summary:This paper deals with oscillatory and asymptotic behaviour of solutions of second order quasilinear difference equation of the form \[ \Delta (a_{n-1}| \Delta y_{n-1}|^{\alpha -1} \Delta y_{n-1})+ F(n, y_n)= G(n, y_n, \Delta y_n), \quad n\in N(n_0) \qquad \mathrm {(E)}\] where $\alpha >0$. Some sufficient conditions for all solutions of (E) to be oscillatory are obtained. Asymptotic behaviour of nonoscillatory solutions of (E) are also considered

Ramuppillai, L.

Thandapani, E.

Czech digital mathematics library

Oscillatory and asymptotic behaviour of perturbed quasilinear second order difference equations

Institute of Mathematics AS CR, v. v. i.

Archivum MathematicumEthiraju Thandapani; L. RamuppillaiOscillatory and asymptotic behaviour of perturbed quasilinear second orderdifference equationsArchivum Mathematicum, Vol. 34 (1998), No. 4, 455--466Persistent URL: http://dml.cz/dmlcz/107673Terms of use:© Masaryk University, 1998Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access todigitized documents strictly for personal use. Each copy of any part of this document must containthese Terms of use.This paper has been digitized, optimized for electronic delivery and stampedwith digital signature within the project DML-CZ: The Czech Digital MathematicsLibrary http://project.dml.czARCHIVUM MATHEMATICUM (BRNO)Tomus 34 (1998), 455 { 466OSCILLATORY AND ASYMPTOTIC BEHAVIOUROF PERTURBED QUASILINEAR SECONDORDER DIFFERENCE EQUATIONSE. Thandapani and L. RamuppillaiAbstract. This paper deals with oscillatory and asymptoticbehaviour of solutionsof second order quasilinear dierence equation of the form(E) (an 1jyn 1j 1yn 1) + F (n; yn) = G(n; yn ;yn); n 2 N(n0)where  > 0. Some sucient conditions for all solutions of (E) to be oscillatoryare obtained. Asymptotic behaviour of nonoscillatory solutions of (E) are alsoconsidered. 1. IntroductionRecently, there has been a lot of interest in the study of oscillation and nonoscil-lation of second order dierence equations see, for example [1{4, 6, 7{13] and thereferences contained therein. Following this trend in this paper, we shall considerthe perturbed quasilinear dierence equation(1) (an 1jyn 1j 1yn 1) + F (n; yn) = G(n; yn;yn);where n 2 N (n0) = fn0; n0 + 1; : : :g, (n0 is a xed nonnegative integer),  isthe forward dierence operator dened by yn = yn+1   yn, fang is a positivereal sequence and  > 0. Moreover, F and G are real valued functions withyn : N (n0)! R, F : N (n0) R! R and G : N (n0) R2 ! R.By a solution of (1) we mean a real sequence fyng satisfying (1) for n 2 N (n0).A nontrivial solution fyng is said to be oscillatory if it is neither eventually positivenor eventually negative, and nonoscillatory otherwise.The purpose of this paper is to establish some new results on the oscillatory andasymptotic behaviour of solutions of (E). The results obtained here dier greatlyfrom those in [6, 7, 8, 11, 12, 13] and other known literature.1991 Mathematics Subject Classication : 39A10.Key words and phrases: perturbed quasilinear dierence equation, oscillatory solution.Received June 19, 1997.456 E. THANDAPANI, L. RAMUPPILLAIThroughout this paper we shall assume that there exist real sequences fqng,fpng and a function f : R! R such that(c1) uf(u) > 0 for all u 6= 0;(c2) f(u)   f(v) = g(u; v)(u   v) for u; v 6= 0,where g is a nonnegative function; and(c3) F (n; u)f(u)  qn, G(n; u; v)f(u)  pn for u; v 6= 0.2. Asymptotic behaviour of nonoscillatory solutionsIn this section, we assume that(2) 1Xj=n0(qj   pj) =1 :Theorem 1. Let condition (c1) { (c3) and (2) hold, then any nonoscillatorysolution of (1) must belong to one of the following two types:Mc : yn ! c 6= 0 for n!1 ;M0 : yn ! 0 for n!1 :Proof. Let fyng be a nonoscillatory solution of (1), then fyng is eventually pos-itive or negative. Thus from (1), we have(3)   an 1jyn 1j 1yn 1f(yn 1)  = (an 1jyn 1j 1yn 1)f(yn) an 1jyn 1j 1(yn 1)2g(yn; yn 1)f(yn 1) f(yn)  [qn   pn]  an 1jyn 1j 1(yn 1)2g(yn; yn 1)f(yn 1) f(yn)<  (qn   pn) :Summing (3) from n0 + 1 to n, we get(4) anjynj 1ynf(yn)  an0 jyn0j 1yn0f(yn0 )   nXs=n0+1(qs   ps) :If fyng is eventually positive, then there exists n1 2 N (n0) such that yn > 0 forn 2 N (n1). Then from (4) and (1), we getyn < 0 for n 2 N (n1) :OSCILLATORY AND ASYMPTOTIC BEHAVIOUR : : : 457Hence fyng is monotonic decreasing, and limn!1 yn = c  0, where c is constant.The proof for the case yn is eventually negative is similar.Thus, any nonoscillatory solution of (1) must belong to either Mc or M0. Thiscompletes the proof of the theorem. Remark 1. In the above theorem, we impose no condition on fang such as1Pn=n0+1 1a1=n is convergent or divergent.Remark 2. If 1Pn=n0+1 1a1=n is convergent or divergent and F (n; yn) = qnf(yn)and G(n; u; v) = 0 for n 2 N (n0), then the possible asymptotic behaviour ofnonoscillatory solutions of equation (1) are given in Lemma 1 of [10] and Lemma1 of [9] respectively.Remark 3. If  = 1, then Theorem 1 reduces to Theorem 1 of [2].Theorem 2. Let conditions (c1) { (c3) and (2) hold.a) If f(u) = juj sgnu, then a necessary condition for equation (1) to have anonoscillatory solution fyng 2Mc is that(5) 1Xn=n1+1  1an nXi=n1+1(qi   pi)! 1= <1where n1 2 N (n0) is suciently large.b) If f(u) = jujm sgnu for 0 < m < , (m > ) then a necessary conditionfor equation (1) to have a nonoscillatory solution fyng 2Mc orM0(fyng 2M0) is also (5).Proof. (a) Let fyng be a nonoscillatory solution of (1) which belongs to Mc. Ifc > 0, then fyng is eventually positive. From the proof of Theorem 1, we havethat yn is eventually negative and from (2), there exists n1 2 N (n0) such thatyn > 0, yn < 0 and nXi=n0+1(q0   p0) > 0 for n 2 N (n1) :Summing (3) from n1 + 1 to n, it follows thatanjynj 1ynyn  an1 jyn1j 1yn1yn1   nXk=n1+1(qk   pk)   nXk=n1+1(qk   pk) ;458 E. THANDAPANI, L. RAMUPPILLAIthat is, 1an nXk=n1+1(qk   pk)! 1=   ynyn :Summing the above inequality from n1 + 1 to n we havenXk=n1+1  1ak kXi=n1+1(qi   pi)! 1=  nXk=n1+1 yk   yk+1yk  yn1+1   cc ;from which letting n!1, we obtain (5).(b) If 0 < m < , let fyng be a solution of (1) which belongs to M0 or Mc, asshown in the proof of case (a), we can obtainynym=n   1an nXk=n1+1(qk   pk)! 1= :Summing the above inequality from n1 + 1 to n, we havenXk=n1+1 " 1ak kXi=n1+1(qi   pi)# 1=  nXk=n1+1 yk   yk+1ym=k  Z yn1+1yn+1 dssm== 11  m h y1 mn1+1   y1 mn+1 i ;from which letting n ! 1 and noting that 0 < m <  and limn!1 yn = 0 orlimn!1 yn = c > 0, we obtain (5).If fyng is eventually negative, similarly, we can show that (5) holds. Thus theproof of Theorem 2 is complete. Remark. If  = 1, then Theorem 2 reduces to Theorem 2 of [2].Example. Consider the dierence equation(2n(+2))jyn 1j 1yn 1) + yn(b(n; yn) + n)= b(n; yn)yn; n  1 ;(8)where  > 0 and b(n; yn) is any function of n and yn and n = 3  (2)n. Choosingf(yn) = yn we haveF (n; yn)f(yn) = b(n; yn) + n = b(n; yn) + 3(2n) = qnand G(n; yn;yn)f(yn) = b(n; yn) = pn. All conditions of Theorem 1 hold, and hencea nonoscillatory solution fyng of (8) must belong either to Mc or M0. In factfyng =  12n 	 2M0 is one such solution.OSCILLATORY AND ASYMPTOTIC BEHAVIOUR : : : 4593. Oscillation of solutionsIn this section we will establish some oscillation criteria for equation (1).Theorem 3. Let f(u) = juj sgnu and the condition (c3) holds. Assume thatfunction H(n; s) satisesH(n; n) = 0 for n  n0 2 N (n0) ;H(n; s) > 0 for n > s  n0 and2H(n; s)  0 for n > s  n0 where2H(n; s) = H(n; s+ 1) H(n; s) :Suppose there exists a real valued function h(n; s) such that 2H(n; s) = h(n; s)[H(n; s)]1= for all n > s  n0where  = + 1 . If for suciently large n1 2 N (n0) such that for n 2 N (n1)qn   pn  0 ;(9) 1Xn=n0+1 1a1=n 1 =1 ;(10)and(11) lim supn!1 1H(n; n0) n 1Xs=n0 " H(n; s)(qs   ps)  as 1  h(n; s)+ 1  +1 # =1then all solutions of equation (1) are oscillatory.Proof. Let fyng be a nonoscillatory solution of equation (1). Without loss ofgenerality we may assume that yn > 0 for n  N0 2 N (n0). From condition (9)and the equation (1), we get(an 1jyn 1j 1yn 1)  0; n  N0and hence an 1jyn 1j 1yn 1 is nonincreasing. We claim thatan 1jyn 1j 1yn 1 > 0 :If an 1jyn 1j 1yn 1  0, then there exists an integer N1 2 N (n0) such thatan 1jyn 1j 1yn 1 < aN1 1jyN1 1j 1yN1 1  0for n > N1.Since an 1jyn 1j 1yn 1 =  an 1( yn 1) ;460 E. THANDAPANI, L. RAMUPPILLAIwe have  an 1( yn 1) <  aN1 1( yN1 1)or yn 1 < a1=N1 1(yN1 1)a1=n 1 ; n > N1 :Summing the above inequality from N1+1 to n and letting n!1, we obtain, inview of condition (10), yn !  1, a contradiction. Hence there exists an integerN0 2 N (n0) such that an 1jyn 1j 1yn 1 > 0 or yn 1 > 0 for all n  N0.Dene zn = an 1(yn 1)yn 1 ; for n  N0 :Then zn   (qn   pn)  an 1(yn 1)yn 1yn 1yn for n  N0 :(12)By the mean value theorem yn 1 =  t 1n yn 1(13)where yn 1 < tn < yn. Using (13) in (12), we getzn <  (qn   pn)   an 1(yn 1)y 1n 1yn 1yn 1 yn for n  N0 :(14)Now using monotonic nature of fyng and fan(yn)g, in (14), we obtainzn   (qn   pn)  z1+1=n+1a1=n 1 for all n  N0 :Then, for all n  N  n0n 1Xs=N H(n; s) (qs   ps) = H(n;N )zN   n 1Xs=N   2H(n; s)zs+1 + H(n; s)zs+1a1=s 1 = H(n;N )zN   n 1Xs=N  h(n; s)H1=(n; s)zs+1 + H(n; s)a1=s 1 zs+1 != H(n;N )zN   n 1Xs=N " h(n; s)H1=(n; s)zs+1 + H(n; s)a1=s 1 zs+1+  h(n; s)+ 1  +1 as 1 # + n 1Xs=N  h(n; s)+ 1  +1 as 1 :OSCILLATORY AND ASYMPTOTIC BEHAVIOUR : : : 461Hence, for all n  N  N0, we haven 1Xs=N " H(n; s)(qs   ps)  as 1  h(n; s)+ 1  +1 # = H(n;N )zN  n 1Xs=N h h(n; s)H1=(n; s)zs+1 + H(n; s)a1=s 1 zs+1 +  h(n; s)+ 1  +1 as 1 i :Since  > 0, we haveh(n; s)H1=(n; s)zs+1 + H(n; s)a1=s 1 zs+1 +  h(n; s)+ 1  +1 as 1  0for all n > s  N0. This implies that for all n  N0n 1Xs=N0 h H(n; s) (qs   ps)  as 1  h(n; s)+ 1  +1 iH(n;N0) zN0H(n; n0) zN0 :Therefore,n 1Xs=n0 [H(n; s) (qs   ps)   as 1  h(n; s)+ 1  +1 #=N0 1Xs=n0 " H(n; s) (qs   ps)  as 1  h(n; s)+ 1  +1 #+ n 1Xs=N0 " H(n; s) (qs   ps)   as 1  h(n; s)+ 1  +1 #H(n; n0)N0 1Xs=n0 (qs   ps) +H(n; n0) zN0=H(n; n0) " N0 1Xs=n0 (qs   ps) + zN0 #for all n  N0. This giveslim supn!1 1H(n; n0) n 1Xs=n0 h H(n; s) (qs   ps)   as 1  h(n; s)+ 1  +1 i N0 1Xs=n0 (qs   ps) + zN0which contradicts (13). This completes the proof of the theorem. In the following theorems, we obtain oscillation criterion for equation (1) byusing a generalized Riccati transformation due to Yu [14].462 E. THANDAPANI, L. RAMUPPILLAITheorem 4. Let  = 1 and the conditions (c1) { (c3), (9) and (10), satises withg(u; v)   > 0 for u; v 6= 0. Let H(n; s) be such that H(n; n) = 0 for n 2 N (n0),H(n; s) > 0 for n > s 2 N (n0) and 2H(n; s) is nonpositive for all n  s 2 N (n0).Suppose there exists a real valued function h(n; s) with 2H(n; s) = h(n; s) p H(n; s) for all n  s 2 N (n0) :If there exists a real positive sequence fng with n+1 < 12 andlim supn!1 1H(n; n0) n 1Xs=n0 h H(n; s)	s   14 as2s+1s h2(n; s)i =1where n = n01 + 2(n   n0)n0 and	n = n h (qn   pn) + an2n+1  (an 1n)ithen all solutions of equation (1) are oscillatory.Proof. Let fyng be a nonoscillatory solution of equation (1). Without loss ofgenerality we may assume that yn > 0 for n  N0 2 N (n0). From the proofof Theorem 3 and from equation (1), we have yn 1 > 0 and fan 1yn 1g isnonincreasing for all n  N1  N0 + 1. Denezn = n h an 1yn 1f(yn) + an 1n i for n  N1 :Then zn = n h anynf(yn 1) + ann+1 i + n h (an 1yn 1)f(yn)  anyng(yn; yn 1)ynf(yn)f(yn 1) + (an 1n)i ;n  N1 :In view of condition (c3) and from the monotonic nature of fyng and fan 1yn 1g,we obtainzn nzn+1n+1 + n h   (qn   pn)   an   zn+1n+1   ann+1  2 +(an 1n)i=  	n   an n2n+1 z2n+1 :OSCILLATORY AND ASYMPTOTIC BEHAVIOUR : : : 463Hence for all n  N  N1, we haven 1Xs=NH(n; s)	s  H(n;N )	N   n 1Xs=N   2H(n; s) zs+1 + H(t; s)as2s+1 sz2s+1 = H(n;N )	N   n 1Xs=N  h(n; s) p H(n; s)zs+1 + as ss+1H(n; s)z2s+1 = H(n;N )	N   n 1Xs=N  s sH(n; s)as2s+1 zs+1 + 12 h(n; s)s+1pasps ! 2+ 14 n 1Xs=N h2(n; s)2s+1ass :Then, for all n  N  N1n 1Xs=N h H(n; s)	s   14 as2s+1h2(n; s)s i  H(n;N )	N  n 1Xs=N  s sH(n; s)as2s+1 zs+1 + 12 h(n; s)s+1pasps ! 2 :This implies that for every n  N1n 1Xs=N1  H(n; s)	s   14 as2s+1h2(n; s)s   H(n;N1)	N1 H(n;N1)j	N1 j H(n; n0)j	N1 j :The rest of the proof is similar to that of Theorem 3 and hence the details areomitted. To prove the next result, we need the following well-known inequality which isdue to Hardy, Littlewood and Polya [5, Theorem 41].Lemma 5. If X and Y are nonnegative, thenX + (   1)Y    XY  1  0;  > 1 ;where equality holds if and only if X = Y .Theorem 6. Let conditions (c1) { (c3) with g(u; v)  0 for u; v 6= 0, and condition(9) and (10) hold. If there exists a positive (nondecreasing for  6= 1) real sequencefcng such that 1Xk=n1+1 ck(qk   pk) =1(15)464 E. THANDAPANI, L. RAMUPPILLAIand 1Xk=n1+1 ak 1(ck 1)+1ck  1Ps=k(qs   ps)  <1(16)then all solutions of equation (1) are oscillatory.Proof. Suppose there exists a nonoscillatory solution fyng. Without loss of gen-erality, we may assume that yn > 0 for n 2 N (n0). From the proof of Theorem 3,we have(17) yn 1 > 0 and an 1(yn 1)  an1(yn1) = M0 :for all n  n1 2 N (n0).Similar to (4), we have  an 1(yn 1)f(yn)  <  (qn   pn) for n  n1 :Summing the last inequality from n 2 N (n1), to N + 1 and letting N ! 1, wehave 0  limN!1 aN (yN )f(yN )  an 1(yn 1)f(yn)   1Xs=n(qs   ps) :Thus(18) 1an 1 1Xs=n(qs   ps)  (yn 1)f(yn) ; for n 2 N (n1) :Now, an 1cn 1(yn 1)f(yn 1)  = cnG(n; yn;xn)   F (n; yn)f(yn)  cnan 1(yn 1)+1g(yn; yn 1)f(yn)f(yn 1) + an 1(yn 1)cn 1f(yn 1)  cn(qn   pn)  cnan 1(yn 1)+1f(yn)f(yn 1) + an 1(yn 1)cn 1f(yn 1)=  cn(qn   pn)  an 1f(yn)f(yn 1) h cn(yn 1)+1f(yn)+1  (yn 1)cn 1f(yn) +   cn 1+1  +1cn i + an 1   +1cn 1  +1cn f(yn)f(yn 1) :Since by lemmacn(yn 1)+1f(yn)+1   (yn 1)cn 1f(yn) +   +1cn 1  +1cn  0OSCILLATORY AND ASYMPTOTIC BEHAVIOUR : : : 465for all n  n1 . Then we have(19)  an 1cn 1(yn 1)f(yn 1)    cn(qn   pn) + an 1   +1cn 1  +1cn f(yn)f(yn 1)=  cn(qn   pn) + an 1   +1cn 1 +1cn (yn 1)2(yn 1)2 f(yn)f(yn 1)=  cn(qn   pn) + a1 n 1   +1cn 1 +1(an 1(yn 1))cn  (yn 1)f(yn)  f(yn1 1) :Using (17), (18) in (19), we obtain(20)   an 1cn 1(yn 1)f(yn 1)    cn(qn   pn) + Man 1(cn 1)+1cn  1Ps=n(qs   ps) where M = M0   +1  +1f(yn1 1) . Summing (20) from n1 + 1 to n, we haveancn(yn)f(yn) an1cn1(yn1)f(yn1 )   nXk=n1+1 ck(qk   pk)+M nXk=n1+1 ak 1(ck 1)+1 ck 1Pi=k(qi   pi)  :Letting n!1 and noting (15) and (16), we getlimn!1 ancn(yn)f(yn) =  1which is a contradiction. This completes the proof of the theorem. Remark. When  = 1 and f(u) = u Theorem 6 reduces to Theorem 4 of B. Liuand J. Yan [2].Example. Consider the dierence equation(21)   1n1+ jyn 1j 1yn  + 2n1+ yn =  2yn4(n+ 1)1+ (yn)2 ; n 2 N (1)where  > 0, 0 <  < 1. Let cn = n, q(n) = 1n1+ and pn = 0, n 2 N (1) andf(yn) = yn, then we nd that all conditions of Theorem 6 are satised. Hence allsolutions of (21) are oscillatory. In fact fyng = f( 1)ng is such a solution. Webelieve that the conclusion is not deducible from the oscillation criteria in [2, 3, 7,8, 12, 13] and other known results.466 E. THANDAPANI, L. RAMUPPILLAIReferences[1] Agarwal, R. P., Dierence Equations and Inequalities, Marcel Dekker, New York, 1992.[2] Bing Liu, Yan, J.,Oscillatory and asymptotic behaviour of second order nonlinear dierenceequations, Proc. Edin. Math. Soc. 39(1996), 525-533.[3] Bing Liu, Cheng, S.S., Positive solutions of second order nonlinear dierence equation, J.Math. Anal. Appl. 204(1996), 482-493.[4] Cheng, S. .,S., Li, H. J., Bounded and zero convergent solutions of second order dierenceequations, J. Math. Anal. Appl. 14(1989), 141-149.[5] Hardy, G.H., Littlewood, J. E., Polya, G., Inequalities, 2nd Edition, Cambridge UniversityPress, 1988.[6] He, H. Z., Oscillatory and asymptotic behaviour of second order nonlinear dierence equa-tions, J. Math. Anal. Appl. 175(1993), 482-498.[7] Szmanda, B., Nonoscillation, oscillation and growth of solutions of nonlinear dierenceequations of second order, J. Math. Anal. Appl. 109(1985), 22-30.[8] Thandapani, E., Oscillation theorems for perturbed nonlinear second order dierence equa-tions, Computers Math. Appl. 28(1994), 309-316.[9] Thandapani, E., Arul, R., Oscillation and nonoscillation theorems for a class of secondorder quasilinear dierence equations, ZAA, 16 (1997), 749-759.[10] Thandapani, E., Arul, R., Oscillation theory for a class of second order quasilinear dier-ence equations, Tamkang J. Math. 28 (1997), 229-238.[11] Trench, W.F., Asymptotic behaviour of solutions of Emden-Fowler dierence equationswith oscillating coecients, J. Math. Anal. Appl. 179(1993), 135-153.[12] Wong, P. J.Y., Agarwal, R. P., Oscillation theorems and existence of positive monotone so-lutions for second order nonlinear dierence equations, Math. Comput. Modelling21(1995),63-84.[13] Wong, P. J. Y., Agarwal, R. P., Oscillation and monotone solutions of second order quasi-linear dierence equations, Funk. Ekva. 39(1996), 491-517.[14] Yu, Y.H., Higher type oscillation criterion and Sturm type comparison theorem, Math.Nachr. 153(1991), 485-496.Department of Mathematics, Madras University P.G.CentreSalem-636 011, Tamil Nadu, INDIA

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Oscillatory and asymptotic behaviour of perturbed quasilinear second order difference equations

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