23.1 Functional equations

E442

[1V9] Note:exercise 1, June 7th 2010.

Prove that there exists one and only one continuous function $f$ on the interval $[-1,1]$ such that

$$f(x)=1+\frac{x}{2}f(x^2)\forall x\in [-1,1].$$

Prove that $f$ is representable as a power series centered at zero; and that the radius of convergence is one.

Hidden solution: [UNACCESSIBLE UUID ’1VB’]

E442

[1VC]Difficulty:*.Note:exercise 3, written exam, June 30th, 2017.

Consider the problem

$$\{\begin{array}{l}{y}^{\prime }(x)=y(x^2)\\ y(0)=1\end{array}$$

(this is not a Cauchy problem).

• Show that, for every $r<1$, there is only one solution defined on $I=(-r,r)$, and deduce that the same is true for $r=1$.

• Show that the solution is representable as the sum of a power series centered in $0$ and converging on the interval $[-1,1]$.

Hidden solution: [UNACCESSIBLE UUID ’1VD’]

E442

[1VF] Note:exercise 3, written exam, June 23th 2012.

Prove that there is one and only one continuous function $f$ on interval $[0,1]$ that satisfies the condition

$$f(x)=\sin (x)+{\int }_0^1\frac{f(t)}{x^2+t^2+1}\mathbb{d}t\forall x\in [0,1].$$

E442

[1VG] Note:exercise 4, written exam, June 23th, 2012.

A function $f(x)={\sum }_{n=0}^{\infty }{a}_n{x}_n$, analytic in a neighborhood of 0, satisfies on its domain the conditions

$$\{\begin{array}{l}{f}^{\prime }(x)=1+f(-x)\\ f(0)=c\end{array};$$

(note that this is not a Cauchy problem!).

• Determine $f$.

• Prove that the function found is the only solution, in the set of all functions that can be derived in a neighborhood of 0.

E442

[1VH]

• Show that there is an unique continuous function$f:(-1,1)\to ℝ$ that satisfies

  $$f(x)=x\cos (f(x)).$$

• Fixed $a,b$, show that there exist a finite number of continuous $f:(-a,b)\to ℝ$ satisfying

  $$f(x)=x\cos (f(x))\forall x\in (a,b).$$

Hidden solution: [UNACCESSIBLE UUID ’1VJ’]