OSCILLATION OF HIGHER ORDER NEUTRAL TYPE DIFFERENCE EQUATIONS A. ZAFER Department of Mathematics Middle East Technical University 06531 Ankara, TURKEY Abstract In this work we are concerned with oscillation of solutions of the neutral difference equation of the form ∆p [xn − pn xn−l ] + δqn f (xn−k ) = hn , n ≥ n0 , δ = ±1, where ∆ is the forward difference operator ∆xn = xn+1 −xn , {pn }, {qn } and {hn } are real sequences, and l and k are integers. A necessary and sufficient condition is obtained under which every bounded solution {xn } is oscillatory when (−1)p δ = −1, and is either oscillatory or such that limn→∞ xn = 0 when (−1)p δ = 1. 1 INTRODUCTION We consider the neutral difference equation ∆p [xn − pn xn−l ] + δqn f (xn−k ) = hn n ≥ n0 (1.1) where δ = ±1 and n ∈ N(n0 ) = {n0 , n0 + 1, . . .}, n0 a fixed positive integer. We assume that l and k are integers with l > 0, x, p, q, and f are real valued functions and f is continuous. By a solution of Eq. (1.1) we mean a real sequence {xn } satisfying Eq. (1.1) so that supn≥m |xn | > 0 for any m ∈ N(n0 ). We always assume that such solutions of Eq. (1.1) exist. A solution of Eq. (1.1) is called oscillatory if there is no end of n1 and n2 (n1 < n2 ) in N(n0 ) such that xn1 xn2 ≤ 0; otherwise it is called nonoscillatory. Clearly, a nonoscillatory solution must be eventually of fixed sign. A. ZAFER The oscillatory behavior of solutions of first and second order difference equations has been extensively studied by many authors (see [1–8] and the references cited therein). However, much less has been done for higher order equations. For some results regarding the oscillation and as well as the asymptotic behavior of higher order difference equations, we refer in particular to [9–13]. In [11] the present author established sufficient conditions for the existence of positive solutions and oscillation of solutions of Eq. (1.1) when pn = c, c %= ±1. Here, an attempt is made to extend to Eq. (1.1) some of the results obtained previously for pn = c > 1. 2 MAIN RESULTS In what follows n(s) denote the usual factorial function; that is, n(0) = 1 and n(s) = n(n − 1) . . . (n − s + 1). Theorem 2.1 Let there exist two real numbers c and C such that 1 < c ≤ pn ≤ C, and assume that for some positive real numbers a and b such that a/b < (c − 1)/C the function f satisfies a Lipschitz condition with Lipschitz constant L on [a, b]. If ∞ ! s(p−1) |qs | < ∞, (2.1) s(p−1) |hs | < ∞, (2.2) s=n ∞ ! s=n then Eq. (1.1) has a nonoscillatory bounded solution {xn }. Proof Let K = max{|f (x)|/|x| : a ≤ x ≤ b} and M = max{K, L}. Due to condition (2.1) and (2.2) we can find a sufficiently large n1 ≥ n0 such that ∞ ! (p − 1)!βc s(p−1) |qs | < , (2.3) 2Mb s=n+l ∞ ! s(p−1) |hs | < s=n+l (p − 1)!βc , 2 (2.4) and n ≥ max{n0 − l, n0 − l + k} for n ≥ n1 , where β is defined as β= (c − 1)b − Ca . c+C NEUTRAL TYPE DIFFERENCE EQUATIONS We introduce the Banach space Y of all functions x : N(n0 ) → R such that ||x|| = sup |xn |. n∈N (n0 ) Define X = {x ∈ Y : a ≤ xn ≤ b, n ∈ N(n0 )}. Clearly X is bounded, convex, and closed subset of Y . Let T be an operator defined on X by T xn = 1 " α + xn+l pn+l ∞ (−1)p ! + (s + p − 1 − n − l)(p−1) δqs f (xs−k ) (p − 1)! s=n+l  ∞ (−1)p−1 ! (s + p − 1 − n − l)(p−1) hs  , + (p − 1)! s=n+l = T xn1 , for n ≥ n1 . for n0 ≤ n ≤ n1 , (2.5) where α = (a + β)C. We shall show that T is a contraction mapping on X. It is easy to see that T maps X into itself. In fact, if x ∈ X, then because of (2.3) and (2.4) it follows that T xn ≤ α 1 β β + ·b+ + =b c c 2 2 and T xn ≥ α β β − − = a. C 2 2 Therefore T X ⊆ X. To show that T is a contraction, let x, y ∈ X. Using (2.3), we easily see from (2.5) that 1 |xn+l − yn+l | c ∞ ! M + s(p−1) |qs ||xs−k − ys−k | c(p − 1)! s=n+l |T xn − T yn | ≤ ≤ M (p − 1)!βc 1 ||x − y|| + ||x − y||, c c(p − 1)! 2Mb A. ZAFER and hence |T xn − T yn | ≤ % & 1 β ||x − y||. + c 2b (2.6) It is easy to verify that 1 β + < 1. (2.7) c 2b In view of (2.6) and (2.7) it follows that T is a contraction on X, and therefore there exists a fixed point x ∈ X such that T x = x. It can easily be seen that x is a bounded positive solution of Eq. (1.1). This completes the proof. Theorem 2.2 Let 1 < c ≤ pn < C, {qn } be eventually positive and xf (x) > 0 for x %= 0. Suppose that there exists an oscillatory function ρ on N such that ∆p ρn = hn and limn→∞ ρn = 0. If ∞ ! s(p−1) qs = ∞, (2.8) then every bounded solution {xn } of Eq. (1.1) is oscillatory when (−1)p δ = −1, and is either oscillatory or else limn→∞ xn = 0 when (−1)p δ = 1. Proof Suppose that Eq. (1.1) has a nonoscillatory solution {xn }. We may assume that {xn } is eventually positive. Set zn = xn − pn xn−l and yn = zn − ρn We claim that zn is eventually negative; otherwise for sufficiently large values of n xn > pn xn−l and by induction xn+ml > cm xn . (2.9) But inequality (2.9) as m → ∞ contradicts the fact that xn is bounded, and so zn must be eventually negative. Since ρn is oscillatory, it follows that yn should also be eventually negative. Furthermore from Eq. (1.1), δ∆p yn = −qn f (xn−k ) < 0. NEUTRAL TYPE DIFFERENCE EQUATIONS Since yn < 0 and δ∆p yn < 0, applying a lemma of Agarwal [9] it follows that there is an integer l ∈ {0, 1} with (−1)p−l δ = 1 so that ∆j yn < 0 for j = 0, 1, . . . , l (−1)j−l ∆j yn < 0 for j = l + 1, . . . , p − 1. (2.10) Multiplying (1.1) by n(p−1) and summing from n1 to n − 1 we obtain n−1 ! s(p−1) δ∆p ys + s=n1 n−1 ! s(p−1) qs f (xs−k ) = 0. (2.11) s=n1 Applying the summation by parts formula [9] to the first term on the left hand side of (2.11) we have n−1 ! s (p−1) p δ∆ ys = s=n1 p−1 ! (−1)k+1 ∆k−1 n(p−1) δ∆p−k yn+k−1 − K k=1 + (−1)p+1δ(p − 1)![yn+p−1 − yn1 +p−1 ] (2.12) where, in view of (2.10), K= p−1 ! (p−1) (−1)k+1 δ∆k−1 n1 ∆p−k yn1 +k−1 > 0. k=1 Using (2.10) and (2.12) in (2.11) leads to n−1 ! s(p−1) qs f (xs−k ) ≤ K + (−1)p δ(p − 1)![yn+p−1 − yn1 +p−1 ]. (2.13) s=n1 Since {yn } is bounded and (2.8) holds it follows from (2.13) that lim inf f (xn−k ) = 0, n→∞ and hence lim inf xn = 0. n→∞ On the other hand, the bounded sequence {yn } being negative and monotone has a finite limit L ≤ 0. Moreover, L = 0 is possible only when l = 0, and L < 0 may hold if l = 1. Suppose L < 0. Clerly limn→∞ zn = L and therefore we can choose n2 ∈ N(n0 ) such that for n ∈ N(n2 ), 2L < xn − pn xn−k < L . 2 A. ZAFER But then −L 2C for n ∈ N(n2 ), a contradiction with lim inf n→∞ xn = 0. Thus, we must have L = 0 and l = 0. Using the arguments developed in [12] one can easily show that lim xn = 0. xn−k > n→∞ This completes the proof. In view of Theorem 2.1 and Theorem 2.2 we obtain the following necessary and sufficient condition for the oscillation of bounded solutions of Eq. (1.1). Theorem 2.3 Let 1 < c ≤ pn < C, {qn } be eventually positive and xf (x) > 0 for x %= 0. Assume that (i) there exists an oscillatory function ρ on N such that ∆p ρn = hn and limn→∞ ρn = 0 (ii) 1 < c ≤ pn ≤ C (iii) (2.2) is satisfied (iv) for some positive real numbers a and b, a/b < (c−1)/C, f satisfies a Lipschitz condition on the interval [a, b]. Then a necessary and sufficient condition for every bounded solution {xn } of Eq. (1.1) to be oscillatory when (−1)p δ = −1, and be either oscillatory or limn→∞ xn = 0 when (−1)p δ = 1 is that ∞ ! s(p−1) qs = ∞. Example. Consider the difference equation ∆4 [xn + (3 + (−1)n )xn−3 ] − xn−4 = 33(−1)n−1 , (2.14) so that p = 4, pn = 3 + (−1)n , qn = 1, δ = −1, and hn = 33(−1)n−1 . It is easy to see that c = 2, C = 4, and the conditions of Theorem 2.3 are satisfied. Therefore, all bounded solutions of (2.14) are oscillatory. Indeed, xn = (−1)n is such a solution of (2.14). NEUTRAL TYPE DIFFERENCE EQUATIONS References [1] M.-P. Chen, B.S. Lalli, and J.S. Yu, Oscillation in neutral delay difference equations with variable coefficients, Computers Math. Applic. 29 5-11 (1995) [2] S. S. Cheng, T. C. Yan, and H. J. Li, Oscillation criteria for second order difference equations, Funkcial. Ekvac. 34 223-239 (1991). [3] X.-Z. He, Oscillatory and asymptotic behavior of second order nonlinear difference equations, J. Math. Anal. Appl. 175 482-498 (1993). [4] J. Jaros and I. P. Stavroulakis, Necessary and sufficient conditions for oscillation of difference equations with several delays, Utilitas Math., to appear [5] B. S. Lalli, B. G. Zhang, and J. Z. Li, On the oscillation of solutions and existence of positive solutions of neutral difference equations, J. Math. Anal. 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