Çaaya Üiversiesi Fe-Edebiya Faülesi, Jural Ars ad Scieces Say : 9 / May s 008 A Ne e Cai Rule ime Scales abe Abdeljawad Absrac I is w, i eeral, a e cai rule eeral ime scale derivaives des beave well as i e case usual derivaive Hwever, we discuss sme special cases were e ime scale derivaive as e usual cai rule e resuls are aalyzed r b e dela ad abla ime scales derivaives Key wrds: ime scales derivaive, -ime scale, H-ime scale, Frward jump perar, Bacward jump perar,rd-ciuus, ld-ciuus, Cai rule Irduci ad Prelimiaries e calculus ime scales was irduced [], [] uiy e ciuus ad discree aalysis As usually expeced we we eeralize sme ery we lse sme ice prperies Oe e basic cceps a researcers as care abu we deal wi diereial euais, variaial calculus ad s, ime scales is cai rule Fr a reas we ry ere summarize ad rmulae sme cases were e cai rule beys e rder, s a i mi be useul use i ex wrs i e uure Fr e ery dela derivaive ime scales we reer [] ad r e e r abla derivaive we reer [] A ime scale is a arbirary empy clsed subse e real lie us real umbers,, ad aural umbers, are examples ime scales ruu is aricle ad llwi [], e ime scale will be deed by Here are dw sme examples ime scales wic we will sudy e cai rule i is aricle Çaaya Üiversiesi, Fe-Edebiya Faülesi, Maemai-Bilisayar Bölümü, Aara e-mail: abe@caayaedur
A Ne O e Cai Rule O ime Scales Examples : N 0 0 Fr, le We 0, e ery - : N calculus is baied [-sur] wi 0 a imprper accumulai pi, were e bacward derivaive is usually deied We e ime scale will ave as a imprper pi, were e rward derivaive is usually deied i :, N 0 Fr ad, le i0 i :, N i0 e rward jump perar : is deied by i s : s wile e bacward jump perar : is deied by sups : s, 3 were, i sup ie i as maximum ad sup i ie i as 0 miimum I is be ed a i e ad 0 Als i, e ad A pi is called ri scaered i, le-scaered i ad islaed i I ceci e rward ad bacward raiess ucis, : 0, are deied, respecively, by ad I rder deie e rward ad bacward ime scale derivaive, we eed e ses ad wic are derived rm e ime scale as llws: I as a le-scaered maximum M, M ad erwise I as a ri-scaered miimum m, e e m ad erwise Assume : ad e e rward ime-scale derivaive is e umber prvided i exiss wi e prpery a ive ay 0, ere exiss a eibrd U ie U, r sme 0 suc a s s s, s U 4 Mrever, we say a is dela diereiable prvided a r all Similarly, e bacward ime-scale derivaive is e umber, prvided i exiss wi e prpery a ive ay 0, ere exiss a eibrd U ie U, r sme 0 suc a s s s, s U, 5
abe ABDELJAWAD ad mrever, we say a is abla diereiable prvided a r all e llwi w erems are valid r e rward ad bacward ime-scale derivaives: erem Assume : is a uci ad le e we ave e llwi: i I is dela diereiable a, e is ciuus a ii I is ciuus a ad is ri-scaered, e is dela diereiable a wi 6 iii I is ri-dese ie e is dela diereiable a i ad ly i e s limi lim exiss as a iie umber I is case s s s lim s s 7 iv I is dela diereiable a, e 8 erem Assume : is a uci ad le e we ave e llwi: i I is abla diereiable a, e is ciuus a ii I is ciuus a ad is le-scaered, e is abla diereiable a wi iii I is le-dese ie e is abla diereiable a i ad ly i e s limi, lim exiss as a iie umber I is case s s s lim s s 9 iv I is abla diereiable a, e 0 e llwi prduc rmulas are useul: I case e ime scales ad ive abve i examples ad all e des are islaed excep r e limi pis ad e jump perars ad are iverses eac er ad ece e llwi relais bewee e dela derivaive ad e abla derivaive ld a b Fr a uci :, will mea e dela derivaive a ad a similar ai ca be assied als r e abla derivaive 3
A Ne O e Cai Rule O ime Scales Deiii A uci : is called rd-ciuus prvided i is ciuus a ridese pis i ad is le-sided limis exis iie a le-dese pis i, ad is called ld-ciuus prvided i is ciuus a le-dese pis i ad is ri-sided limis exis iie a ri-dese pis i Ciuus ucis are clearly rd-ciuus ad ld-ciuus Als i is easy see a is a example a rd-ciuus uci wic is ciuus ad is a example a ld-ciuus uci wic is ciuus is is, curse, rue i e ime scale cais le-dese ri-scaered des ad ri-dese le-scaered des, respecively erem 3 [] Exisece dela aiderivaives Every rd-ciuus uci as dela aiderivaive I paricular, i, e F deied by 4 0 F : s s, r 3 0 is a dela aideivaive erem 4 Exisece abla aiderivaives Every ld-ciuus uci as abla aiderivaive I paricular, i 0, e F deied by F : s s, 0 r 4 is a abla aideivaive e Cai Rule ad Special Cases Recall a i ad are diereiable real-valued ucis deied, e e cai rule saes : d 5 d Quesi: Fr wic ucis :, : ad ime scales e llwi relai is valid: r, 6 were Clearly rm e usual cai rule e abve relai is rue r e ime scale r ay w direiable ucis ad wi Bu i yu le yur ime scale be e aurals N, e : ad :, ad is case yu ca id may examples seueces ad r wic e abve relai is valid Hwever, i ae ay seuece ad le r example e e abve relai i e uesi is valid r e dela derivaive Fr e sae cmpleeess we sae e eeral ime scale dela cai rule erem 5 [] Assume : is ciuus, : is dela diereiable, ad : is ciuusly diereiable e ere exiss a umber c, wi c 7 Deiii [4] A ime scale is said ave e H-prpery i is rward jumpi perar as e rm:, r sme ad We call suc a ime sacle a H-ime scale
abe ABDELJAWAD 5 I is easy see a N,,, ad are all examples H-ime scales Prpsii Le be a H-ime scale, : is w imes dela diereiable uci ad e, 8 Pr By 8 we ave e leads 9 ad applyi 8 aai r e uci we e 0 e e resul llws by i a Similarly, by e elp 0 ad we ca prve Prpsii Le be a H-ime scale, : is w imes abla diereiable uci ad e, Remar I ac, Prpsii ad Prpsii are valid r ime scales wse rward jump perar ad bacward jump perar are dela ad abla diereiable, respecively I paricular, e are valid r e ime scales ad Lemma Le be a a H-ime scale e, r all N ad We w eeralize Prpsii ad Prpsii, r e H-ime scale wi meas e cmpsii imes ad, respecively Prpsii 4 Le be a H-ime scale, : is w imes dela diereiable uci ad, r N e, Pr We llw by iduci e case is rue by Prpsii Assume e resul is rue r e, 8 applied implies a, 3 ad ece by we bai 4 Aai 8 implies a = 5 Frm e iduci sep, i llws a 6
A Ne O e Cai Rule O ime Scales By Lemma ad a is csa we cclude a, 7 ad e pr is cmplee Similarly ad by e elp 0 we ca als sae a cai rule resul r e abla derivaive Prpsii 5 Le be a H-ime scale, : is w imes dela diereiable uci ad, r N e, Fr e purpse cmpariss we sae e llwi cai rule i e ery -calculus, wic is Lemma [3] Prpsii 6 Le x cx, were c ad are csas ad ay uci e D x D x D x 8 were D x meas e abla derivaive e ime scale, 0 As direc cseueces Prpsii 4 ad Prpsii 5, we sae e llwi w useul crllaries Crllary Le be a H-ime scale ad : be a rd-ciuus uci I, ad F s s, N, N e F 9 Crllary Le be a H-ime scale ad : be a ld-ciuus uci I, ad F s s, N, N e F 30 REFERENCES Ber M, Peers A, Dyamic Euais ime Scales: A irduci wi applicais Birauser, Bs, 00 Ber M, Peers A, Advaces i Dyamic Euais ime Scales, Birauser, Bs, 00 Ers, e isry -calculus ad a ew med Liceiae esis, UUDM Repr 000: 6; p://www mauuse/mas/licspd Rui A C Ferreira ad Delim F M rres, Hier-Order Calculus Variais ime ScalesarXiv: 070634v [maoc] 30 Sep 007 6
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