Integral equation and bounded solution, calculus homework help

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Obpun

Mathematics

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I need the proof for the exercise using the similar to the idea of proof this theorem

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L: ㅏ April, ll, 2017 Son Theorem 16: For b>o , The integral equution problem: -xy f(x) d y х I fry, exy at 0 осх
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Explanation & Answer

I wrote the solution. For b>=1 the operator is not a contraction. Probably there is a way to prove that there is a bounded solution for b>=1 but I still don't know it.

𝑏

Prove that integral equation 𝑓(𝑥) = 𝑎 + ∫0 𝑓(𝑦)𝑒 −𝑥𝑦 𝑑𝑦 has bounded solution on [0, 𝑏].
The idea is to prove that the operator
𝑏

𝑇: 𝐶[0, 𝑏] → 𝐶[0, 𝑏], (𝑇(𝑓))(𝑥) = 𝑎 + ∫ 𝑓(𝑦)𝑒 −𝑥𝑦 𝑑𝑦 ,
0

is a contraction mapping in 𝐶[0, 𝑏] and therefore it has a fixed point in 𝐶[0, 𝑏]
(which is a solution of the equation).

Estimate ‖𝐹(𝑓) − 𝐹(𝑔)‖:
𝑏

𝑏

‖𝐹(𝑓) − 𝐹(𝑔)‖ = ‖∫(𝑓(𝑦) − 𝑔(𝑦))𝑒

−𝑥𝑦

𝑑𝑦‖ ≤ ‖𝑓 − 𝑔‖ sup ∫ 𝑒 −𝑥𝑦 𝑑𝑦 =
𝑥∈[0,𝑏]

0

0

1 − 𝑒 −𝑏𝑥
= ‖𝑓 − 𝑔‖ sup (
) = �...


Anonymous
Goes above and beyond expectations!

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