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❈♦r❞✐❛❧ ❱♦❧t❡rr❛ ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥s ♦❢ ✜rst ❦✐♥❞ ●❡♥♥❛❞✐ ❱❛✐♥✐❦❦♦ ❯♥✐✈❡rs✐t② ♦❢ ❚❛rt✉✱ ❊st♦♥✐❛

ˆ t ˆ t � s � � s � 1 1 ( V ϕ u )( t ) = tϕ u ( s ) ds, ( V ϕ,a u )( t ) = tϕ a ( t, s ) u ( s ) ds, 0 < t ≤ T t t 0 0 ❈❖◆❚❊◆❚ ✶✳ ❙♣❛❝❡s C m ❛♥❞ C m,r ⋆ ✷✳ ❈♦r❞✐❛❧ ❱♦❧t❡rr❛ ✐♥t❡❣r❛❧ ♦♣❡r❛t♦rs V ϕ ❛♥❞ V ϕ,a ✿ s♦♠❡ ♠❛♣♣✐♥❣ ♣r♦♣❡rt✐❡s ✸✳ ❘❡❞✉❝t✐♦♥ ♦❢ ❡q✳ V ϕ,a u = f t♦ s❡❝♦♥❞ ❦✐♥❞ ❡q✳ u = V − 1 ϕ V ϕ,b u + V − 1 ϕ f 1 ´ 1 ´ 1 0 x r | ϕ ( x ) | dx < ∞ , 0 x r +1 | ϕ ′ ( x ) | dx < ∞ ✹✳ ■♥✈❡rs✐♦♥ ♦❢ V ϕ ✱ ❝❛s❡ ´ 1 ´ 1 0 x r | ϕ ( x ) | dx < ∞ ✱ 0 x r +1 (1 − x ) | ϕ ′ ( x ) | dx < ∞ ✺✳ ■♥✈❡rs✐♦♥ ♦❢ V ϕ ✱ ❝❛s❡ ✻✳ ❈♦♠♣❛❝t♥❡ss ❝♦♥❞✐t✐♦♥s ❢♦r V ϕ,b ✇✐t❤ b ( t, t ) ≡ 0 ✼✳ ■♥✈❡rs✐♦♥ ♦❢ V ϕ,a ✭t❤❡ ♠❛✐♥ r❡s✉❧ts ♦❢ t❤❡ t❛❧❦✮ ✽✳ ❍♦✇ t♦ s♦❧✈❡ t❤❡ ❡q✉❛t✐♦♥ V ϕ u = f ❄ ✾✳ P♦❧②♥♦♠✐❛❧ ❝♦❧❧♦❝❛t✐♦♥ ❢♦r ❡q✉❛t✐♦♥ V ϕ,a u = f ✶✵✳ ❙♣❧✐♥❡ ❝♦❧❧♦❝❛t✐♦♥ ❢♦r ❡q✉❛t✐♦♥ V ϕ,a u = f ✶

❘❊❋❊❘❊◆❈❊❙ ✭❚❤❡ t❛❧❦ ✐s ❜❛s❡❞ ♦♥ ❬✶✕✺❪ ❛♥❞ ♦♥ ❛ ❝♦♥t✐♥✉❛t✐♦♥ ♦❢ ❬✺❪✮✳ ✶✳ ●✳ ❱❛✐♥✐❦❦♦ ✭✷✵✵✾✮✳ ❈♦r❞✐❛❧ ❱♦❧t❡rr❛ ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥s ✶✳ ◆✉♠❡r✳ ❋✉♥❝t✳ ❆♥❛❧✳ ❖♣t✐♠ ✳✱ ✸✵✱ ✶✶✹✺✕✶✶✼✷✳ ✷✳ ●✳ ❱❛✐♥✐❦❦♦ ✭✷✵✶✵✮✳ ❈♦r❞✐❛❧ ❱♦❧t❡rr❛ ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥s ✷✳ ◆✉♠❡r✳ ❋✉♥❝t✳ ❆♥❛❧✳ ❖♣t✐♠ ✳✱ ✸✶✱ ✶✾✶✕✷✶✾✳ ✸✳ ●✳ ❱❛✐♥✐❦❦♦ ✭✷✵✶✵✮✳ ❙♣❧✐♥❡ ❝♦❧❧♦❝❛t✐♦♥ ❢♦r ❝♦r❞✐❛❧ ❱♦❧t❡rr❛ ✐♥t❡❣r❛❧ ❡q✉❛✲ t✐♦♥s✳ ◆✉♠❡r✳ ❋✉♥❝t✳ ❆♥❛❧✳ ❖♣t✐♠ ✳✱ ✸✶✱ ✸✶✸✕✸✸✽✳ ✹✳ ●✳ ❱❛✐♥✐❦❦♦ ✭✷✵✶✶✮✳ ❙♣❧✐♥❡ ❝♦❧❧♦❝❛t✐♦♥✲✐♥t❡r♣♦❧❛t✐♦♥ ♠❡t❤♦❞ ❢♦r ❧✐♥❡❛r ❛♥❞ ♥♦♥❧✐♥❡❛r ❝♦r❞✐❛❧ ❱♦❧t❡rr❛ ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥s✳ ◆✉♠✳ ❋✉♥❝t✳ ❆♥✳ ❖♣t✐♠ ✳✱ ✸✷✱ ✽✸✕✶✵✾✳ ✺✳ ●✳ ❱❛✐♥✐❦❦♦ ✭s✉❜♠✐tt❡❞✮✳ ❋✐rst ❦✐♥❞ ❝♦r❞✐❛❧ ❱♦❧t❡rr❛ ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥s ✶✳ ✻✳ ❚✳ ❉✐♦❣♦✱ P✳ ▲✐♠❛ ✭✷✵✵✽✮✳ ❙✉♣❡r❝♦♥✈❡r❣❡♥❝❡ ♦❢ ❝♦❧❧♦❝❛t✐♦♥ ♠❡t❤♦❞s ❢♦r ❛ ❝❧❛ss ♦❢ ✇❡❛❦❧② s✐♥❣✳ ❱♦❧t❡rr❛ ✐♥t✳ ❡q✳ ❏✳ ❈♦♠♣✉t✳ ❆♣♣❧✳ ▼❛t❤✳✱ ✷✶✽✱ ✸✵✼✕✸✶✻✳ ✼✳ ❙✳ ●✳ ▼✐❦❤❧✐♥ ✭✶✾✻✶✮✳ ▲✐♥❡❛r ■♥t❡❣r❛❧ ❊q✉❛t✐♦♥s ✳ ❘♦✉t❧❡❞❣❡ P✉❜❧✳ ✽✳ ❑✳ ❊✳ ❆t❦✐♥s♦♥ ✭✶✾✼✹✮✳ ❆♥ ❡①✐st❡♥❝❡ t❤❡♦r❡♠ ❢♦r ❆❜❡❧ ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥s✳ ❙■❆▼ ❏✳ ▼❛t❤✳ ❆♥❛❧ ✳✱ ✺✱ ✼✷✾✕✼✸✻✳ ✷

✶✳ ❙♣❛❝❡s C m ❛♥❞ C m,r ⋆ ❋♦r m ≥ 0 ✱ C m = C m [0 , T ] ✐s t❤❡ st❛♥❞❛r❞ s♣❛❝❡ ✇✐t❤ t❤❡ ♥♦r♠ 0 ≤ t ≤ T | u ( k ) ( t ) | . � u � C m = max 0 ≤ k ≤ m max ❋♦r m ≥ 0 ✱ r ∈ R ✱ s♣❛❝❡ C m,r = C m,r (0 , T ] ❝♦♥s✐sts ♦❢ ❢✉♥❝t✐♦♥s u ∈ C m (0 , T ] ⋆ ⋆ s✉❝❤ t❤❛t ✜♥✐t❡ ❧✐♠✐ts lim t → 0 t k − r u ( k ) ( t ) ✱ k = 0 , ..., m ✱ ❡①✐st❀ t k − r | u ( k ) ( t ) | . � u � C m,r := max 0 ≤ k ≤ m sup ⋆ 0 <t ≤ T ■t ❤♦❧❞s C m,r = t r C m, 0 ✳ ■♥ ♣❛rt✐❝✉❧❛r✱ ❢♦r u ∈ C m ✱ t❤❡ ❢✉♥❝t✐♦♥ t r u ( t ) ❜❡❧♦♥❣s ⋆ ⋆ t♦ C m,r ❜✉t s✉❝❤ ❢✉♥❝t✐♦♥s ❞♦ ♥♦t ❝♦✈❡r ✇❤♦❧❡ C m,r ✳ ⋆ ⋆ ■t ❤♦❧❞s C = C 0 = C 0 , 0 ⋆ ✱ ❛♥❞ C m ⊂ C m, 0 ✱ C m = C m,m ⊕ P m − 1 ❢♦r m ≥ 1 ✳ ⋆ ❈❧❡❛r❧②✱ C m ′ ,r ′ ❢♦r m ′ ≥ m ≥ 0 ✱ r ′ ≥ r ✳ ⊂ C m,r ⋆ ⋆ ✸

✷✳ ❈♦r❞✐❛❧ ❱♦❧t❡rr❛ ✐♥t❡❣r❛❧ ♦♣❡r❛t♦rs ❋♦r ✏❝♦r❡✑ ϕ ∈ L 1 (0 , 1) , t❤❡ ❝♦r❞✐❛❧ ❱♦❧t❡rr❛ ♦♣❡r❛t♦r V ϕ ∈ L ( C ) ✐s ❞❡✜♥❡❞ ❬✶❪ ❛s ˆ t ˆ 1 � s � 1 ( V ϕ u )( t ) = tϕ u ( s ) ds = ϕ ( x ) u ( tx ) dx, 0 ≤ t ≤ T, u ∈ C. t 0 0 ´ 1 0 | ϕ ( x ) | dx ✳ ❋♦r u ∈ C m ✱ ❈❧❡❛r❧②✱ ✐t ❤♦❧❞s t❤❛t � V ϕ � L ( C ) ≤ ˆ 1 ( V ϕ u ) ( k ) ( t ) = ϕ ( x ) x k u ( k ) ( tx ) dx, k = 0 , ..., m, 0 ´ 1 ❛♥❞ ✇❡ ♦❜t❛✐♥ t❤❛t V ϕ ∈ L ( C m ) ✱ � V ϕ � L ( C m ) ≤ 0 | ϕ ( x ) | dx ✳ ❋♦r t❤❡ ♦♣❡r❛t♦r ˆ t � s � 1 ( V ϕ,a u )( t ) = tϕ a ( t, s ) u ( s ) ds t 0 � � ✇✐t❤ t❤❡ ❝♦❡✣❝✐❡♥t ❢✉♥❝t✐♦♥ a ∈ C m ( △ T ) ✱ △ T = ( t, s ) : 0 ≤ s ≤ t ≤ T ✱ ✇❡ st✐❧❧ ❤❛✈❡ V ϕ,a ∈ L ( C m ) ✳ ✹

▲❡ss ♦❜✈✐♦✉s ❢❛❝ts ❬✶✱✷❪ ❝♦♥❝❡r♥ t❤❡ s♣❡❝tr❛✿ σ L ( C m ) ( V ϕ ) = { 0 } ∪ { � ϕ ( k ) : k = 0 , ..., m − 1 } ∪ { � ϕ ( λ ) : λ ∈ C , ❘❡ λ ≥ m } , σ L ( C m ) ( V ϕ,a ) = a (0 , 0) σ L ( C m ) ( V ϕ ) , ✇❤❡r❡ ˆ 1 ϕ ( x ) x λ dx, λ ∈ C , ❘❡ λ ≥ 0 , ϕ ( λ ) = � 0 ✐s t❤❡ ✭s❤✐❢t❡❞✮ ▼❡❧❧✐♥ tr❛♥s❢♦r♠ ♦❢ ϕ ✳ ❍❡♥❝❡ V ϕ ∈ L ( C m ) ❛♥❞ V ϕ,a ∈ L ( C m ) ❛r❡ ♥♦♥❝♦♠♣❛❝t ❢♦r ϕ � = 0 ✱ a (0 , 0) � = 0 ✳ ■t ✐s ❛♥ ❡❧❡♠❡♥t❛r② ❢❛❝t t❤❛t ϕ ( λ ) v λ ✇❤❡r❡ v λ ( t ) = t λ , 0 < t ≤ T, ❘❡ λ ≥ 0 . V ϕ v λ = � ■♥ ♣❛rt✐❝✉❧❛r✱ V ϕ ♠❛♣s ❛ ♣♦❧②♥♦♠✐❛❧ ✐♥t♦ ❛ ♣♦❧②♥♦♠✐❛❧ ♦❢ t❤❡ s❛♠❡ ♦r ❧❡ss ❞❡❣r❡❡✳ ✺

❚♦ tr❡❛t t❤❡ ♦♣❡r❛t♦rs V ϕ ❛♥❞ V ϕ,a ✐♥ s♣❛❝❡s C m,r ✱ ✇❡ ❛ss✉♠❡✱ ✐♥st❡❛❞ ♦❢ ϕ ∈ ⋆ ´ 1 0 x r | ϕ ( x ) | dx < ∞ . ❋♦r u ∈ C m,r L 1 (0 , 1) ✱ t❤❛t ✱ 0 < t ≤ T ✇❡ ❤❛✈❡ ⋆ ˆ 1 ˆ 1 ϕ ( x ) u ( tx ) dx, ( V ϕ u ) ( k ) ( t ) = ϕ ( x ) x k u ( k ) ( tx ) dx, ( V ϕ u )( t ) = 0 0 ˆ 1 t k − r ( V ϕ u ) ( k ) ( t ) = ϕ ( x ) x r ( tx ) k − r u ( k ) ( tx ) dx, 0 ˆ 1 t → 0 t k − r ( V ϕ u ) ( k ) ( t ) = ϕ ( x ) x r dx lim t → 0 ( t k − r u ( k ) ( t )) , k = 0 , ..., m. lim 0 ❲❡ ♦❜s❡r✈❡ t❤❛t V ϕ ♠❛♣s C m,r ✐♥t♦ ✐ts❡❧❢✳ ⋆ ´ 1 0 x r | ϕ ( x ) | dx < ∞ ❢♦r ❛♥ r ∈ R ✳ ❚❤❡♥ ❚❤❡♦r❡♠ ✶✳ ▲❡t a ∈ C m ( △ T ) ❛♥❞ ˆ 1 x r | ϕ ( x ) | dx, V ϕ , V ϕ,a ∈ L ( C m,r ) , � V ϕ � L ( C m,r ) ≤ ⋆ ⋆ 0 σ L ( C m,r ) ( V ϕ ) = { 0 }∪{ � ϕ ( λ ) : λ ∈ C , ❘❡ λ ≥ r } , σ L ( C m,r ) ( V ϕ,a ) = a (0 , 0) σ L ( C m,r ) ( V ϕ ) . ⋆ ⋆ ⋆ ✻

✸✳ ❘❡❞✉❝t✐♦♥ ♦❢ ❡q✳ V ϕ,a u = f t♦ ❛ s❡❝♦♥❞ ❦✐♥❞ ❡q✳ ❲❡ st✉❞② t❤❡ ✉♥✐q✉❡ s♦❧✈❛❜✐❧✐t② ♦❢ t❤❡ ✜rst ❦✐♥❞ ❝♦r❞✐❛❧ ❱♦❧t❡rr❛ ❡q✉❛t✐♦♥ ˆ t � s � 1 V ϕ,a u = f, ♦r tϕ a ( t, s ) u ( s ) ds = f ( t ) , 0 < t ≤ T, ✭✶✮ t 0 ❛♥❞ ❞✐s❝✉ss ♠❡t❤♦❞s ♦❢ s♦❧✈✐♥❣✱ ✐♥ ♣❛rt✐❝✉❧❛r✱ ♣♦❧②♥♦♠✐❛❧ ❛♥❞ s♣❧✐♥❡ ❝♦❧❧♦❝❛t✐♦♥✳ ❊①❛♠♣❧❡s✳ ✶✳ ❋♦r ϕ ( x ) = x − α ✱ α ∈ R ✱ ❡q✉❛t✐♦♥ ✭✶✮ t❛❦❡s t❤❡ ❢♦r♠ ♦❢ ❉✐♦❣♦✬s ❡q✉❛t✐♦♥ t α − 1 ´ t ´ t 0 s − α a ( t, s ) u ( s ) ds = f ( t ) ✱ ♦r 0 a ( t, s ) u ( s ) ds = f ( t ) ✳ ❈❢✳ ❬✻❪✳ ✷✳ ❋♦r ϕ ( x ) = x − α (1 − x γ ) − ν ✱ α ∈ R ✱ γ > 0 ✱ ν ∈ (0 , 1) ✱ ❡q✉❛t✐♦♥ ✭✶✮ t❛❦❡s t❤❡ ❢♦r♠ ♦❢ t❤❡ ❆❜❡❧ ❡q✉❛t✐♦♥ ❬✼✱✽❪ ˆ t ˆ t t α + γν − 1 ( t γ − s γ ) − ν s − α a ( t, s ) u ( s ) ds = f ( t ) , ♦r ( t γ − s γ ) − ν a ( t, s ) u ( s ) ds = f ( t ) . 0 0 ✸✳ ϕ ( x ) = x − α g ( x ) ✱ ϕ ( x ) = x − α (1 − x γ ) − ν g ( x ) ✇✐t❤ g ∈ C [0 , 1] ✱ g ′ ∈ L 1 (0 , 1) ❜♦✉♥❞❡❞ ✐♥ ❛ ✈✐❝✐♥✐t② ♦❢ x = 0 ❛♥❞ ✐♥ ❛ ✈✐❝✐♥✐t② ♦❢ x = 1 ✳ ✼