2004 Conference on Diff. Eqns. and Appl. in Math. Biology, Nanaimo, BC, Canada. Electronic Journal of Differential Equations, Conference 12, 2005, pp. 47–56. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp)

FUNCTIONAL DIFFERENTIAL EQUATIONS OF THIRD ORDER

TUNCAY CANDAN, RAJBIR S. DAHIYA

Abstract. In this paper, we consider the third-order neutral functional differential equation with distributed deviating arguments. We give sufficient conditions for the oscillatory behavior of this functional differential equation.

1. Introduction

The aim of this paper is to develop some oscillation theorems for a third-order equations of the form

[a(t)[b(t)[x(t) + c(t)x(t − τ)]′]′]′ + ∫_{a}^{b} p(t, ξ)x(σ(t, ξ))dξ = 0,  (1.1)

where

(a) a(t), a′(t), b(t), c(t) ∈ C([t_{0}, ∞), (0, ∞)), 0 < c(t) ≤ 1, a′(t) ≥ 0;
(b) ∫^{∞} \frac{dt}{b(t)} = ∞ and ∫^{∞} \frac{dt}{a(t)} = ∞

(c) p(t, ξ) ∈ C([t_{0}, ∞) × [a, b], [0, ∞)), and p(t, ξ) is not eventually zero on any

half line [t_{m}, ∞) × [a, b], t_{m} ≥ t_{0}

(d) σ(t, ξ) ∈ C([t_{0}, ∞) × [a, b], R), σ(t, ξ) + τ ≤ t, σ(t, ξ) is nondecreasing with

respect to t and ξ, respectively, and lim inf_{t→∞ ξ∈[a,b]} σ(t, ξ) = ∞.

As is customary, a solution of equation (1.1) is called oscillatory if it has arbitrarily large zeros. Otherwise, it is called nonoscillatory.

Oscillatory solutions of third-order differential equations

(r_{2}(t)(r_{1}(t)x′(t))′)′ + q(t)f(x(g(t))) = h(t),

and

(b(t)(a(t)y′(t))′)′ + (q_{1}(t)y)′ + q_{2}(t)y′ = f(t)

were considered in [10] and [4], respectively. We refer to [1]-[3] and [7]-[9] for more studies. However, our results are more general with different proofs than those works.

2000 Mathematics Subject Classification. 34K11, 34K15.

Key words and phrases. Oscillation theory; neutral equations.

\begin{matrix}c \\ ◯ \end{matrix}2005 Texas State University - San Marcos.

Published Arpil 20, 2005.

2. Main Results

Let D_{0} = {(t, s)|t > s ≥ t_{0}}, D = {(t, s)|t ≥ s ≥ t_{0}}.

Theorem 2.1. Suppose that there exist \frac{d}{dt}σ(t, a) and H(t, s) ∈ C′(D; R), h(t, s) ∈ C(D_{0}; R) and ρ(t) ∈ C′([t_{0}, ∞), (0, ∞)) such that

(i) H(t, t) = 0, H(t, s) > 0

(ii) H′_{s}(t, s) ≤ 0, and −H′_{s}(t, s) − H(t, s)\frac{ρ′(s)}{ρ(s)} = h(t, s)\sqrt{H(t, s)}.

If

\begin{matrix}lim sup_{t→∞} \frac{1}{H(t, t_{0})} ∫_{t_{0}}^{t} [H(t, s)ρ(s) ∫_{a}^{b} p(s, ξ)[1 − c(σ(s, ξ))]dξ \\ − \frac{a(s)ρ(s)b(σ(s, a))h^{2}(t, s)}{4[σ(s, a) − T]σ′(s, a)}]ds = ∞ \end{matrix}  (2.1)

and

∫_{t_{3}}^{∞} [ ∫_{t_{3}}^{r} \frac{du}{b(u)}( ∫_{u}^{r} \frac{dv}{a(v)})] ∫_{a}^{b} p(r, ξ)dξdr = ∞.  (2.2)

Then every solution of (1.1) is oscillatory or tends to zero as t → ∞.

Proof. Assume, for the sake of contradiction, that equation (1.1) has an eventually positive solution x(t). That is, there exists a t_{0} ≥ 0 such that x(t) > 0 for t ≥ t_{0}. If we put

y(t) = x(t) + c(t)x(t − τ)  (2.3)

then, from (1.1), we obtain

[a(t)[b(t)y′(t)]′]′ = − ∫_{a}^{b} p(t, ξ)x(σ(t, ξ))dξ.  (2.4)

Since x(t) is an eventually positive solution of (1.1) and σ(t, ξ) → ∞ as t → ∞, ξ ∈ [a, b], there exist a t_{1} ≥ t_{0} such that x(t − τ) > 0 and x(σ(t, ξ)) > 0 for t ≥ t_{1}, ξ ∈ [a, b]. Thus in view of (2.3) and (2.4), we have y(t) > 0, t ≥ t_{1} and [a(t)[b(t)y′(t)]′]′ ≤ 0, t ≥ t_{1}. Thus, y(t), y′(t), (b(t)y′(t))′ are monotone and eventually one-signed. We want to show that there is a t_{2} ≥ t_{1} such that

(b(t)y′(t))′ > 0, t ≥ t_{2}.  (2.5)

Suppose on the contrary, (b(t)y′(t))′ ≤ 0. Since the right hand side of (2.4) is not identically zero and a(t) > 0, it is clear that there exists a t_{3} ≥ t_{2} such that a(t_{3})(b(t_{3})y′(t_{3}))′ < 0. Then we have

a(t)(b(t)y′(t))′ ≤ a(t_{3})(b(t_{3})y′(t_{3}))′ < 0, t ≥ t_{3}.  (2.6)

Dividing (2.6) by a(t) and integrating from t_{3} to t, we obtain

b(t)y′(t) − b(t_{3})y′(t_{3}) ≤ a(t_{3})(b(t_{3})y′(t_{3}))′ ∫_{t_{3}}^{t} \frac{ds}{a(s)}.  (2.7)

Letting t → ∞ in (2.7), and because of (b) we see that b(t)y′(t) → −∞ as t → ∞. Thus, there is a t_{4} ≥ t_{3} such that b(t_{4})y′(t_{4}) < 0. By making use of (b(t)y′(t))′ ≤ 0, we obtain

b(t)y′(t) ≤ b(t_{4})y′(t_{4}) < 0, t ≥ t_{4}.  (2.8)

If we divide (2.8) by b(t) and integrate from t_{4} to t with t → ∞, the right-hand side becomes negative. Thus, we have y(t) → −∞. But this is a contradiction, since y(t) is eventually positive, which therefore proves that (2.5) holds. Now we have two possibilities: (I) y′(t) > 0 for t ≥ t_{2}, (II) y′(t) < 0 for t ≥ t_{2}.

(I) Assume y′(t) > 0 for t ≥ t_{2}. From (2.3), y(t) ≥ x(t) for t ≥ t_{2} and

y(σ(t, ξ)) ≥ y(σ(t, ξ) − τ) ≥ x(σ(t, ξ) − τ), t ≥ t_{3} ≥ t_{2}.

Thus from (2.3) and (2.4),

[a(t)[b(t)y′(t)]′]′ = − ∫_{a}^{b} p(t, ξ)[y(σ(t, ξ)) − c(σ(t, ξ))x(σ(t, ξ) − τ)]dξ
≤ − ∫_{a}^{b} p(t, ξ)[y(σ(t, ξ)) − c(σ(t, ξ))y(σ(t, ξ))]dξ
= − ∫_{a}^{b} p(t, ξ)[1 − c(σ(t, ξ))]y(σ(t, ξ))dξ
≤ −y(σ(t, a)) ∫_{a}^{b} p(t, ξ)[1 − c(σ(t, ξ))]dξ.

Then, we have

\frac{[a(t)[b(t)y′(t)]′]′}{y(σ(t, a))} ≤ − ∫_{a}^{b} p(t, ξ)[1 − c(σ(t, ξ))]dξ.

Now set

z(t) = \frac{a(t)(b(t)y′(t))′ρ(t)}{y(σ(t, a))}.

It is obvious that z(t) > 0 for t ≥ t_{3} and the derivative of z(t) is

\begin{aligned}z′(t) & = \frac{[a(t)(b(t)y′(t))′]′ρ(t)}{y(σ(t, a))} + \frac{ρ′(t)}{ρ(t)}z(t) − \frac{[a(t)(b(t)y′(t))′]ρ(t)y′(σ(t, a))σ′(t, a)}{y^{2}(σ(t, a))} \\ & ≤ −ρ(t) ∫_{a}^{b} p(t, ξ)[1 − c(σ(t, ξ))]dξ + \frac{ρ′(t)}{ρ(t)}z(t) − \frac{y′(σ(t, a))σ′(t, a)z(t)}{y(σ(t, a))}. \end{aligned}

(2.9)

On the other hand, since (a(t)(b(t)y′(t))′)′ ≤ 0 and a′(t) ≥ 0, we find that

(b(t)y′(t))′′ ≤ 0.  (2.10)

Using the above inequality and

b(t)y′(t) = b(T)y′(T) + ∫_{T}^{t}(b(s)y′(s))′ds,

we obtain

b(t)y′(t) ≥ (t − T)(b(t)y′(t))′, t ≥ T ≥ t_{3}.

Since (by′)′ is non-increasing, we have

b(σ(t, a))y′(σ(t, a)) ≥ (σ(t, a) − T)(b(t)y′(t))′, t ≥ t_{4} ≥ t_{3}.

Thus, we have

y′(σ(t, a)) ≥ \frac{(σ(t, a) − T)(b(t)y′(t))′}{b(σ(t, a))} .  (2.11)

Then, substituting (2.11) in (2.9), it follows that

z′(t) ≤ −ρ(t) ∫_{a}^{b} p(t, ξ)[1 − c(σ(t, ξ))]dξ + \frac{ρ′(t)}{ρ(t)}z(t) − \frac{[σ(t, a) − T]σ′(t, a)z^{2}(t)}{a(t)ρ(t)b(σ(t, a))},

and

ρ(t) ∫_{a}^{b} p(t, ξ)[1 − c(σ(t, ξ))]dξ ≤ −z′(t) + \frac{ρ′(t)}{ρ(t)}z(t) − \frac{[σ(t, a) − T]σ′(t, a)z^{2}(t)}{a(t)ρ(t)b(σ(t, a))}.

(2.12)

Multiplying both sides of equation (2.12) by H(t, s), and integrating by parts from T^{∗} to t, and using the properties (i) and (ii), we obtain

∫_{T^{∗}}^{t} H(t, s)ρ(s) ∫_{a}^{b} p(s, ξ)[1 − c(σ(s, ξ))]dξds

≤ − ∫_{T^{∗}}^{t} H(t, s)z′(s)ds + ∫_{T^{∗}}^{t} \frac{H(t, s)ρ′(s)z(s)}{ρ(s)}ds

− ∫_{T^{∗}}^{t} \frac{H(t, s)[σ(s, a) − T]σ′(s, a)z^{2}(s)}{a(s)ρ(s)b(σ(s, a))}ds

= H(t, T^{∗})z(T^{∗}) + ∫_{T^{∗}}^{t} [\frac{dH(t, s)}{ds} + H(t, s)\frac{ρ′(s)}{ρ(s)}]z(s)ds

− ∫_{T^{∗}}^{t} \frac{H(t, s)[σ(s, a) − T]σ′(s, a)z^{2}(s)}{a(s)ρ(s)b(σ(s, a))}ds

= H(t, T^{∗})z(T^{∗}) − ∫_{T^{∗}}^{t} h(t, s)\sqrt{H(t, s)}z(s)ds

− ∫_{T^{∗}}^{t} \frac{H(t, s)[σ(s, a) − T]σ′(s, a)z^{2}(s)}{a(s)ρ(s)b(σ(s, a))}ds

= H(t, T^{∗})z(T^{∗}) − ∫_{T^{∗}}^{t} [\sqrt{\frac{H(t, s)[σ(s, a) − T]σ′(s, a)}{a(s)b(σ(s, a))ρ(s)}}z(s)

+ \frac{\sqrt{a(s)b(σ(s, a))ρ(s)}h(t, s)}{2\sqrt{[σ(s, a) − T]σ′(s, a)}}]^{2}ds + ∫_{T^{∗}}^{t} \frac{a(s)ρ(s)b(σ(s, a))h^{2}(t, s)}{4[σ(s, a) − T]σ′(s, a)}ds,

t > T^{∗} ≥ t_{4}. As a result of this, we get

∫_{T^{∗}}^{t} [H(t, s)ρ(s) ∫_{a}^{b} p(s, ξ)[1 − c(σ(s, ξ))]dξ − \frac{a(s)ρ(s)b(σ(s, a))h^{2}(t, s)}{4[σ(s, a) − T]σ′(s, a)}]ds

= H(t, T^{∗})z(T^{∗}) − ∫_{T^{∗}}^{t} [\sqrt{\frac{H(t, s)[σ(s, a) − T]σ′(s, a)}{a(s)b(σ(s, a))ρ(s)}}z(s)

+ \frac{\sqrt{a(s)b(σ(s, a))ρ(s)}h(t, s)}{2\sqrt{[σ(s, a) − T]σ′(s, a)}}]^{2}ds .

From (ii) H′_{s}(t, s) ≤ 0, we have H(t, t_{4}) ≤ H(t, t_{0}), t_{4} ≥ t_{0} and therefore

∫_{t_{4}}^{t} [H(t, s)ρ(s) ∫_{a}^{b} p(s, ξ)[1 − c(σ(s, ξ))]dξ − \frac{a(s)ρ(s)b(σ(s, a))h^{2}(t, s)}{4[σ(s, a) − T]σ′(s, a)}]ds

≤ H(t, t_{4})z(t_{4}) ≤ H(t, t_{0})z(t_{4})

which implies that

\frac{1}{H(t, t_{0})} ∫_{t_{0}}^{t} [H(t, s)ρ(s) ∫_{a}^{b} p(s, ξ)[1 − c(σ(s, ξ))]dξ − \frac{a(s)ρ(s)b(σ(s, a))h^{2}(t, s)}{4[σ(s, a) − T]σ′(s, a)}]ds

= \frac{1}{H(t, t_{0})}[ ∫_{t_{0}}^{t_{4}} + ∫_{t_{4}}^{t} ][H(t, s)ρ(s) ∫_{a}^{b} p(s, ξ)[1 − c(σ(s, ξ))]dξ

− \frac{a(s)ρ(s)b(σ(s, a))h^{2}(t, s)}{4[σ(s, a) − T]σ′(s, a)}]ds

≤ z(t_{4}) + ∫_{t_{0}}^{t_{4}} \frac{H(t, s)}{H(t, t_{0})}ρ(s) ∫_{a}^{b} p(s, ξ)[1 − c(σ(s, ξ))]dξds

≤ z(t_{4}) + ∫_{t_{0}}^{t_{4}} ρ(s) ∫_{a}^{b} p(s, ξ)[1 − c(σ(s, ξ))]dξds.

Now taking upper limits as t → ∞, we obtain

lim sup_{t→∞} \frac{1}{H(t, t_{0})} ∫_{t_{0}}^{t} [H(t, s)ρ(s) ∫_{a}^{b} p(s, ξ)[1 − c(σ(s, ξ))]dξ

− \frac{a(s)ρ(s)b(σ(s, a))h^{2}(t, s)}{4[σ(s, a) − T]σ′(s, a)}]ds

≤ z(t_{4}) + ∫_{t_{0}}^{t_{4}} ρ(s) ∫_{a}^{b} p(s, ξ)[1 − c(σ(s, ξ))]dξds = M < ∞,

where M is a constant. Hence, this result leads to a contradiction to (2). (II) Assume y′(t) < 0 for t ≥ t_{2}. We integrate (1.1) from t to ∞ and since

a(t)(b(t)y′(t))′ > 0, t ≥ t_{2},

we have

−a(t)(b(t)y′(t))′ + ∫_{t}^{∞} ∫_{a}^{b} p(r, ξ)x(σ(r, ξ))dξdr ≤ 0.  (2.13)

Now integrating (2.13) from t to ∞ after dividing by a(t) and using b(t)y′(t) < 0, will lead to

b(t)y′(t) + ∫_{t}^{∞} (∫_{t}^{r} \frac{du}{a(u)}) ∫_{a}^{b} p(r, ξ)x(σ(r, ξ))dξdr ≤ 0.  (2.14)

Dividing (2.14) by b(t) and integrating again from t to ∞ gives

∫_{t}^{∞} [ ∫_{t}^{r} \frac{du}{b(u)}( ∫_{u}^{r} \frac{dv}{a(v)})] ∫_{a}^{b} p(r, ξ)x(σ(r, ξ))dξdr ≤ y(t)  (2.15)

for t ≥ t_{3} ≥ t_{2}. Replacing t by t_{3} in (2.15), we get

∫_{t_{3}}^{∞} [ ∫_{t_{3}}^{r} \frac{du}{b(u)}( ∫_{u}^{r} \frac{dv}{a(v)})] ∫_{a}^{b} p(r, ξ)x(σ(r, ξ))dξ dr ≤ y(t_{3}).  (2.16)

On the other hand, since y is monotonically decreasing function in the interval [t_{3}, ∞], we get lim_{t→∞} y(t) = lim_{t→∞}[x(t) + c(t)x(t − τ)] = K ≥ 0. Suppose that K > 0, then [x(t) + c(t)x(t − τ)] ≥ \frac{K}{2} > 0 for t ≥ t_{4} ≥ t_{3}. From this we can observe that there exists K_{1} such that x(σ(r, ξ)) ≥ K_{1} > 0. Thus from (2.16)

∫_{t_{3}}^{∞} [ ∫_{t_{3}}^{r} \frac{du}{b(u)}( ∫_{u}^{r} \frac{dv}{a(v)})] ∫_{a}^{b} p(r, ξ)K_{1}dξdr ≤ y(t_{3}).

From the previous equation, we have

∫_{t_{3}}^{∞} [ ∫_{t_{3}}^{r} \frac{du}{b(u)}( ∫_{u}^{r} \frac{dv}{a(v)})] ∫_{a}^{b} p(r, ξ)dξdr < ∞.

This is a contradiction to (2.2). Therefore, lim_{t→∞}[x(t) + c(t)x(t − τ)] = 0. Then lim_{t→∞} x(t) = 0, so the proof is complete. □

Example 2.2. Consider the functional differential equation

[x(t) + \frac{1}{2}x(t − π)]′′′ + ∫_{1/5π}^{2/7π} \frac{(2 − e^{−π})e^{1/ξ}}{ξ^{2}} x(t − \frac{1}{ξ})dξ = 0

so that a(t) = b(t) = 1, c(t) = \frac{1}{2}, τ = π, p(t, ξ) = \frac{(2−e^{−π})e^{1/ξ}}{ξ^{2}} , σ(t, ξ) = t − \frac{1}{ξ},

σ(t) = σ(t, b) = t − \frac{7π}{2} , ρ(s) = s, H(t, s) = (t − s)^{2}, h(t, s) = [2 − \frac{(t−s)}{s}].

We can see that the conditions of Theorem 2.1 are satisfied. It is easy to verify that x(t) = e^{t} sin t is a solution of this problem, which is oscillatory.

Theorem 2.3. Suppose that the conditions of Theorem 2.1, and condition (2.2) holds, and

0 < inf_{s≥t_{0}} [ lim inf_{t→∞} \frac{H(t, s)}{H(t, t_{0})}] ≤ ∞,  (2.17)
lim sup_{t→∞} \frac{1}{H(t, t_{0})} ∫_{t_{0}}^{t} \frac{a(s)ρ(s)b(σ(s, a))h^{2}(t, s)}{[σ(s, a) − T)]σ′(s, a)}ds < ∞.  (2.18)

If there exists a function ϕ(t) ∈ C([t_{0}, ∞), R) satisfying

\begin{matrix}lim sup_{t→∞} \frac{1}{H(t, u)} ∫_{u}^{t} [H(t, s)ρ(s) ∫_{a}^{b} p(s, ξ)[1 − c(σ(s, ξ))]dξ \\ − \frac{a(s)ρ(s)b(σ(s, a))h^{2}(t, s)}{4[σ(s, a) − T)]σ′(s, a)}]ds \\ ≥ ϕ(u), u ≥ t_{0}, \end{matrix}  (2.19)

and

lim sup_{t→∞} \frac{1}{H(t, t_{0})} ∫_{t_{0}}^{t} \frac{[σ(u, a) − T]σ′(u, a)ϕ_{+}^{2}(u)du}{a(u)ρ(u)b(σ′(u, a))} = ∞,

ϕ_{+}(u) = max_{u≥t_{0}}{ϕ(u), 0}.

(2.20)

Then every solution of functional differential equation (1.1) is oscillatory or tends to zero as t → ∞.

Proof. Assume, for the sake of contradiction, that equation (1.1) has positive solution, say x(t) > 0, t ≥ t_{0}. Then, proceeding as in the proof of Theorem 2.1, we obtain

\frac{1}{H(t, u)} ∫_{u}^{t} [H(t, s)ρ(s) ∫_{a}^{b} p(s, ξ)[1 − c(σ(s, ξ))]dξ − \frac{a(s)ρ(s)b(σ(s, a))h^{2}(t, s)}{4[σ(s, a) − T]σ′(s, a)}]ds

≤ z(u) − \frac{1}{H(t, u)} ∫_{u}^{t} [\sqrt{\frac{H(t, s)[σ(s, a) − T]σ′(s, a)}{a(s)b(σ(s, a))ρ(s)}}z(s)

+ \frac{\sqrt{a(s)b(σ(s, a))ρ(s)}h(t, s)}{2\sqrt{[σ(s, a) − T]σ′(s, a)}}]^{2}ds,

t > u ≥ t_{1} ≥ t_{0}. Thus taking upper limit as t → ∞ and using (2.19), we have

z(u) ≥ ϕ(u) + lim inf_{t→∞} \frac{1}{H(t, u)} ∫_{u}^{t} [\sqrt{\frac{H(t, s)[σ(s, a) − T]σ′(s, a)}{a(s)b(σ(s, a))ρ(s)}}z(s)

EJDE/CONF/12  FUNCTIONAL DIFFERENTIAL EQUATIONS  53

+ \frac{\sqrt{a(s)b(σ(s, a))ρ(s)}h(t, s)}{2\sqrt{[σ(s, a) − T]σ′(s, a)}}]^{2}ds.

Then from the last inequality we see that z(u) ≥ ϕ(u), and

\begin{matrix}lim inf_{t→∞} \frac{1}{H(t, u)} ∫_{u}^{t} [\sqrt{\frac{H(t, s)[σ(s, a) − T]σ′(s, a)}{a(s)b(σ(s, a))ρ(s)}}z(s) \\ + \frac{\sqrt{a(s)b(σ(s, a))ρ(s)}h(t, s)}{2\sqrt{[σ(s, a) − T]σ′(s, a)}}]^{2}ds \\ ≤ z(u) − ϕ(u) = M < ∞, \end{matrix}  (2.21)

where M is a constant. On the other hand, we have

lim inf_{t→∞} \frac{1}{H(t, t_{1})} ∫_{t_{1}}^{t} [\sqrt{\frac{H(t, s)[σ(s, a) − T]σ′(s, a)}{a(s)b(σ(s, a))ρ(s)}}z(s)

+ \frac{\sqrt{a(s)b(σ(s, a))ρ(s)}h(t, s)}{2\sqrt{[σ(s, a) − T]σ′(s, a)}}]^{2}ds

≥ lim inf_{t→∞} [\frac{1}{H(t, t_{1})} ∫_{t_{1}}^{t} \frac{H(t, s)[σ(s, a) − T]σ′(s, a)z^{2}(s)}{a(s)b(σ(s, a))ρ(s)}ds

+ \frac{1}{H(t, t_{1})} ∫_{t_{1}}^{t} \sqrt{H(t, s)}h(t, s)z(s)ds], t > t_{1}.

(2.22)

Let

v_{1}(t) = \frac{1}{H(t, t_{1})} ∫_{t_{1}}^{t} \frac{H(t, s)[σ(s, a) − T]σ′(s, a)z^{2}(s)}{a(s)b(σ(s, a))ρ(s)}ds,  (2.23)

and

v_{2}(t) = \frac{1}{H(t, t_{1})} ∫_{t_{1}}^{t} \sqrt{H(t, s)}h(t, s)z(s)ds.  (2.24)

Thus, from (2.21) and (2.22), we see that

lim inf_{t→∞}[v_{1}(t) + v_{2}(t)] < ∞.  (2.25)

Now we want to show that

∫_{t_{1}}^{∞} \frac{[σ(s, a) − T]σ′(s, a)z^{2}(s)}{a(s)ρ(s)b(σ(s, a))}ds < ∞, t > t_{1}.  (2.26)

Assume that

∫_{t_{1}}^{∞} \frac{[σ(s, a) − T]σ′(s, a)z^{2}(s)}{a(s)ρ(s)b(σ(s, a))}ds = ∞, t > t_{1}.  (2.27)

Because of (2.17), there exists a constant L > 0 such that

inf_{s≥t_{0}} [ lim inf_{t→∞} \frac{H(t, s)}{H(t, t_{0})}] > L > 0.  (2.28)

From (2.27) one can see that for any positive number λ > 0, there exists a T > t_{1} such that

∫_{t_{1}}^{t} \frac{[σ(s, a) − T]σ′(s, a)z^{2}(s)}{a(s)ρ(s)b(σ(s, a))}ds ≥ \frac{λ}{L}, t > T.  (2.29)

On the other hand,

\begin{aligned}v_{1}(t) & = \frac{1}{H(t, t_{1})} ∫_{t_{1}}^{t} \frac{H(t, s)[σ(s, a) − T]σ′(s, a)z^{2}(s)}{a(s)b(σ(s, a))ρ(s)}ds \\ & = \frac{1}{H(t, t_{1})} ∫_{t_{1}}^{t} H(t, s)d [∫_{t_{1}}^{s} \frac{[σ(u, a) − T]σ′(u, a)z^{2}(u)}{a(u)b(σ(u, a))ρ(u)}] ds \\ & = \frac{1}{H(t, t_{1})} ∫_{t_{1}}^{t} [∫_{t_{1}}^{s} \frac{[σ(u, a) − T]σ′(u, a)z^{2}(u)}{a(u)b(σ(u, a))ρ(u)}du] [−\frac{∂H(t, s)}{∂s}] ds \\ & ≥ \frac{1}{H(t, t_{1})} ∫_{T}^{t} [∫_{t_{1}}^{s} \frac{[σ(u, a) − T]σ′(u, a)z^{2}(u)}{a(u)b(σ(u, a))ρ(u)}du] [−\frac{∂H(t, s)}{∂s}] ds \\ & ≥ \frac{λ}{LH(t, t_{1})} ∫_{T}^{t} [−\frac{∂H(t, s)}{∂s}] ds \\ & = \frac{λ}{L} \frac{H(t, T)}{H(t, t_{1})} ≥ \frac{λ}{L} \frac{H(t, T)}{H(t, t_{0})}, t ≥ T > t_{1}. \end{aligned} (2.30)

Moreover, it follows from (2.28) that

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

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

\frac{H(t, T)}{H(t, t_{0})} ≥ L, t ≥ t_{2}.  (2.31)

It follows from (2.30) and (2.31) that v_{1}(t) ≥ λ, t ≥ t_{2}. Since λ is arbitrary , we have

lim_{t→∞} v_{1}(t) = ∞.  (2.32)

Moreover from (2.25), there exists a convergence subsequence {t_{n}}_{1}^{∞} on [t_{1}, ∞) such that lim_{n→∞} t_{n} = ∞ and

lim_{n→∞}[v_{1}(t_{n}) + v_{2}(t_{n})] = lim inf_{t→∞}[v_{1}(t) + v_{2}(t)] < ∞.  (2.33)

As a result of (2.33) there exists a positive integer n_{1} and constant k such that

v_{1}(t_{n}) + v_{2}(t_{n}) < k, n > n_{1}

and from (2.32), we have

lim_{n→∞} v_{1}(t_{n}) = ∞.  (2.34)

Thus (2.33) and (2.34) will give

lim_{n→∞} v_{2}(t_{n}) = −∞.  (2.35)

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

\frac{v_{2}(t_{n})}{v_{1}(t_{n})} + 1 < ϵ, n > n_{2};

then

\frac{v_{2}(t_{n})}{v_{1}(t_{n})} < ϵ − 1 < 0, n > n_{2}.  (2.36)

Thus (2.35) and (2.36) give

lim_{n→∞} \frac{v_{2}(t_{n})}{v_{1}(t_{n})}v_{2}(t_{n}) = ∞.  (2.37)

EJDE/CONF/12  FUNCTIONAL DIFFERENTIAL EQUATIONS  55

By using the Cauchy-Schwartz inequality, we obtain

0 ≤ v_{2}^{2}(t_{n}) = \frac{1}{H^{2}(t_{n}, t_{1})}[ ∫_{t_{1}}^{t_{n}} \sqrt{H(t_{n}, s)}h(t_{n}, s)z(s)ds]^{2}

≤ [\frac{1}{H(t_{n}, t_{1})} ∫_{t_{1}}^{t_{n}} \frac{H(t_{n}, s)[σ(s, a) − T]σ′(s, a)}{a(s)b(σ(s, a))ρ(s)}z^{2}(s)ds]

[\frac{1}{H(t_{n}, t_{1})} ∫_{t_{1}}^{t_{n}} \frac{a(s)ρ(s)b(σ(s, a))h^{2}(t_{n}, s)}{[σ(s, a) − T]σ′(s, a)}ds]

= v_{1}(t_{n})[\frac{1}{H(t_{n}, t_{1})} ∫_{t_{1}}^{t_{n}} \frac{a(s)ρ(s)b(σ(s, a))h^{2}(t_{n}, s)}{[σ(s, a) − T]σ′(s, a)}ds], t ≥ t_{1}

and

\frac{v_{2}^{2}(t_{n})}{v_{1}(t_{n})} ≤ [\frac{1}{H(t_{n}, t_{1})} ∫_{t_{1}}^{t_{n}} \frac{a(s)ρ(s)b(σ(s, a))h^{2}(t_{n}, s)}{[σ(s, a) − T]σ′(s, a)}ds].  (2.38)

Using (2.31), we see that

\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})}, T > t_{1}. (2.39)

Therefore, we see from (2.38) and (2.39) that

\frac{v_{2}^{2}(t_{n})}{v_{1}(t_{n})} ≤ \frac{1}{LH(t_{n}, t_{0})} ∫_{t_{1}}^{t_{n}} \frac{a(s)ρ(s)b(σ(s, a))h^{2}(t_{n}, s)}{[σ(s, a) − T]σ′(s, a)}ds.  (2.40)

Then, it follows from (2.37) and (2.40) that

lim_{n→∞} \frac{1}{H(t_{n}, t_{0})} ∫_{t_{0}}^{t_{n}} \frac{a(s)ρ(s)b(σ(s, a))h^{2}(t_{n}, s)}{[σ(s, a) − T]σ′(s, a)}ds = ∞,

and then

lim sup_{t→∞} \frac{1}{H(t, t_{0})} ∫_{t_{0}}^{t} \frac{a(s)ρ(s)b(σ(s, a))h^{2}(t, s)}{[σ(s, a) − T]σ′(s, a)}ds = ∞,

which contradicts (2.18). Thus, using z(s) ≥ ϕ(s) with (2.26), we obtain

∫_{t_{1}}^{∞} \frac{[σ(s, a) − T]σ′(s, a)ϕ_{+}^{2}(s)ds}{a(s)ρ(s)b(σ′(s, a))} ≤ ∫_{t_{1}}^{∞} \frac{[σ(s, a) − T]σ′(s, a)z^{2}(s)ds}{a(s)ρ(s)b(σ′(s, a))} < ∞,

which leads to a contradiction to (2.20). This completes the proof. □ Theorem 2.4. Suppose that the conditions of Theorem 2.1, and conditions (2.2), (2.17), (2.20) hold, in addition to

lim inf_{t→∞} \frac{1}{H(t, t_{0})} ∫_{t_{0}}^{t} H(t, s)ρ(s) ∫_{a}^{b} p(s, ξ)[1 − c(σ(s, ξ))]dξds < ∞.  (2.41)

If there exists a function ϕ(t) ∈ C([t_{0}, ∞), R) satisfying

\begin{matrix}lim inf_{t→∞} \frac{1}{H(t, u)} ∫_{u}^{t} [H(t, s)ρ(s) ∫_{a}^{b} p(s, ξ)[1 − c(σ(s, ξ))]dξ \\ − \frac{a(s)ρ(s)b(σ(s, a))h^{2}(t, s)}{4[σ(s, a) − T)]σ′(s, a)}]ds \\ ≥ ϕ(u), u ≥ t_{0}, \end{matrix}  (2.42)

then every solution of (1.1) is oscillatory or tends to zero as t → ∞.

Proof. Assume, for the sake of contradiction, that equation (1.1) has positive solution, say x(t) > 0, t ≥ t_{0}. It follows from (2.42) that

ϕ(t_{0}) ≤ lim inf_{t→∞} \frac{1}{H(t, t_{0})} ∫_{t_{0}}^{t} [H(t, s)ρ(s) ∫_{a}^{b} p(s, ξ)[1 − c(σ(s, ξ))]dξ

− \frac{a(s)ρ(s)b(σ(s, a))h^{2}(t, s)}{4[σ(s, a) − T]σ′(s, a)}]ds

≤ lim inf_{t→∞} \frac{1}{H(t, t_{0})} ∫_{t_{0}}^{t} [H(t, s)ρ(s) ∫_{a}^{b} p(s, ξ)[1 − c(σ(s, ξ))]dξ]ds

− lim sup_{t→∞} \frac{1}{H(t, t_{0})} ∫_{t_{0}}^{t} \frac{a(s)ρ(s)b(σ(s, a))h^{2}(t, s)}{4[σ(s, a) − T]σ′(s, a)}ds.

(2.43)

Then, from (2.41) and (2.43)

lim sup_{t→∞} \frac{1}{H(t, t_{0})} ∫_{t_{0}}^{t} \frac{a(s)ρ(s)b(σ(s, a))h^{2}(t, s)}{4[σ(s, a) − T]σ′(s, a)}ds < ∞.

Thus (2.18) holds in Theorem 2.3. Since the remaining part of the proof is similar to the proof of Theorem 2.3, it is omitted. □

References

[1] R. Agarwal and S. Grace; Interval criteria for oscillation of second order half-linear ordinary differential equations, Oscillation of third order nonlinear differential equations with and without delay, Func. Differ. Eq. 1-2 (2002), 3-16.

[2] T. Candan and R. S. Dahiya; Oscillation of third order nonlinear differential equations with and without delay, Int. J. Differ. Equ. and Appl. 4(2) (2001), 125-139.

[3] T. Candan and R. S. Dahiya; Oscillation of third order functional differential equations with delay, Electron. J. Differ. Equ. Conference 10 (2003), 79-88.

[4] R. S. Dahiya; Oscillation of third order differential equations with and without delay, Differential Equations and Nonlinear Mechanics. 528 (2001), 75-87.

[5] I. Györi and G. Ladas; Oscillation Theory of Delay Differential Equations with Applications, Clarendon Press, Oxford (1991).

[6] G. S. Ladde, V. Lakshmikantham and B. G. Zhang; Oscillation Theory of Differential Equations with Deviating Arguments, Marcel Dekker, Inc. (1987).

[7] V. Sree Hari Rao and R. S. Dahiya; Properties of solutions of a class of third-order linear differential equations, Period. Math. Hungarica 20 (3) (1989), 177-184.

[8] A. S. Tantawy; Oscillation of Third and Higher Order Functional Differential Equations, Ph.D Thesis, ISU (1989).

[9] P. Waltman; Oscillation criteria for third-order nonlinear differential equations, Pasific J. Math. 18 (1966), 385-389.

[10] A. Zafer and R. S. Dahiya; Oscillatory and Asymptotic Behavior of Solutions of Third Order Delay Differential Equations With Forcing Terms, Differential Equations and Dynamical Systems. 1(2), (1993) 123-136.

Tuncay Candan Department of Mathematics, Faculty of Art and Science, Niğde University, Niğde, 51100, Turkey

E-mail address: tcandan@nigde.edu.tr

Rajbir S. Dahiya Department of Mathematics, Iowa State University, Ames, IA 50011, USA

E-mail address: rdahiya@iastate.edu