On the Oscillation of Solutions of Fractional Vector Partial Differential Equations with Deviating Arguments
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1 Aec evew o Mhec d Sc Je 7 ol 5 No 5-57 ISSN: Ole Cogh The AhoAll gh eeved blhed b Aec eech Ie o olc eveloe OI: 56/v55 : h://doog/56/v55 O he Ocllo o Solo o Fcol eco l eel Eqo wh evg Age Sdhv d J vh Abc I h cle we vege he ocllo o olo o cl o col veco l deel eqo wh devg ge o he o F G bec o he bod codo e wll eblh he ce codo o -ocllo o olo o gve e whee veco e lo ovde ele o lle he el ewo: Fcol l ocllo veco deel eqo devg ge Iodco Fcol deel eqo e ow ecogzed ecelle oce o owledge odellg dcl ocee el l d oo ce eleccl ewo obbl d c vco elc eleco che o cooo eleco dc o cole ed ole heolog dl oboc ecooc boecholog ec See he ece oogh [ - 6 9] o heo d lco o col deel eqo Ocllo olo l o ole he qve d qlve heo o col deel eqo Thee e evel e delg wh ocllo o cl col od deel eqo [ ] oweve ol ew el hve eed egdg he ocllo behvo o cl col l deel eqo ee [ 8-6] d he eeece ced hee o Gde d eech ee o MhecThvllv Govee A College Al e vetl Nd Id El : ovdh@glco o Gde d eech ee o MhecThvllv Govee A College Al e vetl Nd Id Coedg ho El : vh@glco
2 6 Aec evew o Mhec d Sc ol 5 Je 7 I 97 ol odced he coce o -ocllo o vege he ocllo o olo o veco deel eqo whee veco e ee he cle [6-7] o veco od deel eqo d [ ] o veco l deel eqo To he ee e hee e lo o lee o ocllo el o veco col od deel eqo d veco col l deel eqo cll o veco col ole l deel eqo Moved b h we e he col ode veco l deel eqo o del eqo Folo o he oble: The ocllo heo o col deel eqo w odced b Gce e l [9] v q l b J q whee q deoe he e-ovlle deel oeo o q whee q Che [] d e l [8] ded he ocllo o he col deel eqo wh ovlle gh ded col devve o ode o he ollowg o q g h e l [8] d Sdhv d vh [] veged he col l deel eqo wh e-ovlle le ded deo o he hl o he o G dv v v q wh he Ne bod codo N F q g G bec o he bod codo To he be o o owledge ohg ow egdg he -ocllo behvo o he ollowg cl o veco col l deel eqo wh oced e o he o G F whee boded do wh ecewe ooh bod co he e-ovlle col devve o ode o wh eec o he lc oeo he Eclde - ce e d he l Eclde o
3 Sdhv & J vh 7 Eqo leeed wh he ollowg bod codo whee he eeo ol veco o d ove coo co o d I wh ollow we lw e who eog h A C ; C ; ; A C ; l l ; co A C G ; d I ; C G; d ; A F C G; d A 5 C ; e cove d o deceg wh o d hee e ove ch h o ll I The d o -ocllo behvo o col l deel eqo ed h e O och o edce l-deol oble o o oe deol ocllo oble o cl col col deel eqle The oe o h e o eblh oe ew -ocllo ce o eqo wh d eqo wh b g geelzed cc echqe d egl vegg ehod O el e eell ew ele I h eco we gve he deo o -ocllo col devve d egl d oe oo whch e el hogho h e Thee e evel o deo o col devve d egl I h e we e he e-ovlle le ded deo o he hl- The ollowg oo wll be ed o he coveece F d whee d eo: B olo o d we e o vl co C G; C G [ˆ ; C G [ ; d e o G d he bod codo d whee ˆ eo: e be ed veco A olo o d o be -ocllo G he e odc h zeo o Ohewe -oocllo eo: The e-ovlle col l devve o ode wh eec o o co gve b : v v dv ovded he gh hd de owe deed o whee he g co eo: The e-ovlle col egl o ode o co : o he hl- gve b
4 8 Aec evew o Mhec d Sc ol 5 Je 7 : o dv v v I ovded he gh hd de owe deed o eo: 5 The e-ovlle col devve o ode o co : o he hl- gve b : o I d d ovded he gh hd de owe deed o whee he celg co o e: [] e be olo o d : d o 5 The d o 6 e: [] I X d Y e oegve he Y X XY 7 whee ove ege -Ocllo o he oble e beg wh he ollowg e e: Ae h 5 A A hold e be ed veco d be olo o I evell ove he e he cl col l eql I evell egve he e he cl col l eql oo e be evell ove Tg he e odc o d we ge F h
5 Sdhv & J vh 9 B A we hve ce C we hve heeoe g we ge Sll le be evell egve we el ob The oo colee The e odc o wh eld he ollowg bod codo e: Ae h A A5 hold e be ed veco I he cl col l eql h o evell ove olo d he cl col l eql h o evell egve olo g he bod codo o he eve olo o he oble o -ocllo G oo Soe o he co h hee -oocllo olo o o G he evell ove o evell egve I evell ove he b e e he bod codo o Th codc he hohe The l oo ollow whe evell egve Theoe: Ae h A A5 d A 6 I A 7 hold I he col deel eql 5 h o evell ove olo d he col deel eql h o evell egve olo he eve olo o d -ocllo G oo Soe o he co h hee e olo o whch o -ocllo G ho lo o geel we e h [ o oe Iegg wh eec o ove we ob 6
6 5 Aec evew o Mhec d Sc ol 5 Je 7 d d d d d 7 g Gee ol d bod codo eld h ds ds d 8 d ds ds d 9 B g Jee eql 7 6 A A d we ge d d d d d d d d Alo b A d I vew o 8-7 eld Theeoe evell ove olo o 5 Th codc he hohe The ce whee [ c be eed ll d we e lo geg codco The oo ow colee Theoe: Soe h he codo 7 A A d hold Fheoe e h hee e ove co ; C ch h l whee e deed 5 A The eve olo o o he oble -ocllo G oo Soe o he co h hee e olo o he oble whch o - ocllo G ho lo o geel we e h [ o oe
7 Sdhv & J vh 5 Th evell ove olo o 5 The hee e ch h d o Theeoe ollow o 5 h [ o 5 Th o o oe e ow cl h o 6 Soe o he d hee e [ ch h Sce cl deceg o [ I cle h : c whee c co o [ Theeoe o 6 we hve [ o c The we ge [ o c Iegg he bove eql o o we hve c [ o c eg we ge c Th codc ece o [ hol ee he co b he geelzed cc bo [ o 7 The we hve o [ Fo 67 5d 5 A ollow h 8 e The The he bove eql becoe
8 5 Aec evew o Mhec d Sc ol 5 Je 7 9 Tg Y X g e d 9 we hve Iegg boh de o he bove eql o o we ob Tg he l e o boh de o he bove eql we ge l whch codc d colee he oo Theoe: Soe h he codo 7 A A d hold Fheoe oe h hee e ove co ; C d co C whee : : ch h o o whee : : d h coo d o-ove l devve o wh eec o he ecod vble d e l whee e deed Theoe The ll he olo o o he oble -ocllo G oo Soe h -oocllo olo o ho lo o geel we e h evell ove olo The evell ove olo o 5 The oceedg he oo o Theoe o ge llg he evo eql b d egg o o o [ we ob Theeoe whch codco o The oo colee Cooll Ae h he codo o Theoe hold wh elced b l
9 Sdhv & J vh 5 l he eve olo o -ocllo G Ne we code he ce whch el h doe o hold I h ce we hve he ollowg el Theoe: Soe h he codo A d hold d h hee e ove A7 co C ; ch h hol Fheoe e h o eve co T whee T T The eve olo o d T o 5 -ocllo o e l oo Soe h -oocllo olo o ho lo o geel we e h evell ove olo The evell ove olo o 5 The oceedg he oo Theoe hee e wo ce o he g o The oo whe evell ove l o h o Theoe d hece oed Ne e h evell egve The hee e ch h o Fo 6 we ge o The o Th we ge l : M M Ae o h M he o A we ge M o [ e 5 The g hee vle he bove eql becoe M o [ d M e cl h Iegg boh de o he l eql o o we hve M M M
10 5 Aec evew o Mhec d Sc ol 5 Je 7 M ece o 6 we ge M Iegg he l eql o o we ge d M eg o we ge l Th codc Theeoe we hve M h l Th l ece he oo -Ocllo o he oble I h eco we eblh ce codo o he ocllo o ll olo o Fo h we eed he ollowg:the lle ege vle o he chle oble o ove d he coeodg ege co ove Theoe: e ll he codo o Theoe d be hold The eve olo o o d -oclle G oo Soe h -oocllo olo o d ho lo o geel we e h [ o oe Mllg boh de o he Eqo b d he egg wh eec o ove we ob o d d d d d g Gee ol d bod codo ollow h d d d d d d d B g d Jee eql 6 A d 7 A we ge d d d d d d Se d d Theeoe d d I 5 B 5 A d 6
11 Sdhv & J vh 55 I vew o -6 el 7 o e o he oo l o h o Theoe d d hece he del e oed Cooll I he eql 7 h o evell ove olo he eve olo o d -ocllo G Cooll e he codo o Cooll hold; he eve olo o d - ocllo G Theoe: e he codo o Theoe hold; The eve olo o 7 -ocllo o e l The oo o Coolle d d Theoe e l o h o Seco d hece he del e oed 5 Ele I h eco we gve ele o lle he el eblhed Seco Ele Code he veco col l deel eqo F 5 G whee G wh he bod codo 5 ee co F d I e o ee h ] [ e e we obeve h e co d
12 56 Aec evew o Mhec d Sc ol 5 Je 7 e d co Te I cle h codo A d hold Theeoe A7 Th ll he codo o Theoe e ed ece ollow h eve olo o 55 e - ocllo G Ic oe ch olo o he oble 5 d 5 e oe h he bove olo o e ocllo G whee e Acowledgee: The ho h oe Thd o h o o colee he e eeece S Abb M Bechoh d JJ Neo Globl Acv o Solo o Nole Fcol Ode e- ovlle ole-sele l Iegl Eqo Elecoc Jol o Qlve Theo o eel Eqo 8-5 Ble ehel EScl d JJ Tllo Fcol Clcl Model d Necl Meho See o Cole Nole d Cho old Scec blhg cec NJ SA 7-85 Ble OG M d O eg A eev Ocllo el o e -ode Fcol eel Eqo Aled Mhec d Coo Xe Che Ocllo Ce o Fcol eel eqo Advce eece Eqo - 8 -Xe Che Ocllo Behvo o Cl o Fcol eel Eqo wh g B Scec Blle See A JI ol O he Ocllo o Solo o eco eel Eqo Sove Mhec old YI ohl Ocllo oee o Solo o eco eel Eqo eel Eqo El Mchev Noo Yohd Ocllo o Solo o eco eel Eqo o bolc Te wh Fcol Age Jol o Cool d Aled Mhec S Gce Agwl JY og AZe O he Ocllo o Fcol eel Eqo Fcol Clcl d Aled Al 5 - G d JE lewood d Gol Ieqle Cbdge Mhecl b Cbdge ve e Cbdge M SA 988 AA lb M Svv d JJ Tllo Theo d Alco o Fcol eel Eqo Eleve Scece B Aed The Nehel 6 JT Mchdo ov d F Md ece o o Fcol Clcl Coco Nole Scece d Necl Slo 6-5 F Md Fcol Clcl d ve e coelc Iel College e odo S Mlle d B o A Iodco o he Fcol Clcl d Fcol eel Eqo Joh le d So New Yo 99 ES No CA Swo Ocllo o Nole eco deel Eqo Al d Mec ed Alc
13 Sdhv & J vh 57 I odlb Fcol eel Eqo Acdec e S ego Cl SA 999 h S h Ocllo o Solo o Ilve eco ebolc eegl Eqo wh el Alcble Al h S h JJ Neo d J Ocllo o Te Fcol l eel Eqo Elecoc Jol o Qlve Theo o eel Eqo 5 - S h h d J J Neo Foced Ocllo o Solo o Nole Fcol l eel EqoAled Mhec d Coo Sdhv d J vh Foced Ocllo o Cl o Fcol bolc l eel Eqo Jol o Advce Mhec Sdhv d J vh Ievl Ocllo Ce o Fcol l eel Eqo h g Te Aled Mhec Scec eech blhg Sdhv d J vh Foced Ocllo o Ode l eel Eqo wh g d Fcol Age Ieol Jol o e d Aled Mhec l E Tov Fcol c ghe edco e Beg d Sge-elg Bel edlbeg Y g Z d S S Coe o O he Ocllo o Fcol-ode el eel Eqo wh Co Coece Coco Nole Scece e N Mo d F e Meg -Ocllo o Solo o Ce eco ebolc eegl Eqo wh evg Age Aled Mhec d Coo e N d ehog Sheg Ocllo oee o Solo o d o l Fcol eel Eqo wh g Te Jol o Nole Scece d Alco B Zheg Ocllo o Cl o Nole Fcol eel Eqo wh g Te Jol o Advced Mhecl Sde Zhel Yge Zho Yg S d Cho Zhg Ocllo o Cl o Fcol eel Eqo cee c Ne d Soce -6 Y Zho Bc Theo o Fcol eel Eqo old Scec Sgoe
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