: Functional equations

24.1 Functional equations

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[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

\[ f(x)=1+\frac{x}{2}f\big(x^ 2\big)\quad ∀ x∈ [-1,1]\quad . \]

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

E443

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

Consider the problem

\[ \begin{cases} y’(x)=y(x^ 2)\\ y(0)=1 \end{cases} \]

(this is not a Cauchy problem).

Show that, for every \(r {\lt} 1\), 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 \([−1, 1]\).

E443

[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

\[ f(x)=\sin (x)+∫_ 0^ 1 \frac{f(t)}{x^ 2+t^ 2+1}\, {\mathbb {d}}t \quad ∀ x∈ [0,1]\quad . \]

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

A function \(f (x) =∑_{n=0}^∞ a_ n x_ n\), analytic in a neighborhood of 0, satisfies on its domain the conditions

\[ \begin{cases} f ’ (x) = 1 + f (−x)\\ f (0) = c \end{cases} \quad ; \]

(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.

E443

[1VH]

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

\[ f(x) = x \cos (f(x))~ ~ . \]
Fixed \(a,b\), show that there exist a finite number of continuous \(f:(-a,b)→ℝ\) satisfying

\[ f(x) = x \cos (f(x))~ ~ ∀ x∈(a,b). \]