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▼❛①✐♠✉♠ ♣r✐♥❝✐♣❧❡ ❢♦r t❤❡ ❢r❛❝t✐♦♥❛❧ ❞✐✛✉s✐♦♥ ❡q✉❛t✐♦♥s ❛♥❞ ✐ts ❛♣♣❧✐❝❛t✐♦♥s ❨✉r✐ ▲✉❝❤❦♦ ❉❡♣❛rt♠❡♥t ♦❢ ▼❛t❤❡♠❛t✐❝s✱ P❤②s✐❝s✱ ❛♥❞ ❈❤❡♠✐str② ❇❡✉t❤ ❚❡❝❤♥✐❝❛❧ ❯♥✐✈❡rs✐t② ♦❢ ❆♣♣❧✐❡❞ ❙❝✐❡♥❝❡s ❇❡r❧✐♥ ❇❡r❧✐♥✱ ●❡r♠❛♥② ❋r❛❝t✐♦♥❛❧ P❉❊s✿ ❚❤❡♦r②✱ ❆❧❣♦r✐t❤♠s ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥s ■❈❊❘▼ ❛t ❇r♦✇♥ ❯♥✐✈❡rs✐t② ❏✉♥❡ ✶✽ ✲ ✷✷✱ ✷✵✶✽

❖✉t❧✐♥❡ ♦❢ t❤❡ t❛❧❦✿ • ▼❛①✐♠✉♠ ♣r✐♥❝✐♣❧❡ ❢♦r ❛ t✐♠❡✲❢r❛❝t✐♦♥❛❧ ❞✐✛✉s✐♦♥ ❡q✉❛t✐♦♥ ✇✐t❤ t❤❡ ❣❡♥❡r❛❧ ❢r❛❝t✐♦♥❛❧ ❞❡r✐✈❛t✐✈❡ • ▼❛①✐♠✉♠ ♣r✐♥❝✐♣❧❡ ❢♦r t❤❡ ✇❡❛❦ s♦❧✉t✐♦♥s ♦❢ ❛ t✐♠❡✲❢r❛❝t✐♦♥❛❧ ❞✐✛✉s✐♦♥ ❡q✉❛t✐♦♥ ✇✐t❤ t❤❡ ❈❛♣✉t♦ ❞❡r✐✈❛t✐✈❡ • ▼❛①✐♠✉♠ ♣r✐♥❝✐♣❧❡ ❢♦r ❛♥ ❛❜str❛❝t s♣❛❝❡✲ ❛♥❞ t✐♠❡✲❢r❛❝t✐♦♥❛❧ ❡✈♦❧✉t✐♦♥ ❡q✉❛t✐♦♥ ✐♥ t❤❡ ❍✐❧❜❡rt s♣❛❝❡ • ❙❤♦rt s✉r✈❡② ♦❢ ♦t❤❡r r❡s✉❧ts ❨✉r✐ ▲✉❝❤❦♦ ✭❇❍❚ ❇❡r❧✐♥✮ ▼❛①✐♠✉♠ ♣r✐♥❝✐♣❧❡ ❏✉♥❡ ✷✶✱ ✷✵✶✽ ✷ ✴ ✸✸

✶✳ ❉✐✛✉s✐♦♥ ❡q✉❛t✐♦♥ ✇✐t❤ t❤❡ ❣❡♥❡r❛❧ ❢r❛❝t✐♦♥❛❧ ❞❡r✐✈❛t✐✈❡ ❲❤❛t ✐s ❛ ♠❛①✐♠✉♠ ♣r✐♥❝✐♣❧❡❄ ▼❛①✐♠✉♠ ♣r✐♥❝✐♣❧❡✿ ❆ ❢✉♥❝t✐♦♥ s❛t✐s✜❡s ❛ ❞✐✛❡r❡♥t✐❛❧ ✐♥❡q✉❛❧✐t② ♦r ❡q✉❛t✐♦♥ ✐♥ ❛ ❞♦♠❛✐♥ D ⇒ ■t ❛❝❤✐❡✈❡s ✐ts ♠❛①✐♠✉♠ ♦♥ t❤❡ ❜♦✉♥❞❛r② ♦❢ D ✳ ❆ ✈❡r② ❡❧❡♠❡♥t❛r② ❡①❛♠♣❧❡✿ f ′′ ( x ) > ✵ , x ∈ ] a , b [ ❛♥❞ f ∈ C ([ a , b ]) ⇒ f ❛❝❤✐❡✈❡s ✐ts ♠❛①✐♠✉♠ ✈❛❧✉❡ ❛t ♦♥❡ ♦❢ t❤❡ ❡♥❞♣♦✐♥ts ♦❢ t❤❡ ✐♥t❡r✈❛❧✳ ❖t❤❡r ❡①❛♠♣❧❡s✿ ▼❛①✐♠✉♠ ♣r✐♥❝✐♣❧❡s ❢♦r ♦r❞✐♥❛r② ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ❛♥❞ ✐♥❡q✉❛❧✐t✐❡s ▼❛①✐♠✉♠ ♣r✐♥❝✐♣❧❡s ❢♦r ♣❛rt✐❛❧ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ❛♥❞ ✐♥❡q✉❛❧✐t✐❡s ❱❡r② r❡❝❡♥t❧②✿ ♠❛①✐♠✉♠ ♣r✐♥❝✐♣❧❡s ❢♦r ❢r❛❝t✐♦♥❛❧ P❉❊s ❨✉r✐ ▲✉❝❤❦♦ ✭❇❍❚ ❇❡r❧✐♥✮ ▼❛①✐♠✉♠ ♣r✐♥❝✐♣❧❡ ❏✉♥❡ ✷✶✱ ✷✵✶✽ ✸ ✴ ✸✸

✶✳ ❉✐✛✉s✐♦♥ ❡q✉❛t✐♦♥ ✇✐t❤ t❤❡ ❣❡♥❡r❛❧ ❢r❛❝t✐♦♥❛❧ ❞❡r✐✈❛t✐✈❡ ●❡♥❡r❛❧ ❢r❛❝t✐♦♥❛❧ ❞❡r✐✈❛t✐✈❡s ▲❡t k ❜❡ ❛ ♥♦♥♥❡❣❛t✐✈❡ ❧♦❝❛❧❧② ✐♥t❡❣r❛❜❧❡ ❢✉♥❝t✐♦♥✳ ❚❤❡ ❣❡♥❡r❛❧ ❢r❛❝t✐♦♥❛❧ ❞❡r✐✈❛t✐✈❡ ♦❢ t❤❡ ❈❛♣✉t♦ t②♣❡✿ � t ( D C k ( t − τ ) f ′ ( τ ) d τ. ( k ) f )( t ) = ✵ ❚❤❡ ❣❡♥❡r❛❧ ❢r❛❝t✐♦♥❛❧ ❞❡r✐✈❛t✐✈❡ ♦❢ t❤❡ ❘✐❡♠❛♥♥✲▲✐♦✉✈✐❧❧❡ t②♣❡✿ � t ( k ) f )( t ) = d ( D RL k ( t − τ ) f ( τ ) d τ. dt ✵ ❋♦r ❛♥ ❛❜s♦❧✉t❡❧② ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥ f ✇✐t❤ t❤❡ ✐♥❝❧✉s✐♦♥ f ′ ∈ L loc ✶ ( I R + ) ✱ ✇❡ ❣❡t � t ( k ) f )( t ) = d ( D C k ( t − τ ) f ( τ ) d τ − k ( t ) f ( ✵ ) = ( D RL ( k ) f )( t ) − k ( t ) f ( ✵ ) dt ✵ ❨✉r✐ ▲✉❝❤❦♦ ✭❇❍❚ ❇❡r❧✐♥✮ ▼❛①✐♠✉♠ ♣r✐♥❝✐♣❧❡ ❏✉♥❡ ✷✶✱ ✷✵✶✽ ✹ ✴ ✸✸

✶✳ ❉✐✛✉s✐♦♥ ❡q✉❛t✐♦♥ ✇✐t❤ t❤❡ ❣❡♥❡r❛❧ ❢r❛❝t✐♦♥❛❧ ❞❡r✐✈❛t✐✈❡ P❛rt✐❝✉❧❛r ❝❛s❡s ✶✮ ❚❤❡ ❝♦♥✈❡♥t✐♦♥❛❧ ❈❛♣✉t♦ ❛♥❞ ❘✐❡♠❛♥♥✲▲✐♦✉✈✐❧❧❡ ❢r❛❝t✐♦♥❛❧ ❞❡r✐✈❛t✐✈❡s✿ τ − α k ( τ ) = Γ( ✶ − α ) , ✵ < α < ✶ . � t ✶ ( D α ( t − τ ) − α f ′ ( τ ) d τ, ∗ f )( t ) = Γ( ✶ − α ) ✵ � t � � ( D α f )( t ) = d ✶ ( t − τ ) − α f ( τ ) d τ . Γ( ✶ − α ) dt ✵ ❨✉r✐ ▲✉❝❤❦♦ ✭❇❍❚ ❇❡r❧✐♥✮ ▼❛①✐♠✉♠ ♣r✐♥❝✐♣❧❡ ❏✉♥❡ ✷✶✱ ✷✵✶✽ ✺ ✴ ✸✸

✶✳ ❉✐✛✉s✐♦♥ ❡q✉❛t✐♦♥ ✇✐t❤ t❤❡ ❣❡♥❡r❛❧ ❢r❛❝t✐♦♥❛❧ ❞❡r✐✈❛t✐✈❡ P❛rt✐❝✉❧❛r ❝❛s❡s ✷✮ ❚❤❡ ♠✉❧t✐✲t❡r♠ ❞❡r✐✈❛t✐✈❡s n τ − α k � k ( τ ) = ✵ < α ✶ < · · · < α n < ✶ a k Γ( ✶ − α k ) , k = ✶ n � ( D C a k ( D α k ( k ) f )( t ) = ∗ f )( t ) , k = ✶ n ( D RL � a k ( D α k f )( t ) . ( k ) f )( t ) = k = ✶ ❨✉r✐ ▲✉❝❤❦♦ ✭❇❍❚ ❇❡r❧✐♥✮ ▼❛①✐♠✉♠ ♣r✐♥❝✐♣❧❡ ❏✉♥❡ ✷✶✱ ✷✵✶✽ ✻ ✴ ✸✸

✶✳ ❉✐✛✉s✐♦♥ ❡q✉❛t✐♦♥ ✇✐t❤ t❤❡ ❣❡♥❡r❛❧ ❢r❛❝t✐♦♥❛❧ ❞❡r✐✈❛t✐✈❡ P❛rt✐❝✉❧❛r ❝❛s❡s ✸✮ ❉❡r✐✈❛t✐✈❡s ♦❢ t❤❡ ❞✐str✐❜✉t❡❞ ♦r❞❡r✿ � ✶ τ − α k ( τ ) = Γ( ✶ − α ) d ρ ( α ) , ✵ ✇❤❡r❡ ρ ✐s ❛ ❇♦r❡❧ ♠❡❛s✉r❡ ♦♥ [ ✵ , ✶ ] ✿ � ✶ ( D C ( D α ( k ) f ) = ∗ f )( t ) d ρ ( α ) , ✵ � ✶ ( D C ( D α f )( t ) d ρ ( α ) . ( k ) f ) = ✵ ❨✉r✐ ▲✉❝❤❦♦ ✭❇❍❚ ❇❡r❧✐♥✮ ▼❛①✐♠✉♠ ♣r✐♥❝✐♣❧❡ ❏✉♥❡ ✷✶✱ ✷✵✶✽ ✼ ✴ ✸✸

✶✳ ❉✐✛✉s✐♦♥ ❡q✉❛t✐♦♥ ✇✐t❤ t❤❡ ❣❡♥❡r❛❧ ❢r❛❝t✐♦♥❛❧ ❞❡r✐✈❛t✐✈❡ ❈♦♥❞✐t✐♦♥s ♦♥ t❤❡ ❦❡r♥❡❧ ❢✉♥❝t✐♦♥ ❑✶✮ ❚❤❡ ▲❛♣❧❛❝❡ tr❛♥s❢♦r♠ ˜ k ♦❢ k ✱ � ∞ k ( t ) e − pt dt , ˜ k ( p ) = ✵ ❡①✐sts ❢♦r ❛❧❧ p > ✵✱ ❑✷✮ ˜ k ( p ) ✐s ❛ ❙t✐❧t❥❡s ❢✉♥❝t✐♦♥✱ ❑✸✮ ˜ k ( p ) → ✵ ❛♥❞ p ˜ k ( p ) → ∞ ❛s p → ∞ ✱ ❑✹✮ ˜ k ( p ) → ∞ ❛♥❞ p ˜ k ( p ) → ✵ ❛s p → ✵✳ ❨✉r✐ ▲✉❝❤❦♦ ✭❇❍❚ ❇❡r❧✐♥✮ ▼❛①✐♠✉♠ ♣r✐♥❝✐♣❧❡ ❏✉♥❡ ✷✶✱ ✷✵✶✽ ✽ ✴ ✸✸

✶✳ ❉✐✛✉s✐♦♥ ❡q✉❛t✐♦♥ ✇✐t❤ t❤❡ ❣❡♥❡r❛❧ ❢r❛❝t✐♦♥❛❧ ❞❡r✐✈❛t✐✈❡ Pr♦♣❡rt✐❡s ♦❢ t❤❡ ❣❡♥❡r❛❧ ❞❡r✐✈❛t✐✈❡s ✭❆✮ ❋♦r ❛♥② λ > ✵✱ t❤❡ ✐♥✐t✐❛❧ ✈❛❧✉❡ ♣r♦❜❧❡♠ ❢♦r t❤❡ ❢r❛❝t✐♦♥❛❧ r❡❧❛①❛t✐♦♥ ❡q✉❛t✐♦♥ ( D C ( k ) f )( t ) = − λ f ( t ) , t > ✵ , u ( ✵ ) = ✶ ❤❛s ❛ ✉♥✐q✉❡ s♦❧✉t✐♦♥ u λ = u λ ( t ) t❤❛t ❜❡❧♦♥❣s t♦ t❤❡ ❝❧❛ss C ∞ ( I R + ) ❛♥❞ ✐s ❛ ❝♦♠♣❧❡t❡❧② ♠♦♥♦t♦♥❡ ❢✉♥❝t✐♦♥✳ ✭❇✮ ❚❤❡r❡ ❡①✐sts ❛ ❝♦♠♣❧❡t❡❧② ♠♦♥♦t♦♥❡ ❢✉♥❝t✐♦♥ κ = κ ( t ) ✇✐t❤ t❤❡ ♣r♦♣❡rt② � t k ( t − τ ) κ ( τ ) d τ ≡ ✶ , t > ✵ . ✵ ✭❈✮ ❋♦r f ∈ L loc ✶ ( I R + ) ✱ t❤❡ r❡❧❛t✐♦♥s ( D C ( D RL ( k ) I ( k ) f )( t ) = f ( t ) , ( k ) I ( k ) f )( t ) = f ( t ) ❤♦❧❞ tr✉❡✱ ✇❤❡r❡ t❤❡ ❣❡♥❡r❛❧ ❢r❛❝t✐♦♥❛❧ ✐♥t❡❣r❛❧ I ( k ) ✐s ❞❡✜♥❡❞ ❜② t❤❡ ❢♦r♠✉❧❛ � t ( I ( k ) f )( t ) = κ ( t − τ ) f ( τ ) d τ. ✵ ❨✉r✐ ▲✉❝❤❦♦ ✭❇❍❚ ❇❡r❧✐♥✮ ▼❛①✐♠✉♠ ♣r✐♥❝✐♣❧❡ ❏✉♥❡ ✷✶✱ ✷✵✶✽ ✾ ✴ ✸✸