Hoai tom tat luan an tieng anh1 final

▼■◆■❙❚❘❨ ❖❋ ❊❉❯❈❆❚■❖◆ ❱■❊❚◆❆▼ ❆❈❆❉❊▼❨ ❆◆❉ ❚❘❆■◆■◆● ❖❋ ❙❈■❊◆❈❊ ❆◆❉ ❚❊❈❍◆❖▲❖●❨ ●❘❆❉❯❆❚❊ ❯◆■❱❊❘❙■❚❨ ❖❋ ❙❈■❊◆❈❊ ❆◆❉ ❚❊❈❍◆❖▲❖●❨ ✳✳✳✳✳✳✳✳✳✳✳✳✯✯✯✳✳✳✳✳✳✳✳✳✳✳✳ ❙❊❱❊❘❆▲ ▼❊❚❍❖❉❙ ❋❖❘ ❙❖▲❱■◆● ❚❍❊ P❘❖❇▲❊▼ ❖❋ ❋■◆❉■◆● ❆ ❩❊❘❖ ❖❋ ▼❆❳■▼❆▲ ▼❖◆❖❚❖◆❊ ❖P❊❘❆❚❖❘ ❆◆❉ ❚❍❊ ▼❯▲❚■P▲❊✲❙❊❚❙ ❙P▲■❚ ❋❊❆❙■❇■▲■❚❨ P❘❖❇▲❊▼ ▼❛❥♦r✿ ❆♣♣❧✐❡❞ ▼❛t❤❡♠❛t✐❝s ❈♦❞❡✿ ✾✳✹✻✳✵✶✳✶✷ ❙❯▼▼❆❘❨ ❖❋ ▼❆❚❍❊▼❆❚■❈❙ ❉❖❈❚❖❘❆▲ ❚❍❊❙■❙ ❍❛ ◆♦✐ ✲ ✷✵✷✷ ❚❤❡ t❤❡s✐s ❤❛s ❜❡❡♥ ❝♦♠♣❧❡t❡❞ ❛t✿ ●r❛❞✉❛t❡ ❯♥✐✈❡rs✐t② ♦❢ ❙❝✐✲ ❡♥❝❡ ❛♥❞ ❚❡❝❤♥♦❧♦❣② ✲ ❱✐❡t♥❛♠ ❆❝❛❞❡♠② ♦❢ ❙❝✐❡♥❝❡ ❛♥❞ ❚❡❝❤✲ ♥♦❧♦❣② ❙✉♣❡r✈✐s♦r✿ Pr♦❢✳ ❉r✳ ◆❣✉②❡♥ ❇✉♦♥❣ ❘❡✈✐❡✇❡r ✶✿ ❘❡✈✐❡✇❡r ✷✿ ❘❡✈✐❡✇❡r ✸✿ ❚❤❡ t❤❡s✐s s❤❛❧❧ ❜❡ ❞❡❢❡♥❞❡❞ ✐♥ ❢r♦♥t ♦❢ t❤❡ ❚❤❡s✐s ❈♦♠♠✐tt❡❡ ❛t ❱✐❡t♥❛♠ ❆❝❛❞❡♠② ♦❢ ❙❝✐❡♥❝❡ ❛♥❞ ❚❡❝❤♥♦❧♦❣② ✲ ●r❛❞✉❛t❡ ❯♥✐✈❡rs✐t② ❖❢ ❙❝✐❡♥❝❡ ❆♥❞ ❚❡❝❤♥♦❧♦❣②✱ ❛t ✳✳✳✳✳✳✳ ❤♦✉r ✳✳✳✳✳✳✳ ✱ ❞❛t❡ ✳✳✳✳✳✳✳ ♠♦♥t❤ ✳✳✳✳✳✳✳ ②❡❛r ✷✵✷✷ ❚❤❡ t❤❡s✐s ❝♦✉❧❞ ❜❡ ❢♦✉♥❞ ❛t✿ ✲ ❚❤❡ ◆❛t✐♦♥❛❧ ▲✐❜r❛r② ♦❢ ❱✐❡t♥❛♠ ✲ ❚❤❡ ▲✐❜r❛r② ♦❢ ●r❛❞✉❛t❡ ❯♥✐✈❡rs✐t② ♦❢ ❙❝✐❡♥❝❡ ❛♥❞ ❚❡❝❤♥♦❧♦❣② ■♥tr♦❞✉❝t✐♦♥ ▼❛♥② ♣r♦❜❧❡♠s ❛r✐③✐♥❣ ❢r♦♠ s❝✐❡♥❝❡✱ t❡❝❤♥♦❧♦❣② ❛♥❞ ♠❛♥② s❡❝t♦rs ✐♥ r❡❛❧ ❧✐❢❡ ❝❛♥ ❜❡ ❢♦r♠✉❧❛t❡❞ ♠❛t❤❡♠❛t✐❝❛❧❧② ❜② ❛ ♣r♦❜❧❡♠ ♦❢ ❢✐♥❞✐♥❣ ❛ ♠✐♥✲ ✐♠✉♠ ♦❢ ❛ ❢✉♥❝t✐♦♥❛❧ ✐♥ ❢✐♥✐t❡ ♦r ✐♥❢✐♥✐t❡ ❞✐♠❡♥s✐♦♥❛❧ s♣❛❝❡✳ ❯♣ t♦ ♥♦✇✱ t❤❡r❡ ❛r❡ ♠❛♥② ✐t❡r❛t✐✈❡ ♠❡t❤♦❞s t♦ ❢✐♥❞ ❛ ♠✐♥✐♠✉♠ ♦❢ t❤❡ ❢✉♥❝t✐♦♥❛❧✳ ❆ ♣❛rt✐❝✉❧❛r❧② ✐♠♣♦rt❛♥t ♠❡t❤♦❞ ❢♦r ❢✐♥❞✐♥❣ ❛ ♠✐♥✐♠✉♠ ♦❢ ❛ ❝♦♥✈❡① ❢✉♥❝✲ t✐♦♥❛❧ ✐s t❤❡ ♣r♦①✐♠❛❧ ♣♦✐♥t ♠❡t❤♦❞ ♣r♦♣♦s❡❞ ❜② ▼❛rt✐♥❡t ✐♥ ✶✾✼✵✳ ■♥ ✶✾✼✻✱ ❘♦❝❦❛❢❡❧❧❛r ❡①t❡♥❞❡❞ t❤❡ ♠❡t❤♦❞ t♦ ❢✐♥❞ ❛ ③❡r♦ ♦❢ ❛ ♠❛①✐♠❛❧ ♠♦♥♦t♦♥❡ ♦♣❡r❛t♦r T ✐♥ ❛ ❍✐❧❜❡rt s♣❛❝❡ H ✱ ♥❛♠❡❧②✿ ❋✐♥❞ p∗ ∈ H s✉❝❤ t❤❛t ∈ T p∗ ✭✵✳✶✮ ❍✐s ♠❡t❤♦❞ ✐s ❞❡s❝r✐❜❡❞ ❛s ❢♦❧❧♦✇s xk+1 = Jk xk + ek ♦r xk+1 = Jk (xk + ek ), k ≥ 1, ✭✵✳✷✮ ✇❤❡r❡ Jk = (I + rk T )−1 ✐s t❤❡ r❡s♦❧✈❡♥t ♦❢ T ✇✐t❤ ♣❛r❛♠❡t❡rs rk > 0✱ ek ✐s ❛♥ ❡rr♦r ✈❡❝t♦r ❛♥❞ I ✐s ❛♥ ✐❞❡♥✐t✐t② ♠❛♣ ♦♥ H ✳ ❍❡ ♣r♦✈❡❞ t❤❡ ✇❡❛❦❧② ❝♦♥✈❡r❣❡♥❝❡ ♦❢ ✭✵✳✷✮ t♦ ❛ ③❡r♦ ♦❢ T ✐❢ t❤❡ s❡t ♦❢ ③❡r♦s ♦❢ T ✐s ♥♦♥✲❡♠♣t②✱ ∞ ∥ek ∥ < ∞ ❛♥❞ rk ≥ ε > ❢♦r ❛❧❧ k ≥ 1✳ k=1 ■♥ ✶✾✾✶✱ ●☎ ✉❧❡r s❤♦✇❡❞ t❤❛t t❤❡ ♣r♦①✐♠❛❧ ♣♦✐♥t ♠❡t❤♦❞ ♦♥❧② ❛❝❤✐❡✈❡s ✇❡❛❦ ❝♦♥✈❡r❣❡♥❝❡ ✐♥ ✐♥❢✐♥✐t❡ ❞✐♠❡♥s✐♦♥❛❧ s♣❛❝❡✳ ■♥ ♦r❞❡r t♦ ♦❜t❛✐♥ str♦♥❣ ❝♦♥✈❡r❣❡♥❝❡✱ s♦♠❡ ♠♦❞✐❢✐❝❛t✐♦♥s ♦❢ t❤❡ ♠❡t❤♦❞ ❤❛✈❡ ❜❡❡♥ ✐♥tr♦❞✉❝❡❞ s✉❝❤ ❛s t❤❡ ♣r♦①✲❚✐❦❤♦♥♦✈ ♠❡t❤♦❞ ♦❢ ▲❡❤❞✐❧✐ ❛♥❞ ▼♦✉❞❛❢✐ ✭✶✾✾✻✮❀ t❤❡ ❝♦♥tr❛❝t✐♦♥✲ ♣r♦①✐♠❛❧ ♣♦✐♥t ♠❡t❤♦❞ ❜② ❑❛♠✐♠✉r❛ ❛♥❞ ❲✳❚❛❦❛❤❛s❤✐ ✭✷✵✵✵✮ ❛♥❞ t❤❡ ✈✐s❝♦s✐t② ❛♣♣r♦①✐♠❛t✐♦♥ ♠❡t❤♦❞ ♦❢ ❲✳ ❚❛❦❛❤❛s❤✐ ✭✷✵✵✼✮✳ ■♥ t❤❡ ♠♦❞✐✲ ❢✐❝❛t✐♦♥s ❛s ✇❡❧❧ ❛s t❤❡ ♣r♦①✐♠❛❧ ♣♦✐♥t ♠❡t❤♦❞ ✐ts❡❧❢✱ ♣❛r❛♠❡t❡rs rk ❛r❡ ❜♦✉♥❞❡❞ ❜❡❧♦✇ ❜② ❛ ♣♦st✐✈❡ ❝♦♥st❛♥t✳ ■♥ ✷✵✶✼✱ ◆✳ ❇✉♦♥❣✱ P✳❚✳❚✳ ❍♦❛✐ ❛♥❞ ◆✳❉✳ ◆❣✉②❡♥ ♣r❡s❡♥t❡❞ ❛ ♥❡✇ ♠♦❞✐❢✐❝❛t✐♦♥ ✇❤❡r❡ rk ❛♣♣r♦❛❝❤❡s ✵✱ ♥❛♠❡❧② rk s❛t✐s❢✐❡s ❝♦♥❞✐t✐♦♥ (A8) ✐s ∞ k=1 rk < +∞✳ ❙♦✱ ✐t ✐s ♥❛t✉r❛❧ t♦ ❛s❦ ✷ ✇❤❡t❤❡r t❤❡r❡ ❡①✐sts ❛ str♦♥❣❧② ❝♦♥✈❡r❣❡♥❝❡ ♠♦❞✐❢✐❝❛t✐♦♥ ✐s t❤❡ ✇❛s t❤❛t {rk } ✐s ❛♥② s❡q✉❡♥❝❡ ♦❢ ♥✉♠❜❡rs ✐♥ (0, ∞)❄ ❲❤❡♥ t❤❡ ♦❜❥❡❝t✐✈❡ ❢✉♥❝t✐♦♥ ✐s ❛ s✉♠ ♦❢ t✇♦ ❝♦♥✈❡① ❢✉♥❝t✐♦♥s✱ t❤✐s ♣r♦❜❧❡♠ ❧❡❛❞s t♦ t❤❡ ♣r♦❜❧❡♠ ♦❢ ❢✐♥❞✐♥❣ ③❡r♦ ♦❢ t❤❡ s✉♠ ♦❢ t✇♦ ♠❛①✐♠❛❧ ♠♦♥♦t♦♥❡ ♦♣❡r❛t♦rs A, B, ✇❤✐❝❤ ✐s t❤❡ ♣r♦❜❧❡♠✿ ❋✐♥❞ p∗ ∈ H s✉❝❤ t❤❛t ∈ (A + B)p∗ ✭✵✳✸✮ Pr♦❜❧❡♠ ✭✵✳✸✮ ❤❛s ❛ttr❛❝t❡❞ t❤❡ ❛tt❡♥t✐♦♥ ♦❢ ♠❛♥② r❡s❡❛r❝❤❡rs ❜❡❝❛✉s❡ ✐t ✐s t❤❡ ❝♦r❡ ♦❢ ♠❛♥② ♣r♦❜❧❡♠s s✉❝❤ ❛s ❝♦♥✈❡① ♣r♦❣r❛♠♠✐♥❣✱ ✈❛r✐❛t✐♦♥❛❧ ✐♥❡q✉❛❧✐t②✱ ❢❡❛s✐❜✐❧✐t② s♣❧✐tt✐♥❣ ♣r♦❜❧❡♠✱ ♠✐♥✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠ ✇✐t❤ ❛♣♣❧✐✲ ❝❛t✐♦♥s ✐♥ ♠❛❝❤✐♥❡ ❧❡❛r♥✐♥❣✱ ✐♠❛❣❡ ♣r♦❝❡ss✐♥❣✱ ❛♥❞ ✐♥✈❡rs❡ ❧✐♥❡❛r✐t② ♣r♦❜✲ ❧❡♠s✳ ❈❧❡❛r❧②✱ ✐❢ A + B ✐s ♠❛①✐♠❛❧ ♠♦♥♦t♦♥❡✱ t❤❡♥ ♠❡t❤♦❞ ✭✵✳✷✮ ✇✐t❤ T = A + B ❝❛♥ ❜❡ ❛♣♣❧✐❡❞ t♦ ❢✐♥❞ ❛ ③❡r♦ ♦❢ ✭✵✳✸✮✳ ■♥ t❤❡ ❝❛s❡ t❤❛t T ✐s ♥♦t ♠❛①✐♠❛❧ ♠♦♥♦t♦♥❡ ♦♥❡ ❝❛♥ ✉s❡ JrA , JrB t♦ ❝♦♥str✉❝t ❛♥ ✐t❡r❛t✐✈❡ ♠❡t❤♦❞✱ ❝❛❧❧❡❞ s♣❧✐tt✐♥❣ ♦♥❡✱ ❢♦r ❢✐♥❞✐♥❣ ❛ ③❡r♦ ♦❢ ✭✵✳✸✮✳ ◆♦t❡ t❤❛t✱ t❤❡ ❝❧❛ss✐❝❛❧ s♣❧✐tt✐♥❣ ♠❡t❤♦❞s ♦❢ P❡❛❝❡♠❛♥✲❘❛❝❤❢♦r❞✱ ❉♦✉❣❧❛s✲ ❘❛❝❤❢♦r❞ ✇❛s ♣r♦♣♦s❡❞ ✐♥ t❤❡ ✶✾✺✵s ❢♦r t❤❡ s♣❡❝✐❛❧ ❝❛s❡ ✇❤❡r❡ ❜♦t❤ A ❛♥❞ B ❛r❡ s✐♥❣❧❡✲✈❛❧✉❡❞ ❧✐♥❡❛r ♦♣❡r❛t♦rs✳ ■♥ ✶✾✼✾✱ ▲✐♦♥s ❛♥❞ ▼❡r❝✐❡r ❡①t❡♥❞❡❞ ❉♦✉❣❧❛s✲❘❛❝❤❢♦r❞✬s s♣❧✐tt✐♥❣ s❝❤❡♠❡ t♦ t❤❡ ❣❡♥❡r❛❧ ❝❛s❡ ✇❤❡r❡ A ❛♥❞ B ❛r❡ ♠✉❧t✐✈❛❧✉❡❞ ♠❛①✐♠❛❧ ♠♦♥♦t♦♥❡ ♦♣❡r❛t♦rs✳ ❆♥♦t❤❡r s♣❧✐tt✐♥❣ ♠❡t❤♦❞✱ ♥❛♠❡❞ t❤❡ ❢♦r✇❛r❞✲❜❛❝❦✇❛r❞✱ ❤❛s ❜❡❡♥ ✐♥tr♦❞✉❝❡❞ ❜② ▲✐♦♥s✱ ▼❡r❝✐❡r ❛♥❞ P❛sst② ✐♥ ✶✾✼✾✱ t❤❛t ❤❛s t❤❡ ❢♦r♠✿ xk+1 = Jk (I − rk A)xk , k ≥ 1, ✭✵✳✹✮ ✇❤❡r❡ A, B ❛r❡ ♠❛①✐♠❛❧ ♠♦♥♦t♦♥❡ ♦♣❡r❛t♦rs ✐♥ H ✱ Jk = (I + rk B)−1 ✐s t❤❡ r❡s♦❧✈❡♥t ♦❢ B ✱ {rk } ✐s ❛ s❡q✉❡♥❝❡ ♦❢ ♣♦s✐t✐✈❡ ♥✉♠❜❡rs✳ ❍♦✇❡✈❡r✱ t❤❡ ✐t❡r❛t✐✈❡ s❡q✉❡♥❝❡ xk ❞❡❢✐♥❡❞ ❜② ✭✵✳✹✮ ♦♥❧② ❝♦♥✈❡r❣❡s ✇❡❛❦❧② t♦ ❛ ③❡r♦ ♦❢ A + B ✳ ❚♦ ♦❜t❛✐♥ ❛ str♦♥❣ ❝♦♥✈❡r❣❡♥t s❡q✉❡♥❝❡✱ s❡✈❡r❛❧ ✐♠♣r♦✈❡❞ ♠♦❞✲ ✐❢✐❝❛t✐♦♥s ♦❢ ✭✵✳✹✮ ✇❡r❡ ❣✐✈❡♥ ✐♥ ❝♦♠❜✐♥❛t✐♦♥s ✇✐t❤ t❤❡ ▼❛♥♥✱ ❍❛❧♣❡r♥ ❛♥❞ ✈✐s❝♦s✐t② ❛♣♣r♦①✐♠❛t✐♦♥ ♠❡t❤♦❞s✳ ❚❤❡ str♦♥❣ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ t❤❡ ♠❡t❤♦❞s r❡q✉✐r❡s t❤❡ s❛♠❡ ❝♦♥❞✐t✐♦♥ ♦♥ rk ❛s t❤❡ ❛❜♦✈❡ ❢♦r t❤❡ r❡s♦❧✈❡♥t ♦❢ T ✳ ❙♦✱ t❤❡ s✐♠✐❧❛r q✉❡st✐♦♥ ❛❜♦✈❡ rk ❢♦r ✭✵✳✹✮ ♥❡❡❞ t♦ ❜❡ st✉❞✐❡❞✳ ❚❤✐s ✐s ❛ s❡❝♦♥❞ ❣♦❛❧ ♦❢ ♦✉r t❤❡s✐s✳ ✸ ❚❤❡ ♥❡①t ♣r♦❜❧❡♠s ✐♥✈❡st✐❣❛t❡❞ ✐♥ t❤✐s ❞✐s❝✉ss✐♦♥✱ ✐s t❤❡ ♠✉❧t✐♣❧❡✲s❡ts s♣❧✐t ❢❡❛s✐❜✐❧✐t② ♣r♦❜❧❡♠ ✭▼❙❙❋P✮✿ ❋✐♥❞ x ∈ C := Ci , s✉❝❤ t❤❛t Ax ∈ Q := i∈J1 Qj , ✭✵✳✺✮ j∈J2 ✇❤❡r❡ {Ci }i∈J1 ❛♥❞ {Qj }j∈J2 ❛r❡ t✇♦ ❝♦✉♥t❛❜❧❡ ✐♥❢✐♥✐t❡ ❢❛♠✐❧✐❡s ♦❢ ❝❧♦s❡❞ ❝♦♥✈❡① s✉❜s❡ts ✐♥ t✇♦ r❡❛❧ ❍✐❧❜❡rt s♣❛❝❡s H1 ❛♥❞ H2 ✱ r❡s♣❡❝t✐✈❡❧②✱ ❛♥❞ A : H1 → H2 ✐s ❛ ❜♦✉♥❞❡❞ ❧✐♥❡❛r ♠❛♣♣✐♥❣✳ ■♥ ✶✾✾✹✱ ❈❡♥s♦r ❛♥❞ ❊❧❢✈✐♥❣ ✐♥tr♦❞✉❝❡❞ ♣r♦❜❧❡♠ ✭✵✳✺✮ ✐♥ t❤❡ ❝❛s❡ t❤❛t J1 ❛♥❞ J2 ❛r❡ ❢✐♥✐t❡ ❛♥❞ ♣r♦♣♦s❡❞ t❤❡ ❈◗✲♠❡t❤♦❞ ❢♦r s♦❧✈✐♥❣ ❛ s♦❧✉t✐♦♥ ♦❢ ✭✵✳✺✮ ✐♥ ❢✐♥✐t❡ ❞✐♠❡♥s✐♦♥❛❧ s♣❛❝❡s✳ Pr♦❜❧❡♠ ✭✵✳✺✮ ❝♦♥t❛✐♥s s❡✈❡r❛❧ ♣r❛❝t✐✲ ❝❛❧ ♦♥❡s s✉❝❤ ❛s t❤❡ ♣❤❛s❡ r❡tr✐❡✈❛❧s ❛♥❞ ✐♠❛❣❡ r❡❝♦♥str✉❝t✐♦♥ ✐♥ ♠❡❞✐❝❛❧✳ ❘❡❝❡♥t❧②✱ t❤❡ ▼❙❙❋P ❝❛♥ ❛❧s♦ ❜❡ ✉s❡❞ t♦ ♠♦❞❡❧ ✐♥t❡♥s✐t②✲♠♦❞✉❧❛t❡❞ r❛✲ ❞✐❛t✐♦♥ t❤❡r❛♣②✳ ❚❤❡ ❈◗✲♠❡t❤♦❞ ❤❛s ❜❡❡♥ ❡①t❡♥❞❡❞ t♦ ✐♥❢✐♥✐t❡ ❞✐♠❡♥s✐♦♥❛❧ ❍✐❧❜❡rt s♣❛❝❡s ❜② ❳✉ ✭✷✵✵✻✮✱ ❍❡ ❡t ❛❧✳ ✭✷✵✶✺✮✱ ❲❡♥ ❡t ❛❧✳ ✭✷✵✶✺✮✳ ■♥ ✷✵✶✼✱ ◆✳ ❇✉♦♥❣ ❝♦♥s✐❞❡r❡❞ t❤❡ ❈◗✲♠❡t❤♦❞ ✐♥ t❤❡ ❝❛s❡ t❤❛t J1 ❛♥❞ J2 ❛r❡ ❝♦✉♥t❛❜❧❡ ✐♥❢✐♥✐t❡ ❢❛♠✐❧✐❡s✳ ◆♦t❡ t❤❛t✱ ✇❤❡♥ J1 = J2 = {1}✱ ❳✉ ✭✷✵✶✵✮ ✐♥✈❡st✐❣❛t❡❞ ❛♥ ✐t✲ ❡r❛t✐✈❡ r❡❣✉❧❛r✐③❛t✐♦♥ ♠❡t❤♦❞ ♦❢ ❇❛❦✉s❤✐♥s❦② ❛♥❞ ❇r✉❝❦ ❢♦r t❤❡ ♣r♦❜❧❡♠✳ ❖✉r t❤✐r❞ ❣♦❛❧ ✐♥ t❤✐s t❤❡s✐s✱ ✐s t♦ st✉❞② t❤❡ ❧❛st ♠❡t❤♦❞ ❢♦r t❤❡ ❝❛s❡ t❤❛t J1 ❛♥❞ J2 ❛r❡ ✐♥❢✐♥✐t❡✳ ❚❤❡ ❣♦❛❧s ♦❢ t❤✐s t❤❡s✐s ❛r❡ t♦ ❛♥s✇❡r t❤❡ t❤r❡❡ q✉❡st✐♦♥ ❛❜♦✈❡ ❛♥❞ ✐s ♣r❡s❡♥t❡❞ ✐♥ ❢♦✉r ❝❤❛♣t❡rs✳ ■♥ ❈❤❛♣t❡r ✶✱ ✇❡ st✉❞② ❢✉♥❞❛♠❡♥t❛❧ ♣r♦♣❡rt✐❡s ♦❢ ❝♦♥✈❡① ❛♥❛❧②s✐s ❢♦r t❤❡ ♣r❡s❡♥t❛t✐♦♥ ♦❢ t❤❡ ♠❛✐♥ r❡s✉❧ts ✐♥ t❤❡ ♥❡①t ❝❤❛♣t❡rs✱ ✐♥❝❧✉❞✐♥❣ s♦♠❡ ❜❛s✐❝ ❝♦♥❝❡♣ts ♦❢ ❝♦♥✈❡① ❛♥❛❧②s✐s✱ s♦♠❡ ♠❡t❤♦❞s ♦❢ ❢✐♥❞✐♥❣ ③❡r♦ ♦❢ ❛ ♠❛①✐✲ ♠❛❧ ♠♦♥♦t♦♥❡ ♦♣❡r❛t♦r✱ t❤❡ s✉♠ ♦❢ t✇♦ ♠❛①✐♠❛❧ ♠♦♥♦t♦♥❡ ♦♣❡r❛t♦rs✱ t❤❡ ♠✉❧t✐♣❧❡✲s❡ts s♣❧✐t ❢❡❛s✐❜✐❧✐t② ♣r♦❜❧❡♠ ❛♥❞ ✐ts s♦❧✉t✐♦♥s✳ ■♥ ❈❤❛♣t❡r ✷✱ ❢♦r ❢✐♥❞✐♥❣ ❛ ③❡r♦ ♦❢ ❛ ♠❛①✐♠❛❧ ♠♦♥♦t♦♥❡ ♦♣❡r❛t♦r ✐♥ ❛ ❍✐❧❜❡rt s♣❛❝❡s ✇❡ ❤❛✈❡ ♣r❡s❡♥t❡❞ ❛ ♠♦❞✐❢✐❝❛t✐♦♥ ♦❢ t❤❡ ♣r♦①✐♠❛❧ ♣♦✐♥t ♠❡t❤♦❞✱ str♦♥❣ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ ✇❤✐❝❤ ✐s ♣r♦✈❡❞ ✇✐t❤♦✉t ❛♥② ❛❞❞✐t✐♦♥❛❧ ❝♦♥❞✐t✐♦♥ ♦♥ t❤❡ r❡s♦❧✈❡♥t ♣❛r❛♠❡t❡r ♦❢ t❤❡ ♦♣❡r❛t♦r✳ ■♥ ❈❤❛♣t❡r ✸✱ ✇❡ ❤❛✈❡ ♦❜t❛✐♥❡❞ ❛ s✐♠✐❧❛r r❡s✉❧t ❢♦r t❤❡ ♠♦♥♦t♦♥❡ ✐♥❝❧✉✲ s✐♦♥ ♣r♦❜❧❡♠✳ ✹ ■♥ ❈❤❛♣t❡r ✹✱ ✇❡ ❤❛✈❡ ✐♥tr♦❞✉❝❡❞ ❛♥ ✐♥t❡r❛t✐✈❡ r❡❣✉❧❛r✐③❛t✐♦♥ ♠❡t❤♦❞ ❢♦r s♦❧✈✐♥❣ t❤❡ ♠✉❧t✐♣❧❡✲s❡ts s♣❧✐t ❢❡❛s✐❜✐❧✐t② ♣r♦❜❧❡♠ ✇✐t❤ t✇♦ ✐♥❢✐♥✐t❡ ❢❛♠✐❧✐❡s ♦❢ ❝❧♦s❡❞ ❝♦♥✈❡① s✉❜s❡ts✳ ❚❤❡ ✐♠♣♦rt❛♥❝❡ ♦❢ ♦✉r ♠❡t❤♦❞ ✐s t❤❛t ❛t ❡❛❝❤ ✐♥t❡r❛t✐♦♥ st❡♣ ✐t ❝♦♥t❛✐♥s ♦♥❧② ❢✐♥✐t❡ ♥✉♠❜❡rs ♦❢ s❡ts ❢r♦♠ t❤❡ ❢❛♠✐❧✐❡s✳ ❚♦ st✉❞② ❛♥❞ s♦❧✈❡ t❤❡ ❣♦❛❧s ♣♦s❡❞✱ ✇❡ ❤❛✈❡ ✉s❡❞ ♠♦❞❡r♥ ♠❡t❤♦❞s ❛♥❞ t♦♦❧s ♦❢ ❢✉♥❝t✐♦♥❛❧ ❛♥❛❧②s✐s✱ ❝♦♥✈❡① ❛♥❛❧②s✐s✱ ♦♣t✐♠✐③❛t✐♦♥ t❤❡♦r② ❛♥❞ ❡①✐st✐♥❣ r❡s✉❧ts ♦♥ ♠❡t❤♦❞s ♦❢ s♦❧✈✐♥❣ ❛❢♦r❡♠❡♥t✐♦♥❡❞ ♣r♦❜❧❡♠s✳ ❈❤❛♣t❡r ✶ ❙♦♠❡ ❜❛s✐❝ ❝♦♥❝❡♣ts ❛♥❞ ♠❡t❤♦❞s ❚❤✐s ❝❤❛♣t❡r ♣r❡s❡♥ts t❤❡ ♥❡❝❡ss❛r② ❦♥♦✇❧❡❞❣❡ t♦ s❡r✈❡ t❤❡ ♣r❡s❡♥t❛t✐♦♥ ♦❢ t❤❡ ♠❛✐♥ r❡s❡❛r❝❤ r❡s✉❧ts ♦❢ t❤❡ t❤❡s✐s ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝❤❛♣t❡rs✳ ❚❤❡ str✉❝t✉r❡ ♦❢ t❤❡ ❝❤❛♣t❡r ❝♦♥s✐sts ♦❢ ❢♦✉r s❡❝t✐♦♥s✿ ❙❡❝t✐♦♥ ✶✳✶ ♣r❡s❡♥ts s♦♠❡ ❜❛s✐❝ ❝♦♥❝❡♣ts ♦❢ ❝♦♥✈❡① ❛♥❛❧②s✐s✱ ♠♦♥♦t♦♥❡ ♦♣❡r❛t♦rs✱ ♠❛①✐♠❛❧ ♠♦♥♦t♦♥❡ ♦♣❡r❛t♦rs ❛♥❞ s✉♠ ♦❢ t✇♦ ♠♦♥♦t♦♥❡ ♦♣❡r❛✲ t♦rs✳ ❙❡❝t✐♦♥ ✶✳✷ ✐♥tr♦❞✉❝❡s ❛♥ ♦✈❡r✈✐❡✇ ♦❢ s♦♠❡ ❜❛s✐❝ ♠❡t❤♦❞s ♦❢ ❢✐♥❞✐♥❣ ③❡r♦s ♦❢ t❤❡ ♠❛①✐♠❛❧ ♠♦♥♦t♦♥❡ ♦♣❡r❛t♦r ❛♥❞ t❤❡ s✉♠ ♦❢ t✇♦ ♠❛①✐♠❛❧ ♠♦♥♦t♦♥❡ ♦♣❡r❛t♦rs✳ ❙❡❝t✐♦♥ ✶✳✸ ♣r❡s❡♥ts t❤❡ ♠✉❧t✐♣❧❡✲s❡ts s♣❧✐t ❢❡❛s✐❜✐❧✐t② ♣r♦❜❧❡♠ ❛♥❞ ✐ts ♠❡t❤♦❞s✳ ❙❡❝t✐♦♥ ✶✳✹ ✐♥tr♦❞✉❝❡s s♦♠❡ ❜❛s✐❝ ❧❡♠♠❛s ✉s❡❞ t♦ ♣r♦✈❡ t❤❡ ♠❛✐♥ r❡s✉❧ts ♦❜t❛✐♥❡❞ ✐♥ t❤❡ ♥❡①t ❝❤❛♣t❡rs ♦❢ t❤❡ t❤❡s✐s✳ ❈❤❛♣t❡r ✷ ■t❡r❛t✐✈❡ ♠❡t❤♦❞s ❢♦r ❢✐♥❞✐♥❣ ❛ ③❡r♦ ♦❢ ❛ ♠❛①✐♠❛❧ ♠♦♥♦t♦♥❡ ♦♣❡r❛t♦r ✐♥ ❍✐❧❜❡rt s♣❛❝❡s ❚❤✐s ❝❤❛♣t❡r ♣r❡s❡♥ts s♦♠❡ ♠♦❞✐❢✐❝❛t✐♦♥s ♦❢ t❤❡ ♣r♦①✐♠❛❧ ♣♦✐♥t ♠❡t❤✲ ♦❞s✱ t❤❡ s❡q✉❡♥❝❡ ♦❢ r❡s♦❧✈❡♥t ♣❛r❛♠❡t❡rs ✐s ❛ s❡q✉❡♥❝❡ ♦❢ ♣♦s✐t✐✈❡ r❡❛❧ ♥✉♠❜❡rs✳ ❆❢t❡r t❤❛t✱ t✇♦ ♥✉♠❡r✐❝❛❧ ❡①❛♠♣❧❡s ❛r❡ ❣✐✈❡♥ ❢♦r ✐❧❧✉str❛t✐♦♥ ❛♥❞ ❝♦♠♣❛r✐s♦♥✳ ❚❤❡ ♣❧♦t ♦❢ t❤✐s ❝❤❛♣t❡r ✐s ♣r❡s❡♥t❡❞ ❜❛s❡❞ ♦♥ ✇♦r❦s [3] ❛♥❞ [5] ✐♥ t❤❡ ▲✐st ♦❢ ♣✉❜❧✐s❤❡❞ ✇♦r❦s✳ ✷✳✶✳ ❙♦♠❡ ♠♦❞✐❢✐❝❛t✐♦♥ ♦❢ ♣r♦①✐♠❛❧ ♣♦✐♥t ♠❡t❤♦❞ ❈♦♥s✐❞❡r t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦❜❧❡♠✿ ❋✐♥❞ p∗ ∈ H s✉❝❤ t❤❛t ∈ T p∗ , ✭✷✳✶✮ ✇❤❡r❡ H ✐s ❛ ❍✐❧❜❡rt s♣❛❝❡ ❛♥❞ T : H → 2H ✐s ❛ ♠❛①✐♠❛❧ ♠♦♥♦t♦♥❡ ♦♣❡r❛t♦rs✳ ❲❡ ❦♥♦✇ t❤❛t t❤❡ ♣r♦①✐♠❛❧ ♣♦✐♥t ♠❡t❤♦❞ ❝♦♥✈❡r❣❡s ♦♥❧② ✇❡❛❦ ✐♥ ✐♥❢✐✲ ♥✐t❡ ❞✐♠❡♥s✐♦♥❛❧ ❍✐❧❜❡rt s♣❛❝❡✳ ■♥ ♦r❞❡r t♦ ♦❜t❛✐♥ str♦♥❣ ❝♦♥✈❡r❣❡♥❝❡✱ ✐♥ ✷✵✶✼✱ ✇❡ ♣r♦♣♦s❡❞ t✇♦ ♥❡✇ ♠♦❞✐❢✐❝❛t✐♦♥s ♦❢ t❤❡ ♣r♦①✐♠❛❧ ♣♦✐♥t ♠❡t❤♦❞ ❢♦r ♣r♦❜❧❡♠ ✭✷✳✶✮ ❤❛✈✐♥❣ t❤❡ s❛♠❡ ❢♦r♠ ❛s t❤❡ ❚✐❦❤♦♥♦✈ r❡❣✉❧❛r✐③❛t✐♦♥ ♣r♦①✐♠❛❧ ♣♦✐♥t ♠❡t❤♦❞ ❛♥❞ t❤❡ ❝♦♥tr❛❝t✐♦♥✲♣r♦①✐♠❛❧ ♣♦✐♥t ♠❡t❤♦❞✱ t❤❛t ❛r❡✿ xk+1 = J k (tk u + (1 − tk )xk + ek ), k ≥ 1, z k+1 = tk u + (1 − tk )J k z k + ek , k ≥ 1, ✇❤❡r❡ J k = J1 J2 · · · Jk ✐s t❤❡ ❝♦♠♣♦s✐t✐♦♥ ♦❢ k r❡s♦❧✈❡♥ts Ji = (I + ri T )−1 , i = 1, 2, , k ✼ ❚❤❡ str♦♥❣ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ t❤❡s❡ ♠❡t❤♦❞s ✐s ❣✉❛r❛♥t❡❡❞ ❜② ❝♦♥❞✐t✐♦♥ ∞ (A8) ✇❤✐❝❤ st❛t❡s t❤❛t ♣❛r❛♠❡t❡rs rk s❛t✐s❢② rk < +∞✳ ❯♥❢♦rt✉♥❛t❧②✱ k=1 t❤❡s❡ ♠❡t❤♦❞s ❛r❡ r❛t❤❡r ❝♦♠♣❧✐❝❛t❡❞ ❜❡❝❛✉s❡ ♦❢ t❤❡ ❤✉❣❡ ♥✉♠❜❡r ♦❢ r❡✲ s♦❧✈❡♥ts ✐♥ ❡❛❝❤ st❡♣ t❤❛t ♠❛❦❡s t❤❡ ❝♦♠♣✉t❛t✐♦♥ ❤❛r❞❡r ❛♥❞ ❝♦♠♣✉t❛t✐♦♥ t✐♠❡ ❧♦♥❣❡r✳ ❋✉t❤❡r♠♦r❡✱ ❝♦♥❞✐t✐♦♥ (A8) ✐s ✈❡r② r❡str✐❝t❡❞✳ ❚♦ ♦✈❡r❝♦♠❡ t❤✐s✱ ✇❡ ✐♥tr♦❞✉❝❡ ❛ ♥❡✇ ♣r♦①✐♠❛❧ ♣♦✐♥t ♠❡t❤♦❞ ✇✐t❤ ❛♥② ♣❛r❛♠❡t❡r s❡✲ q✉❡♥❝❡ ❢♦r s♦❧✈✐♥❣ ♣r♦❜❧❡♠ ✭✷✳✶✮ ❜② r❡♣❧❛❝✐♥❣ t❤❡ ❝♦♠♣♦s✐t✐♦♥ J k ❜② ❛ s✐♠♣❧❡r ❢♦r♠ t❤❛t ✉s❡s ♦♥❧② t✇♦ r❡s♦❧✈❡♥t ♦♣❡r❛t♦rs ❛t ❡❛❝❤ ✐t❡r❛t✐♦♥ ❛♥❞ r❡♣❧❛❝❡ ❝♦♥❞✐t✐♦♥ (A8) ❜② ❛ ❣❡♥❡r❛❧ ❝♦♥❞✐t✐♦♥ t❤❛t {rk } ✐s ❛♥② s❡q✉❡♥❝❡ ✐♥ (0, ∞)✳ ■♥ ♣❛♣❡r [3]✱ ✇❡ ✐♥tr♦❞✉❝❡❞ ♥❡✇ ✐t❡r❛t✐♦♥ s❡q✉❡♥❝❡ xk ❞❡❢✐♥❡❞ ❜②✿ ′ ′ ✭✷✳✷✮ xk+1 = Jk Jc (tk u + (1 − tk )xk ) + ek , ′ ′ xk+1 = tk u + (1 − tk )(Jk Jc xk + ek ), ✭✷✳✸✮ xk+1 = t′k u + βk′ Jc xk + γk′ Jk xk + ek , ✭✷✳✹✮ ✇❤❡r❡ Jc = (I + cT )−1 ❛♥❞ c ✐s ❛♥② ❢✐①❡❞ ♣♦s✐t✐✈❡ r❡❛❧ ♥✉♠❜❡r✳ ❲❡ s❤♦✇ t❤❛t ♠❡t❤♦❞s ✭✷✳✷✮✲✭✷✳✹✮ ❛r❡ t❤❡ s♣❡❝✐❛❧ ❝❛s❡s ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ♠❡t❤♦❞s z k+1 = Jk Jc (I − tk µF )z k + ek , ✭✷✳✺✮ z k+1 = (1 − βk )(I − tk µF )Jc z k + βk Jk z k + ek , ✭✷✳✻✮ ♣r♦♣♦s❡❞ t♦ ❢✐♥❞ s♦❧✉t✐♦♥ p∗ ∈ C ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ✈❛r✐❛t✐♦♥❛❧ ✐♥❡q✉❛❧✐t②✿ p∗ ∈ C : ⟨F p∗ , p∗ − p⟩ ≤ 0, ∀p ∈ C ✭✷✳✼✮ ✇❤❡r❡ C := ZerT ❛♥❞ F : H → H ✐s ❛♥ η ✲str♦♥❣❧② ♠♦♥♦t♦♥❡ L✲▲✐♣s❝❤✐t③ ❝♦♥t✐♥♦✉s ✇✐t❤ η, L > 0✱ µ ✐s ❛ ❢✐①❡❞ ♥✉♠❜❡r ✐♥ (0, 2η/L2 )✳ ❚❤❡♦r❡♠ ✷✳✶ H s✉❝❤ t❤❛t ▲❡t T ❜❡ ❛ ♠❛①✐♠❛❧ ♠♦♥♦t♦♥❡ ♦♣❡r❛t♦r ✐♥ ❛ r❡❛❧ ❍✐❧❜❡rt ZerT ̸= Ø✱ F ❜❡ ❛♥ η ✲str♦♥❣❧② ♠♦♥♦t♦♥❡ ❛♥❞ L✲▲✐♣s❝❤✐t③ ❝♦♥t✐♥✉♦✉s ♦♣❡r❛t♦r ♦♥ H ✇✐t❤ s♦♠❡ ♣♦s✐t✐✈❡ ❢✐①❡❞ ♥✉♠❜❡r ✐♥ (0, 2η/L2 )✳ ❆ss✉♠❡ t❤❛t t❤❡r❡ ∞ ∥ek ∥ ′ k = 0✱ (A1) ∥e ∥ < ∞ ♦r (A1 ) lim k→∞ tk k=1 ❝♦♥st❛♥ts k→∞ (A8′ ) {rk } tk = ∞✱ k=1 ✐s ❛♥② s❡q✉❡♥❝❡ ♦❢ ♥✉♠❜❡rs ✐♥ ❛♥❞ ❤♦❧❞ ❝♦♥❞✐t✐♦♥s✿ ∞ (A5) tk ∈ (0, 1), ∀k ≥ 1, lim tk = 0, η (0, ∞)✳ L✱ µ ❜❡ ❛ ✽ ❚❤❡♥✱ t❤❡ s❡q✉❡♥❝❡ zk ❞❡❢✐♥❡❞ ❜② ✭✷✳✺✮✱ ❛s k → ∞✱ ❝♦♥✈❡r❣❡s str♦♥❣❧② p∗ ✱ s♦❧✈✐♥❣ ✭✷✳✼✮✳ ❚❤❡♦r❡♠ ✷✳✷ ▲❡t H, F, A, tk , rk ❛♥❞ ek ❜❡ ❛s ✐♥ ❚❤❡♦r❡♠ ✷✳✶✳ t♦ t❤❡ ✉♥✐q✉❡ s♦❧✉t✐♦♥ ❛ss✉♠❡ t❤❛t t❤❡ ♣❛r❛♠❡t❡r βk ■♥ ❛❞❞✐t✐♦♥✱ s❛t✐s❢✐❡s ❝♦♥❞✐t✐♦♥s✿ (A6) βk ∈ [a, b] ⊂ (0, 1)✳ z k ❞❡❢✐♥❡❞ s♦❧✉t✐♦♥ p∗ ✱ s♦❧✈✐♥❣ ❚❤❡♥✱ t❤❡ s❡q✉❡♥❝❡ ❜② ✭✷✳✻✮✱ ❛s t♦ t❤❡ ✉♥✐q✉❡ ✭✷✳✼✮✳ k → ∞✱ ❝♦♥✈❡r❣❡s str♦♥❣❧② ❈♦♠♠❡♥t ✷✳✶ ❚❤✐s r❡♠❛r❦ ♣r❡s❡♥ts ❜r✐❡❢❧② ❤♦✇ t♦ ❝❤♦♦s❡ F t♦ ❣❡t ♠❡t❤✲ ♦❞s ✭✷✳✷✮✱ ✭✷✳✸✮ ❢r♦♠ ♠❡t❤♦❞ ✭✷✳✺✮ ❛♥❞ t♦ ❣❡t ♠❡t❤♦❞ ✭✷✳✹✮ ❢r♦♠ ✭✷✳✻✮✳ ❇❛s❡❞ ♦♥ t❤❡ ✐❞❡❛ ♦❢ ❳✉ ✭✷✵✵✷✮✱ ❈❡♥❣ ❡t ❛❧ ✭✷✵✵✽✮✱ ✇❡ ❝♦♥t✐♥✉❡ t♦ ✐♠♣♦✈❡ t❤❡ ♣r♦①✐♠❛❧ ♣♦✐♥t ♠❡t❤♦❞ ❢♦r ♣r♦❜❧❡♠ ✭✷✳✶✮ ✇❤❡r❡ t❤❡r❡ ❛r❡ t✇♦ st❡♣s ✐♥ ❡❛❝❤ ✐t❡r❛t✐♦♥✳ ❚❤❡ str♦♥❣ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ ♥❡✇ ♠❡t❤♦❞s ❤❛s ❜❡❡♥ ♣r♦✈❡❞ ✉♥❞❡r ❛ ❣❡♥❡r❛❧ ❝♦♥❞✐t✐♦♥ (A8′ ) ♦♥ r❡s♦❧✈❡♥t ♣❛r❛♠❡t❡r {rk }✳ ❚❤✐s r❡s✉❧t ✐s ♣r♦✈❡❞ ✐♥ t❤❡ ✇♦r❦ [5]✳ ❚❤❡♦r❡♠ ✷✳✸ ▲❡t H ❜❡ ❛ r❡❛❧ ❍✐❧❜❡rt s♣❛❝❡ ❛♥❞ H s✉❝❤ t❤❛t ZerT ̸= Ø✳ u ∈ H ✳ ❆ss✉♠❡ t❤❛t t❤❡r❡ ❤♦❧❞ T ❜❡ ❛ ♠❛①✐♠❛❧ ♠♦♥♦t♦♥❡ ♦♣❡r❛t♦r ✐♥ ▲❡t ♥✉♠❜❡r✱ ❝♦♥❞✐t✐♦♥s c ❜❡ ❛♥② ❢✐①❡❞ ♣♦ss✐t✐✈❡ r❡❛❧ (A5)✱ (A8′ )✱ (A1) ♦r ∞ k=1 ηk < ∞ (A1′′ ) ∥ek+1 ∥ ≤ ηk ∥˜ xk+1 − xk ∥ ✇✐t❤ ❚❤❡♥✱ t❤❡ s❡q✉❡♥❝❡ {xk } ❞❡❢✐♥❡❞ ❜②  x˜k+1 = J (xk + ek+1 ), k xk+1 = t u + (1 − t )J x˜k+1 , k k c ❝♦♥✈❡r❣❡s str♦♥❣❧② t♦ t❤❡ ♣♦✐♥t ♦♥t♦ ✷✳✷✳ ZerT ❛s p∗ = PZerT u✱ ✭✷✳✽✮ t❤❡ ♠❡tr✐❝ ♣r♦❥❡❝t✐♦♥s ♦❢ u k → ∞✳ ◆✉♠❡r✐❝❛❧ ❡①❛♠♣❧❡ ❈♦♥s✐❞❡r t❤❡ ❢♦r❧❧♦✇✐♥❣ ❝♦♥✈❡① ♦♣t✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠✿ ❢✐♥❞ ❛ ♣♦✐♥t p∗ ∈ E2 s✉❝❤ t❤❛t g(p∗ ) = inf2 g(x), x∈E ✭✷✳✾✮ ✇❤❡r❡ g(x) ✐s ❛ ♣r♦♣❡r ❧♦✇❡r s❡♠✐❝♦♥t✐♥✉♦✉s ❝♦♥✈❡① ❢✉♥❝t✐♦♥✳ ❚❤❡♥ t❤❡ s✉❜❞✐❢❢❡r❡♥t✐❛❧ ∂g ✐s ❛ ♠❛①✐♠❛❧ ♠♦♥♦t♦♥❡ ♦♣❡r❛t♦r✳ ❚❤❡r❡❢♦r❡✱ t❤❡ ♣r♦❜❧❡♠ ✭✷✳✾✮ ✐s ❡q✉✐✈❛❧❡♥t t♦ t❤❡ ♣r♦❜❧❡♠ ♦❢ ❢✐♥❞✐♥❣ ❛ ③❡r♦ ♦❢ ∂g ✳ ❲❡ ✉s❡❞ ♠❡t❤♦❞ ✶✵ ❚❤❡ ♥✉♠❡r✐❝❛❧ ❝❛❧❝✉❧❛t✐♦♥ r❡s✉❧ts ♦❜t❛✐♥❡❞ ✐♥ t❤❡ ❛❜♦✈❡ t❛❜❧❡s s❤♦✇ t❤❛t ♦✉r ♠❡t❤♦❞ ✭✷✳✷✮ ❛❢t❡r ✺✵✵ ✐t❡r❛t✐♦♥s ♦❜t❛✐♥s ❛ s♦❧✉t✐♦♥ t❤❛t ✐s ♠♦r❡ ❛♣♣r♦①✐♠❛t❡ t♦ t❤❡ ❝♦rr❡❝t s♦❧✉t✐♦♥ ♦❢ t❤❡ ♣r♦❜❧❡♠ ✭✷✳✾✮ t❤❛♥ t❤❡ ♠❡t❤♦❞ ✭✷✳✶✷✮ ♦❢ ❳✉✳ ■t s❤♦✉❧❞ ❜❡ ❛❞❞❡❞✱ ❤♦✇❡✈❡r✱ t❤❛t t❤❡s❡ ❝♦♠♣❛r✐s♦♥s ❛r❡ ❜❛s❡❞ s♦❧❡❧② ♦♥ ❡①♣❡r✐♠❡♥t❛❧ r❡s✉❧ts✳ ❖✉r ❛❧❣♦r✐t❤♠ ✇❛s ✐♠♣❧❡♠❡♥t❡❞ ✐♥ ❋r❡❡ P❛s❝❛❧ ■❉❊ s♦❢t✇❛r❡✱ r✉♥ ♦♥ ❛ P❈ ✇✐t❤ ■♥t❡❧✭❘✮ ❈♦r❡✭❚▼✮ ✐✺✲✺✷✵✵❯ ✷✳✷✵●❍③ ♣r♦❝❡ss♦r ❛♥❞ ✹✳✵✵●❇ ♦❢ ♠❡♠♦r②✳ ❈❤❛♣t❡r ✸ ■t❡r❛t✐✈❡ ♠❡t❤♦❞s ❢♦r ③❡r♦s ♦❢ ❛ ♠♦♥♦t♦♥❡ ✈❛r✐❛t✐♦♥❛❧ ✐♥❝❧✉s✐♦♥ ✐♥ ❍✐❧❜❡rt s♣❛❝❡s ❚❤❡ ♠❛✐♥ r❡s✉❧t ♦❢ ❈❤❛♣t❡r ✸ ✐s t♦ ✐♥tr♦❞✉❝❡ ♥❡✇ ❢♦r✇❛r❞✕❜❛❝❦✇❛r❞ s♣❧✐tt✐♥❣ ♠❡t❤♦❞s ❢♦r ❢✐♥❞✐♥❣ ❛ ③❡r♦ ♦❢ t❤❡ s✉♠ ♦❢ t✇♦ ♠❛①✐♠❛❧ ♠♦♥♦t♦♥❡ ♦♣❡r❛t♦rs ✐♥ ❍✐❧❜❡rt s♣❛❝❡s ❜② ❝♦♠❜✐♥✐♥❣ t❤❡ st❡❡♣❡st✲❞❡s❝❡♥t ♠❡t❤♦❞ ✇✐t❤ t❤❡ ❢♦r✇❛r❞✕❜❛❝❦✇❛r❞ s♣❧✐tt✐♥❣ ♦♥❡✳ ❚❤❡ str♦♥❣ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ ♥❡✇ ♠❡t❤✲ ♦❞s ✐s ♣r♦✈❡❞ ✉♥❞❡r ❛ ♥❡✇ ❝♦♥❞✐t✐♦♥ ♦♥ r❡s♦❧✈❡♥t ♣❛r❛♠❡t❡r✱ ♠♦r❡ ♣r❡❝✐s❡❧②✱ {rk } ✐s ❛♥ ❛r❜✐tr❛r② s❡q✉❡♥❝❡ ♦❢ ♥✉♠❜❡rs ✐♥ (0, α) ✇✐t❤ α > 0✳ ◆✉♠❡r✐❝❛❧ ❡①❛♠♣❧❡s ❛r❡ ❣✐✈❡♥ t♦ ✐❧❧✉str❛t❡ t❤❡ ♣r♦♣♦s❡❞ ♠❡t❤♦❞s✳ ❚❤❡ ❝♦♥t❡♥t ♦❢ t❤✐s ❝❤❛♣t❡r ✐s ❝♦♥t❛✐♥❡❞ ✐♥ ✇♦r❦s [1] ❛♥❞ [4] ✐♥ t❤❡ ▲✐st ♦❢ ♣✉❜❧✐s❤❡❞ ✇♦r❦s✳ ✸✳✶✳ ❋♦r✇❛r❞✕❜❛❝❦✇❛r❞ s♣❧✐tt✐♥❣ ♠❡t❤♦❞s ❲✳❚❛❦❛❤❛s❤✐✱ ❲♦♥❣ ❛♥❞ ❨❛♦ ✐♥ ✷✵✶✷ ✐♥tr♦❞✉❝❡❞ ❛ ♠♦❞✐❢✐❝❛t✐♦♥ ♦❢ t❤❡ ❢♦r✇❛r❞✕❜❛❝❦✇❛r❞ s♣❧✐tt✐♥❣ ♠❡t❤♦❞ ❢♦r t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦❜❧❡♠✿ ❋✐♥❞ p∗ ∈ H s✉❝❤ t❤❛t ∈ (A + B)p∗ ✭✸✳✶✮ ❚❤❡② ❞❡❢✐♥❡❞ ❛♥ ✐t❡r❛t✐✈❡ s❡q✉❡♥❝❡ ♦❢ ❍❛❧♣❡r♥✲t②♣❡ ❜② xk+1 = tk u + (1 − tk )Jk (I − rk A)xk , ✇❤❡r❡ u ∈ H, {tk } ⊂ (0, 1)✱ {rk } ⊂ (0, ∞)✳ ❚❤❡② ♣r♦✈❡❞ t❤❛t xk ✭✸✳✷✮ ❝♦♥✲ ✈❡r❣❡s str♦♥❣❧② t♦ PΩ u✱ ✇❤✐❝❤ ✐s t❤❡ ♣r♦❥❡❝t✐♦♥ ♦❢ u ♦♥t♦ Ω✱ t❤❡ s❡t ♦❢ ③❡r♦s ♦❢ A + B ✱ ✐❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❞✐t✐♦♥s ❤♦❧❞✿ (B5) tk ∈ (0, 1), ∀k ≥ 1✱ lim tk = 0✱ k→∞ ∞ (B6) ∞ tk = ∞ ✱ k=0 |tk+1 − tk | < ∞✱ k=0 (B7) < ε ≤ rk ≤ 2α✱ ∞ k=1 |rk+1 − rk | < ∞✳ ✶✷ ■♥ ♦r❞❡r t♦ ✇❡❛❦❡♥ t❤❡ ❛❜♦✈❡ ❝♦♥❞✐t✐♦♥✱ ✐♥ t❤❡ ✇♦r❦ [1] ✐♥ ✷✵✶✽✱ t❤❡ ❛✉t❤♦rs ❝♦♠❜✐♥❡❞ t❤❡ st❡❡♣❡st✲❞❡s❝❡♥t ♠❡t❤♦❞ ✇✐t❤ t❤❡ ❢♦r✇❛r❞✕❜❛❝❦✇❛r❞ s♣❧✐tt✐♥❣ ♦♥❡ ❛♥❞ ♣r♦♣♦s❡ t❤❡ ❢♦r✇❛r❞✕❜❛❝❦✇❛r❞ s♣❧✐tt✐♥❣ ♠❡t❤♦❞s xk = (I − tk F )T k xk , ✭✸✳✸✮ z ∈ H, z k+1 = T k (I − tk F )z k + ek , ✭✸✳✹✮ t♦ ❛♣♣r♦①✐♠❛t❡ t❤❡ s♦❧✉t✐♦♥ p∗ ∈ Ω ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ✈❛r✐❛t✐♦♥❛❧ ✐♥❡q✉❛❧✐t② ⟨F p∗ , p∗ − p⟩ ≤ ∀p ∈ Ω, ✭✸✳✺✮ ✇❤❡r❡ T k = T1 T2 · · · Tk ✇✐t❤ Ti = Ji (I − ri A)✱ ≤ i ≤ k ✱ F : H → H ✐s ❛♥ η ✲str♦♥❣❧② ♠♦♥♦t♦♥❡ ❛♥❞ γ˜ ✲str✐❝t❧② ♣s❡✉❞♦❝♦♥tr❛❝t✐✈❡ ✇✐t❤ η + γ˜ > 1✳ ❆❢t❡r t❤❛t✱ ❢r♦♠ ✭✸✳✹✮ ❜② ❝❤♦♦s✐♥❣ ❛♥ ❛♣♣r♦♣r✐❛t❡ ♠❛♣♣✐♥❣ F ✇❡ ♦❜t❛✐♥ s♦♠❡ ♠♦❞✐❢✐❝❛t✐♦♥s ♦❢ t❤❡ ❢♦r✇❛r❞✕❜❛❝❦✇❛r❞ s♣❧✐tt✐♥❣ ♠❡t❤♦❞ ✐♥❝❧✉❞✐♥❣ t❤❡ ♠❡t❤♦❞ ♣r♦♣♦s❡❞ ❜② ❲✳❚❛❦❛❤❛s❤✐✱ ❲♦♥❣ ❛♥❞ ❨❛♦✳ ❈♦♠♣❛r❡❞ ✇✐t❤ t❤❡ r❡s✉❧ts ♦❢ ❲✳❚❛❦❛❤❛s❤✐✱ ❲♦♥❣ ❛♥❞ ❨❛♦✱ t❤❡ str♦♥❣ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ ♦✉r ♠❡t❤♦❞ ❞♦❡s♥✬t ♥❡❡❞ t❤❡ ❝♦♥❞✐t✐♦♥ (B6) ❛♥❞ r❡♣❧❛❝❡❞ t❤❡ ❝♦♥❞✐t✐♦♥ (B7) ❜② ❛ ♥❡✇ ❝♦♥❞✐t✐♦♥ ❢♦r t❤❡ ♣❛r❛♠❡t❡r s❡q✉❡♥❝❡ {rk } ♦❢ t❤❡ r❡s♦❧✈❡♥t✿ (B9′ ) ∞ rk ∈ (0, α) ✇✐t❤ ❢♦r ❛❧❧ k ≥ ❛♥❞ rk < +∞✳ k=1 ■t ✐s ❡❛s② t♦ s❡❡ t❤❛t ✐❢ t❤❡ ❝♦♥❞✐t✐♦♥ (B9′ ) ✐s s❛t✐s❢✐❡❞ t❤❡♥ rk → ❛s k → ∞✳ ❚❤✐s ❝♦♥❞✐t✐♦♥ ♦♥ ♣❛r❛♠❡t❡r rk ♦❢ t❤❡ r❡s♦❧✈❡♥t ✐s ❞✐❢❢❡r❡♥t ❢r♦♠ ♦♥❡s ♦❢ ♦t❤❡r ♠❡t❤♦❞s✳ ■♥ ♦r❞❡r t♦ ♣r♦✈❡ t❤❡ str♦♥❣ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ ♦✉r ♠❡t❤♦❞s✱ ✇❡ r❡q✉✐r❡ t❤❡ ❢♦❧❧♦✇✐♥❣ r❡s✉❧ts✳ Pr♦♣♦s✐t✐♦♥ ✸✳✶ ▲❡t H ❜❡ ❛ r❡❛❧ ❍✐❧❜❡rt s♣❛❝❡✱ F : H → H ❜❡ ❛♥ η✲ str♦♥❣❧② ♠♦♥♦t♦♥❡ ❛♥❞ ❜❡ ❛ ♥♦♥❡①♣❛♥s✐✈❡ ❜♦✉♥❞❡❞ s❡q✉❡♥❝❡ γ ✲str✐❝t❧② ♣s❡✉❞♦❝♦♥tr❛❝t✐✈❡ ✇✐t❤ η + γ > ❛♥❞ T ♠❛♣♣✐♥❣ ♦♥ H s✉❝❤ t❤❛t F ix(T ) ̸= Ø✳ ❚❤❡♥✱ ❢♦r ❛♥② {xk } ✐♥ H s✉❝❤ t❤❛t limk→∞ ∥xk − T xk ∥ = 0✱ ✇❡ ❤❛✈❡ lim sup ⟨F p∗ , p∗ − xk ⟩ ≤ 0, ✭✸✳✻✮ k→∞ ✇❤❡r❡ p∗ ✐s t❤❡ ✉♥✐q✉❡ s♦❧✉t✐♦♥ ♦❢ t❤❡ ✈❛r✐❛t✐♦♥❛❧ ✐♥❡q✉❛❧✐t② ✭✸✳✺✮✳ ▲❡♠♠❛ ✸✳✶ ▲❡t H ♦❢ ♦❢ H ✳ ▲❡t α > 0✳ C ✐♥t♦ H ❛♥❞ B D(B) ⊆ C ✳ ❜❡ ❛ r❡❛❧ ❍✐❧❜❡rt s♣❛❝❡ ❛♥❞ ▲❡t A ❜❡ ❛♥ α✲✐♥✈❡rs❡ C ❜❡ ❛ ❝❧♦s❡❞ ❝♦♥✈❡① s✉❜s❡t str♦♥❣❧②✲♠♦♥♦t♦♥❡ ♠❛♣♣✐♥❣ ❧❡t ❜❡ ❛ ♠❛①✐♠❛❧ ♠♦♥♦t♦♥❡ ♠❛♣♣✐♥❣ ✐♥ ❙✉♣♣♦s❡ t❤❛t Ω := Zer(A + B) ̸= ∅✱ rk H s✉❝❤ t❤❛t ❜❡ ❛ ♥✉♠❜❡r ✐♥ ✶✸ (0, α) Tk ❛♥❞ ❧❡t TiB = Ji (I − ri A) ▲❡♠♠❛ ✸✳✷ ▲❡t ❜❡ ❛ ♠❛♣♣✐♥❣✱ t❤❛t ❜❡ ❞❡❢✐♥❡❞ ❜② T k = T1 T2 · · · Tk ≤ i ≤ k ✳ ❚❤❡♥✱ F ix(T k ) = Ω H, C, A, B, Ω ✈➔ Ti ❜❡ ❛s ✐♥ ▲❡♠♠❛ ❛♥❞ ❢♦r (B9′ ) ✈➔ |Bx| ≤ φ(∥x∥), ✸✳✶✳ ❆ss✉♠❡ t❤❛t t❤❡r❡ ❤♦❧❞ ❝♦♥❞✐t✐♦♥s✿ (B12) ∥Ax∥ φ(t) ❛♥❞ ✇❤❡r❡ |Bx| = inf{∥y∥ : y ∈ Bx} ✐s ❛ ♥♦♥✲♥❡❣❛t✐✈❡ ❛♥❞ ♥♦♥✲❞❡❝r❡❛s✐♥❣ ❢✉♥❝t✐♦♥ ❢♦r ❛❧❧ ❚❤❡♥✱ ❢♦r ❡❛❝❤ ❢✐①❡❞ x ∈ C ❛♥❞ t ≥ 0✳ ≤ i < k ✱ limk→∞ Tik x ❡①✐sts✱ ❛♥❞ ✇❤❡r❡ Tik = Ti · · · Tk ✳ ▲❡♠♠❛ ✸✳✸ ▲❡t H, C, A, B ❤♦❧❞ ❝♦♥❞✐t✐♦♥s (B9′ ) ❚❤❡♦r❡♠ ✸✳✶ ▲❡t H ❛♥❞ ❧❡t F ♠❛♣♣✐♥❣ ♦♥ ❜❡ H ❛♥❞ ❛♥❞ Ω (B12)✳ H, A, B, Ω ❛♥ η ✲str♦♥❣❧② ❜❡ ❛s ✐♥ ▲❡♠♠❛ ✸✳✶✳ ❆ss✉♠❡ t❤❛t t❤❡r❡ ❚❤❡♥✱ ❛♥❞ rk ❜❡ ❛s ✐♥ ▲❡♠♠❛ ✸✳✶ ✇✐t❤ ♠♦♥♦t♦♥❡ ❛♥❞ s✉❝❤ t❤❛t ❞❡❢✐♥❡❞ ❜② ✭✸✳✸✮✱ F ix(T ∞ ) = Ω✳ η + γ˜ > 1✳ ❚❤❡♥✱ tk ∈ (0, 1) ❛♥❞ tk → 0✱ ❛s γ˜ ✲str✐❝t❧② k → ∞✱ D(A) = ♣s❡✉❞♦❝♦♥tr❛❝t✐✈❡ {xk }✱ p ∗ ∈ Ω✱ t❤❡ s❡q✉❡♥❝❡ ❝♦♥✈❡r❣❡s str♦♥❣❧② t♦ s♦❧✈✐♥❣ ✈❛r✐❛t✐♦♥❛❧ ✐♥❡q✉❛❧✐t② ✭✸✳✺✮✳ Pr♦♣♦s✐t✐♦♥ ✸✳✷ ▲❡t F, H, A, B, Ω, rk ❛♥❞ tk ❜❡ ❛s ✐♥ ▲❡♠♠❛ ✸✳✶✳ ❚❤❡♥✱ {xk } ⊂ H ✱ s❛t✐s❢②✐♥❣ limk→∞ ∥T m xk − xk ∥ = m ≥ 1✱ ✇❡ ❤❛✈❡ ✭✸✳✻✮✳ ❢♦r ❛♥② ❜♦✉♥❞❡❞ s❡q✉❡♥❝❡ ✇✐t❤ ❛♥② ❢✐①❡❞ ✐♥t❡❣❡r ❚❤❡♦r❡♠ ✸✳✷ H, A, B ✱ Ω ❛♥❞ F ❜❡ ❛s ✐♥ ▲❡♠♠❛ ✸✳✶✳ ❆ss✉♠❡ t❤❛t t❤❡r❡ ❤♦❧❞ ❝♦♥❞✐t✐♦♥s✿ (B5)✱ (B9′ )✱ (B12)✱ (B3) ♦r ∥ek ∥ = 0✳ (B3′ ) lim k→∞ tk ❚❤❡♥✱ ❛s k → ∞✱ t❤❡ s❡q✉❡♥❝❡ {z k } ❞❡❢✐♥❡❞ ❜② ✭✸✳✹✮ ❝♦♥✈❡r❣❡s str♦♥❣❧② t♦ t❤❡ ♣♦✐♥t ▲❡t p∗ ∈ Ω✱ s♦❧✈✐♥❣ ✈❛r✐❛t✐♦♥❛❧ ✐♥❡q✉❛❧✐t② ✭✸✳✺✮✳ ❈♦♠♠❡♥t ✸✳✶ ❲❡ ♥♦✇ s❤♦✇ ❤♦✇ t♦ ❝❤♦♦s❡ ❛♥ ❛♣♣r♦♣r✐❛t❡ ♠❛♣♣✐♥❣ F ❢r♦♠ ✭✸✳✹✮ t♦ ♦❜t❛✐♥ s♦♠❡ ♠♦❞✐❢✐❝❛t✐♦♥s ❢♦r t❤❡ ❢♦r✇❛r❞✲❜❛❝❦✇❛r❞ s♣❧✐tt✐♥❣ ♠❡t❤♦❞✱ ✐♥❝❧✉❞✐♥❣ t❤❡ ♠❡t❤♦❞ ♦❢ ❲✳❚❛❦❛❤❛s❤✐✱ ❲♦♥❣ ❛♥❞ ❨❛♦✱ t❤❛t ✐s✿ xk+1 = T k (tk u + (1 − tk )xk ), ✭✸✳✼✮ y k+1 = tk u + (1 − tk )T k y k ✭✸✳✽✮ ❯♥❢♦rt✉♥❛t❡❧②✱ ♠❡t❤♦❞s ✭✸✳✼✮ ❛♥❞ ✭✸✳✽✮ ❛r❡ ❝♦♠♣❧✐❝❛t❡❞ ✇❤❡♥ k ✐s ❧❛r❣❡ ❡♥♦✉❣❤ ❜❡❝❛✉s❡ t❤❡ ♥✉♠❜❡r ♦❢ ❢♦r✇❛r❞✲❜❛❝❦✇❛r❞ ♦♣❡r❛t♦rs ✐♥❝r❡❛s❡s ❛❢✲ t❡r ❡❛❝❤ ✐t❡r❛t✐♦♥✳ ❋✉rt❤❡r♠♦r❡✱ t❤❡ s❡❝♦♥❞ ❝♦♥❞✐t✐♦♥ ♦♥ t❤❡ ♣❛r❛♠❡t❡r s❡q✉❡♥❝❡ {rk } ♦❢ t❤❡ r❡s♦❧✈❡♥t ✐♥ (B9′ ) ❛♥❞ ❝♦♥❞✐t✐♦♥ (B12) r❡❞✉❝❡s t❤❡ ✶✹ ✉s❛❜✐❧✐t② ✉s❡ ♠❡t❤♦❞✳ ❚♦ ♦✈❡r❝♦♠❡ t❤❡ ❛❜♦✈❡ ❧✐♠✐t❛t✐♦♥s✱ ✐♥ t❤❡ ✇♦r❦ [4]✱ ✇❡ ❝♦♥t✐♥✉❡ t♦ ✐♥tr♦❞✉❝❡ ♠♦r❡ ♠♦❞✐❢✐❝❛t✐♦♥s ♦❢ t❤❡ ❢♦r✇❛r❞✲❜❛❝❦✇❛r❞ s♣❧✐tt✐♥❣ ♠❡t❤♦❞ t❤❛t ❡❛❝❤ ✐t❡r❛t✐♦♥ ❝♦♥t❛✐♥s ♦♥❧② t✇♦ ❢♦r✇❛r❞✲❜❛❝❦✇❛r❞ ♦♣❡r❛t♦rs✳ ❚❤❡ str♦♥❣ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ t❤❡ ♠❡t❤♦❞ ♦❜t❛✐♥❡❞ r❡q✉✐r❡s ✇❡❛❦❡r ❝♦♥❞✐t✐♦♥ ♦♥ t❤❡ ♣❛r❛♠❡t❡r rk ❛♥❞ ❛♥❞ r❡♠♦✈❡s ❝♦♥❞✐t✐♦♥ (B12)✳ ❲❡ ❝♦♥✲ s✐❞❡r t❤❡ ❢♦❧❧♦✇✐♥❣ ♠❡t❤♦❞s✿ xk+1 = Tk Tc (t′k u + (1 − t′k )xk + ek ), ✭✸✳✾✮ xk+1 = t′k u + (1 − t′k )Tk Tc xk + ek , ✭✸✳✶✵✮ xk+1 = t′k u + βk′ Tc xk + γk′ Tk xk + ek , ✭✸✳✶✶✮ xk+1 = t′k f (Tc xk ) + βk′ Tc xk + γk′ Tk xk + ek , ✭✸✳✶✷✮ ✇❤❡r❡ Tk = Jk (I − rk A)✱ Tc = (I + cB)−1 (I − cA)✱ rk > 0✱ c ✐s ❛♥ ❛r❜✐tr❛r② ♣♦s✐t✐✈❡ ♥✉♠❜❡r s✉❝❤ t❤❛t < c < α✳ ❲❡ ✇❡r❡ ❛❜❧❡ t♦ s❤♦✇ t❤❛t ♠❡t❤♦❞s ✭✸✳✾✮✱ ✭✸✳✶✵✮ ❛♥❞ ✭✸✳✶✶✮✱ ✭✸✳✶✷✮ ❛r❡ r❡s♣❡❝t✐✈❡❧② t❤❡ s♣❡❝✐❛❧ ❝❛s❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ♠❡t❤♦❞ xk+1 = Tk Tc (I − tk F )xk + ek , ✭✸✳✶✸✮ xk+1 = βk (I − tk F )Tc xk + (1 − βk )Tk xk + ek , ✭✸✳✶✹✮ t♦ ❢✐♥❞ p∗ ∈ Ω s✉❝❤ t❤❛t t❤❡ ✈❛r✐❛t✐♦♥❛❧ ✐♥❡q✉❛❧✐t② ✭✸✳✺✮ ❤♦❧❞s ✇✐t❤ F : H → H ✐s ❛♥ η ✲str♦♥❣❧② ♠♦♥♦t♦♥❡ ❛♥❞ γ˜ ✲str✐❝t❧② ♣s❡✉❞♦❝♦♥tr❛❝t✐✈❡ ✇✐t❤ η + γ˜ > 1✳ ❚❤❡♦r❡♠ ✸✳✸ ▲❡t H ❜❡ ❛ r❡❛❧ ❍✐❧❜❡rt s♣❛❝❡✱ str♦♥❣❧② ♠♦♥♦t♦♥❡ ♦♥❡ ✐♥ ♦♣❡r❛t♦r ✐♥ H ✈➔ →♥❤ ①↕ ❜❡ ❛♥ η + γ˜ > 1✳ ❙✉♣♣♦s❡ Ω ̸= Ø ❛♥❞ t❤❡ s❡q✉❡♥❝❡ ❞❡❢✐♥❡❞ ❜②✿ ✭✸✳✶✺✮ z k+1 = Tk Tc (I − tk F )z k , ✇✐t❤ α✲✐♥✈❡rs❡ H ✇✐t❤ D(A) = H ✱ B ❜❡ ❛ ♠❛①✐♠❛❧ ♠♦♥♦t♦♥❡ F : H → H ❜❡ ❛♥ η ✲str♦♥❣❧② ♠♦♥♦t♦♥❡ ❛♥❞ γ˜ ✲ str✐❝t❧② ♣s❡✉❞♦❝♦♥tr❛❝t✐✈❡ ✇✐t❤ zk α > 0✱ A Tk = Jk (I − rk A)✱ Tc = (I + cB)−1 (I − cA)✳ (B5) ❛♥❞ (B9′′ ) c, rk ∈ (0, α) ❢♦r ❆ss✉♠❡ t❤❛t t❤❡r❡ ❤♦❧❞ ❝♦♥❞✐t✐♦♥s✿ ❚❤❡♥✱ ❛s ❛❧❧ k ≥ 1✳ k → ∞✱ t❤❡ s❡q✉❡♥❝❡ z k ❝♦♥✈❡r❣❡s str♦♥❣❧② t♦ t❤❡ ♣♦✐♥t p∗ ∈ Ω✱ s♦❧✈✐♥❣ ✈❛r✐❛t✐♦♥❛❧ ✐♥❡q✉❛❧✐t② ✭✸✳✺✮✳ ❚❤❡♦r❡♠ ✸✳✹ ▲❡t H, A, B, Ω ❛♥❞ F ❜❡ ❛s ✐♥ ❚❤❡♦r❡♠ ✸✳✸✳ ❚❤❡♥✱ ❛s k → ∞✱ t❤❡ s❡q✉❡♥❝❡ {x } ❞❡❢✐♥❡❞ ❜② ✭✸✳✶✹✮ ✇✐t❤ ❝♦♥❞✐t✐♦♥s (B3) ♦r (B3′ )✱ k ✶✺ (B5)✱ (B8)✱ (B9′′ ) ❝♦♥✈❡r❣❡s str♦♥❣❧② t♦ t❤❡ ♣♦✐♥t p∗ ∈ Ω✱ s♦❧✈✐♥❣ ✈❛r✐❛✲ t✐♦♥❛❧ ✐♥❡q✉❛❧✐t② ✭✸✳✺✮✳ ✸✳✷✳ ◆✉♠❡r✐❝❛❧ ❊①❛♠♣❧❡s ■♥ t❤✐s s❡❝t✐♦♥✱ ♠❡t❤♦❞ ✭✸✳✽✮ ❛♥❞ ❲✳❚❛❦❛❤❛s❤✐✱ ❲♦♥❣ ❛♥❞ ❨❛♦ ❛r❡ ✐♠✲ ♣❧❡♠❡♥t❡❞✳ ❚❤❡s❡ ♠❡t❤♦❞s ❛r❡ ❛♣♣❧✐❡❞ t♦ t❤❡ ❢♦❧❧♦✇✐♥❣ ✈❛r✐❛t✐♦♥❛❧ ✐♥❡q✉❛❧✲ ✐t② ♣r♦❜❧❡♠✿ ❢✐♥❞ p∗ ∈ C s✉❝❤ t❤❛t ⟨Ap∗ , p∗ − p⟩ ≤ 0, ∀p ∈ C, ✭✸✳✶✻✮ ✇❤❡r❡ C ✐s ❛ ❝♦♥✈❡① s✉❜s❡t ✐♥ ❛ ❍✐❧❜❡rt s♣❛❝❡ ❛♥❞ A ✐s ❛♥ α✲✐♥✈❡rs❡ str♦♥❣❧② ♠♦♥♦t♦♥❡ ♠❛♣♣✐♥❣ ♦♥ H ✳ ■t ❝❛♥ ❜❡ ♣r♦✈❡❞ t❤❛t p∗ ✐s ❛ s♦❧✉t✐♦♥ ♦❢ ✭✸✳✶✻✮ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✐t ✐s ❛ ③❡r♦ ♦❢ t❤❡ ✐♥❝❧✉s✐♦♥ ∈ (A + B)x, ✇❤❡r❡ B ✐s t❤❡ ♥♦r♠❛❧ ❝♦♥❡ t♦ C ✱ ❞❡❢✐♥❡❞ ❜② NC x = {w ∈ H : ⟨w, v − x⟩ ≤ 0, ∀v ∈ C} ▲❡t χC ❜❡ t❤❡ ✐♥❞✐❝❛t♦r ❢✉♥❝t✐♦♥ ♦❢ C ✱ t❤❛t ✐s  0, x ∈ C, χC = +∞, x ∈ / C ❚❤❡♥✱ χC ✐s ❛ ♣r♦♣❡r ❧♦✇❡r s❡♠✐❝♦♥t✐♥✉♦✉s ❝♦♥✈❡① ❢✉♥❝t✐♦♥ ♦❢ H ✐♥t♦ (−∞, ∞] ❛♥❞ t❤❡♥ s✉❜❞✐❢❢❡r❡♥t✐❛❧ ∂χC ✐s ❛ ♠❛①✐♠❛❧ ♠♦♥♦t♦♥❡ ♠❛♣♣✐♥❣ C = PC ✳ ❛♥❞ ∂χC = NC ✱ JrBk = Jr∂χ k ❆s ❢♦r ❝♦♠♣✉t❛t✐♦♥✱ ✇❡ ❝♦♥s✐❞❡r t❤❡ ❝❛s❡ t❤❛t n n (xj − aj )2 ≤ r2 }, C = {x ∈ E : ✭✸✳✶✼✮ j=1 ✇❤❡r❡ aj , r ∈ (−∞; +∞) ❢♦r ❛❧❧ ≤ j ≤ n✳ ❲❡ ❢✐rst ✐♥✈❡st✐❣❛t❡ t❤❡ ❝❛s❡ n = 2, a1 = a2 = 2, r = ❛♥❞ Ax = φ′ (x) ✇✐t❤ φ(x) = (x1 −1.5)2 /2✳ ❇② ❝❤♦♦s✐♥❣ u = (2.0; 1.5) ∈ C ✇❡ ♦❜t❛✐♥ p∗ = PΩ u = (1.5; 1.5) ❛s ❛ s♦❧✉t✐♦♥ t♦ ✭✸✳✶✻✮✕✭✸✳✶✼✮ ✇❤❡r❡ Ω = {(1.5; (−∞, ∞))} ∩ C ✐s t❤❡ s♦❧✉t✐♦♥ s❡t ❢♦r t❤❡ st❛t❡❞ ♣r♦❜❧❡♠✳ ❇② ✉s✐♥❣ ❛❧❣♦r✐t❤♠ ✭✸✳✽✮ ✇✐t❤ st❛rt✐♥❣ ♣♦✐♥t x1 = (2.7; 2.7) ∈ C ✱ tk = 1/(k + 1)✱ rk = 1/(k(k + 1)) ✇❡ ❣❡t t❤❡ r❡s✉❧ts ❧✐st❡❞ ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ t❛❜❧❡✳ ✶✻ ❚❛❜❧❡ ✸✳✶✳ ❈❛❧❝✉❧❛t❡❞ r❡s✉❧t ✇❤❡♥ ❛♣♣❧②✐♥❣ t❤❡ ♠❡t❤♦❞ ✭✸✳✽✮✳ y1k k y2k k y1k y2k ✶✵ ✶✳✺✻✼✺✵✺✼✶✾✶ ✶✳✺✹✹✶✵✷✸✹✼✸ ✷✵✵ ✶✳✺✵✸✺✹✹✸✸✸✺ ✶✳✺✵✷✶✶✸✺✹✷✻ ✷✵ ✶✳✺✸✹✺✽✶✾✻✷✻ ✶✳✺✷✶✺✹✹✽✼✼✺ ✸✵✵ ✶✳✺✵✷✸✻✺✶✹✽✽ ✶✳✺✵✶✹✵✽✵✸✽✹ ✸✵ ✶✳✺✷✸✷✺✾✷✸✽✽ ✶✳✺✶✹✷✻✵✹✶✸✶ ✹✵✵ ✶✳✺✵✶✼✼✹✼✶✶✻ ✶✳✺✵✶✵✺✺✻✺✼✼ ✹✵ ✶✳✺✶✼✺✷✸✽✸✼✵ ✶✳✺✶✵✻✺✼✹✸✵✶ ✺✵✵ ✶✳✺✵✶✹✷✵✶✼✽✵ ✶✳✺✵✵✽✹✹✸✹✽✶ ✺✵ ✶✳✺✶✹✵✺✼✽✻✽✸ ✶✳✺✵✽✺✵✼✽✾✸✹ ✶✵✵✵ ✶✳✺✵✵✼✶✵✹✾✽✺ ✶✳✺✵✵✹✷✶✾✾✻✵ ✶✵✵ ✶✳✺✵✼✵✻✽✺✵✵✶ ✶✳✺✵✹✷✸✻✵✶✵✶ ✷✵✵✵ ✶✳✺✵✵✸✺✺✸✺✶✽ ✶✳✺✵✵✷✶✵✾✺✸✺ ■♥ t❤❡ ♠❡❛♥t✐♥❡✱ ✇❤❡♥ ✉s✐♥❣ ❛❧❣♦r✐t❤♠ ✭✸✳✷✮ ✇✐t❤ rk = 0.2 + 1/(k(k + 1)) ∈ (0.2, 1) ❛♥❞ ❦❡❡♣✐♥❣ t❤❡ ♦t❤❡r ♣❛r❛♠❡t❡rs ✉♥❝❤❛♥❣❡❞✱ ✇❡ ♦❜t❛✐♥❡❞ t❤❡ r❡s✉❧ts ❧✐st❡❞ ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ t❛❜❧❡✳ ❚❛❜❧❡ ✸✳✷✳ ❈❛❧❝✉❧❛t❡❞ r❡s✉❧t ✇❤❡♥ ❛♣♣❧②✐♥❣ t❤❡ ♠❡t❤♦❞ ✭✸✳✷✮✳ ❦ xk1 xk2 ❦ xk1 xk2 ✶✵ ✶✳✻✾✷✽✹✻✶✻✵✻ ✶✳✻✵✻✸✺✼✷✶✹✻ ✷✵✵ ✶✳✺✶✷✹✸✻✶✾✼✾ ✶✳✺✶✷✹✸✻✶✽✶✸ ✷✵ ✶✳✻✶✺✹✵✼✾✽✺✻ ✶✳✻✶✹✽✸✾✻✷✷✻ ✸✵✵ ✶✳✺✵✽✸✵✺✶✼✺✶ ✶✳✺✵✽✸✵✺✶✼✶✾ ✸✵ ✶✳✺✼✾✾✵✺✷✷✸✺ ✶✳✺✼✾✽✷✼✺✸✹✶ ✹✵✵ ✶✳✺✵✻✷✸✹✷✶✺✻ ✶✳✺✵✻✷✸✹✹✷✶✹✻ ✹✵ ✶✳✺✻✵✼✷✷✷✾✹✾ ✶✳✺✻✵✼✵✺✷✸✺✹ ✺✵✵ ✶✳✺✵✹✾✽✾✾✶✽✼ ✶✳✺✵✹✾✽✾✾✶✽✸ ✺✵ ✶✳✺✹✽✾✵✵✾✾✺✶ ✶✳✺✹✽✽✾✺✸✶✵✶ ✶✵✵✵ ✶✳✺✵✷✹✾✼✹✽✾✾ ✶✳✺✵✷✹✾✼✹✽✾✾ ✶✵✵ ✶✳✺✷✹✼✸✾✵✾✷✻ ✶✳✺✷✹✼✸✽✽✵✽✻ ✷✵✵✵ ✶✳✺✵✷✹✾✼✹✽✽✾ ✶✳✺✵✷✹✾✼✹✽✽✾ ❖✉r ❛❧❣♦r✐t❤♠ ✇❛s ✐♠♣❧❡♠❡♥t❡❞ ✐♥ ❋r❡❡ P❛s❝❛❧ ■❉❊ s♦❢t✇❛r❡✱ r✉♥ ♦♥ ❛ P❈ ✇✐t❤ ■♥t❡❧✭❘✮ ❈♦r❡✭❚▼✮ ✐✺✲✺✷✵✵❯ ✷✳✷✵●❍③ ♣r♦❝❡ss♦r ❛♥❞ ✹✳✵✵●❇ ♦❢ ♠❡♠♦r②✳ ❈❤❛♣t❡r ✹ ■t❡r❛t✐✈❡ ❘❡❣✉❧❛r✐③❛t✐♦♥ ▼❡t❤♦❞s ❢♦r t❤❡ ▼✉❧t✐♣❧❡✲❙❡ts ❙♣❧✐t ❋❡❛s✐❜✐❧✐t② Pr♦❜❧❡♠ ✐♥ ❍✐❧❜❡rt ❙♣❛❝❡s ■♥ ❈❤❛♣t❡r ✹✱ ✇❡ ❣✐✈❡ ❛♥ ✐t❡r❛t✐✈❡ r❡❣✉❧❛r✐③❛t✐♦♥ ♠❡t❤♦❞ ❢♦r s♦❧✈✐♥❣ t❤❡ ♠✉❧t✐♣❧❡✲s❡ts s♣❧✐t ❢❡❛s✐❜✐❧✐t② ♣r♦❜❧❡♠ ✇✐t❤ t✇♦ ❝♦✉♥t❛❜❧❡ ✐♥❢✐♥✐t❡ ❢❛♠✐❧✐❡s ♦❢ ❝♦♥✈❡① ❝❧♦s❡❞ s✉❜s❡ts ❛♥❞ s❡✈❡r❛❧ ♣❛rt✐❝✉❧❛r ❝❛s❡s✳ ❲❡ ❛❧s♦ ❣✐✈❡ ♥✉♠❡r✐❝❛❧ ❡①❛♠♣❧❡s ❢♦r ✐❧❧✉str❛t✐♥❣ ♦✉r ❛❜♦✈❡ ♠❡t❤♦❞s✳ ❚❤❡ r❡s✉❧ts ♦❢ t❤✐s ❝❤❛♣t❡r ❜❡❧♦♥❣s t♦ ♦✉r ♣❛♣❡r ✐♥ [2] ✐♥ t❤❡ ▲✐st ♦❢ ♣✉❜❧✐s❤❡❞ ✇♦r❦s✳ ✹✳✶✳ ❘❡❣✉❧❛r✐③❛t✐♦♥ ♠❡t❤♦❞s ❛♥❞ ♠✐♥✐♠✉♠✲♥♦r♠ s♦❧✉t✐♦♥ ❚❤❡ s♣❧✐t ❢❡❛s✐❜✐❧✐t② ♣r♦❜❧❡♠ ✭❙❋P✮ x ∈ C s✉❝❤ t❤❛t Ax ∈ Q, ✭✹✳✶✮ ✇❤❡r❡ C, Q ❛r❡ ❝♦♥✈❡① s✉❜s❡t ♦❢ H1 ❛♥❞ H2 , r❡s♣❡❝t✐✈❡❧② ❛♥❞ A : H1 → H2 ✐s ❛ ❜♦✉♥❞❡❞ ❧✐♥❡❛r ♠❛♣♣✐♥❣✳ ❈❧❡❛r❧②✱ ✭✹✳✶✮ ✐s ❡q✉✐✈❛❧❡♥t t♦ t❤❡ ♠✐♥✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠ f (x) := ||Ax − PQ Ax||2 ✭✹✳✷✮ x∈C ❯s✐♥❣ t❤❡ ✐❞❡❛ ♦❢ ❚✐❦❤♦♥♦✈✬s r❡❣✉❧❛r✐③❛t✐♦♥✱ ✐♥ ✷✵✶✵✱ ❳✉ ❝♦♥s✐❞❡r❡❞ t❤❡ ❢♦❧❧♦✇✐♥❣ r❡❣✉❧❛r✐③❛t✐♦♥ ♣r♦❜❧❡♠ fα (x) := ||Ax − PQ Ax||2 + α||x||2 , x∈C ✇❤❡r❡ α > ✐s t❤❡ r❡❣✉❧❛r✐③❛t✐♦♥ ♣❛r❛♠❡t❡r✳ ✭✹✳✸✮ ❚❤❡ ❛✉t❤♦r s❤♦✇❡❞ t❤❛t ♣r♦❜❧❡♠ ✭✹✳✸✮ ❤❛s ❛ ✉♥✐q✉❡ s♦❧✉t✐♦♥✱ ❞❡♥♦t❡❞ ❜② xα ❛♥❞ ✐❢ ❙❋P ✭✹✳✶✮ ❤❛s ❛ s♦❧✉t✐♦♥ t❤❡♥ t❤❡ ❧✐♠✐t lim xα ❡①✐sts ❛♥❞ ✐s t❤❡ α→0 ✶✽ ♠✐♥✐♠✉♠✲♥♦r♠ s♦❧✉t✐♦♥ ♦❢ ❙❋P✳ ■♥ t❤✐s ♣❛♣❡r✱ t❤❡ ❛✉t❤♦r ❛❧s♦ ♣r♦♣♦s❡❞ ❛♥ r❡❣✉❧❛r✐③❛t✐♦♥ ♠❡t❤♦❞ ❜② ❇❛❦✉s❤✐♥s❦② ✭✶✾✼✼✮ ❛♥❞ ❇r✉❝❦ ✭✶✾✼✹✮ ♦❢ t❤❡ ❢♦r♠ xk+1 = PC [I − γk (A∗ (I − PQ )A + αk I)]xk , k ≥ 1, ✭✹✳✹✮ ❛♥❞ ♣r♦✈❡❞ t❤❛t t❤❡ s❡q✉❡♥❝❡ {xk } ❝♦♥✈❡r❣❡s ✐♥ ♥♦r♠ t♦ t❤❡ ♠✐♥✐♠✉♠✲ ♥♦r♠ s♦❧✉t✐♦♥ ♦❢ t❤❡ ❙❋P ✭✹✳✶✮ ✐❢ t❤❡ ♣❛r❛♠❡t❡r s❡q✉❡♥❝❡s {αk }, {γk } s❛t✲ ✐s❢② t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❞✐t✐♦♥s✿ αk , ∀k ❧❛r❣❡ ❡♥♦✉❣❤✱ (C4) < γk ≤ ||A||2 + αk (C5) αk → ❛♥❞ γk → 0, ∞ αk γk = ∞, (C6) k=1 (C7) |γk+1 − γk | + γk |αk+1 − αk | → (αk+1 γk+1 )2 ❨❛♦ ❡t ❛❧✳ ✐♥ ✷✵✶✷ ♣r♦✈❡❞ t❤❡ str♦♥❣ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ ✭✹✳✹✮ t♦ t❤❡ t♦ t❤❡ ♠✐♥✐♠✉♠✲♥♦r♠ s♦❧✉t✐♦♥ t♦ t❤❡ ❙❋P ✭✹✳✶✮ ❜② ✇❡❛❦❡r ❝♦♥❞✐t✐♦♥s ♦♥ {αk } ❛♥❞ {γk }✳ ❈❤✉❛♥❣ ✭✷✵✶✸✮ ✇❡❛❦❡♥ t❤❡s❡ ❝♦♥❞✐t✐♦♥s ♦♥ {αk }, {γk } ❢✉rt❤❡r ❜✉t st✐❧❧ ❣❡t t❤❡ str♦♥❣ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ ✭✹✳✹✮✳ ✹✳✷✳ ■t❡r❛t✐✈❡ ❘❡❣✉❧❛r✐③❛t✐♦♥ ▼❡t❤♦❞s ❢♦r t❤❡ ▼✉❧t✐♣❧❡✲❙❡ts ❙♣❧✐t ❋❡❛s✐❜✐❧✐t② Pr♦❜❧❡♠ ✐♥ ❍✐❧❜❡rt ❙♣❛❝❡s ❚❤❡ ♠✉❧t✐♣❧❡✲s❡ts s♣❧✐t ❢❡❛s✐❜✐❧✐t② ♣r♦❜❧❡♠ ✭▼❙❙❋P✮ ✐s t♦✿ ❋✐♥❞ x ∈ C := Ci s✉❝❤ t❤❛t Ax ∈ Q := i∈J1 Qj , ✭✹✳✺✮ j∈J2 ✇❤❡r❡ {Ci }i∈J1 ❛♥❞ {Qj }j∈J2 ❛r❡ t✇♦ ❝♦✉♥t❛❜❧❡ ✐♥❢✐♥✐t❡ ❢❛♠✐❧✐❡s ♦❢ ❝❧♦s❡❞ ❝♦♥✈❡① s✉❜s❡ts ✐♥ t✇♦ r❡❛❧ ❍✐❧❜❡rt s♣❛❝❡s H1 ❛♥❞ H2 ✱ r❡s♣❡❝t✐✈❡❧②✳ ❉❡♥♦t❡ ❜② Γ t❤❡ s❡t ♦❢ s♦❧✉t✐♦♥s ❢♦r ▼❙❙❋P ✭✹✳✺✮✳ ▼♦t✐✈❛t❡❞ ❜② t❤❡ r❡s✉❧ts ♦❢ ❳✉ ✭✷✵✶✵✮✱ ❨❛♦ ❡t ❛❧✳ ✭✷✵✶✷✮ ❛♥❞ ❇✉♦♥❣ ✭✷✵✶✼✮✱ ✇❡ ❣❡♥❡r❛❧✐③❡ ♠❡t❤♦❞ ✭✹✳✹✮ t♦ ▼❙❙❋P ✭✹✳✺✮ ✐♥ t❤❡ ❝❛s❡ J1 ❛♥❞ J2 ❛r❡ ❝♦✉♥t❛❜❧❡ ✐♥❢✐♥✐t❡ ❢❛♠✐❧✐❡s✳ ❖✉r ♠❡t❤♦❞ ✐s ❞❡❢✐♥❡❞ ❛s ❢♦❧❧♦✇s✿ xk+1 = Uk Tγk ,αk xk , x1 ∈ H1 , ✭✹✳✻✮ ✶✾ ✇❤❡r❡ Uk = β˜k k βi PCi , Tγk ,αk i=1 = I − γk (A (I − Vk )A + αk I), Vk = η˜k k ∗ ηj PQj , j=1 ✭✹✳✼✮ β˜k = β1 + · · · + βk ✱ η˜k = η1 + · · · + ηk ✱ ❛♥❞ t❤❡ ♣❛r❛♠❡t❡rs βi ✱ ηj ✱ αk ❛♥❞ γk s❛t✐s❢② t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❞✐t✐♦♥s✿ ∞ i=1 βi = 1❀ ∞ j=1 ηj = 1❀ (C10) βi > ❢♦r ❛❧❧ i ∈ N+ ❛♥❞ (C11) ηj > ❢♦r ❛❧❧ j ∈ N+ ❛♥❞ (C12) αk ∈ (0, 1)✱ k ∈ N+ s✉❝❤ t❤❛t limk→∞ αk = ❛♥❞ ∞ k=1 αk = ∞❀ (C13) γk ∈ (ε0 , 2/(∥A∥2 + 2)) ❢♦r ❛❧❧ k ∈ N+ ✱ ε0 ✐s ❛ s♠❛❧❧ ♣♦s✐t✐✈❡ ♥✉♠❜❡r✳ ◆♦t❡ t❤❛t✱ ❛t ❡❛❝❤ ✐t❡r❛t✐♦♥✱ ♦✉r ♠❡t❤♦❞ ✉s❡s ♦♥❧② ❢✐♥✐t❡ s✉♠ s♦ ❝❛❧❝✉❧❛✲ t✐♦♥ ❜② t❤❡ ♠❡t❤♦❞ ✐s ❛ s✐♠♣❧❡ ✇♦r❦✳ ❚❤❡ str♦♥❣ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ ♠❡t❤♦❞s ✭✹✳✻✮✲✭✹✳✼✮ ✐s ♣r♦✈❡❞ ✇✐t❤ ❛ss✉♠♣t✐♦♥s (C10), (C11), (C12) ✈➔ (C13)✳ ❆s ❛ ❝♦♥s❡q✉❡♥❝❡✱ ✇❡ ❣❡t s♦♠❡ ♥❡✇ r❡s✉❧ts ❢♦r ❝❛s❡s ✇❤❡r❡ ♦♥❡ ♦❢ t❤❡ s❡ts J1 , J2 ♦r ❜♦t❤ ❛r❡ ❢✐♥✐t❡✳ ■♥ ♦r❞❡r t♦ ♣r♦✈❡ ♦✉r t❤❡♦r❡♠s✱ ✇❡ r❡q✉✐r❡ t❤❡ ❢♦❧❧♦✇✐♥❣ r❡s✉❧ts✳ ▲❡♠♠❛ ✹✳✶ j ∈ J2 ✱ ❧❡t ▲❡t H1 ❛♥❞ H2 ❜❡ t✇♦ r❡❛❧ ❍✐❧❜❡rt s♣❛❝❡s✱ ❧❡t ❜❡ ❛ ♥♦♥❡①♣❛♥s✐✈❡ ♠❛♣♣✐♥❣ ♦♥ H2 ❜❡ ❛ ❜♦✉♥❞❡❞ ❧✐♥❡❛r ♠❛♣♣✐♥❣ ❢r♦♠ A Tj ✱ ❢♦r ❡❛❝❤ ∩j∈J2 ❋✐①(Tj ) ̸= ∅ ❛♥❞ s✉❝❤ t❤❛t H1 ✐♥t♦ H2 ✳ ❚❤❡♥✱ ∩j∈J2 A−1 ❋✐①(Tj ) = ∩j∈J2 ❋✐①(I − γA∗ (I − Tj )A) = A−1 (∩j∈J2 ❋✐①(Tj )), ✇❤❡r❡ γ ✐s ❛ ♣♦s✐t✐✈❡ ♥✉♠❜❡r✳ ▲❡♠♠❛ ✹✳✷ ▲❡t ❡❛❝❤ ❜❡ ❛ ♥♦♥❡①♣❛♥s✐✈❡ ♠❛♣♣✐♥❣ ♦♥ j ∈ N+ ✱ H1 , H2 , A ❛♥❞ γ ❜❡ ❛s ✐♥ t❤❡ ▲❡♠♠❛ ✹✳✶ ❛♥❞ ❧❡t H2 s✉❝❤ t❤❛t Tj ✱ ❢♦r ∩∞ j=1 ❋✐①(Tj ) ̸= ∅✳ ❚❤❡♥✱ C˜ := ∩j∈N+ ❋✐①(I − γA∗ (I − Tj )A) = ❋✐①(T∞ ), ✇❤❡r❡ T∞ = I − γA∗ (I − V∞ )A✱ V∞ = ∞ j=1 ηj Tj ❛♥❞ ηj s❛t✐s❢✐❡s ❝♦♥❞✐t✐♦♥ (C11)✳ ▲❡♠♠❛ ✹✳✸ ▲❡t H ❜❡ ❛ r❡❛❧ ❍✐❧❜❡rt s♣❛❝❡ ❛♥❞ ❧❡t ❛ ❢✐r♠❧② ♥♦♥❡①♣❛♥s✐✈❡ ♠❛♣♣✐♥❣ ♦♥ (C10)✳ ❚❤❡♥✱ t❤❡ ♠❛♣♣✐♥❣s H✳ S∞ := Si ✱ ❢♦r ❡❛❝❤ i ∈ N+ ✱ ❜❡ ❆ss✉♠❡ t❤❛t t❤❡r❡ ❤♦❧❞s ❝♦♥❞✐t✐♦♥ ∞ i=1 βi Si ❛♥❞ I − S∞ ❛r❡ ❛❧s♦ ❢✐r♠❧② ♥♦♥❡①♣❛♥s✐✈❡✳ ▲❡♠♠❛ ✹✳✹ ▲❡t H1 ✱ H2 ❢✐①❡❞ ♥✉♠❜❡r ❛♥❞ A ❜❡ ❛s ✐♥ ▲❡♠♠❛ ✹✳✶✳❚❤❡♥✱ ❢♦r ❛♥ ❛r❜✐tr❛r② γ ∈ (0, 2/(∥A∥ + 2α))✱ t❤❡ ♠❛♣♣✐♥❣ Tγ,α := I − γ(A∗ (I − ✷✵ V )A + αI) ✐s ❛ ❝♦♥tr❛❝t✐♦♥ ✇✐t❤ ❝♦❡❢❢✐❝✐❡♥t ♥♦♥❡①♣❛♥s✐✈❡ ♠❛♣♣✐♥❣ ❛♥❞ Tγ := I − γA∗ (I − V )A ❚❤❡♦r❡♠ ✹✳✶ {Qj }j∈N+ ▲❡t V ✐s ❛ ♣♦s✐t✐✈❡ ♥✉♠❜❡r ✐♥ (0, 1)✳ ❲❤❡♥ ❛♥❞ H1 , H2 A (C13)✳ ✐s ❛ ❢✐r♠❧② α = 0✱ ❜❡ ❛s ✐♥ ▲❡♠♠❛ ✹✳✶✳ ▲❡t {Ci }i∈N+ ❛♥❞ ✐♥ H1 ❛♥❞ H2 ✱ ❜❡ t✇♦ ✐♥❢✐♥✐t❡ ❢❛♠✐❧✐❡s ♦❢ ❝❧♦s❡❞ ❝♦♥✈❡① s✉❜s❡ts ❛♥❞ k → ∞✱ ✇❤❡r❡ ✐s ♥♦♥❡①♣❛♥s✐✈❡✳ r❡s♣❡❝t✐✈❡❧②✳ ❆ss✉♠❡ t❤❛t (C12) α − γα✱ Γ ̸= ∅ ❛♥❞ t❤❡r❡ ❤♦❧❞ ❝♦♥❞✐t✐♦♥s✱ ❚❤❡♥✱ t❤❡ s❡q✉❡♥❝❡ {xk }✱ (C10)✱ (C11)✱ ✭✹✳✻✮✲ ✭✹✳✼✮✱ ❛s ❞❡❢✐♥❡❞ ❜② ❝♦♥✈❡r❣❡s str♦♥❣❧② t♦ t❤❡ ♠✐♥✐♠✉♠✲♥♦r♠ s♦❧✉t✐♦♥ ♦❢ ✭✹✳✺✮ ✇✐t❤ J1 = J2 = N+ ✳ ❚❤❡♦r❡♠ ✹✳✷ {Qj }j∈N+ ▲❡t H1 , H2 ❛♥❞ A ❜❡ ❛s ✐♥ ▲❡♠♠❛ ✹✳✶✳ ▲❡t {Ci }N i=1 ❛♥❞ ❜❡ t✇♦ ❢❛♠✐❧✐❡s ♦❢ ❝❧♦s❡❞ ❝♦♥✈❡① s✉❜s❡ts ✐♥ t✐✈❡❧②✳ ❆ss✉♠❡ t❤❛t Γ ̸= ∅ ❛♥❞ t❤❡r❡ ❤♦❧❞ ❝♦♥❞✐t✐♦♥s H1 ❛♥❞ H2 ✱ r❡s♣❡❝✲ (C11)✱ (C12)✱ (C13) ❛♥❞ (C10′ ) βi > ❚❤❡♥✱ ❛s ❢♦r k → ∞✱ 1≤i≤N N i=1 βi s✉❝❤ t❤❛t t❤❡ s❡q✉❡♥❝❡ {xk }✱ = 1✳ ❞❡❢✐♥❡❞ ❜② N k+1 x k = U Tγk ,αk x , k ≥ 1, x ∈ H1 , U = ✭✹✳✽✮ βi PCi , i=1 ❝♦♥✈❡r❣❡s str♦♥❣❧② t♦ t❤❡ ♠✐♥✐♠✉♠✲♥♦r♠ s♦❧✉t✐♦♥ ♦❢ ❚❤❡♦r❡♠ ✹✳✸ {Qj }M j=1 ▲❡t H1 , H2 ❛♥❞ A ✭✹✳✺✮✳ ❜❡ ❛s ✐♥ ▲❡♠♠❛ ✹✳✶✳ ▲❡t {Ci }i∈N+ ❛♥❞ ❜❡ t✇♦ ❢❛♠✐❧✐❡s ♦❢ ❝❧♦s❡❞ ❝♦♥✈❡① s✉❜s❡ts ✐♥ ❆ss✉♠❡ t❤❛t Γ ̸= ∅ (C11′ ) ηj > ❚❤❡♥✱ ❛s ❢♦r k → ∞✱ ❛♥❞ t❤❡r❡ ❤♦❧❞ ❝♦♥❞✐t✐♦♥s 1≤j≤M M j=1 ηj s✉❝❤ t❤❛t t❤❡ s❡q✉❡♥❝❡ {xk }✱ H1 ❛♥❞ H2 ✱ r❡s♣❡❝t✐✈❡❧②✳ (C10), (C12), (C13) ❛♥❞ = 1✳ ❞❡❢✐♥❡❞ ❜② M x k+1 ∗ k = Uk (I − γk (A (I − V )A + αk I))x , k ≥ 1, x ∈ H1 , V = ηj PQj , i=1 ❝♦♥✈❡r❣❡s str♦♥❣❧② t♦ t❤❡ ♠✐♥✐♠✉♠✲♥♦r♠ s♦❧✉t✐♦♥ ♦❢ ❚❤❡♦r❡♠ ✹✳✹ {Qj }N j=1 ▲❡t H1 , H2 A ✭✹✳✺✮✳ (C13)✳ ❚❤❡♥✱ ❛s {Ci }M i=1 ❛♥❞ H1 ❛♥❞ H2 ✱ r❡✲ ❜❡ ❛s ✐♥ ▲❡♠♠❛ ✹✳✶✳ ▲❡t ❜❡ t✇♦ ❢✐♥✐t❡ ❢❛♠✐❧✐❡s ♦❢ ❝❧♦s❡❞ ❝♦♥✈❡① s✉❜s❡ts ✐♥ s♣❡❝t✐✈❡❧②✳ ❆ss✉♠❡ t❤❛t ❛♥❞ ❛♥❞ ✭✹✳✾✮ Γ ̸= ∅ ❛♥❞ t❤❡r❡ ❤♦❧❞ ❝♦♥❞✐t✐♦♥s (C10′ ), (C11′ ), (C12) k → ∞✱ t❤❡ s❡q✉❡♥❝❡ {xk }✱ ❞❡❢✐♥❡❞ ❜② xk+1 = U (I − γk (A∗ (I − V )A + αk I))xk , x1 ∈ H1 , ✭✹✳✶✵✮ ✷✶ ✇❤❡r❡ U ❛♥❞ V ❛r❡ ❞❡❢✐♥❡❞ ✐♥ ❚❤❡♦r❡♠s ✹✳✷ ❛♥❞ ✹✳✸ r❡s♣❡❝t✐✈❡❧②✱ ❝♦♥✈❡r❣❡s str♦♥❣❧② t♦ t❤❡ ♠✐♥✐♠✉♠✲♥♦r♠ s♦❧✉t✐♦♥ ♦❢ ✹✳✸✳ ✭✹✳✺✮✳ ◆✉♠❡r✐❝❛❧ ❊①❛♠♣❧❡s ❲❡ ❝♦♥s✐❞❡r t❤❡ ▼❙❙❋P ✭✹✳✺✮ ✇✐t❤ C := ❛♥❞ Ci Q := i∈J1 Qj j∈J2 ✇❤❡r❡ ✭✹✳✶✶✮ Ci = {x ∈ En : ai1 x1 + ai2 x2 + · · · + ain xn ≤ bi }, ail , bi ∈ (−∞; +∞)✱ ❢♦r ≤ l ≤ n ❛♥❞ i ∈ J1 ✱ m (yl − ajl )2 ≤ Rj }, Rj > 0, m Qj = {y ∈ E : ✭✹✳✶✷✮ l=1 ajl ∈ (−∞; +∞)✱ ❢♦r ≤ l ≤ m ❛♥❞ j ∈ J2 ✱ ❛♥❞ A ✐s ❛♥ m × n✲♠❛tr✐①✳ ■♥ t❤❡ ❢✐rst ❡①❛♠♣❧❡✱ ✇❡ ❝♦♥s✐❞❡r t❤❡ ❝❛s❡ m = n = 2, A ✐s t❤❡ ✐❞❡♥✐t② ♠❛tr✐①✱ ai1 = 1/i, ai2 = −1✱ bi = 0✱ ∀i ≥ 1✱ Rj = ❛♥❞ aj = (1/j, 0)✱ ∀j ≥ 1✳ ❚❤❡♥✱ ✐t ✐s ♥♦t ❞✐❢❢✐❝✉❧t t♦ ✈❡r✐❢② t❤❛t x∗ = (0; 0) ✐s t❤❡ ✉♥✐q✉❡ ♠✐♥✐♠✉♠✲♥♦r♠ s♦❧✉t✐♦♥ ♦❢ ✭✹✳✶✶✮✲✭✹✳✶✷✮✳ ❙✐♥❝❡ A = I ✱ ❛❧❣♦r✐t❤♠ ✭✹✳✻✮✲ ✭✹✳✼✮ ❤❛s t❤❡ ❢♦r♠ ✭✹✳✶✸✮ xk+1 = Uk ((1 − γk (1 + αk ))xk + γk Vk xk ) ❯s✐♥❣ ♠❡t❤♦❞ ✭✹✳✶✸✮ ✇✐t❤ βi = ηi = 1/(i(i + 1))✱ αk = 1/k ✱ γk = 1/(1 + 0.05 + (1/k) ❛♥❞ ❛ st❛rt✐♥❣ ♣♦✐♥t x1 = (−3.0; 3.0)✱ ✇❡ ♦❜t❛✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ t❛❜❧❡ ♦❢ ♥✉♠❡r✐❝❛❧ r❡s✉❧ts ✭❚❛❜❧❡ ✶✮✳ ❚❛❜❧❡ ✹✳✶✳ ❈♦♠♣✉t❛t✐♦♥❛❧ r❡s✉❧ts ❜② ♠❡t❤♦❞ ✭✹✳✶✸✮✳ k ✶ xk+1 xk+1 k xk+1 xk+1 ✵✳✵✷✹✸✾✵✷✹✸✾ ✵✳✸✻✺✽✺✸✻✺✽✺ ✶✵✵ ✵✳✵✵✶✷✸✾✵✺✵✺ ✵✳✵✵✽✸✾✹✺✷✺✶ ✶✵ ✵✳✵✶✵✷✺✺✸✷✼✹ ✵✳✵✻✾✹✼✾✹✾✻✽ ✺✵✵ ✵✳✵✵✵✷✻✾✺✸✹✼ ✵✳✵✵✶✽✷✻✵✽✽✽ ✷✵ ✵✳✵✵✺✺✸✹✹✾✽✷ ✵✳✵✸✼✹✾✻✵✸✼✻ ✶✵✵✵ ✵✳✵✵✵✶✸✾✹✶✾✷ ✵✳✵✵✵✾✹✹✺✻✵✻ ✸✵ ✵✳✵✵✸✽✶✽✵✹✷✽ ✵✳✵✷✺✽✻✼✶✶✶✷ ✷✵✵✵ ✵✳✵✵✵✵✼✷✵✽✷✹ ✵✳✵✵✵✹✽✽✸✺✺✽ ✹✵ ✵✳✵✵✷✾✷✹✾✽✻✷ ✵✳✵✶✾✽✶✻✻✽✷✼ ✸✵✵✵ ✵✳✵✵✵✵✹✽✾✾✾✹ ✵✳✵✵✵✵✸✸✶✾✻✾ ✷✷ ■♥ t❤❡ s❡❝♦♥❞ ❡①❛♠♣❧❡✱ s❛✈✐♥❣ Ci , βi , ηj , Rj , γk , αk ❛♥❞ t❤❡ st❛rt✐♥❣ ♣♦✐♥t✱ ✇❡ ❝♦♥s✐❞❡r ♥❡✇ Qj = {y ∈ E3 : ∥y −aj ∥ ≤ 1} ✇❤❡r❡ aj = (1/(j +1); 1/(j + 1); 1/(j + 1)) ❛♥❞ A ✐s ❛ × 2✲♠❛tr✐① ✇✐t❤ ❡❧❡♠❡♥ts ai1 = 1✱ ❢♦r i = 1, 2, 3✱ ❛♥❞ ③❡r♦ ❢♦r t❤❡ ♦t❤❡rs✳ ❈❧❡❛r❧②✱ x∗ = (0; 0) ✐s t❤❡ ✉♥✐q✉❡ ♠✐♥✐♠✉♠ ♥♦r♠ s♦❧✉t✐♦♥✳ ❚❤❡ ❝♦♠♣✉t❛t✐♦♥❛❧ r❡s✉❧ts✱ ❜② ✉s✐♥❣ ♠❡t❤♦❞ ✭✹✳✻✮✲ ✭✹✳✼✮✱ ❛r❡ ♣r❡s❡♥t❡❞ ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ♥✉♠❡r✐❝❛❧ t❛❜❧❡✳ ❇↔♥❣ ✹✳✷✳ ❈♦♠♣✉t❛t✐♦♥❛❧ r❡s✉❧ts ❜② ♠❡t❤♦❞ ✭✹✳✻✮✲ ✭✹✳✼✮ xk+1 k ✶ xk+1 k xk+1 xk+1 ✵✳✻✵✶✾✸✽✽✷✼✹ ✶✳✺✸✻✺✽✸✸✻✺✾ ✶✵✵ ✵✳✵✶✹✷✵✹✼✹✶✺ ✵✳✵✸✻✸✵✵✾✽✺✷ ✶✵ ✵✳✶✶✼✻✾✾✹✾✽✶ ✵✳✸✵✵✹✺✹✻✻✶✵ ✺✵✵ ✵✳✵✵✸✵✾✸✹✷✻✽ ✵✳✵✵✼✽✾✻✻✼✸✹ ✷✵ ✵✳✵✻✸✺✶✽✾✺✶✻ ✵✳✶✻✷✶✹✻✺✷✾✵ ✶✵✵✵ ✵✳✵✵✶✻✵✵✶✵✷✹ ✵✳✵✵✹✵✽✹✻✷✹✹ ✸✵ ✵✳✵✹✸✽✶✾✸✹✹✸ ✵✳✶✶✶✽✺✽✽✶✸✾ ✷✵✵✵ ✵✳✵✵✵✽✷✼✷✽✸✹ ✵✳✵✵✷✶✶✶✽✷✽✹ ✹✵ ✵✳✵✸✺✻✾✽✶✶✹✵ ✵✳✵✽✺✻✾✹✺✺✻✻ ✸✵✵✵ ✵✳✵✵✵✺✻✷✸✻✶✺ ✵✳✵✵✶✹✸✺✺✺✺✸ ❚❤r♦✉❣❤ t❤❡ ♥✉♠❡r✐❝❛❧ ❝❛❧❝✉❧❛t✐♦♥ r❡s✉❧ts ♦❜t❛✐♥❡❞ ✐♥ ❚❛❜❧❡ ✹✳✶ ❛♥❞ ❚❛❜❧❡ ✹✳✷✱ ✐t s❤♦✇s t❤❛t ♦✉r ♠❡t❤♦❞s ❛❢t❡r ✸✵✵✵ ✐t❡r❛t✐♦♥s ♦❜t❛✐♥❡❞ ❛ s♦❧✉t✐♦♥ t❤❛t ✐s q✉✐t❡ ❝❧♦s❡ t♦ t❤❡ ❝♦rr❡❝t s♦❧✉t✐♦♥ ♦❢ t❤❡ ♣r♦❜❧❡♠ ✭✹✳✺✮✳ ❖✉r ❛❧❣♦r✐t❤♠ ✇❛s ✐♠♣❧❡♠❡♥t❡❞ ✐♥ ❋r❡❡ P❛s❝❛❧ ■❉❊ s♦❢t✇❛r❡✱ r✉♥ ♦♥ ❛ P❈ ✇✐t❤ ■♥t❡❧✭❘✮ ❈♦r❡✭❚▼✮ ✐✺✲✺✷✵✵❯ ✷✳✷✵●❍③ ♣r♦❝❡ss♦r ❛♥❞ ✹✳✵✵●❇ ♦❢ ♠❡♠♦r②✳ ✷✸ ❈❖◆❈▲❯❙■❖◆❙ ❚❤❡ t❤❡s✐s ❛❝❤✐❡✈❡s t❤❡ ❢♦❧❧♦✇✐♥❣ r❡s✉❧ts✿ ✭✶✮ ❋♦r ❢✐♥❞✐♥❣ ❛ ③❡r♦ ♦❢ ❛ ♠❛①✐♠❛❧ ♠♦♥♦t♦♥❡ ♦♣❡r❛t♦r ✐♥ ❛ ❍✐❧❜❡rt s♣❛❝❡s ✇❡ ❤❛✈❡ ♣r❡s❡♥t❡❞ ❛ ♠♦❞✐❢✐❝❛t✐♦♥ ♦❢ t❤❡ ♣r♦①✐♠❛❧ ♣♦✐♥t ♠❡t❤♦❞✱ str♦♥❣ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ ✇❤✐❝❤ ✐s ♣r♦✈❡❞ ✇✐t❤♦✉t ❛♥ ❛❞❞✐t✐♦♥❛❧ ❝♦♥❞✐t✐♦♥ ♦♥ t❤❡ r❡s♦❧✈❡♥t ♣❛r❛♠❡t❡r ♦❢ t❤❡ ♦♣❡r❛t♦r✳ ✭✷✮ ❆ s✐♠✐❧❛r r❡s✉❧t ✇❛s ♦❜t❛✐♥❡❞ ❢♦r t❤❡ ♠♦♥♦t♦♥❡ ✈❛r✐❛t✐♦♥❛❧ ✐♥❝❧✉s✐♦♥ ♣r♦❜❧❡♠✳ ✭✸✮ ❲❡ ❤❛✈❡ ✐♥tr♦❞✉❝❡❞ ❛♥ ✐♥t❡r❛t✐✈❡ r❡❣✉❧❛r✐③❛t✐♦♥ ♠❡t❤♦❞ ❢♦r s♦❧✈✐♥❣ t❤❡ ♠✉❧t✐♣❧❡✲s❡ts s♣❧✐t ❢❡❛s✐❜✐❧✐t② ♣r♦❜❧❡♠ ✇✐t❤ t✇♦ ✐♥❢✐♥✐t❡ ❢❛♠✐❧✐❡s ♦❢ ❝♦♥✈❡① s✉❜s❡ts✳ ❚❤❡ ✐♠♣♦rt❛♥❝❡ ♦❢ ♦✉r ♠❡t❤♦❞ ✐s t❤❛t ❛t ❡❛❝❤ ✐♥t❡r❛t✐♦♥ st❡♣ ✐t ❝♦♥t❛✐♥s ♦♥❧② ❢✐♥✐t❡ ♥✉♠❜❡rs ♦❢ s❡ts ❢r♦♠ t❤❡ ❢❛♠✐❧✐❡s✳ ✭✹✮ ●✐✈❡ ♥✉♠❡r✐❝❛❧ ❡①❛♠♣❧❡s ❢♦r ✐❧❧✉str❛t✐♥❣ t❤❡ ♣r♦♣♦s❡❞ ♠❡t❤♦❞s✳ ❋✉t✉r❡ r❡s❡❛r❝❤ ✭✶✮ Pr♦♣♦s❡ ❛♥❞ st✉❞② t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ ♥❡✇ ✐t❡r❛t✐✈❡ ♠❡t❤♦❞s t♦ ❢✐♥❞ ③❡r♦s ♦❢ ♠♦♥♦t♦♥❡ ♦♣❡r❛t♦rs✱ s✉♠ ♦❢ t✇♦ ♠♦♥♦t♦♥❡ ♦♣❡r❛t♦rs ✐♥ ❍✐❧❜❡rt s♣❛❝❡s ❛♥❞ ❇❛♥❛❝❤ s♣❛❝❡s✳ ✭✷✮ ❊✈❛❧✉❛t❡ t❤❡ ♦r❞❡r ♦❢ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ ✐t❡r❛t✐✈❡ ♠❡t❤♦❞s t♦ ❢✐♥❞ ③❡r♦s ♦❢ ♠❛①✐♠❛❧ ♠♦♥♦t♦♥❡ ♦♣❡r❛t♦rs✱ t❤❡ s✉♠ ♦❢ t✇♦ ♠❛①✐♠❛❧ ♠♦♥♦t♦♥❡ ♦♣❡r❛✲ t♦rs ✐♥ ❍✐❧❜❡rt s♣❛❝❡ ♣r❡s❡♥t❡❞ ✐♥ ❈❤❛♣t❡r ✷ ❛♥❞ ❈❤❛♣t❡r ✸✳ ✭✸✮ ❈♦♥t✐♥✉❡ t♦ st✉❞② t❤❡ ✐t❡r❛t✐✈❡ r❡❣✉❧❛r✐③❛t✐♦♥ ♠❡t❤♦❞ ❢♦r s♦❧✈✐♥❣ t❤❡ ♠✉❧t✐♣❧❡✲s❡ts s♣❧✐t ❢❡❛s✐❜✐❧✐t② ♣r♦❜❧❡♠ ❢♦r t❤❡ ❝❛s❡ t❤❛t t❤❡ ✐t❡r❛t✐✈❡ ♣❛r❛♠✲ ❡t❡rs ❞♦ ♥♦t ❞❡♣❡♥❞ ♦♥ t❤❡ ♥♦r♠ ♦❢ t❤❡ tr❛♥s❢♦r♠❛t✐♦♥ ♦♣❡r❛t♦r A✳ ✷✹ ▲■❙❚ ❖❋ ❲❖❘❑❙ ❍❆❙ ❇❊❊◆ P❯❇▲■❙❍❊❉ [1] ◆✳ ❇✉♦♥❣✱ P✳❚✳❚✳ ❍♦❛✐✱ ■t❡r❛t✐✈❡ ♠❡t❤♦❞s ❢♦r ③❡r♦s ♦❢ ❛ ♠♦♥♦t♦♥❡ ✈❛r✐✲ ❛t✐♦♥❛❧ ✐♥❝❧✉s✐♦♥ ✐♥ ❍✐❧❜❡rt s♣❛❝❡s✱ ❈❛❧❝♦❧♦✱ ✷✵✶✽✱ ✺✺✱ ❛rt✿✼ ✭❙❈■❊✱ ◗✶✮✳ [2] ◆✳ ❇✉♦♥❣✱ P✳❚✳❚✳ ❍♦❛✐✱ ❑✳❚✳ ❇✐♥❤✱ ■t❡r❛t✐✈❡ ❘❡❣✉❧❛r✐③❛t✐♦♥ ▼❡t❤♦❞s ❢♦r t❤❡ ▼✉❧t✐♣❧❡✲❙❡ts ❙♣❧✐t ❋❡❛s✐❜✐❧✐t② Pr♦❜❧❡♠ ✐♥ ❍✐❧❜❡rt ❙♣❛❝❡s✱ ❆❝t❛ ❆♣♣❧ ▼❛t❤✱ ✷✵✶✾✱ ✶✻✺✱ ✶✽✸✲✶✾✼ ✭❙❈■❊✱ ◗✷✮✳ [3] ◆✳❚✳❚✳ ❚❤✉②✱ P✳❚✳❚✳ ❍♦❛✐✱ ◆✳❚✳❚✳ ❍♦❛✱ ❊①♣❧✐❝✐t ✐t❡r❛t✐✈❡ ♠❡t❤♦❞s ❢♦r ♠❛①✐♠❛❧ ♠♦♥♦t♦♥❡ ♦♣❡r❛t♦rs ✐♥ ❍✐❧❜❡rt s♣❛❝❡s✱ ◆♦♥❧✐♥❡❛r ❋✉♥❝t✐♦♥♥❛❧ ❆♥❛❧②s✐s ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥s✱ ✷✵✷✵✱ ✷✺✭✹✮✱ ✼✺✸✲✼✻✼ ✭❙❈❖P❯❙✮✳ [4] ◆✳❚✳◗✳ ❆♥❤✱ P✳❚✳❚✳ ❍♦❛✐✱ ▼♦❞✐❢✐❡❞ ❢♦r✇❛r❞✲❜❛❝❦✇❛r❞ s♣❧✐tt✐♥❣ ♠❡t❤✲ ♦❞s ✐♥ ❍✐❧❜❡rt s♣❛❝❡s✱ ❊❛st✲❲❡st ❏♦✉r♥❛❧ ♦❢ ▼❛t❤❡♠❛t✐❝s✱ ✷✵✷✵✱ ✷✷✭✶✮✱ ✶✸✲✷✾✳ [5] ◆✳ ❇✉♦♥❣✱ ◆✳❚✳❚✳ ❍♦❛✱ P✳❚✳❚✳ ❍♦❛✐✱ ■t❡r❛t✐✈❡ ♠❡t❤♦❞s ✇✐t❤ ♠❛①✐♠❛❧ ♠♦♥♦t♦♥❡ ♦♣❡r❛t♦rs ✐♥ ❍✐❧❜❡rt s♣❛❝❡s✱ Pr♦❝❡❡❞✐♥❣s ♦❢ t❤❡ ✷✸t❤ ♥❛t✐♦♥❛❧ ✇♦r❦s❤♦♣✿ ❙❡❧❡❝t❡❞ ✐ss✉❡s ♦❢ ✐♥❢♦r♠❛t✐♦♥ t❡❝❤♥♦❧♦❣② ❛♥❞ ❝♦♠♠✉♥✐❝❛t✐♦♥✱ ◗✉↔♥❣ ◆✐♥❤✱ ◆♦✈❡♠❜❡r ✺✲✻✱ ✷✵✷✵✱ ❙❝✐❡♥❝❡ ❛♥❞ ❚❡❝❤♥✐❝s P✉s❤❧✐s❤✐♥❣ ❍♦✉s❡✱ ✷✵✷✵✱ ✶✺✽✲✶✻✹✳