Let:
$$ Tf(t):=\gamma(t)+k\int_0^t\cos(f(t)-f(s))\,ds. $$
How can we show/discuss whether this linear operator is a contraction for some particular $k?$
I proceed by normal step but do not know how to deal with the cosine:
$$ \\||Tg-Tf||\le|k| \int_0^t||\cos(g(t)-g(s))-\cos(f(t)-f(s))||\,ds. $$
I'm stuck here. Please help!