]$ variable and the second equality is $f(x_1) + 3f(x_3) = 8f((x_1+3x_3)/4)$. Or is $x$ some constant and then the last equations leads to $f \equiv c$ for some constant $c$. So: what is variable, what is constant? Mar 4 at 19:19
• @DieterKadelka $f$ must be such that for each $x\in(0,1)$ there exist $x_1,x_3\in[0,1]$ satisfying all three equalities. Mar 4 at 19:22
• @DieterKadelka yes, $x$ is a parameter. Perhaps, I will rename it for more clearance Mar 4 at 20:20

Assume such an $$f$$ exists. For each $$a\in(0,1)$$ fix some solution $$x_1(a),x_3(a)$$ of the system. The first equality can be restated as $$\frac{1}{4}x_1(a) + \frac{3}{4}x_3(a) = a.$$ In other words, $$a$$ is a convex combination of $$x_1(a),x_3(a)$$. Consider a sequence $$a_k \searrow 0$$. As $$x_1(a_k),x_3(a_k) \geq 0$$ it follows that $$x_1(a_k),x_3(a_k) \to 0$$. Taking the limit $$k\to \infty$$ in the second equation, we obtain from the continuity of $$f$$ that $$4 f(0) = 8 f(0).$$ Hence $$f(0) = 0$$. As $$f$$ is monotonically decreasing and takes nonnegative values, it follows that $$f$$ is constant.