Europen Journl of Mhemcs nd Compuer Scence Vol 4 No, 7 SSN 59-995 THE EXSTENCE OF SOLUTONS FOR A CLASS OF MPULSVE FRACTONAL Q-DFFERENCE EQUATONS Shuyun Wn, Yu Tng, Q GE Deprmen of Mhemcs, Ynbn Unversy, Yn, Jln, CHNA Correspondence should be ddressed o Q GE, ge9688@6com ABSTRACT n hs pper, we prove he exsence nd unueness of soluons for clss of nl vlue problem for mpulsve frconl -dfference euon of order by pplyng some well-nown fxed pon heorems Some exmples re presened o llusre he mn resuls MSC: 6A; 9A; 4A7 Keywords: -clculus ; mpulsve frconl -dfference euons; exsence; unueness NTRODUCTON n recen yers, he opc of -clculus hs rced he enon of severl reserchers nd vrey of new resuls on -dfference nd frconl -dfference euons cn be found n he ppers [-] nd he references ced heren n [4] he noons of -dervve nd - negrl of funcon f : J : [, ] R hve been nroduced nd her bsc properes ws proved As pplcons exsence nd unueness resuls for nl vlue problems for frs nd second order mpulsve -dfference euons re proved n [5], he uhors ppled he conceps of unum clculus developed n [4] o sudy clss of boundry vlue problem of ordnry mpulsve -negro-dfference euons, some exsence nd unueness resuls for hs problem were proved by usng vrey of fxed pon heorems n [6] he uhors used he -shfng operor o develop he new conceps of frconl unum clculus such s he Remnn Louvlle frconl dervve nd negrl nd her properes They lso formuled he exsence nd unueness resuls for some clsses of frs nd second orders mpulsve frconl -dfference euons nspred by[6], n hs pper, we sudy he exsence nd unueness of soluons for he followng nl vlue problem for mpulsve frconl -dffer- ence euon of order he form D ( (, (,, x f x J Δ x( ( x(,,,, m, ( Δ x( ( x(,,,, m, x(, D x( (, D x where J T m m T J J nd [, ],, [, ], (, ],,,, m D respecvely re he Remnn-Louvlle frconl -dfference of order nd on nervl J, for,,, m, f : JRR s connuous funcon,, ( RR, for,,, m The noon Δ x ( nd Δ x ( re defned by C where nd Δ x( x( x(,,,, m, Δ x( x( x(,,,, m, ( respecvely re he Remnn-Louvlle frconl -negrl of order nd on J R, {,,, m}, (, ] D Progressve Acdemc Publshng, UK Pge 6 wwwdpublconsorg
Europen Journl of Mhemcs nd Compuer Scence Vol 4 No, 7 SSN 59-995 Prelmnres Ths secon s devoed o some bsc conceps such s -shfng operor, Remnn Louvlle frconl -negrl nd -dfference on gven nervl The presenon here cn be found n, for exmple, [6,7] We defne -shfng operor s Φ ( m m ( The power of -shfng operor s defned s More generlly, f R, hen ( ( ( n m, ( n m ( n Φ ( m, { } Φ ( mn Φ ( mn ( ( n ( n m n n N, Defnon The frconl -dervve of Remnn Louvlle ype of order v on nervl [ b, ] s defned by ( D f ( f ( nd ( v ( ( l l D f D v f (, v, where l s he smlles neger greer hn or eul o ν Defnon Le nd f be funcon defned on [ b, ] The frconl -negrl of Remnn Louvlle ype s gven by ( f ( f ( nd ( ( f ( ( Φ ( s f ( s ds,, [, b] ( From [6], we hve he followng formuls for [, b],, R : ( ( D ( (, ( ( ( ( Lemm Le, R nd f be connuous funcon on[, b], The Remnn Louvlle frconl -negrl hs he followng sem-group propery f ( f ( f ( Lemm 4 Le f be -negrble funcon on[ b, ] Then he followng euly holds D f ( f ( For, [, b] Lemm 5 Le nd p be posve neger Then for [, b] he followng euly holds p p p p ( D f ( D f ( D f ( ( p Lemm 6 ([8]Le E be Bnch spce Assume h s n open bounded subse of E wh nd le T: E be compleely connuous operor such h Tu u, u Then T hs fxed pon n Lemm 7 ([8] Le E be Bnch spce Assume h T : E E s compleely connuous operor nd he se V { ue u Tu, } s bounded Then T hs fxed pon n E Le PC( J, R { x : J R: x( s connuous everywhere excep for some whch x ( nd x ( Progressve Acdemc Publshng, UK Pge 7 wwwdpublconsorg
Europen Journl of Mhemcs nd Compuer Scence Vol 4 No, 7 SSN 59-995 - + exs nd x( = x(,,,,, m} For R, we nroduce he spce C, ( J, R { x : J R: : ( x( C( J, R } wh he norm for ech J nd x sup J ( x( nd PC ( J, R { x : J R: : C, ( x( C( J, R,,,, m} wh he norm x mx{sup ( x( :,,, m} Clerly PC ( J, R s Bnch spce Lemm8 f xpc( J, R s soluon of (, hen for ny J,,,, m, m( m( x( = (, ( f x, ( ( ( Where m [ f ( s, x( s( f ( s, x( s( ( x( ] PC J, ( f ( s, x( s( ( x( + f ( s, x( s( ( x( ( m [ f ( s, x( s( f ( s, x( s( ( x( ] f ( s, x( s( ( x(, ( Wh ( < Proof For J, ng he Remnn-Louvlle frconl -negrl of order for he frs euon of ( nd usng Defnon wh Lemm 5, we ge x( = C C f (, x( (4 ( ( where C= x( nd C = x( The frs nl condon of ( mples h C = Tng he Remnn-Louvlle frconl -dervve of order for (4 on J, we hve D x( C f (, x(, And D x( C Therefore, (4 cn be wren s x( = C f (, x( (5 ( Applyng he Remnn-Louvlle frconl -dervve of orders nd =, we hve x( C f ( s, x( s(, x( C f ( s, x( s(, (6 For J =(, ], Remnn-Louvlle frconl -negrng (, we obn ( ( for (5 ( ( x( = x( x( f (, x(, (7 Usng he ump condons of euon ( wh (6-(7 for J, we ge ( ( x( = [ C f ( s, x( s( ( x( ] [ C f ( s, x( s( ( x( ] f (, x( ( ( Repeng he bove process, for J =(, ], we obn Progressve Acdemc Publshng, UK Pge 8 wwwdpublconsorg
Europen Journl of Mhemcs nd Compuer Scence Vol 4 No, 7 SSN 59-995 ( x( = [ C ( f ( s, x( s( ( x( ( ( + f ( s, x( s( ( x( ] [ C ( (, ( ( ( ( ] (, ( f s x s x f x, Tng he Remnn-Louvlle frconl -dervve of order (, follows h D x( = C f ( s, x( s( ( x( f (, x( (8 for (8 nd usng For {,,, m}, (, ], we hve D ( = (, ( ( ( ( (, ( ( x C f s x s x f s x s The nl condon D x( ( D x leds o C = [ (, ( ( ( ( (, ( ( ] f s x s x f s x s Subsung he vlue of C n (8, we obn ( Conversely, ssume h x s soluon of he mpulsve frconl negrl euon (, hen by drec compuon, follows h he soluon gven by(ssfes euon ( Ths complees he proof Mn resuls Ths secon dels wh he exsence nd unueness of soluons for he euon ( n vew of Lemm 8, we defne n operor A: PC( J, R PC( J, R by where m, m re gven by ( nd ( m( m( ( Ax( = (, ( f x, ( ( f (, x ( x ( x Theorem Le lm,lm nd lm (,,, m, hen euon x x x x x x ( hs les one soluon Proof To show h Ax PC ( J, R for xpc ( J, R, we suppose, J, nd, hen ( Ax( ( Ax( m ( m ( = ( [ (, ( ( ] f s x s ( ( Progressve Acdemc Publshng, UK Pge 9 wwwdpublconsorg
Europen Journl of Mhemcs nd Compuer Scence Vol 4 No, 7 SSN 59-995 m ( m ( ( [ (, ( ( ] f s x s ( ( ( ( { [ f ( s, x( s ( f ( s, x( s ( ( x( ] ( ( f ( s, x( s ( ( x( + f ( s, x( s ( ( x( } ( ( { [ f ( s, x( s ( f ( s, x( s ( ( x( ] ( f ( s, x( s ( ( x( } ( ( Φ ( s f ( s, x( s d s ( ( ( ( ( [( ( Φ ( s ( ( Φ ( s ] f ( s, x( s d s As, we hve ( Ax( ( Ax( for ech,,,, m Therefore, we ge Ax PC ( J, R Now we show h he operor A: PC ( J, R PC ( J, R s compleely connuous Noe h A s connuous n vew of connuy of f, nd Le B PC ( J, R be bounded Then, here exs posve consns L (,, such h f (, x L, ( x L, ( x L, x B Thus, x B, We hve m [ L ( f ( s, x( s ( ( x( ] f s x s x f s x s x ( ( (, ( ( ( ( + ( (, ( ( ( ( T ( L L L ( L ( ( L L m ( L L ( L L, Therefore, ( L f (, x( ( (, ( T ( ( ( Ax( [ ( L L L ( whch mples h ( L ( ( L L ] ( L ( [ ( L L L L ] ( ( T T m TL [ ( ] L ml ml LmT ( T LT ( (, [ ( L ml LT ml ] ( Progressve Acdemc Publshng, UK Pge wwwdpublconsorg
Europen Journl of Mhemcs nd Compuer Scence Vol 4 No, 7 SSN 59-995 T T m TL ( Ax( [ ( L ml ml L mt ] ( T LT L ml L T ml L ( ( [ ( ] : On he oher hnd, for ny, J,wh, m,we hve ( ( Ax( ( ( Ax( ( ( TL [ ( ] T L L L L T ( ( ( [ ( L L LT L ] ( ( f ( s, x( s( ( f ( s, x( s( (, Ths mples h A s euconnuous on ll he subnervls J,,,,, m Thus, by Arzel Ascol Theorem, follows h A: PC ( J, R PC ( J, R s compleely connuous f (, x ( x ( x Now, n vew of lm,lm nd lm (,,, m, here exss x x x x x x consn r such h f (, x x, ( x x, ( x x, for x r, where (,, ssfy T T m T [ ( m m mt ] ( T T [ ( m T m ] ( ( Defne ={ x PC ( J, R : x r} nd e xpc ( J, R such h x = rso h x Then, by he process used o obn (, we hve T T mt ( ( Ax( { [ ( m m mt ] ( T T m T m x x, ( ( [ ( ] } whch mples h ( Ax( x, x Therefore, by Lemm 6, he operor A hs les one fxed pon, whch n urn mples h ( hs les one soluon x Ths complees he proof Theorem Assume h (H here exs posve consns L ( =,, such h f (, x L, ( x L, ( x L for J, xr nd,,, m Then euon ( hs les one soluon Proof As shown n Theorem, he operor A: PC( J, R PC( J, R s compleely connuous Now, we show he se V { xpc ( J, R x Ax, } s bounded Le x V, hen x Ax, For ny J, we hve m( m( x( = (, ( f x, ( ( ( where m, m re gven by ( nd ( Combnng (H nd (, we obn Progressve Acdemc Publshng, UK Pge wwwdpublconsorg
Europen Journl of Mhemcs nd Compuer Scence Vol 4 No, 7 SSN 59-995 m ( m ( ( x( ( f (, x( ( ( T T m TL [ ( L ml ml LmT ] ( LT L ml LT ml L ( ( :=, T [ ( ] Thus, for ny J, follows h x L So, he se V s bounded Therefore, by he concluson of Lemm 7, he operor A hs les one fxed pon Ths mplesh ( hs les one soluon Ths complees he proof Theorem Assume h (H here exs posve consns N ( =,, such h f (, x f (, y N x y, ( x ( y N x y, ( x ( y N x y for J, xr nd,,, m Then euon ( hs unue soluon f T m T = [ ( N mn ( T mn N( T mt N( m ], ( Where T =mx{ T, T, T } = mn{ (, (, ( } Proof For x, ypc ( J, R, we hve ( T ( ( Ax( ( Ay( { [ f ( s, x( s f ( s, y( s ( ( (, ( (, ( ( ( ( ( ( ] f s x s f s y s x y ( f ( s, x( s f ( s, y( s ( ( ( ( ( x y + f ( s, x( s f ( s, y( s ( ( x( ( y( } ( { [ (, ( (, ( ( f s x s f s y s ( f s x s f s y s x y (, ( (, ( ( ( ( ( ( ] f s x s f s y s x y (, ( (, ( ( ( ( ( ( } ( f ( s, x( s f ( s, y( s ( ( T TN { [ ( N N N NT ] ( ( + [ ( N N N T N ] } x y ( ( NT PC Progressve Acdemc Publshng, UK Pge wwwdpublconsorg
Europen Journl of Mhemcs nd Compuer Scence Vol 4 No, 7 SSN 59-995 T T [ ( N N( T N N( T T N( ] x y T m T [ ( N mn( T mn N( T mt N( m ] x y x y PC where s gven by ( Thus, Ax Ay x y As, herefore, A s PC PC conrcon Hence, by he conrcon mppng prncple, euon ( hs unue soluon Exmples Exmple 4 Consder he followng mpulsve frconl -dfference nl vlue problem: D x( rcn x( e x (, [, ],, 7 4 8 Δ x( cos x(,,,,,, Δ x( sn x(,,,,,, x(, D7 x( D x(, 4 8 5 4 4 Here, ( 7 ( 8,,,,, m, T,,, 4, f (, x( rcn x( e x (, ( x( cos x(, ( x( sn x(, Clerly, ll he ssumpons of Theorem re ssfed Thus, by he concluson of Theorem, he mpulsve frconl -dfference nl vlue problem 4 hs les one soluon Exmple 4 Consder he followng mpulsve frconl -dfference nl vlue problem: e sn x( D x(, [,],, ( 5 7 4 x 4 8 Δ x( cos x(,,,,9,, x ( Δ x( sn(4 e,,,,9,, x(, D7 x( D x(, 4 8 5 4 4 Here, ( 7 ( 8,,,,9, m 9, T,,, 4, f (, x( e 5 sn x( 4 x (, x x ( ( ( cos (, x ( x( sn(4 e, Clerly L e, L 6, L 9 nd he condons of Theorem cn redly be verfed Therefore, he concluson of Theorem pples o he mpulsve frconl -dfference nl vlue problem 4 REFERENCES []Jcson, FH: -Dfference euons Am J Mh, 5-4 (9 PC PC Progressve Acdemc Publshng, UK Pge wwwdpublconsorg
Europen Journl of Mhemcs nd Compuer Scence Vol 4 No, 7 SSN 59-995 [] Al-Slm, WA: Some frconl -negrls nd -dervves Proc Ednb Mh Soc 5(, 5-4 (966/967 [] Agrwl, RP: Cern frconl -negrls nd -dervves Proc Cmb Phlos Soc 66, 65-7 (969 [4] Bngerezo, G: Vronl -clculus J Mh Anl Appl 89, 65-665 (4 [5] Dobrogows, A, Odzewcz, A: Second order -dfference euons solvble by fcorzon mehod J CompuAppl Mh 9, 9-46 (6 [6]Gsper, G, Rhmn, M: Some sysems of mulvrble orhogonl -Rch polynomls Rmnun J, 89-45(7 [7]sml, MEH, Smeonov, P: -Dfference operors for orhogonl polynomls J Compu Appl Mh, 749-76(9 [8] Bohner, M, Gusenov, GS: The h-lplce nd -Lplce rnsforms J Mh Anl Appl 65, 75-9 ( [9]El-Shhed, M, Hssn, HA: Posve soluons of -dfference euon Proc Am Mh Soc 8,7-78 ( []Ahmd, B: Boundry-vlue problems for nonlner hrd-order -dfference euons Elecron J Dffer Eu, 94( [] Ahmd, B, Alsed, A, Nouys, SK: A sudy of second-order -dfference euons wh boundry condons AdvDffer Eu, 5 ( []Ahmd, B, Neo, JJ: On nonlocl boundry vlue problems of nonlner -dfference euons Adv Dffer Eu, 8 ( []Yu, C, Wng, J: Exsence of soluons for nonlner second-order -dfference euons wh frs-order -dervves Adv Dffer Eu, 4 ( [4]Trboon, J, Nouys, SK: Qunum clculus on fne nervls nd pplcons o mpulsve dfference euons Adv Dffer Eu, 8 ( [5] C Thpryoon, J Trboon, SK Nouys: Sepred boundry vlue problems for second-order mpulsve -negro-dfference euons, Adv Dffer Eu 4,88 (4 [6]J Trboon, SK Nouys, P Agrwl: New conceps of frconl unum clculus nd pplcons o mpulsve frconl -dfference euons Adv Dffer Eu 5,8 (5 [7]Bshr Ahmd, Sors K Nouys, Jessd Trboon, Ahmed Alsed, Hmed H Alsulm: mpulsve frconl -negro-dfference euons wh sepred boundry condons Appled Mhemcs nd Compuon, 8,99 (6 [8]JX Sun:Nonlner Funconl Anlyss nd s Applcon, Scence Press, Beng, 8 Progressve Acdemc Publshng, UK Pge 4 wwwdpublconsorg