Oscillation of difference causal operator equations

New sufficient conditions are provided to guarantee that the (nontrivial) solutions of a discrete equation of the form ∆x(n) + F (n, x) = 0, n = 0, 1, 2, · · · , (where F (n, ·) is a causal operator) either oscillate, or converge monotonically to zero.

We denote by s(n), t(n) the infimum and the supremum, respectively, of the set M n satisfying the condition (C1). The set M n is the memory, while the interval [t(n), n] is the amnesia of the equation. In discrete equations the set M n consists of integer numbers. For technical reasons we shall consider the integer τ (n) := min{j ∈ N : t(n) ≤ j}. Also, we shall denote by I n the interval [s(n), τ (n)] and let k n := n − τ (n). Now let us see how the machine (2) works. Given a initial function φ : [s(0), 0] → R, we find x(1) by the relation x(1) = φ(0)+F (0; φ). Next, in order to obtain the value x(2) we need to know the values of x on the set I 1 ∪ {1}. Inductively we need obtain all values x(n). But here there is a problem: In order to go from n to n+1 and it holds [(−∞, n) \ Z] ∩ [∪ j≤n I(j)] = ∅, we have to know some values of x in the open intervals between the integers which are less than or equal to n. To overcome these uncertainties one could give any value of x on these intervals (and produce an uncountable set of solutions), but, for our purpose, we suggest that the values of x between the integers are defined as convex combination of the ends of the corresponding intervals. Namely, for our convenience, we accept x to be linear between the positive integers and so it must have the value x((1 − λ)(n − 1) + λn) := (1 − λ)x(n − 1) + λx(n), for all λ ∈ [0, 1] and n = 1, 2, · · · Apart of the fact that the most usual and traditional approximation of a C 1 -arc is its chord, the suggestion to use "straight bridges" agrees with the following two facts: 1) Monotonicity of x on the interval [0, +∞) is equivalent with the same kind of monotonicity of the sequence x(0), x(1), · · · . 2) Oscillation of x in the usual sense 2 , is equivalent with the oscillation of the sequence x(0), x(1), · · · To shorten the text we shall say that a function has the property (P) if it is either oscillating, or it converges monotonically to zero.
Our purpose in this work is to provide sufficient conditions for all solutions of equation (2) to have property (P). To our knowledge, this general class of discrete equations (which, obviously, contains the discrete equations) is the first time introduced for further study. Difference and discrete equations were studied by a great number of authors. See, e.g., [1] - [27] and the references therein. There are, moreover, very good books on difference equations, as, for instance, [1,3,9,11]. In particular the book [3] presents an extensive list of references on the subject.
Searching the literature we can find a great number of sufficient conditions which guarantee oscillation, concerning even higher order difference equations, see, e.g., [26,27] and the references therein. Let us focus on conditions which guarantee the fact that all solutions of the linear difference equations (which is a special case of equation (2)), are oscillating. Information about these conditions can be found elsewhere, see, e.g. [24]. Nevertheless, we focus our attention on the first sufficient condition for oscillation of all solutions of (3), which were given by Ladas et al. [15]: This condition was extended in [20] concerning the equation ∆x(n) + p(n)x(h(n)) = 0, where h(n) is a increasing integer-delay-sequence converging to +∞. This condition was improved in [21].
Concerning the difference equation with several delays for oscillation of all solutions were given in [10]. (Here k i := n−τ i (n), i = 1, 2, · · · , m.) In this work we state and prove two theorems: In the first one we give a sufficient condition which resembles to (4) and in the second one we give a new sufficient condition for the property (P). In the last section enough applications are presented to show that these two conditions are independent and they do not imply each other.

The main results
Consider the difference equation (2) and assume that the response operator F satisfies the following basic condition: (C2) There is a sequence (b(n)) of positive real numbers, such that It is clear that if F (n; ·) is a positive linear operator then it satisfies condition (C2). Other examples of such operators will be given in the last section. There are cases where only one of these conditions is true. For example, the function F (n; x) := n + x(n − 2)e x(n−1) , satisfies (6), (with b n := 1,) but not (7).
We observe that these two mathematical relations can be written in a unified form as min{F (n; Therefore, if F (n, ·) is odd, then the two conditions are equivalent.
The first result of this paper concerns an extension of the validity of condition (4) to this general setting. (2), where the operator F satisfies conditions (C1) and (C2). Assume that it holds τ (n) ≤ n − 1, for all n and the sequence (k n ) is bounded and moreover lim inf n s(n) = +∞.

Theorem 2.1 Consider equation
If the condition is satisfied, then any solution has property (P).
Proof: Assume that x is a (nontrivial) non-oscillating solution of equation (2). If x is eventually negative, the second relation in (C2) is satisfied for all large n. Then the function y := −x is eventually positive and it satisfies the difference equation ∆y(n) + F * (n, y) = 0, where the new function F * (n, u) := −F (n, −u), obviously, satisfies the first condition in (C2) for all large n. Thus, we can assume that x is eventually positive, i.e. there is some t 0 such that x(r) ≥ 0, for all r ≥ t 0 . Due to (8), there is some t 1 such that t 0 ≤ s(t), for all t ≥ t 1 . Then, for all n ≥ n 1 := [t 1 ] + 1 and r ∈ I n , we have x(r) > 0, and therefore This relation implies that the solution x is an eventually monotonically decreasing function and therefore the following inequalities hold: From (2) and the first condition in (C2) we get Summing up this inequality by parts we obtain for all n ≥ n 2 , where n 2 is a positive integer such that n 2 ≥ min{j : s(j) ≥ n 1 }.
By using the Arithmetic Mean-Geometric Mean inequality we obtain Then form (13) we get Therefore we have lim inf n→+∞ This relation and (9) permit us to choose a constant γ ∈ (0, 1) such that lim inf Thus there is some n 3 ≥ n 2 such that for all n ≥ n 3 it holds Assume that x does not converge to 0. Hence, due to its monotonicity, it must stay away from zero, i.e. there is some µ > 0 such that x(r) ≥ µ, for all r ≥ n 3 . Letting n 4 ≥ n 3 such that s(n) ≥ n 3 , for all n ≥ n 4 , we have t(n) ≥ n 3 , for all n ≥ n 4 . Then, due to (15), the quantity ζ := lim inf n→+∞ Obviously we have 1 1−γ =: δ > 1 and from (17) it follows that because, due to our assumptions, it holds k n ≥ 1. This relation implies that the number ζ belongs to the bounded interval (1, x(τ (n 3 )) µ ].
Then, for some n 5 ≥ n 4 and for all n ≥ n 5 , we have On the other hand, for some subsequence (n m ) converging to +∞, it holds for all m. From (12) and (19), we get that Summing up by parts this inequality we obtain for all n ≥ n 6 , where n 6 is a positive integer such that for all n ≥ n 6 it holds s(n) ≥ n 5 .
By using (21), the Arithmetic Mean-Geometric Mean inequality (14) and relation (17), we obtain From (22) we conclude that 1 − ε < ζ − ε < 1 γ . Then from (20) for all large m, where where −ε ≤ c ≤ 1 γ + ε. The function ψ m is positive on the interior of the interval of its definition and it vanishes at the end points. Notice that the number 1 belongs to this set. The first derivative of ψ m vanishes only at the point Therefore ψ m takes its maximum at this point. Assuming that θ 0 < 1, the function ψ m would be decreasing on the interval [1, 1 γ + ε]. Then, it is clear that, its maximum on this interval is achieved at the point 1 and it must be equal to But this quantity is smaller than or equal to which, due to the second condition in (18), is smaller than 1. Hence we must have θ 0 ≥ 1 and then Since the sequence (k n ) is bounded we have Thus, from (23), (24) and (25) If there is a term B λ of the sequence defined by then property (P) keeps in force.
Proof: Assume that x is a nonoscillating solution. Then, again, as in Theorem (2.1), we can assume that x is an eventually positive function satisfying relation (11). Let us define which, in case the solution does not converge to zero, is bounded above. Obviously, z ≥ 1. Let 0 < ε << 1, be small enough. Corresponding to this ε, there is some n(ε) 0 such that x(τ (n)) for all n ≥ n 0 (ε). Then due to (2), (11) and (6) we have for all n ≥ n 0 (ε). So, there is some n 1 (ε), such that for all n ≥ n 1 (ε) it holds s(n) ≥ n 0 (ε). Then, for all n ≥ n 1 (ε), we get Therefore it holds and so By the monotonicity of the logarithm and the fact that z ≥ 1, we obtain ln z ≥ lim inf n→+∞ n j=τ (n) Next we are going to eliminate the parameter ε from (27) and (28). (For any future reference to this fact we call the procedure Main Step.) To do that, define the sequences a ε,n := n j=τ (n) ln(1 + (1 − ε)b j ) and a n := n j=τ (n) and observe that 0 ≤ |a ε,n − a n | = a n − a ε,n = n j=τ (nn) This fact together with (26) implies that lim ε→0 (a ε,n − a n ) = 0, uniformly for all large n. Therefore, given δ > 0 there is ε 0 such that, for all ε ∈ (0, ε 0 ) it holds 0 ≤ a n − a ε,n ≤ δ, uniformly for all large n and so, for all such ε, it holds 0 ≤ lim inf n→+∞ [a n − a ε,n ] ≤ δ. This relation means that lim ε→0 lim inf n→+∞ [a n − a ε,n ] = 0 (29) and so, from (28), we can get [a ε,n − a n ] + lim inf ε→0 lim inf n→+∞ a n .
Therefore, because of (29), we conclude that ln z ≥ lim inf n→+∞ a n , which implies that and the Basic Step is done. Now, we observe that relations (30) and (27) give which, by following the Basic Step, as previously, implies that z ≥ lim inf n→+∞ n j=τ (n) Continue in this way and obtain a increasing sequence (B i ), with B 0 = 1, such that for any i = 0, 1, 2, · · · Now, fix a λ ∈ {0, 1, · · · }. By using the fact that z ≥ B λ , from Due to the strict inequality, we can take a constant number c ∈ (0, 1) such that Therefore it holds Therefore we have Getting the limits in both parts as ε tends to zero and following the previous Basic Step we can eliminate ε, and finally, obtain contrary to (E λ ). The proof is complete.

Applications
First we give four examples where both theorems are applicable.
Application 3.1 Motivated from [7] we consider the equation Now we see that τ (n) = n − 1, k n = k = 1 and b n = 1 e . Thus condition (9) is satisfied and so all solutions have property (P). On the other hand condition (2.3) of [7] is not true and therefore Theorem 2.1 of [7] is not applicable. Notice that condition (E λ ) is also satisfied for at least λ = 1. Here we have l > 0, τ (n) = n − 1, b n = 1 a and k(n) = k = 1. Observe that condition (9) takes the form 1 a > 1 4 and it is true for a < 4. Also condition (E λ ), for λ = 0, holds for a < 1 + √ 2. We can observe that when we increase the parameter a but keeping it smaller than 4, condition (E λ ) holds for large values of λ.
Next we give a difference equation where Theorem 2.2 is applicable, but not Theorem 2.1. So property (P) is satisfied.
Application 3.6 Finally we shall discuss the simple linear difference equation x n+1 − x n + px n−1 = 0, where p > 0. Here we have τ (n) = n − 1, k n = 1 and b n = p. Let us compute the first terms of the sequence (B l ). Indeed, we obtain that relation (E i ), for i = 0, 1, 2, · · · , 35 holds as equality for the values of the parameter p described in the following tabular: In this tabular we see that the sequence of the minimum values of p approaches the value 0, 25, which is the exact value of p generating oscillatory solutions. Indeed, as it is known from the basic theory of linear difference equations, if the discriminant of the characteristic equation x 2 − x + px = 0 is negative, the solutions are oscillatory