Kocaeli Üniversitesi 41380, Kocaeli, Türkiye

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Ἀβραάμ Βουγιουκλάκης
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1 ĐKĐCĐ ETEBEDE EUTA DEAY DĐFEASĐYE DEKEEĐ KAAIIĞI AĐ FUAT YEĐÇEĐOĞU BAIŞ DEĐ Koali Ünirii 438 Koali Türi Ö: B maald abi aaılı doğral iini mrbdn nral dla difranil dnlmlrin daranışları ürind ml bir orm rilmişir. B ormin onçlarından ararlanara ararlılı rirlri ld dilmişir. O THE STABĐĐTY OF SECOD ODE EUTA DEAY DĐFFEETĐA EQUATĐO Abra: In i ar a bai orm on baior of olion of linar ond ordr nral dla diffrnial aion i ablid. A a onn of i orm a abili ririon i obaind. Kword: ral dla diffrnial aion Cararii aion Sabili Triial olion.. Inrodion and Prliminari onidr iniial al roblm for ond ordr dla diffrnial aion wr ar ral nmbr i oii ral nmbr and i a gin oninol diffrniabl iniial fnion on inral [ ]. In man fild of onmorar in and nolog m wi dlaing lin ar ofn m and dnamial ro in ar dribd b m of dla diffrnial aion [45]. T dla aar in omliad m wi logial and oming di wr rain im for informaion roing i ndd. T or of linar dla diffrnial aion a bn dlod in fndamnal monogra [] [4-6] [8].

2 T aion of form of i of inr in biolog in laining lf-balaning of man bod and in roboi in onring bid robo [] []. T ar illraion of inrd ndlm roblm. A ial aml i balaning of a i [3]. Eaion of form of an b d a of aion for nmrial mod [7] [4]. In [3] i a bn ablid bonddn ndr ondiion b > > and r < b on [ of olion of ond ordr aion b r r for wr r > ogr wi a gin oninol diffrniabl iniial fnion on [ r]. > nl Calon and Smid al. [] a ablid abili riria for a ond ordr dla diffrnial aion of form wi and. Ponragin or of ai-olnomial i d in i d. < Ti ar dal wi abili of riial olion for a ond ordr linar nral dla diffrnial aion wi onan dla. An ima of olion i ablid. T ffiin ondiion for abili amoi abili and inabili of riial olion ar gin. Or rl ar drid b of ral roo wi an aroria ror of orronding in a n ararii aion. T ni alid in obaining or rl ar originad in a ombinaion of mod d in [9] and []. A al a wi oninol diffrniabl ral-ald fnion dfind on inral [ i aid o b a olion of iniial al roblm and if aifi for all and for all. I i nown a for aml [4] for an gin iniial fnion r i a ni olion of iniial roblm - or mor brifl olion of -. Bfor loing i ion w will gi wo wll-nown dfiniion for aml [5]. T riial olion of i aid o b abl a if for r ε > r i a nmbr l l ε > a for an iniial fion wi ma < l olion of - aifi < ε for all [.

3 3 Orwi riial olion of i aid o b nabl a. oror riial olion of i alld amoiall abl a if i i abl in abo n and in addiion r i a nmbr > l a for an iniial fnion wi <l olion of - aifi lim.. Samn of main rl and ommn If w loo for a olion of of form for I w a i a roo of fir ararii aion. 3 now b olion of -. Dfin for [ wr i a ral roo of ararii aion 3. Tn for r w a or [ ]. 4 oror iniial ondiion an b ialnl wrin for ] [. 5 Frrmor b ing fa a i a roo of 3 and aing ino aon 5 w an rif a 4 i ialn o d d

4 4 d d d d d d d 6 wr. d. 7 and w dfin for. Tn w an a 6 rd o following ialn aion d. 8

5 5 If w loo for a olion of 8 of form for I w a i a roo of ond ararii aion. 9 On or and iniial ondiion 5 an b ialnl wrin ] [. F dno ararii fnion of 9 i.. F Sin i a rmoabl inglari of F w an rgard F a a nir fnion wi F. B b dfiniion of a roo of ararii aion 9 m bom. now b olion of 8- and b a ral roo of ararii aion 9. Dfin for for all [. Tn for r w a d. oror iniial ondiion an b ialnl wrin for ] [. Frrmor b ing fa a i a ral roo of 9 and aing ino aon w an rif a i ialn o

6 6 d d d d d d d d d d d d d d d d d d d d d d d d d d d 3

7 7 wr d d d. 4 w dfin w for 5 wr.. 6 Tn w an a 3 rd o following ialn aion d w w w d d w. 7 On or and iniial ondiion an b ialnl wrin w for ] [. 8 W a following bai orm. Torm. and b ral roo of ararii aion 3 and 9. Am a roo and a following ror µ < 9 and

8 8. Tn for an I C ] [ olion of - aifi µ for all wr and wr gin in 7 4 and 6 ril and ma. Proof. I a o a ror 9 garan a >. Aling dfiniion of and w w an obain a i ialn o w µ. So w will ro. From 8 and i follow a w for ] [. 3 W will ow a i a bond of w on wol inral [. aml w for all [. 4 To i nd l onidr an arbirar nmbr > ε. W laim a ε < w for r [. 5 Orwi b 3 r i a * > a ε < w for * < and ε * w. Tn ing 7 w obain * * * * d w w w ε d d w * * { }[ ] ε [ ] ε <

9 9 wi in iw of 9 lad o a onradiion. So or laim i r. Sin 5 old for r ε > i follow a 4 i alwa aifid. B ing 4 and 7 w dri w w w { µ for all. Ta man old. d w dd } Torm. and Conidr aifi b ral roo of ararii aion 3 and 9. a in Torm. Tn for an C [ ] I lim wr and olion of - wr gin in 7 4 and 6 ril. Proof. B dfiniion of and w w a o ro a lim w. 6 In nd of roof w will abli 6. B ing 7 and aing ino aon and 4 on an ow b an a indion a w aifi n w µ for all n n B 9 garan a µ. T from 7 i follow immdial a w nd o ro a i.. 6 old. < < T roof of Torm i omld.

10 Torm 3. and b ral roo of ararii aion 3 and 9 and alo ondiion in Torm and µ b roidd. Tn for an I C ] [ olion of - aifi for all µ 8 wr wa gin in { } ma 3 and { } ma ma ma ma ma. 3 oror riial olion of i abl if i i amoiall abl if < < and i i nabl if > >. Proof. B Torm i aifid wr and ar dfind b 7 and ril. From i follow a µ. 33 Frrmor b ing and 3 from 7 4 and w obain d

11 d d d { } { } ma ma. Hn from 33 w onld a for all µ 34 and onnl 8 old. ow l am a and. Dfin ma. I follow a. From 34 i follow a

12 µ µ for r. Sin > b aing ino aon fa a w a µ µ > µ µ for r [ wi man a riial olion of i abl a. if < and < n 8 garan a lim and o riial olion of i amoiall abl a. Finall if > > n riial olion of i nabl a. Orwi r i a nmbr l > of roblm - aifi Dfin l a for an C [ ] I wi < l olion < for all. 35 for [ ]. Frrmor b dfiniion of and b ing 9 w a d

13 3 d d d >. I C ] [ b dfind b l wr l i a nmbr wi l l < <. oror l b olion of -. From Torm i follow a aifi lim lim l / > l l. B w a l l < and n from 35 and ondiion > > i follow a lim. Ti i a onradiion. T roof of Torm 3 i omld.

14 4 3. Eaml Eaml. Conidr wr i an arbirar oninol diffrniabl iniial fnion on inral [ ]. In i aml w al ararii aion 3 and 9. Ta i ararii aion 3 i and w a i a roo of 37. Tn for ararii aion 9 i 6. 6 Trfor i a roo and ondiion of Torm 3 ar aifid. Ta i µ < and. 6 6 µ Sin and ro olion of 36 i amoiall abl. < < Eaml. Conidr 38 wr i an arbirar oninol diffrniabl iniial fnion on [ ]. T ararii aion 3 i 39 and w ail a i a roo of 39. Taing ararii aion 9 i

15 5. d Trfor w find a i a roo. Corronding o roo and ondiion of Torm 3 ar aifid. Ta i 5 µ < and. µ Sin and < ro olion of 38 i abl. Eaml 3. Conidr π π π π π π wr i an arbirar oninol diffrniabl iniial fnion on [ ]. T ararii aion 3 i π π π and w ail a i a roo of 4. Taing ararii aion 9 i π π. Trfor w find a i a roo. Corronding o roo and ondiion of Torm 3 ar aifid. Sin > and > ro olion of 4 i nabl.

16 6 4. frn []. Bllman and K. Coo Diffrnial-Diffrn Eaion. Aadmi Pr w Yor 963. [] B. Calon and D. Smid Sabili riria for rain ond-ordr dla diffrnial aion wi mid offiin Jornal of Comaional and Alid amai [3].D. Drir Eonnial da in om linar dla diffrnial aion JSTO: Amrian amaial onl: ol. 85 o [4].D. Drir Ordinar and Dla Diffrnial Eaion Al. a. in Sringr Brlin 977. [5] El gol.e. and orin S.B. Inrodion o Tor and Aliaion of Diffrnial Eaion wi Diaing Argmn Aadmi Pr w Yor ondon 973. [6] J.K. Hal and S.. Vrdn nl Inrodion o Fnional Diffrnial Eaion Sringr Brlin Hidlbrg w Yor 993. [7] G.D. H and T. ii Sabili of nmrial mod for m of nral dla diffrnial aion BIT mrial amai [8] V. Kolmanoi A. i Alid Tor of Fnional Diffrnial Eaion Klr Aadmi Dordr 99. [9] I.-G.E. Kordoni and C.G. Pilo T Baior of olion of linar ingro-diffrnial aion wi nbondd dla Comr & amai wi Aliaion []. adonald Biologial Dla Sm: inar Sabili Tor Cambridg Uniri Pr Cambridg w Yor 989. [] C.G. Pilo and I.K. Prnara Priodi fir ordr linar nral dla diffrnial aiıon Alid amai and Comaion7 3-. [] C.. Sl Sdi of ar r in Alid amai Vol.7 Amrian amaial Soi I [3] S.A. Tobia ain Tool Vibraion Blai ondon 965. [4] S. Yalçınbaş F. Yniçrioğl Ea and Aroima Solion of Sond Ordr Inlding Fnion Dla Diffrnial Eaıon wı Variabl Coffıın Alid amai and Comaion ol:

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