Electronic Journal of Qualitative Theory of Differential Equations 2009, No. 25, 1-18; http://www.math.u-szeged.hu/ejqtde/

(1)

Electronic Journal of Qualitative Theory of Differential Equations 2009, No. 25, 1-18; http://www.math.u-szeged.hu/ejqtde/

! " # "

$%&'(&) *+,+

-./0.1 23/454647 .8 -63505

9:; <

7=>507/

:

3? @.3/4=6045.3A B

: 06

; 4C .8

D5 EF7= -

: 4F7G

:

450/A H6//5

:

&IJKL%MNOPQRPSTUNVOWQX

$%&'(&)K Y+*+

-./0.1 Z70F350

:; [

35>7=/54C .8

@.GG6350

: 45.3/

: 3?

238

.=G

:

450/AH6//5

:

&IJKL%M\V]^N_N]XQT`UNVOW

a RU

b& cK''d (ef gh&cf'K% KiK%dgLg (j (i& c%Kgg (j Lif&k'K% (hI

&'Kf('g Kgg(cLKf&l mLfn j'KcfL(iK% ('l&' lLo&'&ifLK% &peKfL(ig fnKf K'Lg&g

Li J&cnKiLcg+ b& &gfKq%Lgn K c(ii&cfL(i q&fm&&i fn& &Lk&i)K%e&g (j fn&g&

(h&'Kf('gKilfn& r&'(g(jsLffKkIt&u&'fdh&jeicfL(ig+ b& kL)&gevcL&if

c(ilLfL(igj('c(Jh%&f&i(ig&%jKl w(Lifi&ggKil c(Jh%&f&i&ggfn&gdgf&Jg(j

fn& &Lk&i)K%e&g+

x y

zi {|} fn& gh&cf'K% KiK%dgLg (j (h&'Kf('g(j fn& j('J

A ^[α,β] _γ u(x) = c α

Z x 0

(x − t) ^α ¹ ⁻ ¹ u(t)dt + c β,γ

Z 1 0

x ¹ ^β ⁻ ¹ (1 − t) ^γ ¹ ⁻ ¹ u(t)dt.

mKg cK''L&l (ef {| }+ Y&'&

α, β, γ, c α , c β,γ

K'& '&K% ieJq&'g~Kil

α, β, γ

^K'&

h(gLfL)&+ +s+ eq'&&) c(igLl&'&l gLJL%K' (h&'Kf('g Li fn& hKh&' { }+

*n&g& (h&'Kf('g K'Lg& Li fn& gfeld (j q(eilK'd )K%e& h'(q%&Jg j(' lLo&'I

&ifLK% &peKfL(ig(jj'KcfL(iK%('l&' g&& {} Kil'&j&'&ic&gfn&'&Li~Li mnLcn

c(''&gh(ilLik '&&i jeicfL(ig K'& c(igf'ecf&l+

zi hK'fLce%K' Lf mKg gn(mi Li {| } ~fnKf fn&(h&'Kf('

A ^ρ u(x) = A ^[ρ,ρ] _ρ u(x) = 1 Γ(ρ ⁻ ¹ )

Z x 0

(x−t) ¹ ^ρ ⁻ ¹ u(x)dt− 1 Γ(ρ ⁻ ¹ )

Z 1 0

x ¹ ^ρ ⁻ ¹ (1−t) ¹ ^ρ ⁻ ¹ u(t)dt

Lg K%J(gf i(iIg&%jIc(iwekKf& g&& {} j('

ρ > 1

^~ ^m^nL^%& ^j('

0 < ρ < 1

^~ ^fn&

&'i&% (j fn& (h&'Kf('

A ^ρ

^L^gⁱ^(iI^i&kK^fL^)& ^Kil ^fn^&^'^&ln^(%J ^gh^&cf(' ^(j ^fn&

(h&'Kf('

A ^ρ

^c(Li^cL^l&g^m^Lfn^fn^&^g&f^(j^'(⁽^fg⁽^j^fn^&^mn^(%&^jei^cfL(ⁱ^(j^s^LffKk^I

t&u&' fdh&

E ρ (λ; ρ ⁻ ¹ ) = X ∞ k=0

λ ^k

Γ(ρ ⁻ ¹ + kρ ⁻ ¹ ) .

(2)

A ^ρ

^K^'& ^c(^J^h^%& ^j(^'

ρ > 1,

^mⁿ^L%& ^j('

0 < ρ < 1,

^fn^& ⁽^h&'Kf(^'

A ^ρ

^nK^g ^'&K^%

&Lk&iieJq&'g Li jKcf~ Lj

1 2 < ρ < 1

^fn& ^g&f ^(j ^'&K% ^&Lk^&i^ieJ^q^&'g ^L^g^iLf&~

L+&+ K%% fn& r&'(g (j fn& jeicfL(i

E ρ (λ; ρ ⁻ ¹ )

^K'& ^c(^J^h%& ^j(^'

ρ > 1,

^m^nL%&

j('

0 < ρ < 1

^fn& ^je^icfL⁽ⁱ

E ρ (λ; ρ ⁻ ¹ )

^nK^g ^'&K^% ^r&'(^g+ ^*^nLg ^h'()&g ^fn&

KggdJhfL(i Kq(ef fn& & Lgf&ic& (j K '&K% r&'(g (j fn& jeicfL(i

E ρ (λ; ρ ⁻ ¹ )

j('

1 2 < ρ < 1,

^Kg ^gfKf&l ^Li ^fn& ^J⁽ⁱ⁽^k'K^hn ^{ ^hk⁺ ^}⁺

*n&kL)&ihKh&'Lgl&)(f&lKgm&%%f(gfeldfn&q(eilK'd)K%e&h'(q%&Jg

j('fn&lLo&'&ifLK%&peKfL(ig(jj'KcfL(iK%('l&'Kilfn&Kcc(JhKidLikfn&J

Lif&k'Kf&l (h&'Kf('g(jfn& j('J

A ^[α;β] γ .

zi ('l&' f( gfKf& fn& h'(q%&Jg Li c(ic&'i m& Jegf J&ifL(i g(J& c(iI

c&hfg j'(J j'KcfL(iK% cK%ce%eg+

t&f

f (x) ∈ L 1 (0, 1).

^*^n&i^~^fn& ^jei^cfL(ⁱ

d ⁻ ^α

dx ⁻ ^α f (x) ≡ 1 Γ(α)

Z x 0

(x − t) ^α ⁻ ¹ f (t)dt ∈ L 1 (0, 1)

Lg cK%%&lfn&j'KcfL(iK%Lif&k'K%(j('l&'

α > 0

^mLfn ^gfK'fLik ^h^(L^if

x = 0

^{}⁺

$il fn&jeicfL(i

d ⁻ ^α

d(1 − x) ⁻ ^α f (x) ≡ 1 Γ(α)

Z 1 x

(t − x) ^α ⁻ ¹ f (t)dt ∈ L 1 (0, 1)

Lg cK%%&l fn& j'KcfL(iK% Lif&k'K% (j ('l&'

α > 0

^mLfn ^&ilLi^k ^h(^Lif

x = 1

{}+ Y&'&

Γ(α)

^L^g ^e%&'^g ^k^KJ^J^KIjei^cfL(ⁱ⁺ ^zf ^Lg ^c%&K^' ^fnKf ^mⁿ^&i

α = 0,

^m&

Ll&ifLjd q(fn fn& j'KcfL(iK% Lif&k'K%g mLfn fn& jeicfL(i

f (x).

^$^g ^m& ^i(m

{}~fn&jeicfL(i

g(x) ∈ L 1 (0, 1)

Lg cK%%&l fn& j'KcfL(iK% l&'L)KfL)&(jfn& jeicfL(i

f (x) ∈ L 1 (0, 1)

(j ('l&'

α > 0

^m^Lfn ^gfK'fLi^k ^h(^Lif

x = 0,

^Lj

f (x) = d ⁻ ^α

dx ⁻ ^α g(x).

*n&i l&i(fLik

g(x) = d ^α dx ^α f (x),

m&gnK%% J&Ki Li fn& jefe'&~qd

d ^α

dx ^α ,

(3)

α < 0 α > 0.

^*^n& ^j'K^cfL(^iK^% l&'L)KfL)&

d ^α d(1 − x) ^α

(jfn&jeicfL(i

f (x) ∈ L 1 (0, 1)

⁽^j⁽^'l&'

α > 0,

^m^Lfn ^fn&^&il^Lik ^h^(Lif

x = 1

Lg l&i&l Li K gLJL%K' mKd+

t&f

{γ k } 0 n

^q&^K^id^g&f⁽^j^'&K%^ie^J^q&'g^~^gKfLgjdLik^fn^&^c(i^lLfL(ⁱ

0 < γ j ≤ 1, (j = 0, 1, .., n).

^b& l&i(f&

σ k = X k j=0

; µ k = σ k + 1 = X k j=0

γ j , (k = 0, 1, ..., n)

Kil m& KggeJ&fnKf

1 ρ =

X n j=0

γ j − 1 = σ n = µ n − 1 > 0.

(%%(mLik {}~m& c(igLl&' fn& lLo&'&ifLK% (h&'Kf('g~

D ^(σ ⁰ ⁾ f (x) ≡ d ⁻ ⁽¹ ⁻ ^γ ⁰ ⁾ dx ⁻ ⁽¹ ⁻ ^γ ⁰ ⁾ f (x),

mnLcn K'&~k&i&'K%%d~(jj'KcfL(iK% ('l&'

D ^(σ ¹ ⁾ f (x) ≡ d ⁻ ⁽¹ ⁻ ^γ ¹ ⁾ dx ⁻ ⁽¹ ⁻ ^γ ¹ ⁾

d ^γ ⁰ dx ^γ ⁰ f (x), D ^(σ ⁿ ⁾ f (x) ≡ d ⁻ ⁽¹ ⁻ ^γ ⁿ ⁾

dx ⁻ ⁽¹ ⁻ ^γ ⁿ ⁾ d ^γ ⁿ⁻ ¹ dx ^γ ⁿ⁻¹ ... d ^γ ⁰

dx ^γ ⁰ .

Y&'&m&i(f&~ fnKf Lj

γ 0 = γ 1 = ... = γ n = 1

^fn&i ⁽^q)L^(e^g%d

D ^(σ ^k ⁾ f (x) = f ^(k) (x), (k = 0, 1, 2, ..., n).

*( l&J(igf'Kf& fn& qKgLc Ll&Kg~ m& gfK'f qd Li)&gfLkKfLik J(gf gLJh%&

cKg&g+ *neg~m&hef~

γ 3 = γ 4 = ... = γ n = 0,

^Ki^l ^c(^igLl&'^K ^h^'(^q%&J ^m^nLcn

LgKiKiK%(ke&(jfn&m&%%Ii(mih'(q%&J (j,f('JItL(e)L%%&+ *nLgh'(q%&J

mnLcn m& cK%% h'(q%&J $~Kg Li { }~J&Kigfn&j(%%(mLikM

zi c%Kgg

L 2 (0, 1)(orL 1 (0, 1))

il K i(if'L)LK% g(%efL(i (j fn& &peKfL(i

D ^(σ ² ⁾ y − [λ + q(x)]y = 0

^|

(4)

x ∈ (0, 1],

D ^(σ ⁰ ⁾ y

x=0 cos α + D ^(σ ¹ ⁾ y

x=0 sin α = 0 D ^(σ ⁰ ⁾ y

x=1 cos β + D ^(σ ¹ ⁾ y

x=1 sin β = 0,

mn&'&

λ, α, β; (Imα = Imβ = 0),

^K'& ^K^'qLf'K'd ^hK^'KJ&f&'g~

q(x) ∈ L 2 (0, 1).

^s^+s⁺^rn^'qK^gn^cnw^Ki ^m^'Lf&g ^{ ^}^M

”

^pe^&gfL(^ig^(j^c(Jh%&f&i&gg (jgdgI f&J (j fn& &Lk&ijeicfL(ig (j h'(q%&J $~ (' K J('& l&%LcKf& pe&gfL(i~

mn&fn&' fn&g& jeicfL(ig j('J K qKgLg Li

L 2 (0, 1),

^K'& ^ei^l(e^qf%d Lif&'&gfI Lik+ ef fn&L' g(%efL(ig~ KhhK'&if%d~ K'& jKc&l mLfn gLkiLcKif KiK%dfLcK%

lLvce%fL&g+

”

zi {} g&&K%g( {}~LfnKgq&&i h'()&lfnKffn&gdgf&J (j&Lk&ijeicfL(ig

(j h'(q%&J $ Lg c(Jh%&f& Li

L 2 (0, 1),

^mn^&i

q(x) ≡ 0.

tKf&'~ s+s+sK%KJel Kil nLg lLgcLh%&g {|} I {||} K%g( nK)&&gfKq%Lgn&l

fn& c(Jh%&f&i&gg (j gdgf&J (j fn& &Lk&ijeicfL(ig (j gLJL%K' h'(q%&Jg fn&

cKg& (j mn&i

q(x)

^Lg ^Kⁱ ^Ki^K%dfLcK% ^jei^cfL(i⁺

zifnLghKh&'m&kL)&fn&c(Jh%&f&g(%efL(i(jfn&h'(q%&J (i c(Jh%&f&I

i&gg(jfn& gdgf&J (j &Lk&ijeicfL(ig(jh'(q%&J |Ih'()LlLik

q(x) > 0

⁺

b&gnK%%c(igLl&')K'L(eg)K'LKifg(j&peKfL(i |+ $f

γ 0 = γ 1 = 1,

^&pe^KfL(i

|Lg f'Kigj('J&l Lif( fn& &peKfL(i

1 Γ(1 − γ 2 )

Z x 0

u ⁰⁰ (t)

(x − t) ^γ ² dt − (λ + q(x))u(x) = 0,

mnLcnLgcK%%&lKj'KcfL(iK%(gcL%%KfL(iK%&peKfL(i {}~Kilfn&(h&'Kf('

D ^(σ ² ⁾

Lg cK%%&l I fn& (h&'Kf(' (jj'KcfL(iK% lLo&'&ifLKfL(i Li Khef( g&ic& {}+ $f

γ 0 = γ 2 = 1

^~^fn& &peKfL(i |Lg f'Kigj('J&l Lif( fn& &peKfL(i

1 Γ(1 − γ 1 )

d dx

Z x 0

u ⁰ (t)

(x − t) ^γ ¹ dt − (λ + q(x))u(x) = 0.

*n& &peKfL(i Kg fn& J(l&%%Lik &peKfL(i (j j'KcfL(iK% ('l&'

1 < σ < 2

nKg q&&i Li)&gfLkKf&l g&& {} Kil '&j&'&ic&g fn&'&Li+

x74

q(x)

⁷ ^4F⁷ ^F^:;⁸^;^5G^547? ⁸⁶³^045.³ ^ZF⁷³ ^4F⁷ ^/C/4^7G ^.8

4F7 75 E73863045.3/ .8

9

=.

;

7G

1 Γ(1 − γ 1 )

d dx

Z x 0

u ⁰ (t)

(x − t) ^γ ¹ dt − (q(x) + λ)u(t)dt = 0, u(0) = 0, u(1) = 0

^5/ ^0.^G⁹^;⁷⁴⁷ ⁵³

L 2 (0, 1).

*n& h'((j(j*n&('&J | Lg qKg&l (i t&JJK |+

(5)

xZF7 .

9 7=

:

4.=

S,

⁵³^?607? ^C ^4F⁷ ^?5 ⁷⁼⁷³⁴⁵^:^; ⁷⁹^=7//5.3

l(u) = 1 Γ(1 − γ 1 )

d dx

Z x 0

u ⁰ (t)

(x − t) ^γ ¹ dt − q(x)u(x),

: 3?

.63?

:

=C 0.3?545.3/

u(0) = 0, u(1) = 0,

F

: /

:

3 53>7=/7 .

9 7=

: 4.=

x

t&JJK | mL%% q& h'()&l Lj m& cKi gn(m~ fnKf

h'(q%&J $~Kf

q(x) > 0

ⁿ^Kg ^fn&^e^iLpe&^f'L^)L^K% ^g(^%efL(i

u(0) = 0

^+*( ^h^'()&

fn& %Kgf gfKf&J&if m& l( fn& j(%%(mLik+ b& Je%fLh%d q(fn gLl&g (j fn&

&peKfL(i

1 Γ(1 − γ 1 )

d dx

Z x 0

u ⁰ (t)

(x − t) ^γ ¹ dt = q(x)u(x)

qd

(x − t) ^γ ¹ ,

^Ki^l ^Lif&k^'Kf&^j'(J

0

^f(

x.

^b& ^(qfK^Li

1 Γ(1 − γ 1 )

Z x 0

(x − t) ^γ ¹ d dt

Z t 0

u ⁰ (ξ)

(t − ξ) ^γ ¹ dξdt = Z x 0

(x − t) ^γ ¹ q(t)u(t)dt.

t&fg cK%ce%Kf& fn& Lif&k'K%

Iu = 1

Γ(1 − γ 1 ) Z x 0

(x − t) ^γ ¹ ( d dt

Z t 0

u ⁰ (ξ)

(t − ξ) ^γ ¹ dξ)dt.

b& nK)&

Iu = 1

Γ(1 − γ 1 ) Z x 0

(x − t) ^γ ¹ d(

Z t 0

u ⁰ (ξ) (t − ξ) ^γ ¹ dξ)

= 1

Γ(1 − γ 1 ) (x − t) ^γ ¹ Z t 0

u ⁰ (ξ) (t − ξ) ^γ ¹ dξ

x

0

− 1

Γ(1 − γ 1 ) Z x 0

[(x−t) ^γ ¹ ] ⁰ Z t

0

u ⁰ (ξ) (t − ξ) ^γ ¹ dξdt

= −x ^γ ¹ D ^(σ ¹ ⁾ u

t=0 − γ 1

Γ(1 − γ 1 ) Z x

0

(x − t) ^γ ¹ ⁻ ¹ ( Z t 0

u ⁰ (ξ) (t − ξ) ^γ ¹ dξ)dt

= −x ^γ ¹ D ^(σ ¹ ⁾ u

t=0 − γ 1

Γ(1 − γ 1 ) Z x 0

u ⁰ (ξ)(

Z x ξ

(x − t) ^γ ¹ ⁻ ¹ (t − ξ) ⁻ ^γ ¹ dt)dξ

(6)

= x ^γ ¹ D ^(σ ¹ ⁾ u

t=0 − Γ(γ 1 )γ 1

Γ(1) Z x 0

u ⁰ (ξ)dξ.

*neg

Iu = −x ^γ ¹ D ^(σ ¹ ⁾ u

t=0 − c[u(x) − u(0)].

*n&'&j('&

cu(x) = x ^γ ¹ D ^(σ ¹ ⁾ u

t=0 + Z x 0

(x − 1) ^γ ¹ q(t)u(t)dt,

c = Γ(1 + γ 1 ).

^Y^&i^c&^~^fn& k&i&'K% g(%efL(i (j fn& &peKfL(i

1 Γ(1 − γ 1 )

d dx

Z x 0

u ⁰ (t)

(t − ξ) ^γ ¹ dξ − q(x)u = 0,

gKfLgjdLik fn& c(ilLfL(i

u(0) = 0,

^nK^g^fn^&^j('J

u(x) = cx ^γ ¹ + Z x 0

(x − t) ^γ ¹ q(t)u(t)dt = cx ^γ + Au(x).

*n& %Kgf &peKfL(i %&Klg~f(M

u(x) = c(I − A) ⁻ ¹ x ^γ ¹ = c(x ^γ ¹ + Ax ^γ ¹ + A ² x ^γ ¹ + ... + A ⁿ x ^γ ¹ + ...).

,Lic&

Ax ^γ ¹ = Z x 0

(x − t) ^γ ¹ q(t)t ^γ ¹ dt,

fneg

A ² x ^γ ¹ = A(Ax ^γ ¹ ) = Z x 0

(x − t) ^γ ¹ q(t)(

Z t 0

(t − ξ) ^γ ¹ q(ξ)ξ ^γ ¹ dξ)dt.

(ig&pe&if%d~fn&&'i&% (jfn& (h&'Kf('

A ²

^Lg ^&peK^%^f(

K 2 (t; s) = Z t

s

(t − τ) ^γ ¹ q(τ)(τ − s) ^γ ¹ q(s)dτ.

*n& &'i&% (j fn&(h&'Kf('

A ⁿ

^Lg ^l&i^&l ^qd ^fn& ^i(mⁱ ^j('J^e^%K

K n (t; s) = Z t

s

K(t; s)K n − 1 (τ; s)dτ.

(7)

A, A ² , ..., A ⁿ

K'&h(gLfL)&~fn& jeicfL(i

u(x) = c(x ^γ ¹ + Ax ^γ ¹ + A ² x ^γ ¹ + ... + A ⁿ x ^γ ¹ + ...),

Kf

x = 1,

^cKⁱⁱ⁽^f^q^& ^&pe^K% ^f( ^~^m^nLcn ^h'()&g ^*^n&(^'&J ⁺ ^(m ^m^&^k^L)&^fn&

c(Jh%&f& h'((j(j*n&('&J |+

x

*n&h'((j(j*n&('&J |~LgqKg&l(i + +tLlgLLg

i(mi fn&('&J

C

{|}+ $cc('lLik f( + +tLlgLLg {|} fn&('&J~ Lf Lg

&i(ekn f( &gfKq%Lgn~fnKf fn& (h&'Kf('

S ⁻ ¹

^Lg ^lLggLh^KfL)&^K^il ^f'K^c& ^c%K^gg⁺

t&f

u(t)

^q& ^K^id^jeicfL(i ^j'(^J ^fn& ^l(^J^K^Li ^(j

D(S) ⊂ L 2 (0, 1).

^*n^&i

Re(Su, u) = 1 Γ(1 − γ 1 ) Re

Z 1 0

( d dx

Z x 0

u ⁰ (t)

(x − t) ^γ ¹ dt + q(x))u(x)dx

= 1

Γ(1 − γ) Re Z 1 0

( d dx

Z x 0

u ⁰ (t)

(x − t) ^γ ¹ ) dtu(x)dx+ 1 Γ(1 − γ 1 ) Re

Z 1 0

q(x)u(x)u(x)dx.

,Lic&

1 Γ(1 − γ 1 ) Re

Z 1 0

d dx

Z x 0

u ⁰ (t)

(x − t) ^γ dt)u(x)dx

= 1

Γ(1 − γ 1 ) Re Z 1 0

u(x)dx Z x 0

u ⁰ (ξ) (x − ξ) ^γ ¹ dξ

= 1

Γ(1 − γ 1 ) Re[u(x) Z x 0

u ⁰ (ξ) (x − ξ) ^γ ¹ dξ

1

0

− Z 1 0

{ Z t 0

u ⁰ (ξ)

(t − ξ) ^γ ¹ dξ}u ⁰ (ξ)dx]

= − 1

Γ(1 − γ 1 ) Re



 Z x 0

u ⁰ (ξ)

(x − ξ) ^γ ¹ dξ, u ⁰ (x)



 .

(8)

(A ^ν f ; f ), (0 ≤ ν ≤ 1),

0 ≤ arg λ ≤ πν

^x

*n&('&J (jsKfgK&)I (%KifgfKf&g~fnKffn&)K%e&(jfn&j('J

( _dx ^d ^−α −α f, f )

%L&g Li fn& Kik%&

|argz| < ^πα ₂ .

^,(^~^L^f ^j(^%%(m^g ^j'(J ^fn& fn&('&J (j sKfgK&)I (%KiffnKf fn& (h&'Kf('

T u =

 



1 Γ(1 − γ 1 )

d dx

R x 0

u ⁰ (t) (x − t) ^γ ¹ dt u(0) = 0, u(1) = 0

Lgi(f(i%d lLggLhKfL)&~qef K%g( g&cf('LK%fn&)K%e&g(jfn&j('J

(Su, u)

%L&g Li fn& Kik%&

|argz| ≤ ^πγ ₂ ¹ .

^(m ^fn^& ^lL^ggL^hKfL)Lfd ^(j ^fn^& ⁽^h^&'K^f('

S

LJh%L&g fn& lLggLhKfL)Lfd (j fn& (h&'Kf('

S ⁻ ¹ .

^& ^f ^m^& &gfKq%Lgn~ fnKf fn&

(h&'Kf('

S ⁻ ¹

^Lg ^f'K^c& ^c%K^gg+ ^t&f

λ 1 , λ 2 , ..., λ n , ..

^K^il

µ 1 , µ 2 , ..., µ n ...

^q^& ^fn&

&Lk&i)K%e&g (j fn& (h&'Kf('g

S

^Ki^l

T

^c(^''&gh⁽ⁱ^lLik^%d~ ^ieJ^&'^Kf&l ^Li ^fn&

i(iIl&c'&KgLik ('l&' (j J(le%eg+ *n&i {} m& i(m~ fnKf

|λ n − µ n | ≤

||q(x)||.

^(m^~^fKLik ^Lif( ^Kcc(^eif^fnK^f^fn& ^&L^k&iieJ^q^&'g^(j ^fn^&⁽^h&'Kf(^'

T

c(LicLl&~mLfn r&'(g (j fn& jeicfL(i

E ρ (λ; ρ ⁻ ¹ )([1], [8])

^K^il ^fn& ^KgdJ^hf(fLc

(j r&'(g(jfn&sLffKkIt&u&'jeicfL(i

E ρ (z; µ) = X ∞ k=0

z ^k

Γ(µ + kρ ⁻ ¹ ) , ρ > 0

Lgm&%%gfelL&l g&&{|}h+|~m&(qfKLi~fnKffn&(h&'Kf('

S ⁻ ¹

^nK^g^K^iLf&

gh&cf'K% f'Kc&~L+&+ fn& (h&'Kf('

S ⁻ ¹

^L^g^f'K^c& ^c%Kgg⁺

,Lic& fn& (h&'Kf('

S ⁻ ¹

^Lg ^lL^ggLh^KfL)& ^K^il ^Lg ^f'Kc& ^c%K^gg ^K^cc(^'lLik ^f(

+ +tLlgLLgfn&('&J~fn&(h&'Kf('

S ⁻ ¹

ⁿ^Kg ^fn&^c(Jh%&f&gdgf&J (j &Lk&iI jeicfL(ig Li

L 2 (0; 1).

x

;;

75 E73>

:;

67/ .8 4F7.

9 7=

: 4.=

S

^;⁵⁷⁵³ ^4F⁷^:^3E^;⁷

|argz| ≤

πγ 1

2 ;

⁵³ ⁹^:⁼⁴⁵⁰⁶^;:⁼ ^:;; ^7=./ ^.8 ^4F⁷⁸⁶^3045.3

E ρ (z; ρ ⁻ ¹ )

^:;^/^. ^;⁵⁷ ⁵³ ^4F⁷ ^:^3E^;⁷

|argz| ≤ ^πγ ₂ ¹ .

zi K gLJL%K' mKd m& cKi Li)&gfLkKf& Kid )K'LKifg (j h'(q%&J $+ b&

gn(m n(m f( l( Lf+ b& c(igLl&' fn& L'Lcn%&f h'(q%&J j(' fn& j'KcfL(iK%

(gcL%%KfL(iK%&peKfL(i~L+&+ h'(q%&J

( A f 1 ) 1

Γ(1 − γ 2 ) Z x 0

u ⁰⁰ (t)

(x − t) ^γ ² dt − (q(x) + λ)u = 0 u(0) = 0, u(1) = 0.

1 ⁰

^ZF⁷ ^/C/4^7G ^.8 ^{75 E738}⁶³^045.3^/ ^.8 ⁹^=.^;^7G

( A) e

^5/ ^0.G⁹^;⁷⁴⁷

53

L 2 (0; 1).

(9)

1 ⁰

t&JJK

1 ⁰ .

^ZF⁷ ^.⁹⁷⁼^:^4.=A ⁵³^?6^07? ^C ^4F⁷ ^?5 ⁷⁼⁷³⁴⁵^:; ⁷⁹^=7//5.3

`(u) = e 1 Γ(1 − γ 2 )

Z x 0

u ⁰⁰ (t)

(x − t) ^γ ² dt − (q(x) + λ)u

: 3?

.63?

:

=C 0.3?545.3/

u(0) = 0, u(1) = 0

F

: /

:

3 53>7=/7 .

9 7=

: 4.=

1 ⁰ .

^t&J^J^K

1 ⁰

^mL^%%^q&^h'()&l^Lj^m&^gn(m^fnK^f^h^'(q^%&J

( A f 1 ),

^j('

q(x) > 0,

^nK^g ^fn& ^ei^Lpe& ^f'L)LK^% ^g(^%efL(ⁱ

u(x) = 0.

^*( ^h'(^)& ^fn&

eiLpe&i&gg(jfn& f'L)LK% g(%efL(i~m& Je%fLh%d q(fn gLl&g (j fn& &peKfL(i

1 Γ(1 − γ 2 )

Z x 0

u ⁰⁰ (t)

(x − t) ^γ ² dt = (q(x)u(x))

qd

(x − t) ^γ ¹

^Kil ^fn&i ^Lif&k'^Kf&^j'(J

0

^f(

x.

^*^n&i ^m& ^k&f

1 Γ(1 − γ 2 ) Z x

0

(x − t) ^γ ² Z t

0

( u ⁰⁰ (ξ)

(t − ξ) ^γ ² dξ)dt = Z x 0

(x − t) ^γ ² q(t)u(t)dt.

gLik fn& L'Lcn%&fh&'JefKfL(i j('Je%Kg~m& f'Kigj('J fn& Lif&k'K%

1 Γ(1 − γ 2 )

Z x 0

(x − t) ^γ ² ( Z t 0

u ⁰⁰ (ξ) (t − ξ) ^γ ² dξ)dt.

%&K'%d~

1 Γ(1 − γ ² )

Z x 0

(x − t) ^γ ² ( Z t

0

u ⁰⁰ (ξ) (t − ξ) ^γ ² dξ)dt

= 1

Γ(1 − γ 2 ) Z x

0

u ⁰⁰ (ξ){

Z x ξ

(x − t) ^γ ¹ (t − ξ) ⁻ ^γ ¹ dt}dξ

= Γ(1 + γ 2 )Γ(1 − γ 2 ) Γ(1 − γ 2 )

Z x 0

u ⁰⁰ (ξ)(x − ξ)dξ

= u ⁰ (0)Γ(1 + γ 1 )x + u ⁰ (x)Γ(1 + γ 2 ) − Γ(1 + γ 2 )u(0).

(10)

1 Γ(1 − γ 2 )

Z x 0

u ⁰⁰ (t)

(x − t) ^γ ² dt − q(x)u(x) = 0,

gKfLgjdLik fn& c(ilLfL(i

u(0) = 0,

^nK^g^fn^&^j('J

u(x) = cx + Z x 0

(x − t) ^γ ¹ q(t)u(t)dt = cx + Au. e

(ig&pe&if%d

u(x) = c(I − A) e ⁻ ¹ x = c(x + Ax e + A e ² x + ...).

,Lic&

q(x) > 1

^Lf ^Lg ^c%&K' ^fnKf

y(1) 6= 0,

^K^il ^j'(^J ⁿ&'& j(%%(mg fn& h'((j (j t&JJK

1 ⁰ .

^(m ^m& ^kL^)& ^fn& ^h'(⁽^j ^(j ^*n&('&J

1 ⁰ .

^b& ^fK^& ^K^l)KifK^k&

(j + +tLlgLLg fn&('&J {|} KkKLi+ b& mL%% h'()&~fnKf fn& (h&'Kf('

S e

^Lg

lLggLhKfL)&+ t&f

u(x)

^q^& ^Kid ^jei^cfL(ⁱ ^j'(^J ^fn& ^l^(J^KLi ⁽^j ^fn& ^(h^&'Kf('

S D(S) ⊂ L 2 (0, 1).

^*n^&^jeicfL(i

u ε (x) =

v(x), x ∈ [0, ε]

u(x), x ∈ [ε, 1] ,

mn&'&

ε > 0,

^K^il ^fn& ^je^icfL(ⁱ

v(x),

^L^g ^gecn^~^fn^Kf

v ⁰ (x)| _x=0 = 0

Kil Lg geJJKq%& mLfn gpeK'& Li

[0; ε].

zf Lg h(ggLq%& f( gn(m~fnKf

ε lim → 0 ( Su e ε ; u ε ) = ( Su, u). e

('fnLghe'h(g&LfLg&i(ekn f( fK&Li Kcc(eif {|} h+ fn&j('Je%K

d ^α

dx ^α f (x) = f ⁰ (0)

Γ(1 − α) x ⁻ ^α + 1 Γ(1 − α)

Z x 0

(x − t) ⁻ ^α f ⁰ (t)dt.

Y&'&

f ⁰ (x) ∈ L(0, 1), (0 < α < 1).

$jf&'fnKf~'&h&KfLikfn&h'((j(j*n&('&J |m('lqdm('l~m&(qfKLi~fnKf

fn& )K%e&g (j fn& j('J

( Su, u) e

^%L& ^Li ^fn^& ^K^ik%&

|argz| ≤ ^πγ ₂ ² .

^fn& ^& ^Lgf&i^c&

(jfn&&'i&%Li (h&'Kf('

S e ⁻ ¹

^cKⁱ ^q&^&gfKq^%Lgn&l ^Li ^K ^mKd ^gLJ^L%K^'^f( ^fnKf ^(j

(11)

fn& & Lgf&ic& (j fn& &'i&% Li (h&'Kf('

S ⁻ ¹ .

^(' ^fnLg ^h^e'h^(g& ^Lf ^Lg ^&i(^ekn

f( '&J&Jq&'~ fnKf ieJq&'

λ j

Lg fn& &Lk&i)K%e& (j fn& (h&'Kf('

T e

⁽ⁱ^%d ^Li

fn& cKg&~ mn&i

λ j

Lg K r&'( (j fn& jeicfL(i

E ρ (λ, 2).

^*ne^g ^fn& ⁽^h^&'K^f('

( S e ⁻ ¹ )

^~ ^m^n&i ^Lf ^Lg ^f'K^c& ^c%Kgg ^Ki^l ^l^LggLhKfL^)&^~ ^nK^g ^K ^c(^J^h%&f& gdgf&J (j

&Lk&ijeicfL(ig+

1”.

^;^;^{75 E73>}^:^;⁶^7/ ^.8 ^4F⁷^.⁹⁷⁼^:⁴^.=

S e

^;⁵⁷⁵³ ^4F⁷^:^3E^;⁷

|argz| ≤

πγ 2

2 .

²³ ⁹^:⁼⁴⁵⁰⁶^;:^=A ^:;^; ^7=./ ^.8 ^4F⁷ ⁸⁶³^045.³

E ρ (z; 2)

^:^;^/^. ^;⁵⁷ ⁵³ ^4F⁷ ^:^3E^;⁷

|argz| ≤ ^πγ ₂ ² .

d J&Kig (j (h&'Kf('g

A ^[α,β] γ

Lf Lg h(ggLq%& f( (qfKLi gfKf&J&ifg gLJLI

%K' f( t&JJKg | Kil

1 ⁰ ,

^Li ^K ^lLo&'&if mKd+ b& j('Je%Kf& Kil h'()& fn&

j(%%(mLik gfKf&J&if+ (' gLJh%LcLfd m&hef

||q(x)|| ≤ _Γ(1+γ ² ₁ ₎ .

ZF7 .

9 7=

: 4.=

S

^5/ ^53>7=45^;⁷ ^:³^? ^0.G⁹^:^04A ^:³^?

||S ⁻ ¹ || L 2 ≤ 1

Γ(1 − γ 1 ) − ||q(x)|| .

'((j(j*n&('&J

cKiq&(qfKLi&lj'(J fn&j(%%(mLik i(mifn&('&J+

x

74

T

^:³^?

A

⁷ ^:^3C ^.⁹⁷⁼^:^4.^=/ ⁸^=.G

X

^534.

Y.

⁷ ^:^//^6G⁷ ^4F^:⁴

T ⁻ ¹

⁷^5/4/ ^:³^? ⁷^;^.3E/ ^4.

B(Y, X).

⁷ ^:;^/^. ^:^//6G^7A

||A|| ≤ a||u|| + b||T u||

u ∈ D(T ),

1F7=74F7 0.3/4

: 34/

a

^:³^?

b

^/^:^45/8^C ^4F⁷⁵³⁷⁶^:;⁵⁴^C

a||T ⁻ ¹ || +b < 1.

^ZF⁷³ ^4F⁷

.

9 7=

: 4.=

S = T +A

¹⁵^;^;⁷⁰^;^./^7? ^:³^?^G^:^C ⁷⁵³^>7=45^;^7A^:^3?

S ⁻ ¹ ∈ B(Y, X)

: 3?

||S ⁻ ¹ || ≤ ||T ⁻ ¹ ||

1 − a||T ⁻ ¹ || − b ,

||S ⁻ ¹ − T ⁻ ¹ || ≤ ||T ⁻ ¹ ||(a||T ⁻ ¹ || + b) 1 − a||T ⁻ ¹ || − b.

B6=4F7=G.=7A 58 4F7 .

9 7=

: 4.=

T ⁻ ¹

^5/ ^0.G⁹^:^04A4F⁷³

S ⁻ ¹

^5/ ^0.G⁹^:⁰⁴ ⁴^..

h&'Kf('

S

^~^Li ⁽^e' ^cKg&^~^nK^g^fn& ^j('J

Su =

 



1 Γ(1 − γ 1 )

d dx

R x 0

u ⁰ (t)

(x − t) ^γ ¹ dt + q(x)u(x) u(0) = 0, u(1) = 0

zf Lg h(ggLq%& f( '&h'&g&if fn& (h&'Kf('

S

^Li ^fn^& ^j('J

Su = T u + Au

mn&'&

T u

^Lg ^i(f ^K lLgfe'q&l (h&'Kf('~

T u =

 



1 Γ(1 − γ 1 ) d

dx

R x 0

u ⁰ (t)

(x − t) ^γ1 dt

u(0) = 0, u(1) = 0

(12)

Au

Au =

q(x)u(x)

u(0) = 0, u(1) = 0 .

$gLf mKgK%'&Kld i(f&l fn& (h&'Kf('

A ^ρ u(t) = 1 Γ(ρ ⁻ ¹ )



 Z x 0

(x − t) ¹ ^ρ ⁻ ¹ u(t)dt − Z 1 0

x ¹ ^ρ ⁻ ¹ (1 − t) ¹ ^ρ ⁻ ¹ u(t)dt



 ,

Lg fn& Li)&'g&(jfn& (h&'Kf('

T

^m^n&i

¹ _ρ = 1 + γ 1

{}~L+&+

T ⁻ ¹ u = A ^ρ u

^Ki^l

Lf Lg (q)L(eg fnKf

||A ^ρ u|| ≤ 2

Γ(ρ ⁻ ¹ ) = 2 Γ(1 + γ 1 ) ,

||Au|| ≤ ||q(x)||||u|| ≤ 2 Γ(1 + γ 1 )

(m

||S ⁻ ¹ || ≤

2 Γ(1+γ 1 ) 1 − ||q(x)||

2 Γ(1 + γ 1 ) = 2

Γ(1 + γ 1 ) − 2||q(x)||

zifn&h'((j(jt&JJK |~LfLgc&'fKLi%d&gg&ifLK%fnKf

q(x)

^L^g^h⁽^gLfL)&⁺ ^zi

fn& h'((j (j fn& *n&('&J ~

q(x)

^Lg ^i(f '&peL'&l f( q&h(gLfL)&+ zi & Kcf%d fn&gKJ&mKdm&cKi (qfKLi gLJL%K' '&ge%fgj('fn&h'(q%&J

( A), e

^K^gg(^gLKf&l

mLfn h'(q%&J

(A).

^*( ^gfKf& ^h'(q^%&J

( A) e

^m& l&i&

e σ k =

X k j=0

γ 2 − j − 1,

e

µ k = e σ k + 1 = X k j=0

γ 2 − j

(k = 0, 1, 2),

D ⁽ ₁ ^f ^σ ⁰ ⁾ f (x) ≡ d ⁻ ⁽¹ ⁻ ^γ ² ⁾

d(1 − x) ⁻ ⁽¹ ⁻ ^γ ² ⁾ f (x), D ₁ ⁽ ^σ ^f ¹ ⁾ f (x) ≡ − d ⁻ ⁽¹ ⁻ ^γ ¹ ⁾

d(1 − x) ⁻ ⁽¹ ⁻ ^γ ¹ ⁾ d ^γ ²

d(1 − x) ^γ ² f (x), D ₁ ⁽ ^σ ^f ² ⁾ f (x) ≡ d ⁻ ⁽¹ ⁻ ^γ ⁰ ⁾

d(1 − x) ⁻ ⁽¹ ⁻ ^γ ⁰ ⁾ d ^γ ¹ d(1 − x) ^γ ¹

d ^γ ²

d(1 − γ) ^γ ² f (x).

(13)

$ili(m~h'(q%&J

( A) e

^J^Kd^q^&^h^ef^Kg^j(%%^(m^g+ ^zi ^c%K^gg

L 2 (0, 1)

⁽^'

L 1 (0, 1)

m&il i(if'L)LK% g(%efL(i (jfn&&peKfL(i

D ⁽ ^f ^σ ² ⁾ z − {λ + q(x)}z = 0, x ∈ [0, 1),

gKfLgjdLik fn&q(eilK'd c(ilLfL(ig

D ⁽ ₁ ^f ^σ ⁰ ⁾ z

x=0 cos α + D ₁ ⁽ ^σ ^f ¹ ⁾ z

x=0 sin α = 0 D ₁ ⁽ ^σ ^f ⁰ ⁾ z

x=0 cos β + D ₁ ⁽ ^σ ^f ¹ ⁾ z

x=0 sin β = 0.

fn& Kgg(cLKf&l h'(q%&J kL)&g &gg&ifLK%%d~ i&m '&ge%fg~Li fn& cKg& mn&i

fn& ('l&' (j fn& j'KcfL(iK% lLo&'&ifLK% &peKfL(i Lg %&gg fnKi (i&+ *( fnLg

cKg&m&gnK%% l&)(f&g&hK'Kf&hKh&'+ *( gn(m~n(m f(f'Kigj&'fn&(qfKLi&l

'&ge%fg (i K cKg& (j fn& lLo&'&ifLK% &peKfL(ig (j ('l&'nLkn&' fnKi fm(~ m&

c(igLl&'fn& j(%%(mLik h'(q%&J+

zi c%Kgg

L 2 (0, 1)

^m^&ⁱ^l ⁱ⁽^iIf'L)LK^% ^g(%efL(ⁱ ^(j ^fn& &peKfL(i

D ^(σ ³⁾ y − {λ + q(x)y = 0, }(7)

gKfLgjdLik fn& c(ilLfL(ig

D ^(σ ⁰ ⁾ y| x=0 = 0; D ^(σ ¹ ⁾ y| x=0 = 0; D ^(σ ⁰ ⁾ y| x=1 = 0; (8)

b& hef

γ 0 = γ 2 = γ 3 = 1.

σ 3 = 1

ρ = γ 0 + γ 1 + γ 2 + γ 3 − 1 = 2 + γ 1

zi fnLg cKg& h'(q%&J

(7) − (8)

^m^L%% ^q& '&m'Lff&i Kg j(%%(mg

1 Γ(1 − γ 1 )

d ² dx ²

Z x 0

y ⁰ (t)

(x − t) ^γ ¹ − {λ + q(x)}y = 0(7 ⁰ ) y(0) = 0; D ^(σ ¹ ⁾ y| x=0 = 0; y(1) = 0(8 ⁰ )

b& i(m j'(J {}~ fnKf fn& (h&'Kf('~ Li)&'g& f( fn& (h&'Kf(' Lilec&l qd

lLo&'&ifLK% & h'&ggL(i

ly = 1 Γ(1 − γ 1 )

d ² dx ²

Z x 0

y ⁰ (t) (x − t) ^γ ¹ dt

Kil gKfLgjdLik f( q(eilK'd c(ilLfL(ig

(8 ⁰ )

^Lg ^&peK^% ^f( ^fn^& ^(h^&'Kf('

A ^ρ

^j(^'

1 < ρ < 2

^Li ^h^Kh^&' ^{|}^~ ^fn& ^&Lk^&i^ie^J^q^&'g ^Ki^l ^fn& &Lk&i)K%e&g (j fn& & I h'&ggL(i j('

A ^ρ

^K^f&)&'d

A ^ρ .

^(e^ilK^'dI)K%e^&^h'(^q%&J^g^j(^'^fn&&peKfL(ig (j

(14)

'&JK'Kq%& hKh&' {|} Li fnLg f(hLc+ *n&'&j('& fn& Kefn(' m(e%l %L& f(

lm&%% (i fnLgh'(q%&J Li J('& l&fKL%g+

74

q(x) ≡ 0,

^4F⁷³ ^4F⁷ ³^6G⁷⁼

λ j

5/ 4F7 75 E73>

:;

67 .8

9

=.

;

7G

(7 ⁰ ) − (8 ⁰ ),

⁵

λ

^5/ ^:^7=. ^.8 ^4F⁷⁸⁶³^045.3

E ρ (λ; ρ ⁻ ¹ ),

^:³^?⁸^63045.3/

x ¹ ^ρ ⁻ ¹ E ρ (λ j x ¹ ^ρ ; 1 ρ )

:

=7 75 E73863045.3/ .8 4F5/

9

=.

;

7G

*n& k&i&'K% g(%efL(i (j fn& &peKfL(i

(7 ⁰ )

^gK^fLgjdLi^k ^fn^& ^c(il^LI

fL(ig

y(0) = 0, D ^(σ ¹ ⁾ y| x=0 = 0; D ^(σ ² ⁾ y| x=0 = y ₂ ⁰

(q)L(eg%d gKfLg&gfn& &peKfL(i

y(x; λ) = y ₂ ⁰

Γ(2 + γ 1 ) x ^1+γ ¹ + λ Γ( ¹ _ρ )

Z x 0

(x − t) ¹ ^ρ ⁻ ¹ y(t; λ)dt; (9)

b& m'Lf&(ef fn& g(%efL(i (j fn& &peKfL(i Kcc('lLik f( fn& i(mi j('I

Je%K {{|} h+|}

y(x; λ) = y ₂ ⁰

Γ(2 + γ 1 ) x ^1+γ ¹ +λ Z x 0

(x−t) ¹ ^ρ ⁻ ¹ E ρ (λ(x−t) ¹ ^ρ ; 1

ρ )( y ⁰ ₂

Γ(1 + γ 1 ) t ^2+γ ¹ )dt

Kil cK%cce%Kf& fn& Lif&k'K%

1 Γ(1 + γ 1 )

Z x 0

(x − t) ¹ ^ρ ⁻ ¹ E ρ (λ(x − t) ^ρ ¹ ; 1

ρ )t ^1+γ ¹ dt

Kcc('lLik f( s+s+rn'qKgncnwKigj('Je%K { h+||}

Z l 0

x ^α ⁻ ¹ E ρ (λx ¹ ^ρ ; α)(l − x) ^β ⁻ ¹ E ρ (λ ^∗ (l − x) ¹ ^ρ ; β )dx

= λE ρ (l ¹ ^ρ λ; α + β) − λ ^∗ E ρ (l ¹ ^ρ λ ^∗ ; α + β)

λ − λ ^∗ l ^α+β ⁻ ¹

(α > 0, β > 0) 1

Γ(ρ ⁻ ¹ ) Z x 0

(x − t) ¹ ^ρ ⁻ ¹ E ρ (λ(x − t) ¹ ^ρ ; 1

ρ )t ¹ ^ρ ⁻ ¹ dt

(15)

= E ρ (x ¹ ^ρ ; 2 ρ )x ² ^ρ ⁻ ¹ .

'(J n&'& m& k&f

y(x; λ) = y 2 0

Γ(ρ − 1) x ¹ ^ρ ⁻ ¹ + λy ⁰ ₂ E ρ (λx ¹ ^ρ ; 2 ρ )x ² ^ρ ⁻ ¹

= y ₂ ⁰ x ^ρ ¹ ⁻ ¹ [ 1

Γ(ρ ⁻ ¹ ) + λx ^ρ ¹ E ρ (λx ¹ ^ρ ; 2 ρ )]

= y ₂ ⁰ x ¹ ^ρ ⁻ ¹ E ρ (λ x ¹ ^ρ ; 1 ρ ).

*neg~m&nK)&h'()&l~fnKffn& k&i&'K% g(%efL(i (j&peKfL(i

(7 ⁰ )

^~^gKfL^gjdLik

y(o) = 0; D ^(σ ¹ ⁾ y| x=0 = 0; D ^(σ ² ⁾ y| x=0 = y ⁰ ₂ ,

nKg fn&j('J

y(x; λ) = y ⁰ ₂ x ¹ ^ρ ⁻ ¹ E ρ (λx ¹ ^ρ ; 1 ρ ).

'(J n&'& Lf j(%%(mg fnKf fn& ieJq&'

λ

^L^g ^Ki &Lk&iieJq&' (j h'(q%&J

(7 ⁰ ) − (8 ⁰ )

^m^n&i ^Ki^l ⁽ⁱ^%d ^~^m^n&i

y(1; λ) = y ₂ ⁰ E ρ (λ; 1 ρ ) = 0,

L+&+ &Lk&iieJq&'g (j h'(q%&J

(7 ⁰ ) − (8 ⁰ )

^c(Li^cL^l&^m^Lfn ^r&'( ⁽^j^fn& ^jeicfL(i

E ρ (λx ¹ ^ρ ; ¹ _ρ ),

^K^il ^&Lk^&ijei^cfL(^ig ^nK)&^fn& ^j(^'J

E ρ (λx ¹ ^ρ ; 1 ρ ),

mnLcnh'()&g*n&('&J + zfLgLif&'&gfLikn(m K%%fnLgnKgKh'&cLg&weicfL(i

mLfn fn& cKg&(j

1 2 < ρ < 1,

^m^nL^cn ^m& Li)&gfLkKf&l Kq()&+

ZF7 .

9 7=

:

4.=

D ¹ ^ρ

⁵³^?6^07? ^C ^4F⁷ ^?5 ⁷⁼⁷³⁴⁵^:; ⁷⁹^=7//5.3

1 Γ(1 − γ 0 ) d ² dx ²

Z x 0

y ⁰ (t)

(x − t) ^γ dt + q(x)y

: 3? 4F7

.63?

:

=C 0.3?545.3/

(8 ⁰ )

^5/ ^?^5//5⁹^:^45>7

(D ¹ ^ρ y, y) =



 1 Γ(1 − γ 1 )

d ² dx ²

Z x 0

y ⁰ (t)

(x − t) ^γ ¹ dt, y(x)



 + (q(x)y, y)

(16)

= 1 Γ(1 − γ)

Z 1 0



 Z x 0

y ⁰ (t) (x − t) ^γ dt





00

y(x)dx

+(q(x)y, y) = 1 Γ(1 − γ)

Z 1 0

yd



 Z x 0

y ⁰ (t) (x − t) ^γ ⁰ dt





0

+ (q(x)y, y)

= 1

Γ(1 − γ)



y(x) Z x 0

y ⁰ (t) (x − t) ^γ dt





b

a

= 1

Γ(1 − γ) Z 1 0

d dx

Z x 0

y ⁰ (t) (x − t)γ 1

y ⁰ (x)dx + (q(x)y, y)

= − dγ

dx ^γ z, z

+ (q(x)y, y),

mn&'&

z = y ⁰ .

^,Lic&

z ∈ A 0 γ [0, 1]

^fn& ^g&f ^(j ^K%% ^jeicfL(i^g

z(x)

^~ ^nK)Lik

Kqg(%ef&%d c(ifLie(eg (i {~| } j'KcfL(iK% Lif&k'K% (j ('l&'

1 − γ

^mLfn ^fn&

gfK'fLik h(Lif

0

^K^il ^fn&^&i^lLik ^h^(Lif

x

^~^m^nL^cn ^)Ki^Lgng ^Kf

x = 0

^~fn&'&j('&

Kcc('lLik f( *KJK'Lig fn&('&J {|} h+

fn&'& Lg K eiLpe& jeicfL(i

u ∈ L(0, 1)

^gecn^~^fnK^f

z(x) = _dx ^d ⁻ ₋ ⁽¹ (1 ^−γ) −γ) u.

^Y^&i^c&

d ^γ dx ^γ z, z

=

u, d ⁻ ⁽¹ ⁻ ^γ) dx ⁻ ⁽¹ ⁻ ^γ) u

,

Kil fn& )K%e&g (j fn& %Kgf j('J~ Kcc('lLik f( sKfgK&)I (%Kifg fn&('&J~

J&ifL(i&l Kq()& %Kdg Li fn& c(''&gh(ilLik g&cf('+ '(J fnLg j(%%(mg fn&

h'((j(j (e' fn&('&J+ (m nK)Lik K lLggLhKfL)Lfd~m& cKi h'()& c(Jh%&f&I

i&gg (j fn& gdgf&J (j fn& &Lk&ijeicfL(ig (j h'(q%&J

(7 ⁰ ) − (8 ⁰ )

^~'&)&'g&Kg Kq()&~Lj fn&'&Lg Ki (h&'Kf('~Li)&'g&f( fn& (h&'Kf('

D ¹ ^ρ u,

^Ki^l ^&pe^K% ^f(

D ¹ ^ρ u =

 



1 Γ(1 − γ) d ²

dx ²

R x 0

u ⁰ (t)

(x − t) ^γ dt, 0 < γ < 1 u(0) = 0; D ^(σ ¹ ⁾ u

x=0 = 0; u(1) = 0

*n& (h&'Kf('

A ^ρ

^Kf

¹ _ρ = 2 + γ

^Lg ^fn& Li)&'g& f( fn& (h&'Kf('

D ^ρ ¹

^{}⁺

$il Li k&i&'K%~ fn& (h&'Kf(' Li)&'g& f( fn& (h&'Kf('~ k&i&'Kf&l qd fn&

lLo&'&ifLK% & h'&ggL(i

lu = 1 Γ(1 − γ)

d ⁿ ⁻ ¹ dx ⁿ ⁻ ¹

Z x 0

u ⁰ (t)

(x − t) ^γ dt, (γ 0 = 1, γ 1 = γ, γ 2 = 1, ..., γ n = 1

(17)

u(0) = 0; D ^(σ ¹ ⁾ u

x=0 = 0, ..., D ^(σ ⁿ⁻¹ ⁾ u

x=0 = 0, u(1) = 0

Lg &peK% f(

A ^ρ

^Kf

¹ _ρ = n − 1 + γ.

^*n^Lg^cKi ^q^& cn&c&l lL'&cf%d Kg Li {} Lf Lg i&c&ggK'df( gn(m~ fnKf

A ^ρ D ¹ ^ρ u = u

^Ki^l

D ¹ ^ρ A ^ρ u = u.

^$îl î(m^~^wêgf ^K^g

mKgl(i&Li {}~(i&cKi Li)&gfLkKf&~qdJ&Kig(j(h&'Kf('g(jLil

A ^α,β _γ

^~^fn&

pe&gfL(ig (j gLJh%LcLfd (j fn& c(''&gh(ilLik &Lk&i)K%e&g Kil JK'Lik (ef

fn&l(JKLigmn&'&fn&g&&Lk&i)K%e&gl( i(f& Lgf('~&peK%%d~fn&pe&gfL(ig

(jgLJh%LcLfd(jr&'(g(jfn&c(''&gh(ilLikjeicfL(ig(jsLffKkIt&u&'gfdh&

Kil l(JKLig Li fn&c(Jh%& h%Ki& mn&'& fn&g& r&'(g l( i(f & Lgf+

LiK%%d~ fn& i& f heq%LcKfL(i mL%% q& l&)(f&l f( pe&gfL(ig (j & Lgf&ic&

(j qKgLg~ fn& c(igf'ecf&l gdgf&Jg (j fn& &Lk&ijeicfL(ig Kil fn& h'((j (j

fn& (gcL%%KfLik h'(h&'fd (j fn& (h&'Kf('

A ρ [ρ,ρ]

^(j ^cKg&g

0 < ρ < 1/2.

^zf ^Lg

Kqg(%ef&%di&mc%Kgg(jfn&(gcL%%KfLik(h&'Kf('glLo&'&ifj'(J fn&(h&'Kf('g

(j s+ + '&Li {| }+

{| } $%&'(&) *+ ,+ $q(ef (i& c%Kgg (j (h&'Kf('g Kgg(cLKf&l mLfn lLo&'&ifLK%

&peKfL(ig(jj'KcfL(iK%('l&'+,Lq&'LKisKfn&JKfLcK%(e'iK%+~ (%I

eJ& ~ieJq&' ~hh+|| I| +

{ } eq'&&) +s+ zr)+ (g+ $Kl+ Ke+ $ g&'L&g (j fn& JKfn&JKfLcLKi

)(%+ i(+| hh+| I|~+

{} $%&'(&) *+,+ *n& q(eilK'd h'(q%&Jg j(' lLo&'&ifLK% &peKfL(ig mLfn

j'KcfL(iK%l&'L)KfL)&g+Lgg&'fKfL(i~l(cf('(jhndgLcK%KilJKfn&JKfLcK%

gcL&ic&g~~s(gc(m~s(gc(m ,fKf& iL)&'gLfd+

{} '(lgLL s+ ,+ *'LKike%K' Kil ('lKi '&h'&g&ifKfL(ig (j (h&'Kf('g s+M

KeK |

{ } Kcnegn&) $+s+*n&j'KcfL(iK%cK%ce%KfLik KilLfg Khh%LcKfL(is+M LrI

JKf%Lf~~h+ +

{} rn'qKgnwKi s+s+~ &'g&gdKi $+ + zr)+ $ $'J,,~ g&'L&g (j hndg+I

JKfn+ gcL&ic&g |~)(%+|~i(+ +

{ } rn'qKgnwKis+s+zr)+$$'J,,~| ~)(%eJ& ~ieJq&'~hh+ | I

+

{} KcnK&)$+s+*n&q(eilK'dh'(q%&Jgj('lLo&'&ifLK%&peKfL(ig(jj'KcI

fL(iK% ('l&'+Lgg&'fKfL(i~K%cnL~eggLK++

(18)

fL(iK% ('l&'+ *n& '&h('fg (j L'cKggLKi zif&'iKfL(iK% $cKl&Jd (j ,cLI

&ic&g (%Kld $ldkg(d n&'&gg(d s&r wleiK'(li(d $Kl&JLL

Ke~ |

+ (%eJ& |~ ieJq&' |+

{|} sK%KJel s+s+ *n& '&h('fg (j KfL(iK% $cKl&Jd (j gcL&ic&g (j

'KLi& (%Kld iKfL(iK%i(L KKl&JLL iKe 'KLiL+ |~ ieJq&'

~hh+I

+

{|| } sK%KJel s+s+ eggLKi w(e'iK% (j JKfn&JKfLcK% hndgLcg )(%+ i(+

|

{|} tLlgLL + +~*'+s(g+ sKf+ I)K~ |~)(%+~hh+I|+

{|} sKfgK&) +z+~ (%Kif +$+ '+JKf+w(e'iK%+|~)(%eJ& |~ieJq&'

~hh+I +

{|} (nq&'k z++~ '&Li s+ +*n&fn&('d (j (%f&''K (h&'Kf('gLi L%q&'f

ghKc&Kil Lfg Khh%LcKfL(ig+s+M | +

{|} rn'qKgnwKis+s+zif&k'K%*'Kigj('JKfL(igKilh'&g&ifKfL(ig(jjeicI

fL(ig Li c(Jh%& ghKc&+s+M KeK ~ |~h+ +

{|} *+Kf(+*n& fn&('d (j LilLkiKfL(i (j %Li&K'(h&'Kf('g+s+ | +

{| } KifJKn&' ++~ '&Li s++

gcL%%KfL(i JKf'L &g~ &'i&%g Kil gJK%%

ecfeKfL(ig (j J&cnKiLcK%gdgf&Jg+(gf&nLrlKf J+I%+~ |+

&c&L)&l KieK'd |~

Electronic Journal of Qualitative Theory of Differential Equations 2009, No. 25, 1-18; http://www.math.u-szeged.hu/ejqtde/