y ∆ (t) + p(t)y σ (t) ∈ F (t, y(t)), t ∈ J := [0, b] ∩ T , t 6= t k , k = 1, . . . , m, y(t + k ) − y(t − k ) = I k (y(t − k )), k = 1, . . . , m,

(1)

Electronic Journal of Qualitative Theory of Differential Equations 2005, No. 12, 1-22; http://www.math.u-szeged.hu/ejqtde/

! "$#% '&)(*,+- .!,/,0-,

- 123 5467+

89;:<>=@?BAC:<>D*E%<>=@AFDG9IHKJIL2MONP;AF?QE%<>RISUTMOTDGA!VWAFRI:X89;:<B=ZYOTAFRIH\[]NAFTAF9

^_a`Obdce_UfebdghcjiWkFi-lm_UfenFop*_fjgrqtsFiu3vCwyxFghzai3cju{ghfjo|k}i~>grkFg0i30`}`!u

7Qd}vddd}v~>grkFg0i30`}`!u3vChdo3c{gri

i$p6_agr_aiy`Oi3r_acj`}g_an}bBbFZc3v,`Oi3xFn}bdnFce_ds}xFghzu{`C_}kFavbdsC_anF_a`WsFxFg zt¡uj`C_B¢kF

£%¤j¥a¦¨§j©aª¤j¥6ª¬«]ªC¤|®*¤®*«U¯$°«¬±²¯«¬±³¤´´µ«U¯W¶W¤¯$·O¥-¸¹Bº »©>§

¼-½!¾$¿$ÀÁÂµ¿

Ã@Ä-ÅÆÇÉÈÊ>Ë3Ê}ÌeÍ$ÎÏIÌÇÐÄUÑGÌÈÅÇÐÒUË3Å¬Ì/ÅÆtÌ/ÌjÓaÇÉÈ¬Å¬ÌeÄtÔÌÖÕ3×OÈ¬Õ³ØÐÙÅÇÉÕ³ÄtÈÚË3ÄtÛ|ÌjÓdÅÍ¬ÌeÜ|Ë3ØCÈ¬Õ³ØÐÙÅÇÉÕ³ÄtÈ

×ÝÕ³ÍË%ÞtÍ¬ÈÅÕ³Í¬ÛÌeÍyÇÐÜÊÙØÉÈÇÐÑGÌÛdßÄ>Ë3ÜÇÉÔ|ÇÐÄtÔeØÐÙtÈÇÉÕ³ÄàÕ³ÄÅÇÐÜÌ|ÈKÔ$Ë3ØÉÌÈ$á%âIßÙtÈÇÐÄÒ*ÈÙÇÐÅKË3ãØÉÌ

ÞÓdÌÛ]Ê}Õ³ÇÐÄÅÖÅÆtÌÕ³Í¬ÌeÜÈ$ÎBÏIÌÈÅÙtÛdß%ÅÆtÌÔ$Ë³È¬Ì\ÏÚÆtÌeÄ*ÅÆtÌÍÇÐÒµÆUÅ/Æ>Ë3ÄtÛ]È¬ÇÉÛÌÆ>Ë³È/ÔÕ³ÄUÑGÌjÓ6Ë³È

ÏIÌeØÐØ!Ë³È;ÄtÕ³ÄtÔÕ³ÄUÑGÌjÓäÑµË3ØÐÙtÌÈ$á

å <æçMODU:è7ACR:méTDGACè3<>èaêÚë p]ìFsFhujghzaiÚk}>xC_ap]grÚghxF3hsFujghbdxFu3vGkFihfe_7kFi3c{ghz_feg zdivUbdxfecj_afjgrbdxv

ií>fjcji3p6_a0u{bdrs}fjgrbdx}u3vBîBí}ikmìObdgrxf3v}feghp6iWu{³_ahi3u

8LðïòñjLð[ïó]ïN9ô3<>Saõ,ö%=ZACè³è3HZ÷SdA}õGHZMORIè ùøaúdWødûBvCøaúdüd}v}øaú7øû>

ý

!þ0]ÿ,,

\nFghu-ìC_ì!icgru|3baxF3icjxFikghfen fenFi]iíBghu{feixF3i*ba\ujbars}fjgrbdxFu|_ax}k ií>fec{i3p6_a;ujbdhs}feghbdxFuZbdc-_

3r_auju-ba\ghxFghfjgÝ_az_ahsFi]ìFcjbd`}ri3p Zbac-ghp6ìFs}rujg zdiäk}>xC_ap]gr]grx}3rs}ujgrbaxFu-bax feghp6iäu{³_ahi3u3àlbdc{i

ìFc{i33ghujihdvCghx~Bi3fegrbaxøBv/i|3baxFujghkFi3c\fen}iZbarrbghxFìFc{bd`Fhi3p

y ^∆ (t) + p(t)y ^σ (t) ∈ F (t, y(t)), t ∈ J := [0, b] ∩ T , t 6= t k , k = 1, . . . , m, y(t ⁺ _k ) − y(t ⁻ _k ) = I k (y(t ⁻ _k )), k = 1, . . . , m,

y(0) = η,

^ø

nFicji

T

gru/_-feghp6iu{³_ahiav

F : [0, b] ×

→ P (

)

gru_bdp6ìF_afÖz_arsFikp%sF fegÉ¡z_ars}i3käp*_ìv

P (

)

grufjnFiþ@_ap6ghhba\_rùxFbaxFi3p]ì}f ujs}`Fujifeub

, I k ∈ C(

,

), η ∈

, 0 = t 0 <

t ₁ < ... < t m < t _m+1 = b,

_axFkZbdc\i3_an

k = 1, . . . , m

^v

y(t ⁺ _k ) = lim

h→0 ⁺ y(t k + h)

^_axFk

y(t ⁻ _k ) = lim

h→0 ⁻ y(t k + h)

c{i3ìFc{i3ujixf7fenFiycjghdnf_axFk*hifÖrghp6g feu ba

y(t)

^_f

t = t k

ghxmfenFi-ujixFujiþbafegrp]iuj3_ariu3v fjnC_f gru3v

t k + h ∈ [0, b] ∩ T

Zbdcyi³_n

h

^ghxà_

xFigrdnt`ObdcjnFb>b>k6ba

0

_ax}k*grx*_akFkFg feghbdxvgh

t k

ghuc{grdnfÖuj3_fjfji3cjikvBfjnFi3x

y(t ⁺ _k ) = y(t k )

vnFi3c{i³_auv !"$#&%'() *

+-,./102+v ad4365 bF >vCì

(2)

gh

t k

ghu\rif\uj3_fjfji3cjikvFfen}i3x

y(t ⁻ _k ) = y(t k ), σ

gru_Zs}xFfjgrbdxfenF_f grh `OikFiîFxFi3kÝ_fji3c\_xFk

y ^σ (t) = y(σ(t)).

ë p6ì}sFru{ghzdi*kFg i3c{i3xfegr_a/iq>sF_feghbdxFuþnC_³zdi,`Oi3bdp6i*grp]ì!bdcKf_axfgrx cji3ixtfäai³_ac{u]ghx p6_fenB

i3p6_fjgr³_ p]bBk}i3ru6ba|cji3_aWìFc{bBi3uju{i3u_axFk fjnFi_acjghujiàghxìFnFixFbdp]i3xC_ u{fjsFkFghi3kXgrxì}nt>u{gr3uv

nFip6gh³_a;feinFxFbdhbdadvIì!baìFsFÝ_Ufegrbax k}>xC_p6gh3u3v`Fgrbafji3nFx}bdrbd _axFk i3bdxFbap6gh3u3 \nFicjinC_u

`Oi3i3x_ ujghdxFghîF³_axf6kFizai3rbaìFp6ixfgrx grp]ìFsFhuji,fenFibdc{av\grxQc{i3i3xfdi3_acjuvi3ujìOi3gÝ_ahh ghx fen}i

_ac{i³_ bagrp]ìFsFhujg zdikFg icji3xfjgÝ_a7iqtsC_feghbdxFu ghfjn î}íBikp6bap6ixtfjuujiifjnFip]bdxFbadce_aì}nFuþba

^_atujnFp]grU_axfjnC_ap ¤ª©aºa¨vC~}_ap]bdgrhi3xFbä_axFkIicji3uKf>sF43 _axFkfen}i|c{iZicji3x}3i3ufen}i3cjigrx

ë x c{i3i3xfai³_ac{uk}>xC_ap]gräiq>sF_feghbdxFu|bdx fjgrp]iþuj³_ri3u-nC_³zdi6cji3i3g zdi3k p%sFn _Ufjfeixfegrbax i

cjiZi3cfjb6fjnFi%`ObBbdtu `tm/bdn}xFi3c_xFk Iifeicjujbax}v

v^_atujn}p6gh_xtfjnC_ap¤ª©aº¢aø_axFkmfjb

fen}ic{iZi3c{i3xFi3ughfji3k fjnFi3c{i3ghx \nFi,feghp6i,uj3_ariu63_arsFrs}u]nF_au]fjcjip6ixFkFbdsFu6ì!bfei3xfjgÝ_a7Zbdc

_aìFì}rgr3_feghbdxFugrx,p*_UfenFip*_fjgr3_a!p]b>kFi3hu7baIcji³_ìFcjb>3iuju{i3u _axFkìFnFi3x}bdp6ixC_}v}Zbac\ií}_ap]ìFrighx

ìFn>ujgh3u3v>nFip6gh³_a}fji3nFx}bdrbddvìObdìFsFr_feghbdxþk}>xC_ap]gru3v`FghbafeinFxFbdhbda_axFk]i33baxFbdp]gr3uvxFi3s>

ce_BxFif /bacjtu3vtujb>3gr_atujgri3x}3i3uvujii7fenFip6bdx}bddce_ìFnFubsFh`C_anþ_ax}k ghrdic¢ v/bdn}xFi3cÚ_xFk

Iifeicjujbax}v

vÚ^_atujnFp]grU_axfjnC_ap ¤ª©aºø\_xFk febfenFic{iZicji3x}3i3u%fjnFi3c{i3ghxi33ixfeh

ixFkFicjujbax

W_axFkX/ixFnFbdn}ce_ò¤ª]©aº

v\ûBv/WnC_³zai grxFg fegÝ_Ufei3k fen}i u{fes}k}ba|ghp6ìFs}rujg zdi

k}>xC_ap]gr]i3qtsC_UfegrbaxFu%bdx fjgrp]i6u{³_ahi3u \nFiîCc{u{fìC_aìOi3c-Zbacgrp]ìFsFhujg zdi]k}>xC_ap]gr]grxFrsFu{grbdxFu

/_auWìFc{bdì!bauji3km`tm/iÝ_ac{`Fg¨vO/ixFnFbdnFcj__axFkWsC_nC_a`ú ë xmfjnFgruyìC_aìOi3cvi%3baxtfjgrxtsFi|fenFghu

u{fjsFk}Q`tQ3bdx}ujgrk}i3cjghxF p]bdcjidi3x}i3ce_W3r_auju{i3ubaghp6ìFs}rujg zdik}>xC_ap]grmghxF3hsFujghbdxFu6bdxQfeghp6i

uj3_ariu3 i-ujnC_rìFcjbµz>grkFi-iíBgruKfeixF3i%c{i3u{sFhfju ZbdcfenFi-ìFcjba`Frip

ø

1\nFiîCc{u{fbdxFic{i3hgri3u

bdx fenFiäxFbaxFrghxFi³_ac_hfeicjxC_Ufeghzaiþba/^ice_³K~BnF_asFkFic-f>ì!i

ü nFix fen}i]c{grdnfnC_xFk u{grkFiäghu

3baxtzaií]z_ars}i3kvdfjnFiuji3bdxFk*_axFkäfenFi\fen}grcjkäc{i3h%_aru{bbdx]fjnFixFbdxFhgrxFi3_acÚ_a feicjxC_fjghzai\ba^0ice_³

~BnC_sFkFi3c fBìOi`}s}fWsFx}kFi3c /i3_aaic3bdxFk}ghfeghbdxFubdxmfjnFi-Zs}xFfjgrbdxFu

I k (k = 1, ..., m)

_axFkmfjnFi p]g íBi3k di3x}i3ce_rgri3k ^0grìFu{nFghfj_axFk7_ce_fjnFo3b>kFbdcKu]3bdxFk}ghfeghbdxFuä_ax}k fen}ir_au{f]bdx}ibax fen}i

î}íBi3k ìObdgrxf fenFibdcjip)Zbdc 3bdxfecj_afjgrbdxmp%sFhfjg ¨za_rsFikp6_aìFuWk}sFifeb /bµz>ghfjþ_axFk 5 _akFhi3c!

ø

nFixäfjnFi7c{grdnfùnF_axFkþujghkFi7ghuxFbafùxFi3i3ujuj_acjghh-3bdxzaiíþz_ahsFi3k \nFi7r_au{fuji3feghbdxþgruI3bdxFi3c{xFi3k

ghfjn fjnFi6i$íBgru{fji3xFiba/ií>fec{i3p6_a;ujbdhs}feghbdxFuba7fjnFi*_`!bµzdi6p6ixfegrbaxFi3k ìFcjbd`}ri3p.` sFu{grxF_

cji3ixtfî}íBikìObdghxtf/fenFibdcjip kFsFiWfjb 0nC_aai"

ú Zbdc/fjnFiujs}p baI_þ3bdxfjce_afegrbaxp%sFhfjghz_ahsFi3k

p6_aì]_axFk6_|bdp6ì}rifji3 bdxfegrxtsFbasFubaxFikFiîCx}i3k*bax6bdc{kFi3c{i3k67_axC_n*ujìF_a3iu3 \nFiujic{i3u{sFhfju

3bap6ìFhi3p]i3xf\fen}iWZi*2i$í}ghu{fji3xFi-c{i3ujs}hfeukFizdbafji3kfebäk}>xC_ap]grghxF3hsFujghbdxFu7bdxfjgrp]iuj³_ri3u

# $

I \3 ,+I

i6grrI`Fcjghi&%Fmcji³_ahujbap6iþ`C_ujgrkFiîCx}ghfeghbdxFu_xFk @_afjuWZcjbdpfegrp]i3uuj³_ri3u3_arsFrs}ufenC_Uf

i ghr s}ujigrxfen}i|u{i3qtsFi¨

feghp6iuj³_ri

T

ghuÚ_xFbdxFip6ìBf%3hbdujikäujsF`Fu{ifÖba

.

^ë fÚZbarrbufjnC_fùfjnFi('{sFp]ìþbdìOi3ce_Ufebdc{u

σ, ρ : T → T

k}iîCxFikm`

σ(t) = inf {s ∈ T : s > t}

^_axFk

ρ(t) = sup{s ∈ T : s < t}

u{sFìFìFhi3p]i3xfeikm`

inf ∅ := sup T

^_xFk

sup ∅ := inf T

^_ac{i i3hkFiîCx}i3k \n}iìObdgrxf

t ∈ T

gru-rif{k}i3xFu{iavùhif{¡uj3_fjfji3cjikvùc{grdnf{¡kFi3x}ujiavcjgrantfKu{³_f{fei3c{i3k g

ρ(t) = t, ρ(t) < t, σ(t) =

+-,./102+v ad4365 bF >vCìÖ

(3)

t, σ(t) > t

cjiujìOi3fjghzai3ha ë

T

nC_au_%c{grdnf{¡uj3_fjfji3cjikp6ghxFgrpsFp

m

^vFkFiîCxFi

T ^k := T − {m}

bafjnFi3c grujiv u{if

T ^k = T .

^ë

T

nF_au_6hifKuj3_fjfji3c{i3kàp6_UíBgrp%s}p

M

^vkFiîCxFi

T ^k := T − {M }

bafjnFi3c grujivÚujif

T ^k = T .

\n}ixFbafe_feghbdxFu

[0, b], [0, b),

^_ax}k ^ujb ^bdxv ghr;kFixFbafei6fegrp]i*u{³_ahi3u grxfji3c{z_ahu

[0, b] = {t ∈ T : a ≤ t ≤ b},

nFicji

0, b ∈ T

^ghfjn

0 < ρ(b).

<÷ùRHZõµH@MOR ¤ª

X

^»¤© ^¶ ©a·O©>§ ´ }©>§¤C¤±$¹>· §ª@¦¨«U·

f : T → X

¦ºº»¤à§j©aººr¤j¥

rd−

§«U·Cª@¦·C¹«U¹´ O¯e«µ¦@¥>¤j¥*¦ª¦r´ä§«U·Cª@¦·C¹«U¹´©aª¤j©>§¯¦U>ªK¥>¤·}´µ¤ C«U¦·Cª7©a·O¥]}©U´©*ºr¤Z±$ª¨´¦@¥>¤j¥

ºÐ¦®þ¦ª/©aª¤j©>§ C«U¦·Cª

¤

¯$¦ª¤

f ∈ C rd ( T ) = C rd ( T , X ).

t ∈ T ^k ,

^ªZC¤

∆

¥>¤¯¦U©aª@¦a¤m«¬±

f

^©aª

t,

^¥>¤· ^«Uª¤j¥

f ^∆ (t),

»¤ªZC¤·C¹B®

»¤¯ O¯e«µ¦@¥>¤j¥]¦ª\¤ d¦r´ªZ´!]¦±ù±³«U¯-©aºº

ε > 0

ªC¤¯e¤þ¤ d¦r´ªZ´|©*· ¤¦U}»«U¯eC«µ«3¥

U

^«¬±

t

´¹§ªZ}©aª

|f (σ(t)) − f (s) − f ^∆ (t)[σ(t) − s]| ≤ ε|σ(t) − s|

±³«U¯-©aºº

s ∈ U,

^©aª#"$

t

2ZsFxFfegrbax

F

gru\³_rrik,_axfeghkFi3c{ghz_feg zdiWb

f : T → X

ìFcjbµz>grkFik

F ^∆ (t) = f (t)

Zbdc\i³_n

t ∈ T ^k .

% <'& AFDG?() ¡¦*!,+Z±

f

¦r´þ§«U·Cª@¦·C¹«U¹´-;ªC¤·

f rd−

§«U·Cª@¦·C¹«U¹´

¡¦¦*!.+±

f

¦r´-¥>¤ºÐª¨©¥a¦/W¤¯e¤·Cª¦@©»3ºr¤©aª

t

^ªC¤·

f

¦r´%§«U·Cª@¦·C¹C«U¹t´©aª

t

2ZsFxFfjgrbdx

p : T →

ghu\³_ahrik ¯¤0a¯e¤´´¦a¤gh

1 + µ(t)p(t) 6= 0

^Zbdc_r

t ∈ T ,

nFicji

µ(t) = σ(t) − t

^v ^nFghn gru3_arhi3kfenFi1a¯{©a¦·C¦· ¤´´Ö±$¹B· §ª@¦¨«U·C i%kFi3x}bafeiþ`

R ⁺

^fen}i

ujifb/fen}i6c{i3dc{i3uju{ghzai6ZsFxFfeghbdxFu3 \nFi]di3xFice_ahgri3k iíBì!baxFi3xfegr_aÚZsFxFfegrbax

e p

grukFiîFxFi3k _u

fen}isFxFghq>s}i*ujbars}fjgrbdx ba\fenFi*grxFg fegÝ_z_arsFi6ìFcjba`Frip

y ^∆ = p(t)y, y(0) = 1

^v ^nFi3c{i

p

^gru_

cjidcjiuju{ghzdiZsFxFfeghbdxÚxiíBìFhgrghf7Zbdc{p%sFr_-Zbdc

e p (t, 0)

gru\aghzdix`

e p (t, s) = exp Z t

s

ξ _µ(τ) (p(τ ))∆τ

ghfen

ξ h (z) =

( Log(1 + hz) h

gh

h 6= 0,

z

^gh

h = 0.

2CbdcÚp]bdcji\kFif_aghruvuji3i ¨ /ri³_cjhav

e p (t, s)

xFizdicÖz_axFghujnFiu3 ixFb dghzaiujbdp]i\ZsFxFkC_ap]i3x>

f_IìFc{bdìOi3c{fjgriu baÚfjnFii$íBì!bdx}i3xfegr_aZs}xFfjgrbdx ^0if

p, q : T →

^f bcjidcjiujujg zdiZs}xFfjgrbdxFu

i|k}iîCxFi

p ⊕ q = p + q + µpq, p := − p

1 + µp , p q := p ⊕ ( q).

+-,./102+v ad4365 bF >vCì;ø

(4)

T<tMDU<'& ¸´´¹B®*¤ª}©aª

p, q : T →

©a¯e¤¯e¤0a¯e¤´´¦a¤ù±$¹>· §ª¦¨«U·}´-;ªC¤·mªC¤Ú±³«Uººr«

¦·'äC«Uº ¥

¡¦*!

e 0 (t, s) ≡ 1

^©a·O¥

e p (t, t) ≡ 1

¡¦¦*!

e p (σ(t), s) = (1 + µ(t)p(t))e p (t, s);

¡¦¦¦*!

1 e p (t, s) = e _p (t, s);

¡¦ !

e p (t, s) 1

e p (s, t) = e _p (s, t);

!

e p (t, s)e p (s, r) = e p (t, r);

µ¦*!

e p (t, s)e q (t, s) = e p⊕q (t, s);

µ¦¦*!

e p (t, s)

e q (t, s) = e _{p q} (t, s).

C([0, b],

)

gru/fjnFi|7_axC_nu{ìC_a3iba_ah3bdxfjgrxtsFbdsFu7Zs}xFfjgrbdxFu\Zc{bdp

[0, b]

^ghxfeb ^ghfen

fen}i|x}bdcjp

kyk ∞ = sup{|y(t)| : t ∈ [0, b]}.

L ¹ ([0, b],

)

k}i3xFbafji6fjnFi*u{ìC_a3i6ba7ZsFx}feghbdxFuZcjbdp

[0, b]

^grxfjb ^n}grn _ac{i^0i3`Oi3u{dsFi grxfji3dcj_a`FrigrxfenFifjgrp]iWuj3_ariujixFuji|xFbdc{p6ik,`t

kyk L ¹ = Z b

0

|y(t)|∆t

Zbdc\i3_an

y ∈ L ¹ ([0, b],

) AC((0, b),

)

ghufjnFi7u{ìC_a3i\bkFg

i3c{i3xfegr_a`Fri;ZsFxFfeghbdxFu

y : (0, b) →

nFbauji7îCc{u{fÚkFihfe_

kFicjghz_fjghzaiav

y ^∆

vCghu_a`Fu{bdrs}fji3 3baxtfjgrxtsFbds}u3

^0if

(X, | · |)

`Oi_xFbdcjp]i3k ujìC_aiav

P (X) = {Y ⊂ X : Y 6= ∅}

^v

P cl (X) = {Y ∈ P (X) : Y

^3hbdujik

}

^v

P b (X) = {Y ∈ P(X) : Y

`!bds}xFkFi3k

}, P c (X) = {Y ∈ P (X) : Y

3baxtzaií

}, P cp (X) = {Y ∈ P (X) : Y

3bdp]ìC_af

}.

p%s}hfeg z_arsFik p*_aì

N : [0, b] → P cl (

)

^gruue_grk feb,`Oi®*¤j©U´¹>¯{©»3ºr¤vgh;Zbdcizdic{

y ∈

vfjnFiþZsFxFfjgrbdx

t 7−→ d(y, N (t)) = inf{|y − z| : z ∈ N (t)}

gru-p6i3_aujs}ce_a`Fhi nFicji

d

gru|fenFi*p6ifecjgh*ghxFkFsFi3k ` fjnFi7_xC_an ujìF_a3i

ë x nC_fZbdhrbu3v i ghr0_ujujs}p6ifenC_UfyfjnFiZsFxFfegrbax

F : [0, b] ×

→ P(

)

^gru

7_acj_fenFobBk}bdc{avCg¨ia

g

t → F (t, x)

gru\p]i³_au{sFce_a`}riZbdc\i3_an

x ∈

^v

ghg

x → F (t, x)

ghu\sFìFìOi3cujip6gh3bdxfjgrxtsFbdsFu7Zbac_arp]bdu{f\_ar

t ∈ [0, b]

^v

2Cbdc\i³_n

y ∈ C([0, b],

)

^vChif

S F,y

fenFiujifybau{i3hi3fjgrbdx}uyb

F

k}iîCxFikm`

S F,y = {v ∈ L ¹ ([0, b],

) : v(t) ∈ F (t, y(t)), a.e. t ∈ [0, b]}.

\nFiZbdhrbgrx}^0i3p]p6_|ghu/3c{sF3gr_agrx*fjnFiì}cjb>ba0babds}c/p6_agrx]cjiujsF feu n}i3xfen}iWp%sF feghz_aÉ

sFikp6_aì,nC_au3baxtzaiíz_ars}i3u3

+-,./102+v ad4365 bF >vCìÚú

(5)

< & & A, ¤ª

X

^»¤ ^© ^¶ ©a·O©>§ ´ }©>§¤ ¤ª

F : J × X −→ P cp,c (X)

^»¤ ^©

©a¯{©aªZ «3¥>«U¯$°|®þ¹BºÐª@¦U©aºÐ¹C¤j¥þ®]© m©a·O¥ºr¤ª

Γ

^{»¤W©%ºÐ¦·} ¤j©a¯W§«U·Cª¦·C¹C«U¹´®]© O¦·';±$¯«U®

L ¹ (J, X)

ª«

C(J, X)

^;ªC¤· ªC¤þ«- C¤¯j©aª«U¯

Γ ◦ S F : C(J, X) −→ P cp,c (C(J, X)),

y 7−→ (Γ ◦ S F )(y) := Γ(S F (y) )

¦r´©,§ºr«µ´µ¤j¥a¯{© Cà«- C¤¯{©aª«U¯¦·

C(J, X ) × C(J, X).

*3 ,/ ' ,$

i grr!_au{ujsFp]iWZbdc/fjnFicjip*_aghxFkFic/bafen}gru\ìC_aìOi3cfenC_f3v}Zbdc7i3_an

k = 1, . . . , m,

fen}iì!bdghxfeu baCghp6ì}sFru{i

t k

_ac{ic{grdnfIkFi3xFu{ia ë xbdcjkFicfjbkFiîCxFi/fjnFi/u{bdrsBfegrbax|b

ø

v /i7u{nC_ah>bdxFu{grkFic

fen}iZbarrbghxF%ujìF_a3i

P C = {y : [0, b] −→

: y k ∈ C(J k ,

), k = 0, . . . , m,

_axFk,fen}i3cjii$í}ghu{f

y(t ⁻ _k )

_axFk

y(t ⁺ _k )

^ghfjn

y(t ⁻ _k ) = y(t k ), k = 1, . . . , m},

nFghnghu_]/_axC_anujìC_3i ghfenfjnFixFbdcjp

kyk P C = max{ky k k J k , k = 0, . . . , m},

nFicji

y k

ghuùfenFicjiu{fjcjgrfeghbdxþba

y

^feb

J k = (t k , t _k+1 ] ⊂ [0, b], k = 1, . . . , m

^v>_ax}k

J ₀ = [t 0 , t ₁ ].

^0ifysFuu{fe_ac{fy`kFiîCxFgrx} nC_Uf /i|p6i3_ax,`_äu{bdrsBfegrbaxbaì}cjbd`Fhi3p

ø

<÷ùRHZõµH@MOR) ¸ ±$¹>· §ª¦¨«U·

y ∈ P C ∩ AC(J \{t ₁ , . . . t m },

)

¦r´y´3©a¦@¥äª«]»¤©þ´µ«UºÐ¹>ª@¦¨«U· «¬±

! !ä¦±ªC¤¯e¤þ¤ d¦r´ªZ´|© ±$¹B· §ª@¦¨«U·

v ∈ L ¹ ([0, b],

)

´¹C§,ªZ}©aª

y ^∆ (t) + p(t)y ^σ (t) = v(t)

^© ^@¤«U·

J \{t k }, k = 1, . . . , m,

©a·O¥,±³«U¯ ¤j©>§

k = 1, . . . , m

*ªC¤þ±$¹B· §ª@¦¨«U·

y

^{´3©aª@¦r´} ^"7¤´ ^ªC¤ §«U·O¥a¦ª@¦¨«U·

y(t ⁺ _k ) − y(t ⁻ _k ) = I k (y(t ⁻ _k )),

©a·O¥6ªC¤|¦·Cª@¦@©aºI§«U·O¥a¦ª¦¨«U·

y(0) = η.

ixFii3kfenFiZbdhrbgrxF%_asBíBgrhgÝ_acK6c{i3u{sFhf

uji3i ¢û

< & & A,) ¤ª

p : T →

^»¤

rd−

§«U·Cª@¦·C¹«U¹´*©a·O¥m¯e¤0a¯e¤´´¦a¤¹ C«µ´µ¤

f : T →

rd−

§«U·Cª@¦·C¹«U¹´ ¤ª

t 0 ∈ T ,

^©a·O¥

y 0 ∈

.

^C¤·

y

¦r´ªC¤%¹>·C¦3¹¤|´µ«UºÐ¹>ª¦¨«U· «¬±ªC¤þ¦·C¦ª@¦@©aº U©aºÐ¹C¤ O¯e«a»3ºr¤®

y ^∆ (t) + p(t)y ^σ (t) = f(t), t ∈ [0, b] ∩ T , t 6= t k , k = 1, . . . , m

^ú

y(t ⁺ _k ) − y(t ⁻ _k ) = I k (y(t ⁻ _k )), k = 1, . . . , m,

³

y(0) = y 0 ,

^ü

¦±-©a·O¥,«U·CºÐ°]¦±

y(t) = e _p (t, 0)y ₀ + Z t

0

e _p (t, s)f(s)∆s + X

0<t ^k <t

e _p (t, t k )I k (y(t ⁻ _k )).

^û

+-,./102+v ad4365 bF >vCì-3

(6)

\nFiZsFx}feghbdx

F : [0, b] ×

→ P (

)

^gru 7_acj_fenFobBk}bdc{a

W

\nFicji|iíBgruKf3bdx}u{f_xtfju

c k > 0

u{sFnfjnC_f

|I k (x)| ≤ c k

Zbdc\i³_n

k = 1, . . . , m

_xFkZbac_ar

x ∈

.

ø

\nFicjiþi$í}ghu{f_63bdxfeghxtsFbdsFuWx}bdxBk}i33c{i³_au{grxFZs}xFfjgrbdx

ψ : [0, ∞) −→ (0, ∞),

^_]ZsFxF

fjgrbdx

p ∈ L ¹ ([0, b],

+ )

_ax}km_þ3baxFu{fe_axf

M > 0

u{sFnfjnC_f

kF (t, x)k _P = sup{|v| : v ∈ F (t, x)} ≤ p(t)ψ(|x|)

Zbac\i³_an

(t, x) ∈ [0, b] ×

,

_ax}k

M

|η| sup

t∈[0,b]

e p (t, 0) +

m

X

k=1

c k sup

t∈[0,b]

e p (t, t k ) + sup

(t,s)∈[0,b]×[0,b]

e p (t, s)ψ(M ) Z b

0

p(s)∆s

> 1.

T<tMDU<'& )) ¹ C«µ´µ¤ª}©aªÖ>°- C«UªC¤´µ¤´ ! !þC«Uº ¥ C¤·àªC¤¦® O¹>º´¦a¤-¥a°U·O©a®þ¦¨§

¦· §ºÐ¹´¦¨«U·}´1 ! !}©U´©aªÖºr¤j©U´ª7«U· ¤´µ«UºÐ¹Bª@¦¨«U· «U·

[0, b]

DGMM ce_axFuKZbdcjp fjnFi]ìFcjbd`}ri3p

ø

ghxtfjb_î}íBi3k ì!bagrxf|ìFcjbd`}ri3p/bdxFujghkFi3cfen}i

bdìOi3cj_febdc

N : P C −→ P (P C)

k}iîCxFikm`

N (y) = {h ∈ P C : h(t) = e p (t, 0)η + Z t

0

e p (t, s)v (s)∆s

+ X

0<t k <t

e p (t, t k )I k (y(t ⁻ _k )), v ∈ S F,y }.

% <'& AFDG?() ºr¤j©a¯ºÐ° B±$¯«U® ¤®þ®]© ÚªC¤ "$B¤j¥ C«U¦·CªZ´|«¬±

N

©a¯¤y´µ«UºÐ¹>ª¦¨«U·}´ª¬« ! ! iujnF_arujnFb fenF_f

N

ue_fjgruKîCi3uyfenFiþ_ujujs}p6ì}fjgrbdx}u baÚfjnFix}bdxFrghxFi³_c _hfeicjxC_Ufeghzai-bÖ^ice_³

~BnC_sFkFi3cf>ì!i \nFi-ì}cjb>ba grr `!idg zdixghxu{izai3ce_uKfeiìFu3

ï õ³<>é Cê

N (y)

ghu\3bdxzdi$íZbdc\i³_an

y ∈ P C

ë xFkFii3kvgh

h 1 , h 2

`Oi3rbaxFfeb

N (y)

^vOfenFix fenFicjiþiíBghu{f

v 1 , v 2 ∈ S F,y

ujsFn fenF_fWZbdcWi3_an

t ∈ [0, b]

^/inC_³zdi

h i (t) = e p (t, 0)η + Z t

0

e p (t, s)v i (s)∆s + X

0<t k <t

e p (t, t k )I k (y(t ⁻ _k )) (i = 1, 2).

^0if

0 ≤ d ≤ 1

^\nFi3x vZbac\i³_an

t ∈ [0, b]

^inC_³zdi

(dh 1 + (1 − d)h 2 )(t) = e p (t, 0)η +

Z t 0

e p (t, s)[dv 1 (s) + (1 − d)v 2 (s)]∆s

+ X

0<t k <t

e _p (t, t k )I k (y(t ⁻ _k )).

+-,./102+v ad4365 bF >vCì;ü

(7)

~BghxF3i

S F,y

gru\bdxzdi$í

`!i³_as}uji

F

nC_au\bdxzdiíz_ahsFi3u

vFfen}i3x

dh 1 + (1 − d)h 2 ∈ N (y).

ï õ³<>é

N

^®]© C´-»«U¹B·O¥>¤j¥þ´µ¤ªZ´¦·Cª«*»«U¹>·O¥>¤j¥ä´µ¤ªZ´|¦·

P C.

^0if

B q = {y ∈ P C : kyk P C ≤ q}

`!iy_|`!bds}xFkFi3k6ujif7grx

P C

^_axFk

y ∈ B q

vtfen}i3x6ZbacÖi3_an

h ∈ N (y)

v}fenFicjiiíBghu{feu

v ∈ S F,y

u{sFnfjnC_f\Zbdc\i3_an

t ∈ [0, b]

^v

h(t) = e p (t, 0)η + Z t

0

e p (t, s)v(s)∆s + X

0<t k <t

e p (t, t k )I k (y(t ⁻ _k )).

2Ccjbdp

_axFk

ø

/i|nC_³zdi

|h(t)| ≤ |η| sup

t∈[0,b]

e _p (t, 0) + sup

(t,s)∈[0,b]×[0,b]

e _p (t, s) Z b

0

|v(s)|∆s +

m

X

k=0

e p (t, t k )c k

≤ |η| sup

t∈[0,b]

e _p (t, 0) + sup

(t,s)∈[0,b]×[0,b]

e _p (t, s) Z b

0

ψ (q)p(s)∆s +

m

X

k=0

sup

t∈[0,b]

e _p (t, t k )c k

≤ |η| sup

t∈[0,b]

e _p (t, 0) + sup

(t,s)∈[0,b]×[0,b]

e _p (t, s)ψ(q)kpk L ¹

+

m

X

k=0

sup

t∈[0,b]

e p (t, t k )c k .

ï õ³<>é )

N

^®]© C´-»«U¹B·O¥>¤j¥þ´µ¤ªZ´¦·Cª«¤ 3¹>¦¨§«U·Cª¦·C¹C«U¹t´´µ¤ª´%«¬±

P C

^0if

u ₁ , u ₂ ∈ J, u ₁ < u ₂

^_ax}k

B q

`Oi|_þ`!bds}xFkFi3kujifyba

P C

_uyghx~>feiìàþ_axFk

y ∈ B q

2Cbdc\i³_n

h ∈ N (y)

v}fen}i3cjii$í}ghu{fju

v ∈ S F,y

u{sFnfjnC_fZbdc\i3_an

t ∈ [0, b]

^v

h(t) = e _p (t, 0)η + Z t

0

e _p (t, s)v(s)∆s + X

0<t k <t

e _p (t, t k )I k (y(t ⁻ _k )).

+-,./102+v ad4365 bF >vCìÖû

(8)

|h(u ₂ ) − h(u ₁ )| ≤ |e _p (u ₂ , 0) − e _p (u ₁ , 0)||η|

+ψ(q)kpk L ¹

Z u 1

0

|e p (u 2 , s) − e p (u 1 , s)|∆s +ψ(q)kpk L ¹

Z u 2

u 1

e p (u 2 , s)∆s

+ X

0≤t k <u 1

|e _p (u 2 , t k ) − e _p (u 1 , t k )|c k

+ X

u 1 ≤t k <u 2

e _p (u ₂ , t k )c k .

\nFiäcjghdnfnC_ax}k u{grkFi%fji3xFk}uWfeb,3icjb_au

u 2 − u 1 → 0

u_3bdx}uji3qtsFixF3i6ba/~>feiìFu

feb,ø

febadifjnFi3c-ghfen]fenFi c{3i3U ujbdrg \nFibdcjipv /i ³_ax*3bdxFrsFk}i fenF_f

N : P C −→ P(P C )

^gru

3bap6ìFhifji3h63baxtfjgrxtsFbds}u3

ï õ³<>é

N

}©U´-©§ºr«µ´µ¤j¥a¯{© C

^0if

y n → y ∗ , h n ∈ N (y n )

^_axFk

h n → h ∗

ixFii3kfebäujnFb fjnC_f

h ∗ ∈ N (y ∗ )

h n ∈ N (y n )

p6i3_axFu\fenF_f\fenFicjiiíBghu{feu

v n ∈ S F,y n

ujsFnfenC_Uf\Zbdc\i³_an

t ∈ [0, b]

^v

h n (t) = e _p (t, 0)η +

Z t 0

e _p (t, s)v n (s)∆s + X

0<t k <t

e _p (t, t k )I k (y n (t ⁻ _k )).

i|psFu{fyujn}b fenC_UfyfjnFi3c{iiíBgruKfeu

h _∗ ∈ S F,y _∗

ujsFnfenC_f\Zbac\i³_an

t ∈ [0, b]

^v

h ∗ (t) = e p (t, 0)η +

Z t 0

e p (t, s)v ∗ (s)∆s + X

0<t k <t

e p (t, t k )I k (y ∗ (t ⁻ _k )).

/ri3_acj dvFu{grxFi

I k , k = 1, . . . , m,

_ac{i3bdxfjgrxtsFbdsFuv /inC_³zdi

h n − X

0<t k <t

e p (t, t k )I k (y n (t ⁻ _k ))

−

h ∗ − X

0<t k <t

e p (t, t k )I k (y ∗ (t ⁻ _k ))

_{P C} −→ 0,

^_au

n → ∞.

/bdxFu{grkFic\fenFi3baxtfjgrxtsFbds}uyhgrxFi3_ac7bdìOi3cj_febac

Γ : L ¹ ([0, b],

) → C([0, b],

)

dg zdi3x`

v 7−→ (Γv)(t) = Z t

0

e _p (t, s)v (s)ds.

2Ccjbdp ^0i3p]p6_mB3BvIghf|Zbdrhb ufjnC_f

Γ ◦ S F

gru-_rbdu{i3k dcj_aìFn baì!ice_fjbdc3lbdc{i3bµzdic3v i

nC_³zdi

h n (t) − X

0<t k <t

e _p (t, t k )I k (y n (t ⁻ _k ))

∈ Γ(S F,y n ).

+-,./102+v ad4365 bF >vCì;

(9)

~BghxF3i

y n → y _∗ ,

ghf7Zbarrbu7Zc{bdp^0i3p]p6_äB3fjnC_f\Zbdc\i3_an

t ∈ [0, b]

^v

h _∗ (t) = e _p (t, 0)η +

Z t 0

e _p (t, s)v ∗ (s)∆s + X

0<t k <t

e _p (t, t k )I k (y ∗ (t ⁻ _k )),

Zbdc\u{bdp6i

v _∗ ∈ S F,v _∗

ï õ³<>é ê ¸ O¯$¦¨«U¯¦Ú»«U¹B·O¥U´%«U·m´µ«UºÐ¹>ª@¦¨«U·}´

^0if

y

`OiujsFn,fenF_f

y ∈ λN (y)

Zbdc\u{bdp6i

λ ∈ (0, 1)

\nFi3xvFfjnFi3c{iiíBgruKfeu

v ∈ S F,y

u{sFn

fenF_f\Zbdc\i³_n

t ∈ [0, b]

^v

y(t) = λe p (t, 0)η + λ Z t

0

e p (t, s)v(s)∆s + λ X

0<t k <t

e p (t, t k )I k (y(t ⁻ _k )).

\nFghuyghp6ì}rgriu/`

W

_ax}k

ø

fjnC_f³vFZbac\i³_an

t ∈ [0, b]

^v

|y(t)| ≤ |η| sup

t∈[0,b]

e _p (t, 0) +

m

X

k=1

c k sup

t∈[0,b]

e _p (t, t k )

+ sup

(t,s)∈[0,b]×[0,b]

e p (t, s) Z b

0

p(s)ψ(|y(s)|)∆s

≤ |η| sup

t∈[0,b]

e _p (t, 0) +

m

X

k=1

c k sup

t∈[0,b]

e _p (t, t k )

+ sup

(t,s)∈[0,b]×[0,b]

e _p (t, s)ψ(kyk P C ) Z b

0

p(s)∆s.

/bdxFu{i3qtsFixtfjh

kyk P C

|η| sup

t∈[0,b]

e p (t, 0) +

m

X

k=1

c k sup

t∈[0,b]

e p (t, t k ) + sup

(t,s)∈[0,b]×[0,b]

e p (t, s)ψ(kyk P C ) Z b

0

p(s)∆s

≤ 1.

\nFixm`

ø

vBfenFicji|iíBgruKfeu

M

u{sFnfjnC_f

kyk P C 6= M.

^0if

U = {y ∈ P C : kyk P C < M }.

\nFibaì!ice_fjbdc

N : U → P (P C )

ghu-s}ìFì!icu{i3p]gr3baxtfjgrxtsFbds}u%_axFk 3bdp]ìFrifeih bdxfegrxtsFbasFu3

2Ccjbdp fen}i%nFbdgh3i%ba

U

^v fjnFi3c{iþgru xFb

y ∈ ∂U

^ujsFn ^fjnC_f

y ∈ λN (y)

Zbacujbap6i

λ ∈ (0, 1).

u_6bdxFujiqtsFi3xFiþbafjnFix}bdxFrghxFi³_c_ahfji3cjxF_feg zdi|baÚ^0i3cj_³t¬~BnC_as}kFi3cf>ìOi

üv /i-kFi3kFs}3i

fenF_f

N

nC_auy_þî}íBi3kì!bdghxf

y

^grx

U

n}grnmghu_þujbdhs}feghbdxbafenFiìFc{bd`Frip

ø

ix}b ìFc{i3ujixf f bäbafenFic7iíBghu{feixF3i|cjiujsF feu/Zbac/fenFiì}cjbd`Fhi3p

ø

nFi3x,fen}icjgrantf

nC_ax}kujgrk}iWnC_au7bdxzdi$íz_arsFiu7sFxFk}i3c i³_ai3c\3baxFkFghfjgrbdx}u/bdxfjnFiZsFxFfeghbdxFu

I k (k = 1, ..., m)

_uys}uji3kgrx

4

Zbac\grp]ìFsFru{ghzaiWkFg i3c{i3xfegr_aghxF3hsFujghbdxFu

+-,./102+v ad4365 bF >vCì;

(10)

! C¤¯e¤%¤ d¦r´ª\§«U·}´ª¡©a·Cª´

c k > 0

´¹§ª}©aª

|I k (x)| ≤ c k |x|

±³«U¯%¤j©>§

k = 1, ..., m

^{©a·O¥©aºº}

x ∈

.

!

H d (F (t, y), F (t, y)) ≤ l(t)|y − y|

±³«U¯¤j©>§

t ∈ [0, b]

^©a·O¥ ^©aºº

y, y ∈

^C¤¯e¤

l ∈ L ¹ ([0, b],

+ ) ∩ R ⁺

^©a·O¥

d(0, F (t, 0)) ≤ l(t)

^© ^@¤

t ∈ [0, b].

+Z±

sup

(t,s)∈[0,b]×[0,b]

e p (t, s)klk L ¹ +

m

X

k=1

sup

t∈[0,b]

e p (t, t k )c k < 1,

ªC¤· ªZC¤ O¯e«a»3ºr¤® ! !}©U´-©aªºr¤j©U´ª\«U· ¤W´µ«UºÐ¹>ª¦¨«U· «U·

[0, b]

DGMM ^0if

y

`!iujsFnfjnC_f

y ∈ λN (y)

Zbdc]ujbdp]i

λ ∈ (0, 1)

\nFixv7fen}i3cjimiíBgruKf

v ∈ S F,y

t ∈ [0, b]

^v

y(t) = λe p (t, 0)η + λ

Z t 0

e p (t, s)v(s)∆s + λ X

0<t k <t

e p (t, t k )I k (y(t ⁻ _k )).

\nFghuyghp6ì}rgriu/`

ú _ax}k

3

fjnC_fZbdc\i³_n

t ∈ [0, b]

^v

|y(t)| ≤ |η| sup

t∈[0,b]

e _p (t, 0) +

m

X

k=1

sup

t∈[0,b]

e _p (t, t k )c k |y(t ⁻ _k )|

+ sup

(t,s)∈[0,b]×[0,b]

e p (t, s) Z b

0

|v(s)|∆s.

≤ |η| sup

t∈[0,b]

e _p (t, 0) +

m

X

k=1

sup

t∈[0,b]

e _p (t, t k )c k |y(t ⁻ _k )|

+ sup

(t,s)∈[0,b]×[0,b]

e _p (t, s) Z b

0

|l(s)y(s) + l(s)|∆s

≤ |η| sup

t∈[0,b]

e _p (t, 0) +

m

X

k=1

sup

t∈[0,b]

e _p (t, t k )c k kyk P C

+ sup

(t,s)∈[0,b]×[0,b]

e _p (t, s)kyk P C klk _L ¹

+ sup

(t,s)∈[0,b]×[0,b]

e p (t, s)klk L ¹ .

/bdxFu{i3qtsFixtfjh

kyk P C ≤

|η| sup

t∈[0,b]

e p (t, 0) + sup

(t,s)∈[0,b]×[0,b]

e p (t, s)klk L ¹

1 − sup

(t,s)∈[0,b]×[0,b]

e p (t, s)klk L ¹ −

m

X

k=1

sup

t∈[0,b]

e p (t, t k )c k

:= M.

+-,. 10 +\v!ad365ybF BvFì

(11)

U = {y ∈ P C : kyk P C < M + 1}.

\nFibaì!ice_fjbdc

N : U → P (P C )

2CcjbdpfenFi|nFbdgh3iba

U

vfjnFi3c{igruxFb

y ∈ ∂U

ujs}nfenF_f

y ∈ λN (y)

Zbdc\u{bdp]i

λ ∈ (0, 1).

^u

_3bdxFu{i3qtsFixF3iba!fenFiyxFbdx}rgrx}i³_ac;_a feicjxC_fjghzai\ba0^0i3cj_³t¬~BnC_as}kFi3cÖfBìOi!

ü ¨v /iykFikFsF3iyfenC_Uf

N

nC_auy_%î}íBi3kì!bagrxf

y

^ghx

U

nFgrngru_äujbdhs}feghbdxbafenFiìFc{bd`Fhi3p ø

T<tMDU<'& )

+·*©¥¥a¦ª¦¨«U·äª« !©a·O¥ ! ©U´´¹B®*¤ª}©aª0ªZC¤ ±³«Uººr«

¦·'|§«U·O¥a¦ª¦¨«U·}´ÖC«Uº ¥

-!

lim

|x|→+∞

I k (x)

x = 0

±³«U¯%¤j©>§

k = 1, ..., m

+Z±

sup

(t,s)∈[0,b]×[0,b]

e p (t, s)klk L ¹ +

m

X

k=1

sup

t∈[0,b]

e p (t, t k )ε k < 1,

C¤¯e¤

ε k , k = 1, . . . , m

^©a¯e¤ C«µ´¦ª@¦a¤-§«U·}´ª¨©a·Cª´Wª}©aª

¦ººC´ C¤§¦"7¤j¥þº ©aª¤¯ ÚªC¤·mªC¤ O¯e«a»3ºr¤®

! !%}©U´-©aªÖºr¤j©U´ª\«U· ¤W´µ«UºÐ¹Bª@¦¨«U· «U·

[0, b]

DGMM

^0if

y

`!iujsFnfjnC_f

y ∈ λN (y)

Zbdc]ujbdp]i

λ ∈ (0, 1)

\nFixv7fen}i3cjimiíBgruKf

v ∈ S F,y

t ∈ [0, b]

^v

y(t) = λe _p (t, 0)η + λ

Z t 0

e _p (t, s)v(s)∆s + λ X

0<t ^k <t

e _p (t, t k )I k (y(t ⁻ _k )).

ü

grp]ìFhgri3uÖfenC_UfyZbac\i³_an

ε k > 0

v}fjnFi3c{iiíBgruKfeuy_]bdxFuKf_axf

A > 0

ujsFnfenC_Uf

|x| ≥ A ⇒ |I k (x)| ≤ ε k |x|.

^0if

E 1 = {t; t ∈ [0, b] : |x(t)| < A}, E 2 = {t; t ∈ [0, b] : |x(t)| ≥ A}

_axFk

C ₁ = max{|I k (x(t))|, t ∈ E ₁ }.

(12)

3 _xFk

ü

v}Zbdc\i3_an

t ∈ [0, b]

^v

|y(t)| ≤ |η| sup

t∈[0,b]

e _p (t, 0) + X

t k ∈E 1

e _p (t, t k )|I k (y(t ⁻ _k ))|

+ X

t k ∈E 2

e p (t, t k )|I k (y(t ⁻ _k ))|

+ sup

(t,s)∈[0,b]×[0,b]

e _p (t, s)kyk P C klk L ¹

+ sup

(t,s)∈[0,b]×[0,b]

e p (t, s)klk L ¹

≤ |η| sup

t∈[0,b]

e _p (t, 0) + C ₁

m

X

k=1

sup

t∈E 1

e _p (t, t k ) +

m

X

k=1

sup

t∈E 2

e _p (t, t k )ε k kyk P C

+ sup

(t,s)∈[0,b]×[0,b]

e _p (t, s)kyk P C klk L ¹

+ sup

(t,s)∈[0,b]×[0,b]

e p (t, s)klk L ¹ .

/bdxFu{i3qtsFixtfjh

kyk P C ≤

|η| sup

t∈[0,b]

e _p (t, 0) + C ₁

m

X

k=1

sup

t∈E 1

e _p (t, t k ) + sup

(t,s)∈[0,b]×[0,b]

e _p (t, s)klk _L ¹

1 − sup

(t,s)∈[0,b]×[0,b]

e _p (t, s)klk _L ¹ −

m

X

k=1

sup

t∈E 2

e _p (t, t k )ε k

:= ¯ M .

^0if

U = {y ∈ P C : kyk P C < M ¯ + 1}.

\nFibaì!ice_fjbdc

N : U → P (P C )

2Ccjbdp fen}i%nFbdgh3i%ba

U

^v fjnFi3c{iþgru xFb

y ∈ ∂U

^ujsFn ^fjnC_f

y ∈ λN (y)

Zbacujbap6i

λ ∈ (0, 1).

u_6bdxFujiqtsFi3xFiþbafjnFix}bdxFrghxFi³_c_ahfji3cjxF_feg zdi|baÚ^0i3cj_³t¬~BnC_as}kFi3cf>ìOi

üv /i-kFi3kFs}3i

fenF_f

N

nC_auy_þî}íBi3kì!bdghxf

y

^grx

U

n}grnmghu_þujbdhs}feghbdxbafenFiìFc{bd`Frip

ø

i\ì}cji3u{i3xf;xFbQ_cjiujsF fùZbdcIfjnFi\ìFc{bd`Frip

ø

ghfjnþ_WxFbdx}3bdxzdi$íþz_ars}i3kþcjghdnfùnC_xFk

ujghkFiaÖ^0if

(X, d)

`Oi|_þp6ifecjghujìC_3igrxFk}sF3ikZc{bdp fen}i|x}bdcjp]i3k,ujìF_a3i

(X, | · |)

/bdxFu{grkFic

H d : P (X) × P (X) −→

₊ ∪ {∞}

dghzai3x`

H d (A, B) = max

sup

a∈A

d(a, B), sup

b∈B

d(A, b)

,

nFicji

d(A, b) = inf

a∈A d(a, b), d(a, B ) = inf

b∈B d(a, b)

^\nFi3x

(P b,cl (X), H d )

gruÖ_|p6ifecjghujìC_ai _axFk

(P cl (X), H d )

ghu_äai3xFice_ahgr3ik,p6ifecjghWujìF_a3i

u{i3i

(13)

<÷ùRHZõµH@MOR) ¸ ®þ¹BºÐª@¦U©aºÐ¹¤j¥«- C¤¯{©aª«U¯

N : X → P cl (X)

¦r´þ§j©aººr¤j¥

© !

γ

¦* C´µ§>¦ª-¦±-©a·O¥,«U·CºÐ°ä¦±ªZC¤¯e¤þ¤ d¦r´ª´

γ > 0

´¹C§ªZ}©aª

H d (N (x), N(y)) ≤ γd(x, y)

±³«U¯%¤j©>§

x, y ∈ X,

» ! ©§«U·Cª@¯{©>§ª¦¨«U·à¦±©a·O¥«U·CºÐ°]¦±¦ªÖ¦r´

γ

¦* C´µ§>¦ª

¦ªZ

γ < 1

N

^nC_au6© ^"$B¤j¥ C«U¦·Cªùg fjnFi3c{i]ghu

x ∈ X

^u{sFn ^fjnC_f

x ∈ N (x).

\nFiäî}íBi3k ì!bdghxfujif|ba

fen}iypsFhfjghz_ars}i3k%bdìOi3cj_febac

N

grr}`!ikFixFbafeik6`

F ixN

2Cbdcùp]bdcji\kFif_aghruÚbdx]p%sF feg za_rsFik p6_aìFu /ic{iZic|fjbfen}i*`ObBbdtu-ba10igrp]rgrx}

3 ¨vbdc{xFgrigr

û v ys _axFk _aìC_di3bdc{dgrbas

¢a_ax}k bdruKfebdx}bddbµz¢aü

WsFcW3baxFujghkFi3cj_feghbdxFu _acji%`F_aujik bdxmfen}iZbdrhbgrxFäî}íBi3k ìObdgrxf fenFibdcjipZbdcy3baxtfjce_afeghbdx

p%sF feg za_rsFik6bdìOi3cj_febacju7dg zdix`t /bµzBg fe_ax}k 5_akFhi3cgrx

û

ø

ujii_aru{b 0i3ghp6hgrxFFv 3

\nFibdcjip

4

< & & A,) ¤ª

(X, d)

^{»¤%©§«U®} Oºr¤ª¤®*¤ª@¯$¦¨§´ }©>§¤ +Z±

N : X → P cl (X)

¦r´%©m§«U·Cª¯{©>§

ª@¦¨«U· ÖªZC¤·

F ixN 6= ∅

^0ifysFugrxfec{bBk}sF3in>ì!bafjnFi3u{i3u1nFgrnm_acji_ujujs}p6iknFicji3_feic

F : [0, b] ×

−→ P cp (

)

nC_auäfen}iì}cjbdìOi3cKf fenC_f

F (·, y) : [0, b] → P cp (

)

^ghu

p]i³_au{sFce_`FriWZbdc\i3_an

y ∈

\nFicji|iíBgruKf3bdx}u{f_xtfju

d k ≥ 0

ujs}nfenF_f

|I k (y) − I k (y)| ≤ d k |y − y|

Zbdc\i3_an

y, y ∈

.

T<tMDU<'& ) ¸W´´¹>®*¤|ªZ}©aª ! ! !*©a¯e¤´3©aª¦r´ "7¤j¥ ¤ª

τ > 1

^+Z±

1 τ +

m

X

k=1

G _∗ d k < 1,

^C¤¯e¤

G _∗ = sup

(t,s)∈[0,b]×[0,b]

e _p (t, s),

ªC¤· ªZC¤ +² ! !%}©U´|©aªºr¤j©U´ª\«U· ¤W´µ«UºÐ¹>ª¦¨«U· «U·

[0, b]

% <'& AFDG?() «U¯¤j©>§

y ∈ P C,

^{ªC¤´µ¤ª}

S F,y

¦r´,· «U· ¤® Oª@°à´¦· §¤»3°( !

F

^}©U´©

®*¤j©U´¹>¯j©»3ºr¤´µ¤ºr¤§ª¦¨«U· ´µ¤¤ C¤«U¯e¤®+ + +-!

DGMM i%ujnF_arujn}b fjnC_f

N

ue_UfegruKîCi3uyfenFi%_auju{sFp6ìBfegrbaxFu bÖ^ip6p6_]ø}B \nFiìFcjb>ba

grh `Oidghzai3xgrxf b*u{fji3ìFu

ï õ³<>é

N (y) ∈ P cl (P C)

±³«U¯¤j©>§

y ∈ P C

ë xFkFii3kvÚhif

(y n ) n≥0 ∈ N (y)

^u{sFn ^fenC_f

y n −→ y ˜

^grx

P C

^\nFi3x

y ˜ ∈ P C

^_axFk ^fjnFi3c{i

iíBghu{feu

v n ∈ S F,y

ujs}nfenF_fZbdc\i³_an

t ∈ [0, b]

y n (t) = e _p (t, 0)η + Z t

0

e _p (t, s)v n (s)∆s + X

0<t k <t

e _p (t, t k )I k (y(t ⁻ _k )).

+-,. 10 +\v!ad365ybF BvFì ø

(14)

w u{grxF fenFi/@_ffjnC_f

F

nC_uù3bdp]ìC_afz_ahsFi3u_axFk%Zcjbap 3

v i\p*_³|ìC_au{ufebW_ujsF`}uji3qtsFixF3i

gh xFi3i3u{ue_acK6fjbdiffjnC_f

v n

3baxtzai3c{di3u7fjb

v

^ghx

L ¹ ([0, b],

)

_axFk*nFi3xFi

v ∈ S F,y

\nFixvBZbdc

i³_n

t ∈ [0, b]

^v

y n (t) −→ y(t) = ˜ e _p (t, 0)η + Z t

0

e _p (t, s)v(s)∆s + X

0<t k <t

e _p (t, t k )I k (y(t ⁻ _k )).

~BbFv

˜

y ∈ N (y)

ï õ³<>é C¤¯e¤%¤ d¦r´ª´

γ < 1

´¹C§,ªZ}©aª

H d (N (y), N(y)) ≤ γky − yk

Zbdc\i³_n

y, y ∈ P C.

^0if

y, y ∈ P C

^_axFk

h ₁ ∈ N (y)

&\n}i3xvfenFicji-iíBgruKfeu

v ₁ (t) ∈ F (t, y(t))

ujsFnmfjnC_fZbdci³_n

t ∈ [0, b]

^v

h ₁ (t) = e _p (t, 0)η + Z t

0

e _p (t, s)v 1 (s)∆s + X

0<t k <t

e _p (t, t k )I k (y(t ⁻ _k )).

2Ccjbdp

23

ghf\Zbdhrbu/fjnC_f\Zbdc\i3_an

t ∈ [0, b]

^v

H d (F (t, y(t)), F (t, y(t))) ≤ l(t)|y(t) − y(t)|.

ixF3ivCfenFicjiiíBgruKfeu

w ∈ F (t, y(t))

t ∈ [0, b]

^v

kv ₁ (t) − wk ≤ l(t)|y(t) − y(t)|.

/bdxFu{grkFic

U : [0, b] → P (

)

dghzai3x`t

U(t) = {w ∈

: kv 1 (t) − wk ≤ l(t)|y(t) − y(t)|}.

~BghxF3iyfenFiWpsFhfjghz_ars}i3k]bdì!ice_fjbdc

V (t) = U(t) ∩ F (t, y(t))

grup6i3_aujsFcj_a`Fhi

uji3iÚc{bdì!baujghfjgrbdx

ëjë{ë ¢ú,ghx

vfenFicjiäiíBghu{feu_ZsFx}feghbdx

v 2 (t)

^n}grn gru_p]i³_au{sFce_a`}ri%u{i3rifjgrbdxàZbdc

V

^þ~Bb}v

v ₂ (t) ∈ F (t, y(t))

_xFkZbdc\i³_an

t ∈ [0, b]

^v

kv ₁ (t) − v ₂ (t)k ≤ l(t)|y(t) − y(t)|.

^0ifysFukFiîCxFiZbdc\i³_an

t ∈ [0, b]

^v

h 2 (t) = e p (t, 0)η +

Z t 0

e p (t, s)v 2 (s)∆s + X

0<t k <t

e p (t, t k )I k (y(t ⁻ _k ))

i|k}iîCxFi|bdx

P C

_ax,i3qtsFg z_arixtfyxFbacjp `

kyk _∗ = sup

t∈J

e _{(τ G} _∗ _l) (t, 0)|y(t)|

^Zbdc_r

y ∈ P C,

+-,. 10 +\v!ad365ybF BvFì ú

(15)

nFicji

e _{(τ G} _∗ l )(t, 0)

ghu7fenFisFx}grqtsFiujbdhs}feghbdxbafenFiìFc{bd`Fhi3p

y ^∆ (t) = τ G _∗ l(t)y(t), y(0) = 1,

nFicji

τ G _∗ l

gru_äcjidcjiuju{ghzdiZsFxFfeghbdx

2Ccjbdp

23

_axFk

vFZbdc\i3_an

t ∈ [0, b]

^v

|h 1 (t) − h 2 (t)| ≤ Z 1

0

e p (t, s)kv 1 (s) − v 2 (s)k∆s +

m

X

k=1

e _p (t, s)|I k (y(t ⁻ _k )) − I k (y(t ⁻ _k ))|

≤ Z t

0

G _∗ l(s)|y(s) − y(s)|∆s +

m

X

k=1

G ∗ d k |y(t ⁻ _k ) − y(t ⁻ _k )|

≤ ¹ _τ Z t

0

τ G ∗ l(s)|y(s) − y(s)|∆s +

m

X

k=1

d k G _∗ |y(t ⁻ _k ) − y(t ⁻ _k )|

≤ ¹ _τ Z t

0

τ G _∗ l(s)e τ G _∗ l (s, 0)e τ G _∗ l (0, s)|y(s) − y(s)|∆s + τ

τ

m

X

k=1

d k G ∗ |y(t ⁻ _k ) − y(t ⁻ _k )|

≤ 1 τ

Z t 0

[e τ G ∗ l (s, 0)] ^∆ e (τ G _∗ l) (s, 0)|y(s) − y(s)|∆s + τ

τ

m

X

k=1

d k G _∗ |y(t ⁻ _k ) − y(t ⁻ _k )|

≤ 1

τ e τ G ∗ l (t, 0)ky − yk ∗ +

m

X

k=1

d k G ∗ e τ G ∗ l (t, 0)ky − yk ∗ .

\ntsFu

kh 1 − h 2 k ∗ ≤ 1 τ +

m

X

k=1

G ∗ d k

!

ky − yk ∗ .

_ax _axC_rbddbasFu-c{i3r_feghbdxvba`}f_aghxFi3k `t grxfeicjnC_xFdgrx}mfenFi*cjbdhi3u%ba

y

^_ax}k

y,

ghf|Zbdrhbu fenF_f

H d (N (y), N (y)) ≤ 1 τ +

m

X

k=1

G ∗ d k

!

ky − yk ∗ .

~BbFv

N

gru;_3baxtfjce_afeghbdx]_axFk6fjntsFu3v>`]^ip6p6_ø}¢}v

N

nF_auÖ_î}íBik*ìObdgrxf

y

n}grn*ghu;ujbdhs}feghbdx feb ø

+-,. 10 +\v!ad365ybF BvFì 3

(16)

iþi3qtsFgrìàfjnFiþujìF_a3i

P C

g fenmfjnFiþbdc{kFi3cWc{i3r_feghbdx

” ≤ ”

^k}iîCxFik `tfenFiþbdxFi

K

^grx

P C

^v

fenF_fgru

K = {y ∈ P C : y(t) ≥ 0, t ∈ [0, b]}.

ë

fgruÖ>x}b xfjnC_fÖfenFiy3baxFi

K

ghux}bdcjp6_aCgrx

P C.

^\nFi kFife_agrhu;ba03baxFi3u/_axFk6fjnFi3ghcì}cjbdìOi3cKfegriu p6_µ`!iZbdsFx}kghxyi3gh>tghÝ_þ_axFk^_atujnFp]grU_axfjnC_ap

^if

α, β ∈ P C

ujsFnmfjnC_f

α ≤ β.

fen}i3x`t_xbacjkFicyghxfei3cKz_a

[α, β]

ip6i3_ax_äu{if baIì!bdghxfeu\grx

P C

dg zdix`

[α, β] = {y ∈ P C : α ≤ y ≤ β}.

^0if

D, Q ∈ P cl (P C )

^\nFixà`

D ≤ Q

^i|p6i3_ax

α ≤ β

^Zbac_ar

α ∈ D

^_axFk

β ∈ Q.

^\ntsFu

α ≤ D

grp]ìFhgri3u;fenF_f

α ≤ y

Zbdc\_ah

y ∈ Q

grxìF_ac{fjgr3s}Ý_acvBg

D ≤ D,

fen}i3x,ghf7ZbarrbuÖfenC_f

D

gru_þu{grxFdhifjbdx,ujif3

X

N : X → P cl (X)

^¦r´

x, y ∈ X

^¦ª

x < y,

N (x) ≤ N(y).

T<tMDU<'&

¤ª

[a, b]

A, B : [a, b] → P cl (X)

¡¦*!

A

¡¦¦*!

B

^¦r´§«U® Oºr¤ª¬¤ºÐ°§«U·Cª¦·C¹C«U¹´-

¡¦¦¦*!

A

^©a·O¥

B

¡¦ !

A(x) + B (x) ⊂ [a, b]

±³«U¯-©aºº

x ∈ [a, b].

¹B¯$ªC¤¯]¦±äªC¤§«U· ¤

K

^¦·

X

x ∈ A(x) + B(x)

x _∗

x ^∗

^¦·

[a, b].

^{«U¯e¤«a¤¯}

x _∗ = lim

n→∞ x n

©a·O¥

x ^∗ = lim

n∞ y n ,

^C¤¯e¤

{x n }

^©a·O¥

{y n }

[a, b]

^¥>¤ ^";· ^¤j¥»3°

x n+1 ∈ A(x n ) + B (x n ), x 0 = a

^©a·O¥

y n+1 ∈ A(y n ) + B(y n ), y 0 = b.

\nFiWZbdhrbgrxF|3baxF3iì}fbarbi3c\_xFksFìFìOi3c\u{bdrs}fjgrbdx}uÖZbdc

ø

nC_u7`!ii3x,grxfjcjb>kFsF3ik

` /ixFnFbdnFcj_}v ixFkFi3c{ujbdx v_ax}k 5febdsB_au üùZbdcWìOi3c{grb>kFgr`!bds}xFkC_acKmz_ars}i%ìFcjba`Frip6uWZbdc

grp]ìFsFhujg zdi]kFg

i3c{i3xfegr_a;grxFrsFu{grbdxFu_fîBí}ik p6bap6ixtfju

uji3i,_rujb 3

ë f6grh;`!i]fenFi`F_aujgh

feb>bdghxfenFi|_aìFì}cjb_anfjnC_fZbdrhbu3

<÷ùRHZõµH@MOR ) ¸ ±$¹B· §ª@¦¨«U·

α ∈ P C

¤¯´µ«UºÐ¹Bª@¦¨«U· «¬± ! !¦±þªZC¤¯e¤

¤ d¦r´ª´

v 1 ∈ L ¹ (J,

)

´¹C§àª}©aª

v 1 (t) ∈ F (t, α(t))

^© ^@¤«U·

J

α ^∆ (t) + p(t)α ^σ (t) ≤ v 1 (t)

J

t 6= t k , α(t ⁺ _k ) − α(t ⁻ _k ) ≤ I k (α(t ⁻ _k )), t = t k , k = 1, . . . , m

^©a·O¥

α(0) ≤ η

β ∈ P C

v 2 ∈ L ¹ (J,

)

´¹C§,ª}©aª

v 2 (t) ∈ F (t, β (t))

^© ^@¤«U·

J

β ^∆ (t) + p(t)β ^σ (t) ≥ v 2 (t)

^© ^@¤«U·

J

t k 6= t k

β(t ⁺ _k ) − β(t ⁻ _k ) ≥ I k (β(t ⁻ _k ))

t = t k

k = 1, . . . , m

^Ö©a·O¥

β(0) ≥ η

+-,. 10 +\v!ad365ybF BvFì ü

(17)

\nFip%s}hfeg ZsFxFfegrbax

F (t, y)

gru\x}bdxFkFi3cji3_aujghxF]grx

y

_ahp6bau{f\izdi3cK nFi3c{i-Zbdc

t ∈ [0, b]

\nFiZsFx}feghbdxFu

I k , k = 1, . . . , m

_ac{i|bdxfeghx>s}bdsFuy_axFk,xFbdx}kFi3cji³_ujgrx}F

Wø

\nFicji7i$íBgru{f

α

^_axFk

β ∈ P C,

cjiujìOi3feghzai3 -hb/icù_axFk-sFìFìOi3cIujbdhs}feghbdxFu0ZbdcfenFiìFcjbd`}ri3p ø

u{sFnfjnC_f

α ≤ β.

[0, b]

DGMM 0 iîCxFif b6p%s}hfeg z_arsFikp6_aìFu

A, B : P C → P (P C )

^`

A(y) = {h ∈ P C : h(t) = e p (t, 0)η + Z t

0

e p (t, s)v(s)∆s, v ∈ S F,y },

_axFk

B (y) = {h ∈ P C : h(t) = X

0<t k <t

e _p (t, t k )I k (y(t ⁻ _k ))}.

ë f|³_ax `Oi]u{nFbxvù_u|ghx fenFiäìFc{bBbaZub \nFibdcjip6u|ø}¢ø,_axFk ø}BvfjnC_f

A

^_axFk

B

kFiîCx}i]fjnFi p%sF fegÉ¡z_ars}i3k bdìOi3cj_febdc{u

A : [α, β] → P cl,cv,bd (P C)

^_xFk

B : [α, β ] → P cp,cv (P C )

^ë ^f

³_x `Oiujghp6ghÝ_ac{h u{nFbx fenC_f

A

^_xFk

B

_acjic{i3u{ì!ifeg zdih psFhfjg ¡z_ahsFi3k 3bdxfjce_afegrbax _axFk

3bap6ìFhifji3hm3bdxfjgrxtsFbdsFubdx

[α, β ]

^" i]ujnC_ahùujn}b ðfenC_f

A

^_ax}k

B

_acjiäghujbafjbdxFiägrxFcji3_aujghxF

bdx

[α, β ]

^{;^if}

x, y ∈ [α, β ]

`!iu{sFnfjnC_f

x < y.

^\nFi3x`t v}Zbdc\i3_an

t ∈ [0, b]

^v

A(x) = {h ∈ P C : h(t) = e _p (t, 0)η +

Z t 0

e _p (t, s)v (s)∆s, v ∈ S F,x }

≤ {h ∈ P C : h(t) = e p (t, 0)η + Z t

0

e p (t, s)v (s)∆s, v ∈ S F,y }

= A(y).

ixF3i

A(x) ≤ A(y)

Ö~Bghp6ghÝ_ac{hþ`

W

vBZbdc\i³_an

t ∈ [0, b], B(x) ≤ B(y).

\ntsFu

A

^_xFk

B

_ac{ighujbafjbdxFighxF3c{i³_au{grxF6bdx

[α, β]

2ghxC_ahh iþìFc{bGzaifenF_f

A(y) + B(y) ⊂ [α, β]

Zbdci3_an

y ∈ [α, β]

^{^if}

h ∈ A(y) + B(y)

^`!i*_ax iri3p]i3xf3 \nFixvùfenFicjii$íBgru{fju

v ∈ S v,y

t ∈ [0, b]

^v

h(t) = e _p (t, 0)η +

Z t 0

e _p (t, s)v(s)∆s + X

0<t k <t

e _p (t, t k )I k (y(t ⁻ _k )).

^0if

t i = max{t k : t k < t}.

+-,. 10 +\v!ad365ybF BvFì û

(18)

t ∈ [0, b]

^v

h(t) ≤ e _p (t, 0)β(0) + Z t 1

0

e _p (t, s)[β ^∆ (s) + p(t)β ^σ (s)]∆s +

Z t 2

t ⁺ ₁

e _p (t, s)[β ^∆ (s) + p(t)β ^σ (s)]∆s + . . . +

Z t t i

e _p (t, s)[β ^∆ (s) + p(t)β ^σ (s))]∆s +

k=i

X

k=1

e _p (t, t k )I k (β(t ⁻ _k )).

\ntsFu3vCZbdc\i3_an

t ∈ [0, b]

e p (t, 0)h(t) ≤ β(0) + Z t 1

0

[e p (s, 0)β(s)] ^∆ ∆s + Z t 2

t ⁺ ₁

[e p (s, 0)β(s)] ^∆ ∆s

+ . . . + Z t

t ⁺ i

[e p (s, 0)β(s)] ^∆ ∆s +

k=i

X

k=1

e p (t k , 0)I k (β(t ⁻ _k ))

= β(0) + e p (t ⁻ ₁ , 0)β(t ⁻ ₁ ) − e p (0, 0)β(0) + e p (t ⁻ ₂ , 0)β(t ⁻ ₂ )

−e p (t ⁺ ₁ , 0)β(t ⁺ ₁ ) + . . . + e p (t, 0)β(t) − e p (t ⁺ _i , 0)β(t ⁺ _i ) +

k=i

X

k=1

e p (t k , 0)I k (β(t ⁻ _k ))

= −e p (t 1 , 0)[β(t ⁺ ₁ ) − β(t ⁻ ₁ )] − e p (t 2 , 0)[β(t ⁺ ₂ ) − β(t ⁻ ₂ )]

− . . . − e p (t i , 0)[β(t ⁺ _i ) − β(t ⁻ _i )] + e p (t, 0)β(t) +

k=i

X

k=1

e p (t k , 0)I k (β(t ⁻ _k ))

≤ −e p (t ₁ , 0)I ₁ (β(t ⁻ ₁ )) − e p (t ₂ , 0)I ₂ (β(t ⁻ ₂ ))

− . . . − e p (t i , 0)I i (β(t ⁻ _i )) + e p (t, 0)β(t) +

k=i

X

k=1

e p (t k , 0)I k (β(t ⁻ _k ))

= e p (t, 0)β(t).

ixF3i

h(t) ≤ β(t)

Zbdc\i3_an

t ∈ [0, b].

~Bghp6ghÝ_ac{hdv`,c{i3ìFr_a3ghxF

β

^g fen

α

_axFk,cjizdi3c{ujghxF]fjnFibdcjk}i3c3v i|³_axìFcjbµzaifjnC_f

h(t) ≥ α(t),

Zbac\i³_an

t ∈ [0, b].

\nFix

α ≤ N (y) ≤ β

^Zbac_ar

y ∈ [α, β].

(19)

u_ 3baxFujiq>s}i3xFi ba \nFi3bacji3p ú}Bv /i kFikFsFi fjnC_f

N

nC_auhi³_auKf_axFk dcji3_feiu{f,î}íBi3k ìObdgrxfÖgrx

[α, β].

\nFgru;ZsFc{fjnFi3cÖgrp]ìFhgri3uIfenC_f;fjnFi ì}cjbd`Fhi3p ø

nF_aup]grx}grp6_aF_ax}k6p6_UíBgrp6_a

ujbars}fjgrbdxFu/bdx

[0, b].

+- ",

~BsFì}ì!bdu{i

T = [0, 1] ∪ [2, 3] ∪ [4, 5]

^_axFk

p

_|cji3acji3u{ujg zdiZsFx}feghbdx ibdxFu{grkFicÖfjnFiiq>sF_feghbdx

y ^∆ (t) = p(t)y(t), y(0) = 1.

i|3_axi³_au{grh6ujn}b òfenC_f\fjnFisFxFghq>s}i|u{bdrsBfegrbaxbaIfjnFi|_a`ObGzai-iqtsC_feghbdx,gru\dg zdix`

y(t) = e p (t, 0) =



 



 



e

⁰ ^t ^p(s)∆s ,

^gh

t ∈ [0, 1],

exp( R 1

0 p(s)∆s + R t

2 p(s)∆s),

^gh

t ∈ [2, 3], exp( R 1

0 p(s)∆s + R 3

2 p(s)∆s + R t

4 p(s)∆s),

^gh

t ∈ [4, 5].

hujb i-bdxFujghkFi3c7fjnFiZbdrhbgrxF%kBBxF_ap6ghgrxFrsFu{grbdxbIfjnFiZbdcjp

y ^∆ (t) + p(t)y ^σ (t) ∈ F (t, y(t)), t ∈ [0, 1], t 6= 1

2 ,

y 1 2

+

− y 1 2

−

= I ₁ y 1

2

−

,

y(0) = 0,

nFicji

F : [0, 1] ×

→ P (

)

gru7fjnFip%sF feghz_ahsFi3k*p*_aìk}iîCxFikm`

(t, x) → F (t, x) := h x ²

x ² + 2 + t, x ²

x ² + 1 + t + 1 i .

ë fgru3ri3_ac fjnC_f

F

gru_bdp6ìF_afbdxzdi$í za_rsFik p%sFhfjghz_ahsFi3kp*_aìà_axFk ba7_ace_UfenFobBkFbac{d

y ∆ (t) + p(t)y σ (t) ∈ F (t, y(t)), t ∈ J := [0, b] ∩ T , t 6= t k , k = 1, . . . , m, y(t + k ) − y(t − k ) = I k (y(t − k )), k = 1, . . . , m,

Electronic Journal of Qualitative Theory of Differential Equations 2005, No. 12, 1-22; http://www.math.u-szeged.hu/ejqtde/