Iterative oscillation tests for difference equations with several non-monotone arguments
Elena Braverman, George E. Chatzarakis, Ioannis P. Stavroulakis
aa r X i v : . [ m a t h . D S ] F e b ITERATIVE OSCILLATION TESTS FOR DIFFERENCEEQUATIONS WITH SEVERAL NON-MONOTONE ARGUMENTS
E. BRAVERMAN , G. E. CHATZARAKIS, AND I. P. STAVROULAKIS Abstract.
We consider difference equations with several non-monotone deviating arguments andnonnegative coefficients. The deviations (delays and advances) are, generally, unbounded. Sufficientoscillation conditions are obtained in an explicit iterative form. Additional results in terms of lim infare obtained for bounded deviations. Examples illustrating the oscillation tests are presented.
Keywords : difference equations, non-monotone arguments, retarded arguments, advanced ar-guments, bounded delays, bounded advances, oscillation. : 39A10, 39A21 INTRODUCTION
The paper deals with the difference equation with several variable retarded arguments of theform ∆ x ( n ) + m X i =1 p i ( n ) x ( τ i ( n )) = 0, n ∈ N , (E R )where N is the set of nonnegative integers, and the (dual) difference equation with several variableadvanced arguments of the form ∇ x ( n ) − m X i =1 p i ( n ) x ( σ i ( n )) = 0, n ∈ N . (E A )Equations (E R ) and (E A ) are studied under the following assumptions: everywhere ( p i ( n )), 1 ≤ i ≤ m , are sequences of nonnegative real numbers , ( τ i ( n )), 1 ≤ i ≤ m , are sequences of integerssuch that either τ i ( n ) ≤ n − , ∀ n ∈ N , and lim n →∞ τ i ( n ) = ∞ , 1 ≤ i ≤ m , (1.1)or τ i ( n ) ≤ n − , ∀ n ∈ N and ∃ M i > n − τ i ( n ) ≤ M i , 1 ≤ i ≤ m (1.1 ′ )and ( σ i ( n )), 1 ≤ i ≤ m , are sequences of integers such that either σ i ( n ) ≥ n + 1 , ∀ n ∈ N , 1 ≤ i ≤ m , (1.2)or σ i ( n ) ≥ n + 1 , ∀ n ∈ N and ∃ µ i > σ i ( n ) − n ≤ µ i , 1 ≤ i ≤ m . (1.2 ′ )Here, ∆ denotes the forward difference operator ∆ x ( n ) = x ( n + 1) − x ( n ) and ∇ corresponds tothe backward difference operator ∇ x ( n ) = x ( n ) − x ( n − τ i ( n ) ≤ n − σ i ( n ) ≥ n + 1) is not essential for lim sup-typetests; all the results of Section 2 apply to the case when τ i ( n ) ≤ n and there is a non-delay term τ j ( n ) = n . However, the requirement lim n →∞ τ i ( n ) = ∞ is significant, as Example 2.2 illustrates.A similar remark applies to advanced difference equation (E A ). Corresponding author. E-mail [email protected], phone (403)-220-3956, fax (403)-282-5150 n the special case m = 1 equations (E R ) and (E A ) reduce to the equations∆ x ( n ) + p ( n ) x ( τ ( n )) = 0, n ∈ N (1.3)and ∇ x ( n ) − p ( n ) x ( σ ( n )) = 0, n ∈ N , (1.4)respectively.Set w = − min n ≥ ≤ i ≤ m τ i ( n ).Clearly, w is a finite positive integer if (1.1) holds.By a solution of (E R ), we mean a sequence of real numbers ( x ( n )) n ≥− w which satisfies (E R ) forall n ≥ . It is clear that, for each choice of real numbers c − w , c − w +1 , ..., c − , c , there exists aunique solution ( x ( n )) n ≥− w of (E R ) which satisfies the initial conditions x ( − w ) = c − w , x ( − w +1) = c − w +1 , ..., x ( −
1) = c − , x (0) = c .By a solution of (E A ), we mean a sequence of real numbers ( x ( n )) n ≥ which satisfies (E A ) for all n ≥ x ( n )) n ≥− w (or ( x ( n )) n ≥ ) of (E R ) (or (E A )) is called oscillatory, if the terms x ( n )of the sequence are neither eventually positive nor eventually negative. Otherwise, the solution issaid to be nonoscillatory . An equation is oscillatory if all its solutions oscillate.In the last few decades, the oscillatory behavior and the existence of positive solutions of differ-ence equations with several deviating arguments have been extensively studied, see, for example,papers [1-5,7-15] and references cited therein. More precicely, in 1999, Zhang et al. [14] studied theexistence and nonexistence of positive solutions and in 1999 and 2001, Tang et al. [10,11] considered(E R ) with constant delays, introduced some new techniques to analyze generalized characteristicequations and obtained several sharp oscillation conditions including infinite sums. In 2002, Zhanget al. [15] presented some oscillation criteria involving lim sup for (E R ) with constant delays, whileLi et al. [7] investigated only (E A ) under the assumption that the advances are constant and es-tablished a new oscillation condition involving lim inf. In 2003, Wang [12] considered (E R ) withconstant delays and obtained some new oscillation criteria involving lim sup, while Luo et al. [8]studied a nonlinear difference equation with several constant delays and established some sufficientoscillation conditions. In 2005, Yan et al. [13] got oscillation criteria for (E R ) with variable delays.In 2006, Berezansky et al. [1] mostly investigated non-oscillation but also obtained the followingsufficient oscillation test (Theorem 5.16): Iflim sup n →∞ m X i =1 p i ( n ) > n →∞ m X i =1 n − X j = τ ( n ) p i ( j ) > e ,where τ ( n ) = max ≤ i ≤ m τ i ( n ), ∀ n ≥
0, then all solutions of (E R ) oscillate. In the recent papers[3-5], Chatzarakis et al. studied equations (E R ) and (E A ) and presented some oscillation conditionsinvolving lim sup and lim inf.However, most oscillation results presented in the previous papers, require that the argumentsbe monotone increasing. While this condition is satisfied by a variety of differential equations withvariable delays, for difference equations, due to the discrete nature of the arguments, if retardedarguments are strictly increasing, then the deviations are eventually constant. This is one ofmotivations to investigate difference equations with non-monotone arguments. The challenge ofthis study is illustrated by the fact that, according to [2, Theorem 3], there is no constant A > uch that the inequalitieslim sup n →∞ ( n − τ ( n )) p ( n ) > A and lim inf n →∞ n − X j = τ ( n ) p i ( j ) > A guarantee oscillation of (1.3).The paper is organized as follows. Section 2 contains oscillation conditions for both retardedand advanced equations in terms of lim sup, where deviations of the argument, generally, are notassumed to be bounded. In Section 3 conditions in terms of lim inf are obtained in the case whendeviations of the argument are bounded; however, a weaker result is valid for unbounded delays.Section 4 illustrates the results of the present paper with examples where oscillation could not havebeen established using previously known results.2. UNBOUNDED DEVIATIONS OF ARGUMENTS AND ITERATIVE TESTS
In 2011, Braverman and Karpuz [2] and in 2014, Stavroulakis [10], established the followingresults for the special case of Eq. (1.3) under the assumption that the argument τ ( n ) is non-monotone, (1.1) holds and ξ ( n ) = max ≤ s ≤ n τ ( s ) . Theorem 2.1 ([2, Theorem 5]) . If lim sup n →∞ n X j = ξ ( n ) p ( j ) ξ ( n ) − Y i = τ ( j ) − p ( i ) > , (2.1) then all solutions of (1.3) oscillate. Theorem 2.2 ( [9, Theorem 3.6] ). Assume that a = lim inf n →∞ n − X i = τ ( n ) p ( i ) andlim sup n →∞ n X j = ξ ( n ) p ( j ) ξ ( n ) − Y i = τ ( j ) − p ( i ) > − c ( a ) , (2.2) where c ( a ) = (cid:16) − a − √ − a − a (cid:17) , if 0 < a ≤ /e, (cid:16) − a − √ − a + a (cid:17) , if 0 < a ≤ − √ p ( n ) ≥ α . Then all solutions of equation (1.3) oscillate.
In this section, we study oscillation of (E R ) and (E A ). We establish new sufficient oscillationconditions involving lim sup under the assumption that the arguments are non-monotone and (1.1)holds. Even for equation (1.3) with one delay, our results improve (2.1) and (2.2). The method weuse is based on the iterative application of the Gronwall inequality.2.1. Retarded difference equations.
Let ϕ i ( n ) = max ≤ s ≤ n τ i ( s ) , n ≥ ϕ ( n ) = max ≤ i ≤ m ϕ i ( n ) , n ≥
0. (2.4)Clearly, the sequences of integers ϕ ( n ), ϕ i ( n ), 1 ≤ i ≤ m are non-decreasing and ϕ ( n ) ≤ n − ϕ i ( n ) ≤ n −
1, 1 ≤ i ≤ m for all n ≥ he following simple result is cited to explain why we can consider only the case m X i =1 p i ( n ) < , ∀ n ≥
0. (2.5)
Theorem 2.3.
Assume that there exists a subsequence θ ( n ), n ∈ N of positive integers suchthat m X i =1 p i ( θ ( n )) ≥ , ∀ n ∈ N . (2.6) Then all solutions of (E R ) oscillate . Proof.
Assume, for the sake of contradiction, that ( x ( n )) n ≥− w is a nonoscillatory solution of (E R ).Then it is either eventually positive or eventually negative. As ( − x ( n )) n ≥− w is also a solution of(E R ), we may restrict ourselves only to the case where x ( n ) > n. Let n ≥ − w bean integer such that x ( n ) > n ≥ n . Then, there exists n ≥ n such that x ( τ i ( n )) > ∀ n ≥ n , 1 ≤ i ≤ m . In view of this, Eq.(E R ) becomes∆ x ( n ) = − m X i =1 p i ( n ) x ( τ i ( n )) ≤ ∀ θ ( n ) ≥ n ,which means that the sequence ( x ( n )) is eventually decreasing.Taking into account the fact that (2.6) holds, equation (E R ) gives x ( θ ( n ) + 1) = x ( θ ( n )) − m X i =1 p i ( θ ( n )) x ( τ i ( θ ( n ))) ≤ x ( θ ( n )) − x ( θ ( n )) m X i =1 p i ( θ ( n ))= x ( θ ( n )) − m X i =1 p i ( θ ( n )) ! ≤
0, for all θ ( n ) ≥ n , where θ ( n ) → ∞ as n → ∞ , which contradicts the assumption that x ( n ) > n ≥ n . (cid:3) In a series of further computations, we try to extend the iterative result, established by Ko-platadze and Kvinikadze in [6] for a differential equation with a single delay, to difference equationswith several retarded arguments. The following lemma provides an estimation of a positive solutionrate of decay, which is a useful tool for obtaining oscillation conditions.
Lemma 2.1.
Assume that (2.5) holds and x ( n ) is a positive solution of (E R ). Set a ( n, k ) := n − Y i = k " − m X ℓ =1 p ℓ ( i ) (2.7) and a r +1 ( n, k ) := n − Y i = k " − m X ℓ =1 p ℓ ( i ) a − r ( i, τ ℓ ( i )) , r ∈ N . (2.8) Then x ( n ) ≤ a r ( n, k ) x ( k ) , r ∈ N . (2.9) Proof.
Since x ( n ) is a positive solution of (E R ),∆ x ( n ) = − m X i =1 p i ( n ) x ( τ i ( n )) ≤ ∀ n ∈ N ,which means that the sequence ( x ( n )) is decreasing. rom (E R ) and the decreasing character of ( x ( n )) , we have∆ x ( n ) + x ( n ) m X i =1 p i ( n ) ≤ ∀ n ∈ N .Applying the discrete Gronwall inequality, we obtain x ( n ) ≤ x ( k ) n − Y i = k " − m X ℓ =1 p ℓ ( i ) = a ( n, k ) x ( k ) , ∀ n ≥ k ≥ r = 1.Assume that (2.9) holds for some r >
1. Substituting x ( τ ℓ ( n )) ≥ x ( n ) a − r ( n, τ ℓ ( n ))into (E R ) leads to the inequality∆ x ( n ) + m X ℓ =1 p ℓ ( n ) x ( n ) a − r ( n, τ ℓ ( n )) ≤
0, for sufficiently large n .Again, applying the discrete Gronwall inequality, we obtain x ( n ) ≤ x ( k ) n − Y i = k " − m X ℓ =1 p ℓ ( i ) a − r ( i, τ ℓ ( i )) = a r +1 ( n, k ) x ( k ) , due to the definition of a r +1 in (2.8), which concludes the induction step. The proof of the lemmais complete. (cid:3) It is interesting to see how the estimate developed in Lemma 2.1 works in the case of autonomousequations.
Example 2.1.
For the equation∆ x ( n ) + 14 x ( n −
1) = 0 , inequality (2.9) for k = n − x ( n ) ≤ a r ( n, n − x ( n − . Here a ( n, n −
1) = , a ( n, n −
1) = , a ( n, n −
1) = , a ( n, n −
1) = , a ( n, n −
1) = , . . . ,a k − = k +14 k , a k = k +12 k +1 , thus the best possible estimate of the rate of decrease of a nonoscillatorysolution is x ( n + 1) ≤ x ( n ), or x ( n ) ≤ x (0)( ) n , which is in compliance with the fact that λ = is the (double) root of the characteristic equation λ − λ + = 0. Theorem 2.4.
Assume that ( p i ( n )), 1 ≤ i ≤ m , are sequences of nonnegative real numbers, (1.1) and (2.5) hold, ϕ ( n ) is defined by (2.4) and a r ( n, k ) are denoted in (2.7),(2.8). If lim sup n →∞ n X j = ϕ ( n ) m X i =1 p i ( j ) a − r ( ϕ ( n ) , τ i ( j )) > , (2.10) then all solutions of (E R ) oscillate . Proof.
Assume, for the sake of contradiction, that ( x ( n )) n ≥− w is a nonoscillatory solution of (E R ).Then it is either eventually positive or eventually negative. Similarly to the proof of Theorem 2.3,we may restrict ourselves only to the case x ( n ) > n. Let n ≥ − w be an integer uch that x ( n ) > n ≥ n . By (1.1), there exists n ≥ n such that x ( τ i ( n )) > ∀ n ≥ n ,1 ≤ i ≤ m . In view of this, Eq.(E R ) becomes∆ x ( n ) = − m X i =1 p i ( n ) x ( τ i ( n )) ≤ ∀ n ≥ n ,which means that the sequence ( x ( n )) is eventually decreasing.Summing up (E R ) from ϕ ( n ) to n , and using the fact that the function x is non-increasing, whilethe function ϕ (as defined by (2.4)) is non-decreasing, and taking into account that τ i ( j ) ≤ ϕ ( n ) and x ( τ i ( j )) ≥ x ( ϕ ( n )) a − r ( ϕ ( n ) , τ i ( j )),we obtain, for sufficiently large n , x ( ϕ ( n )) = x ( n + 1) + n X j = ϕ ( n ) m X i =1 p i ( j ) x ( τ i ( j )) ≥ n X j = ϕ ( n ) m X i =1 p i ( j ) x ( τ i ( j )) ≥ x ( ϕ ( n )) n X j = ϕ ( n ) m X i =1 p i ( j ) a − r ( ϕ ( n ) , τ i ( j )).Consequently, x ( ϕ ( n )) − n X j = ϕ ( n ) m X i =1 p i ( j ) a − r ( ϕ ( n ) , τ i ( j )) ≤ n →∞ n X j = ϕ ( n ) m X i =1 p i ( j ) a − r ( ϕ ( n ) , τ i ( j )) ≤ (cid:3) To establish the next theorem we need the following lemma.
Lemma 2.2. (cf. [3, Lemma 2.1]) . Assume that (1.1) holds, ( x ( n )) is an eventually positivesolution of (E R ) , and α = min { α i : 1 ≤ i ≤ m } , (2.11) where α i = lim inf n →∞ n − X j = ϕ i ( n ) p i ( j ) . (2.12) If < α ≤ − √ then lim inf n →∞ x ( n + 1) x ( ϕ ( n )) ≥ − α − √ − α − α Theorem 2.5.
Assume that ( p i ( n )), 1 ≤ i ≤ m , are sequences of nonnegative real numbers, (1.1) and (2.5) hold, and ϕ ( n ) is defined by (2.4). If < α ≤ /e , where α is denoted by (2.11),and lim sup n →∞ n X j = ϕ ( n ) m X i =1 p i ( j ) a − r ( ϕ ( n ) , τ i ( j )) > − − α − √ − α − α where a r ( n, k ) is as in (2.7), (2.8), then all solutions of (E R ) oscillate . roof. Assume, for the sake of contradiction, that ( x ( n )) n ≥− w is a nonoscillatory solution of(E R ). Then, as in the proof of Theorem 2.4, for sufficiently large n , we obtain x ( ϕ ( n )) = x ( n + 1) + n X j = ϕ ( n ) m X i =1 p i ( j ) x ( τ i ( j )) ≥ x ( n + 1) + x ( ϕ ( n )) n X j = ϕ ( n ) m X i =1 p i ( j ) a − r ( ϕ ( n ) , τ i ( j )).that is, n X j = ϕ ( n ) m X i =1 p i ( j ) a − r ( ϕ ( n ) , τ i ( j )) ≤ − x ( n + 1) x ( ϕ ( n )) ,which gives lim sup n →∞ n X j = ϕ ( n ) m X i =1 p i ( j ) a − r ( ϕ ( n ) , τ i ( j )) ≤ − lim inf n →∞ x ( n + 1) x ( ϕ ( n )) .Assume that 0 < α ≤ /e (clearly, α < − √
2) and, by Lemma 2.2, inequality (2.13) holds, andso the last inequality leads tolim sup n →∞ n X j = ϕ ( n ) m X i =1 p i ( j ) a − r ( ϕ ( n ) , τ i ( j )) ≤ − − α − √ − α − α Remark 2.1.
Observe that conditions (2.1) and (2.2) are special cases of (2.10) and (2.14)respectively, when r = 1. Remark 2.2.
The following example illustrates the significance of the condition lim n →∞ τ i ( n ) = ∞ ,1 ≤ i ≤ m , in Theorems 2.4 and 2.5. Example 2.2.
Consider the retarded difference equation (1.3) with p ( n ) = (cid:26) / , if n = 3 k, / , if n = 3 k, τ ( n ) = (cid:26) − , if n = 3 k,n − , if n = 3 k, k ∈ N . Obviously (2.5) is satisfied. Also, by (2.4), ϕ ( n ) = max ≤ s ≤ n τ ( s ) = (cid:26) n − , if n = 3 k,n − , if n = 3 k, k ∈ N . If n = 3 k , then n X j = ϕ ( n ) p ( j ) = n X j = n − p ( j ) = 310 + 310 + 12 = 1110 . Therefore lim sup n →∞ n X j = ϕ ( n ) p i ( j ) ≥ > , which means that (2.10) is satisfied for any r .Also, if n = 3 k , then n − X j = ϕ ( n ) p i ( j ) = n − X j = n − p i ( j ) = 310 + 310 = 35 nd, if n = 3 k , then n − X j = ϕ ( n ) p ( j ) = n − X j = n − p ( j ) = (cid:26) / , if n = 3 k + 1 , / , if n = 3 k + 2 , k ∈ N . Therefore α = lim inf n →∞ n − X j = ϕ ( n ) p i ( j ) = min (cid:26) , , (cid:27) = 310 < e and n X j = ϕ ( n ) p i ( j ) = 1110 > − − α − √ − α − α ≃ . r . Observe, however, that equation (1.3) has a nonoscil-latory solution x ( −
1) = − , x (0) = 1 , x (1) = 137 , x (2) = 10970 ,x (3) = 1 , x (4) = 137 , x (5) = 10970 , x (6) = 1 , . . . which illustrates the significance of the condition lim n →∞ τ ( n ) = ∞ in Theorems 2.4 and 2.5.2.2. Advanced difference equations.
Similar oscillation theorems for the (dual) advanced dif-ference equation (E A ) can be derived easily. The proofs of these theorems are omitted, since theyfollow a similar procedure as in Subsection 2.1.Denote ρ i ( n ) = min s ≥ n σ i ( s ) , n ≥ ρ ( n ) = min ≤ i ≤ m ρ i ( n ) , n ≥
0. (2.16)Clearly, the sequences of integers ρ ( n ), ρ i ( n ), 1 ≤ i ≤ m are non-decreasing and ρ ( n ) ≥ n + 1, ρ i ( n ) ≥ n + 1, 1 ≤ i ≤ m for all n ≥ Theorem 2.3 ′ . Assume that there exists a subsequence θ ( n ) , n ∈ N of positive integers suchthat m X i =1 p i ( θ ( n )) ≥ ∀ n ∈ N . Then all solutions of (E A ) oscillate . Theorem 2.4 ′ . Assume that ( p i ( n )), 1 ≤ i ≤ m , are sequences of nonnegative real numbers, (1.2) and (2.5) hold, ρ ( n ) is defined by (2.16) and b ( n, k ) := k Y i = n +1 " − m X ℓ =1 p ℓ ( i ) (2.17)and b r +1 ( n, k ) := k Y i = n +1 " − m X ℓ =1 p ℓ ( i ) b − r ( i, σ ℓ ( i )) , r ∈ N . (2.18) f lim sup n →∞ ρ ( n ) X j = n m X i =1 p i ( j ) b − r ( ρ ( n ) , σ i ( j )) >
1, (2.19) then all solutions of (E A ) oscillate . Lemma 2.2 ′ (cf. [3, Lemma 2.1]). Assume that (1.2) holds, ( x ( n )) is a n eventually positivesolution of (E R ) , and α = min { α i : 1 ≤ i ≤ m } , (2.20) where α i = lim inf n →∞ ρ i ( n ) X j = n +1 p i ( j ) . (2.21) If < α ≤ − √ then lim inf n →∞ x ( n − x ( ρ ( n )) ≥ − α − √ − α − α Theorem 2.5 ′ . Assume that ( p i ( n )), 1 ≤ i ≤ m are sequences of nonnegative real numbers, (1.2) and (2.5) hold, ρ ( n ) is defined by (2.16) , and a is denoted by (2.20) . If < α ≤ /e , and lim sup n →∞ ρ ( n ) X j = n m X i =1 p i ( j ) b − r ( ρ ( n ) , σ i ( j )) > − − α − √ − α − α where b r ( n, k ) is defined in (2.17), (2.18), then all solutions of (E A ) oscillate .3. BOUNDED DEVIATIONS OF ARGUMENTS
In this section, new sufficient oscillation conditions, involving lim inf, under the assumption thatall the delays (advances) are bounded and (1.1 ′ ) (cid:0) or (1.2 ′ ) (cid:1) holds, are established for equations(E R ) and (E A ). These conditions improve Theorem 5.16 in [1] and extend the following oscillationresults by Chatzarakis et al. [5] to the case of non-monotone arguments. Theorem 3.1 ( [5, Theorem 2.2] ). Assume that the sequences ( τ i ( n )) , ≤ i ≤ m are increasing, (1.1) holds, lim sup n →∞ m X i =1 p i ( n ) > n →∞ m X i =1 n − X j = τ i ( n ) p i ( j ) > e . (3.1) Then all solutions of Eq. (E R ) oscillate. Theorem 3.2 ( [5, Theorem 3.2] ). Assume that the sequences ( σ i ( n )) , ≤ i ≤ m are increasing, (1.2) holds, lim sup n →∞ m X i =1 p i ( n ) > n →∞ m X i =1 σ i ( n ) X j = n +1 p i ( j ) > e . (3.2) Then all solutions of Eq. (E A ) oscillate. .1. Retarded difference equations.
We present a new sufficient oscillation condition for (E R ),under the assumption that all the delays are bounded and (1.1 ′ ) holds. Theorem 3.3.
Assume that ( p i ( n )), 1 ≤ i ≤ m , are sequences of nonnegative real numbers, (1.1 ′ ) holds and for each k = 1 , . . . , m ,lim inf n →∞ m X i =1 n − X j = τ k ( n ) p i ( j ) > (cid:18) M k M k + 1 (cid:19) M k +1 , (3.3) then all solutions of (E R ) oscillate.Proof. Assume, for the sake of contradiction, that ( x ( n )) n ≥− w is a nonoscillatory solution of (E R ).Without loss of generality, we can assume that x ( n ) > n ≥ n . Then, there exists n ≥ n such that x ( τ i ( n )) > ∀ n ≥ n , 1 ≤ i ≤ m . In view of this, Eq.(E R ) becomes∆ x ( n ) = − m X i =1 p i ( n ) x ( τ i ( n )) ≤ ∀ n ≥ n ,which means that the sequence ( x ( n )) n ≥ n is non-increasing. The sequences b k ( n ) = (cid:18) n − τ k ( n ) n − τ k ( n ) + 1 (cid:19) n − τ k ( n )+1 satisfy the inequality 14 ≤ b k ( n ) ≤ (cid:18) M k M k + 1 (cid:19) M k +1 , n ≥ , ≤ k ≤ m. (3.4)Due to (3.3), for each k = 1 , . . . , m , we can choose n ( k ) ≥ n and ε k > n ≥ n ( k ), m X i =1 n − X j = τ k ( n ) p i ( j ) > (cid:18) M k M k + 1 (cid:19) M k +1 + ε k and d k := (cid:18) M k M k + 1 (cid:19) − M k − "(cid:18) M k M k + 1 (cid:19) M k +1 + ε k > . Denoting ε := min ≤ k ≤ m ε k > , d := min ≤ k ≤ m d k > , we obtain m X i =1 n − X j = τ k ( n ) p i ( j ) > (cid:18) M k M k + 1 (cid:19) M k +1 + ε and (cid:18) M k M k + 1 (cid:19) − M k − "(cid:18) M k M k + 1 (cid:19) M k +1 + ε ≥ d > , which together with (3.3) immediately implies for any k = 1 , . . . , m m X i =1 n − X j = τ k ( n ) p i ( j ) b k ( n ) ≥ (cid:18) M k M k + 1 (cid:19) − M k − m X i =1 n − X j = τ k ( n ) p i ( j ) > d > . ividing (E R ) by x ( n ) we have x ( n + 1) x ( n ) = 1 − m X i =1 p i ( n ) x ( τ i ( n )) x ( n ) , n ≥ n . Multiplying and taking into account that x ( τ i ( n )) /x ( n ) ≥
1, we obtain the estimate x ( n ) x ( τ k ( n )) = n − Y j = τ k ( n ) x ( j + 1) x ( j ) ≤ n − Y j = τ k ( n ) − m X i =1 p i ( j ) ! . Using this estimate and the relation between the arithmetic and the geometric means, we have x ( n ) x ( τ k ( n )) ≤ − n − τ k ( n ) m X i =1 n − X j = τ k ( n ) p i ( j ) n − τ k ( n ) . (3.5)Observe that the function f : (0 , → R defined as f ( y ) := y (1 − y ) ρ , ρ ∈ N , attains its maximum at y = ρ , which equals f max = ρ ρ (1 + ρ ) ρ . In the inequality y (1 − y ) ρ ≤ ρ ρ (1 + ρ ) ρ , y ∈ (0 , , ρ ∈ N ,assuming ρ = n − τ k ( n ), y = x/ρ , where x = m X i =1 n − X j = τ k ( n ) p i ( j ), we obtain from (3.5) x ( τ k ( n )) x ( n ) ≥ m X i =1 n − X j = τ k ( n ) p i ( j ) (cid:18) n − τ k ( n ) + 1 n − τ k ( n ) (cid:19) n − τ k ( n )+1 = m X i =1 n − X j = τ k ( n ) p i ( j ) b k ( n ) > d for any n ≥ n ( k ). Denote n = max { n (1) , . . . , n ( m ) } . If we continue this procedure assuming n ≥ n + M , where M = max ≤ i ≤ m M i , and using the properties of the geometric and the algebraicmean, we have x ( n ) x ( τ k ( n )) = n − Y j = τ k ( n ) x ( j + 1) x ( j ) = n − Y j = τ k ( n ) − m X i =1 p i ( j ) x ( τ i ( j )) x ( j ) ! ≤ n − Y j = τ k ( n ) − d m X i =1 p i ( j ) ! ≤ − dn − τ k ( n ) m X i =1 n − X j = τ k ( n ) p i ( j ) n − τ k ( n ) . Applying the same argument, we obtain x ( τ k ( n )) x ( n ) ≥ d m X i =1 n − X j = τ k ( n ) p i ( j ) b k ( n ) > d , n ≥ n + 2 M, k = 1 , . . . , m,x ( τ k ( n )) x ( n ) > d r , n ≥ n + rM, k = 1 , . . . , m. Due to (3.3), we observe that for any k lim sup n →∞ m X i =1 p i ( n ) ≥ M k (cid:18) M k M k + 1 (cid:19) M k +1 . ince the function h ( x ) := 1 x x x +1 ( x + 1) x +1 = x x ( x + 1) x +1 , x ≥
1, is decreasing and M k ≤ M , weconclude that lim sup n →∞ m X i =1 p i ( n ) ≥ c := 1 M (cid:18) MM + 1 (cid:19) M +1 (3.6)and choose a subsequence ( θ ( n )) of N such that m X i =1 p i ( θ ( n )) ≥ c > . Since 0 < x ( n + 1) x ( n ) = 1 − m X i =1 p i ( n ) x ( τ i ( n )) x ( n ) , we have m X i =1 p i ( n ) x ( τ i ( n )) x ( n ) < . In particular, min ≤ k ≤ m x ( τ k ( θ ( n ))) x ( θ ( n )) m X i =1 p i ( θ ( n )) < . Choosing r ∈ N such that d r > c , where c was defined in (3.6), θ ( n ) ≥ n + rM and noticing that d r < min ≤ k ≤ m x ( τ k ( θ ( n ))) x ( θ ( n )) ≤ m X i =1 p i ( θ ( n )) ! − ≤ c , we obtain a contradiction, which concludes the proof. (cid:3) The following result is valid as (cid:18) nn + 1 (cid:19) n +1 < e . In (3.7) a non-strict inequality is also sufficient. Theorem 3.4.
Assume that ( p i ( n )), 1 ≤ i ≤ m , are sequences of nonnegative real numbersand (1.1) holds. If lim inf n →∞ m X i =1 n − X j = τ k ( n ) p i ( j ) > e , , k = 1 , . . . , m, (3.7) then all solutions of (E R ) oscillate. Similar oscillation theorems for the (dual) advanceddifference equation (E A ) can be derived easily. The proofs of these theorems are omitted, sincethey follow the schemes of Subsection 3.1. Theorem 3.3 ′ . Assume that ( p i ( n )), 1 ≤ i ≤ m , are sequences of nonnegative real numbers, allthe advances are bounded, (1.2 ′ ) holds. lim inf n →∞ m X i =1 σ k ( n ) X j = n +1 p i ( j ) > (cid:18) µ k µ k + 1 (cid:19) µ k +1 , k = 1 , . . . , m, (3.8) then all solutions of (E A ) oscillate. heorem 3.4 ′ . Assume that ( p i ( n )), 1 ≤ i ≤ m , are sequences of nonnegative real numbers,all the advances are bounded and (1.2) holds. If lim inf n →∞ m X i =1 σ k ( n ) X j = n +1 p i ( j ) > e , k = 1 , . . . , m, (3.9) then all solutions of (E A ) oscillate. EXAMPLES
In this section, we present examples illustrating the significance of our results. Observe that mostof the relevant oscillation results cited in the introduction cannot be applied because they assumethat the delays (advances) are constant and consequently the deviating arguments are increasing.When possible, we compare our results to the known ones, for variable deviations of the argumentand non-monotone arguments.
Example 4.1.
Consider the retarded difference equation∆ x ( n ) + px ( τ ( n )) = 0 , n ∈ N , (4.1)where p < τ ( n ) = n − k − , if n = 3 kn − k − , if n = 3 k + 1 n − k − , if n = 3 k + 2 , k ∈ N .Obviously (1.1) and (2.5) are satisfied. Also, by (2.4) we have ϕ ( n ) = max ≤ s ≤ n τ ( s ) = n − k − , if n = 3 kn − k − , if n = 3 k + 1 n − k − , if n = 3 k + 2 , k ∈ N .If n = 3 k , then ϕ ( n ) = n − τ ( n ) and, in view of (2.7), the left-hand side in (2.10) is n X j = ϕ ( n ) p ( j ) a − ( ϕ ( n ) , τ ( j )) = n X j = n − p ( j ) a − ( n − , τ ( j ))= p a ( n − , τ ( n − p a ( n − , τ ( n ))= p a ( n − , n −
5) + p a ( n − , n − p n − Y i = n − (cid:16) − p a ( i,τ ( i )) (cid:17) + p , here n − Y i = n − (cid:18) − p a ( i, τ ( i )) (cid:19) = (cid:18) − p a ( n − , τ ( n − (cid:19) (cid:18) − p a ( n − , τ ( n − (cid:19) ×× (cid:18) − p a ( n − , τ ( n − (cid:19) (cid:18) − p a ( n − , τ ( n − (cid:19) = (cid:18) − pa ( n − , n − (cid:19) (cid:18) − pa ( n − , n − (cid:19) (cid:18) − pa ( n − , n − (cid:19) (cid:18) − pa ( n − , n − (cid:19) = (cid:18) − pa ( n − , n − (cid:19) (cid:18) − pa ( n − , n − (cid:19) (cid:18) − pa ( n − , n − (cid:19) (cid:18) − pa ( n − , n − (cid:19) = − p (cid:18) − p − p (1 − p )4 (cid:19) (cid:18) − p − p − p (cid:19) − p − p − p (1 − p )4 ×× − p (cid:18) − p − p (1 − p )2 (cid:19) (cid:18) − p − p (1 − p )4 (cid:19) (cid:18) − p − p − p (cid:19) Therefore n X j = ϕ ( n ) p ( j ) a − ( ϕ ( n ) , τ ( j )) = p ++ p − p − p − p (1 − p )4 !(cid:18) − p − p − p (cid:19) − p − p − p (1 − p )2 ! − p − p (1 − p )4 !(cid:18) − p − p − p (cid:19) − p − p − p (1 − p )4 ! . The computation immediately implies that, if p ∈ (0 . , . n →∞ n X j = ϕ ( n ) p ( j ) a − ( ϕ ( n ) , τ ( j )) > p = 0 .
175 inequality (2.10) holds for r = 3, which means that all solutions of (4.1)oscillate. Nevertheless, for p ∈ (0 . , . r = 3, sincelim sup n →∞ n X j = ϕ ( n ) p ( j ) a − ( ϕ ( n ) , τ ( j )) > − (cid:16) − p − p − p − p (cid:17) .Observe, however, that a = lim inf n →∞ n − X i = τ ( n ) p ( i ) = lim inf n →∞ n − X i = n − p ( i ) = p and c ( a ) = 12 (cid:16) − a − p − a − a (cid:17) = 12 (cid:16) − p − p − p − p (cid:17) . lso, n X j = ϕ ( n ) p ( j ) ϕ ( n ) − Y i = τ ( j ) − p ( i ) = p − p ) + p, if n = 3 kp − p ) + 2 p, if n = 3 k + 1 p − p ) + 2 p + p − p , if n = 3 k + 2 , k ∈ N . Thusmax (cid:26) p − p ) + p, p − p ) + 2 p, p − p ) + 2 p + p − p (cid:27) = p − p ) + 2 p + p − p .If p < . p − p ) + 2 p + p − p ≤ p < . p − p ) + 2 p + p − p ≤ − (cid:16) − p − p − p − p (cid:17) .Therefore, if p ∈ (0 . , . p ∈ (0 . , . r = 3 while (2.10 )and also (2.1) and (2.2) are not satisfied. For example, for p = 0 . r = 3.At this point we should remark that in condition (2.14) an extra requirement is needed: 0 < α ≤ /e. Thus, when α → Example 4.2.
Consider the delay difference equation∆ x ( n ) + 18 x ( τ ( n )) + 112 x ( τ ( n )) = 0 , n ≥
0, (4.2)with τ ( n ) = (cid:26) n − n is even n − n is odd and τ ( n ) = (cid:26) n − n is even n − n is odd .Here, it is clear that (1.1) and (2.5) are satisfied. Also, by (2.3) and (2.4) we have ϕ ( n ) = ϕ ( n ) = (cid:26) n − n is even n − n is oddand ϕ ( n ) = max ≤ i ≤ ϕ i ( n ) = (cid:26) n − n is even n − n is odd .If n is even, then ϕ ( n ) = n − τ ( n ) = n − τ ( n ) = n − n X j = ϕ ( n ) m X i =1 p i ( j ) a − ( ϕ ( n ) , τ i ( j )) = n X j = n − X i =1 p i ( j ) a − ( n − , τ i ( j ))= 18 · " (cid:0) − (cid:1) + 1 + 11 − + 112 · " (cid:0) − (cid:1) + 1 + 1 (cid:0) − (cid:1) ≃ . n X j = ϕ ( n ) m X i =1 p i ( j ) a − ( ϕ ( n ) , τ i ( j )) = n X j = n − X i =1 p i ( j ) a − ( n − , τ i ( j )) / h − · − − · − i (cid:20) − · ( − ) − · ( − ) (cid:21) + 1 / (cid:20) − · ( − ) − · ( − ) (cid:21) h − · − − · − i + 18 + 112 + 1 / h − · − − · − i + 1 / (cid:20) − · ( − ) − · ( − ) (cid:21) h − · − − · − i i.e., n X j = ϕ ( n ) m X i =1 p i ( j ) a − ( ϕ ( n ) , τ i ( j )) ≃ . n →∞ n X j = ϕ ( n ) m X i =1 p i ( j ) a − ( ϕ ( n ) , τ i ( j )) ≥ . > a = lim inf n →∞ n − X i = ϕ ( n ) p ( i ) = 18 and a = lim inf n →∞ n − X i = ϕ ( n ) p ( i ) = 112and therefore α = min { α i : 1 ≤ i ≤ } = 112 . Hencelim sup n →∞ n X j = ϕ ( n ) m X i =1 p i ( j ) a − ( ϕ ( n ) , τ i ( j )) > − (cid:16) − a − p − a − a (cid:17) ≃ . n − τ ( n ) ≤ M and n − τ ( n ) ≤ M and therefore M = max ≤ i ≤ M i = 4.Observe that lim inf n →∞ X i =1 n − X j = τ i ( n ) p i ( j ) = 18 + 112 = 524 < (cid:18) MM + 1 (cid:19) M +1 = 0 . Figure 1: Two solutions of (4.2) for (left) − ≤ n ≤ ≤ n ≤
65, (right) 60 ≤ n ≤ Example 4.3.
Consider the delay difference equation∆ x ( n ) + a ( n ) x ( τ ( n )) + 1125 x ( τ ( n )) = 0 , n ≥
0, (4.3)with τ ( n ) = (cid:26) n − n is even, n − n is odd, τ ( n ) = (cid:26) n − n is even, n − n is odd, a ( n ) = if n is even, if n is odd.Evidently n − τ ( n ) ≤ M and n − τ ( n ) ≤ M and therefore for k = 1lim inf n →∞ X i =1 n − X j = τ ( n ) p i ( j ) = 37125 + 1125 = 38125 = 0 . > (cid:18) M M + 1 (cid:19) M +1 ≈ . , while for k = 2lim inf n →∞ X i =1 n − X j = τ ( n ) p i ( j ) = 37125 + 3125 + 2125 = 0 . > (cid:18) M M + 1 (cid:19) M +1 ≈ . Acknowledgments
The first author was partially supported by NSERC, grants RGPIN/261351-2010 and RGPIN-2015-05976.
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E-mail address : [email protected] Department of Electrical and Electronic Engineering Educators, School of Pedagogical andTechnological Education (ASPETE), 14121, N. Heraklio, Athens, Greece
E-mail address : [email protected], [email protected] Department of Mathematics, University of Ioannina, 451 10 Ioannina, Greece
E-mail address : [email protected]@cc.uoi.gr