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Oscillation of certain third order nonlinear functional differential equations.

Abstract

In this paper we shall investigate the oscillatory properties of the equations

[d.sup.2]/d[t.sup.2] (1/a(t) [(dx(t)/dt).sup.[alpha]]) + q(t)f(x[g(t)]) = 0

and

[d.sup.2]/d[t.sup.2] (1/a(t) [(dx(t)/dt).sup.[alpha]]) = q(t)f (x[g(t)]) + p(t)h(x[[sigma](t)]),

where [alpha] is the ratio of two positive odd integers.

AMS subject classification: 34C10.

Keywords: Functional differential equations, oscillation, nonoscillation, comparison.

1. Introduction

Consider the third order nonlinear functional differential equations of the form

[d.sup.2]/d[t.sup.2] (1/a(t)[(dx(t)/dt).sup.[alpha]]) + q(t)f (x[g(t)]) = 0 (1.1)

and

[d.sub.2]/d[t.sup.2] (1/a(t)[(dx(t)/dt).sup.[alpha]]) = q(t)f (x[g(t)]) + p(t)h(x[[sigma](t)]), (1.2)

where

(i) [alpha] is the ratio of two positive odd integers,

(ii) a, p, q [member of] C([[t.sub.0],[infinity]), [0,[infinity])) such that sup{p(t) : t [greater than or equal to] T} > 0 and sup{q(t) : t [greater than or equal to] T } > 0 for any T [greater than or equal to] [t.sub.0] [greater than or equal to] 0, and a(t) > 0, t [greater than or equal to] [t.sub.0],

(iii) g, [sigma] [member of] [C.sup.1]([[t.sub.0],[infinity]),R) satisfying g'(t) [greater than or equal to] 0, [sigma]'(t) [greater than or equal to] 0, g(t) < t, [sigma](t) > t and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(iv) f, h [member of] C(R,R), xf(x) > 0, xh(x) > 0, f'(x) [greater than or equal to] 0 and h'(x) [greater than or equal to] 0 for x [not equal to] 0.

By a solution of equation (1.1) or (1.2) we mean a function x [member of] [C.sup.1]([[T.sub.x],[infinity]),R), [T.sub.x] [greater than or equal to] [t.sub.0] which has the property that (1/a)[(x').sup.[alpha]] [member of] [C.sup.2]([[T.sub.x],[infinity]),R) and satisfies equation (1.1) or (1.2) for all large t [greater than or equal to] [T.sub.x]. A solution is said to be oscillatory if it has a sequence of zeros clustering at t = [infinity], otherwise, a solution is said to be nonoscillatory. An equation is said to be oscillatory if all its solutions are oscillatory.

In the last three decades there has been an increasing interest in studying the oscillatory and nonoscillatory behavior of solutions of functional differential equations. Most of the work on the subject, however, has been restricted to first and second order equations, equations of type (1.1) and (1.2) with [alpha] = 1 as well as higher order equations and half-linear equations of the form

d/dt (1/a(t)[(dx(t)/dt).sup.[alpha]] + [delta]q(t)f (x[g(t)]) = 0,

where [delta] = [+ or -]1. For recent contributions, we refer to [1, 4] and the references cited therein.

It appears that only little is known regarding the oscillation of equations (1.1) and (1.2). Therefore, the main goal here is to present asymptotic study of the oscillation of all solutions of equations (1.1) and (1.2). Moreover, we shall establish some new criteria for the oscillation of similar type equations, namely,

d/dt (1/a(t)[([d.sup.2]x(t)/d[t.sup.2]).sup.[alpha]]) + q(t)f (x[g(t)]) = 0 (1.3)

and

d/dt (1/a(t)[([d.sup.2]x(t)/d[t.sup.2].sup.[alpha]]) = q(t)f (x[g(t)]) + p(t)h(x[[sigma](t)]). (1.4)

The obtained results extend, improve and correlate many known criteria which have appeared recently in the literature.

2. Oscillation of Equation (1.1)

We shall assume throughout that

[[integral].sup.[infinity]] [a.sup.1/[alpha]](s)ds =[infinity]. (2.1)

We define the operators

[L.sub.0]x(t) = x(t), [L.sub.1]x(t) = 1/a(t) [(d/dt [L.sub.0]x(t)).sup.[alpha]], [L.sub.2]x(t) = d/dt [L.sub.1]x(t), [L.sub.3]x(t) = d/dt [L.sub.2]x(t). (2.2)

Thus, equation (1.1) can be written as

[L.sub.3]x(t) + q(t)f (x[g(t)]) = 0.

If we let x be an eventually positive solution of equation (1.1), then [L.sub.3]x(t) [less than or equal to] 0 eventually, and hence [L.sub.i]x(t), i = 0, 1, 2 are eventually of one sign.

There are two possibilities to consider:

(I) [L.sub.i]x(t) > 0, i = 0, 1, 2 eventually, or

(II) [L.sub.0]x(t) > 0, [L.sub.1]x(t) < 0 and [L.sub.2]x(t) > 0 eventually.

Case (I) Let [L.sub.i]x(t) > 0, i = 0, 1, 2 for t [greater than or equal to] [t.sub.0] [greater than or equal to] 0. Then, it follows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

or

x'(t) [greater than or equal to] [a.sup.1/[alpha]](t)[(t - [t.sub.0]).sup.1/[alpha]][L.sup.1/[alpha].sub.2] x(t) for t [greater than or equal to] [t.sub.0].

Integrating the above inequality from [t.sub.0] to t, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.3)

Case (II) Let [L.sub.0]x(t) > 0, [L.sub.1]x(t) < 0 and [L.sub.2]x(t) > 0 for t [greater than or equal to] [t.sub.0] [greater than or equal to] 0. Then, for t [greater than or equal to] s [greater than or equal to] [t.sub.0],

[L.sub.1]x(t) - [L.sub.1]x(s) = [[integral].sup.t.sub.s] [L.sub.2]x(u)du

and so,

-[L.sub.1]x(s) = - 1/a(s) [(x'(s)).sup.[alpha]] [greater than or equal to] (t - s)[L.sub.2]x (t),

or

-x'(s) [greater than or equal to] [a.sup.1/[alpha]](s)[(t - s).sup.1/[alpha]][L.sup.1/[alpha].sub.2] x(t).

Thus, it follows that

x(s) [greater than or equal to] ([[integral].sup.t.sub.s] [a.sup.1/[alpha]](u)[(t - u).sup.1/[alpha]]du) [L.sup.1/[alpha].sub.2] x(t) for t [greater than or equal to] s [greater than or equal to] [t.sub.0]. (2.4)

In what follows we shall use the following notations. For t [greater than or equal to] s [greater than or equal to] T [greater than or equal to] [t.sub.0] [greater than or equal to] 0, we let

[A.sub.1][t, T] = [[integral].sup.t.sub.T] [a.sup.1/[alpha]](s)[(s - [t.sub.0]).sup.1/[alpha]]ds

and

[B.sub.1][t, s] = [[integral].sup.t.sub.s] [a.sup.1/[alpha]](u)[(t - u).sup.1/[alpha]]du.

Thus, inequalities (2.3) (of Case (I)) and (2.4) (of Case (II)) can be written as

x(t) [greater than or equal to] [A.sub.1][t, [t.sub.0]][L.sup.1/[alpha].sub.2] x(t), t [greater than or equal to] [t.sub.0] (2.5)

and

x(s) [greater than or equal to] [B.sub.1][t, s][L.sup.1/[alpha].sub.2] x(t), t [greater than or equal to] s [greater than or equal to] [t.sub.0]. (2.6)

For equation (1.3), we define

[[bar.L].sub.0]x(t) = x(t), [[bar.L].sub.1]x(t) = d/dt [[bar.L].sub.0]x(t), [[bar.L].sub.2]x(t) = 1/a(t) [(d/dt [L.sub.1]x(t)).sup.[alpha]], [[bar.L].sub.3]x(t) = d/dt [[bar.L].sub.2]x(t). (2.7)

Then, the equation (1.3) takes the form

[[bar.L].sub.3]x(t) + q(t)f (x[g(t)]) = 0.

If x is an eventually positive solution of equation (1.3), then the cases (I) and (II) are to be considered with L replaced by [bar.L]. For Case (I) one can easily conclude that

x(t) [greater than or equal to] [A.sub.3][t, [t.sub.0]][[bar.L].sup.1/[alpha].sub.2] x(t) for t [greater than or equal to] [t.sub.0], (2.8)

where

[A.sub.3][t, T] = [[integral].sup.t.sub.T] ([[integral].sup.s.sub.T] [a.sup.1/[alpha]](u)du)ds for t [greater than or equal to] T [greater than or equal to] [t.sub.0]

and if Case (II) holds, then for t [greater than or equal to] s [greater than or equal to] [t.sub.0], we see that

x(s) [greater than or equal to] [B.sub.3][t, s][[bar.L].sup.1/[alpha].sub.2] x(t), (2.9)

where

[B.sub.3][t, s] = [[integral].sup.t.sub.s] ([[integral].sup.t.sub.u] [a.sup.1/[alpha]]([tau])d[tau])du.

We are now ready to prove oscillatory criteria for the equation (1.1). For this, we shall assume that

-f (-xy) [greater than or equal to] f (xy) [greater than or equal to] f (x)f (y) for xy > 0, (2.10)

f ([u.sup.1/[alpha]])/u [greater than or equal to] k > 0, k is a real constant, u [not equal to] 0 (2.11)

and

[[integral].sup.[+ or -][epsilon].sub.0] du/f([u.sup.1/[alpha]]) < [infinity] for every [epsilon] > 0. (2.12)

Theorem 2.1. Let conditions (i)-(iv), (2.1), (2.10) and (2.11) hold. If for t [greater than or equal to] [t.sub.0] [greater than or equal to] 0,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.13)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (2.14)

then equation (1.1) is oscillatory.

Proof. Let x be an eventually positive solution of equation (1.1). Then, [L.sub.3]x(t) [less than or equal to] 0 eventually and hence [L.sub.i]x(t), i = 1, 2, 3 are eventually of one sign. This leads to the two possibilities (I) and (II). For Case (I), we obtain (2.5). Now there exists a T [greater than or equal to] [t.sub.0] such that

x[g(t)] [greater than or equal to] [A.sub.1][g(t), [t.sub.0]][L.sup.1/[alpha].sub.2] x[g(t)] for t [greater than or equal to] T. (2.15)

Integrating equation (1.1) from g(t) to t ([greater than or equal to] T), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Thus, it follows that

[L.sub.2]x[g(t)] [greater than or equal to] f ([L.sup.1/[alpha].sub.2] x[g(t)]) [[integral].sup.t.sub.g(t)] q(s)f ([A.sub.1][g(s), [t.sub.0]])ds,

or

[L.sub.2]x[g(t)]/f([L.sup.1/[alpha].sub.2] x[g(t)]) [greater than or equal to] [[integral].sup.t.sub.g(t)] q(s)f ([A.sub.1][g(s), [t.sub.0]])ds.

Taking lim sup of both sides of the above inequality as t [right arrow][infinity],we arrive at a contradiction to condition (2.13).

Next for the Case (II), we obtain (2.6). Substituting g(s) and g(t) for s and t respectively, we have

x[g(s)] [greater than or equal to] [B.sub.1][g(t), g(s)][L.sup.1/[alpha].sub.2] x[g(t)] for t [greater than or equal to] s [greater than or equal to] [t.sub.0]. (2.16)

Integrating equation (1.1) from g(t) to t, we find

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

or

[L.sub.2]x[g(t)]/f([L.sup.1/[alpha].sub.2] x[g(t)]) [greater than or equal to] [[integral].sup.t.sub.g(t)] q(s)f ([B.sub.1][g(t), g(s)])ds.

Taking lim sup of both sides of the above inequality as t [right arrow] [infinity], we obtain a contradiction to condition (2.14). This completes the proof. []

The following corollary is immediate.

Corollary 2.2. Let conditions (i)-(iv), (2.1), (2.10) and (2.11) hold. If condition (2.14) holds, then all bounded solutions of equation (1.1) are oscillatory.

Theorem 2.3. Let conditions (i)-(iv), (2.1), (2.10) and (2.12) hold. If for t [greater than or equal to] [t.sub.0] [greater than or equal to] 0,

[[integral].sup.[infinity]] q(s)f ([A.sub.1][g(s), [t.sub.0]])ds =[infinity] (2.17)

and

[[integral].sup.[infinity]] q(s)f ([B.sub.1][g(t), g(s)])ds =[infinity], (2.18)

then equation (1.1) is oscillatory.

Proof. Let x be an eventually positive solution of equation (1.1). Proceeding as in the proof of Theorem 2.1 we obtain (2.15) for Case (I) and (2.16) for Case (II). Now for Case (I), from equation (1.1), we obtain

- d/dt [L.sub.2]x(t) = q(t)f (x[g(t)]) [greater than or equal to] q(t)f ([A.sub.1][g(t), [t.sub.0][L.sup.1/[alpha].sub.2] x[g(t)]) [greater than or equal to] q(t)f ([A.sub.1][g(t), [t.sub.0])f ([L.sup.1/[alpha].sub.2] x(t)),

or

-d/dt[L.sub.2]x(t)/f([L.sup.1/[alpha].sub.2] x(t)) [greater than or equal to] q(t)f ([A.sub.1][g(t), [t.sub.0]]) for t [greater than or equal to] T [greater than or equal to] [t.sub.0].

Integrating the above inequality from T to t, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Taking limit of both sides of the above inequality as t [right arrow] [infinity], we obtain a contradiction to condition (2.17).

Next for Case (II), from equation (1.1), we see that

-[L.sub.3]x(s) = q(s)f (x[g(s)]) [greater than or equal to] q(s)f ([B.sub.1][g(t), g(s)])f ([L.sup.1/[alpha].sub.2] x(s)) for t [greater than or equal to] s [greater than or equal to] T [greater than or equal to] [t.sub.0],

or

-d/ds[L.sub.2]x(s)/f([L.sup.1/[alpha].sub.2] x(s)) [greater than or equal to] q(s)f ([B.sub.1][g(t), g(s)]).

The rest of the proof is similar to that of Case (I) and hence omitted. This completes the proof. []

The following corollary is immediate.

Corollary 2.4. Let conditions (i)-(iv), (2.1) and (2.10) hold. If

U/f ([u.sup.1/[alpha]]) [right arrow] 0 as u [right arrow] 0 (2.19)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.20)

then all bounded solutions of equation (1.1) are oscillatory.

Next, we present the following comparison result for the oscillation of equation (1.1).

Theorem 2.5. Let conditions (i)-(iv), (2.1) and (2.10) hold. If the first order delay equations

y'(t) + q(t)f ([A.sub.1][g(t), [t.sub.0]])f ([y.sup.1/[alpha]][g(t)]) = 0, [t.sub.0] [greater than or equal to] 0 (2.21)

and

z'(t) + q(t)f ([B.sub.1] [t + g(t)/2, g(t)]) f ([z.sup.1/[alpha]] [t + g(t)/2]) = 0 (2.22)

are oscillatory, then equation (1.1) is oscillatory.

Proof. Let x be an eventually positive solution of equation (1.1). Proceeding as in the proof of Theorem 2.1 we obtain (2.15) for Case (I) and (2.6) for Case (II). Now for Case (I), using (2.10) and (2.15) in equation (1.1), we have

-[L.sub.3]x(t) = q(t)f (x[g(t)]) [greater than or equal to] q(t)f ([A.sub.1][g(t), [t.sub.0]][L.sup.1/[alpha].sub.2] x[g(t)]) [greater than or equal to] q(t)f ([A.sub.1][g(t), [t.sub.0]])f ([L.sup.1/[alpha].sub.2] x[g(t)]) for t [greater than or equal to] T [greater than or equal to] [t.sub.0].

Setting y(t) = [L.sub.2]x(t) > 0 for t [greater than or equal to] T, we obtain

y'(t) + q(t)f ([A.sub.1][g(t), [t.sub.0]])f ([y.sup.1/[alpha]][g(t)]) [less than or equal to] 0 for t [greater than or equal to] T.

Integrating the above inequality from t ([greater than or equal to] T) to u and letting u [right arrow] [infinity], we have

y(t) [greater than or equal to] [[integral].sup.[infinity].sub.t] q(s)f ([A.sub.1][g(s), [t.sub.0]])f ([y.sup.1/[alpha]][g(s)])ds, t [greater than or equal to] T.

As in [5] it is easy to conclude that there exists a positive solution y of equation (2.21) with l[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], which contradicts the fact that equation (2.21) is oscillatory.

Next for Case (II), substituting g(t) and (t +g(t))/2 for s and t, respectively in (2.6), we have

x[g(t)] [greater than or equal to] [B.sub.1] [t + g(t)/2, g(t)][L.sup.1/[alpha].sub.2]x [t + g(t)/2] for t [greater than or equal to] T.

Using this inequality in equation (1.1) and proceeding as in Case (I), we obtain

z'(t) + q(t)f ([B.sub.1] [t + g(t)/2, g(t)]) f ([z.sup.1/[alpha]] [t + g(t)/2]) [less than or equal to] 0 for t [greater than or equal to] T,

where z(t) = [L.sup.1/[alpha].sub.2] x(t), t [greater than or equal to] T. The rest of the proof is similar to that of Case (I) above and hence omitted. []

The following corollary is immediate.

Corollary 2.6. Let conditions (i)-(iv), (2.1), (2.10) and (2.11) hold. If

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.23)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.24)

then equation (1.1) is oscillatory.

Remark 2.7. We note that identical results as those presented above for the oscillation of equation (1.3) can be easily obtained by replacing [A.sub.1] and [B.sub.1] with [A.sub.3] and [B.sub.3], respectively. The details are left to the reader.

3. Oscillation of Equation (1.2)

In this section, we shall give some new criteria for the oscillation of equations (1.2) and (1.4). For t [greater than or equal to] s [greater than or equal to] T [greater than or equal to] [t.sub.0], we let

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[B.sub.4][t, T] = [[integral].sup.t.sub.T] ([[integral].sup.t.sub.s] [a.sup.1/[alpha]](u)du)ds.

Using (2) in equation (1.2), we get

[L.sub.3]x(t) = q(t)f (x[g(t)]) + p(t)h(x[[sigma](t)]).

Now, if x is an eventually positive solution of equation (1.2), then [L.sub.3]x(t) [greater than or equal to] 0 eventually and hence [L.sub.i]x(t), i = 0, 1, 2 are eventually of one sign. We shall distinguish the following two cases:

([I.sub.1]) [L.sub.i]x(t) > 0, i = 0, 1, 2 eventually,

([II.sub.1]) [L.sub.0]x(t) > 0, [L.sub.1]x(t) > 0 and [L.sub.2]x(t) < 0 eventually.

Case ([I.sub.1]) Let [L.sub.i]x(t) > 0, i = 0, 1, 2 for t [greater than or equal to] [t.sub.0] [greater than or equal to] 0. Then, for t [greater than or equal to] s [greater than or equal to] [t.sub.0],

[L.sub.1]x(t) - [L.sub.1]x(s) = [[integral].sup.t.sub.s] [L.sub.2]x(u)du

and so

[L.sub.1]x(t) [greater than or equal to] (t - s)[L.sub.2]x(s),

or

x'(t) [greater than or equal to] [a.sup.1/[alpha]](t)[(t - s).sup.1/[alpha]][L.sup.1/[alpha].sub.2] x(s).

Thus, it follows that

x(t) [greater than or equal to] ([[integral].sup.t.sub.s] [(a(u)(u - s)).sup.1/[alpha]] du) [L.sup.1/[alpha].sub.2] x(s) = [A.sub.2][t, s][L.sup.1/[alpha].sub.2] x(s), t [greater than or equal to] s [greater than or equal to] [t.sub.0]. (3.1)

Case ([II.sub.1]) Let [L.sub.0]x(t) > 0, [L.sub.1]x(t) > 0, [L.sub.2]x(t) < 0 for t [greater than or equal to] [t.sub.0] [greater than or equal to] 0. Then, for t [greater than or equal to] s [greater than or equal to] [t.sub.0],

-[L.sub.1]x(s) [less than or equal to] [L.sub.1]x(t) - [L .sub.1]x(s) = [[integral].sup.t.sub.s] [L.sub.2]x(u)du

and so,

[L.sub.1]x(s) [greater than or equal to] [[integral].sup.t.sub.s] (-[L.sub.2]x(u))du,

or

x'(s) [greater than or equal to] [(a(s)(t - s)).sup.1/[alpha]](-[L.sup.1/[alpha].sub.2] x(t)).

Thus, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.2)

Next using (2) in equation (1.4), we see that

[[bar.L].sub.3]x(t) = q(t)f (x[g(t)]) + p(t)h(x[[sigma](t)]).

Now if x is an eventually positive solution of equation (1.4), then the Cases ([I.sub.1]) and ([II.sub.1]) are considered with L replacing L.

Now for Case ([I.sub.1]) one can easily see that for t [greater than or equal to] s [greater than or equal to] [t.sub.0],

x(t) [greater than or equal to] [A.sub.4][t, s][L.sup.1/[alpha].sub.2] x(s) (3.3)

and for Case ([II.sub.1]), we obtain

x(t) [greater than or equal to] [B.sub.4][t, [t.sub.0]](-[L.sup.1/[alpha].sub.2] x(t)) for t [greater than or equal to] [t.sub.0]. (3.3)

We shall assume that

-h(-xy) [greater than or equal to] h(xy) [greater than or equal to] h(x)h(y) for xy > 0 (3.4)

and

h([u.sup.1/[alpha]])/u [greater than or equal to] [k.sub.1] > 0, where [k.sub.1] is a constant, u [not equal to] 0. (3.5)

Theorem 3.1. Let conditions (i)-(iv), (2.1), (2.10), (2.11), (3.4) and (3.5) hold. If

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.6)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.7)

then equation (1.2) is oscillatory.

Proof. Let x be an eventually positive solution of equation (1.2). Then, [L.sub.3]x(t) [greater than or equal to] 0 eventually and hence [L.sub.i]x(t), i = 0, 1, 2 are eventually of one sign. Next, we distinguish the two Cases ([I.sub.1]) and ([II.sub.1]). For Case ([I.sub.1]) we obtain (3.1). Letting s = [sigma](t) and t = [sigma](s) in (3.1), we have

x[[sigma](s)] [greater than or equal to] [A.sub.2][[sigma](s), [sigma](t)][L.sup.1/[alpha].sub.2] x[[sigma](t)], s [greater than or equal to] t. (3.8)

From equation (1.2), we find that

[L.sub.3]x(s) [greater than or equal to] p(s)h(x[[sigma](s)]) [greater than or equal to] p(s)h([A.sub.2][[sigma](s), [sigma](t)][L.sup.1/[alpha].sub.2] x[[sigma](t)]) [greater than or equal to] p(s)h([A.sub.2][[sigma](s), [sigma](t)])h([L.sup.1/[alpha].sub.2] x[[sigma](t)]).

Integration of the above inequality on [t, [sigma](t)] yields

[L.sub.2]x[[sigma](t)] [greater than or equal to] (([[integral].sup.[sigma](t).sub.t] p(s)h([A.sub.2][[sigma](s), [sigma](t)])ds)h([L.sup.1/[alpha].sub.2] x[[sigma](t)]),

or

[L.sub.2]x[[sigma](t)]/h([L.sup.1/[alpha].sub.2] x[[sigma](t)]) [greater than or equal to] [[integral].sup.[sigma](t).sub.t] p(s)h([A.sub.2][[sigma](s), [sigma](t)])ds.

Taking lim sup of both sides as t [right arrow] [infinity], we have a contradiction to condition (3.6).

Next for Case ([II.sub.1]), we obtain (3.2) for t [greater than or equal to] [t.sub.0] [greater than or equal to] 0. There exists a T [greater than or equal to] [t.sub.0] such that

x[g(t)] [greater than or equal to] [B.sub.2][g(t), [t.sub.0]](-[L.sup.1/[alpha].sub.2] x[g(t)]) for t [greater than or equal to] T. (3.9)

It follows from equation (1.2) that

[L.sub.3]x(t) [greater than or equal to] q(t)f (x[g(t)]) [greater than or equal to] q(t)f ([B.sub.2][g(t), [t.sub.0]])f (-[L.sup.1/[alpha].sub.2] x[g(t)]).

Integrating the above inequality on [g(t), t], we find

-[L.sub.2]x[g(t)] [greater than or equal to] ([[integral].sup.t.sub.g(t)] q(s)f ([B.sub.2][g(s), [t.sub.0]]ds) f(-[L.sup.1/[alpha].sub.2] x[g(t)]).

The rest of the proof is similar to that of Case (I1) above and hence omitted. []

Next, we replace conditions (2.10) and (2.11) by

[f.sup.1/[alpha]](u)/u [greater than or equal to] m > 0, m is a constant, for u [not equal to] 0 (3.10)

and conditions (3.4) and (3.5) by

[h.sup.1/[alpha]](u)/u [greater than or equal to] [m.sub.1] > 0, [m.sub.1] is a constant, for u [not equal to] 0 (3.11)

and prove the following result.

Theorem 3.2. Let conditions (i)-(iv), (2.1), (3.10) and (3.11) hold. If

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.12)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.13)

then equation (1.2) is oscillatory.

Proof. Let x be an eventually positive solution of equation (1.2). As in the proof of Theorem 3.1 the Cases ([I.sub.1]) and ([II.sub.1]) are considered. Suppose ([I.sub.1]) holds. It follows from equation (1.2) that for [beta] [greater than or equal to] t [greater than or equal to] [t.sub.0],

[L.sub.2]x([beta]) [greater than or equal to] [[integral].sup.[beta].sub.t] p(s)h(x[[sigma](s)])ds [greater than or equal to] ([[integral].sup.[beta].sub.t] p(s)ds) h(x[[sigma](t)])

and for [eta] [greater than or equal to] t [greater than or equal to] [t.sub.0], we have

[L.sub.1]x([eta]) [greater than or equal to] [[integral].sup.[eta].sub.t] [L.sub.2]x([beta])d[beta] [greater than or equal to] ([[integral].sup.[eta].sub.t] [[integral].sup.[beta].sub.t] p(s)dsd[beta])h(x[[sigma](t)]),

or

x'([xi]) [greater than or equal to] [(a([eta]) [[integral].sup.[eta].sub.t] [[integral].sup.[beta].sub.t] p(s)dsd[beta]).sup.1/[alpha]] [h.sup.1/[alpha]](x[[sigma](t)]).

Now for [xi] [greater than or equal to] t [greater than or equal to] [t.sub.0], we find

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Putting [xi] = [sigma](t) in the above inequality, we get

x[[sigma](t)]/[h.sup.1/[alpha]](x[[sigma](t)]) [greater than or equal to] [[integral].sup.[sigma](t).sub.t] [(a([eta]) [[integral].sup.[eta].sub.t] [[integral].sup.[beta].sub.t] p(s)dsd[beta]).sup.1/[alpha]] d[eta].

Taking lim sup of the above as t [right arrow] [infinity], we obtain a contradiction to condition (3.12).

Suppose ([II.sub.1]) holds. It follows from equation (1.2) that

-[L.sub.2]x(t) [greater than or equal to] [[integral].sup.[infinity].sub.t] q(s)f (x[g(s)])ds [greater than or equal to] [[integral].sup.[infinity].sub.t] q(s)ds) f (x[g(t)]), t [greater than or equal to] T [greater than or equal to] [t.sub.0].

Using inequality (3.9) and the fact that -[L.sub.2]x(t) is nonincreasing, we have

x[g(t)] [greater than or equal to] [B.sub.2][g(t), [t.sub.0]](-[L.sup.1/[alpha].sub.2] x[g(t)]) [greater than or equal to] [B.sub.2][g(t), [t.sub.0]](-[L.sup.1/[alpha].sub.2] x(t)) [greater than or equal to] [B.sub.2][g(t), [t.sub.0]] [([[integral].sup.[infinity].sub.t] q(s)ds).sup.1/[alpha]] [f.sup.1/[alpha]](x[g(t)]),

or

x[g(t)]/[f.sup.1/[alpha]](x[g(t)]) [greater than or equal to] [B.sub.2][g(t), [t.sub.0]] [([[integral].sup.[infinity].sub.t] q(s)ds).sup.1/[alpha]], t [greater than or equal to] T [greater than or equal to] [t.sub.0].

The rest of the proof is similar to that of Case (I1) and hence omitted. []

In what follows for t [greater than or equal to] s [greater than or equal to] T [greater than or equal to] [t.sub.0], we let

C[t, s] = [[integral].sup.t.sub.s] [a.sup.1/[alpha]](u)du.

Now we shall prove the following comparison results.

Theorem 3.3. Let conditions (i)-(iv), (2.1), (2.10) and (3.4) hold. If all unbounded solutions of the second order advanced equation

[d.sup.2]y(t)/d[t.sup.2] - p(t)h(C [[sigma](t), t + [sigma](t)/2]) h ([y.sup.1/[alpha]] [t + [sigma](t)/2]) = 0 (3.14)

and all bounded solutions of the second order delay equation

[d.sup.2]z(t)/d[t.sup.2] - q(t)f (C[g(t), [t.sub.0]])f ([z.sup.1/[alpha]][g(t)]) = 0, [t.sub.0] [greater than or equal to] 0 (3.15)

are oscillatory, then equation (1.2) is oscillatory.

Proof. Let x be an eventually positive solution of equation (1.2). As in the proof of Theorem 3.1, we have the Cases ([I.sub.1]) and ([II.sub.1]) to consider. For Case ([I.sub.1]) we have [L.sub.i]x(t) > 0, i = 1, 2 for t [greater than or equal to] [t.sub.0]. Thus, for s [greater than or equal to] t [greater than or equal to] [t.sub.0] it follows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Let y(t) = [L.sub.1]x(t). Substituting [sigma](t) and (t + [sigma](t))/2 for s and t respectively in the above inequality, we obtain

x[[sigma](t)] [greater than or equal to] C [[sigma](t), t + [sigma](t)/2] [y.sup.1/[alpha]] [t + [sigma](t)/2] for t [greater than or equal to] T [greater than or equal to] [t.sub.0]. (3.16)

Using (3.16) in equation (1.2), we have

[d.sup.2]y(t)/d[t.sup.2] [greater than or equal to] p(t)h(x[[sigma](t)]) [greater than or equal to] p(t)h(C [[sigma](t), t + [sigma](t)/2]) h([y.sup.1/[alpha]] [t + [sigma](t)/2]) for t [greater than or equal to] T.

By a comparison theorem in [2, 3], we see that equation (3.14) has an unbounded eventually positive solution, which is a contradiction.

For Case ([II.sub.1]) we have [L.sub.0]x(t) > 0, [L.sub.1]x(t) > 0 and [L.sub.2]x(t) < 0 for t [greater than or equal to] [t.sub.0]. Thus, for t [greater than or equal to] [t.sub.0] [greater than or equal to] 0 it follows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Let z(t) = [L.sub.1]x(t). Substituting g(t) for t in the above inequality, we get

x[g(t)] [greater than or equal to] C[g(t), [t.sub.0]][z.sup.1/[alpha]][g(t)] for t [greater than or equal to] [T.sub.1] [greater than or equal to] [t.sub.0]. (3.17)

Using (2.10) and (3.17) in equation (1.2), we obtain

[d.sup.2]z(t)/d[t.sup.2] [greater than or equal to] q(t)f (x[g(t)]) [greater than or equal to] q(t)f (C[g(t), [t.sub.0]])f ([z.sup.1/[alpha]][g(t)]) for t [greater than or equal to] [T.sub.1].

Once again, by a comparison result in [2, 3], one can easily see that equation (3.15) has a bounded eventually positive solution, which is a contradiction. This completes the proof. []

Theorem 3.4. Let conditions (i)-(iv), (2.1), (2.10) and (3.4) hold. If the first order advanced equation

dy(t)/dt - p(t)h([A.sub.2] [[sigma](t), t + [sigma](t)/2]) h ([y.sup.1/[alpha]] [t + [sigma](t)/2]) = 0 (3.18)

and the first order delay equation

dz(t)/dt + q(t)f ([B.sub.2][g(t), [t.sub.0]])f ([z.sup.1/[alpha]][g(t)]) = 0, [t.sub.0] [greater than or equal to] 0 (3.19)

are oscillatory, then equation (1.2) is oscillatory.

Proof. Let x be an eventually positive solution of equation (1.2). As in the proof of Theorem 3.1, we consider the Cases ([I.sub.1]) and ([II.sub.1]). If Case ([I.sub.1]) holds, then from (3.1), we have

x[[sigma](t)] [greater than or equal to] [A.sub.2] [[sigma](t), t + [sigma](t)/2] [L.sup.1/[alpha].sub.2] x [t + [sigma](t)/2] for t [greater than or equal to] T [greater than or equal to] [t.sub.0]. (3.20)

Using (3.20) and (3.4) in equation (1.2), we get

[L.sub.3]x(t) [greater than or equal to] p(t)h(x[[sigma](t)]) [greater than or equal to] p(t)h ([A.sub.2] [[sigma](t), t + [sigma](t)/2]) h ([L.sup.1/[alpha].sub.2] x [t + [sigma](t)/2]).

Setting [L.sub.2]x(t) = y(t) for t [greater than or equal to] T, we find

dy(t)/dt [greater than or equal to] p(t)h ([A.sub.2] [[sigma](t), t + [sigma](t)/2]) h ([y.sup.1/[alpha]] [t + [sigma](t)/2]) for t [greater than or equal to] T.

As in [2, 3], we see that the equation (3.18) has an eventually positive solution, which is a contradiction.

Next if ([II.sub.1]) holds, then as in the proof of Theorem 3.1, we obtain (3.9) for t [greater than or equal to] T. Now using (3.9) and (2.10) in equation (1.2), we find

[L.sub.3]x(t) [greater than or equal to] q(t)f (x[g(t)]) [greater than or equal to] q(t)f ([B.sub.2][g(t), [t.sub.0]])f (-[L.sup.1/[alpha].sub.2] x[g(t)]), t [greater than or equal to] T.

Putting z(t) = -[L.sub.2]x(t), t [greater than or equal to] T we have

dz(t)/dt + q(t)f ([B.sub.2][g(t), [t.sub.0]])f ([z.sup.1[alpha]][g(t)]) [less than or equal to] 0.

The rest of the proof is similar to that of Theorem 2.5 and hence omitted. []

From Theorem 3.4, one can easily deduce the following corollaries.

Corollary 3.5. Let conditions (i)-(iv), (2.1), (2.10), (2.11), (3.4) and (3.5) hold. If

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.21)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.22)

then equation (1.2) is oscillatory.

Corollary 3.6. Let conditions (i)-(iv), (2.1), (2.10), (2.12) and (3.4) hold and

[[integral].sup.[+ or -][infinity].sub.[+ or -][epsilon]] du/h([u.sup.1/[alpha]]) < [infinity] for [epsilon] > 0. (3.23)

If

[[integral].sup.[infinity]] p(s)h([A.sub.2] [[sigma](s), s + [sigma](s)/2]) ds =[infinity] (3.24)

and

[[integral].sup.[infinity]] q(s)f ([B.sub.2][g(s), [t.sub.0]])ds =[infinity], [t.sub.0] [greater than or equal to] 0, (3.25)

then equation (1.2) is oscillatory.

Remark 3.7. We note that identical results as those obtained above for the oscillation of equation (1.4) can be easily established by replacing [A.sub.2] and [B.sub.2] by [A.sub.4] and [B.sub.4] respectively and equations (3.14) and (3.15) in Theorem 3.3 by

d/dt (1/a(t) [(dy(t)/dt).sup.[alpha]]) - p(t)h(C [[sigma](t), t + [sigma](t)/2]) h ([y.sup.1/[alpha]] [t + [sigma](t)/2]) = 0 (3.26)

and

d/dt (1/a(t) [(dz(t)/dt).sup.[alpha]])- q(t)f (C[g(t), [t.sub.0]])f ([z.sup.1/[alpha]][g(t)]) = 0, [t.sub.0] [greater than or equal to] 0 (3.27)

respectively. The details are left to the reader.

4. Examples and Remarks

Remark 4.1. By applying Theorem 2.1 to the equation

[d.sup.2]/d[t.sup.2] [(dx(t)/dt).sup.[alpha]] + q[x.sup.[alpha]][t - [tau]] = 0, (4.1)

where [alpha] is as in (i), q and [tau] are positive constants, we find that equation (4.1) is oscillatory if

q > [alpha] + 2/[[tau].sup.2] [([alpha] + 1/[alpha][tau]).sup.[alpha]]. (4.2)

Also, we see that the equation

[d.sup.2]/d[t.sup.2] [(dx/dt).sup.[alpha]] + q[x.sup.[beta]][t - [tau]] = 0 (4.3)

is oscillatory by Theorem 2.3 provided that [alpha], q and [tau] are as in equation (4.1) and [beta] is the ratio of two positive odd integers with 0 < [beta] < [alpha].

Remark 4.2. By applying Theorem 3.1 to the equation

[d.sup.2]/d[t.sup.2] [(dx(t)/dt).sup.[alpha]] = q[x.sup.[alpha]][t - [tau]] + p[x.sup.[alpha]][t + [sigma]], (4.4)

where [alpha] is as in (i), p, q, [tau] and [sigma] are positive real constants, it follows that equation (4.4) is oscillatory if

p > [alpha] + 2/[[sigma].sup.2] [([alpha] + 1/[alpha][sigma]).sup.[alpha]]. (4.5)

Remark 4.3. By applying Theorem 3.3 to equation (4.4), one may conclude that equation (4.4) is oscillatory if all unbounded solutions of the advanced second order equation

y" (t) - p ([sigma]/2).sup.[alpha]] y [t + [sigma]/2] = 0 (4.6)

and all bounded solutions of the second order delay equation

z" (t) - q[(t - [tau] - [t.sub.0]).sup.[alpha]]z[t - [tau]] = 0, [t.sub.0] [greater than or equal to] 0 (4.7)

are oscillatory.

We note that if we apply Theorem 3.4 to equation (4.4), then we can easily see that equation (4.4) is oscillatory if the first order equation with advanced argument

y' (t) - [sigma]/2 p [([alpha][sigma]/2([alpha] + 1)).sup.[alpha]] y [t + [sigma]/2] = 0 (4.8)

and the first order delay equation

z' (t) + q [([alpha]/[alpha] + 1).sup.[alpha]] [(t - [tau] - [t.sub.0]).sup.[alpha]+1]z[t - [tau]] = 0, [t.sub.0] [greater than or equal to] 0 (4.9)

are oscillatory.

Next, by applying Corollary 3.5 to equation (4.4), we see that equation (4.4) is oscillatory for any q > 0 and

p > 1/e [(2/[sigma]).sup.2] [(2([alpha] + 1)/[alpha][sigma]).sup.[alpha]],

and by Corollary 3.6, we see that the equation

[d.sup.2]/d[t.sup.2] [(dx(t)/dt.sup.[alpha]] = q[x.sup.[beta]][t - [tau]] + p[x.sup.[gamma]] [t + [sigma]] (4.10)

is oscillatory provided that p, q, [tau] and [sigma] are positive constants, [alpha], [beta] and [gamma] are ratios of two positive odd integers with 0 < [beta] < [alpha] < [gamma].

Remark 4.4. Similar oscillation results as those presented above can be obtained for equations (1.3) and (1.4) with constant coefficients and deviations. The details are left to the reader.

We note that our results in this paper are new even for the special cases of the equations considered with constant coefficients and deviations.

Received March 30, 2007; Accepted April 4, 2007

References

[1] Ravi P. Agarwal, Said R. Grace, and Donal O'Regan. Oscillation theory for second order linear, half-linear, superlinear and sublinear dynamic equations, Kluwer Academic Publishers, Dordrecht, 2002.

[2] Ravi P. Agarwal, Said R. Grace, and Donal O'Regan. Oscillation theory for second order dynamic equations, volume 5 of Series in Mathematical Analysis and Applications, Taylor & Francis Ltd., London, 2003.

[3] I. Gyori and G. Ladas. Oscillation theory of delay differential equations, Oxford Mathematical Monographs. The Clarendon Press Oxford University Press, New York, 1991.With applications, Oxford Science Publications.

[4] Takasi Kusano and B. S. Lalli. On oscillation of half-linear functional-differential equations with deviating arguments, Hiroshima Math. J., 24(3):549-563, 1994.

[5] Ch. G. Philos. On the existence of nonoscillatory solutions tending to zero at [infinity] for differential equations with positive delays, Arch. Math. (Basel), 36(2):168-178, 1981.

Ravi P. Agarwal (1), Said R. Grace (2) and Patricia J.Y. Wong (3)

(1) Department of Mathematical Sciences, Florida Institute of Technology, Melbourne, FL 32901, U.S.A. E-mail: agarwal@fit.edu

(2) Department of Engineering Mathematics, Faculty of Engineering, Cairo University, Orman, Giza 12221, Egypt E-mail: srgrace@eng.cu.edu.eg

(3) School of Electrical and Electronic Engineering, Nanyang Technological University, 50 Nanyang Avenue, Singapore 639798, Singapore E-mail: ejywong@ntu.edu.sg
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Date:Jun 1, 2007
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