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関数方程式のあやしい世界
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Yoriyuki Yamagata
April 13, 2019
Science
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710
関数方程式のあやしい世界
関数方程式の闇
Yoriyuki Yamagata
April 13, 2019
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Transcript
͜ͷΑ͏ͳfΛͯ͢ٻΊΑ f(x) = x͕Ұͭͷղ͕ͩ… ؔํఔࣜ f(x + y) = f(x)
+ f(y) f(1) = 1
ղ ଞʹղʁ f(x) + f(y) = f(x + y +
2f(xy)) f: ℝ≥0 → ℝ≥0 f(x) = 0 and f(x) = x
ఆཧ f͕࿈ଓͳΒɺղ f(x) = 0 ·ͨ f(x) = x
ิ̍ f͕୯ࣹͳΒf(x)=√x f(x) + f(y) + f(1) = f(x +
y + 1 + 2f(y) + 2f(xy + x + 2xf(y)) f(x) + f(y) + f(1) = f(x + y + 1 + 2f(xy) + 2f(x) + 2f(y)) ূ໌ɿ f͕୯ࣹΑΓ f(xy + x + 2xf(y)) = f(xy) + f(x) = f(xy + x + 2f(x2y)) ∴ f(x2y) = xf(y) ∴ f(x) = f(1) x
ิ̎ f(x)x͕૿Ճͨ͠ͱ͖ݮগ͠ͳ͍ t ↦ t + 2f(xt) ҙͷਖ਼ͷ࣮Λͭ ূ໌ɿ Αͬͯy>xʹରͯ͋͠Δt͕ଘࡏͯ͠
f(y) = f(x + t + 2f(xt)) = f(x) + f(t) ≥ f(x)
ఆཧͷূ໌ ͋͠ΔaͰf(a) = 0ͳΒ f(2a) ≤ f(2a + 2f(a2)) =
2f(a) = 0 fݮগ͠ͳ͍͔Βɺf߃తʹ̌ɻ ͦ͏Ͱͳ͍ͱ͢Δɻิ̎ͷূ໌ͱಉ༷ʹ ҙͷy>xʹ͍ͭͯɺ͋Δt͕͋ͬͯy-x = t + f(tx) f(y) = f(x + t + f(tx)) = f(x) + f(t) > f(x) Αͬͯf୯ௐ૿େɻิ̍ΑΓ
ະղܾ f͕ҰൠͷؔͩͬͨΒʁ