Question

1.2 Let $(X, d)$ be a complete metric space and let $F: X \to X$ be such that $F^N: X \to X$ is a contraction for some positive integer $N$. Show that $F$ has a unique fixed point $u \in X$ and that for each $x \in X$, $\lim_{n \to \infty} F^n(x) = u$. 1.3 Using the result obtained in Exercise 1.2, give an alternative proof for the Picard-Lindelöf theorem (Theorem 1.7).

          1.2 Let $(X, d)$ be a complete metric space and let $F: X \to X$ be such
that $F^N: X \to X$ is a contraction for some positive integer $N$.
Show that $F$ has a unique fixed point $u \in X$ and that for each
$x \in X$, $\lim_{n \to \infty} F^n(x) = u$.
1.3 Using the result obtained in Exercise 1.2, give an alternative proof
for the Picard-Lindelöf theorem (Theorem 1.7).

1.2 Let (X, d) be a complete metric space and let F: X → X be such
that F^N: X → X is a contraction for some positive integer N.
Show that F has a unique fixed point u ∈ X and that for each
x ∈ X, limn →∞ F^n(x) = u.
1.3 Using the result obtained in Exercise 1.2, give an alternative proof
for the Picard-Lindelöf theorem (Theorem 1.7).

Added by Kelly A.

Elementary and Intermediate Algebra

Alan S. Tussy, R. David Gustafson 5th Edition

Instant Answer

Solved by Expert Sri K

02/25/2023

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2, we know that F has a unique fixed point u in X and that for each x in X, lim(n->∞) F^n(x) = u. Show more…

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1.3 Using the result obtained in Exercise 1.2, give an alternative proof for the Picard-Lindelöf theorem (Theorem 1.7). Exercise 1.2: Let (X, d) be a complete metric space and let F: X -> X be such that F^N: X -> X is a contraction for some positive integer N. Show that F has a unique fixed point u in X and that for each x in X, lim(n->∞) F^n(x) = u.