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Question: Continuous Solution of Integral Equation In a Metric Space

Question:Continuous Solution of Integral Equation In a Metric Space

clm55 8 Maple
example integral student
March 29 2009
0
So here's my problem: Find a continuous solution to the integral equation: f(x) = (1/2)*[integral_[0,1](x*y/(y+1))*f(y)dy + x^3] by finding a fixed point of the fucntion psi:(C([0,1]),d_inf) --> (C[0,1]),d_inf) psi(f(x)) = (1/2)*[integral_[0,1](x*y/(y+1))*f(y)dy + x^3] . Here, C([0,1]) is the set of all continuous functions over [0,1] and d_inf is the supremum metric. I honestly have no idea how to do this. My professor said this... A fixed point for the map psi is a solution of the integral equation. The (proof of) the Banach Fixed Point Theorem tells you how to find a fixed point of a contraction: fix any starting point (so choose some f_0 in C([0,1]) - for example f_0(x)=x) calculate f_1 = psi (f_0) calculate f_2 = psi (f_1) and repeat. He recommended doing the calculations in Maple. So here I am. Thanks for any help!
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