BOUNDED SOLUTION OF THIRD ORDER NONL INEAR GENERALIZEDDIFFERENCE EQUATIONSAbstract.In the paper, we discuss the non linear generalized difference equation ( ( ) ( ( ) ( ))) ( ) ( ( 2 )), [0, ) a k b k u k q k f uk k + l l ll l(1)where( ), ( ), ( ) a k b k q kare positive real valuedfunctions,fis real valuedfunctions with( ) 0 uf u >for all0 u . Also we otain the sufficient condition for the oundedness of all nonoscillator! solutions of (1). "uitale e#a$ples are given to illustrate the $ain results. 1. Introducton1. %he theor! ofdifference equations is ased on theoperator defined as !. ( ) ( 1) ( ), . uk uk uk k N + (2)". where N&'0,1,2,(,)* . +venthough$an! authors [1,11,havesuggestedthe definition of as#. ( ) ( ) ( ) uk uk uk + l,, , k N l(()$. no significant progress too- place onthis line. .hen we too- up thedefinitionofasgivenin(2), thetheor! of difference equations aredeveloped in a different direction..e otained so$e interesting resultsin /u$er %heor!. 0or convenience,welaeledtheoperatordefined! (() asland ! defining itsinverse 1 l, $an! interesting resultson/u$er %heor!were otained.1! e#tending the stud! for co$ple#function and lto e real, so$e newqualitative properties li-e rotator!,e#pandingandshrin-ing, spiral andweli-e were studied for thesolutions of difference equationsinvolvingl. %he results otainedcan e found in [(23,.%.In 2004, 5. 5aria "usai 5anuelet.al, e#tend the theor! ofgeneralized difference operator thefirst -indlto the generalizeddifference operator of the third -ind1 2 (, ,l l lfor the positive reals1 2 (, , l l land otain so$esignificant results, relations, thediscrete version of 6einitz theore$,ino$ial theore$ and /ewton7sfor$ulaaccordingto1 2 (, ,l l l.Also,find the for$ulae for the su$ofsecond partial su$ of higher powersof arith$etic progression, the su$ ofsecondpartial su$s of consecutiveter$sofarith$eticprogressionandsu$of all secondpartial su$s ofarith$etic2geo$etric progression arederived ! using the solutions ofthird order generalized differenceequation in the field&. of /u$erical 5ethods [4,.'. 8ence, in this paper, we discussthe sufficient conditions for theoundedness of all non oscillator!solutions of the third order nonlineardifference equation. (. !.)AIN RESULTS1*. L+,,a !.1.An! eventuall!positive solution( ) ukof equation(1)elongstooneofthefollowingfour classes911.( )1( ) 0, ( ) 0, ( ( ) ( )) 0: M uk uk b k uk > > >l l l( )2( ) 0, ( ) 0, ( ( ) ( )) 0: M uk uk b k uk > > < >l l l( );( ) 0, ( ) 0, ( ( ) ( )) 0: M uk uk b k uk > < l lfor large. k8ence, we otain( ( ) ( )), b k u k l l( ) uk l,( ) uk are eventuall! one ofsign. %hus, we have proved ourle$$a.1". /ow that if we assu$e 1#.1 11 1.( ) ( )k ka k b k 1$. %hen, ! and1&. 111 1 11 1( ) .( ) ( )krk r sqsb k a r 1 1 ] < l(;)1'. %hen ever!1M2 t!pe solution ofequation (1) is ounded.1(. Proof.6et( ) ukeanunoundedsolution of equation (1) of 1M2t!pe.1!le$$a 2.1, we have( ) 0, uk >( ) 0 uk >land( ( ) ( )) 0 b k u k >l lfor [0, ,, k fro$ (1) !*.( ( ) ( ( ) ( )))( )( ( 2 ))a k b k ukq kf uk +l l ll l !1.( ) ( ( ) ( ))( ( 2 ))( ) ( ( ) ( ))( ( 2 ))a k b k ukf uka k b k u kf uk+ + ++ +l ll ll l ll ll l!!.( ) ( ( ) ( ))( ( 2 ))( ) ( ( ) ( ))( ( ))a k b k ukf uka k b k ukf uk+ + ++ +l ll ll l ll ll l(>) !".( ) ( ( ) ( ))( ( ))a k b k ukf uk 1 1+ ]ll ll lfor[0, ). k !#. "u$$ingothsides of (>) fro$1, N to r we otain !$.1( ) ( ( ) ( ))( )( ( ))( ) ( ( ) ( ))( ( ))rs NaN bN uNqsf uNa r b r urf ur ++ +l ll ll ll l!%. and therefore !&.1( ) ( ( ) ( ))1( )( ) ( ) ( ( ))rs NaN bN uNq sa r a r f uN ++lll l !'.( ( ) ( )) ( ) ( )( ( )) ( ( ))b r ur b r urf ur f ur + + + +l l ll ll l l l!(.( ) ( )( ) ( ).( ( )) ( ( ))b r urb r urf ur f ur 1 1 ]l lll l"*. "u$$ing once again, fro$1,kr N to 1 1 ]l"1.we otain"!.1 11( ) ( ( ) ( )) 1( )( ) ( ) ( ( ))( ) ( ) ( ) ( ).( ( )) ( ( ))k krr N s N r NaN bN uNqsa r a r f u rb k uk bN uNf u k f uN 11 11 ] ] ++ l ll ll ll ll l8ence"".111( )1 1( )( ( )) ( ) ( )( ) ( ( ) ( )) 1( ) ( ( )) ( )( ) ( ). (?)( ) ( ( ))krr N s Nkr Nu kqsf uk b k a raN bN uNb k f uN a rbN uNb k f uN 1 1 ] 1 1 ] +++ llll llll ll"#. "ince( ) f u uisnonincreasingfor0, u > fro$ (?) we have"$.111( )( ) ( ( ))( ) ( ) ( ( ))( ( )) 1 1( )( ) ( ) ( )( ) ( ( ) ( )) ( ( )) 1( ) ( ( )) ( ) ( )( ) ( ). (3)( ) ( )krr N s Nkr Nukuk f uNuk uN f ukf uNqsuN b k a raN bN uN f uNuN f uN b k a rbN uNuNb k 1 1 ] 1 1 ] +++ l llll lll lll ll6et( )( ) ( ) ( ( ))t kgt uk uk + llfor. k t k + l %hen( )@( )ukg tlland( ) ( ) gt uk for. k t k + l8ence"%.( ) @( ) @( )( ) ( ) ( )ln[ ( ) ( ), ln[ ( ),ln[ ( ), ln[ ( ),. (A)k kk kuk g t g tdt dtuk uk g tu k uk uku k uk+ + + + l lllll"&. /ow, su$$ing oth sides of (A)fro$ k N to 1 n , we otain"'.1 1( )[ln ( ) ln ( ),( )ln ( ) ln ( ).n nk N k Nukuk ukukuk uN + lll"(.8ence, ! (3) we get #*.11 1ln ( ) ln ( )( ( )) 1 1( )( ) ( ) ( )( ) ( ( ) ( )) ( ( ))( ) ( ( ))kn rk N r N s Nuk uNf uNqsuN b k a raN bN uN f uNuN f uN 1 1 ] + + ll lll l#1.11 1( ) ( ) 1 1 1.( ) ( ) ( ) ( )kn nk N r N k NbN uNb k a r uN b k 1 1 ] _+ , lll0ro$ (;) there follows the convergence of theseries11 11 1( ) ( )kk rb k a r 1 1 ] land 11,( )kb k

and therefore ln[ ( ), uk is ounded. %his is contradiction co$pletes our proof. #!. T-+or+, !.". Assu$e the condition(1) holds. %hen ever!2M2t!pesolution of equation (1) is ounded.#". Proof . If ( ) uk is a solution of the equation (1) of 2M2t!pe then there e#ists K such that( ) 0 uk >,( ) 0 uk >land ( ( ) ( )) 0 b k u k l."o the solution( ) uk$ust eounded. %his co$pletes our proof.#&. Bonsequentl! ! le$$a 2.1 ,%heore$ 2.2 , and %heore$ 2.( thefollowing theore$ is otained.#'. T-+or+, !.#. Assu$e f is a non decreasing function, ( ) = f u uis non increasing for 0 u >and #(.111 1 11 1( )( ) ( )krk r sqsb k a r 1 1 ] < l .$*. %hen ever! eventuall! positivesolution of equation (1) is ounded.$1. If the assu$ption, that( ) f uis nondecreasing and( ) = f u uis nonincreasing for 0 u >, is replaced !the assu$ption that( ) f uis nonincreasing and( ) = f u uis nondecreasingfor0 u and(1)holds.%hen ever!1M2t!pe solution ofequation (1) is ounded.$&. Proof.6et( ) ukeasolutionof1M2t!peequation(1). %hentheree#istssuchareal nu$er Kthat( ) 0 uk >,( ) 0 uk >land( ( ) ( ) ) 0 b k uk >l lfor. k K "ince$'.( ( )) ( ( )) f uk f uk + l for k K , wehave$(.( ( ) ( ( ) ( )))( ) ( ( 2 )) ( ) ( ( )),a k b k ukq k f u k q k f uK k K + l l ll l l%*. "u$$ing oth sides of the aoveinequalit!fro$s K to1 r weget%1.( ) ( ( ) ( )) ( ) ( ( ) ( )) a r b r ur aK bK uK l l l l62.1( ( )) ( ).rs Kf uK q sl 8ence 63.11( ( ) ( )) ( ) ( ( ) ( ))( )( ( )) ( ). (10)( )rs Kb r ur aK bK uKa rf uK qsa r +l l l ll"u$$ing (10) fro$r K to1,k1 1 ]l %#. we otain, ( )( ) ( )( )bKuk uKb k l l%$.1111 1( ) ( ( ) ( ))( ) ( )( ( )) 1( ).( ) ( )kr Kkrr K s KaK bK uKb k a rf uKq sb k a r 1 1 ] 1 1 ] + + ll lll%%. 0inall! we get%&.11111 11( ) ( ) ( ) ( )( )1 1( ) ( ( ) ( ))( ) ( )1 1( ( )) ( ).( ) ( )kk Kkkk K r Kkk rk K r K s Kuk uK b K uKb kaK bK uKb k a rf uK qsb k a r 1 1 ] 1 1 ] + + + lll lll0ro$ (;) we have11 11 1( ) ( )kk rb k a r 1 1 ] < land11( )kb k< , and which !ields ( ) ukis ounded.Bonsequentl!!le$$a2.1,%heore$2.(and %heore$2.?,we get the following theore$.%'. T-+or+, !.&. 6et f e non increasing function for 0 u and111 1 11 1( ) .( ) ( )krk r sqsb k a r 1 1 ] < l%(. %hen ever! eventuall! positivesolution of equation (1) is ounded.&*. E.a,/0+ !.'.Bonsider the thirdorder generalized difference equation2 22(( 2 ) ( ( )))1, [0, ), (11)( )( 2 ) ( 2 )k k ukkk k u k + + + +l l lll l l l&1. where.kj k 1 1 ]ll8ere allconditions of theore$ 2.3 aresatisfied. 8ence ever! eventuall!positive solution of (11) is ounded.Cne such solution is 1( ) . ukk 1 1 ]l &!. Iftheassu$ptionthat( ) f uisnonincreasingfor0 u >isreplaced!the assu$ption that( ) f uis nondecreasing for0 u ) '!. %hen ever!2M2t!pe solution ofequation (1) is unounded.%he proofis si$ilar totheproof of %heore$2.4. If the condition (1>) is notsatisfied, thentheaoveresult $a!fail. It is shown in +#a$ple (.2.'". E.a,/0+ ".!. Bonsider theequation(( ) (( ) ( )))2( 2 ), [0, ). (1?)( )k k ukkuk kkk + + + + +l l ll llll lAll conditionsof%heore$(.1aresatisfied e#cept condition (1>),na$el! '#.1 11 1 1 11 1 1 1.( ) ( ) ( )k kk r k rb k a r k a r 11 11 ] ] + + l ll l'$. "o we cannot sa! ever! 2M2t!pe solution is unounded. In fact, equation (1?) got the oundedsolution 1( ) 1 ukk 1 1 ]l .0ro$ theore$ (1) in [2, we have the following result.'%. T-+or+, ".". Iffis nondecreasing and'&.1 11 1( ) ( )k ka k b k 1 1 11 1( )( ) ( )k rk r sq ka r b s .''. %henever!nonoscillator!solutionof equation (1) is unounded.'(. E.a,/0+ ".#. Bonsider theequation (*.2 (1 1( )? (13 2 (( 2 ), [0, ). (13)( > )k kk kuku k kk 11 11 ] ] 11 11 ] ] _ _ , , + +ll ll ll lll lAll conditions of %heore$(.(aresatisfied. 8ence ever! nonoscillator! solution of (13) isunounded. Cne such solutions is( ) ( ( ). uk k +l

(1. T-+or+, ".$.+quation (1) cannothave a quic-l! oscillator! solution. (!. Proof. 6et ( ) 0 zk > for all k K

and suppose that ( ) ( 1) ( )kuk zk 1 1 ]l is a solution of equation (1).%hen( ( ) ( ( ) ( ))) a k b k u k l l l93.1( 1) [ ( ) ( 2 ) ( ( )( ( ) ( 2 )( ) ( )) ( 2 ).( ( ) ( ) ( ) ( )( ) ( )) ( )( ) ( ) ( ),ka k b k zka k b ka k b k zka k b k a kb ka kb k zka kb kzk 1+ 1 ] + + + + + + + + + + ' ; + + + + + + + + ll l ll ll l ll l ll%herefore equation (1) can e writtenas 94.1( 1) [ ( ) ( 2 ) ( ( )( ( ) ( 2 )( ) ( )) ( 2 )( ( ) ( ) ( ) ( )ka k b k zka k b ka k b k zka k b k a kb k 1+ 1 ] + + ++ + ++ + + ++ + + + +ll l ll ll l ll l l($.2( ) ( )) ( ) ( ) ( ) ( ),( ) (( 1) ( 2 )).ka kb k zk a kb kzkq k f zk 1+ 1 ]+ + + +lll l(%. 1! ta-inglis even andk '$ultiple ofl* we have,[ ( ) ( 2 ) ( ( ) a k b k zk + + + l l l(&.2( ( ) ( 2 ) ( ) ( )) ( 2 )( ( ) ( ) ( ) ( )( ) ( )) ( ) ( ) ( ) ( ),( ) (( 1) ( 2 )),ka k b k a k b k zka k b k a kb ka kb k zk a kb kzkq k f zk 1+ 1 ]+ + + + + + ++ + + + ++ + + +ll l l l ll l lll lwhere[ ( ) ( 2 ) ( ( ) a k b k zk + + + l l l('.( ( ) ( 2 ) ( ) ( )) ( 2 )( ( ) ( ) ( ) ( )( ) ( )) ( ) ( ) ( ) ( ), 0a k b k a k b k zka k b k a kb ka kb k z k a kb kzk+ + + + + + ++ + + + ++ + + ( ) ( ( 2 )) 0. q k f zk + > l l ((. Cn the other hand, ! ta-ing lis oddand k {multiple of l},we fnd 1**.[ ( ) ( 2 ) ( ( ) ( ( ) ( 2 )( ) ( )) ( 2 )( ( ) ( ) ( ) ( )( ) ( )) ( ) ( ) ( ) ( ),( ) ( ( 2 )).a k b k zk a k b ka k b k z ka k b k a kb ka kb k zk a kb kz kq k f zk+ + + + + ++ + + ++ + + + ++ + + +l l l l ll l ll l lll l%he left side of the aove equation isalwa!s positive, and the right side is alwa!s negative. %his contradiction proves our theore$.1*1. REFERENCES1. D.E Agarwal, Fifference+quationsandInequalities, 5arcel Fe--er, /ew Gor-,2000.2. ".".Bheng and 8.H. 6i, 1ounded"olutions of /onlinear Fifference+quations,%a$-angH.5ath, 21(1440),1(321;2.(. 5.5aria "usai 5anuel, I.1ritto Anton!Javier and +.%handapani, %heor! ofIeneralized Fifference Cperator and ItsApplications, 0ar+ast Hour.of5athcl."ciences, 20(2) (200?), 1?( 2 131.;. 5.5aria "usai 5anuel, I.1ritto Anton!Javier and +.%handapani, KualitativeEroperties of "olutions of Bertain BlassofFifference+quations, 0ar+ast Hour.of 5athe$atical. "cins, 2((() (200?),24>2(0;.>. 5.5aria "usai 5anuel, I.1ritto Anton!Javier and +.%handapani, Ieneralized1ernoulli Eol!no$ials %hrough.eighted Eochha$$er "!$ols, 0ar+ast Hournal of Applied 5athe$atics,2?(() (2003), (212(((.?. 5.5aria "usai 5anuel, A.Ieorge 5aria"elva$and I.1ritto Anton! Javier,Dotator!and 1oundednessof "olutionsof Bertain Blass of Fifference+quations, Intrnl. Hrnl. of Eure andApplied 5aths., (((() (200?), (((2(;(.3. 5.5aria "usai 5anuel and I.1rittoAnton! Javier, Decessive, Fo$inantand "piral 1ehaviors of "olutions ofBertain Blass of Ieneralized Fifference+quations, International Hournal ofFifferential +quations and Applications,10(;) (2003), ;2(2;((.A. 5.5.". 5anuel, I.1.A. Javier, L.Bhandrase-ar, Ieneralized FifferenceCperatorof%he"econd432?0?.1*!.1*".1,2Fepart$ent of 5athe$atics,1*#. "