EXERCISE 1.40. (i) Show there is exactly one u C([1,1]) that satisfies the integral equation...

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EXERCISE 1.40. (i) Show there is exactly one u € C([−1,1]) that satisfies the integral equation X u(x) = x ["* 1² cos(u(1)) dt, x € [−1, 1]. Hint: Use the contraction principle in the space C([-1,1]). (ii) Show that u is differentiable. Is u' differentiable? EXERCISE 1.40. (i) Show there is exactly one u € C([−1,1]) that satisfies the integral equation X u(x) = x ["* 1² cos(u(1)) dt, x € [−1, 1]. Hint: Use the contraction principle in the space C([-1,1]). (ii) Show that u is differentiable. Is u' differentiable?

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