A class of fourth--order neutral type difference equations with
quasidifferences and deviating arguments is considered. Our approach is based
on studying the considered equation as a system of a four--dimensional
difference system. The sufficient conditions under which the considered
equation has no quickly oscillatory solutions are given

Jankowski, Robert

Schmeidel, Ewa

Zonenberg, Joanna

Opuscula Mathematica

English

arXiv

A class of fourth-order neutral type difference equations with quasidifferences and deviating arguments is considered. Our approach is based on studying the considered equation as a system of a four-dimensional difference system. The sufficient conditions under which the considered equation has no quickly oscillatory solutions are given. Finally, the sufficient conditions under which the equation is almost oscillatory are presented

Jankowski, R.

Schmeidel, E.

Zonenberg, J.

Biblioteka Nauki - repozytorium artykuÅÃ³w

Oscillatory properties of solutions of the fourth order difference equations with quasidifferences

Robert Jankowski

Ewa Schmeidel

Joanna Zonenberg

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Opuscula Math. 34, no. 4 (2014), 789–797http://dx.doi.org/10.7494/OpMath.2014.34.4.789 Opuscula MathematicaThis work is dedicated to Professor Leon Mikołajczykon the occasion of his 85th birthday.OSCILLATORY PROPERTIES OF SOLUTIONSOF THE FOURTH ORDER DIFFERENCE EQUATIONSWITH QUASIDIFFERENCESRobert Jankowski, Ewa Schmeidel, and Joanna ZonenbergCommunicated by Marek GalewskiAbstract. A class of fourth-order neutral type diﬀerence equations with quasidiﬀerencesand deviating arguments is considered. Our approach is based on studying the consideredequation as a system of a four-dimensional diﬀerence system. The suﬃcient conditions underwhich the considered equation has no quickly oscillatory solutions are given. Finally, thesuﬃcient conditions under which the equation is almost oscillatory are presented.Keywords: fourth-order diﬀerence equation, neutral type, quickly oscillatory solutions,almost oscillatory.Mathematics Subject Classiﬁcation: 39A21, 39A10.1. INTRODUCTIONIn this article, we consider a class of fourth-order nonlinear diﬀerence equations ofthe form:nanhbn ((cn((xn + pnxn ))))io+ dnf(xn ) = 0; (1.1)where ,  and  are the ratios of odd positive integers, integers ; are deviatingarguments,  6= minf 4;  4g. Moreover, (pn) is a real sequence, (dn) is of one sign,and (an);(bn);(cn) are positive real sequences deﬁned for n 2 Nn0 = fn0;n0+1;:::g,n0 2 N = f0;1;2;:::g, n0  maxf1;;g. Function f : R ! R, where R denotes theset of real numbers. Here  is the forward diﬀerence operator deﬁned for any realsequence (yn) by yn = yn+1  yn. If  > minf 4; 4g, then obviously there existsa solution of equation (1.1) that can be found recursively for given initial conditions.If  < minf 4;   4g, then in order to ensure the existence of a unique solution ofequation (1.1) for given initial conditions, we assume in this paper that the functionf is invertible.c  AGH University of Science and Technology Press, Krakow 2014 789790 Robert Jankowski, Ewa Schmeidel, and Joanna ZonenbergIn the last few years, great attention has been paid to the study of fourth-ordernonlinear diﬀerence equations. It is interesting to extend the known oscillation criteriafor a larger class of fourth order nonlinear diﬀerence equations with quasidiﬀerences.The oscillatory and asymptotic properties of fourth order nonlinear diﬀerence equa-tions were investigated, among many others, by: Agarwal, Grace and Manojlović[3,4], Došlá and Krejčová [5], Migda, Musielak, Popenda, Schmeidel and Szmanda[8–14], Smith and Taylor [15], and Arockiasamy, Dhanasekaran, Graef, Pandian andThandapani [16–18], and references therein.The background for diﬀerence equations can be found in some well known mono-graphs, see, for example: Agarwal, Bohner, Grace and O’Regan [2], Agarwal [1], andKelly and Peterson [6].Setzn = xn + pnxn : (1.2)The sequence (zn) is called a companion sequence of a sequence (xn) relative to (pn).By a solution of equation (1.1) we mean a real sequence (xn) satisfying equation(1.1) for n 2 Nn0. A nontrivial solution (xn) of (1.1) is called nonoscillatory if itis either eventually positive or eventually negative, and it is otherwise oscillatory.Equation (1.1) is called almost oscillatory if all its solutions are oscillatory orlimn!1xn = 0:A solution (xn) of equation (1.1) is called quickly oscillatory ifxn = ( 1)nqn;where qn is of one sign for n 2 Nn0.If pn  0 and f(x) = x, then equation (1.1) takes the following form:(an(bn(cn(xn)))) + dnxn+ = 0:Since in our consideration  2 Z we see that this special case of equation (1.1) withnegative  was studied in [5]. The results presented in this paper have improved andgeneralized those obtained by Došlá and Krejčová.We consider (1.1) as a four-dimensional system. Ifyn = cn(zn); wn = bn(yn); tn = an(wn);then equation (1.1) can be written as the nonlinear system8> > > > <> > > > :(xn + pnxn ) = Cny1n ;yn = Bnw1n ;wn = Ant1n ;tn =  dnf(xn );(1.3)whereAn = a  1n ; Bn = b  1n ; Cn = c  1n : (1.4)Oscillatory properties of solutions of the fourth order diﬀerence equations... 791In this paper we study the properties of solutions of equation (1.1). Firstly, weshow the inﬂuence of the deviating arguments  and  on the existence of quicklyoscillatory solutions of (1.1). Next, some monotonic properties of the solution of theconsidered equation written as system (1.3) are given. Finally, the suﬃcient conditionsunder which equation (1.1) is almost oscillatory are given. The results are illustratedby examples.2. SOME BASIC LEMMASIn 2005, Migda and Migda presented the following result which will be used in thesequel (see [7, Lemma 1]).Lemma 2.1. Let (xn), (pn) be a real sequences and (zn) be a sequence deﬁne by (1.2),for n  . Assume that (xn) is bounded, limn!1 zn = l 2 R, limn!1 pn = p 2 R. Ifjpj 6= 1, then (xn) is convergent and limn!1 xn = l1+p.The next lemma is a simple generalization of Lemma 2 from the Migda and Migdapaper [7].Lemma 2.2. Assume that x: N ! R andlimn!1pn = p; where jpj < 1: (2.1)If sequence (zn) deﬁned by (1.2) is bounded, then sequence (xn) is bounded too.Proof. Let L > 0 be such that jznj  L for n  1. Set P : =1+jpj2 . From (2.1), weget 0 < P < 1, and there exists n1 2 N such that jpnj  P, for n  n1. SetK = maxfjxn1j;jxn1+1j;:::;jxn1++1jg: (2.2)Let n  n1 be an arbitrary integer. Then there exists m 2 N such that n1  n m n1 + . From (1.2), we havejxnj  jznj + Pjxn j  L + Pjxn j:Analogously,jxn j  L + Pjxn 2j;andjxnj  L + PL + P2jxn 2j:After m steps we obtainjxnj  L + PL + P2L +  + Pm 1L + Pmjxn mj:Using (2.2), we have jxn mj  K. Since 0 < P < 1, we obtain Pmjxn mj < K.From the above,jxnj  L(1 + P + P2 +  + Pm 1) + K  K +L1   P:This completes the proof.Note that, putting pn  p in the above lemma, we get Lemma 2 from paper [7].792 Robert Jankowski, Ewa Schmeidel, and Joanna Zonenberg3. EXISTENCE OF QUICKLY OSCILLATORY SOLUTIONSIn this section necessary conditions for the existence of a quickly oscillatory solutionof equation (1.1) are presented.Theorem 3.1. Assume that pn  0, dn > 0 for n 2 Nn0,  is even, andxf(x) > 0; for x 6= 0: (3.1)If  is even, then equation (1.1) has no quickly oscillatory solutions with positive eventerms. If  is odd, then equation (1.1) has no quickly oscillatory solutions with positiveodd terms.Proof. Letxn = ( 1)nqn be a quickly oscillatory solution of (1.1). (3.2)Assume also that (qn) is a positive sequence. Since  is even, we getzn = (xn + pnxn ) = xn+1 + pn+1xn+1    xn   pnxn = ( 1)n+1(qn+1 + qn + pn+1qn +1 + pnqn ):Hence (zn) is a quickly oscillatory sequence and we can write zn = ( 1)n+1sn,where sn = qn+1 + qn + pn+1qn +1 + pnqn  > 0. From the ﬁrst equation of system(1.3) we haveyn =znCn= ( 1)n+1rn;where rn = ( snCn) > 0. From the second equation of (1.3) we getwn =ynBn= ( 1)nln;where ln =rn+1Bn + rnBn> 0. Repeating the argument, we get from the thirdequation of (1.3) the following equalitytn =wnAn= ( 1)n+1gn;where gn =ln+1An + lnAn> 0. Consequently, from the fourth equation we havetn = ( 1)n(gn+1 + gn) =  dnf(xn ):Hence( 1)n+1(gn+1 + gn) = dnf(xn ): (3.3)By (3.1), we have xn f(xn ) > 0. From (3.2) we see that xn  = ( 1)n qn .Since  is even we get f(xn ) is positive for even n. The left-hand side of equality(3.3) is negative for even n, whereas the right-hand side is positive and vice versa.This contradiction ended the proof in the case of even .For odd  the proof is analogous and hence is omitted.Oscillatory properties of solutions of the fourth order diﬀerence equations... 793Remark 3.2. Assume that pn  0, dn < 0 for n 2 Nn0,  is even, and condition(3.1) is satisﬁed. If  is even, then equation (1.1) has no quickly oscillatory solutionswith positive odd terms. If  is odd, then equation (1.1) has no quickly oscillatorysolutions with positive even terms.The following examples illustrate Theorem 3.1. Here  is even and an equation ofthe form (1.1) has a quickly oscillatory solution with positive even terms.Example 3.3. Consider the equation22xn +12nxn 2+ dn sgn(xn ) = 0; (3.4)where ,  are some positive integers, and  is odd,dn = (2n+232 + 22 2) + 2(2n+132 + 22 2) + (2n32 + 22 2) > 0;and f(x) = sgnx. Equation (3.4) has a quickly oscillatory solution xn = ( 1)n2n.Example 3.4. Consider the equation22xn +13nxn 2+ dnxn  = 0; (3.5)where  is some positive integer,  is an odd integer anddn =4 +423n+4+ 24 +423n+3+4 +423n+2> 0and f(x) = x. Equation (3.5) has a quickly oscillatory solution xn = ( 1)n.4. ALMOST OSCILLATORY PROPERTYIn this section we assume that there exists a ﬁnite limit of sequence (pn). After somelemmas concerning the behavior of solutions of system (1.3), the suﬃcient conditionsunder which equation (1.1) is almost oscillatory are presented.Lemma 4.1. Assume that conditions (2.1) and (3.1) are satisﬁed. If (x;y;w;t) is asolution of system (1.3) with bounded ﬁrst component such that one of its componentsis of one sign, then one of the following two cases hold:1) all its components are of one sign for enough large n,2) sequence (yn) is of one sign for enough large n and limn!1 xn = 0.Proof. Firstly, we assume that sequence (xn) is positive. (For negative (xn) the proofis analogous and hence omitted.) Since (dn) is of one sign for n  n0, and by (3.1),we have that (tn) is also of one sign. It means that if tn > 0, then sequence (tn)is increasing, and if tn < 0, then sequence (tn) is decreasing. In both cases sequence794 Robert Jankowski, Ewa Schmeidel, and Joanna Zonenberg(tn) is of one sign for large n. This implies, by the third equation of system (1.3), thatwn is of one sign for large n. Repeating the same arguments as before, we obtainthat sequence (wn) is of one sign, and consequently (yn) is of one sign for large n.Hence, condition 1) of the thesis is fulﬁlled.Next, we assume that (yn) is of one sign for n 2 Nn0. By the ﬁrst equation ofsystem (1.3), we get that sequence (zn) deﬁned by (1.2) is monotonic for large n.Hence limn!1 zn exists. Let us consider three possible cases:(i) limn!1 zn = 1,(ii) limn!1 zn = l 6= 0,(iii) limn!1 zn = 0.Case (i). If (zn) is unbounded, then (xn) is unbounded too. By the assumption ofthe Lemma this case is excluded.Case (ii). Since sequence (zn) is bounded, by condition (2.1) and Lemma 2.2, sequence(xn) is bounded too. By Lemma 2.1, we have limn!1 xn = l1+p 6= 0. It means thatsequence (xn) is of one sign for large n.Case (iii). By Lemmas 2.1 and 2.2, we get limn!1 xn = 0. It means that condition 2)of the thesis is satisﬁed.Assuming that (tn) or (wn) is of one sign, on virtue of analogous arguments asabove, we also get the thesis.Corollary 4.2. Assume that (2.1) and (3.1) are satisﬁed. If (x;y;w;t) is a solutionof system (1.3) with bounded ﬁrst component such that one of its components is ofone sign, then there exists a limit of sequence (xn) and exactly one of the followingtwo cases are held:1) limn!1 xn 6= 0, and sequences y, w and t are monotonic for enough large n, or2) sequence (yn) is of one sign for enough large n and limn!1 xn = 0.Corollary 4.3. Assume that conditions (2.1) and (3.1) are satisﬁed, and1 Xn=n0An =1 Xn=n0Bn =1 Xn=n0Cn = 1: (4.1)If (x;y;w;t) is a solution of system (1.3) such thatlimn!1xn 2 R; (4.2)thenlimn!1yn = limn!1wn = limn!1tn = 0:Proof. Since conditions (2.1) and (4.2) hold, limn!1 zn is ﬁnite. From the ﬁrst equa-tion of system (1.3), we getzn = zn0 +n 1 Xi=n0Ciy1i : (4.3)Oscillatory properties of solutions of the fourth order diﬀerence equations... 795Without loss the generality, for the sake of contradiction, assume that limn!1 yn > 0.Hence limn!1 y1n > 0. Since Cn is positive, letting n to inﬁnity, the right hand sideof equality (4.3) tends to inﬁnity whereas the left hand side has a ﬁnite limit. Itfollows that limn!1 yn = 0. In a similar manner the remaining part of the thesis isobtained.Theorem 4.4. If conditions (2.1), (3.1) and (4.1) are satisﬁed, f : R ! R is acontinuous function, and series1 Xi=1di is divergent; (4.4)then any bounded solution of equation (1.1) is almost oscillatory.Proof. For the sake of contradiction assume that equation (1.1) has a nonoscillatorybounded solution which does not approach zero. Without loss of generality we assumethat xn > 0 for large n. From the above and Corollary 4.2, we have that the ﬁnitelimit limn!1 xn exists. Set limn!1 xn = c 2 (0;1). So, by (3.1), we have f(c) > 0.Then there exists a positive integer n1 such that f(xn ) f(c)2 for n  n1. Thisimplies that:– if (dn) is a positive sequence, then1 Xi=n1dif(xi ) f(c)21 Xi=n1di = +1;– if (dn) is a negative sequence, then1 Xi=n1dif(xi ) f(c)21 Xi=n1di =  1:Summing the last equation of system (1.3) from 1 into n   1, we havetn   t1 =  n 1 Xi=1dif(xi ):By Corollary 4.3, we get that sequence (tn) tends to zero as n tends to 1. Hence,letting n to 1, we obtaint1 =1 Xi=1dif(xi ):The left-hand side of the above equation is a constant whereas the right-hand is  1or +1. This contradiction ended the proof.The following two examples show equations of type (1.1) the coeﬃcients of whichfulﬁll the assumptions of Theorem 4.4 and both of them have an almost oscillatorysolution. The ﬁrst one has an asymptotically zero solution, and the second one hasan oscillatory solution.796 Robert Jankowski, Ewa Schmeidel, and Joanna ZonenbergExample 4.5. Consider the equationn3xn +14xn 2+ (1   n)xn+3 = 0:The above equation has solution xn =   12n which tends to zero.Example 4.6. Consider the equationn3xn +14xn 2+ 10(2n + 1)xn+3 = 0:This equation has oscillatory solution xn =( 1)n10 .REFERENCES[1] R.P. Agarwal, Diﬀerence equations and inequalities. Theory, methods and applications,Marcel Dekker, Inc., New York, 1992.[2] R.P. Agarwal, M. Bohner, S.R. Grace, D. O’Regan, Discrete Oscillation Theory, Hin-dawi Publishing Corporation, New York, 2005.[3] R.P. Agarwal, S.R Grace, J.V Manojlović On the oscillatory properties of certain fourthorder nonlinear diﬀerence equations, Math. Anal. Appl. 322 (2006), 930–956.[4] R.P. Agarwal, J.V. Manojlović, Asymptotic behavior of nonoscillaory solutions of fourthorder nonlinear diﬀerence equations, Dyn. Contin. Discrete Impuls. Syst. Ser. A, Math.Anal. 16 (2009), 155–174.[5] Z. Došlá, J. Krejčová, Oscillation of a class of the fourth-order nonlinear diﬀerenceequations, Adv. Diﬀerence Equ., doi:10.1186/1687-1847-2012-99, (2012).[6] W.G. Kelly, A.C. Peterson, Diﬀerence equations, Academic Press, Inc., Boston-SanDiego, 1991.[7] M. Migda, J. Migda, Asymptotic properties of solutions of second-order neutral diﬀer-ence equations, Nonlinear Anal. 63 (2005), e789–e799.[8] M. Migda, E. Schmeidel, Asymptotic properties of fourth order nonlinear diﬀerenceequations, Math. Comput. Modelling 39 (2004), 1203–1211.[9] M. Migda, A. Musielak, E. Schmeidel, On a class of fourth order nonlinear diﬀerenceequations, Adv. Diﬀerence Equ. 1 (2004), 23–36.[10] M. Migda, A. Musielak, E. Schmeidel, Oscillatory of fourth order nonlinear diﬀerenceequations with quasidiﬀerences, Opuscula Math. 26 (2006), 371–380.[11] J. Popenda, E. Schmeidel, On the solution of fourth order diﬀerence equations, RockyMt. J. Math. 25 (1995), 1485–1499.[12] E. Schmeidel, Oscillation and nonoscillation theorems for fourth order diﬀerence equa-tions, Rocky Mt. J.Math. 33 (2003), 1083–1094.Oscillatory properties of solutions of the fourth order diﬀerence equations... 797[13] E. Schmeidel, Nonscillation and oscillation properties for fourth order nonlinear diﬀer-ence equations, New Progress in Diﬀerence Equations, B. Aulbach, S. Elaydi, G. Ladas(eds.), CRC, Boca Raton, FL, (2004), 531–538.[14] E. Schmeidel, B. Szmanda, Oscillatory and asymptotic behavior of certain diﬀerenceequation, Nonlinear Anal. 47 (2001), 4731–4742.[15] B. Smith, W.E. Taylor, Oscillatory and asymptotic behavior of certain fourth orderdiﬀerence equations Rocky Mt. J. Math. 16 (1986), 403–406.[16] E. Thandapani, J.R. Graef, Oscillatory and asymptotic behavior of fourth order non-linear delay diﬀerence equations Fasc. Math. 31 (2001), 23–36.[17] E. Thandapani, S. Pandian, R. Dhanasekaran, J. Graef, Asymptotic results for a class offourth order quasilinear diﬀerence equations J. Diﬀer. Equ. Appl. 13 (2007), 1085–1103.[18] E. Thandapani, I.M. Arockiasamy, On fourth order nonlinear oscillations of diﬀerenceequations Comput. Math. Appl. 42 (2001), 357–368.Robert Jankowskirjjankowski@math.uwb.edu.plLodz University of TechnologyInstitute of MathematicsWólczańska 215, 90–924 Łódź, PolandEwa Schmeideleschmeidel@math.uwb.edu.plUniversity of BialystokFaculty of Mathematics and Computer ScienceAkademicka 2, 15–267 Białystok, PolandJoanna Zonenbergjzonenberg@math.uwb.edu.plUniversity of BialystokFaculty of Mathematics and Computer ScienceAkademicka 2, 15–267 Białystok, PolandReceived: January 1, 2014.Revised: March 5, 2014.Accepted: April 1, 2014.

Lodz University of Technology Repository

Opuscula Math. 34, no. 4 (2014), 789–797http://dx.doi.org/10.7494/OpMath.2014.34.4.789 Opuscula MathematicaThis work is dedicated to Professor Leon Mikołajczykon the occasion of his 85th birthday.OSCILLATORY PROPERTIES OF SOLUTIONSOF THE FOURTH ORDER DIFFERENCE EQUATIONSWITH QUASIDIFFERENCESRobert Jankowski, Ewa Schmeidel, and Joanna ZonenbergCommunicated by Marek GalewskiAbstract. A class of fourth-order neutral type difference equations with quasidifferencesand deviating arguments is considered. Our approach is based on studying the consideredequation as a system of a four-dimensional difference system. The sufficient conditions underwhich the considered equation has no quickly oscillatory solutions are given. Finally, thesufficient conditions under which the equation is almost oscillatory are presented.Keywords: fourth-order difference equation, neutral type, quickly oscillatory solutions,almost oscillatory.Mathematics Subject Classification: 39A21, 39A10.1. INTRODUCTIONIn this article, we consider a class of fourth-order nonlinear difference equations ofthe form:∆{an[∆(bn (∆(cn(∆(xn + pnxn−δ))γ))β)]α}+ dnf(xn−τ ) = 0, (1.1)where α, β and γ are the ratios of odd positive integers, integers τ, δ are deviatingarguments, τ 6= min{−4, δ− 4}. Moreover, (pn) is a real sequence, (dn) is of one sign,and (an), (bn), (cn) are positive real sequences defined for n ∈ Nn0 = {n0, n0 + 1, . . . },n0 ∈ N = {0, 1, 2, . . . }, n0 ≥ max{1, δ, τ}. Function f : R → R, where R denotes theset of real numbers. Here ∆ is the forward difference operator defined for any realsequence (yn) by ∆yn = yn+1− yn. If τ > min{−4, δ− 4}, then obviously there existsa solution of equation (1.1) that can be found recursively for given initial conditions.If τ < min{−4, δ − 4}, then in order to ensure the existence of a unique solution ofequation (1.1) for given initial conditions, we assume in this paper that the functionf is invertible.c© AGH University of Science and Technology Press, Krakow 2014 789790 Robert Jankowski, Ewa Schmeidel, and Joanna ZonenbergIn the last few years, great attention has been paid to the study of fourth-ordernonlinear difference equations. It is interesting to extend the known oscillation criteriafor a larger class of fourth order nonlinear difference equations with quasidifferences.The oscillatory and asymptotic properties of fourth order nonlinear difference equa-tions were investigated, among many others, by: Agarwal, Grace and Manojlović[3, 4], Došlá and Krejčová [5], Migda, Musielak, Popenda, Schmeidel and Szmanda[8–14], Smith and Taylor [15], and Arockiasamy, Dhanasekaran, Graef, Pandian andThandapani [16–18], and references therein.The background for difference equations can be found in some well known mono-graphs, see, for example: Agarwal, Bohner, Grace and O’Regan [2], Agarwal [1], andKelly and Peterson [6].Setzn = xn + pnxn−δ. (1.2)The sequence (zn) is called a companion sequence of a sequence (xn) relative to (pn).By a solution of equation (1.1) we mean a real sequence (xn) satisfying equation(1.1) for n ∈ Nn0 . A nontrivial solution (xn) of (1.1) is called nonoscillatory if itis either eventually positive or eventually negative, and it is otherwise oscillatory.Equation (1.1) is called almost oscillatory if all its solutions are oscillatory orlimn→∞xn = 0.A solution (xn) of equation (1.1) is called quickly oscillatory ifxn = (−1)nqn,where qn is of one sign for n ∈ Nn0 .If pn ≡ 0 and f(x) = xλ, then equation (1.1) takes the following form:∆(an(∆bn(∆cn(∆xn)γ)β)α) + dnxλn+τ = 0.Since in our consideration τ ∈ Z we see that this special case of equation (1.1) withnegative τ was studied in [5]. The results presented in this paper have improved andgeneralized those obtained by Došlá and Krejčová.We consider (1.1) as a four-dimensional system. Ifyn = cn(∆zn)γ , wn = bn(∆yn)β , tn = an(∆wn)α,then equation (1.1) can be written as the nonlinear system∆(xn + pnxn−δ) = Cny1γn ,∆yn = Bnw1βn ,∆wn = Ant1αn ,∆tn = −dnf(xn−τ ),(1.3)whereAn = a− 1αn , Bn = b− 1βn , Cn = c− 1γn . (1.4)Oscillatory properties of solutions of the fourth order difference equations. . . 791In this paper we study the properties of solutions of equation (1.1). Firstly, weshow the influence of the deviating arguments δ and τ on the existence of quicklyoscillatory solutions of (1.1). Next, some monotonic properties of the solution of theconsidered equation written as system (1.3) are given. Finally, the sufficient conditionsunder which equation (1.1) is almost oscillatory are given. The results are illustratedby examples.2. SOME BASIC LEMMASIn 2005, Migda and Migda presented the following result which will be used in thesequel (see [7, Lemma 1]).Lemma 2.1. Let (xn), (pn) be a real sequences and (zn) be a sequence define by (1.2),for n ≥ δ. Assume that (xn) is bounded, limn→∞ zn = l ∈ R, limn→∞ pn = p ∈ R. If|p| 6= 1, then (xn) is convergent and limn→∞ xn = l1+p .The next lemma is a simple generalization of Lemma 2 from the Migda and Migdapaper [7].Lemma 2.2. Assume that x : N→ R andlimn→∞ pn = p, where |p| < 1. (2.1)If sequence (zn) defined by (1.2) is bounded, then sequence (xn) is bounded too.Proof. Let L > 0 be such that |zn| ≤ L for n ≥ 1. Set P : = 1+|p|2 . From (2.1), weget 0 < P < 1, and there exists n1 ∈ N such that |pn| ≤ P , for n ≥ n1. SetK = max{|xn1 |, |xn1+1|, . . . , |xn1+δ+1|}. (2.2)Let n ≥ n1 be an arbitrary integer. Then there exists m ∈ N such that n1 ≤ n−mδ ≤n1 + δ. From (1.2), we have|xn| ≤ |zn|+ P |xn−δ| ≤ L+ P |xn−δ|.Analogously,|xn−δ| ≤ L+ P |xn−2δ|,and|xn| ≤ L+ PL+ P 2|xn−2δ|.After m steps we obtain|xn| ≤ L+ PL+ P 2L+ · · ·+ Pm−1L+ Pm|xn−mδ|.Using (2.2), we have |xn−mδ| ≤ K. Since 0 < P < 1, we obtain Pm|xn−mδ| < K.From the above,|xn| ≤ L(1 + P + P 2 + · · ·+ Pm−1) +K ≤ K + L1− P .This completes the proof.Note that, putting pn ≡ p in the above lemma, we get Lemma 2 from paper [7].792 Robert Jankowski, Ewa Schmeidel, and Joanna Zonenberg3. EXISTENCE OF QUICKLY OSCILLATORY SOLUTIONSIn this section necessary conditions for the existence of a quickly oscillatory solutionof equation (1.1) are presented.Theorem 3.1. Assume that pn ≥ 0, dn > 0 for n ∈ Nn0 , δ is even, andxf(x) > 0, for x 6= 0. (3.1)If τ is even, then equation (1.1) has no quickly oscillatory solutions with positive eventerms. If τ is odd, then equation (1.1) has no quickly oscillatory solutions with positiveodd terms.Proof. Letxn = (−1)nqn be a quickly oscillatory solution of (1.1). (3.2)Assume also that (qn) is a positive sequence. Since δ is even, we get∆zn = ∆(xn + pnxn−δ) = xn+1 + pn+1xn+1−δ − xn − pnxn−δ= (−1)n+1(qn+1 + qn + pn+1qn−δ+1 + pnqn−δ).Hence (∆zn) is a quickly oscillatory sequence and we can write ∆zn = (−1)n+1sn,where sn = qn+1 + qn + pn+1qn−δ+1 + pnqn−δ > 0. From the first equation of system(1.3) we haveyn =(∆znCn)γ= (−1)n+1rn,where rn = ( snCn )γ > 0. From the second equation of (1.3) we getwn =(∆ynBn)β= (−1)nln,where ln =(rn+1Bn+ rnBn)β> 0. Repeating the argument, we get from the thirdequation of (1.3) the following equalitytn =(∆wnAn)α= (−1)n+1gn,where gn =(ln+1An+ lnAn)α> 0. Consequently, from the fourth equation we have∆tn = (−1)n(gn+1 + gn) = −dnf(xn−τ ).Hence(−1)n+1(gn+1 + gn) = dnf(xn−τ ). (3.3)By (3.1), we have xn−τf(xn−τ ) > 0. From (3.2) we see that xn−τ = (−1)n−τqn−τ .Since τ is even we get f(xn−τ ) is positive for even n. The left-hand side of equality(3.3) is negative for even n, whereas the right-hand side is positive and vice versa.This contradiction ended the proof in the case of even τ .For odd τ the proof is analogous and hence is omitted.Oscillatory properties of solutions of the fourth order difference equations. . . 793Remark 3.2. Assume that pn ≥ 0, dn < 0 for n ∈ Nn0 , δ is even, and condition(3.1) is satisfied. If τ is even, then equation (1.1) has no quickly oscillatory solutionswith positive odd terms. If τ is odd, then equation (1.1) has no quickly oscillatorysolutions with positive even terms.The following examples illustrate Theorem 3.1. Here δ is even and an equation ofthe form (1.1) has a quickly oscillatory solution with positive even terms.Example 3.3. Consider the equation∆2(∆2(xn +12nxn−2λ))β+ dn sgn(xn−τ ) = 0, (3.4)where τ , λ are some positive integers, and τ is odd,dn = (2n+232 + 22−2λ)β + 2(2n+132 + 22−2λ)β + (2n32 + 22−2λ)β > 0,and f(x) = sgnx. Equation (3.4) has a quickly oscillatory solution xn = (−1)n2n.Example 3.4. Consider the equation∆2(∆2(xn +13nxn−2λ))β+ dnxn−τ = 0, (3.5)where λ is some positive integer, τ is an odd integer anddn =(4 +423n+4)β+ 2(4 +423n+3)β+(4 +423n+2)β> 0and f(x) = x. Equation (3.5) has a quickly oscillatory solution xn = (−1)n.4. ALMOST OSCILLATORY PROPERTYIn this section we assume that there exists a finite limit of sequence (pn). After somelemmas concerning the behavior of solutions of system (1.3), the sufficient conditionsunder which equation (1.1) is almost oscillatory are presented.Lemma 4.1. Assume that conditions (2.1) and (3.1) are satisfied. If (x, y, w, t) is asolution of system (1.3) with bounded first component such that one of its componentsis of one sign, then one of the following two cases hold:1) all its components are of one sign for enough large n,2) sequence (yn) is of one sign for enough large n and limn→∞ xn = 0.Proof. Firstly, we assume that sequence (xn) is positive. (For negative (xn) the proofis analogous and hence omitted.) Since (dn) is of one sign for n ≥ n0, and by (3.1),we have that (∆tn) is also of one sign. It means that if ∆tn > 0, then sequence (tn)is increasing, and if ∆tn < 0, then sequence (tn) is decreasing. In both cases sequence794 Robert Jankowski, Ewa Schmeidel, and Joanna Zonenberg(tn) is of one sign for large n. This implies, by the third equation of system (1.3), that∆wn is of one sign for large n. Repeating the same arguments as before, we obtainthat sequence (wn) is of one sign, and consequently (yn) is of one sign for large n.Hence, condition 1) of the thesis is fulfilled.Next, we assume that (yn) is of one sign for n ∈ Nn0 . By the first equation ofsystem (1.3), we get that sequence (zn) defined by (1.2) is monotonic for large n.Hence limn→∞ zn exists. Let us consider three possible cases:(i) limn→∞ zn = ±∞,(ii) limn→∞ zn = l 6= 0,(iii) limn→∞ zn = 0.Case (i). If (zn) is unbounded, then (xn) is unbounded too. By the assumption ofthe Lemma this case is excluded.Case (ii). Since sequence (zn) is bounded, by condition (2.1) and Lemma 2.2, sequence(xn) is bounded too. By Lemma 2.1, we have limn→∞ xn = l1+p 6= 0. It means thatsequence (xn) is of one sign for large n.Case (iii). By Lemmas 2.1 and 2.2, we get limn→∞ xn = 0. It means that condition 2)of the thesis is satisfied.Assuming that (tn) or (wn) is of one sign, on virtue of analogous arguments asabove, we also get the thesis.Corollary 4.2. Assume that (2.1) and (3.1) are satisfied. If (x, y, w, t) is a solutionof system (1.3) with bounded first component such that one of its components is ofone sign, then there exists a limit of sequence (xn) and exactly one of the followingtwo cases are held:1) limn→∞ xn 6= 0, and sequences y, w and t are monotonic for enough large n, or2) sequence (yn) is of one sign for enough large n and limn→∞ xn = 0.Corollary 4.3. Assume that conditions (2.1) and (3.1) are satisfied, and∞∑n=n0An =∞∑n=n0Bn =∞∑n=n0Cn =∞. (4.1)If (x, y, w, t) is a solution of system (1.3) such thatlimn→∞xn ∈ R, (4.2)thenlimn→∞ yn = limn→∞wn = limn→∞ tn = 0.Proof. Since conditions (2.1) and (4.2) hold, limn→∞ zn is finite. From the first equa-tion of system (1.3), we getzn = zn0 +n−1∑i=n0Ciy1γi . (4.3)Oscillatory properties of solutions of the fourth order difference equations. . . 795Without loss the generality, for the sake of contradiction, assume that limn→∞ yn > 0.Hence limn→∞ y1γn > 0. Since Cn is positive, letting n to infinity, the right hand sideof equality (4.3) tends to infinity whereas the left hand side has a finite limit. Itfollows that limn→∞ yn = 0. In a similar manner the remaining part of the thesis isobtained.Theorem 4.4. If conditions (2.1), (3.1) and (4.1) are satisfied, f : R → R is acontinuous function, and series∞∑i=1di is divergent, (4.4)then any bounded solution of equation (1.1) is almost oscillatory.Proof. For the sake of contradiction assume that equation (1.1) has a nonoscillatorybounded solution which does not approach zero. Without loss of generality we assumethat xn > 0 for large n. From the above and Corollary 4.2, we have that the finitelimit limn→∞ xn exists. Set limn→∞ xn = c ∈ (0,∞). So, by (3.1), we have f(c) > 0.Then there exists a positive integer n1 such that f(xn−τ ) ≥ f(c)2 for n ≥ n1. Thisimplies that:– if (dn) is a positive sequence, then∞∑i=n1dif(xi−τ ) ≥ f(c)2∞∑i=n1di = +∞,– if (dn) is a negative sequence, then∞∑i=n1dif(xi−τ ) ≤ f(c)2∞∑i=n1di = −∞.Summing the last equation of system (1.3) from 1 into n− 1, we havetn − t1 = −n−1∑i=1dif(xi−τ ).By Corollary 4.3, we get that sequence (tn) tends to zero as n tends to ∞. Hence,letting n to ∞, we obtaint1 =∞∑i=1dif(xi−τ ).The left-hand side of the above equation is a constant whereas the right-hand is −∞or +∞. This contradiction ended the proof.The following two examples show equations of type (1.1) the coefficients of whichfulfill the assumptions of Theorem 4.4 and both of them have an almost oscillatorysolution. The first one has an asymptotically zero solution, and the second one hasan oscillatory solution.796 Robert Jankowski, Ewa Schmeidel, and Joanna ZonenbergExample 4.5. Consider the equation∆(n∆3(xn +14xn−2))+ (1− n)xn+3 = 0.The above equation has solution xn = − 12n which tends to zero.Example 4.6. Consider the equation∆(n∆3(xn +14xn−2))+ 10(2n+ 1)xn+3 = 0.This equation has oscillatory solution xn =(−1)n10 .REFERENCES[1] R.P. Agarwal, Difference equations and inequalities. Theory, methods and applications,Marcel Dekker, Inc., New York, 1992.[2] R.P. Agarwal, M. Bohner, S.R. Grace, D. O’Regan, Discrete Oscillation Theory, Hin-dawi Publishing Corporation, New York, 2005.[3] R.P. Agarwal, S.R Grace, J.V Manojlović On the oscillatory properties of certain fourthorder nonlinear difference equations, Math. Anal. Appl. 322 (2006), 930–956.[4] R.P. Agarwal, J.V. Manojlović, Asymptotic behavior of nonoscillaory solutions of fourthorder nonlinear difference equations, Dyn. Contin. Discrete Impuls. Syst. Ser. A, Math.Anal. 16 (2009), 155–174.[5] Z. Došlá, J. Krejčová, Oscillation of a class of the fourth-order nonlinear differenceequations, Adv. Difference Equ., doi:10.1186/1687-1847-2012-99, (2012).[6] W.G. Kelly, A.C. Peterson, Difference equations, Academic Press, Inc., Boston-SanDiego, 1991.[7] M. Migda, J. Migda, Asymptotic properties of solutions of second-order neutral differ-ence equations, Nonlinear Anal. 63 (2005), e789–e799.[8] M. Migda, E. Schmeidel, Asymptotic properties of fourth order nonlinear differenceequations, Math. Comput. Modelling 39 (2004), 1203–1211.[9] M. Migda, A. Musielak, E. Schmeidel, On a class of fourth order nonlinear differenceequations, Adv. Difference Equ. 1 (2004), 23–36.[10] M. Migda, A. Musielak, E. Schmeidel, Oscillatory of fourth order nonlinear differenceequations with quasidifferences, Opuscula Math. 26 (2006), 371–380.[11] J. Popenda, E. Schmeidel, On the solution of fourth order difference equations, RockyMt. J. Math. 25 (1995), 1485–1499.[12] E. Schmeidel, Oscillation and nonoscillation theorems for fourth order difference equa-tions, Rocky Mt. J.Math. 33 (2003), 1083–1094.Oscillatory properties of solutions of the fourth order difference equations. . . 797[13] E. Schmeidel, Nonscillation and oscillation properties for fourth order nonlinear differ-ence equations, New Progress in Difference Equations, B. Aulbach, S. Elaydi, G. Ladas(eds.), CRC, Boca Raton, FL, (2004), 531–538.[14] E. Schmeidel, B. Szmanda, Oscillatory and asymptotic behavior of certain differenceequation, Nonlinear Anal. 47 (2001), 4731–4742.[15] B. Smith, W.E. Taylor, Oscillatory and asymptotic behavior of certain fourth orderdifference equations Rocky Mt. J. Math. 16 (1986), 403–406.[16] E. Thandapani, J.R. Graef, Oscillatory and asymptotic behavior of fourth order non-linear delay difference equations Fasc. Math. 31 (2001), 23–36.[17] E. Thandapani, S. Pandian, R. Dhanasekaran, J. Graef, Asymptotic results for a class offourth order quasilinear difference equations J. Differ. Equ. Appl. 13 (2007), 1085–1103.[18] E. Thandapani, I.M. Arockiasamy, On fourth order nonlinear oscillations of differenceequations Comput. Math. Appl. 42 (2001), 357–368.Robert Jankowskirjjankowski@math.uwb.edu.plLodz University of TechnologyInstitute of MathematicsWólczańska 215, 90–924 Łódź, PolandEwa Schmeideleschmeidel@math.uwb.edu.plUniversity of BialystokFaculty of Mathematics and Computer ScienceAkademicka 2, 15–267 Białystok, PolandJoanna Zonenbergjzonenberg@math.uwb.edu.plUniversity of BialystokFaculty of Mathematics and Computer ScienceAkademicka 2, 15–267 Białystok, PolandReceived: January 1, 2014.Revised: March 5, 2014.Accepted: April 1, 2014.

Oscillatory Properties of Solutions of the Fourth Order Difference
  Equations with Quasidifferences

Oscillatory Properties of Solutions of the Fourth Order Difference Equations with Quasidifferences

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Oscillatory Properties of Solutions of the Fourth Order Difference Equations with Quasidifferences

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