擬単調写像に関する変分不等式問題の飯塚-高橋型近似列のもつ性質

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ٖ୯ௐࣸ૾ʹؔ͢Δม෼ෆ౳ࣜ໰୊ͷ൧௩

-

ߴڮܕۙࣅྻͷ΋

ͭੑ࣭

Properties of Iiduka-Takahashi type iterations of variational inequality problems for pseudomonotone mappings

๛ాণ࢙ Masashi Toyoda

ۄ઒େֶ޻ֶ෦ϚωδϝϯταΠΤϯεֶՊ, 194–8610 ౦ژ౎ொాࢢۄ઒ֶԂ 6–1–1 College of Engineering, Tamagawa University,

6–1–1 Tamagawa-gakuen, Machida-shi, Tokyo 194–8610

Abstract

In this paper, we introduce an iteration for finding an element of the set of solutions of the variational inequality for a psuedomonotone mapping in a Hilbert space. Moreover, we consider strong convergence of the iteration to the solution of variational inequality problem.

Keywords: Variational inequality, pseudomonotone mapping, strong convergence.

ୈ1અ ͸͡Ίʹ H Λ Hilbert ۭؒͱ͠, C Λ H ͷดತू ߹ͱ͢Δ. A Λ C ͔Β H ΁ͷࣸ૾ͱ͢Δ. x∗ ∈ C ͕ม෼ෆ౳ࣜ໰୊ͷղͰ͋Δͱ͸, ೚ ҙͷ x_{∈ C} ʹରͯ͠ ⟨Ax∗, x− x∗� ≥ 0 ΛΈͨ͢ͱ͖Λ͍͏. ม෼ෆ౳ࣜ໰୊ͷղશମ ͷू߹ΛVI(C, A) ͱ͋ΒΘ͢. ͋Δ α > 0͕ ଘࡏͯ͠,೚ҙͷx, y_{∈ C} ʹରͯ͠

⟨Ax − Ay, x − y� ≥ α∥Ax − Ay∥2 ΛΈͨ͢ͱ͖, ࣸ૾ A Λٯڧ୯ௐ (inverse-strongly monotone) ͱ͍͏. A ͕ٯڧ୯ௐࣸ ૾ͷ৔߹,ม෼ෆ౳ࣜ໰୊ͷղʹڧऩଋ͢Δ݁ Ռ͕طʹಘΒΕ͍ͯΔ. ࣮ࡍ, 2004೥,൧௩-ߴ ڮ[4]͸n = 1, 2, 3, . . . ʹରͯ͠ xn+1= αnx + (1− αn)PC(xn− λnAxn) ͰఆΊΔۙࣅྻ(Ҏ߱,൧௩-ߴڮܕۙࣅྻͱݺ Ϳ)Λಋೖ͠,ม෼ෆ౳ࣜ໰୊ͷղʹڧऩଋ͢Δ ͜ͱΛࣔͨ͠. ͜͜Ͱ{αn} ͸(0, 1) ͷ਺ྻͰ lim n→∞αn= 0, ∞ ∑ n=1 αn=∞ ͓Αͼ ∞ ∑ n=1 |αn+1− αn| < ∞ ΛΈͨ͢΋ͷͱ͢Δ. {λn} ͸ (0, 1) ͷ෦෼۠ ؒ[a, b]ͷ਺ྻͰ ∞ ∑ n=1 |λn+1− λn| < ∞ ΛΈͨ͢΋ͷͱ͢Δ. PC ͸ H ͔Β C ͷ্΁ ͷڑ཭ࣹӨͰ͋Δ. [8, 9]΋ࢀর͞Ε͍ͨ. ͞ Βʹ, 2011೥, ੨ࢁ[1]͸, {αn} ͱ {λn} ͷ৚ ݅Λ lim n→∞αn= 0, ∞ ∑ n=1 αn=∞ ͓Αͼ 0 < inf n λn≤ sup_n λn< 2α ʹऑΊͨͱͯ͠΋,൧௩-ߴڮܕۙࣅྻ͕,ม෼ ෆ౳ࣜ໰୊ͷղʹڧऩଋ͢Δ͜ͱΛࣔͨ͠. ͜ ͜Ͱα ͸ٯڧ୯ௐࣸ૾ A ͷ܎਺Ͱ͋Δ. ͜Ε

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Βͷݚڀ͕ࣔ͢Α͏ʹ, A Λٯڧ୯ௐࣸ૾ͱ͢ Δͱ͖,൧௩-ߴڮܕۙࣅྻ͸{αn}, {λn}ʹର ͯ͠े෼ͳԾఆΛ͢Ε͹,ม෼ෆ౳ࣜ໰୊ͷղ ʹڧऩଋ͢Δ͜ͱ͕஌ΒΕ͍ͯΔ. Ұํ,ࣸ૾ A͕༗քͰ,͞Βʹ x ʹऑऩଋ ͢Δ఺ྻ{xn} ͕ lim sup n→∞ ⟨Axn, xn− x⟩ ≤ 0 ΛΈͨ͢ͳΒ͹,೚ҙͷ y_{∈ C} ʹରͯ͠ ⟨Ax, x − y⟩ ≤ lim inf_n

→∞ ⟨Ax, xn− y⟩ ͕੒Γཱͭͱ͖, ࣸ૾ A Λٖ୯ௐ (psue-domonotone) ͱ͍͏. A ͕ٖ୯ௐࣸ૾ͷ৔߹ ͷม෼ෆ౳ࣜ໰୊͸,ྫ͑͹, [3, 5, 6, 2]ͰѻΘ Ε͍ͯΔ. A͕ٖ୯ௐࣸ૾ͷ৔߹,ม෼ෆ౳ࣜ ໰୊ͷղʹऑऩଋ͢Δ݁Ռ͸ಘΒΕ͍ͯΔ΋ͷ ͷ([3, 6]), චऀͷ஌ΔݶΓ, ͍·ͩڧऩଋ͢Δ ݁ՌΛಘΒΕ͍ͯͳ͍. A͕ٖ୯ௐࣸ૾ͷ৔߹ Ͱ΋,ม෼ෆ౳ࣜ໰୊ͷղʹڧऩଋ͢Δۙࣅྻ ͸ߏ੒Ͱ͖ΔͰ͋Ζ͏͔. ٯڧ୯ௐࣸ૾ʹ৔߹Ͱ͸ڧऩଋΛಘΒΕͯ ͍Δ൧௩-ߴڮܕۙࣅྻΛར༻͢Ε͹,ٖ୯ௐࣸ ૾ͷ৔߹ͷม෼ෆ౳ࣜ໰୊ͷղ΁ͷڧऩଋఆཧ ΛಘΒΕΔ͔΋͠Εͳ͍. ͦ͜Ͱຊ࿦จʹ͓͍ ͯ, A ͕ٖ୯ௐࣸ૾ͷ৔߹,൧௩-ߴڮܕۙࣅྻ ͕ͲͷΑ͏ͳੑ࣭Λ΋ͭͷ͔Λߟ࡯͢Δ. ·ͨ, ڧ͍ԾఆͰ͸͋Δ͕,൧௩-ߴڮܕۙࣅྻ͕ڧऩ ଋ͢ΔͨΊͷे෼৚݅Λࣔ͢. ୈ2અ ४උ H Λ Hilbert ۭؒͱ͠, C Λ H ͷดತू ߹ͱ͢Δ. ೚ҙͷ x_{∈ H} ʹରͯ͠ ∥x − z∥ = min{∥x − y∥ | y ∈ C} ͱͳΔΑ͏ͳz_{∈ C}͕Ұҙʹଘࡏ͢Δ. PCx = z Ͱද͠, PC Λ C ͔ΒH ΁ͷ্΁ͷڑ཭ࣹ Өͱ͍͏. ڑ཭ࣹӨPC ʹରͯ͠, z = PCx Ͱ ͋ΔͨΊͷඞཁे෼৚݅͸ ⟨x − z, z − y⟩ ≥ 0 ͕೚ҙͷy_{∈ C} ʹରͯ͠੒Γཱͭ͜ͱͰ͋Δ. ৄ͘͠͸[7]Λࢀর͞Ε͍ͨ. AΛC͔ΒH ΁ͷࣸ૾ͱ͢Δ. ࣸ૾A͕ ίΞγϒ(coercive)Ͱ͋Δͱ͸,೚ҙͷ x_{∈ C} ʹରͯ͠ ⟨Ax, x⟩ ≥ ∥x∥2 ͕੒Γཱͭͱ͖Λ͍͏. ࣸ૾A͕ٖ୯ௐ (psue-domonotone) Ͱ͋Δͱ͸, A ͕࣍ͷ(I)(II)Λ Έͨ͢ͱ͖Λ͍͏. (I) A ͸༗քͰ͋Δ. (II) xʹऑऩଋ͢Δ఺ྻ{xn} ͕ lim sup n→∞ ⟨Axn, xn− x⟩ ≤ 0 ΛΈͨ͢ͱ͖,೚ҙͷ y_{∈ C} ʹରͯ͠ ⟨Ax, x − y⟩ ≤ lim inf_n

→∞ ⟨Ax, xn− y⟩ ͕੒Γཱͭ. ࣸ૾A͕ϙςϯγϟϧ(potential)Ͱ͋Δͱ͸, ೚ҙͷx, y∈ C ʹରͯ͠ ∫ 1 0

(⟨A(t(x + y)), x + y⟩ − ⟨A(tx), x⟩) dt =

∫ 1

0 ⟨A(x + ty), y⟩dt

ΛΈͨ͢ͱ͖Λ͍͏. ࣸ૾A͕Ϧϓγοπ࿈ଓ

(Lipschitz continuous)Ͱ͋Δͱ͸,͋ΔL > 0 ͕ଘࡏͯ͠,೚ҙͷ x, y∈ C ʹରͯ͠

∥Ax − Ay∥ ≤ L∥x − y∥

͕੒Γཱͭͱ͖Λ͍͏. C ͷ఺ྻ{xn}Λ࣍ͷΑ͏ʹఆٛ͢Δ. x∈ C ͱ͢Δ. x1 = xͱ͢Δ. n = 1, 2, 3, . . .ʹର ͯ͠ xn+1= αnx + (1− αn)PC(xn− λnAxn) ͱ͢Δ. ͜͜Ͱ {αn} ͸ (0, 1) ͷ਺ྻͰ͋Δ. ·ͨ{λn} ͸(0, 1) ͷ෦෼۠ؒ[a, b] ͷ਺ྻͰ ͋Δ. ҎԼ,֤nʹରͯ͠ yn= PC(xn− λnAxn) ͱ͓͘. ิॿ໋୊ 1. ࣸ૾ A ͸ٖ୯ௐͱ͢Δ. ఺ྻ {xn}, {yn} ͸༗քͰ͋Δͱ͠ xn− yn→ 0 ͱ͢Δ. ͜ͷͱ͖, ఺ྻ {xn} ͷ෦෼ྻ {xni} ͷऑऩଋઌ x∗ ͸ม෼ෆ౳ࣜ໰୊ͷղͰ͋Δ. ͢ͳΘͪ x∗ _{∈ VI(C, A)}Ͱ͋Δ.

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ূ໌. ·ͣ,೚ҙͷ y_{∈ C} ʹରͯ͠ lim inf n→∞ ⟨Axn, xn− y⟩ ≤ 0 ͓Αͼ lim sup n→∞ ⟨Axn, xn− y⟩ ≤ 0 ͕੒Γཱͭ͜ͱΛࣔ͢. yn= PC(xn− λnAxn) ΑΓ,೚ҙͷ y∈ C ʹରͯ͠ ⟨yn− (xn− λnAxn), y− yn⟩ ≥ 0 Ͱ͋Δ. ͕ͨͬͯ͠ ⟨yn− xn, xn− yn⟩ ≥ ⟨−λnAxn, y− yn⟩, ͢ͳΘͪ −_λ1 n∥xn− yn∥∥v − yn∥ ≥ ⟨Axn , yn− y⟩ ΛಘΔ. y∈ C ͱ͢Δ. ͜ͷͱ͖ ⟨Axn, xn− y⟩

=_⟨Axn, xn− yn⟩ + ⟨Axn, yn− y⟩

≤ ⟨Axn, xn− yn⟩ + 1 λn∥yn− xn∥∥y − yn∥ ≤ ∥Axn∥∥xn− yn∥ + 1 a∥xn− yn∥∥y − yn∥ Ͱ͋Δ. ͜͜Ͱ {λn} ⊂ [a, b] Ͱ͋Δ͜ͱʹ஫ ҙ͞Ε͍ͨ. {xn}, {yn} ͸༗ք, ࣸ૾ A ͸༗ քͰ, xn− yn→ 0 Ͱ͋Δ͔Β lim inf n→∞ ⟨Axn, xn− y⟩ ≤ 0 ͓Αͼ lim sup n→∞ ⟨Axn, xn− y⟩ ≤ 0 ΛಘΔ. ্هͷ y ͷ୅ΘΓʹ x∗ Ͱߟ͑Ε͹ಉ༷ʹ ෆ౳ࣜ ⟨Axni, xni− x∗⟩ ≤ ∥Axni∥∥xni− yni∥ + 1 a∥xni− yni∥∥x∗− yni∥ ΛಘΔ. Αͬͯ lim sup i→∞ ⟨Axni , xni− x∗⟩ ≤ 0 ͕੒Γཱͭ. y_{∈ C} ͱ͢Δ. A ͸ٖ୯ௐͰ͋Δ ͔Β

⟨Ax∗, x∗− v⟩ ≤ lim inf

i→∞ ⟨Axni, xni− y⟩

Ͱ͋Δ. ͜ΕΑΓ

⟨Ax∗, x∗_{− y⟩ ≤ lim inf}

i→∞ ⟨Axni, xni − y⟩ ≤ 0 ΛಘΔ. ͢ͳΘͪ ⟨Ax∗, y− x∗⟩ ≥ 0 Ͱ͋Δ. □ ໋୊ 2. H Λ Hilbert ۭؒͱ͠, C Λ H ͷด ತू߹ͱ͢Δ. AΛC ͔ΒH ΁ͷٖ୯ௐࣸ૾ ͱ͢Δ. C ͷ఺ྻ{xn}Λ࣍ͷΑ͏ʹఆٛ͢Δ. x∈ C ͱ͢Δ. x1 = x ͱ͢Δ. n = 1, 2, 3, . . . ʹରͯ͠ xn+1= αnx + (1− αn)PC(xn− λnAxn) ͱ͢Δ. ͜͜Ͱ {αn} ͸ (0, 1) ͷ਺ྻͰ͋ Δ. ·ͨ {λn} ͸ (0, 1) ͷ෦෼۠ؒ [a, b] ͷ ਺ྻͰ͋Δ. yn = PC(xn− λnAxn) ͱ͓͘. x∗ = P_VI(C,A)x ͱ͓͘. ఺ྻ {xn}, {yn} ͸ ༗քͰ͋Δͱ͠, xn − yn → 0 ͱ͠, ͞Βʹ n = 1, 2, 3, . . . ʹରͯ͠ ∥yn− x∗∥ ≤ ∥xn− x∗∥ ΛΈͨ͢ͱ͢Δ. ·ͨ ∞ ∑ n=1 αn=∞ ΛΈͨ͢ͱ͢Δ. ͜ͷͱ͖ {xn} ͸ x∗ ʹڧऩ ଋ͢Δ. ূ໌. ·ͣ lim sup n→∞ ⟨x − x ∗_{, x} n− x∗⟩ ≤ 0 ͕੒Γཱͭ͜ͱΛࣔ͢. {xn} ͸༗քͰ͋Δ͔ Β,෦෼ྻ{xni} ͕ଘࡏͯ͠x∗ ʹऑऩଋ͢Δ.

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ิॿ໋୊ 1 ΑΓ x∗ _{∈ VI(C, A)} Ͱ͋Δ. ͨ͠ ͕ͬͯ lim sup n→∞ ⟨x − x ∗_{, x} n− x∗⟩ = lim i→∞⟨x − x ∗_{, x} ni − x∗⟩ =_{⟨x − x}∗, x∗_{− x}∗_{⟩ ≤ 0} ΛಘΔ. ϵ > 0 ͱ͢Δ. lim sup_n_→∞⟨x − x∗_{, x}_n₋ x∗_{⟩ ≤ 0} ΑΓ,͋Δࣗવ਺ n0 ͕ଘࡏͯ͠,೚ҙ ͷn_{≥ n}0 ʹରͯ͠ ⟨x − x∗, xn− x∗⟩ < ϵ ͕੒Γཱͭ. ·ͨ αnx + (1− αn)yn− (αnx + (1− αn)x∗) = xn+1− x∗+ αn(x∗− x) Ͱ͋Δ͔Β ∥αnx + (1− αn)yn− (αnx + (1− αn)x∗)∥2 ≥ ∥xn+1− x∗∥2+ 2αn⟨x∗− x, xn+1− x∗⟩ ΛಘΔ. Αͬͯn≥ n0 ʹରͯ͠ ∥xn+1− x∗∥2 ≤ 2αn⟨x − x∗, xn+1− x∗⟩ + (1_{− α}n)2∥yn− x∗∥2 ≤ 2αn⟨x − x∗, xn+1− x∗⟩ + (1− αn)2∥xn− x∗∥2 ≤ 2αnϵ + (1− αn)∥xn− x∗∥2 = 2ϵ (1_{− (1 − α}n)) + (1− αn)∥xn− x∗∥2 Ͱ͋Δ. ͕ͨͬͯ͠ ∥xn+1− x∗∥2 ≤ 2ϵ  1 − n ∏ k=n0 (1_{− α}k)   + n ∏ k=n0 (1_{− α}k)∥xn0− x∗∥ 2 ͕੒Γཱͭ. ͜ΕΑΓ lim sup n→∞ ∥xn+1− x ∗_∥2 _{≤ 2ϵ} Ͱ͋Δ. ϵ ͸೚ҙͷਖ਼਺Ͱ͋Δ͔Β lim sup n→∞ ∥xn+1− x ∗_∥2_{≤ 0} Ͱ͋Δ. ͕ͨͬͯ͠xn→ x∗ ΛಘΔ. □ ୈ3અ ͓ΘΓʹ ໋୊ 2 Ͱ͸, _{xn} ͕ม෼ෆ౳ࣜ໰୊ͷղ x∗ ʹڧऩଋ͢ΔͨΊʹ,఺ྻ {xn}, {yn}͸༗ քͰ͋Δͱ͠, xn− yn→ 0 ͱ͠ ∥yn− x∗∥ ≤ ∥xn− x∗∥ ΛΈͨ͢ͱԾఆͨ͠. ࠓޙͷ՝୊͸, ͜ΕΒͷ ԾఆΛΈͨ͢{αn}, {λn} ͷ৚݅ΛݟۃΊΔ͜ ͱͰ͋Δ. ·ͩ,ղܾͰ͖͍ͯͳ͍͕,͜͜Ͱ͸ [3, 6]Λࢀߟʹ,ߟ࡯ΛਐΊ͓ͯ͘. ิॿ໋୊ 3. ࣸ૾ A ͸ϙςϯγϟϧͱ͠, Ϧ ϓγοπ࿈ଓͱ͢Δ. A Λ༻͍ͨࣸ૾ Φ Λ, x∈ C ʹରͯ͠ Φ(x) = ∫ 1 0 ⟨A(tx), x⟩dt ͰఆΊΔ. ͜ͷͱ͖, ೚ҙͷ n = 1, 2, 3, . . . ʹ ରͯ͠ Φ(yn) + ( 1 b − L 2 ) ∥xn− yn∥2 ≤ Φ(xn) ͕੒Γཱͭ. ূ໌. yn= PC(xn− λnAxn) ΑΓ ⟨yn− (xn− λnAxn), xn− yn⟩ ≥ 0 Ͱ͋Δ. Αͬͯ ⟨yn− xn, xn− yn⟩ ≥ ⟨−λnAxn, xn− yn⟩ Ͱ͋Δ. ͞Βʹ −_λ1 n∥xn− yn∥ 2_{≥ ⟨−Ax} n, xn− yn⟩ Ͱ͋Δ. A ͸Ϧϓγοπ࿈ଓͰ͋Δ͔Β |⟨A(xn+ t(yn− xn))− Axn, y−xn⟩|

≤ ∥A(xn+ t(yn− xn))− Axn∥∥yn− xn∥

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Ͱ͋Δ. ͕ͨͬͯ͠, A͸ϙςϯγϟϧͰ͋Δ ͔Β Φ(yn)− Φ(xn) ∫ 1 0 ⟨A(ty n), yn⟩dt − ∫ 1 0 ⟨A(tx n), xn⟩dt = ∫ 1 0 ⟨A(xn + t(yn− xn)), yn− xn⟩dt = ∫ 1 0 A(xn+ t(yn− xn))− Axn, yn− xn⟩dt +⟨Axn, yn− xn⟩ ≤ ∫ 1 0 Lt_∥yn− xn∥2dt− 1 λn∥xn− yn∥ 2 = ( L 2 − 1 λn ) ∥xn− yn∥2 ≤ ( L 2 − 1 b ) ∥xn− yn∥2 Ͱ͋Δ. Αͬͯ Φ(yn) + ( 1 b − L 2 ) ∥xn− yn∥2 ≤ Φ(xn) ΛಘΔ. □ ิॿ໋୊3ʹ,͞Βʹ{Φ(xn)}, {Φ(yn)}ͷ ۃݶ஋͕ଘࡏͯ͠, Ұக͢ΔͱԾఆ͢Δ. ͜ͷ ͱ͖ Φ(yn) + ( 1 b − L 2 ) ∥xn− yn∥2 ≤ Φ(xn) ΑΓ xn− yn→ 0 Ͱ͋Δ. ͔͠΋ A ͕ίΞγϒͱ͠, Φ(x) ≥ Φ(xn), Φ(x) ≥ Φ(yn) ͕੒ΓཱͭͱԾఆ͢Δ. ͜ͷͱ͖,_{xn}, {yn} ͸༗քͰ͋Δ. ࣮ࡍ Φ(x) ≥ Φ(xn) = ∫ 1 0 ⟨A(txn ), txn⟩ 1 tdt ≥ ∫ 1 0 1 t∥txn∥ 2_dt = ∫ 1 0 1 t∥xn∥ 2_{dt =} 1 2∥xn∥ 2 ΑΓ,_{xn} ͸༗քͱͳΔ. ಉ༷ʹ,{yn}΋༗ք ͱͳΔ. Ҏ্ͷߟ࡯͔Β, AΛίΞγϒ,ϙςϯγϟ ϧͰϦϓγοπ࿈ଓͳٖ୯ௐࣸ૾ͱ͢Δͱ͖, ࣍ͷ(I)(II)ΛΈͨ͢ {αn}, {λn} ͷे෼৚݅ ͸Կ͔, ͱ͍͏ͷ͕࢒͞Εͨ՝୊ͱͳΔ. (I) {Φ(xn)}, {Φ(yn)}ͷۃݶ஋͕ଘࡏͯ͠,Ұக͢ Δ. (II) Φ(x)≥ Φ(xn), Φ(x) ≥ Φ(yn) ͕੒Γ ཱͭ. ࢀߟจݙ

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[8] ๛ాণ࢙,ώϧϕϧτۭؒͰͷۃେ୯ௐ࡞ ༻ૉʹؔ͢Δऩଋఆཧ,ژ౎େֶ਺ཧղੳ ݚڀॴߨڀ࿥, 1755 (2011), 46–52. [9] ๛ాণ࢙, ٯڧ୯ௐࣸ૾ʹؔ͢Δม෼ෆ ౳ࣜ໰୊ΛѻͬͨBadriev ͱZadvornov ͷ݁ՌͷҰߟ࡯, ۄ઒େֶ޻ֶ෦لཁ, 47 (2012), 81–87. 2015೥3݄13೔ݪߘड෇ɹ Received, March 13, 2015