Oscillation results for difference equations with oscillating coefficients

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1 Chatzarakis et al. Advances in Difference Equations :53 DOI /s R E S E A R C H Open Access Oscillation results f difference equations with oscillating coefficients Gege E Chatzarakis 1, Hajnalka Péics 2*, Sandra Pinelas 3 and Ioannis P Stavroulakis 4 * Crespondence: peics@gf.uns.ac.rs 2 Faculty of Civil Engineering, University of Novi Sad, Kozaračka 2/A, Subotica, 24000, Serbia Full list of auth infmation is available at the end of the article Abstract Sufficient conditions which guarantee the oscillation of all solutions of difference equations with oscillating coefficients and several deviating arguments are presented. The cresponding difference equations of both retarded and advanced type are studied. Examples illustrating the results are also given. 1 Introduction In the present paper, we study the oscillaty behavi of the solutions of the difference equation xn+ m p i nx τ i n =0, n N 0, E R and the dual advanced difference equation xn m p i nx σ i n =0, n N, E A where m N, {p i n} n N0,1 i m, are real sequences with oscillating terms, {τ i n} n N0, 1 i m, are sequences of integers such that τ i n n 1, n N 0, and lim n τ in=, 1 i m, 1.1 and {σ i n} n N,1 i m, are sequences of integers such that σ i n n +1, n N,1 i m. 1.2 Here, as usual, denotes the fward difference operat xn=xn +1 xn and denotes the backward difference operat xn=xn xn 1. By a solution of E R, we mean a sequence of real numbers {xn} n w which satisfies E R f all n N 0. Here, { w = min τi n:1 i m }. n Chatzarakis et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the iginal wk is properly credited.

2 Chatzarakis et al. Advances in Difference Equations :53 Page 2 of 17 It is clear that, f each choice of real numbers c w, c w+1,...,c 1, c 0,thereexistsaunique solution {xn} n w of E R which satisfies the initial conditions x w =c w, x w +1= c w+1,...,x 1 = c 1, x0 = c 0. By a solution of E A, we mean a sequence of real numbers {xn} n N0 which satisfies E A f all n N. Asolution{xn} n w {xn} n N0 ofe R E A is called oscillaty, ifthetermsxn of the sequence are neither eventually positive n eventually negative. Otherwise, the solution is said to be non-oscillaty. In the last few decades, the oscillaty behavi of all solutions of difference equations has been extensively studied when the coefficients p i n are nonnegative. However, f the general case when p i n are allowed to oscillate, it is difficult to study the oscillation of E R E A, since the difference xn xn of any non-oscillaty solution of E R E A is always oscillaty. Therefe, the resultson oscillation of difference and differential equations with oscillating coefficients are relatively scarce. Thus, a small number of paper are dealing with this case. See, f example, [1 17] and the references cited therein. F E R ande A with oscillating coefficients, Bohner et al. [2, 3] established the following theems. Theem 1.1 [2, Theem 2.4] Assume that 1.1 holds and that the sequences {τ i n} n N0 areincreasingfalli {1,...,m}. Suppose also that f each i {1,...,m} there exists a sequence {n i j} such that lim n i j= and where p k n 0, n A = { [ τ τ ni j, n i j ] } N, 1 k m, 1.3 τn= max 1 i m τ in, n N If, meover, m nj q=τnj p i q > 1, 1.5 where nj=min{n i j:1 i m}, then all solutions of E R oscillate. Theem 1.2 [2, Theem 3.4] Assume that 1.2 holds and that the sequences {σ i n} n N areincreasingfalli {1,...,m}. Suppose also that f each i {1,...,m} there exists a sequence {n i j} such that lim n i j= and where p k n 0, n B = { [ ni j, σ σ n i j ] } N, 1 k m, 1.6 σ n= min 1 i m σ in, n N. 1.7

3 Chatzarakis et al. Advances in Difference Equations :53 Page 3 of 17 If, meover, m σ nj q=nj p i q > 1, 1.8 where nj=max{n i j:1 i m}, then all solutions of E A oscillate. Theem 1.3 [3, Theem 2.1] Assume that 1.1 holds and that the sequences {τ i n} n N0 areincreasingfalli {1,...,m}. Suppose also that f each i {1,...,m} there exists a sequence {n i j} such that lim n i j=, and p k n 0, n C = n If, meover, { [ τi τi ni j, n i j ] } N, 1 k m, 1.9 m p i n>0, f all n C m n i j 1 q=τ i n i j then all solutions of E R oscillate. p i q> 1 e, 1.11 Theem 1.4 [3, Theem 3.1] Assume that 1.2 holds and that the sequences {σ i n} n N areincreasingfalli {1,...,m}. Suppose also that f each i {1,...,m} there exists a sequence {n i j} such that lim n i j=, and p k n 0, n D = n If, meover, { [ ni j, σ i σi ni j ] } N, 1 k m, 1.12 m p i n>0 f all n D m σ i n i j q=n i j+1 then all solutions of E A oscillate. p i q> 1 e, 1.14 Recently, Berezansky et al. [1]andChatzarakiset al. [4] established the following theems. Theem 1.5 [1, Theem 8] and [4, Theem 2.1] Assume that 1.1 holds and the sequences {τ i n} n N0 are increasing f all i {1,...,m} and the sequence τ is defined by

4 Chatzarakis et al. Advances in Difference Equations :53 Page 4 of Suppose also that f each i {1,...,m} there exists a sequence {n i j} such that lim n i j=, Set p k n 0, n A = α := m nj 1 q=τnj { [ τ τ ni j, n i j ] } N, 1 k m. where nj=min{n i j:1 i m}. If 0<α 1/2, and m nj q=τnj m nj q=τnj p i q, 1.15 p i q>1 then all solutions of E R oscillate. α 2 41 α, 1.16 p i q> α 1 2α, 1.17 Theem 1.6 [1, Theem 9] and [4, Theem 3.1] Assume 1.2 holds and the sequences {σ i n} n N are increasing f all i {1,...,m} and the sequence σ is defined by 1.7. Suppose also that f each i {1,...,m} there exists a sequence {n i j} such that lim n i j=, Set p k n 0, n B = α := m σ nj q=nj+1 { [ ni j, σ σ n i j ] } N, 1 k m. where nj=max{n i j:1 i m}. If 0<α 1/2, and m m σ nj q=nj σ nj q=nj p i q>1 then all solutions of E A oscillate. p i q, 1.18 α 2 41 α, 1.19 p i q> α 1 2α, 1.20

5 Chatzarakis et al. Advances in Difference Equations :53 Page 5 of 17 The auths study further E R ande A and derive new sufficient oscillation conditions. These conditions are the analogues of the oscillation conditions f the cresponding difference equations with positive coefficients, which were studied by Chatzarakis et al. [5]. Examples illustrate cases when the results of the paper imply oscillation while previously known results fail. 2 Oscillation criteria 2.1 Retarded difference equations We present new sufficient conditions f the oscillation of all solutions of E R, under the assumption that the sequences τ i are increasing f all i {1,...,m}. Theem 2.1 Assume that 1.1 holds, the sequences {τ i n} n N0 areincreasingfalli {1,...,m} and the sequence τ is defined by 1.4. Suppose also that f each i {1,...,m} there exists a sequence {n i j} such that lim n i j=, with p k n>0, n A = { [ τ τ ni j, n i j ] } N, 1 k m, { pk n:n A } >0, 1 k m. 2.1 n If, meover, [ m m l=1 nj 1 k=τ l nj ] 1/m p i k > 1 e, 2.2 where nj=min{n i j:1 i m}, then all solutions of E R oscillate. Proof Assume, f the sake of contradiction, that {xn} n w is an eventually positive solution of E R. Then there exists j 0 N such that p k n > 0 x τ k n >0 f all n [ τ τ ni j 0, n i j 0 ] N, 1 k m, falln Therefe, by E R wehave xn +1 xn= [ τ τ ni j 0, n i j 0 ] N, 1 k m. m p i nx τ i n <0, f every n m [ττn ij 0, n i j 0 ] N. This guarantees that the sequence x is strictly decreasing on m [ττn ij 0, n i j 0 ] N. Set nj 0 =min { n i j 0 :1 i m }

6 Chatzarakis et al. Advances in Difference Equations :53 Page 6 of 17 and with z i nj0 = xτ inj 0, 1 i m, 2.3 xnj 0 ϕ i = z i nj0, 1 i m. 2.4 It is obvious that z i nj0 >1 and ϕ i 1 fi =1,2,...,m. Dividing both sides of E R byxnj 0, we obtain xnj 0 xnj 0 + m p i nj0 xτ i nj 0 xnj 0 =0, xnj 0 xnj 0 + m p i nj0 z i nj0 = Summing up 2.5fromτ ρ nj 0 to fρ =1,2,...,m,wefind j=τ ρ nj 0 xj xj + m j=τ ρ nj 0 p i jz i j= But j=τ ρ nj 0 xj xj = j=τ ρ nj 0 j=τ ρ nj 0 xj +1 xj 1 xj +1 1+ln xj 1 = ln xnj 0 xτ ρ nj 0, j=τ ρ nj 0 xj xj ln z ρ nj Combining 2.6and2.7, we obtain ln z ρ nj0 + m j=τ ρ nj 0 p i jz i j 0,

7 Chatzarakis et al. Advances in Difference Equations :53 Page 7 of 17 ln z ρ nj0 m j=τ ρ nj 0 p i jz i j, ρ =1,2,...,m. 2.8 Nowwewillshowthatϕ i < f i =1,2,...,m. Indeed, assume that ϕ i = f some i, i =1,2,...,m.Byusing2.5wehave xnj 0 +1 xnj 0 1+ m p i nj0 z i nj0 =0, xnj 0 +1 xnj 0 + m p i nj0 z i nj0 =1. Consequently, [ xnj 0 +1 xnj 0 + m p i nj0 z i nj0 ] =1 and therefe xnj 0 +1 xnj 0 + m pi nj0 z i nj0 1, xnj 0 +1 xnj 0 + m p inj 0 z i nj0 1. In view of 2.1 and taking into account the fact that ϕ i = f some i, the above inequality is false. Hence ϕ i < f i =1,2,...,m. Taking the limit inferis on both sides of the above inequalities 2.8, we obtain ln ϕ ρ m ϕ i k=τ ρ nj 0 p i k, ρ =1,2,...,m, 2.9 and by adding we find m ln ϕ i m m ϕ i j=1 k=τ j nj 0 p i k. Set f ϕ 1, ϕ 2,...,ϕ m m ln ϕ i m m ϕ i j=1 k=τ j nj 0 p i k.

8 Chatzarakis et al. Advances in Difference Equations :53 Page 8 of 17 Clearly f ϕ 1, ϕ 2,...,ϕ m 0 f all ϕ 1, ϕ 2,...,ϕ m 1. Since f f ϕ i = 1 ϕ i m j=1 k=τ j nj 0 p i k=0, ϕ i = 1 m j=1 nj0 1 k=τ j nj 0 p ik, i =1,2,...,m, the function f has a maximum at the critical point 1 m j=1 nj0 1 k=τ j nj 0 p 1k,..., 1 m j=1 nj0 1 k=τ j nj 0 p mk because the quadratic fm m i,j=1 2 f ϕ i ϕ j α i α j = m αi 2 ϕi 2 <0. Since f ϕ 1, ϕ 2,...,ϕ m 0, the maximum of f at the critical point should be nonnegative. That is, [ m m ln j=1 k=τ j nj 0 ] p i k m 0, i.e., max f ϕ 1, ϕ 2,...,ϕ m = ln ϕ i 1 m m j=1 k=τ j nj 0 p i k m. Hence m m j=1 [ m m j=1 k=τ j nj 0 k=τ j nj 0 p i k 1 e, m ] 1/m p i k 1 e, which contradicts 2.2. The proof of the theem is complete.

9 Chatzarakis et al. Advances in Difference Equations :53 Page 9 of 17 Theem 2.2 Assume that 1.1 holds, the sequences {τ i n} n N0 are increasing f all i {1,...,m} and the sequence τ is defined by 1.4. Suppose also that f each i {1,...,m} there exists a sequence {n i j} such that lim n i j=, with p k n>0, n A = { [ τ τ ni j, n i j ] } N, 1 k m, { pk n:n A } >0, 1 k m. n If, meover, 1 m m + 2 m m i<l i,l=1 nj 1 k=τ i nj p i k nj 1 k=τ l nj p i k nj 1 k=τ i nj where nj=min{n i j:1 i m}, then all solutions of E R oscillate. 1/2 p l k > 1 e, 2.10 Proof Assume, f the sake of contradiction, that {xn} n w is an eventually positive solution of E R. Then there exists j 0 N such that p k n > 0 x τ k n >0 f all n [ τ τ ni j 0, n i j 0 ] N, 1 k m, falln Therefe, by E R wehave xn +1 xn= [ τ τ ni j 0, n i j 0 ] N, 1 k m. m p i nx τ i n <0, f every n m [ττn ij 0, n i j 0 ] N. This guarantees that the sequence x is strictly decreasing on m [ττn ij 0, n i j 0 ] N. Taking into account the fact that ϕ i < f i =1,2,...,m see Theem 2.1, by using 2.9andthefactthat 1 e > ln ϕ ρ ϕ ρ, ρ =1,2,...,m, we obtain 1 m e > k=τ l nj 0 ϕ i p i k, ϕ l l =1,2,...,m.

10 Chatzarakis et al. Advances in Difference Equations :53 Page 10 of 17 Adding these inequalities we have m m e + [ m i<l i,l=1 m +2 [ m i<l i,l=1 k=τ i nj 0 k=τ i nj 0 p i k k=τ l nj 0 p i k k=τ l nj 0 ϕ i p i k + ϕ l p i k k=τ i nj 0 k=τ i nj 0 p l k ϕ l ϕ i p l k] 1/2, ] m k=τ i nj 0 p i k+2 m i<l i,l=1 k=τ l nj 0 p i k k=τ i nj 0 p l k m e, which contradicts The proof of the theem is complete. A slight modification in the proofs of Theem 2.1 Theem 2.2 leads to the following result about retarded difference inequalities. Theem 2.3 Assume that all conditions of Theem 2.1 Theem 2.2 hold. Then i the difference inequality xn+ m p i nx τ i n 0, n N 0, has no eventually positive solutions; ii the difference inequality xn+ m p i nx τ i n 0, n N 0, has no eventually negative solutions. 2.2 Advanced difference equations Similar oscillation theems f the dual advanced difference equation E A canbederived easily. The proofs of these theems are omitted, since they follow a similar procedure as in Section 2.1. Theem 2.4 Assume that 1.2 holds, the sequences {σ i n} n N areincreasingfalli {1,...,m} and the sequence σ is defined by 1.7. Suppose also that f each i {1,...,m}

11 Chatzarakis et al. Advances in Difference Equations :53 Page 11 of 17 there exists a sequence {n i j} such that lim n i j=, { [ p k n 0, n B = ni j, σ σ n i j ] } N, 1 k m, with { pk n:n B } >0, 1 k m n If, meover, [ m m l=1 σ l nj k=nj+1 ] 1/m p i k > 1 e, 2.12 where nj=max{n i j:1 i m}, then all solutions of E A oscillate. Theem 2.5 Assume that 1.2 holds, the sequences {σ i n} n N areincreasingfalli {1,...,m} and the sequence σ is defined by 1.7. Suppose also that f each i {1,...,m} there exists a sequence {n i j} such that lim n i j=, with p k n 0, n B = { [ ni j, σ σ n i j ] } N, 1 k m, { pk n:n B } >0, 1 k m. n If, meover, 1 m m + 2 m m i<l i,l=1 σ i nj k=nj+1 p i k σ l nj k=nj+1 p i k σ i nj k=nj+1 where nj=max{n i j:1 i m}, then all solutions of E A oscillate. 1/2 p l k > 1 e, 2.13 A slight modification in the proofs of Theems 2.4 Theem 2.5 leads to the following result about advanced difference inequalities. Theem 2.6 Assume that all conditions of Theem 2.4 Theem 2.5 hold. Then i the difference inequality xn m p i nx σ i n 0, n N, has no eventually positive solutions;

12 Chatzarakis et al. Advances in Difference Equations :53 Page 12 of 17 ii the difference inequality xn m p i nx σ i n 0, n N, has no eventually negative solutions. 2.3 Special cases Inthecasewherep i, i =1,2,...,m, are oscillating real constants and τ i are constant retarded arguments of the fm τ i n=n k i,σ i are constant advanced arguments of the fm σ i n=n + k i, k i N, i =1,2,...,m,E R E A takes the fm xn+ m p i xn k i =0, n N 0 xn m p i xn + k i =0, n N. E F this equation, as a consequence of Theem 2.1 Theem 2.4andTheem2.2 Theem 2.5, we have the following collary. Collary 2.1 Assume that [ m ] 1/m m p i k i > 1 e, 2.14 m 2 1 pi k i > 1 m e Then all solutions of E oscillate. 3 Examples The following two examples illustrate that the conditions f oscillations 2.14and2.15 are independent. They are chosen in such a way that only one of them is satisfied. Also, in Example 3.1, Theem1.1 [2], Theem 1.3 [3] andtheem1.5 [1, 4] donotimply oscillation, and neither do Theem 1.2 [2], Theem 1.4 [3] andtheem1.6 [1, 4] in Example 3.2. Accding to conditions 2.14and2.15oscillation is established in Example 3.1 and Example 3.2,respectively. Example 3.1 Consider the retarded difference equation xn+p 1 nxn 1+p 2 nxn 2=0, n N 0, 3.1 where p 1 nandp 2 n are oscillating coefficients, as shown in Figure 1. In view of 1.4, it is obvious that τn=n 1.Observethatf n 1 j=12j +5, j N,

13 Chatzarakis et al. Advances in Difference Equations :53 Page 13 of 17 Figure 1 Coefficients of Example 3.1. we have p 1 n>0feveryn A,where A = [ τ τ n1 j, n 1 j ] N = [12j +3,12j +5] N. Also, f n 2 j=12j +4, j N, we have p 2 n>0feveryn B,where B = [ τ τ n2 j, n 2 j ] N = [12j +2,12j +4] N. Therefe p 1 n>0 and p 2 n > 0 f all n A B = +3,12j +4] N, [12j that is, 1.3 is satisfied. Meover, the computation immediately implies that [ 2 ] 1/ p i k i = , > 1 e, that is, condition 2.14 of Collary 2.1 is satisfied and therefe all solutions of 3.1 oscillate. Observe, however, that pi k i = , < 1 e, that is, condition 2.15of Collary 2.1 is not satisfied.

14 Chatzarakis et al. Advances in Difference Equations :53 Page 14 of 17 Also, nj=min { n i j:1 i 2 } =12j +4, j N, α = = m nj 1 q=τnj [ 12j+3 2 = Observe that q=12j+3 nj q=τnj [ 12j+4 q=12j+3 p i q p 1 q+ p i q p 1 q+ 12j+3 q=12j+3 12j+4 q=12j+3 ] p 2 q = ,000 =0.259, ] p 2 q = ,000 = < 1, < 1 α 2 41 α , < α 1 2α Therefe the conditions 1.5, 1.16, and 1.17are not satisfied. On the other hand, p 1 n>0feveryn A = A and p 2 n>0feveryn B,where Hence B = [ τ2 τ2 n2 j, n 2 j ] N = [12j,12j +4] N. p 1 n>0 and p 2 n > 0 f all n A B = A B, that is, condition 1.9is satisfied. In view of this,we have 2 n i j 1 q=τ i n i j p i q= that is, condition 1.11is not satisfied. [ 12j+4 q=12j+4 p 1 q+ 12j+3 q=12j+2 = ,000 =0.348<1 e, Example 3.2 Consider the advanced difference equation p 2 q ] xn p 1 nxn +1 p 2 nxn +2=0, n N, 3.2 where p 1 nandp 2 n are oscillating coefficients, as shown in Figure 2.

15 Chatzarakis et al. Advances in Difference Equations :53 Page 15 of 17 Figure 2 Coefficients of Example 3.2. In view of 1.7, it is obvious that σ n=n +1.Observethatf n 1 j=16j +3, j N, we have p 1 n>0feveryn A,where A = [ n1 j, σ σ n 1 j ] N = [16j +3,16j +5] N. Also, f n 2 j=16j +1, j N, we have p 2 n>0feveryn B,where B = [ n2 j, σ σ n 2 j ] N = [16j +1,16j +3] N. Therefe p 1 n>0 and p 2 n > 0 f all n A B = {16j +3}, that is, 1.6 is satisfied. Meover, the computation immediately implies that pi k i 2 = , > 1 e,

16 Chatzarakis et al. Advances in Difference Equations :53 Page 16 of 17 that is, condition 2.15 of Collary 2.1 is satisfied and therefe all solutions of 3.2 oscillate. Observe, however, that [ 2 ] 1/2 2 p i k i = 87 1, < 1 e, that is, condition 2.14of Collary 2.1 is not satisfied. Observe that nj=max { n i j:1 i 2 } =16j +3, j N. Now α = = 2 σ nj q=nj+1 [ 16j+4 q=16j+4 p i q p 1 q+ 16j+4 q=16j+4 ] p 2 q = 87 1, = Also 2 = σ nj q=nj [ 16j+4 p i q q=16j+3 p 1 q+ 16j+4 q=16j+3 ] p 2 q =2 87 1, = Observe that < 1, < 1 α 2 41 α , < α 1 2α Therefe the conditions 1.8, 1.19, and 1.20arenotsatisfied. On the other hand, p 1 n>0feveryn A = A and p 2 n>0feveryn B,where B = [ n2 j, σ 2 σ2 n2 j ] N = [16j +1,16j +5] N. Therefe p 1 n>0 and p 2 n > 0 f all n A B = [16j +3,16j +5],

17 Chatzarakis et al. Advances in Difference Equations :53 Page 17 of 17 that is, condition 1.12is satisfied. In view of this,we have m = σ i n i j q=n i j+1 [ 16j+4 q=16j+4 p i q p 1 q+ 16j+3 q=16j+2 that is, condition 1.14is not satisfied. ] p 2 q = 87 1, , =0.36<1 e, Competing interests The auths declare that they have no competing interests. Auths contributions This paper is the result of the joint effts of all the auths. All auths read and approved the final manuscript. Auth details 1 Department of Electrical and Electronic Engineering Educats, School of Pedagogical and Technological Education ASPETE, N. Heraklio, Athens, 14121, Greece. 2 Faculty of Civil Engineering, University of Novi Sad, Kozaračka 2/A, Subotica, 24000, Serbia. 3 Departamento de Ciências Exactas e Naturais, Av. Conde Castro Guimarães, Amada, , Ptugal. 4 Department of Mathematics, University of Ioannina, Ioannina, , Greece. Acknowledgements The research of Hajnalka Péics is suppted by the Serbian Ministry of Science, Technology and Development f Scientific Research Grant no. III Received: 29 September 2014 Accepted: 28 January 2015 References 1. Berezansky, L, Chatzarakis, GE, Domoshnitsky, A, Stavroulakis, IP: Oscillations of difference equations with several oscillating coefficients. Abstr. Appl. Anal. 2014,ArticleID Bohner, M, Chatzarakis, GE, Stavroulakis, IP: Qualitative behavi of solutions of difference equations with several oscillating coefficients. Arab. J. Math. 3, Bohner, M, Chatzarakis, GE, Stavroulakis, IP: Oscillation in difference equations with deviating arguments and variable coefficients. Bull. Kean Math. Soc. 521, Chatzarakis, GE, Lafci, M, Stavroulakis, IP: Oscillation results f difference equations with several oscillating coefficients. Appl. Math. Comput. 251, Chatzarakis, GE, Péics, H, Stavroulakis, IP: Oscillation in difference equations with deviating arguments and variable coefficients. Abstr. Appl. Anal. 2014, Article ID Fukagai, N, Kusano, T: Oscillation they of first der functional-differential equations with deviating arguments. Ann. Mat. Pura Appl. 136, Grammatikopoulos, MK, Koplatadze, R, Stavroulakis, IP: On the oscillation of solutions of first-der differential equations with retarded arguments. Gegian Math. J. 10, Kulenovic, MR, Grammatikopoulos, MK: First der functional differential inequalities with oscillating coefficients. Nonlinear Anal. 8, Ladas, G, Sficas, G, Stavroulakis, IP: Functional-differential inequalities and equations with oscillating coefficients. In: Trends in They and Practice of Nonlinear Differential Equations. Lecture Notes in Pure and Appl. Math., vol. 90, pp Dekker, New Yk Ladas, G, Stavroulakis, IP: Oscillations caused by several retarded and advanced arguments. J. Differ. Equ. 44, Li, X, Zhu, D, Wang, H: Oscillation f advanced differential equations with oscillating coefficients. Int. J. Math. Math. Sci. 33, Qian, C, Ladas, G, Yan, J: Oscillation of difference equations with oscillating coefficients. Rad. Mat. 8, Tang, XH, Cheng, SS: An oscillation criterion f linear difference equations with oscillating coefficients. J. Comput. Appl. Math. 132, Xianhua, T: Oscillation of first der delay differential equations with oscillating coefficients. Appl. Math. J. Chin. Univ. Ser. B 15, Yan, W, Yan, J: Comparison and oscillation results f delay difference equations with oscillating coefficients. Int. J. Math. Math. Sci. 19, Yu, JS, Tang, XH: Sufficient conditions f the oscillation of linear delay difference equations with oscillating coefficients. J. Math. Anal. Appl. 250, Yu, JS, Zhang, BG, Qian, XZ: Oscillations of delay difference equations with oscillating coefficients. J. Math. Anal. Appl. 177,