Electronic Journal of Qualitative Theory of Differential Equations 2010, No. 61, 1-15; http://www.math.u-szeged.hu/ejqtde/

OSCILLATION CRITERIA OF SECOND ORDER NEUTRAL DELAY

DYNAMIC EQUATIONS WITH DISTRIBUTED DEVIATING ARGUMENTS

E. THANDAPANI^{1} AND V. PIRAMANANTHAM^{2}

Abstract. In this paper we establish some oscillation theorems for second order neutral dynamic equations with distributed deviating arguments. We use the Riccati transformation technique to obtain sufficient conditions for the oscillation of all solutions. Further, some examples are provided to illustrate the results.

1. Introduction

In this paper we are concerned with the oscillatory behavior of solutions of second order neutral type dynamic equations with distributed deviating arguments of the form

(r(t)(x(t) + p(t)x(t − τ))^{Δ})^{Δ} + ∫_{a}^{b} q(t, ξ)f(x(g(t, ξ)))Δξ = 0,  t ∈ 𝕋  (1.1)

subject to the conditions:

(A_{1}) r(t), p(t) are positive real valued rd-continuous functions on time scales with

0 ≤ p(t) < 1; (A_{2}) f ∈ C(ℝ, ℝ) such that uf(u) > 0 for u ≠ 0, and f(−u) = −f(u); (A_{3}) g(t, ξ) ∈ C_{rd}(𝕋 × [a, b]_{𝕋}, 𝕋), g(t, ξ) ≤ t, ξ ∈ [a, b]_{𝕋},where [a, b]_{𝕋} = {t ∈ 𝕋 : a ≤ t ≤

b}, g is strictly increasing with respect to t and decreasing with respect to ξ, and the integral of equation (1.1) is in the sense of Riemann (see [7]).

By a solution of equation (1.1), we mean a nontrivial real valued function x(t) which has the properties x(t) + p(t)x(t − τ) ∈ C_{rd}^{1}([t_{y}, ∞)_{𝕋} and r(t)[x(t) + p(t)x(t − τ)]^{Δ} ∈ C_{rd}^{1}([t_{y}, ∞)_{𝕋} and satisfying equation (1.1) for all t ∈ [t_{0}, ∞)_{𝕋}.

We restrict our attention to nontrivial solutions of equation (1.1) that exist on some half-line [t_{y}, ∞)_{𝕋}, and satisfying sup{| x(t) |: t ∈ [t_{1}, ∞)_{𝕋}} > 0 for any t_{1} ∈ [t_{y}, ∞)_{𝕋}. A solution x(t) of equation (1.1) is said to be oscillatory if it is neither eventually positive nor eventually negative; otherwise it is called nonoscillatory. The equation itself is

EJQTDE, 2010 No. 61, p. 1 called oscillatory if all its solutions are oscillatory. Since we are interested in oscillatory behavior of solutions, we will suppose that the time scale 𝕋 under considerations is not bounded above, that is, it is a time scale interval of the form [t_{0}, ∞)_{𝕋}.

1991 Mathematics Subject Classification. Primary 34K11; Secondary 39A10, 34C15, 34C10, 39A99.

Key words and phrases. Oscillation, distributed deviating arguments, neutral dynamic equations, time scales.

1Ramanujan Institute for Advanced Study in Mathematics, University of Madras, Chennai, India E-mail: ethandapani@yahoo.co.in

2Department of Mathematics, Bharathidasan University, Tiruchirappalli, India, Email:

piramanantham@yahoo.co.in.

We note that if 𝕋 = ℝ, then we have f^{Δ}(t) = f′(t), and equation (1.1) becomes the second order neutral differential equation with distributed deviating arguments of the form

(r(t)(x(t) + p(t)x(t − τ))′)′ + ∫_{a}^{b} q(t, ξ)f(x(g(t, ξ)))dξ = 0,  t ≥ t_{0}.  (1.2)

If 𝕋 = ℕ, we have f^{Δ}(n) = f(n + 1) − f(n), and the equations (1.1) becomes the second order neutral difference equation with distributed deviating arguments of the form

Δ(r_{n}Δ(x_{n} + p_{n}x_{n−τ})) + ∑_{s=a}^{b} q_{s}(ξ)f(x(g_{s}(ξ))) = 0,  t ∈ ℕ.  (1.3)

Recently there has been an increasing interest in studying the oscillation of solutions of dynamic equations with continuous deviating arguments, see for example [1,6,14,16-18] and the references cited therein. To the best of our knowledge no paper has been published in dynamic equations with distributed deviating arguments. This motivated us to study the oscillatory behavior of equation (1.1).

The purpose of this paper is to derive some sufficient conditions for the solutions of the equation (1.1) to be oscillatory under the conditions

(C_{1})  ∫_{t_{0}}^{∞} \frac{1}{r(s)}Δs = ∞, and (C_{2})  ∫_{t_{0}}^{∞} \frac{1}{r(s)}Δs < ∞.

In Section 2, we present some basic lemmas , and in Section 3 we will use the Riccati transformation technique to prove our oscillation results of the equations (1.1) under the condition (C_{1}). Also we derive sufficient condition for the equation (1.1) to be oscillatory under the condition (C_{2}). In Section 4, we present some examples to illustrate our main results.

2. Some basic lemmas

In this section, we give some preliminary lemmas which are useful to prove the main results.

Lemma 2.1. [4]Assume that v ∈ 𝕋 → ℝ is strictly increasing, and 𝕋̃ = v(𝕋) is a time scale. Let w : 𝕋̃ → ℝ. If v^{Δ}(t), and w𝕋̃(v(t) exist for t ∈ 𝕋^{k}, then

(w ∘ v)^{Δ} = (wΔ̃ ∘ v)v^{Δ}.

EJQTDE, 2010 No. 61, p. 2 Lemma 2.2. If a > 0, b ≥ 0, then

−ax^{2} + bx ≤ −\frac{1}{2}x^{2} + \frac{b^{2}}{2a}, x ∈ ℝ.

Proof. The proof is obvious. □

Lemma 2.3. Assume that condition (C_{1}) holds. Let x(t) be an eventually positive solution of equation (1.1), and let y(t) = x(t) + p(t)x(t − τ). Then there exists a t_{1} ≥ t_{0} such that y(t) > 0, y^{Δ}(t) > 0 and (r(t)y^{Δ}(t))^{Δ} ≤ 0, t ∈ [t_{1}, ∞)_{𝕋}.

Proof. Suppose that x(t) is an eventually positive solution of equation (1.1) with x(t) > 0, x(t − τ) > 0 and x(g(t, ξ)) > 0 for all t ∈ [t_{1}, ∞)_{𝕋} and ξ ∈ [a, b]_{𝕋}. Set y(t) = x(t) + p(t)x(t − τ). Then y(t) > 0 for all t ∈ [t_{1}, ∞)_{𝕋}. In view of equation (1.1) we have (r(t)y^{Δ}(t))^{Δ} ≤ 0, and this implies that r(t)y^{Δ}(t) is an eventually decreasing function, since q(t, ξ) > 0. We claim that r(t)y^{Δ}(t) is eventually nonnegative on [t_{1}, ∞)_{𝕋}. Suppose not, there is a t_{2} ∈ [t_{1}, ∞)_{𝕋} such that r(t_{2})y^{Δ}(t_{2}) =: α < 0. Then

r(t)y^{Δ}(t) ≤ r(t_{2})y^{Δ}(t_{2}) = α, t ∈ [t_{2}, ∞)_{𝕋}.

Hence

y(t) ≤ y(t_{2}) + α ∫_{t_{2}}^{t} \frac{1}{r(s)}Δs

which implies by condition (C_{1}) that y(t) → −∞ as t → ∞. This contradicts the fact that y(t) > 0 for all t ∈ [t_{i}, ∞)_{𝕋}. Hence r(t)y^{Δ}(t) > 0 eventually. □

3. Oscillation results

In this section, first we derive some sufficient conditions for the solutions of equation (1.1) to be oscillatory when the condition (C_{1}) holds. We begin with the following theorem.

Theorem 3.1. Assume that condition (C_{1}) holds, and further assume that there exist g^{Δ}(t, b), H(t, s) ∈ C_{rd}^{1}(D; ℝ), h(t, s) ∈ C_{rd}^{1}(D; ℝ), and α(t) ∈ C_{rd}^{1}([t_{0}, ∞), (0, ∞)), where D_{0} = {(t, s)/t > s > t_{0}}, D = {(t, s)/t ≥ s ≥ t_{0}}, such that

(A_{4}) H(σ(t), t) = 0, H(t, s) > 0;

(A_{5}) H^{Δ_{s}}(t, s) ≤ 0 and −H^{Δ_{s}}(t, s) − H(t, s)\frac{α^{Δ}(s)}{α^{σ}(s)} = h(t, s)\sqrt{H(t, s)}. If

lim sup_{t→∞} \frac{1}{H(t, t_{0})} ∫_{t_{0}}^{t} [H(t, s)α(s) ∫_{a}^{b} Mq(s, ξ)[1 − p(g(s, ξ))]Δξ
−\frac{r(g(s, ξ))(α^{σ}(s))^{2}h^{2}(t, s)}{4α(s)g^{Δ}(s, a)}]Δs = ∞, (3.1)

then equation (1.1) is oscillatory.

EJQTDE, 2010 No. 61, p. 3 Proof. Suppose to the contrary that equation (1.1) has a nonoscillatory solution x(t). Without loss of generality we may assume that x(t) is an eventually positive solution of equation (1.1). Then there is a t_{1} ≥ t_{0} such that x(t) > 0, x(t−τ) > 0, and x(g(t, ξ)) > 0 for all t ∈ [t_{1}, ∞)_{𝕋}, ξ ∈ [a, b]_{𝕋}. Now we define y(t) = x(t) + p(t)x(t − τ) for t ∈ [t_{1}, ∞)_{𝕋}.

Then using Lemma 2.3, there exists a t_{2} ≥ t_{0} such that

y(t) > 0, y^{Δ}(t) > 0 and (r(t)y^{Δ}(t))^{Δ} ≤ 0, t ∈ [t_{2}, ∞)_{𝕋}.  (3.2)

From Lemma 2.3 and using y(t) ≥ x(t), t ∈ [t_{1}, ∞)_{𝕋}, we get y(g(t, ξ)) ≥ y(g(t, ξ) − τ) ≥ x(g(t, ξ) − τ), and we write

0 = (r(t)(x(t) + p(t)x(t − τ))^{Δ})^{Δ} + ∫_{a}^{b} q(t, ξ)f(x(g(t, ξ)))Δξ
≥ (r(t)(x(t) + p(t)x(t − τ))^{Δ})^{Δ} + ∫_{a}^{b} q(t, ξ)Mx(g(t, ξ))Δξ
= (r(t)(x(t) + p(t)x(t − τ))^{Δ})^{Δ}
+ ∫_{a}^{b} Mq(t, ξ)[y(g(t, ξ)) − p(g(t, ξ))x(g(t, ξ) − τ)Δξ
≥ (r(t)(x(t) + p(t)x(t − τ))^{Δ})^{Δ} + ∫_{a}^{b} Mq(t, ξ)[1 − p(g(t, ξ))]y(g(t, ξ))Δξ,

or

(r(t)(x(t) + p(t)x(t − τ))^{Δ})^{Δ} + ∫_{a}^{b} Mq(t, ξ)[1 − p(g(t, ξ))]y(g(t, ξ))Δξ ≤ 0. (3.3)

Using the fact that g(t, ξ) is decreasing with respect to ξ, we have from (3.3) that

(r(t)(x(t) + p(t)x(t − τ))^{Δ})^{Δ} + y(g(t, b)) ∫_{a}^{b} Mq(t, ξ)[1 − p(g(t, ξ))]Δξ ≤ 0. (3.4)

Define

w(t) = α(t)\frac{r(t)(y^{Δ}(t))^{γ}}{y(g(t, b))} .  (3.5)

Then clearly w(t) ≥ 0 for all t ∈ [t_{2}, ∞)_{𝕋}, and

w^{Δ}(t) = (r(t)y^{Δ}(t))^{σ}[\frac{α(t)}{y(g(t, b))}]^{Δ} + \frac{α(t)}{y(g(t, b))}(r(t)y^{Δ}(t))^{Δ}
= \frac{α(t)}{y(g(t, b))}(r(t)y^{Δ}(t))^{Δ} + (ry^{Δ})^{σ} \frac{α^{Δ}(t)y(g(t, b)) − α(t)[y(g(t, b))]^{Δ}}{y(g(t, b))y(g(σ(t), b))}
≤ −α(t) ∫_{a}^{b} Mq(t, ξ)[1 − p(g(t, ξ))]Δξ + α^{Δ}(t)\frac{(ry^{Δ})^{σ}}{y(g(σ(t)b))}
−\frac{α(t)(ry^{Δ})^{σ}}{y(g(t, b))y(g(σ(t), b))}\{y(g(t, b))\}^{Δ}.  (3.6)

Let g_{b}(t) = g(t, b). Then using Lemma 2.1, we have

(y ∘ g_{b})^{Δ}(t) = y^{Δ}(g_{b}(t))g_{b}^{Δ}(t).  (3.7)

From (3.6) and (3.7), we obtain

w^{Δ}(t) ≤ −α(t) ∫_{a}^{b} Mq(t, ξ)[1 − p(g(t, ξ))]Δξ + \frac{α^{Δ}(t)}{α^{σ}(t)}w^{σ}(t)
−\frac{α(t)(ay^{Δ})^{σ}}{y(g(t, b))y(g(σ(t), b))}y^{Δ}(g(t, b))g^{Δ}(t, a).  (3.8)

Since y^{Δ}(t) ≥ 0, and (r(t)y^{Δ}(t))^{Δ} ≤ 0, t ≥ t_{1}, we have

y(g(t, b)) ≤ y(g(t, b))^{σ}, and r(t)y^{Δ}(t) ≤ r(g(t, b)y^{Δ}(g(t, b)).  (3.9)

From (3.8) and (3.9), we obtain

α(t) ∫_{a}^{b} Mq(t, ξ)[1 − p(g(t, ξ))]Δξ ≤ −w^{Δ}(t) + \frac{α^{Δ}(t)}{α^{σ}(t)}w^{σ}(t)
−\frac{α(t)g^{Δ}(t, b)}{a(g(t, b))(α^{σ}(t))^{2}} \{w^{σ}(t)\}^{2}. (3.10)

Multiplying (3.10) by H(t, s) and then integrating from T to t , for any t ≥ T ≥ t_{2}, we have

∫_{T}^{t} H(t, s) α(s) ∫_{a}^{b} Mq(s, ξ)[1 − p(g(s, ξ))]ΔξΔs ≤ − ∫_{T}^{t} H(t, s)w^{Δ}(s)Δs

+ ∫_{T}^{t} H(t, s)\frac{α^{Δ}(s)}{α^{σ}(s)}w^{σ}(s)Δs − ∫_{T}^{t} H(t, s)\frac{α(s)g^{Δ}(s, b)}{a(g(s, b))α^{2}(σ(s))}\{w^{σ}(s)\}^{2}Δs.

Using integrating by parts, we have

∫_{T}^{t} H(t, s)w^{Δ}(s)Δs = H(t, s)w(s)|_{T}^{t} − ∫_{T}^{t} H^{Δ_{s}}(t, s)w^{σ}(s)Δs
= H(t, T)w(T) − ∫_{T}^{t} H^{Δ_{s}}(t, s)w^{σ}(s)Δs.  (3.11)

EJQTDE, 2010 No. 61, p. 5 In view of conditions (A_{4}), (A_{5}), and (3.11), we have

∫_{T}^{t} H(t, s)α(s) ∫_{a}^{b} Mq(s, ξ)[1 − p(g(s, ξ))]ΔξΔs

≤ H(t, T)w(T) + ∫_{T}^{t} (H^{Δ_{s}}(t, s) + H(t, s)\frac{α^{Δ}(s)}{α^{σ}(s)})w^{σ}(s)Δs

− ∫_{T}^{t} H(t, s)\frac{α(s)g^{Δ}(s, b)}{a(g(s, b))α^{2}(σ(s))}\{w^{σ}(s)\}^{2}Δs

= H(t, T)wΔ(T) − ∫_{T}^{t} h(t, s)\sqrt{H(t, s)}w^{σ}(s)Δs

− ∫_{T}^{t} H(t, s)\frac{α(s)g^{Δ}(s, b)}{a(g(s, b))α^{2}(σ(s))}\{w^{σ}(s)\}^{2}Δs

≤ H(t, T)w(T) − ∫_{T}^{t} [\sqrt{\frac{H(t, s)α(s)g^{Δ}(s, b)}{r(g(s, b))(α^{σ}(t))^{2}}}w^{σ}(s) + \frac{\sqrt{r(g(s, b))}α^{σ}(s)h(t, s)}{2\sqrt{α(s)g^{Δ}(s, b)}}]Δs

+ ∫_{T}^{t} \frac{r(g(s, b))(α^{σ}(s))^{2}h^{2}(t, s)}{4α(s)g^{Δ}(s, b)}Δs,

or

∫_{T}^{t} [H(t, s)α(s) ∫_{a}^{b} Mq(s, ξ)[1 − p(g(s, ξ))]Δξ − \frac{r(g(s, ξ))(α^{σ}(s))^{2}h^{2}(t, s)}{4α(s)g^{Δ}(s, b)}]Δs

≤ H(t, T)w(T) − ∫_{T}^{t} [\sqrt{\frac{H(t, s)α(s)g^{Δ}(s, b)}{r(g(s, b))(α^{σ}(t))^{2}}}w^{σ}(s) + \frac{\sqrt{r(g(s, b))}α^{σ}(s)h(t, s)}{2\sqrt{α(s)g^{Δ}(s, b)}}]^{2}Δs

which implies that

\frac{1}{H(t, t_{0})} ∫_{t_{0}}^{t} [H(t, s)α(s) ∫_{a}^{b} Mq(s, ξ)[1 − p(g(s, ξ))]Δξ − \frac{r(g(s, b))(α^{σ}(s))^{2}h^{2}(t, s)}{4α(s)g^{Δ}(s, b)}]Δs

≤ w(t_{0}) + ∫_{t_{0}}^{T} {α(s) ∫_{a}^{b} q(t, ξ)[1 − p(g(s, ξ))]Δξ}Δs = M < ∞,

(3.12)

where M is a constant, which contradicts (3.1).

Suppose that x(t) is an eventually negative solution of equation (1.1). Then by taking z(t) = −x(t), we have that z(t) is eventually positive solution of the equation (1.1), since f(−u) = −f(u). Similar to the proof as above we obtain a contradiction. This completes the proof. □

As a consequence of Theorem 3.1, we obtain the following corollary.

EJQTDE, 2010 No. 61, p. 6 Corollary 3.2. Assume that the hypotheses of Theorem 3.1 hold. If

lim sup_{t→∞} \frac{1}{H(t, t_{0})} ∫_{t_{0}}^{t} H(t, s)α(s) ∫_{a}^{b} Mq(s, ξ)[1 − p(g(s, ξ))]ΔξΔs = ∞,

and

lim sup_{t→∞} \frac{1}{H(t, t_{0})} ∫_{t_{0}}^{t} H(t, s)\frac{r(g(s, b))(α^{σ}(s))^{2}h^{2}(t, s)}{4α(s)g^{Δ}(s, b)}]Δs < ∞,

then every solution of equation (1.1) is oscillatory.

Remark 3.3. By introducing different choices of H(t, s) and α(t) in Theorem 3.1, we can obtain several different oscillation criteria for the equation (1.1). For instance, let H(t, s) = 1 and H(t, s) = (t − s)^{m}, t ≥ s ≥ t_{0}, in which m > 1 is an integer, we obtain the following criteria respectively.

Corollary 3.4. Assume that condition (C_{1}) holds. Further assume that there exist g^{Δ}(t, b), and α(t) as in Theorem 3.1 such that

lim sup_{t→∞} ∫_{t_{0}}^{t} [α(s) ∫_{a}^{b} Mq(s, ξ)[1 − p(g(s, ξ))]Δξ − \frac{r(g(s, b))(α^{Δ}(s))_{+}^{2}}{4α(s)g^{Δ}(s, b)} ]Δs = ∞. (3.13)

Then every solution of equation (1.1) is oscillatory.

If we choose H(t, s) = (t − s)^{m}, m > 1, then it is easy to verify that H^{Δ_{s}} ≤ −m(t − s)^{m−1}, t > s ≥ t_{0}. FromTheorem 3.1, we have

Corollary 3.5. Assume that condition (C_{1}) holds, and there exist g^{Δ}(t, b) and α(t) as in Theorem 3.1. If, for m > 1,

lim sup_{t→∞} \frac{1}{t^{m}} ∫_{t_{0}}^{t} [(t − s)^{m}α(s) ∫_{a}^{b} Mq(s, ξ)[1 − p(g(s, ξ))]Δξ

−\frac{r(g(s, ξ))(α^{σ}(s))^{2}h^{2}(t, s)}{4α(s)g^{Δ}(s, b)}]Δs = ∞,

where h(t, s) = (t − s)^{m/2−1}(m − (t − s)\frac{(α^{Δ}(s))_{+}^{2}}{(α^{σ}(s))^{2}} ), then every solution of equation (1.1) is oscillatory.

Next, if we consider α(t) = t and α(t) = 1 for t ≥ t_{0} in Corollary 3.4, we can obtain few more oscillation criteria as corollaries of Theorem 3.1.

Corollary 3.6. Assume that condition (C_{1}) holds, and there exists g^{Δ}(t, b) as in Theorem 3.1. If

lim sup_{t→∞} ∫_{t_{0}}^{t} [s ∫_{a}^{b} Mq(s, ξ)[1 − p(g(s, ξ))]Δξ − \frac{r(g(s, b))}{4sg^{Δ}(s, b)}]Δs = ∞,  (3.14)

Corollary 3.7. Assume that condition (C_{1}) holds, and there exists g^{Δ}(t, b) as in Theorem 3.1. If

lim sup_{t→∞} ∫_{t_{0}}^{t} ∫_{a}^{b} Mq(s, ξ)[1 − p(g(s, ξ))]ΔξΔs = ∞,  (3.15)

then every solution of equation (1.1) is oscillatory.

Theorem 3.8. Assume that the hypotheses of Theorem 3.1 hold. Further assume that

0 < lim inf_{s≥t_{0}} [ lim inf_{t→∞} \frac{H(σ(t), s)}{H(σ(t), t_{0})}] ≤ ∞,  (3.16)

and there exists a positive delta differentiable function α(t) such that

lim_{t→∞} \frac{1}{H(σ(t), t_{0})} ∫_{T_{0}}^{t} \frac{r(g(s, b))(α^{σ}(s))^{2}}{α(s)g^{Δ}(s, b)} Δs < ∞,  (3.17)

hold. If there exists a function φ ∈ C_{rd}([t_{0}, ∞), ℝ) such that

∫_{t_{1}}^{∞} \frac{α(s)g^{Δ}(s, b)}{a(g(s, b))α^{2}(σ(s))}φ_{+}^{2}(s)Δs = ∞,  (3.18)

and for every T ≥ t_{0},

lim inf_{t→∞} \frac{1}{h(t, u)} ∫_{u}^{t} [H(t, s)α(s) ∫_{a}^{b} Mq(s, ξ)[1 − p(g(s, ξ))]Δξ

−\frac{r(g(s, ξ))(α^{σ}(s))^{2}h^{2}(t, s)}{4α(s)g^{Δ}(s, b)}]Δs ≥ φ(T),  (3.19)

where φ_{+}(t) = max{φ(t), 0}, then every solution of equation (1.1) is oscillatory.

Proof. On the contrary, we assume that (1.1) has a nonoscillatory solution x(t). We suppose without loss of generality that x(t) > 0 for all t ∈ [t_{0}, ∞)_{𝕋}. Proceeding as in the proof of Theorem 3.1, for t > u ≥ t_{1} ≥ t_{0}, we have

\frac{1}{H(t, u)} ∫_{u}^{t} [H(t, s)α(s) ∫_{a}^{b} Mq(s, ξ)[1 − p(g(s, ξ))]Δξ

−\frac{r(g(s, ξ))(α^{σ}(s))^{2}h^{2}(t, s)}{4α(s)g^{Δ}(s, b)}]Δs ≤ w(u) − \frac{1}{H(t, u)} ∫_{u}^{t} [\sqrt{\frac{H(t, s)α(s)g^{Δ}(s, b)}{r(g(s, b))(α^{σ}(t))^{2}}}w^{σ}(s)

+\frac{\sqrt{r(g(s, b))}α^{σ}(s)h(t, s)}{2\sqrt{α(s)g^{Δ}(s, b)}}]^{2}Δs.

Let t → ∞ and taking the upper limit, we have

lim sup_{t→∞} \frac{1}{h(t, u)} ∫_{u}^{t} [H(t, s)α(s) ∫_{a}^{b} Mq(s, ξ)[1 − p(g(s, ξ))]Δξ

−\frac{r(g(s, ξ))(α^{σ}(s))^{2}h^{2}(t, s)}{4α(s)g^{Δ}(s, b)}]Δs ≤ w(u) − lim inf_{t→∞} \frac{1}{h(t, u)} ∫_{T}^{u} [\sqrt{\frac{H(t, s)α(s)g^{Δ}(s, b)}{r(g(s, b))(α^{σ}(t))^{2}}}w^{σ}(s)

+\frac{\sqrt{r(g(s, b))}α^{σ}(s)h(t, s)}{2\sqrt{α(s)g^{Δ}(s, b)}}]^{2}Δs.

From (3.19), we have

w(u) ≥ φ(u) for all u ≥ t_{0},  (3.20)

and

lim inf_{t→∞} \frac{1}{h(t, u)} ∫_{u}^{t} [\sqrt{\frac{H(t, s)α(s)g^{Δ}(s, b)}{r(g(s, b))(α^{σ}(t))^{2}}}w^{σ}(s) + \frac{\sqrt{r(g(s, b))}α^{σ}(s)h(t, s)}{2\sqrt{α(s)g^{Δ}(s, b)}}]^{2}Δs
≤ w(u) − φ(u) = M < ∞,  (3.21)

where M is a constant. Let

F (t) = \frac{1}{H(t, t_{1})} ∫_{t_{1}}^{t} \frac{H(t, s)α(s)g^{Δ}(s, b)}{a(g(s, b))α^{2}(σ(s))}(w^{σ}(s))^{2}Δs,  (3.22)

and

G(t) = \frac{1}{H(t, t_{1})} ∫_{t_{1}}^{t} h(t, s)\sqrt{H(t, s)}w^{σ}(s)]Δs  (3.23)

for t > t_{0}. Then by (3.21), (3.22) and (3.23), we obtain

lim inf_{t→∞}[G(t) + F (t)] < ∞.  (3.24)

Now we claim that

∫_{t_{1}}^{∞} \frac{α(s)g^{Δ}(s, b)}{a(g(s, b))α^{2}(σ(s))}(w^{σ}(s))^{2}Δs < ∞.  (3.25)

Suppose to the contrary that

∫_{t_{1}}^{∞} \frac{α(s)g^{Δ}(s, b)}{a(g(s, b))α^{2}(σ(s))}(w^{σ}(s))^{2}Δs = ∞.  (3.26)

From (3.16), there is a positive η > 0 satisfying

lim inf_{s≥t_{0}} [ lim inf_{t→∞} \frac{H(σ(t), s)}{H(σ(t), t_{0})}] > η > 0.  (3.27)

On the other hand by (3.26) for any positive number μ > 0, there exists T ≥ t_{1} such that

∫_{t_{1}}^{∞} \frac{α(s)g^{Δ}(s, b)}{a(g(s, b))α^{2}(σ(s))}(w^{σ}(s))^{2}Δs ≥ \frac{μ}{η} for all t ≥ T.  (3.28)

EJQTDE, 2010 No. 61, p. 9 So, for all t ≥ T > t_{1}, integration by parts yields that

F (t) = \frac{1}{H(t,)} ∫_{t_{1}}^{t} \frac{H(t, s)α(s)g^{Δ}(s, b)}{a(g(s, b))α^{2}(σ(s))}(w^{σ}(s))^{2}Δs
= \frac{1}{H(t, t_{1})} ∫_{t_{1}}^{t} H(t, s)Δ[ ∫_{t_{1}}^{s} \frac{α(u)g^{Δ}(u, b)}{a(g(u, b))α^{2}(σ(u))}(w^{σ}(u))^{2}Δu]
= \frac{1}{H(t, t_{1})} ∫_{T}^{t} [ ∫_{t_{1}}^{s} \frac{α(u)g^{Δ}(u, b)}{a(g(u, b))α^{2}(σ(u))}(w^{σ}(u))^{2}Δu]H^{Δ_{s}}(t, s)Δs
≥ \frac{μ}{η} \frac{1}{H(t, t_{1})} ∫_{T}^{t} H^{Δ_{s}}(σ(t), s)Δs
= \frac{μ}{η} \frac{H(t, T)}{H(t, t_{1})} ≥ \frac{μ}{η} \frac{H(t, T)}{H(t, t_{0})}.  (3.29)

It follows from (3.27) that

lim inf_{t→∞} \frac{H(t, s)}{H(t, t_{0})} ≥ η > 0,  s ≥ t_{0}.  (3.30)

Therefore, there exists a t_{2} ≥ T such that

\frac{H(t, T)}{H(t, t_{0})} ≥ η  (3.31)

for all t ≥ t_{2}. From (3.29) and (3.31),we have

G(t) ≥ η for t ≥ t_{2}.  (3.32)

Since μ is arbitrary, we conclude that

lim_{t→∞} F (t) = ∞.  (3.33)

From (3.24), there is a sequence {t_{n}}_{n=1}^{∞} in [t_{1}, ∞)_{𝕋} with lim_{n→∞} t_{n} = ∞, and such that

lim_{n→∞}(F (t_{n}) + G(t_{n})) = lim_{t→∞}(F (t_{n}) + G(t_{n})) < ∞.

Thus there exist constants N_{1} and M such that

F (t_{n}) + G(t_{n}) ≤ M  for n > N_{1}.  (3.34)

It follows from (3.33) that

lim_{n→∞} F (t_{n}) = ∞.  (3.35)

Further, from (3.34) and (3.35), we obtain

lim_{n→∞} G(t_{n}) = −∞.  (3.36)

Then for any ϵ ∈ (0, 1), there exists a positive integer N_{2} such that

\frac{G(t_{n})}{F (t_{n})} + 1 < ϵ, for n > N_{2},

EJQTDE, 2010 No. 61, p. 10 or

\frac{G(t_{n})}{F (t_{n})} < ϵ − 1 < 0.  (3.37)

From (3.36) and (3.37),we have

lim_{n→∞} \frac{G(t_{n})}{F (t_{n})}G(t_{n}) = ∞.  (3.38)

On the other hand, by Schwarz inequality,we have

0 ≤ G^{2}(t_{n}) = \frac{1}{H^{2}(t_{n}, t_{1})}\{∫_{t_{1}}^{t_{n}} \sqrt{H(t_{n}, s)}h(t_{n}, s)w^{σ}(s)Δs\}^{2}

≤ \{\frac{1}{H(t_{n}, t_{1})} ∫_{t_{1}}^{t_{n}} \frac{H(t_{n}, s)α(s)g^{Δ}(s, b)}{a(g(s, b))α^{2}(σ(s))}(w^{σ}(s))^{2}Δs\}^{2}

×\{\frac{1}{H(t_{n}, t_{1})} ∫_{t_{1}}^{t_{n}} \frac{a(g(s, b))α^{2}(σ(s))}{α(s)g^{Δ}(s, b)}h(t_{n}, s)(Δs\}

= F (t_{n})\frac{1}{H(t_{n}, t_{1})} ∫_{t_{1}}^{t_{n}} \frac{a(g(s, b))α^{2}(σ(s))}{α(s)g^{Δ}(s, b)}h(t_{n}, s)(Δs,

\frac{G^{2}(t_{n})}{F (t_{n})} ≤ \frac{1}{H(t_{n}, t_{1})} ∫_{t_{1}}^{t_{n}} \frac{a(g(s, b))α^{2}(σ(s))}{α(s)g^{Δ}(s, b)}h(t_{n}, s)Δs,  (3.39)

for all large n. In view of (3.31), we obtain

\frac{1}{H(t_{n}, t_{1})} ≤ \frac{1}{H(t_{n}, T)} = \frac{H(t_{n}, t_{0}}{H(t_{n}, T)} \frac{1}{H(t_{n}, t_{0})} ≤ \frac{1}{LH(t_{n}, t_{0})},  (3.40)

and therefore, from (3.39) and (3.40), we have

\frac{G^{2}(t_{n})}{F (t_{n})} ≤ \frac{1}{LH(t_{n}, t_{1})} ∫_{t_{1}}^{t_{n}} \frac{a(g(s, b))α^{2}(σ(s))}{α(s)g^{Δ}(s, b)}h(t_{n}, s)Δs.  (3.41)

From (3.41) and(3.38), we have

lim_{n→∞} \frac{1}{H(σ(t_{n}), t_{0})} ∫_{t_{0}}^{t_{n}} \frac{a(g(s, b))α^{2}(σ(s))}{α(s)g^{Δ}(s, b)}h(t_{n}, s)Δs = ∞.  (3.42)

This implies that

lim sup_{t→∞} \frac{1}{H(σ(t), t_{0})} ∫_{t_{0}}^{t} \frac{a(g(s, b))α^{2}(σ(s))}{α(s)g^{Δ}(s, b)}h(t, s)Δs = ∞,  (3.43)

which contradicts (3.17). Therefore, from (3.25) and w^{σ}(s) ≥ φ(s), we have

∫_{t_{1}}^{∞} \frac{α(s)g^{Δ}(s, b)}{a(g(s, b))α^{2}(σ(s))}φ_{+}^{2}(s)Δs ≤ ∫_{t_{1}}^{∞} \frac{α(s)g^{Δ}(s, b)}{a(g(s, b))α^{2}(σ(s))}w^{2}(σ(s))Δs < ∞, (3.44)

EJQTDE, 2010 No. 61, p. 11 which contradicts (3.18). This completes the proof. □

Next we obtain sufficient condition for the equation (1.1) to be oscillatory when condition (C_{2}) holds.

Lemma 3.9. Assume condition (C_{2}) holds. If x(t) is an eventually positive solution of equation (1.1) and y(t) > 0, then y(t) satisfies the following inequality

r(t)π(t)y^{Δ}(t) + y(t) ≥ 0  (3.45)

where

π(t) = ∫_{t}^{∞} \frac{1}{r(s)}Δs

for all sufficiently large t ≥ t_{0}.

Proof. From (3.3), it is clear that r(t)y^{Δ}(t) is nonincreasing on [t_{0}, ∞)_{𝕋} for some t_{0} > 0.. Then

r(u)y^{Δ}(u) ≤ r(s)y^{Δ}(s) for all u > s > t_{0}.

Dividing the last inequality by r(u) and integrating over [s, t]_{𝕋}, we obtain

0 < y(t) ≤ y(s) + r(s)y^{Δ}(s) ∫_{s}^{t} \frac{1}{r(u)}Δu, t > s > t_{0}.  (3.46)

Then we consider the following two cases: Case(I): y^{Δ}(t) ≥ 0. From (3.46), we find

0 < y(t) ≤ y(s) + r(s)y^{Δ}(s) ∫_{s}^{t} \frac{1}{r(u)}Δu,  (3.47)
≤ y(s) + r(s)y^{Δ}(s) ∫_{s}^{∞} \frac{1}{r(u)}Δu  (3.48)

which implies (3.45). Case(II): y^{Δ}(t) < 0. From (3.46), the condition (C_{2}) implies that y(t) is bounded from above. Letting t → ∞ in (3.46), we have

0 ≤ r(t)π(t)(y^{Δ}(t)) + y(t)

which gives (3.45). □

Theorem 3.10. Assume that condition (C_{2}) holds. If

∫ ^{∞} π(σ(t)) ∫_{a}^{b} kQ(t, ξ)π(g(t, ξ))ΔξΔs = ∞,  (3.49)

Proof. Suppose to the contrary that x(t) > 0 on [t_{0}, ∞)_{𝕋}. Then there exists a point t_{1} in [t_{0}, ∞)_{𝕋} such that x(t) > 0, x(t − τ) > 0 and x(g(t, ξ)) > 0 for all ξ ∈ [a, b]_{𝕋}, and for all t ∈ [t_{1}, ∞)_{𝕋}. From (3.3), we write (r(t)y^{Δ}(t)) ≤ 0. Therefore r(t)y^{Δ}(t) ≥ 0 or r(t)y^{Δ}(t) < 0 eventually. Since r(t) > 0, we have y^{Δ}(t) ≥ 0 or y^{Δ}(t) < 0. Now we consider the case y^{Δ}(t) ≥ 0. We see that y(t) ≥ y(t_{2}) for some t_{2} ∈ [t_{1}, ∞)_{𝕋}. Since lim_{t→∞} π(t) = 0, we find that y(t_{2}) ≥ π(t) for all t ∈ [t_{2}, ∞)_{𝕋.} Hence y(t) ≥ π(t) for all t ∈ [t_{2}, ∞)_{𝕋}. Suppose that y^{Δ}(t) < 0. Then since (r(t)y^{Δ}(t))^{Δ} ≤ 0 for all t ∈ [t_{2}, ∞)_{𝕋},

r(t)y^{Δ}(t) ≤ r(t_{2})y^{Δ}(t_{2}) = −c_{0} < 0, t ∈ [t_{2}, ∞)_{𝕋}.  (3.50)

Substituting (3.45) in the above inequality we obtain

y(t) ≥ c_{0}π(t) for t ∈ [t_{2}, ∞)_{𝕋},  (3.51)

or

y(g(t, ξ)) ≥ c_{0}π(g(t, ξ)) for t ∈ [t_{2}, ∞)_{𝕋}.  (3.52)

Then from equation (1.1), we have

(r(t)y^{Δ}(t))^{Δ} + ∫_{a}^{b} Mc_{0}q(t, ξ)[1 − p(g(t, ξ))]π(g(t, ξ))Δξ ≤ 0.  (3.53)

Now multiplying the last inequality by π(σ(t)) and then integrating, we obtain

π(t)r(t)y^{Δ}(t) + y(t) − π(t_{2})r(t_{2})y^{Δ}(t_{2}) − y(t_{2})
+ ∫_{t_{2}}^{t} π(σ(t)) ∫_{a}^{b} Mc_{0}q(s, ξ)[1 − p(g(s, ξ))]π(g(s, ξ))ΔξΔs ≤ 0.

This implies that

∫_{t_{2}}^{t} π(σ(t)) ∫_{a}^{b} Mc_{0}q(s, ξ)[1 − p(g(s, ξ))]π(g(s, ξ))ΔξΔs ≤ π(t_{2})r(t_{2})y^{Δ}(t_{2}) + y(t_{2}).

This contradicts (3.49) as t → ∞. Hence every solution of equation (1.1) is oscillatory. The proof is now complete. □

4. Examples

In this section, we present some examples to illustrate our main results when the conditions (C_{1}) and (C_{2}) hold.

Example 4.1. Consider the dynamic equation

((x(t) + \frac{1}{2}x(t − 1))^{ΔΔ} + ∫_{0}^{1} x(t − ξ)Δξ = 0.  (4.1)

where r(t) = 1, p(t) = 1, a = 0, b = 1, τ = 1, q(t, ζ) = 1. By Corollary 3.6, every solution of equation (4.1) is oscillatory.

EJQTDE, 2010 No. 61, p. 13 Example 4.2. Consider the dynamic equation

((x(t) + \frac{1}{t + 1}x(t − 1)))^{ΔΔ} + ∫_{0}^{1} γ \frac{t − ζ + 1}{t(t − ξ)}x(t − ξ)Δξ = 0.  (4.2)

Here p(t) = \frac{1}{t+1}, r(t) = 1, τ = 1, a = 1/2, b = 1, f(u) = u, M = 1, q(t, ξ) = \frac{t−ξ+1}{t(t−ξ)} and g(t, ξ) = t − ξ. Then by Corollary 3.7, we have

lim sup_{t→∞} ∫_{t_{0}}^{t} ∫_{a}^{b} Mq(s, ξ)[1 − p(g(s, ξ))] ΔξΔs
= lim sup_{t→∞} ∫_{t_{0}}^{t} ∫_{1/2}^{1} \frac{t − ξ + 1}{t(t − ξ)}[1 − \frac{1}{t − ξ + 1}]ΔξΔs
= lim sup_{t→∞} ∫_{t_{0}}^{t} \frac{1}{t}Δξ
= ∞.

Therefore every solution of equation (4.2) is oscillatory.

Example 4.3. Consider the dynamic equation

(tσ(t)(x(t) + p(t)x(t − τ))^{Δ})^{Δ} + ∫_{0}^{1} \frac{σ(t)}{t}(σ(t) − ξ)x(t − ξ)Δξ = 0, t ∈ [1, ∞)_{𝕋}. (4.3)

Here p(t) = 1/2, r(t) = tσ(t), q(t, ξ) = \frac{σ(t)}{t}(σ(t) − ξ), and g(t, ξ) = t − ξ. Since (C_{2}) holds, from Theorem 3.10, we have

lim_{t→∞} ∫_{1}^{t} π(s) ∫_{0}^{1} q(s, ξ)π(g(s, ξ))ΔξΔs = lim_{t→∞} ∫_{1}^{t} \frac{1}{σ(s)} ∫_{0}^{1} \frac{σ(s)}{s}(σ(s) − ξ)(g(t, ξ))ΔξΔs
= lim_{t→∞} ∫_{1}^{t} \frac{1}{s}Δs = ∞.

Hence every solution of equation (1.1) is oscillatory.

Acknowledgements. The authors thank the referee for his constructive suggestions which improved the content of the paper.

References

[1] R. P. Agarwal, S. R. Grace and D. O’Regan, Oscillation Theory for Second Order Dynamic Equations, Taylor and Francis, London, 2003.

[2] R. P. Agarwal, M. Bohner and W. T. Li, Nonoscillation and Oscillation: Theory of Functional Differential Equations, Marcell Dekker, New York, 2004.

[3] R. P. Agarwal, M. Bohner and A. Peterson, Dynamic equations on time scales: A survey. J.Comp.Appl.Math., Special Issue on Dynamic Equations on Time Scales, edited by R. P. Agarwal, M. Bohner and D. O’Regan (Preprint in Ulmer Seminare 5), 141(1-2)(2002), 1-26.

EJQTDE, 2010 No. 61, p. 14

[4] M. Bohner and A. Peterson, Dynamic Equations on Time Scales, An Introduction with Applications, Birkhauser, Boston, 2001.

[5] M. Bohner and A. Peterson, Advances in Dynamic Equations on Time Scales, Birkhauser, Boston, 2003.

[6] L. Erbe, A. Peterson and S. H. Saker,, Oscillation criteria for second order nonlinear delay dynamic equations, J.Math.Anal.Appl., 333(2007), 505-522.

[7] M. Huang and W. Feng, Oscillation for forced second order nonlinear dynamic equations on time scales, Elect.J.Diff.Eqn., 145 (2005), 1-8.

[8] S. Hilger, Analysis on measure chains - A unified approach to continuous and discrete calculus, Results Math. 8 (1990), 18-56.

[9] G. H. Hardy, J. E. Littlewood and G. Polya, Inequalities, Second ed., Cambridge University Press, Cambridge,1952.

[10] J. C. Jiang and X. P. Li, Oscillation of second order nonlinear neutral differential equations, Appl.Math.Comput., 135(2005), 531-540.

[11] J. Jiang, Oscillation of second order nonlinear neutral delay difference equations, Appl.Math.Comput., 146(2003), 791-801.

[12] H. J. Li and W. L. Liu, Oscillation criteria for second order neutral differnetial equation, Canad.J.Math., 48 (2006), 871-886.

[13] Ch. G. Philos, I. K. Purnaras and Y. G. Sficas, Oscillation in second order linear differential equation with distributed type deviating arguments, J.Math.Anal.Appl. 176(1993), 458-492.

[14] S. H. Saker, Oscillation of second order nonlinear neutral delay dynamic equations, J.Comp.Appl.Math. 187(2006), 123-141.

[15] S. H. Saker, Oscillation of second order neutral linear differential equations, J.Math,Anal.Appl., 194(1995), 312-321.

[16] S. H. Samir, D. O’Regan and R. P. Agarwal, Oscillation theorems for second order nonlinear neutral delay dynamic equations on time scales, Acta.Math.Sinica, English Series, 24 (2007), 1409-1432.

[17] Y. Sahiner, Oscillation of second order delay differential equations on time scales, Nonlinear Analysis: Theory, Methods and Applications, 63(2005), 173-180.

[18] H. Wu, R. K. Zhuang, and R. M. Mathsen, Oscillation criteria for second-order nonlinear neutral variable delay dynamic equations, Appl.Math.Comp., 178(2006), 321-331.

[19] J. Zho and F. Meng , Oscillation criteria for second-order neutral equations with distributed deviating argument, Appl.Math.Comp. 206 (2008), 485-493.

(Received February 1, 2010)