GEORGIAN MATHEMATICAL JOURNAL: Vol. 5, No. 1, 1998, 49-54

MONOTONE SOLUTIONS OF A HIGHER ORDER NEUTRAL DIFFERENCE EQUATION

SUI SUN CHENG AND GUANG ZHANG

Abstract. A real sequence {x_{k}} is said to be (∗)-monotone with respect to a sequence {p_{k}} and a positive integer σ if x_{k} > 0 and (−1)^{n}Δ^{n}(x_{k} − p_{k}x_{k−σ}) ≥ 0 for n ≥ 0. This paper is concerned with the existence of (∗)-monotone solutions of a neutral difference equation. Existence criteria are derived by means of a comparison theorem and by establishing explicit existence criteria for positive and/or bounded solutions of a majorant recurrence relation.

§ 1. Introduction

A sequence {x_{k}} is said to be (∗)-monotone if it satisfies

x_{k} > 0, Δx_{k} ≥ 0, Δ^{2}x_{k} ≤ 0, Δ^{3}x_{k} ≥ 0, …, (−1)^{n}Δ^{n+1}x_{k} ≥ 0.

A typical example is the sequence defined by x_{k} = 1 − λ^{k}, 0 < λ < 1, k = 0, 1, 2, . . .. Given a positive integer σ and a sequence {p_{k}}, we can generalize the concept of a (∗)-monotone sequence {x_{k}} by requiring

x_{k} > 0, x_{k} − p_{k}x_{k−σ} ≥ 0,
Δ (x_{k} − p_{k}x_{k−σ}) ≤ 0, …, (−1)^{n}Δ^{n} (x_{k} − p_{k}x_{k−σ}) ≥ 0.

Such a sequence will again be called (∗)-monotone with respect to the integers n, σ and the sequence {p_{k}}.

This note is concerned with the (∗)-monotone solutions of a class of nonlinear recurrence relations of the form

Δ^{n} (y_{k} − p_{k}y_{k−σ}) + q_{k}f(y_{k−τ}) = 0, k = 0, 1, 2, …,  (1)

where n is a positive odd integer, σ is a positive integer, τ is a non-negative integer, {p_{k}} and {q_{k}} are non-negative sequences such that {q_{k}} does not vanish identically for all large k, and f is a real function defined on R such

49 that f is positive nondecreasing for x > 0. The forward difference operator is defined as usual, i.e., Δx_{k} = x_{k+1} − x_{k}.

1991 Mathematics Subject Classification. 39A10.

Key words and phrases. Neutrai difference equation, positive solution, monotone solution, existence theorem.

1072-947X/98/0100-0049$12.50/0 \begin{matrix}c \\ ◯ \end{matrix} 1998 Plenum Publishing Corporation

Let μ = max{σ, τ}. Then by a solution of (1) we mean a real sequence {y_{k}} which is defined for k ≥ −μ and which satisfies equation (1) for k ≥ 0. By writing equation (1) in the form of a recurrence relationy_{n+k} = F (y_{n+k−1}, …, y_{k}, y_{k−σ}, y_{k−τ}) it is clear that an existence and uniqueness theorem for the solutions of (1) satisfying appropriate initial conditions can easily be formulated and proved by induction. A solution {y_{k}} of (1) is said to be eventually positive if y_{k} > 0 for all large k, and eventually negative if y_{k} < 0 for all large k. It is said to be oscillatory if it is neither eventually positive nor eventually negative. Finally, it is said to be eventually (∗)-monotone if it is (∗)-monotone for all large k.

We will be concerned with the existence of eventually (∗)-monotone solutions of (1). Similar problems related to both differential and difference equations have been considered in [1–4].

§ 2. Comparison Theorem

We first establish a Sturmian type criterion which has not been explored before. To be more precise, we will assume the existence of an eventually (∗)-monotone solution of a majorant relation of the form

Δ^{n} (x_{k} − P_{k}x_{k−σ}) + Q_{k}F (x_{k−τ}) ≤ 0, k = 0, 1, 2, …,  (2)

and then show that (1) also has an eventually (∗)-monotone solution. The correct assumptions will be stated later, for now, we will assume in the sequel that the sequences {P_{k}} and {Q_{k}}, and the function F satisfy the same assumptions that have respectively been imposed on {p_{k}}, {q_{k}} and f.

Let {x_{k}} be an eventually (∗)-monotone solution of (2) such that x_{k} > 0 for k ≥ N − μ and the sequence {z_{k}} defined by

z_{k} = x_{k} − P_{k}x_{k−σ}, k ≥ 0,  (3)

satisfiesz_{k} ≥ 0, Δz_{k} ≤ 0, …, Δ^{n}z_{k} ≤ 0, for k ≥ N − μ. Then in view of (2),Δ^{n}z_{k} ≤ −Q_{k}F (x_{k−τ}), k ≥ N. By summing the above inequality n times from k to infinity, we obtain respectively

Δ^{n−1}z_{k} ≥ ∑_{j=k}^{∞} Q_{j}F (x_{j−τ}), Δ^{n−2}z_{k} ≤ − ∑_{j=k}^{∞} ∑_{k=i}^{∞} Q_{i}F (x_{i−τ}),

…

x_{k} − P_{k}x_{k−σ} = z_{k} ≥ ∑_{j=k}^{∞} \frac{(j − k + n − 1)^{(n−1)}}{(n − 1)!} Q_{j}F (x_{j−τ}), k ≥ N,

MONOTONE SOLUTIONS OF NEUTRAL DIFFERENCE EQUATION 51

where the factorial function h^{(m)}(i) is defined by h(i)h(i−1)···h(i−m+1). As a consequence, by assuming P_{k} ≥ p_{k} and Q_{k} ≥ q_{k} for k ≥ 0, F (x) ≥ f(x) for x > 0, as well as f is positive nondecreasing on (0, ∞), we see that

x_{k} ≥ p_{k}x_{k−σ} + ∑_{j=k}^{∞} \frac{(j − k + n − 1)^{(n−1)}}{(n − 1)!} q_{j}f(x_{j−τ}), k ≥ N.  (4)

Let Ω be the set of all real sequences w = {w_{k}}_{n=N−μ}^{∞}. Define an operator T : Ω → Ω by(T w)_{k} = 1, N − μ ≤ k ≤ N − 1, and

(T w)_{k} = \frac{1}{x_{k}} \{p_{k}w_{k−σ}x_{k−σ} +∑_{j=k}^{∞} \frac{(j−k+n−1)^{(n−1)}}{(n − 1)!} q_{j}f(w_{j−τ} x_{j−τ})\}, k ≥N.

Consider the following successive approximations: w^{(0)} ≡ 1, w^{(j+1)} = T w^{(j)} for j = 0, 1, 2, . . .. By means of (4) and induction, it is easily seen that0 ≤ w_{k}^{(j+1)} ≤ w_{k}^{(j)} ≤ 1, k ≥ N, j ≥ 0. Thus, as m → ∞, w^{(m)} converges (pointwise) to some non-negative sequence w^{∗} which satisfies w_{k}^{∗} = 1 for N − μ ≤ k ≤ N − 1, and furthermore, by means of the Lebesgue dominated convergence theorem, we may take limits on both sides of w^{(j+1)} = T w^{(j)} to obtain

w_{k}^{∗}x_{k} − p_{k}w_{k−τ}^{∗} x_{k−τ} = ∑_{j=k}^{∞} \frac{(j − k + n − 1)^{(n−1)}}{(n − 1)!} q_{j}f(w_{k−τ}^{∗} x_{k−τ}),; k ≥ N.

Taking differences on both sides of the above equality, we see that {y_{k}} defined byy_{k} = w_{k}^{∗}x_{k}, k ≥ N − μ, is an eventually non-negative solution of the recurrence relation (1). It will be a (∗)-monotone solution if we can show that {y_{n}} is eventually positive. Indeed, note that y_{k} > 0 for N −μ ≤ k ≤ N −1. Suppose to the contrary that y_{k} > 0 for N −μ ≤ k < k^{∗} and y_{k^{∗}} = 0, then

0 = y_{k^{∗}} = p_{k^{∗}}y_{k^{∗}−τ} + ∑_{j=k^{∗}}^{∞} \frac{(j − k + n − 1)^{(n−1)}}{(n − 1)!} q_{j}f(y_{j−τ}).

Since y_{k^{∗}−σ} > 0, (j − k^{∗} + n − 1)^{(n−1)} > 0 for j ≥ k^{∗}, we must have p_{k^{∗}} = 0 and q_{j}f(y_{j−τ}) = 0 for j ≥ k^{∗}. In other words, if τ = 0, we may prevent this from happening by imposing the condition p_{k} > 0 for all k ≥ 0, or, if τ > 0, by imposing the condition p_{k} > 0, or, the vector (q_{k}, q_{k+1}, …, q_{k+τ−1}) ≠ 0 for all k ≥ 0.

We summarize these results.

Theorem 1. Suppose that either (H1) τ = 0 and p_{k} > 0 for k ≥ 0, or (H2) τ > 0, and, p_{k} > 0 or the vector (q_{k}, …, q_{k+τ−1}) ≠ 0 for k ≥ 0. Suppose that {P_{k}} and {Q_{k}} are two sequences such that P_{k} ≥ p_{k} ≥ 0 and Q_{k} ≥ q_{k} ≥ 0 for k ≥ 0. Suppose further that F and f are real functions such that F (x) ≥ f(x) for x > 0. If (2) has a (∗)-monotone solution with respect to σ, n and {P_{k}}, then (1) will have a (∗)-monotone solution with respect to σ, n and {p_{k}}.

§ 3. Existence Theorems

Given a real sequence {u_{k}} with sign conditions on its difference sequences {Δ^{r}u_{k}} and {Δ^{t}u_{k}}, the in-between difference sequences {Δ^{s}u_{k}} will also satisfy certain sign conditions. Similar results are well known in the theory of ordinary differential equations (see, for example, Kiguradze and Chanturia [7]). We will make use of these criteria to derive the existence of (∗)-monotone solutions.

Lemma 1 (Zhou and Yan [5]). Let {u_{k}} be a real bounded sequence of fixed sign. Suppose u_{k}Δ^{t}u_{k} ≤ 0 for some odd integer t > 1 and all large k. Then (−1)^{s}u_{k}Δ^{s}u_{k} ≥ 0 for s = 1, 2, …, t and all large k.

Lemma 2 (Zafer and Dahiya [6]). Let m be a positive integer. Let {y_{k}}_{k=0}^{∞} be a real sequence such that the sequences {y_{k}}, …, {Δ^{m−1}y_{k}} are of constant sign. Suppose further that y_{k}Δ^{m}y_{k} ≥ 0 for k ≥ 0. Then either

(i) y_{k}Δ^{j}y_{k} ≥ 0 for each j ∈ {1, 2, …, m − 1} and all large k, or (ii) there is an integer t ∈ {1, 2, …, m − 2} such that (−1)^{m−t} = 1

and for each j ∈ {1, …, t}, y_{k}Δ^{j}y_{k} > 0 for all large k, and for each j ∈ {t + 1, …, m − 2}, (−1)^{j−t}y_{k}Δ^{j}y_{k} > 0 for all large k.

Lemma 3. Suppose there is an integer N such thatP_{N+jσ} ≤ 1, j = 0, 1, 2, . . .. Then for any eventually positive solution {x_{k}} of (2), the sequence {z_{k}} defined by (3) is also eventually positive.

Proof. Suppose x_{k−τ} > 0 for all large k. Then in view of (2) we see that Δ^{n}z_{k} ≤ −Q_{k}F (x_{k−τ}) ≤ 0 for all large k. Since {Q_{k}} is not identically zero for all large k, thus for each j ∈ {0, 1, …, n − 1}, {Δ^{j}z_{k}} is of constant sign for all large k. In particular, {z_{k}} is eventually positive or eventually negative. Suppose to the contrary that {z_{k}} is eventually negative, then in view of Lemma 2 and the fact that n is an odd integer, we see that Δz_{k} < 0 for all large k. Thus there is a positive number α such that z_{k} ≤ −α for all large k. Therefore x_{k} ≤ −α + P_{k}x_{k−σ} for k greater than or equal to some integer, which, without loss of generality, may be taken to be N+σ N^{∗}. Then we have x_{N+σ} ≤ −α + P_{N+σ}x_{N} ≤ −α + x_{N}, x_{N+2σ} ≤ −α + P_{N+2σ}x_{N+σ} ≤ −2α+x_{N}, …,x_{N+jσ} ≤ −jσ+x_{N}. But for sufficiently large j, the right-hand side is negative, while the right-hand side remains positive. A contradiction is obtained.

We remark that the proof of the above lemma is similar to that of Lemma 3.4 in [8] but is included here for the sake of completeness.

Theorem 2. Suppose {P_{k}} is bounded and there is an integer N such that P_{N+jσ} ≤ 1 for j ≥ 0. Then any eventually positive and bounded solution {x_{k}} of (2) is eventually (∗)-monotone.

Proof. In view of Lemma 3, the sequence {z_{k}} defined by z_{k} = x_{k} −P_{k}x_{k−σ} is eventually positive. Since {x_{k}} and {P_{k}} is bounded, we see further that {z_{k}} is bounded. But then by means of Lemma 1, (−1)^{j}Δ^{j}z_{k} ≥ 0 for j = 1, 2, …, n and all large k.

We also have a result which removes the boundedness condition in Theorem 2.

Theorem 3. Suppose there is an integer N such that P_{N+jσ} ≤ 1 for j ≥ 0. Suppose further that lim inf_{x→∞} F (x) ≥ d > 0 and that

∑_{i=0}^{∞} Q_{i} = ∞.  (5)

Then an eventually positive solution {x_{k}} of (2) is also a (∗)-monotone solution.

Proof. Let {x_{k}} be an eventually positive solution of (2). Then the sequence {z_{k}} defined by (3) will satisfy Δ^{n}z_{k} ≤ −Q_{k}F (x_{k−τ}) ≤ 0 for all large k. Furthermore, since {Q} does not vanish identically for all large k, thus either

lim_{k→∞} Δ^{n−1}z_{k} = −∞,  (6)

or

lim_{k→∞} Δ^{n−1}z_{k} = c.  (7)

If (6) holds, then it is easy to see that lim_{k→∞} z_{k} = −∞. This implies that x_{k} ≤ −α+P_{k}x_{k−σ} for some positive number and all large k. As we have seen in the proof of Lemma 3, this is impossible. Therefore (7) must hold. We assert further that c = 0. Otherwise, it is easy to see that lim_{k→∞} z_{k} = −∞ (which is impossible) or lim_{k→∞} z_{k} = +∞. If z_{k} → +∞, then in view of (3) it is clear that lim_{k→∞} x_{k} = +∞. By summing both sides of (2) from 0 to

∞ we obtain c + ∑_{i=0}^{∞} Q_{i}F (x_{i−σ}) = Δ^{n−1}z_{0}, so that d ∑_{i=0}^{∞} Q_{i} < ∞, which is

contrary to (5). We have thus shown that Δ^{n−1}z_{k} > 0 for all large k and strictly decreases to zero as k → ∞. By repeating the above arguments it is not difficult to see that (−1)^{j}Δ^{j}z_{k} > 0, j = 1, 2, …, n − 1, for all large k, and also, lim_{k→∞} Δ^{j}z_{k} = 0, j = 1, 2, …, n − 1.

In view of Theorems 2 and 3, we need to find some explicit existence criteria for eventually positive and/or bounded solutions of recurrence relations of form (2) so that our comparison Theorem 1 can be used to produce existence criteria for (∗)-monotone solutions of (1). For the linear equationΔ^{n} (x_{k} − P x_{k−σ}) + Q_{k}x_{k−τ} = 0, k = 0, 1, 2, …, such existence criteria can be found in [8].

Acknowledgement

The first author S. S. Cheng is partially supported by the National Science Council of R.O.C.

References

1. S. S. Cheng, Oscillation theorems for fourth order differential and difference equations. Proceedings of the International Conference on Functional Differential Equations, Publishing House of Electronic Industry, Guangzhou, China, 1993, 37–46.

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3. M. P. Chen, J. S. Yu, and L. H. Huang, Oscillations of first order neutral differential equations with variable coefficients. J. Math. Anal. Appl. 185(1994), 288–301.

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5. X. L. Zhou and J. R. Yan, Oscillatory properties of higher order nonlinear difference equations. Comput. Math. Appl. 31(12)(1996), 61–68.

6. A. Zafer and R. S. Dahiya, Oscillation of a neutral difference equation. Appl. Math. Lett. 6(1993), 71–74.

7. I. T. Kiguradze and T. A. Chanturia, Asymptotic Properties of Solutions of Nonautonomous Ordinary Differential Equations. Kluwer Academic Publishers, 1993.

8. S. S. Cheng, G. Zhang, and W. T. Li, On a higher order neutral difference equation. Recent Trends in Mathematical Analysis (Ed. T. M. Rassias), to appear.

(Received 15.08.1995)

Authors’ addresses:

Sui Sun Cheng

Department of Mathematics

Tsing Hua University

Hsinchu, Taiwan 30043, R.O.C.

Guang Zhang

Department of Mathematics Datong Advanced College Datong, Shanxi 037008, P. R. China