Pacific Journal of Mathematics

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1 Pacific Journal of Mathematics OPTIMAL OSCILLATION CRITERIA FOR FIRST ORDER DIFFERENCE EQUATIONS WITH DELAY ARGUMENT GEORGE E. CHATZARAKIS, ROMAN KOPLATADZE AND IOANNIS P. STAVROULAKIS Volume 235 No. 1 March 2008

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3 PACIFIC JOURNAL OF MATHEMATICS Vol. 235, No. 1, 2008 OPTIMAL OSCILLATION CRITERIA FOR FIRST ORDER DIFFERENCE EQUATIONS WITH DELAY ARGUMENT GEORGE E. CHATZARAKIS, ROMAN KOPLATADZE AND IOANNIS P. STAVROULAKIS Consider the first order linear difference equation uk + pk uτk = 0, k N, where uk = uk + 1 uk, p : N +, τ : N N, τk k 2 and lim τk = +. Optimal conditions for the oscillation of all proper solutions of this equation are established. The results lead to a sharp oscillation condition, when k τk + as k +. Examples illustrating the results are given. 1. Introduction The first systematic study for the oscillation of all solutions to the first order delay differential equation 1-1 u t + pt uτt = 0, where p L loc + ; +, τ C + ; +, τt t for t + and lim τt = +, t + in the case of constant coefficients and constant delays was made by Myshkis [1972]. For the differential equation 1-1 the problem of oscillation is investigated by many authors. See, for example, [Elbert and Stavroulakis 1995; Koplatadze and Chanturiya 1982; Koplatadze and Kvinikadze 1994; Ladas et al. 1984; Sficas and Stavroulakis 2003] and the references cited therein. Theorem 1.1 [Koplatadze and Chanturiya 1982]. Assume that 1-2 lim inf t + t τt Then all solutions of Equation 1-1 oscillate. ps ds > 1 e. MSC2000: primary 39A11; secondary 39A12. Keywords: difference equation, proper solution, positive solution, oscillatory. 15

4 16 G. E. CHATZARAKIS, R. KOPLATADZE AND I. P. STAVROULAKIS It is to be emphasized that condition 1-2 is optimal in the sense that it cannot be replaced by the condition 1-3 lim inf t + t τt ps ds 1 e. For example, if τt = t δ or τt = α t or τt = t α, where δ > 0, α 0, 1, examples can be given such that condition 1-3 is satisfied, but 1-1 has a nonoscillatory solution. The discrete analogue of the first order delay differential equation 1-1 is the first order difference equation 1-4 uk + pk uτk = 0, where 1-5 uk = uk + 1 uk, p : N +, τ : N N, τk k 1, lim τk = +. By a proper solution of 1-4 we mean a function u : N n0 with n 0 = min{τk : k N n } and N n = {n, n + 1,... }, which satisfies 1-4 on N n and sup{ ui : i k} > 0 for k N n0. A proper solution u : N n0 of 1-4 is said to be oscillatory around zero if for any positive integer n N n0 there exist n 1, n 2 N n such that un 1 un 2 0. Otherwise, the proper solution is said to be nonoscillatory. In other words, a proper solution u is oscillatory if it is neither eventually positive nor eventually negative. Oscillatory properties of the solutions of 1-4, in the case of a general delay argument τk, have been recently investigated in [Chatzarakis et al. 2008a; 2008b], while the special case when τk = k n, n 1, has been studied rather extensively. See, for example, [Agarwal et al. 2005; Baštinec and Diblik 2005; Chatzarakis and Stavroulakis 2006; Domshlak 1999; Elaydi 1999; Ladas et al. 1989] and the references cited therein. In this particular case, 1-4 becomes 1-6 uk + pk uk n = 0, k N. For this equation Ladas, Philos and Sficas established the following theorem. Theorem 1.2 [Ladas et al. 1989]. Assume that 1-7 lim inf k 1 n Then all proper solutions of 1-6 oscillate. n n+1. pi > n + 1

5 OSCILLATION CRITERIA FOR DIFFERENCE EQUATIONS WITH DELAY ARGUMENT 17 This result is sharp in the sense that the inequality 1-7 cannot be replaced by the nonstrong one for any n N. Hence, Theorem 1.2 is the discrete analogue of Theorem 1.1 when τt = t δ. An interesting question then arises whether there exists the discrete analogue of Theorem 1.1 for 1-4 in the case of a general delay argument τk when lim k τk = +. In the present paper optimal conditions for the oscillation of all proper solutions of 1-4 are established and a positive answer to the above question is given. 2. Some auxiliary lemmas Let k 0 N. Denote by U k0 the set of all proper solutions of 1-4 satisfying the condition uk > 0 for k k 0. Remark 2.1. We will suppose that U k0 =, if 1-4 has no solution satisfying the condition uk > 0 for k k 0. Lemma 2.2. Assume that k 0 N, U k0 =, u U k0, τk k 1, τ is a nondecreasing function and 2-1 lim inf Then k 1 i=τk 2-2 lim sup pi = c > 0. uτk uk c 2. Proof. By 2-1, for any ε 0, c, it is clear that 2-3 k 1 i=τk pi c ε for k N k0. Since u is a positive proper solution of 1-4, then there exists k 1 N k0 such that Thus, from 1-4 we have uτk > 0 for k N k1. uk + 1 uk = pkuτk 0 and so u is an eventually nonincreasing function of positive numbers. Now from inequality 2-3 it is clear that, there exists k k such that 2-4 k 1 pi < c ε 2 and k pi c ε 2.

6 18 G. E. CHATZARAKIS, R. KOPLATADZE AND I. P. STAVROULAKIS This is because in the case where pk < c ε 2, it is clear that there exists k > k such that 2-4 is satisfied, while in the case where pk c ε 2, then k = k, and therefore and k 1 k 1 pi = k pi by which we mean = 0 < c ε 2 pi = k pi = pk c ε 2. That is, in both cases 2-4 is satisfied. Now, we will show that τk k 1. Indeed, in the case where pk c ε 2, since k = k, it is obvious that τk k 1. In the case where pk < c ε 2, then k > k. Assume, for the sake of contradiction, that τk > k 1. Hence, k τk k 1 and then k 1 i=τk k 1 pi pi < c ε 2. This, in view of 2-3, leads to a contradiction. Thus, in both cases, we have τk k 1. Therefore, it is clear that 2-5 k 1 i=τk pi = k 1 i=τk k 1 pi pi c ε c ε 2 = c ε 2. Now, summing up 1-4 first from k to k and then from τk to k 1, and using that the function u is nonincreasing and the function τ is nondecreasing, we have or uk uk + 1 = k k piuτi 2-6 uk c ε 2 uτk, and then uτk uk = or k 1 i=τk k 1 piuτi i=τk 2-7 uτk c ε uτk 1. 2 pi uτk c ε 2 uτk, pi uτk 1 c ε 2 uτk 1,

7 OSCILLATION CRITERIA FOR DIFFERENCE EQUATIONS WITH DELAY ARGUMENT 19 Combining inequalities 2-6 and 2-7, we obtain uτk 1 uk 4 c ε 2 and, for large k, we have uτk uk c ε 2. Hence, uτk lim sup uk c ε 2, which, for arbitrarily small values of ε, implies 2-2. Lemma 2.3. Assume that k 0 N, U k0 =, u U k0, τk k 1, τ is a nondecreasing function and condition 2-1 is satisfied. Then k lim uk exp λ pi = + for any λ > 4 c 2. Proof. Since all the conditions of Lemma 2.2 are satisfied, for any γ > 4/c 2, there exists k 1 N k0 such that 2-9 Also, for any n N k1 n k=k 1 or uk n uk + 1 = k=k 1 n k 1 uτk uk + 1 γ for k N k 1. 1 uk uk + 1 n k=k ln n k=k 1 Moreover, from 1-4, we have n k=k 1 = n k 1 uk uk + 1 uk uk + 1 uk uk + 1 = = n k=k 1 exp n k=k 1 ln un + 1 ln. uk 1 ln n pk uτk uk + 1. k=k 1 Combining 2-9 with the last two relations, we obtain n un + 1 uk 1 exp γ pk. k=k 1 uk uk + 1 uk uk + 1 un + 1 = ln, uk 1

8 20 G. E. CHATZARAKIS, R. KOPLATADZE AND I. P. STAVROULAKIS Now, by 2-1, it is obvious that + pi = +. Therefore, for λ > 4/c 2, the last inequality yields n lim un + 1 exp λ pk = +, n + k=k 1 or which implies 2-8, since lim uk exp λ k 1 pi = +, 1 k 1 pi Next, consider the difference inequality k 1 1 pi uk + qk uσ k 0, where q : N +, σ : N N and lim σ k = +. In the sequel the following lemma will be used, which has recently been established in [Chatzarakis et al. 2008a]. Lemma 2.4. Assume that 2-1 is satisfied, and for sufficiently large k σ k τk k 1, pk qk and u : N k0 0, + is a positive proper solution of Then, there exists k 1 N k0 such that U k1 = and u U k1 is the solution of 1-4, which satisfies the condition 0 < u k uk for k N k1. By virtue of Lemma 2.4, we can formulate Lemma 2.3 in the following more general form, where the function τ is not required to be nondecreasing. Lemma 2.5. Assume that k 0 N, U k0 =, u U k0, τk k 1 and condition 2-1 is satisfied. Then, for any λ > 4/c 2, condition 2-8 holds. Proof. Since u : N k0 0, + is a solution of 1-4, it is clear that u is a solution of the inequality uk + pk uσ k 0 for k N k1, where σ k = max{τi : 1 s k, s N} and k 1 > k 0 is a sufficiently large number.

9 OSCILLATION CRITERIA FOR DIFFERENCE EQUATIONS WITH DELAY ARGUMENT 21 First we will show that 2-11 lim inf k 1 i=σ k pi = c. Assume that 2-11 is not satisfied. Then there exists a sequence {k i } + of natural numbers such that σ k i = τk i i = 1, 2,... and 2-12 lim inf k j 1 j + i=σ k j pi = c 1 < c. Also, from the definition of the function σ, and in view of σ k i = τk i, for any k i, there exists k i < k i such that σ k = σ k i for k i k k i, lim i + k i = + and σ k i = τk i. Thus k i 1 j=τk i p j = k i 1 j=σ k i p j = and, by the virtue of 2-12, we have lim inf i + k i 1 k i 1 j=σ k i j=τk i p j lim inf i + p j k i 1 j=σ k i k i 1 j=σ k i p j i = 1, 2,..., p j = c 1 < c. In view of 2-1, the last inequality leads to a contradiction. Therefore 2-11 holds. Now, by Lemma 2.4, we conclude that the equation uk + pk uσ k = 0 has a solution u which satisfies the condition < u k uk for k N k1, where k 1 > k 0 is a sufficiently large number. Hence, taking into account that the function σ is nondecreasing, in view of Lemma 2.3, we have lim u k 1 k exp λ pi = +, where λ > 4/c 2. Therefore, by 2-13, we get k 1 lim uk exp λ pi = + for any λ > 4 c 2.

10 22 G. E. CHATZARAKIS, R. KOPLATADZE AND I. P. STAVROULAKIS Lemma 2.6 Abel transformation. Let {a i } + and {b i} + be sequences of nonnegative numbers and 2-14 Then + a i < +. k a i b i = A 1 b 1 A k+1 b k+1 where A i = + j=i a j. Proof. Since 2-14 is satisfied, we have or k A i+1 b i b i+1 = k k A i+1 b i b i+1, k+1 A i+1 b i i=2 A i b i = A 2 b 1 A k+1 b k+1 + = A 2 b 1 A k+1 b k+1 = A 1 b 1 A k+1 b k+1 k a i b i = A 1 b 1 A k+1 b k+1 k A i+1 A i b i i=2 k a i b i i=2 k a i b i, k A i+1 b i b i+1. Koplatadze, Kvinikadze and Stavroulakis established the following lemma. For completeness, we present the proof here. Lemma 2.7 [Koplatadze et al. 2002]. Let ϕ, ψ : N 0, +, ψ be nonincreasing and suppose lim ϕk = +, lim inf ψk ϕk = 0, where ϕk = inf{ϕs : s k, s N}. Then there exists an increasing sequence of natural numbers {k i } + such that lim k i = +, ϕk i = ϕk i, ψk ϕk ψk i ϕk i i + k = 1, 2,..., k i ; i = 1, 2,....

11 OSCILLATION CRITERIA FOR DIFFERENCE EQUATIONS WITH DELAY ARGUMENT 23 Proof. Define the sets E 1 and E 2 by k E 1 ϕk = ϕk, k E 2 ϕs ψs ϕk ψk for s {1,..., k}. According to 2-15 and 2-16, it is obvious that 2-17 sup E i = + i = 1, 2. Show that 2-18 sup E 1 E 2 = +. Let k 0 E 2 be such that k 0 / E 1. By 2-16 there is k 1 > k 0 such that ϕk = ϕk 1 for k = k 0, k 0 + 1,..., k 1 and ϕk 1 = ϕk 1. Since ψ is nonincreasing, we have ϕk ψk ϕk 1 ψk 1 for k = 1,..., k 1. Therefore k 1 E 1 E 2. The above argument together with 2-17 imply that 2-18 holds. Remark 2.8. The analogue of this lemma for continuous functions ϕ and ψ was proved first in [Koplatadze 1994]. 3. Necessary conditions of the existence of positive solutions The results of this section play an important role in establishing sufficient conditions for all proper solutions of 1-4 to be oscillatory. Theorem 3.1. Assume that k 0 N, U k0 =, 1-5 is satisfied, 3-1 lim inf and k 1 i=τk 3-2 lim sup k 1 i=τk Then there exists λ [1, 4/c 2 ] such that 3-3 lim sup lim inf exp k 1 λ + ε ε 0+ + pi pi = c > 0, pi < +. τi 1 pi exp λ + ε pl 1.

12 24 G. E. CHATZARAKIS, R. KOPLATADZE AND I. P. STAVROULAKIS Proof. Since U k0 =, Equation 1-4 has a positive solution u : N k0 0, +. First we show that k lim sup uk exp pi < +. Indeed, if k 1 N k0, we have or k ui = ui 1 k ui + 1 ui 1 k ln k k 1 = k 1 exp ln ui + 1 uk + 1 k k 1 = ln ui uk 1, k ui ui 1 ln uk + 1 uk 1. ui + 1 k k 1 ui By 1-4, and taking into account that the function u is nonincreasing, we have k ui k ui = pi uτi k pi. ui Combining the last two inequalities, we obtain k uk + 1 exp pi uk 1, 1 that is, 3-4 is fulfilled. On the other hand, since all the conditions of Lemma 2.5 are satisfied, we conclude that condition 2-8 holds for any λ > 4/c 2. Denote by the set of all λ for which τk lim uτk exp λ pi = + and λ 0 = inf. In view of 1-5, 2-8 and 3-4, it is obvious that λ 0 [1, 4/c 2 ]. Thus, it suffices to show, that for λ = λ 0 the inequality 3-3 holds. First, we will show that for any ε > 0 τk lim uτk exp λ 0 + ε pi = +.

13 OSCILLATION CRITERIA FOR DIFFERENCE EQUATIONS WITH DELAY ARGUMENT 25 Indeed, if λ 0, it is obvious from 3-5 that condition 3-6 is fulfilled. If λ 0, according to the definition of λ 0, there exists λ k > λ 0 such that λ k λ 0 when k + and λ k, k = 1, 2,.... Thus, condition 3-5 holds for any λ = λ k. However, for any ε > 0, there exists λ k = λ k ε such that λ 0 < λ k λ 0 +ε. This insures the validity of 3-5 and 3-6 for any ε > 0. Similarly, we show that for any ε > 0, τk lim inf uτk exp λ 0 ε pi = 0. Hence, by virtue of 1-5, 3-6 and 3-7, it is clear that for any ε > 0, the functions τk ϕk = uτk exp λ 0 + ε pi and ψk = exp 2ε k 1 pi satisfy the conditions of Lemma 2.7 for sufficiently large k. Hence, there exists an increasing sequence {k i } + of natural numbers satisfying lim i + k i = +, 3-9 ψk i ϕk i ψk ϕk for k k k i, where k is a sufficiently large number, and 3-10 ϕk i = ϕk i i = 1, 2,..., Now, given that uτi exp λ 0 +ε Equation 1-4 implies uτk j + i=τk j pl τi 1 { τs 1 inf uτs expλ 0 +ε = ϕi, pi uτi + i=τk } pl : s i, s N τi 1 pi ϕi exp λ 0 + ε pl

14 26 G. E. CHATZARAKIS, R. KOPLATADZE AND I. P. STAVROULAKIS that is, uτk j k j 1 i=τk j pi ϕi exp 2ε i 1 pl exp i 1 2ε pl exp λ 0 + ε τi pi ϕi exp λ 0 + ε j τi 1 pl pl, for j = 1, 2,.... Thus, by 3-9, and using the fact that the function ϕ is nondecreasing, the last inequality yields k j uτk j ϕk j exp 2ε k j 1 i=τk j Also, in view of Lemma 2.6, we have 3-12 I k j, ε = pl i 1 pi exp 2ε + + ϕk j pi exp λ 0 + ε j k j 1 i=τk j pi exp 2ε i 1 τk j 1 + = exp 2ε pi exp 2ε k j 1 k j 1 + exp 2ε j + i=τk j τi 1 pl exp λ 0 + ε τi 1 pl τi 1 pl exp λ 0 + ε pl j = 1, 2,.... pl τi 1 pi exp λ 0 + ε pl + pi pi exp j i pl exp 2ε τi 1 pi exp λ 0 + ε pl λ 0 + ε τi 1 i 1 pl pl j = 1, 2,....

15 OSCILLATION CRITERIA FOR DIFFERENCE EQUATIONS WITH DELAY ARGUMENT 27 Given that exp 2ε inequality 3-12 becomes i τk j 1 + I k j, ε exp 2ε pi exp 2ε Therefore, by 3-11, we take k j 1 uτk j ϕk j exp 2ε Thus, 3-8 and 3-10 imply τk j 1 + exp λ 0 + ε pi i=τk j pl exp 2ε i=τk j k j 1 i 1 pl 0, τi 1 pi exp λ 0 + ε pl + pi pi exp j τk j 1 pl exp 2ε + i=τk j pi exp λ 0 + ε pl λ 0 + ε τi 1 pi exp λ 0 + ε pl τi 1 τi 1 k j 1 exp 2ε i=τk j pl. pl. pi. From the last inequality, and taking into account that 3-2 is satisfied, we have τk j lim sup exp λ 0 +ε pi j + where Hence, for any ε > 0, 3-13 gives lim inf exp k 1 λ 0 + ε M = lim sup + pi i=τk j k 1 i=τk τi 1 pi exp λ 0 +ε pl pi. τi 1 pi exp λ 0 + ε exp2εm, pl exp2εm,

16 28 G. E. CHATZARAKIS, R. KOPLATADZE AND I. P. STAVROULAKIS which implies lim sup lim inf exp k 1 λ 0 + ε ε 0+ + pi τi 1 pi exp λ 0 + ε pl 1. Remark 3.2. Condition 3-2 is not a limitation since, as proved in [Chatzarakis et al. 2008a], if τ is a nondecreasing function and then U k0 =, for any k 0 N. lim sup k i=τk pi > 1, Remark 3.3. In 3-1, without loss of generality, we may assume that c 1. Otherwise, for any k 0 N, we have U k0 = [Chatzarakis et al. 2008a]. Theorem 3.4. Assume that all the conditions of Theorem 3.1 are satisfied. Then 3-14 lim inf k 1 i=τk pi 1 e. Proof. Since all the conditions of Theorem 3.1 are satisfied, there exists λ = λ 0 [1, 4/c 2 ] such that the inequality 3-3 holds. Assume that the condition 3-14 does not hold. Then, there exists k 1 N and ε 0 > 0 such that Therefore, for any ε > 0, k 1 i=τk pi 1 + ε 0 e k I k, ε = exp λ 0 + ε + pi for k N k1. τi 1 pi exp λ 0 + ε λ0 + ε1 + ε 0 k 1 exp exp λ 0 + ε e + pi exp Defining i 1 pl = a i 1, we will show that + lim inf expλ 0 + εa k 1 pi i 1 λ 0 + ε pl pi exp λ 0 + εa i 1 1 λ 0 + ε. pl for k N k1.

17 OSCILLATION CRITERIA FOR DIFFERENCE EQUATIONS WITH DELAY ARGUMENT 29 Indeed, since lim inf k 1 i=τk pi = c > 0, it is obvious that + pi = +, that is, lim i + a i = +. Therefore + expλ 0 + εa k 1 Hence, by 3-15, we obtain pi exp λ 0 + εa i 1 + = expλ 0 + εa k 1 a i a i 1 exp λ 0 + εa i 1 = expλ 0 + εa k 1 expλ 0 + εa i 1 = expλ 0 + εa i 1 lim sup lim inf I k, ε 1 exp ε 0+ λ ai + ai exp λ 0 + εa i 1 a i 1 exp λ 0 + εsds a i 1 ds a i 1 exp λ 0 + εsds = 1 λ 0 + ε. λ0 1 + ε 0 e 1 + ε 0. This contradicts 3-3 for λ = λ Sufficient conditions of the proper solutions to be oscillatory Theorem 4.1. Assume that conditions 1-5, 3-1, 3-2 are satisfied and that, for any λ [1, 4/c 2 ], 4-1 lim sup ε 0+ k 1 lim inf exp λ+ε + pi Then all proper solutions of Equation 1-4 oscillate. τi 1 pi exp λ+ε pli > 1. Proof. Assume that u : N k0 0, + is a positive proper solution of 1-4. Then U k0 =. Thus, in view of Theorem 3.1, there exists λ 0 [1, 4/c 2 ] such that the condition 3-3 is satisfied for λ = λ 0. But this contradicts 4-1. Using Theorem 3.4, we can similarly prove:

18 30 G. E. CHATZARAKIS, R. KOPLATADZE AND I. P. STAVROULAKIS Theorem 4.2. Assume that conditions 1-5 and 3-2 are satisfied and 4-2 lim inf k 1 i=τk pi > 1 e. Then all proper solutions of Equation 1-4 oscillate. Remark 4.3. It is to be pointed out that Theorem 4.2 is the discrete analogue of Theorem 1.1 for the first order difference equation 1-4 in the case of a general delay argument τk. Remark 4.4. The condition 4-2 is optimal for 1-4 under the assumption that lim k τk = +, since in this case the set of natural numbers increases infinitely in the interval [τk, k 1] for k +. Now, we are going to present two examples to show that the condition 4-2 is optimal, in the sense that it cannot be replaced by the nonstrong inequality. Example 4.5. Consider 1-4, where 4-3 with [αk] the integer part of αk. It is obvious that Therefore τk = [αk], pk = k λ k + 1 λ [αk] λ, α 0, 1, λ = ln 1 α, k 1+λ k λ k + 1 λ λ for k kk λ k + 1 λ [αk] λ λ e Hence, in view of 4-3 and 4-4, we have for k +. or lim inf k 1 i=τk pi = λ lim inf e = λ e lim inf k 1 i=[αk] k 1 i=[αk] lim inf k 1 i=τk e λ ii λ i + 1 λ [αi] λ 1 i 1 i = λ e ln 1 α = 1 e pi = 1 e.

19 OSCILLATION CRITERIA FOR DIFFERENCE EQUATIONS WITH DELAY ARGUMENT 31 Observe that all the conditions of Theorem 4.2 are satisfied except the condition 4-2. In this case, it is not guaranteed that all solutions of 1-4 oscillate. Indeed, it is easy to see that the function u = k λ is a positive solution of 1-4. Example 4.6. Consider 1-4, where 4-5 τk = [k α ], pk = ln λ k ln λ k + 1 ln λ [k α ], α 0, 1, λ = ln 1 α, with [k α ] the integer part of k α. It is obvious that k ln 1+λ kln λ k ln λ k + 1 λ for k +. Therefore 4-6 k ln k ln λ [k α ]ln λ k ln λ k + 1 λ e for k +. On the other hand, k 1 i=[k α ] k 1 1 i ln i i=[k α ] i+1 which tends to ln1/α as k +, and k 1 i=[k α ] k 1 1 i ln i i=[k α ] i i i 1 ds k s ln s = [k α ] ds s ln s = ln ln k ln[k α ], ds k 1 s ln s = ds lnk 1 = ln [k α ] 1 s ln s ln[k α ] 1, which also tends to ln1/α as k +. Together these two bounds imply lim k 1 i=[k α ] Hence, in view of 4-5 and 4-6, we obtain lim inf k 1 i=[k α ] pi = lim inf = λ e = λ e k 1 i=[k α ] lim inf k 1 i=[k α ] lim inf k 1 i=[k α ] 1 i ln i = ln 1 α. ln λ [i α ]ln λ i ln λ i + 1 e λ i ln i lnλ [i α ]ln λ i ln λ 1 i + 1 i ln i 1 i ln i = λ e ln 1 α = 1 e.

20 32 G. E. CHATZARAKIS, R. KOPLATADZE AND I. P. STAVROULAKIS We again observe that all the conditions of Theorem 4.2 are satisfied except 4-2. In this case, it is not guaranteed that all solutions of 1-4 oscillate. Indeed, it is easy to see that the function u = ln λ k is a positive solution of 1-4. References [Agarwal et al. 2005] R. P. Agarwal, M. Bohner, S. R. Grace, and D. O Regan, Discrete oscillation theory, Hindawi, New York, MR 2006e:39001 Zbl [Baštinec and Diblik 2005] J. Baštinec and J. Diblik, Remark on positive solutions of discrete equation uk + n = pk uk, Nonlinear Anal , e2145 e2151. [Chatzarakis and Stavroulakis 2006] G. E. Chatzarakis and I. P. Stavroulakis, Oscillations of first order linear delay difference equations, Aust. J. Math. Anal. Appl. 3:1 2006, Art. 14, 11 pp. electronic. MR 2007a:39010 Zbl [Chatzarakis et al. 2008a] G. E. Chatzarakis, R. Koplatadze, and I. P. Stavroulakis, Oscillation criteria of first order linear difference equation with delay argument, Nonlinear Anal , [Chatzarakis et al. 2008b] G. E. Chatzarakis, Ch. G. Philos, and I. P. Stavroulakis, On the oscillation of the solutions to linear difference equations with variable delay, To appear. [Domshlak 1999] Y. Domshlak, What should be a discrete version of the Chanturia Koplatadze lemma?, Funct. Differ. Equ. 6: , MR 2002f:39032 Zbl [Elaydi 1999] S. N. Elaydi, An introduction to difference equations, 2nd ed., Undergraduate Texts in Mathematics, Springer, New York, MR 2001g:39001 Zbl [Elbert and Stavroulakis 1995] Á. Elbert and I. P. Stavroulakis, Oscillation and nonoscillation criteria for delay differential equations, Proc. Amer. Math. Soc. 123:5 1995, MR 95f: Zbl [Koplatadze 1994] R. Koplatadze, On oscillatory properties of solutions of functional-differential equations, Mem. Differential Equations Math. Phys , 179 pp. MR 97g:34090 Zbl [Koplatadze and Chanturiya 1982] R. G. Koplatadze and T. A. Chanturiya, Oscillating and monotone solutions of first-order differential equations with deviating argument, Differentsial nye Uravneniya 18:8 1982, , MR 83k:34069 Zbl [Koplatadze and Kvinikadze 1994] R. Koplatadze and G. Kvinikadze, On the oscillation of solutions of first-order delay differential inequalities and equations, Georgian Math. J. 1:6 1994, MR 95j:34103 Zbl [Koplatadze et al. 2002] R. Koplatadze, G. Kvinikadze, and I. P. Stavroulakis, Oscillation of secondorder linear difference equations with deviating arguments, Adv. Math. Sci. Appl. 12:1 2002, MR 2003f:39039 Zbl [Ladas et al. 1984] G. Ladas, Y. G. Sficas, and I. P. Stavroulakis, Nonoscillatory functional-differential equations, Pacific J. Math. 115:2 1984, MR 86g:34099 Zbl [Ladas et al. 1989] G. Ladas, Ch. G. Philos, and Y. G. Sficas, Sharp conditions for the oscillation of delay difference equations, J. Appl. Math. Simulation 2:2 1989, MR 90g:39004 Zbl [Myshkis 1972] A. D. Myshkis, Lineinye differencialnye uravneni s zapazdyva wim argumentom, 2nd ed., Nauka, Moscow, MR 50 #5135 Zbl

21 OSCILLATION CRITERIA FOR DIFFERENCE EQUATIONS WITH DELAY ARGUMENT 33 [Sficas and Stavroulakis 2003] Y. G. Sficas and I. P. Stavroulakis, Oscillation criteria for first-order delay equations, Bull. London Math. Soc. 35:2 2003, MR 2003m:34160 Zbl Received May 30, Revised July 10, GEORGE E. CHATZARAKIS DEPARTMENT OF MATHEMATICS UNIVERSITY OF IOANNINA IOANNINA GREECE ROMAN KOPLATADZE DEPARTMENT OF MATHEMATICS UNIVERSITY OF TBILISI UNIVERSITY STREET 2 TBILISI 0143 GEORGIA roman@rmi.acnet.ge IOANNIS P. STAVROULAKIS DEPARTMENT OF MATHEMATICS UNIVERSITY OF IOANNINA IOANNINA GREECE ipstav@cc.uoi.gr

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