Electronic Journal of Qualitative Theory of Differential Equations 2009, No. 16, 1-9; http://www.math.u-szeged.hu/ejqtde/

ASYMPTOTIC AND OSCILLATORY BEHAVIOR OF SECOND ORDER NEUTRAL QUANTUM EQUATIONS WITH MAXIMA

DOUGLAS R. ANDERSON AND JON D. KWIATKOWSKI

Abstract. In this study, the behavior of solutions to certain second order quantum (q-difference) equations with maxima are considered. In particular, the asymptotic behavior of non-oscillatory solutions is described, and sufficient conditions for oscillation of all solutions are obtained.

1. introduction

Quantum calculus has been utilized since at least the time of Pierre de Fermat [8, Chapter B.5] to augment mathematical understanding gained from the more traditional continuous calculus and other branches of the discipline; see Kac and Cheung [4], for example. In this study we will analyze a second order neutral quantum (qdifference) equation

D_{q}^{2}(x(t) + p(t)x(q^{−k}t)) + r(t) max_{s∈{0,··· ,ℓ}} x(q^{−s}t) = 0,  (1.1)

where the real scalar q > 1 and the q-derivatives are given, respectively, by the difference quotient

D_{q}y(t) = \frac{y(qt) − y(t)}{qt − t}, and D_{q}^{2}y(t) = D_{q} (D_{q}y(t)) .

Equation (1.1) is a quantum version of

Δ^{2}(x_{n} + p_{n}x_{n−k}) + q_{n} max_{{n−ℓ,··· ,ℓ}} x_{s} = 0,  (1.2)

studied by Luo and Bainov [5]; there the usual forward difference operator Δy_{n} := y_{n+1} − y_{n} was used. For more results on differential and difference equations related to (1.1) and (1.2), please see the work by Bainov, Petrov, and Proytcheva [1, 2, 3], Luo and Bainov [5], Luo and Petrov [6], and Petrov [7]. The particular appeal of (1.1) is that it is still a discrete problem, but with non-constant step size between domain points.

EJQTDE, 2009 No. 16, p. 1

2000 Mathematics Subject Classification. 34A30, 39A13, 34C10.

Key words and phrases. quantum calculus, second order equations, q-difference equations, oscillation, delay.

2. preliminary results

For q > 1, define the quantum half line by

(0, ∞)_{q} := {⋯, q^{−2}, q^{−1}, 1, q, q^{2}, ⋯ }.

Let k, ℓ be non-negative integers, r : (0, ∞)_{q} → [0, ∞), p : (0, ∞)_{q} → ℝ, and consider the second order neutral quantum (q-difference) equation

D_{q}^{2}(x(t) + p(t)x(q^{−k}t)) + r(t) max_{s∈{0,··· ,ℓ}} x (q^{−s}t) = 0,  (2.1)

where we assume

∑_{η∈[t_{0},∞)_{q}} ηr(η) = ∞, t_{0} ∈ (0, ∞)_{q}.  (2.2)

Definition 2.1. A function f : (0, ∞)_{q} → ℝ eventually enjoys property P if and only if there exists t^{∗} ∈ (0, ∞)_{q} such that for t ∈ [t^{∗}, ∞)_{q} the function f enjoys property P. A solution x of (2.1) is non-oscillatory if and only if x(t) < 0 or x(t) > 0 eventually; otherwise x is oscillatory.

Define the function z : (0, ∞)_{q} → ℝ via

z(t) := x(t) + p(t)x(q^{−k}t).  (2.3)

Then from (2.1) we have that

D_{q}^{2}z(t) = −r(t) max_{s∈{0,··· ,ℓ}} x(q^{−s}t),  (2.4)

and

D_{q}z(t) = D_{q}z(t_{0}) − (q − 1) ∑_{η∈[t_{0},t)_{q}} ηr(η) max_{s∈{0,··· ,ℓ}} x(q^{−s}η).  (2.5)

We will use these expressions involving z in the following lemmas.

Lemma 2.2. Assume x is a solution of (2.1), r satisfies (2.2), z is given by (2.3), and

p ≤ p(t) ≤ P < −1 for all t ∈ [t_{0}, ∞)_{q}.  (2.6)

(a) If x(t) > 0 eventually, then either

z(t) < 0, D_{q}z(t) < 0, and D_{q}^{2}z(t) ≤ 0 eventually and  (2.7)
lim_{t→∞} z(t) = lim_{t→∞} D_{q}z(t) = −∞,  (2.8)

or

z(t) < 0, D_{q}z(t) > 0, and D_{q}^{2}z(t) ≤ 0 eventually and  (2.9)

EJQTDE, 2009 No. 16, p. 2

lim_{t→∞} z(t) = lim_{t→∞} D_{q}z(t) = 0.  (2.10)

(b) If x(t) < 0 eventually, then either

z(t) > 0, D_{q}z(t) > 0, and D_{q}^{2}z(t) ≥ 0 eventually and  (2.11)
lim_{t→∞} z(t) = lim_{t→∞} D_{q}z(t) = ∞,  (2.12)

or

z(t) > 0, D_{q}z(t) < 0, and D_{q}^{2}z(t) ≥ 0 eventually and  (2.13)

(2.10) holds.

Proof. We will prove (a); the proof of (b) is similar and thus omitted. Since x(t) > 0 eventually and r(t) ≥ 0, it follows from (2.4) that D_{q}^{2}z(t) ≤ 0 eventually and D_{q}z is an eventually nonincreasing function. Then either there exists an L := lim_{t→∞} D_{q}z(t) ∈ ℝ, or lim_{t→∞} D_{q}z(t) = −∞. If lim_{t→∞} D_{q}z(t) = −∞, then lim_{t→∞} z(t) = −∞ and (2.7) and (2.8) hold. So, let L := lim_{t→∞} D_{q}z(t) ∈ ℝ; then one of the following three cases holds: (i) L < 0; (ii) L > 0; (iii) L = 0.

(i) If L < 0, then lim_{t→∞} z(t) = −∞. From (2.3) it follows that the inequality

z(t) > p(t)x(q^{−k}t) ≥^{(2.6)} px(q^{−k}t)

holds. Thus lim_{t→∞} x(t) = ∞. From (2.2) and (2.5) we see that lim_{t→∞} D_{q}z(t) = −∞, a contradiction.

(ii) If L > 0, we arrive at a contradiction analogous to (i).

(iii) Assume L = 0. Since D_{q}z is an eventually decreasing function, D_{q}z(t) > 0 eventually and z is an eventually increasing function. Thus either lim_{t→∞} z(t) = M ∈ ℝ, or lim_{t→∞} z(t) = ∞. If M > 0, then x(t) > z(t) > M/2 for large t ∈ (0, ∞)_{q}, and from assumption (2.2) and equation (2.5) it follows that lim_{t→∞} D_{q}z(t) = −∞, a contradiction. Using a similar argument we reach a contradiction if lim_{t→∞} z(t) = ∞. Therefore we assume there exists a finite limit, lim_{t→∞} z(t) = M ≤ 0. If M < 0, then

M > z(t) > p(t)x(q^{−k}t) ≥ px(q^{−k}t).

Thus for large t we have

M/p < x(q^{−k}t),

and again from assumption (2.2) and equation (2.5) we have that lim_{t→∞} D_{q}z(t) = −∞ = L, a contradiction of L = 0. Consequently lim_{t→∞} z(t) = 0, and since z is an eventually increasing function, z(t) < 0 eventually and (2.9) and (2.10) hold. □

EJQTDE, 2009 No. 16, p. 3 Lemma 2.3. Assume x is a solution of (2.1), r satisfies (2.2), z is given by (2.3), and

−1 ≤ p(t) ≤ 0, t ∈ (0, ∞)_{q}.  (2.14)

Then the following assertions are valid.

(a) If x(t) < 0 eventually, then relations (2.10) and (2.13) hold.

(b) If x(t) > 0 eventually, then relations (2.9) and (2.10) hold.

Proof. We will prove (a); the proof of (b) is similar and thus omitted. From (2.4) it follows that D_{q}^{2}z(t) ≥ 0 eventually, and D_{q}z is an eventually nondecreasing function. Assumption (2.2) implies that r(t) ≠ 0 eventually, and thus either D_{q}z(t) > 0 eventually or D_{q}z(t) < 0. Suppose that D_{q}z(t) > 0. Since D_{q}z is a nondecreasing function, there exists a constant c > 0 such that D_{q}z(t) ≥ c eventually. Then lim_{t→∞} D_{q}z(t) = ∞. From (2.3) we obtain the inequality

z(t) < p(t)x(q^{−k}t) ≤ −x(q^{−k}t)

and therefore lim_{t→∞} x(t) = −∞. On the other hand, from (2.3) again and from the inequality z(t) > 0 there follows the estimate

x(t) > −p(t)x(q^{−k}t) ≥ x(q^{−k}t).

The inequalities x(t) < 0 and x(t) > x(q^{−k}t) eventually imply that x is a bounded function, a contradiction of the condition lim_{t→∞} x(t) = −∞ proved above. Thus D_{q}z(t) < 0, and z is an eventually decreasing function. Let L = lim_{t→∞} D_{q}z(t). Then lim_{t→∞} z(t) = −∞. From the inequality x(t) < z(t) it follows that lim_{t→∞} x(t) = −∞, and then (2.5) implies the relation lim_{t→∞} D_{q}z(t) = ∞. The contradiction obtained shows that L = 0, that is lim_{t→∞} D_{q}z(t) = 0. Suppose that z(t) < 0 eventually. Since z is a decreasing function, there exists a constant c < 0 such that z(t) ≤ c eventually. The inequality z(t) > x(t) implies that x(t) ≤ c eventually. From (2.5) it follows that lim_{t→∞} D_{q}z(t) = ∞. The contradiction obtained shows that z(t) > 0, and since z is an eventually decreasing function, then there exists a finite limit M = lim_{t→∞} z(t). If M > 0, then z(t) > M eventually. From (2.3) it follows that

M < z(t) < p(t)x(q^{−k}t) ≤ −x(q^{−k}t),

that is x(q^{−k}t) < −M. From (2.5) we obtain that lim_{t→∞} D_{q}z(t) = ∞, a contradiction. Hence, M = 0, in other words lim_{t→∞} z(t) = 0. Since z is a decreasing function, z(t) > 0 eventually, and we have shown that if x is an eventually negative solution of (2.1), then (2.10) and (2.13) are valid. □

EJQTDE, 2009 No. 16, p. 4 Lemma 2.4. The function x is an eventually negative solution of (2.1) if and only if −x is an eventually positive solution of the equation

D_{q}^{2}(y(t) + p(t)y(q^{−k}t)) + r(t) min_{s∈{0,··· ,ℓ}} y (q^{−s}t) = 0.

Lemma 2.4 is readily verified.

3. main results

In this section we present the main results on the oscillatory and asymptotic behavior of solutions to (2.1).

Theorem 3.1. Assume r satisfies (2.2), and

−1 < p ≤ p(t) ≤ 0, t ∈ (0, ∞)_{q}.  (3.1)

If x is a nonoscillatory solution of (2.1), then lim_{t→∞} x(t) = 0.

Proof. Let x(t) > 0 eventually. Then Lemma 2.3 implies that z(t) < 0 eventually and lim_{t→∞} z(t) = 0. From (3.1) we have that

x(t) < −p(t)x(q^{−k}t) < x(q^{−k}t),

so that x is bounded. Let c = lim sup_{t→∞} x(t), and suppose that c > 0. Choose an increasing quantum sequence of points {t_{i}} from (0, ∞)_{q} such that lim_{i→∞} t_{i} = ∞ and lim_{i→∞} x(t_{i}) = c. Set d = lim sup_{i→∞} x(q^{−k}t_{i}), and note that d ≤ c. Choose a subsequence of points {t_{j}} ⊂ {t_{i}} such that d = lim_{j→∞} x(q^{−k}t_{j}), and pass to the limit in the inequality z(t_{j}) ≥ x(t_{j}) + px(q^{−k}t_{j}) as j → ∞. We then see that

0 ≥ c + pd ≥ c + pc = c(1 + p) > 0,

a contradiction. Thus lim sup_{t→∞} x(t) = 0 and lim_{t→∞} x(t) = 0. The case where x(t) < 0 eventually is similar and is omitted. □

Theorem 3.2. Assume r satisfies (2.2), and condition (2.6) holds. If x is a bounded nonoscillatory solution of (2.1), then lim_{t→∞} x(t) = 0.

Proof. Let x(t) > 0 eventually; the case where x(t) < 0 eventually is similar and is omitted. Since x is bounded, it follows from (2.3) that z is also bounded. Since (2.6) holds, Lemma 2.2 implies that z(t) < 0 eventually and lim_{t→∞} z(t) = 0. As in the proof of Theorem 3.1, let c = lim sup_{t→∞} x(t), and suppose that c > 0. Choose an increasing quantum sequence of points {t_{i}} from (0, ∞)_{q} such that lim_{i→∞} t_{i} = ∞ and lim_{i→∞} x(t_{i}) = c. Set d = lim sup_{i→∞} x(q^{−k}t_{i}), and note that d ≤ c. Choose a

EJQTDE, 2009 No. 16, p. 5 subsequence of points {t_{j}} ⊂ {t_{i}} such that d = lim_{j→∞} x(q^{−k}t_{j}), and pass to the limit in the inequality

z(t_{j}) ≤ x(t_{j}) + P x(q^{−k}t_{j})

as j → ∞. We then see a contradiction, so that lim sup_{t→∞} x(t) = 0 and lim_{t→∞} x(t) = 0. □

Theorem 3.3. Assume condition (2.6) holds, and the coefficient function r satisfies

0 < r ≤ r(t) ≤ R, t ∈ [t_{0}, ∞)_{q}.  (3.2)

If x is an eventually positive solution of (2.1), then either lim_{t→∞} x(t) = ∞ or lim_{t→∞} x(t) = 0.

Proof. Lemma 2.2 implies that either lim_{t→∞} z(t) = −∞ or lim_{t→∞} z(t) = 0. First, we consider lim_{t→∞} z(t) = −∞. Then

z(t) > p(t)x(q^{−k}t) ≥ px(q^{−k}t),

so that lim_{t→∞} x(t) = ∞. Next, we consider lim_{t→∞} z(t) = 0. In this case Lemma 2.2 implies that z is an eventually negative increasing function. If the solution x does not vanish at infinity, then there exist a constant c > 0 and an increasing quantum sequence of points {t_{i}} from (t_{0}, ∞)_{q} such that t_{i+1} > q^{ℓ}t_{i} and x(t_{i}) > c/2 for each i ∈ ℕ. Then, we have

max_{s∈{0,··· ,ℓ}} x (q^{−s}t) > c/2, t ∈ [t_{i}, t_{i+ℓ}]_{q}.

From this last inequality and (3.2) we obtain the estimate

(q − 1) ∑_{η∈[t_{i},q^{ℓ}t_{i}]_{q}} ηr(η) max_{s∈{0,··· ,ℓ}} x (q^{−s}η) > (q − 1)(ℓ + 1)t_{0}rc/2.  (3.3)

It then follows from (3.3) and the choice of the quantum sequence {t_{i}} that

(q − 1) ∑_{η∈[t_{0},∞)_{q}} ηr(η) max_{s∈{0,··· ,ℓ}} x (q^{−s}η) ≥ (q − 1) ∑_{i=1}^{∞} ∑_{η∈[t_{i},q^{ℓ}t_{i}]_{q}} ηr(η) max_{s∈{0,··· ,ℓ}} x (q^{−s}η)
> (q − 1) ∑_{i=1}^{∞}(ℓ + 1)t_{0}rc/2 = ∞.

From (2.5) we then see that lim_{t→∞} D_{q}z(t) = −∞. On the other hand, Lemma 2.2 implies that D_{q}z(t) > 0 eventually, a contradiction. Thus lim_{t→∞} x(t) = 0. □

Theorem 3.4. Assume condition (3.2) holds, and p(t) ≡ −1. If x is an eventually positive solution of (2.1), then lim_{t→∞} x(t) = 0.

EJQTDE, 2009 No. 16, p. 6 Proof. Lemma 2.3 implies that lim_{t→∞} z(t) = 0, where z is an eventually increasing negative function. Suppose that the solution x does not vanish at infinity. From (2.3) and the fact that z(t) < 0, it follows that x(t) < x(q^{−k}t) eventually, so that x is bounded. Let c = lim sup_{t→∞} x(t) > 0. Choose an increasing quantum sequence of points {t_{i}} from (0, ∞)_{q} such that such that t_{i+1} > q^{ℓ}t_{i} and x(t_{i}) > c/2 for each i ∈ ℕ. Then, we have

max_{s∈{0,··· ,ℓ}} x (q^{−s}t) > c/2, t ∈ [t_{i}, t_{i+ℓ}]_{q}.

The proof is then completed in a way identical to the proof of Theorem 3.3. □

We now present a few sufficient conditions for the oscillation of all solutions of (2.1).

Theorem 3.5. Assume r satisfies (2.2), and at least one of the following conditions

1 < p ≤ p(t) ≤ P,  (3.4)
0 ≤ p(t) ≤ P < 1,  (3.5)
p(t) ≡ 1,  (3.6)

holds for all t ∈ [t_{0}, ∞)_{q}. Then each solution of (2.1) oscillates.

Proof. Assume to the contrary that x is a nonoscillatory solution of (2.1). Let x(t) > 0 eventually; the case where x(t) < 0 eventually is similar and is omitted.

First, let (3.4) hold. By (2.4), D_{q}^{2}z(t) ≤ 0 eventually and D_{q}z(t) is nonincreasing. From (2.2) we know that D_{q}z(t) ≠ 0 eventually, and since x(t) > 0 and p(t) > 0 in this case, z(t) > 0 and D_{q}z(t) > 0 eventually. Suppose that lim_{t→∞} z(t) = c < ∞; we will show that lim inf_{t→∞} x(t) > 0. To this end, assume instead that lim inf_{t→∞} x(t) = 0. Choose an increasing quantum sequence of points {t_{i}} from (0, ∞)_{q} such that lim_{i→∞} t_{i} = ∞ and lim_{i→∞} x(q^{−k}t_{i}) = 0. It then follows from (2.3) that lim_{i→∞} x(t_{i}) = c. Using (2.3) and (3.4) we have that

z(q^{k}t_{i}) = x(q^{k}t_{i}) + p(q^{k}t_{i})x(t_{i}) > p(q^{k}t_{i})x(t_{i}) ≥ px(t_{i});

letting i → ∞ we see that c ≥ pc > c, a contradiction. Thus lim inf_{t→∞} x(t) > 0, so that there exists a positive constant d with x(t) ≥ d > 0 eventually. From (2.2) and (2.5) it follows that lim_{t→∞} D_{q}z(t) = −∞, a contradiction of D_{q}z(t) > 0 eventually. Consequently, lim_{t→∞} z(t) = ∞. By (2.3) and (3.4), we must have lim_{t→∞} x(t) = ∞, which again implies by (2.2) and (2.5) that lim_{t→∞} D_{q}z(t) = −∞, a contradiction. We conclude that if (3.4) holds, then (2.1) has no eventually positive solutions.

EJQTDE, 2009 No. 16, p. 7

Next, let (3.5) hold. As in the previous case, through two contradictions we arrive at the result.

Finally, let (3.6) hold. As in the first case, D_{q}^{2}z(t) ≤ 0, D_{q}z(t) > 0, and z(t) > 0 eventually. Using (2.3) twice, we see that

x(q^{k}t) − x(q^{−k}t) = z(q^{k}t) − z(t);

as z is eventually increasing, it follows that x(q^{k}t) > x(q^{−k}t) eventually. Thus lim inf_{t→∞} x(t) > 0. As in the first case, this leads to a contradiction and the result follows. □

4. example

In this section we offer an example related to the results of the previous section. Note that in Theorem 3.1, in the case where p(t) < 0 eventually, we do not consider the oscillatory behavior of solutions of (2.1) because there always exists a nonoscillatory solution. This is shown in the following example.

Example 4.1. Consider the quantum equation

D_{q}^{2}(x(t) + p(t)x(q^{−k}t)) + r(t) max_{s∈{0,··· ,ℓ}} x(q^{−s}t) = 0,  t ∈ (t_{0}, ∞)_{q},  (4.1)

where q = 2, k = 2, r(t) = 1/t, t_{0} = 8, ℓ is a positive integer, and

p(t) = \frac{1 − 6t + 4t^{2} + 4t^{3}}{−8t^{2}(4 − 6t + t^{2})} ∈ [\frac{−2257}{10240}, 0) , t ∈ [t_{0}, ∞)_{q}.

Then (4.1) has a negative solution x that satisfies lim_{t→∞} x(t) = 0.

Proof. Since r(t) = 1/t,

∑_{η∈[t_{0},∞)_{q}} ηr(η) = ∑_{η∈[2^{3},∞)_{2}} 1 = ∞,

so that (2.2) is satisfied. Using a computer algebra system, one can verify that

x(t) = −\sqrt{t} exp (\frac{− ln^{2}(t)}{2 ln(2)})

is a negative, increasing solution of (4.1) that vanishes at infinity, as guaranteed by Theorem 3.1. □

EJQTDE, 2009 No. 16, p. 8

References

[1] D. Bainov, V. Petrov, and V. Proytcheva, Oscillatory and asymptotic behavior of second order neutral differential equations with maxima, Dynamic Systems Appl., 4 (1995) 137–146.

[2] D. Bainov, V. Petrov, and V. Proytcheva, Oscillation and nonoscillation of first order neutral differential equations with maxima, SUT J. Math., 31 (1995) 17–28.

[3] D. Bainov, V. Petrov, and V. Proytcheva, Existence and asymptotic behavior of nonoscillatory solutions of second order neutral differential equations with maxima, J. Comput. Appl. Math., 83 (1997) 237–249.

[4] V. Kac and P. Cheung, Quantum Calculus, Springer, New York, 2002.

[5] J. W. Luo and D. D. Bainov, Oscillatory and asymptotic behavior of second-order neutral difference equations with maxima, J. Comput. Appl. Math., 131 (2001) 333–341.

[6] J. W. Luo and V. A. Petrov, Oscillation of second order neutral differential equations with maxima, J. Math. Sci. Res. Hot-Line, 3:11 (1999) 17–22.

[7] V. A. Petrov, Nonoscillatory solutions of neutral differential equations with maxima, Commun. Appl. Anal., 2 (1998) 129–142.

[8] G. F. Simmons, Calculus Gems: Brief Lives and Memorable Mathematics, McGraw-Hill, New York, 1992.

(Received January 19, 2009)

Department of Mathematics and Computer Science, Concordia College, Moorhead, MN 56562 USA

    E-mail address: andersod@cord.edu, jdkwiatk@cord.edu

EJQTDE, 2009 No. 16, p. 9