On the Nonlinear Difference Equation

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Joural of Appled Mathemats ad Phss 6 4-9 Pulshed Ole Jauar 6 SRes http://wwwsrporg/joural/jamp http://ddoorg/436/jamp644 O the Nolear Dfferee Equato Elmetwall M Elaas Adulmuhaem A El-Bat Departmet of

Iteratoal Lese (CC BY http://reatveommosorg/leses//4/ Astrat I ths paper we vestgate some qualtatve ehavor of the solutos of the dfferee equato + = a+ = where the oeffets a ad are postve real umers {

1 Joural of Appled Mathemats ad Phss Pulshed Ole Jauar 6 SRes O the Nolear Dfferee Equato Elmetwall M Elaas Adulmuhaem A El-Bat Departmet of Mathemats Fault of See Masoura Uverst Masoura Egpt Reeved 7 Novemer 5; aepted Jauar 6; pulshed 6 Jauar 6 Coprght 6 authors ad Setf Researh Pulshg I Ths wor s lesed uder the Creatve Commos Attruto Iteratoal Lese (CC BY Astrat I ths paper we vestgate some qualtatve ehavor of the solutos of the dfferee equato + = a+ = where the oeffets a ad are postve real umers { } ad where the tal odtos + are artrar postve real umers Kewords Dfferee Equato Stalt Perodt Boudedess Gloal Stalt Itroduto Our am ths paper s to stud wth some propertes of the solutos of the dfferee equato + = a+ = ( where the oeffets a ad ad where the tal odtos + are artrar postve real umers There s a lass of olear dfferee equatos ow as the ratoal dfferee equatos eah of whh ossts of the rato of two polomals the sequee terms the same form There has ee a lot of wor oerg the gloal asmptots of solutos of ratoal dfferee equatos []-[8] Ma researhers have vestgated the ehavor of the soluto of dfferee equato For eample: Amleh et al [9] has studed the gloal stalt oudedess ad the perod harater of solutos of the equato are postve real umers { } How to te ths paper: Elaas EM ad El-Bat AA (6 O the Nolear Dfferee Equato Joural of Appled Mathemats ad Phss 4-9

2 E M Elaas A A El-Bat = + α + Our am ths paper s to eted ad geeralze the wor [9] [] ad [] That s we wll vestgate the gloal ehavor of ( ludg the asmptotal stalt of equlrum pots the estee of ouded soluto the estee of perod two soluto of the reursve sequee of Equato ( Now we reall some well-ow results whh wll e useful the vestgato of ( ad whh are gve [] Let I e a terval of real umers ad let : + F I I where F s a otuous futo Cosder the dfferee equato = F = ( + wth the tal odto + I Defto (Equlrum Pot A pot I s alled a equlrum pot of Equato ( f = f That s = for s a soluto of Equato ( or equvaletl s a fed pot of f Defto (Stalt Let ( e equlrum pot of Equato ( the A equlrum pot of Equato ( s alled loall stale f for ever ε > there ests δ > suh that f ( + wth < δ the < ε for all A equlrum pot of Equato ( s alled loall asmptotall stale f s loall stale ad there ests γ > suh that f ( wth < γ the lm = + 3 A equlrum pot of Equato ( s alled a gloal attrator f for all ( have lm = + we 4 A equlrum pot of Equato ( s alled gloall asmptotall stale f s loall stale ad a gloal attrator 5 A equlrum pot of Equato ( s alled ustale f s ot loall stale Defto 3 (Permaee Equato ( s alled permaet f there ests umers m ad M wth < m< M < suh that for a tal odtos + ( there ests a postve teger N whh depeds o the tal odtos suh that Defto 4 (Perodt A sequee { } m M for all s sad to e perod wth perod p f = + p= for all A sequee { } s sad to e perod wth prme perod p f p s the smallest postve teger havg ths propert The learzed equato of Equato ( aout the equlrum pot s defed the equato = where z = pz (3 +

3 E M Elaas A A El-Bat ( F p = = The haraterst equato assoated wth Equato (3 s λ λ λ λ (4 + p p p p = Theorem [3] Let [ pq ] e a terval of real umers ad assume that g: [ pq ] + [ pq ] s a otuous futo satsfg the followg propertes: (a g( + s o-reasg the frst ( terms for eah the last term for eah [ pq ] for all = mm pq pq s a soluto of the sstem ( If [ ] [ ] mples + [ ] ( ( M= g mmm mm ad m= g M M M M m m= M pq ad o-dereasg Theorem [] Assume that F s a C -futo ad let e a equlrum pot of Equato ( The the followg statemets are true: If all roots of Equato (4 le the ope ut ds λ < the he equlrum pot s loall asmptotall stale If at least oe root of Equato (4 has asolute value greater tha oe the the equlrum pot s ustale 3 If all roots of Equato (4 have asolute value greater tha oe the the equlrum pot s a soure Theorem 3 [4] Assume that p R = The p < s a suffet odto for the asmptotall stale of Equato (5 Loal Stalt of Equato ( + + p p= = (5 I ths seto we vestgate the loal stalt harater of the solutos of Equato ( Equato ( has a uque ozero equlrum pot = a+ Let The we get + Let f : ( ( = a+ e a futo defed G = = a+ G

4 E M Elaas A A El-Bat ad ad Therefore t follows that The we see that ( u f ( u u u = a+ ( u f u u u u j = j = u j u ( ( f u u u u = u u f j = = Pj j = uj ag + G ( The the learzed equato of ( aout s Theorem Assume that f G = = P u ag + G z + = pz ( ( ag G < The the equlrum pot of Equato ( s loall stale Proof It s follows Theorem (3 that Equato ( s loall stale f the That s Ths mples that Thus Hee the proof s ompleted p + + p + p < ( G < ag + G ag + G ag + G ag + G + < ag + G ag + G ( G ( ag + G ( G < ( ag G < 3

5 E M Elaas A A El-Bat 3 Perod Solutos I ths seto we vestgate the perod harater of the postve solutos of Equato ( Theorem 3 Equato ( has postve prme perod-two soluto ol f Proof Assume that there ests a prme perod-two soluto pq pq odd ad α β aβ aα + > 4 aαβ (3 of ( Let = q + = p Se odd we have = p Thus from Equato ( we get p p = a+ q+ p+ q+ + p ad ad ad ad that Let The The Sutratg (3 from (33 gves Se p q we have q q = a+ p+ q+ p+ + q α = β = p p = a+ αq+ β p q q = a+ αp+ βq αpq + β p = aαq + aβ p + p (3 αpq + βq = aαp + aβq + q (33 ( p q = ( a a + ( p q β β α Also se p ad q are postve ( aβ aα It follows (34 (35 ad the relato aβ aα + p+ q = (34 β + should e postve Aga addg (3 ad (33 elds αpq+ β p + q = aα + aβ + p+ q (35 p q p q pq p q + = + R ( + β( α β aα aβ aα pq = (36 4

6 E M Elaas A A El-Bat Assume that p ad q are two dstt real roots of the quadrat equato ad so whh s equvalet to Thus the proof s ompleted 4 Bouded Soluto ( + ( α β aα aβ aα βt aβ aα + t + = ( a a ( + ( α β aα aβ aα β α + 4β > ( α β ( aβ aα + > 4 aαβ Our am ths seto we vestgate the oudedess of the postve solutos of Equato ( Theorem 4 The solutos { } = Proof Let { } = The of Equato ( are ouded e a soluto of Equato ( We see from Equato ( that = + = a a O the other had we see that the hage of varales = trasforms Equato ( to the followg form: = a Hee we ota Thus + a+ = M for all (4 = a = = ( ( + a a a a a a + a + a + + ( ( a a a a 5

7 E M Elaas A A El-Bat ad so It follows that Thus we ota From (4 ad (4 we see that ( + ( + + ( ( + + ( a a a a a a a + a a + = + = a a = a a ( + + ( + a a = E for all a a = m for all E = (4 m M for all Therefore ever soluto of Equato ( s ouded 5 Gloal Stalt of Equato ( Our am ths seto we vestgate the gloal asmptot stalt of Equato ( Theorem 5 If ag = a + the the equlrum pot of Equato ( s gloal attrator + Proof Let f : ( ( e a futo defed u f ( u u u = a+ u the we a see that the futo f ( u u u Suppose that ( mm s a soluto of the sstem m= f( M M M M m M= f( mmm mm s dereasg the rest of argumets ad reasg u ad The from Equato ( we see that m m = a+ M = a+ M M + m m+ M the Thus m M m = a+ M = a+ G M + m G m+ M G Mm + m = a G M + a m + m G Mm+ M = a G m+ a M + M ( = ( + ( m M a ag m M m= M It follows Theorem ( that s a gloal attrator of Equato ( ad the the proof s omplete 6

8 E M Elaas A A El-Bat 6 Numeral Eamples For ofrmg the results of ths seto we osder umeral eamples whh represet dfferet tpes of soluto of Equato ( Eample 6 Cosder the dfferee equato + 9 = where = a = 5 = 9 α = = 6 β = = 5 Fgure shows that the equlrum pot of Equato ( has loall stale wth tal data = = 3 (see Tale Eample 6 Cosder the dfferee equato + 4 = where odd = a = 5 = 4 α = = β = = Fgure shows that Equato ( whh s perod wth perod two Where the tal data satsfes odto (3 of Theorem (3 = = 3 (see Tale Fgure Ref Tale The equlrum pot of Equato ( ( ( ( (

9 E M Elaas A A El-Bat Fgure Ref 4 Tale The tal data satsfes odto (3 of Theorem (3 ( ( ( ( ( Remar 6 Note that the speal ases of Equato ( have ee studed [9] whe = = = = ad [] whe = = = = ad [] whe = = = Referees [] Elaas EM El-Metwall H ad Elsaed EM (5 O the Perod Nature of Some Ma-Tpe Dfferee Equatos Iteratoal Joural of Mathemats ad Mathematal Sees Ad- [] Elaas EM El-Metwall H ad Elsaed EM (6 O the Dfferee Equato + = a d vaes Dfferee Equatos -(6 Artle ID:

10 E M Elaas A A El-Bat [3] Elaas EM El-Metwall H ad Elsaed EM (7 Qualtatve Behavor of Hgher Order Dfferee Equato Soohow Joural of Mathemats [4] El-Moeam MA ad Zaed E (4 Dams of the Ratoal Dfferee Equato l + = A + B + C l+ DCDIS Seres A: Mathematal Aalss d e l [5] Elad SN (996 A Itroduto to Dfferee Equatos Udergraduate Tets Mathemats Sprger New Yor [6] Ko VL ad Ladas G (993 Gloal Behavor of Nolear Dfferee Equatos of Hgher Order wth Applatos Kluwer Aadem Pulshers Dordreht [7] Stev S (5 O the Reursve Sequee + α + β = f ( + Tawaese Joural of Mathemats Mathema- α + α + α [8] Zaed E ad EI-Moeam MA ( O the Ratoal Reursve Sequee + = β + B + β ta Bohema l l [9] Amleh AM Grove EA Georgou DA ad Ladas G (999 O the Reursve Sequee = + α + Joural of Mathematal Aalss ad Applatos [] Hamza AE (6 O the Reursve Sequee + = α + Joural of Mathematal Aalss ad Applatos [] Saleh M ad Aloqel M (5 O the Ratoal Dfferee Equato + = A+ Appled Mathemats ad Computato [] Grove EA ad Ladas G (5 Perodtes Nolear Dfferee Equatos Vol 4 Chapma ad Hall/CRC Boa Rato [3] Elaas EM El-Metwall H ad Elsaed EM (7 O the Dfferee Equatos = of Corete ad Applale Mathemats α β + γ Joural [4] Kuleov MRS ad Ladas G ( Dams of Seod Order Ratoal Dfferee Equatos wth Ope Prolems ad Cojetures Chapma & Hall/CRC Florda 9

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Ruin Probability-Based Initial Capital of the Discrete-Time Surplus Process Ru Probablty-Based Ital Captal of the Dsrete-Tme Surplus Proess by Parote Sattayatham, Kat Sagaroo, ad Wathar Klogdee AbSTRACT Ths paper studes a surae model uder the regulato that the surae ompay has