関数方程式のあやしい世界

関数方程式の闇

1. ͜ͷΑ͏ͳfΛ͢΂ͯٻΊΑ f(x) = x͕Ұͭͷղ͕ͩ… ؔ਺ํఔࣜ f(x + y) = f(x) + f(y) f(1) = 1
2. ͸ղ ଞʹղ͸ʁ f(x) + f(y) = f(x + y + 2f(xy)) f: ℝ≥0 → ℝ≥0 f(x) = 0 and f(x) = x
3. ఆཧ f͕࿈ଓͳΒ͹ɺղ͸ f(x) = 0 ·ͨ͸ f(x) = x
4. ิ୊̍ f͕୯ࣹͳΒ͹f(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
5. ิ୊̎ f(x)͸x͕૿Ճͨ͠ͱ͖ݮগ͠ͳ͍ t ↦ t + 2f(xt) ͸೚ҙͷਖ਼ͷ࣮਺஋Λ΋ͭ ূ໌ɿ Αͬͯy>xʹରͯ͋͠Δt͕ଘࡏͯ͠ f(y) = f(x + t + 2f(xt)) = f(x) + f(t) ≥ f(x)
6. ఆཧͷূ໌ ΋͋͠ΔaͰf(a) = 0ͳΒ͹ f(2a) ≤ f(2a + 2f(a2)) = 2f(a) = 0 f͸ݮগ͠ͳ͍͔Βɺf͸߃౳తʹ̌ɻ ͦ͏Ͱͳ͍ͱ͢Δɻิ୊̎ͷূ໌ͱಉ༷ʹ ೚ҙͷy>xʹ͍ͭͯɺ͋Δt͕͋ͬͯy-x = t + f(tx) f(y) = f(x + t + f(tx)) = f(x) + f(t) > f(x) Αͬͯf͸୯ௐ૿େɻิ୊̍ΑΓ
7. ະղܾ໰୊ f͕Ұൠͷؔ਺ͩͬͨΒʁ