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sequence of integral

Source: Romania 1997

August 27, 2005
calculusintegrationfunctionreal analysisreal analysis unsolved

Problem Statement

Suppose that (fn)n∈N(f_n)_{n\in N} be the sequence from all functions fn:[0,1]→R+f_n:[0,1]\rightarrow \mathbb{R^+} s.t. f0f_0 be the continuous function and ∀x∈[0,1],∀n∈N,fn+1(x)=∫0x11+fn(t)dt\forall x\in [0,1] , \forall n\in \mathbb {N} , f_{n+1}(x)=\int_0^x \frac {1}{1+f_n (t)}dt. Prove that for every x∈[0,1]x\in [0,1] the sequence of (fn(x))n∈N(f_n(x))_{n\in N} be the convergent sequence and calculate the limitation.
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