On the Asymptotic Behavior of Solutions for a Class of Second Order Nonlinear Difference Equations

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1 IOSR Joral o Matemati IOSR-JM e-issn: p-issn: X. Volme Ie 6 Ver. IV Nov. - De.6 PP 6- O te Aymptoti Beavior o Soltio or a Cla o Seod Order Noliear Dieree Eqatio V. Sadaivam Po. Sdar ad A. Sati 3 3 P ad Reear Departmet o MatematiTirvallvar overmet Art Collee Raipram Namaal Tamil Nad Idia. Om Mra Collee o Art ad Siee Salem Tamil Nad Idia. Abtrat: We tdy te aymptoti beavior o oltio or a la o eod order oliear dieree eqatio. Ui Biari ieqality we obtai oditio der wi all oltio are aymptoti to a b a were a ad b are real otat. Keyword: Aymptoti beavior Biari ieqality eod order oliear dieree eqatio. I. Itrodtio I ti paper we are oered wit te oliear dieree eqatio wi i ed i modeli o a lare mber o pyial ytem. Here i te orward dieree operator deied by x x x were N { } ad Z. By a oltio o eqatio we mea a eqee o real mber wi i deied or N ad wi atiie eqatio. A oltio { x } i aid to be oillatory i it i eiter evetally poitive or evetally eative ad it i ooillatory oterwie. Aymptoti beavior o oltio o eod order dieree eqatio ave bee tdied by may ator or example ee [5 7-6] ad te reeree ited tere i. For a eeral barod o dieree eqatio ee te moorap [ 7]. II. Mai Relt We bei wit te ollowi relt. Teorem.. Sppoe tat te tio v atiie te ollowi oditio. i i otio i D { v/ N v R} were ; ii Tere exit two eqee ad tat v v were or > te eqee i poitive ad odereai < ad i we deote ad. Te every oltio o eqatio i aymptoti to a b were a ad b are real otat. Proo. It ollow rom i by tadard exitee Teorem ee or example[] tat eqatio a oltio orrepodi to te iitial data ad. Smmi two time rom to we et or ad DOI:.979/ Pae. 3

2 O te Aymptoti Beavior o Soltio or a Cla o Seod Order Noliear Dieree Eqatio DOI:.979/ Pae It ollow rom ad 3 or. So applyi ii i te above ieqalitie we et or 4 5 Deote by te rit ad ide o ieqality 5 Hee 4 ad 5 imply tat ad 6 Te eqee i oderai or > ad ee we et by 6 o we olde tat or. 7 Applyi direte Biari ieqality [7] to 7 we et or were i te oltio o ad i te ivere tio o wi i deied o x N N. Now pt. < K Sie i ireai we et < o it ollow rom 6 tat. ad By ii we ave

3 O te Aymptoti Beavior o Soltio or a Cla o Seod Order Noliear Dieree Eqatio tereore exit a well a tere exit a lim a. a R tat Frter i te ame way a i [ 7] we a ere tat tere exit a oltio wit te property lim. Fially by te L Hopital rle we olde tat lim lim a ad t tere exit a b R tat lim a b o te proo i ow omplete. Example.. Coider te oliear dieree eqatio [ a b] a E were < ad. Tereore eqatio E atiie all oditio o Teorem.. Tereore we dede tat or ay oltio o eqatio E tere exit real a b tat a b. Teorem.. Sppoe tat apart rom amptio i o Teorem. te tio v atiie te ollowi oditio ii tere exit two poitive eqee tat v v were or > te eqee i poitive ad oderai < ad i we deote te. Te every oltio o eqatio i aymptoti to a b were a b are real otat. Te proo o te teorem i aaloo to tat o Teorem. ad t it i omitted. Corollary.. Coider te eqatio a 8 were a <. Te lim exit ad te eeral oltio o eqatio 8 i aymptoti to d d a were d may be ero or d may be ero bt ot bot imltaeoly. Proo. Te olio o orollary ollow rom Teorem. wit a ad. DOI:.979/ Pae

4 O te Aymptoti Beavior o Soltio or a Cla o Seod Order Noliear Dieree Eqatio Example.. Coider te oliear dieree eqatio a b a E All oditio o Teorem. are atiied. Hee it ollow rom Teorem. tat or ay oltio o eqatio E tere exit real otat a b tat a b a. Fially ari i te ame way a i Teorem. we a alo prove te ollowi eeral relt wi wit ive exatly Teorem. ad wit ive exatly Teorem.. Teorem.3. Sppoe tat apart rom te amptio i o Teorem. te tio v atiie te ollowi oditio ii tere exit poitive eqee ad tat v v were or > te eqee ad are poitive ad odereai < ad i we deote te. Te every oltio o eqatio i aymptoti to a b were a b are real otat. Corollary.. Coider te well-ow Emde-Fowler eqatio m 9 wit < m < ad <. Te lim exit ad all oltio o eqatio 9 i aymptoti to d d a. Proo. Te olio o orollary ollow rom Teorem.3 wit m ad. Example.3. Coider te oliear dieree eqatio E 3 were ad a. All oditio o Teorem.3 are atiied. Te or ay 3 oltio o eqatio E 3 te exitee o real mber a b tat a b o a. Oberve tat 5 i te oe oltio o eqatio E 3 wit te iitial data ad. 6 Example.4. Coider te bliear dieree eqatio / E4 were < m < ad <. All oditio o Corollary. are atiied ad all oltio o eqatio 9 i aymptoti to d d a. Remar.. It i importat to ote tat by Teorem.-.3 ad Corollarie. ad. all oltio o eqatio 8 ad 9 repetively are aymptoti to a b a to te retritio o te tio may eem to be artiiial. It i poible to relax tee amptio or a ertai rater wide la o eqee to te prie to be paid or it i te deired aymptoti beavior oly or a part o oltio o eqatio wit iitial data atiyi a ertai etimate. DOI:.979/ Pae

5 O te Aymptoti Beavior o Soltio or a Cla o Seod Order Noliear Dieree Eqatio Aowledemet Te ator wi to expre teir iere ta to te reeree or valable ommet ad etio. Reeree []. Aarwal. R.P. Dieree Eqatio ad Ieqalitie Teory Metod ad Appliatio Marel Deer New Yor 99. []. Aarwal. R.P ad Wo P.J.Y. Advaed Topi i Dieree Eqatio Klwer Aademi Pblier rop Dordret 997. [3]. Bellma. R. Stability Teory o Dieretial Eqatio Mraw-Hill Lodo 953. [4]. Biari. I. A eeraliatio o a lemma o Bellma ad it appliatio to iqee problem o dieretial eqatio Ata. Mat. Aad. Si. H [5]. Ce. S.S. Li. H.J ad Patla. W.T. Boded ad ero overet oltio o eod order dieree eqatio J. Mat. Aal. Appl [6]. Croi. J Dieretial Eqatio Itrodtio ad Qalitative Teory Deer New Yor 98. [7]. Dai. B.X. ad Ha. L. H. Aymptoti beavior o oltio or a la o oliear dieree eqatio Compt. Mat. Appl [8]. He. X.Z Oillatory ad aymptoti beavior o eod order oliear dieree eqatio J. Mat. Aal. Appl [9]. Hooer. J.W ad Patla. W.T. A eod order oliear dieree eqatio. oillatio ad aymptoti beavior J. Mat. Aal. Appl []. Media. R. Aymptoti beavior o oltio o eod order dieree eqatio J. Compt. Appl. Mat []. Tadapai. E. Aymptoti ad oillatory beavior o oltio o a eod order oliear etral delay dieree eqatio Riv. Mat. Uiv. Varma []. Tadapai.E. Sdar. P. rae. J.R ad Spie. P.W. Aymptoti propertie o oltio o oliear eod order etral delay dieree eqatio Dyami. Syt. Appl [3]. Tadapai. E ad Maria.S.I Te aymptoti beavior o oltio o oliear eod order dieree eqatio Appl. Mat. Lett [4]. Wa. J. Me. F ad Li.J. Aymptoti beavior o oltio o eod order oliear dieree eqatio Kodai Mat. J [5]. Wei. Aymptoti beavior relt or oliear etral delay dieree eqatio Appl. Mat. Comp [6]. Za. B. ad Zo. Y. Oillatio ad ooillatio or eod order liear dieree eqatio Comp. mat. Appl [7]. Laiatam. V ad Triiate Teory o Dieree Eqatio Nmerial Metod ad Appliatio Aademi Pre INC New Yor 988. DOI:.979/ Pae