Existence of solution math homework help

Show that the integral equation

x(t) = log(1 + t) + (1/5)(integral of e^(-t)cos^2(ts)(x(s))^2ds), integration going from 0 to 1

has a solution in C[0,1]. Justify your answer carefully.

Hint: Use the contraction mapping theorem. You may need to work with a suitable subset of C[0,1] rather than C[0,1]. Identifying such a subset is part of the problem.

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