Let f be a function which satisfied following conditions:
1) f is of class C^1(ℝ), and ∣f'(x)∣< 1, ∀ x ∊ ℝ
2) f(x+1)=f(x), ∀ x ∊ ℝ (f is periodical with basic period ω=1)
Let p be a function defined with, p(x)=x+f(x), ∀ x ∊ ℝ
Prove that
\(\displaystyle \lim_{n\to\infty}{\frac {x+p(x)+p(p(x))+...+p(p(...p(x)...))}{n^2} }\) (composition in last term is taken n-1 times)
exists, and doesn't depend on x. ...............................Latex edited
P.S. This is my first post here, I don't know how to tape this quotient. I ask admin, please, to type for me this correctly, just this time. (And delee this P.S.)
Thanks in advance
1) f is of class C^1(ℝ), and ∣f'(x)∣< 1, ∀ x ∊ ℝ
2) f(x+1)=f(x), ∀ x ∊ ℝ (f is periodical with basic period ω=1)
Let p be a function defined with, p(x)=x+f(x), ∀ x ∊ ℝ
Prove that
\(\displaystyle \lim_{n\to\infty}{\frac {x+p(x)+p(p(x))+...+p(p(...p(x)...))}{n^2} }\) (composition in last term is taken n-1 times)
exists, and doesn't depend on x. ...............................Latex edited
P.S. This is my first post here, I don't know how to tape this quotient. I ask admin, please, to type for me this correctly, just this time. (And delee this P.S.)
Thanks in advance
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