ASYMPTOTIC BEHAVIOR OF SOLUTIONS OF DISCRETE EQUATIONS ON DISCRETE REAL TIME SCALES

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1 ASYPTOTI BEHAVIOR OF SOLUTIONS OF DISRETE EQUATIONS ON DISRETE REAL TIE SALES J. Dlí B. Válvíová 2 Bro Uversy of Tehology Bro zeh Repul 2 Deprme of heml Alyss d Appled hems Fuly of See Uversy of Zl Žl Slov Repul ABSTRAT. Ths oruo dels wh he vesgo of sympo ehvor of soluos of dsree equos o dsree rel me sles. Sysem of dsree equos u ( + ) = F ( u ( )) wh F : T R R s osdered o dsree rel me sle T. We develop geerl prple whh gves guree h he grph of les oe soluo sys presred dom. Exsee of ouded soluos of oler sysem s gve o llusre our resuls. Key words d phrses: Dsree equo sympo ehvor of soluo ouded soluo rero. hems Suje lssfos: 39A 39A. Iroduo. We use followg oo: for egers s q s q we defe q Zs := { ss + q } where possly s = or q = s dmed oo. Throughou q hs pper usg oo Z s or oher oe wh ouple of egers sq we suppose uomlly s q. We osder he sysem of dsree equos u ( + ) = F ( u ( )) () wh F : T R R u = ( u u ) where T = T( ) s rrry dsree me sle. e. T ( ) := { } wh R Z s eger d < + for y Z. We suppose h he mppg F s ouous wh respe o he seod rgume. We osder he l prolem () (2) where s u ( + s) = u R (2) wh fxed posve eger s. The exsee d uqueess of soluo of he l prolem () (2) o T ( + s) := { } Z + ss ovous. oreover due o ouy of he fuo F wh respe o he seod rgume hs l prolem depeds ouously o l d. The sequee {( u( ))} Z + ss lled he grph of soluo u = u( ) for Z + sof he l prolem () (2). We defe se Ω T R s Ω:= {( u) : T Z u ( )} where Ω ( ) s ope ouded d oeed se he spe 5-98

2 S ( ) {( ) u := u : u R } for every T. Ovously Ω= ( ) TΩ. The oudry Ω ( ) of Ω ( ) s defed he spe S ( ) orde wh usul defos. u Defo. We sy h po s he frs oseque po o po = ( u ) T R d we wre = [ ] f s frs oorde s shfed o + d he seod oe s he resul of he mppg of he po y mes of ().e. = [ ] = ( u( )) + + wh u ( + ) = F ( ). We sy h po s s he s -h oseque po of po = ( u ) T R f [ ] := = s where provded 2 s T R. Defo 2. A ouous fuo Vu ( ):[ ) R R s lled oeg fuo for he ses Ω ( ) T( ) f for every T( ) : Ω( ) {( u) : V( u) < }. Wh he d of he oeg fuo we defe he ses h wll e used he followg. Le oss α β R α < β e gve. We defe uxlry se Vαβ := {( u) : α β V( u) } wh oudry Vαβ :={( u) : α β V( u) = } d uxlry se Vα := {( u) : α < V( u) } wh oudry V :={( u) : α < V( u) = }. α Defo 3. If A B re y wo ses of opologl spe d π : B A s ouous mppg from B oo A suh h π ( p) = p for every p A he π s sd o e rero of B oo A. Whe here exss rero of B oo A A s lled rer of B. Prolem. Suppose h he ove uxlry supposos hold. We wll ry o fd suffe odos wh respe o he rgh-hd sde of he sysem () order o guree he exsee of les oe soluo u = u ( ) T Z ssfyg ( u ( )) Ω ( ) for every T. Exsee Resuls I hs seo we formule m resuls of he vesgo. Theorem. Suppose F : T R R s ouous wh respe o he seod rgume. oreover he ses Ω ( ) T d orrespodg oeg fuo Vu ( ) re gve suh h he followg properes hold:. The se s ovex for every T. V

3 2. For every T d Ω ( ) he le segme oeg po d s frs oseque po hs oly oe po of erseo wh he se mely he po self. 3. There exss rero π of he se The here exss soluo u u ( ) u for every T. V V + oo he se Ω ( ). = T of () ssfyg he relo ( ( )) Ω ( ) (3) PROOF. Le us suppose h he l d ( u ( )) ( ) geerg soluo u = u ( ) of he sysem () wh he propery (3) does o exs. Th mes o he oher hd for every = ( u ) ( ) here s rel umer Z + suh h for he orrespodg soluo u = u ( ) of he sysem () ssfyg he l odo u ( ) = u we hve ( u ( )) / ( ) d ( ) l + l l =. I mes h he ( )-h oseque po of he po does o elog o Ω ( ) u ll preedg oseque pos l = elog l o he orrespodg ses Ω ( ). oreover f = ( u ( )) Ω ( ) he + + ( ) ( ) + l / ( ). I hs wy we reformule hese resog: For y po = u here exss eger Z + suh h he orrespodg soluo u = u ( ) Z of he sysem () ssfyg he l odo u ( ) = u ssfes d for mddle oseque pos = ( u ( )) / ( ) (4) 2 l Z l ( ( )) ( ) l + l u + l + l = ( u ) wh = ( u ( )) ( ). (5) (f y) he relos = (6) hold. oreover le us remr h for u Ω ( ) (7) we hve Ω ( + ). Now we prove h uder ove desred properes here exss rero of he se Ω ( ) oo he se Ω ( ). I oher words hs suo ouous mppg-rero P of se (whh s opologlly equvle o losed -dmesol ll; our se - wh he se Ω ( )) oo s oudry ( our se - oo he se Ω ( ) ) for whh oudry pos re sory pos wll exs. Ths resuls ordo se well ow f ses h he oudry of -dmesol ll o e s rer. We wll osru suh rero P wh he d of wo uxlry mppgs P d P d he gve rero π. Le us defe mppg P of po = ( u ) wh u ( ) s 5-

4 P : ( u ( )) Ω ( ) where he vlue ws well defed ove y he relos d properes (4) (7). Ovously f u Ω ( ) he P ( ) =. Noe h he mppg P s ouous (due o ouous depedee o l d). Suppose = ( u ). The P( ) =. Le us defe mppg P : P ( ) = P ( P ( )) = wh eg he po of erseo of he le segme of he le oeg pos d d he se V. Le us show h due o he ovexy propery he po s defed uquely. Le hs o e he se. The here re les wo erseo pos d lyg o he le L oeg pos d he order ded (we suppose whou loss of geerly h s erer o h s). Se s he oudry po of he se V every s suffely smll eghourhood U( ) [ ) R os oempy se S ossg of exeror pos wh respe o V.e. for every s S we hve V. s Whou loss of geerly we wll suppose h S os ll exeror pos wh U( Δ ). Le us e po L V lyg ewee d d e suffely smll eghourhood U( Δ ) of suh h Δ U( ) U( ) =. Le us oe y po A U( Δ ) wh y le segme LA. Due o suffely smlless of eghourhood U( Δ ) we hve V. Bu o he oher hd here exss suh po A U( Δ ) h LA L Δ S. We ge ordo se S / V A. So d he uqueess of he po s proved. Le us prove he ouy of he ompose mppg P o P. I s ovous h P( P( )) = f Ω ( ). We show h due o he ouous depedee of soluo o he l d he ompose mppg P o P s ouous. There s o dou h P o P s ouous eghourhood of f Ω ( ). Suppose ow ( ) Ω. The P( P ( )) = d he resul of he mppg P( P( U( ))) where U( ) Ω ( ) s suffely smll eghourhood of he po gves smll eghourhood of he po V V due 2 o ouy of he mp F d ouous depedee of soluos o l d. Ths mes h he ompose mppg P o P s ouous. Defe he resulg rero wh he d of mppgs P P d π s 5-

5 P= π op o P. Ovously P:Ω( ) Ω ( ). As omposo of hree ouous mppgs P s ouous oo d P ( ) = f Ω ( ). So P s he desred rero. Ths f leds o ordom. Le uxlry fuos ( ) ( ): T( ) R = wh ( ) < ( ) e gve. We defe he fuos d he ses B ( u) := u + ( ) = ( u) := u ( ) = Ω :={( u) : T( ) B( u) = B B ( u) ( u) forll j s = d j } j s Ω :={( u) : T( ) ( u) = B j( u) s( u) forll j s = d s } for every =. Suppose h he se Ω s wre he form Ω= {( u) : T( ) B( u) < j( u) < j = }. (8) Followg resul s osequee of Theorem. We om s proof. Theorem 2. Le = e rel fuos defed o T ( ) suh h ( ) < ( ) d F : T( ) R R s ouous wh he respe o he seod rgume. If moreover he se Ω hs he form (8) F( u) < ( + ) for every = d every ( u) B F( u) > ( + ) for every = d every ( u) he here exss soluo u = u ( ) of he sysem () ssfyg he equles ( ) < u ( ) < ( ) for every T( ) d =. Exsee of Bouded Soluos of Noler Sysem Le us de suffe odos uder whh here exss ouded soluo of sysem of dsree equos u( + ) = μ( ) u( ) + ω( u( )) = (9) wh T( ) u = ( u u 2 u ) R μ = ( μ μ2 μ) : T( ) R d ω = ( ω ω2 ω) : T( ) R R. Bsed o Theorem 2 oe prove he followg Theorem 3. Le = e rel fuos defed o T ( ) suh h ( ) < ( ) ω : T ( ) Ω R wh Ω defed y (9) s ouous wh respe o he seod rgume. If moreover ( + ) μ( ) ( ) > ω( u) for every = d every ( u) d B 5-2

6 ( + ) μ( ) ( ) < ω( u) for every = d every ( u) he here exss soluo u = u ( ) of he sysem (2) ssfyg he equles ( ) < u ( ) < ( ) for every N( ) d =. Defo 4. Le posve umer δ e gve. We sy h soluo u = u( ) T( ) of he sysem (9) s δ -ouded f he equly u ( ) < δ wh u ( ) = mx = { u( ) } holds for every T( ). The ls heorem s osequee of Theorem 3. We om s proof. Theorem 4. Le posve umer δ e gve. Le ω : T ( ) R R s ouous wh respe o he seod rgume. If moreover + μ ( ) > ω ( u) / δ for every = d every ( u) d B μ ( ) < ω ( u) / δ for every = d every ( u) he here exss soluo u = u( ) N( ) of he sysem (2) ssfyg for every N( ). u ( ) < δ Aowledgme Ths wor ws suppored y he Gr /3238/6 of he Gr Agey of Slov Repul (VEGA). REFERENES:. WAŻEWSKI T. Sur u prpe opologque de l exme de l llure sympoque des égrles des équos dfféreelles ordres A. So. Polo. h. (947) BORSUK K. Theory of Rers PWN Wrsw DIBLÍK J. Dsree rer prple for sysems of dsree equos ompu. h. Appl. (2) DIBLÍK J. Asympo ehvour of soluos of sysems of dsree equos v Lypuov ype ehque ompu. h. Appl. (23)