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◆❛❦❤✉s❤❡✈ ❡①tr❡♠✉♠ ♣r✐♥❝✐♣❧❡ ❢♦r ❛ ❝❧❛ss ♦❢ ✐♥t❡❣r♦✲❞✐✛❡r❡♥t✐❛❧ ♦♣❡r❛t♦rs ■♥t❡r♥❛t✐♦♥❛❧ ❈♦♥❢❡r❡♥❝❡ ♦♥ ❋r❛❝t✐♦♥❛❧ ❈❛❧❝✉❧✉s ✾✲✶✵ ❏✉♥❡ ✷✵✷✵ ●❤❡♥t ❆♥❛❧②s✐s ✫ P❉❊ ❈❡♥t❡r ●❤❡♥t ❯♥✐✈❡rs✐t②✴❩❖❖▼ ❆rs❡♥ Ps❦❤✉ ■♥st✐t✉t❡ ♦❢ ❆♣♣❧✐❡❞ ▼❛t❤❡♠❛t✐❝s ❛♥❞ ❆✉t♦♠❛t✐♦♥ ❑❇❙❈ ❘❆❙✱ ◆❛❧❝❤✐❦✱ ❘✉ss✐❛

■♥tr♦❞✉❝t✐♦♥ ❋❡r♠❛t✬s ❡①tr❡♠✉♠ t❤❡♦r❡♠ f ( x ) ∈ C 1 ( a, b ) y ∈ ( a, b ) f ( y ) ≥ f ( x ) ⇒ f ′ ( y ) = 0 a < x < b ( D α f ) ( y ) = 0 �⇒ α ∈ (0 , 1) f ( y ) ≥ f ( x ) ⇒ f ′ ( y ) ≥ 0 a < x < y ( D α f ) ( y ) ≥ 0 ⇒

■♥tr♦❞✉❝t✐♦♥ ❊①tr❡♠✉♠ ♣r✐♥❝✐♣❧❡ ❢♦r t❤❡ ❘✐❡♠❛♥♥✲▲✐♦✉✈✐❧❧❡ ❞❡r✐✈❛t✐✈❡ � x f ( t )( x − t ) − α ax f ) ( x ) = d ( D α Γ(1 − α ) dt α ∈ (0 , 1) dx a ❚❤❡♦r❡♠ ❬ ❆✳ ▼✳ ◆❛❦❤✉s❤❡✈✱ ❉✐✛❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥s ✱ ✶✾✼✹✱ ✈♦❧✳ ✶✵ ❪ f ( x ) ∈ H λ [ y − δ, y ] ▲❡t f ( x ) ∈ L ( a, y ) ✱ ( λ > α ) ✱ ❛♥❞ f ( y ) ≥ f ( x ) ∀ x ∈ ( a, y ) ✳ ax f ) ( y ) ≥ f ( y )( y − a ) − α ( D α ❚❤❡♥ Γ(1 − α ) . ( D α ■♥ ❛❞❞✐t✐♦♥✱ ✐❢ f ( y ) > 0 t❤❡♥ ax f ) ( y ) > 0 ✳

■♥tr♦❞✉❝t✐♦♥ ❆♣♣❧✐❝❛t✐♦♥ ◮ ▼✐①❡❞ t②♣❡ P❉❊ ◮ ▲♦❛❞❡❞ ✐♥t❡❣r❛❧ ❛♥❞ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ◮ ❉❡❣❡♥❡r❛t❡ P❉❊ ◮ Pr♦❜❧❡♠s ✇✐t❤ s❤✐❢t ❢♦r P❉❊ ◮ ❋r❛❝t✐♦♥❛❧ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s

◆❛❦❤✉s❤❡✈ ❆✳▼✳ ❉✐✛❡r✳ ❊q✉✳ ✶✾✼✹ ❑❤✉❜✐❡✈ ❑✳❯✳ ❘❡♣✳ ❆■❆❙ ✷✵✶✹ ◆❛❦❤✉s❤❡✈ ❆✳▼✳ ❉✐✛❡r✳ ❊q✉✳ ✶✾✼✺ ▼❛s❛❡✈❛ ❖✳❑❤✳ ❘❡♣✳ ❆■❆❙ ✷✵✶✹ ◆❛❦❤✉s❤❡✈ ❆✳▼✳ ❉♦❦❧✳ ❆❦❛❞✳ ◆❛✉❦ ✶✾✼✼ ▼❛s❛❡✈❛ ❖✳❑❤✳ ❉✐✛❡r✳ ❊q✉✳ ✷✵✶✹ ◆❛❦❤✉s❤❡✈ ❆✳▼✳ ❊q✉❛t✐♦♥s ♦❢ ▼❛t❤❡♠❛t✐❝❛❧ ❇✐♦❧♦❣②✱ ▼♦s❝♦✇✱ ✶✾✾✺ Ps❦❤✉ ❆✳❱✳ ❉✐✛❡r✳ ❊q✉✳ ✷✵✶✺ ◆❛❦❤✉s❤❡✈ ❆✳▼✳ ❋r❛❝t✐♦♥❛❧ ❈❛❧❝✉❧✉s ❛♥❞ ✐ts ▲✉❝❤❦♦ ❨✉✳✱ ❨❛♠❛♠♦t♦ ▼✳ ❋❈❈❆ ✷✵✶✻ ❆♣♣❧✐❝❛t✐♦♥✱ ▼♦s❝♦✇✱ ✷✵✵✸ ❈❛♦ ▲✳✱ ❑♦♥❣ ❍✳✱ ❩❡♥❣ ❙❤✳✲❉✳ ❏◆❙❆ ✷✵✶✼ ◆❛❦❤✉s❤❡✈ ❆✳▼✳ Pr♦❜❧❡♠s ✇✐t❤ ❙❤✐❢t ❢♦r P❉❊✱ ❊❢❡♥❞✐❡✈ ❇✳■✳ ▼❛t❤✳ ◆♦t❡s ✷✵✶✽ ▼♦s❝♦✇✱ ✷✵✵✻ ❇♦r✐❦❤❛♥♦✈ ▼✳✱ ❑✐r❛♥❡ ▼✳✱ ❚♦r❡❜❡❦ ❇✳❚✳ ▼❛t❤✳ ▲✉❝❤❦♦ ❨✉✳ ❏✳ ▼❛t❤✳ ❆♥❛❧✳ ❆♣♣❧✳ ✷✵✵✾ ▲❡tt✳ ✷✵✶✽ ▼❛♠❝❤✉❡✈ ▼✳❖✳ ❘❡♣✳ ❆■❆❙ ✷✵✶✵ ❇♦r✐❦❤❛♥♦✈ ▼✳✱ ❚♦r❡❜❡❦ ❇✳❚✳ ▼❛t✳ ❩❤✳ ✷✵✶✽ ◆❛❦❤✉s❤❡✈ ❆✳▼✳ ▲♦❛❞❡❞ ❊q✉❛t✐♦♥s ❛♥❞ ❚❤❡✐r ❑✐r❛♥❡ ▼✳✱ ❚♦r❡❜❡❦ ❇✳❚✳ ❋❈❈❆ ✷✵✶✾ ❆♣♣❧✐❝❛t✐♦♥✱ ▼♦s❝♦✇✱ ✷✵✶✷ ▲✉❝❤❦♦ ❨✉✳✱ ❨❛♠❛♠♦t♦ ▼✳ ▼❛①✐♠✉♠ ♣r✐♥❝✐♣❧❡ ❆❧✲❘❡❢❛✐ ▼✳ ❊❏◗❚❉❊ ✷✵✶✷ ❢♦r t❤❡ t✐♠❡✲❢r❛❝t✐♦♥❛❧ P❉❊s ❆♥❛t♦❧② ❑♦❝❤✉❜❡✐✱ ◆❛❦❤✉s❤❡✈ ❆✳▼✳ ❘❡♣✳ ❆■❆❙ ✷✵✶✹ ❨✉r✐ ▲✉❝❤❦♦ ✭❊❞s✳✮ ❍❛♥❞❜♦♦❦ ♦❢ ❋r❛❝t✐♦♥❛❧ ❈❛❧❝✉❧✉s ✇✐t❤ ❆♣♣❧✐❝❛t✐♦♥s ✷ ✷✵✶✾ ❆❧✲❘❡❢❛✐ ▼✳✱ ▲✉❝❤❦♦ ❨✉✳ ❋❈❈❆ ✷✵✶✹

▼❛✐♥ r❡s✉❧ts ■♥t❡❣r♦✲❞✐✛❡r❡♥t✐❛❧ ♦♣❡r❛t♦rs � x ( D f ) ( x ) = d k ( x, t ) f ( t ) dt, dx a k : S → R f : ( a, y ) → R ( D f ) : ( b, y ) → R S = { ( x, t ) : b < x < y, a < t < x } a, b, y ∈ R ∪{−∞} −∞ ≤ a ≤ b < y f ( t ) ∈ C ( a, y ] ∩ L ( a, y ) [ k ( x, t )] x = y ∈ L ( a, y ) � � k ( x, t ) , k x ( x, t ) ∈ C S \ { x = t }

▼❛✐♥ r❡s✉❧ts ❊①tr❡♠✉♠ ♣r✐♥❝✐♣❧❡ ❚❤❡♦r❡♠ ▲❡t [ k x ( x, t )] x = y ≤ 0 ❛♥❞ � x − ε [ f ( t ) − f ( x )] k x ( x, t ) dt ⇒ h 0 ( x ) a ε → 0 � x − ε d k ( x, t ) dt ⇒ h 1 ( x ) x ∈ ( y − δ, y ] dx a [ f ( x − ε ) − f ( x )] k ( x, x − ε ) ⇒ 0

❚❤❡♦r❡♠ ❝♦♥t✐♥✉❛t✐♦♥ ❚❤❡♥ f ( y ) = sup a<t<y f ( t ) = ⇒ ( D f ) ( y ) ≥ f ( y ) ( D 1 ) ( y ) f ( t ) ≡ const ( D f ) ( y ) = f ( y ) ( D 1 ) ( y ) ⇐ ⇒ ♦r [ k x ( x, t )] x = y ≡ 0 ❛♥❞ f ( t ) �≡ const [ k x ( x, t )] x = y �≡ 0 = ⇒ ( D f ) ( y ) > 0 f ( y ) ( D 1 ) ( y ) ≥ 0

Pr♦♦❢ s❦❡t❝❤ � x − ε K ε ( x ) = k ( x, t ) f ( t ) dt ( ε > 0) . a � x − ε � x − ε K ε ( x ) = [ f ( t ) − f ( x )] k ( x, t ) dt + f ( x ) k ( x, t ) dt a a � x − ε � x − ε d [ f ( t ) − f ( x )] k x ( x, t ) dt + f ( x ) d dxK ε ( x ) = k ( x, t ) dt dx a a � x d ( D f ) ( x ) = lim dxK ε ( x ) = [ f ( t ) − f ( x )] k x ( x, t ) dt + f ( x ) ( D 1 ) ( x ) . ε → 0 a

❘❡♠❛r❦ D RL = D I ❘✐❡♠❛♥♥✲▲✐♦✉✈✐❧❧❡ t②♣❡ D C = I D ❈❛♣✉t♦ t②♣❡ D C u + Lu = F u ∈ C (Ω) D RL u + Lu = F u �∈ C (Ω) ϕ · u ∈ C (Ω)

❲❡✐❣❤t❡❞ ❡①tr❡♠✉♠ ♣r✐♥❝✐♣❧❡ ▲❡t ϕ ( t ) ∈ C ( a, y ] ∩ L ( a, y ) ❛♥❞ f ( t ) = ϕ ( t ) g ( t ) g ( t ) ∈ C [ a, y ] ϕ ( t ) [ k x ( x, t )] x = y ≤ 0 ❛♥❞ � x − ε ϕ ( t ) [ g ( t ) − g ( x )] k x ( x, t ) dt ⇒ h 0 ( x ) a ε → 0 � x − ε d ϕ ( t ) k ( x, t ) dt ⇒ h 1 ( x ) x ∈ ( y − δ, y ] dx a [ g ( x − ε ) − g ( x )] k ( x, x − ε ) ⇒ 0

❲❡✐❣❤t❡❞ ❡①tr❡♠✉♠ ♣r✐♥❝✐♣❧❡ ❝♦♥t✐♥✉❛t✐♦♥ ❚❤❡♥ g ( y ) = sup a<t<y g ( t ) = ⇒ ( D f ) ( y ) ≥ g ( y ) ( D ϕ ) ( y ) g ( t ) ≡ const ( D f ) ( y ) = g ( y ) ( D ϕ ) ( y ) ⇐ ⇒ ♦r ϕ ( t ) [ k x ( x, t )] x = y ≡ 0 ❛♥❞ g ( t ) �≡ const ϕ ( t ) [ k x ( x, t )] x = y �≡ 0 = ⇒ ( D f ) ( y ) > 0 g ( y ) ( D ϕ ) ( y ) ≥ 0

Pr♦♦❢ s❦❡t❝❤ ❇② f ( t ) = ϕ ( t ) g ( t ) ✇❡ ❣❡t � x � x � ˜ ( D f ) ( x ) = d f ( t ) k ( x, t ) dt = d g ( t ) ˜ � k ( x, t ) dt = D g ( x ) , dx dx a a ✇❤❡r❡ ˜ k ( x, t ) = ϕ ( t ) k ( x, t ) .

❲❡✐❣❤t❡❞ ❡①tr❡♠✉♠ ♣r✐♥❝✐♣❧❡ ❢♦r t❤❡ ❘✐❡♠❛♥♥✲▲✐♦✉✈✐❧❧❡ ❢r❛❝t✐♦♥❛❧ ❞❡r✐✈❛t✐✈❡ ▲❡t α ∈ (0 , 1) ❛♥❞ t 1 − α f ( t ) ∈ C [0 , y ] f ( t ) ∈ H λ [ y − δ, y ] ( λ > α, δ > 0) ❛♥❞ t 1 − α f ( t ) ≤ y 1 − α f ( y ) ∀ t ∈ (0 , y ) ❚❤❡♥ ( D α 0 x f ) ( y ) ≥ 0 ❛♥❞ ( D α f ( t ) = const · t α − 1 0 x f ) ( y ) = 0 ⇐ ⇒

Pr♦♦❢ s❦❡t❝❤ ❚❛❦✐♥❣ k ( x, t ) = ( x − t ) − α ϕ ( t ) = t α − 1 ❛♥❞ Γ(1 − α ) ✇❡ ❣❡t g ( t ) = t 1 − α f ( t ) ( D f ) ( x ) = ( D α ( D α 0 x f ) ( x ) ❛♥❞ 0 x ϕ ) ( x ) = 0 ❚❤❡r❡❢♦r❡ ( D α 0 x f ) ( y ) = ( D α 0 x ϕg ) ( y ) ≥ g ( y ) ( D α 0 x ϕ ) ( y ) = 0

❆♣♣❧✐❝❛t✐♦♥ ❡①❛♠♣❧❡ Pr♦❜❧❡♠ st❛t❡♠❡♥t � ∂ α ∂y α − ∂ 2 � u ( x, y ) = f ( x, y ) (0 < α < 1) ∂x 2 Ω = { ( x, y ) : z 1 ( y ) < x < z 2 ( y ) , 0 < y < T } u ( z 1 ( y ) , y ) = ϕ 1 ( y ) u ( z 2 ( y ) , y ) = ϕ 2 ( y ) 0 < y < T y → 0 y 1 − α u ( x, y ) = τ ( x ) lim z 1 (0) < x < z 2 (0) y 1 − α u ( x, y ) ∈ C (Ω)

❆♣♣❧✐❝❛t✐♦♥ ❡①❛♠♣❧❡ ❉♦♠❛✐♥ Ω = { ( x, y ) : z 1 ( y ) < x < z 2 ( y ) , 0 < y < T } z 1 ( y ) ր z 2 ( y ) ց z 1 ( y ) < z 2 ( y )