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On the oscillation of fourth order nonlinear dynamic equations.

1. INTRODUCTION

This paper deals with the oscillatory behavior of the fourth order nonlinear dynamic equation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.1)

on an arbitrary time scale T [subset or equal to] IR with sup T = [infinity], subject to the following hypothesis:

(i). the function f:[[t.sub.0], [infinity]) x IR [right arrow] IR is continuous, satisfying sgn f(t,x) = sgn x and f(t,x) [less than or equal to] f(t,y) for x [less than or equal to] y and t [greater than or equal to] [t.sub.0].

By a solution of equation (1.1), we mean a nontrivial real-valued function x satisfying equation (1.1) for t [greater than or equal to] [t.sub.x] [greater than or equal to] [t.sub.0]. A solution x of equation (1.1) is called nonoscillatory if there exists a to G T such that x(t)x([sigma](t)) > 0 for all t [member of] [[t.sub.o], [infinity]) [intersection] T, otherwise it is called oscillatory. Equation (1.1) is called oscillatory if all its solutions are oscillatory.

In the last two decades, there has been much interest in studying the oscillatory behavior of first and second order dynamic equations on time scales, and for recent contributions we refer the reader to [1,2,5,6]. With respect to dynamic equations on time scales, it is a fairly new topic, and for general basic ideas and background, we refer the reader to [3,4]. It seems that nothing is known regarding the oscillation of the nonlinear equation (1.1). Therefore, our main goal here is to establish some new criteria for the oscillation of strongly superlinear and strongly sublinear dynamic equation (1.1). Many of our results are new for the corresponding fourth order nonlinear differential and difference equations.

In order to prove our results, we shall employ the formula:

[([(x(t)).sup.[alpha]]).sup.[DELTA]] = [alpha] [[integral].sup.1.sub.0][[h[x.sup.[sigma]](t) + (1 - h)x(t)].sup.[gamma]-1][x.sup.[DELTA]](t)[DELTA]h, (1.2)

where x(t) is a delta-differentiable and eventually positive or eventually negative, which is a simple consequence of Keller's chain rule (see [3, Theorem 1.90]).

The following two lemmas are needed in the proofs of our main results.

Lemma 1.1. Suppose that x(t) is an eventually positive solution of equation (1.1).Then, there exists a to [member of] T such that one of the following two cases holds:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.3)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.4)

Lemma 1.2. Suppose [[absolute value of x].sup.[DELTA]] is of one sign on [[t.sub.0], [infinity]) and [gamma] > 0. Then,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.5)

For a proof see [6, Lemma 2.1].

We shall obtain results for strongly superlinear and strongly sublinear equations according to the following classification.

Equation (1.1) (or the function f) is said to be strongly superlinear if there exists a constant [alpha] > 1 such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.6)

and it is said to be strongly sublinear if there exists a constant [beta], 0 < [beta] < 1 such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.7)

If (1.6) holds with [alpha] = 1, then equation (1.1) is called superlinear and if (1.7) holds with [beta] = 1, then equation (1.1) is called sublinear.

We shall also employ the Taylor monomials [{[h.sub.n](t,s)}.sup.[infinity].sub.n=0] (see [3, Sec. 1.6]) which are defined recursively as

[h.sub.0](t,s) = 1,[h.sub.n+1](t,s) = [[integral].sup.t.sub.s][h.sub.n](u,s)[DELTA]u, t,s [member of] [[t.sub.0], [infinity]) [intersection] T, n [greater than or equal to] 1.

2. OSCILLATION OF STRONGLY SUPERLINEAR EQUATION (1.1)

Our first result is embodied in the following theorem.

Theorem 2.1. Suppose that equation (1.1) is strongly superlinear. If

[[integral].sup.[infinity]] s[absolute value of f(s,c)][DELTA]s = [infinity] (2.1)

for any constant c [not equal to] 0, then equation (1.1) is oscillatory.

Proof. Let x(t) be a nonoscillatory solution of equation (1.1), say, x(t) > 0 for t [greater than or equal to] [t.sub.0]. By Lemma 1.1, there are two cases to consider:

Case (I). Let (1.3) hold. Then there exist a constant [c.sub.1] > 0 and a [t.sub.1] [greater than or equal to] [t.sub.0] such that

[x.sup.[sigma]](t) [greater than or equal to] [c.sub.1] and [x.sup.[sigma]](t) [greater than or equal to] [c.sub.1]t for t [greater than or equal to] [t.sub.1]. (2.2)

Integrating equation (1.1) from [t.sub.1] [greater than or equal to] 1 to t [greater than or equal to] [t.sub.1], we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Thus, it follows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.3)

Using (2.2) and the strong superlinearity of f, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which contradicts condition (2.1).

Case (II). Let (1.4) hold. Then, there exist a constant [c.sub.2] > 0 and a [[bar.t].sub.1] [greater than or equal to] [t.sub.0] such that

x(t) [greater than or equal to] [c.sub.2] for t [greater than or equal to] [[bar.t].sub.1]. (2.4)

Multiplying equation (1.1) by t and integrating from [[bar.t].sub.1] to t, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Thus, we find

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.5)

Using (2.4) and the strong superlinearity of f and proceeding as in Case (I), we obtain a contradiction to (2.1). This completes the proof.

Next, we present the following sharper criterion than Theorem 2.1 for the oscillation of strongly superlinear dynamic equation (1.1).

Theorem 2.2. Suppose that equation (1.1) is strongly superlinear. If

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.6)

for any constant c [not equal to] 0 and [t.sub.0] [greater than or equal to] 0 and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.7)

for any constant [bar.c] [not equal to] 0, then equation (1.1) is oscillatory.

Proof. Let x(t) be a nonoscillatory solution of equation (1.1), say, x(t) > 0 for t [greater than or equal to] [t.sub.0]. By Lemma 1.1, we need to consider the following two cases:

Case (I). Let (1.3) hold. Then there exist constants [c.sub.1] [member of] (0,1) and [c.sub.2] > 0 and a [t.sub.1] [greater than or equal to] [t.sub.0] such that

y(t) [greater than or equal to] [c.sub.1]t[y.sup.[DELTA]](t) for t [greater than or equal to] [t.sub.1] (2.8)

and

y(t) [greater than or equal to] [c.sub.2] for t [greater than or equal to] [t.sub.1], (2.9)

where y(t) = [x.sup.[DELTA][DELTA]](t) for t [greater than or equal to] [t.sub.1]. From (2.9) one can easily see that there exists a constant [[bar.c].sub.2] > 0 such that

[x.sup.[sigma]](t) [greater than or equal to] [[bar.c].sub.2][h.sub.2](t,[t.sub.1]) for t [greater than or equal to] [t.sub.1]. (2.10)

Also, from (2.8) and the fact that [y.sup.[DELTA]] is decreasing for t [greater than or equal to] [t.sub.1], we see that there exists a constant [[bar.c].sub.1] > 0 such that

[x.sup.[DELTA]](t) [greater than or equal to] [[bar.c].sub.1][h.sub.2](t,[t.sub.1])[y.sup.[DELTA]](t) for t [greater than or equal to] [t.sub.1]. (2.11)

Integrating equation (1.1) from t [greater than or equal to] [t.sub.1] to u > t and letting u [right arrow] [infinity], we find

[y.sup.[DELTA]](t) [greater than or equal to] [[integral].sup.[infinity].sub.t]f(s,[x.sup.[sigma]](s))[DELTA]s. (2.12)

Using (1.6) and (2.10) in (2.12), we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.13)

Using (2.8) in (2.13), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for t [greater than or equal to] [t.sub.1], or

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Using Lemma 1.2 (the left-hand side), we find

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Integrating this inequality from [t.sub.1] to t [greater than or equal to] [t.sub.1], we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which contradicts (2.6).

Case (II). Let (1.4) hold. Then, there exist constants [b.sub.1] [member of] (0,1) and [b.sub.2] > 0 and a [[bar.t].sub.1] [greater than or equal to] [t.sub.0] such that

[x.sup.[sigma]](t) [greater than or equal to] x(t) [greater than or equal to] [b.sub.1]t[x.sup.[DELTA]](t) for t [greater than or equal to] [t.sub.1] (2.14)

and

[x.sup.[sigma]](t) [greater than or equal to] x(t) [greater than or equal to] [b.sub.2] for t [greater than or equal to] [t.sub.1]. (2.15)

Integrating equation (1.1) three times from t [greater than or equal to] [t.sub.1] to u [greater than or equal to] t and letting u [right arrow] [infinity] and using (1.6) and (2.15), we find

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

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

Next, we present the following result.

Theorem 2.3. Suppose that equation (1.1) is strongly superlinear. If

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.16)

for any constant c [not equal to] 0 and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.17)

for any constant [bar.c] [not equal to] 0, then equation (1.1) is oscillatory.

Proof. Let x(t) be a nonoscillatory solution of equation (1.1), say, x(t) > 0 for t [greater than or equal to] [t.sub.0]. By Lemma 1.1, we consider the following two cases:

Case (I). Let (1.3) hold. As in the proof of Theorem 2.2, we obtain (2.10)-(2.13) for t [greater than or equal to] [t.sub.1]. Integrating (2.11) from [t.sub.1] to t [greater than or equal to] [t.sub.1], we have

[x.sup.[sigma]](t) [greater than or equal to] x(t) [greater than or equal to] [[??].sub.1][h.sub.3](t,[t.sub.1])[y.sup.[DELTA]](t) for t [greater than or equal to] [t.sub.1], (2.18)

where [[??].sub.1] is a positive constant. Using (2.18) in (2.13) we find

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

or

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Taking lim sup of both sides of (2.19) as t [right arrow] [infinity], we obtain a contradiction to (2.16).

Case (II). Let (1.4) hold. As in the proof of Theorem 2.2-Case (I), we obtain (2.14), (2.15) and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.19)

Using (2.14) in (2.19), we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.20)

Taking lim sup of both sides of (2.20) as t [right arrow] [infinity], we obtain a contradiction to (2.17). This completes the proof.

The following result deals with the oscillation of superlinear equation (1.1).

Theorem 2.4. Suppose that equation (1.1) is superlinear (i.e., f satisfies (1.6) with [alpha] = 1). If

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.21)

for any constants c [not equal to] 0, [c.sub.1] > 0 and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.20)

for any constants [bar.c] [not equal to] 0, [c.sub.2] > 0, then equation (1.1) with [alpha] = 1 is oscillatory.

Proof. Follows from Theorem 2.3 by letting [alpha] = 1. We omit the details.

As illustrative examples, first we let T = R. In this case equation (1.1) becomes the differential equation

[x.sup.(4)](t) + f(t,x(t)) = 0 (2.23)

and when Theorem 2.2 applied to equation (2.23), we get

Corollary 2.1. Assume that equation (2.23) is strongly superlinear. If

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.24)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.25)

for any constants c [not equal to] 0 and [bar.c] [not equal to] 0, then equation (2.23) is oscillatory.

Next, we let T = Z. In this case, equation (1.1) takes the form

[[DELTA].sup.4]x(t) + f (t,x(t + 1)) = 0 (2.26)

and Theorem 2.3 becomes

Corollary 2.2. Assume that equation (2.26) is strongly superlinear. If

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for any constants c [not equal to] 0 and [bar.c] [not equal to] 0, then equation (2.25) is oscillatory.

3. OSCILLATION OF STRONGLY SUBLINEAR EQUATION (1.1)

First, we present the following result.

Theorem 3.1. Suppose that equation (1.1) is strongly sublinear. If

[[integral].sup.[infinity]][([h.sub.3](s,[t.sub.0])/[h.sub.2](s,[t.sub.0])).sup.[beta]][absolute value of f(s,c[h.sub.3](s,[t.sub.0]))][DELTA]s = [infinity] (3.1)

for any constant c [not equal to] 0 and

[[integral].sup.[infinity]][[h.sub.1](s,[t.sub.0])/[([h.sub.3](s,[t.sub.0])).sup.[beta]]][absolute value of f(s,[bar.c][h.sub.3](s,[t.sub.0]))][DELTA]s = [infinity] (3.2)

for any constant [bar.c] [not equal to] 0, then equation (1.1) is oscillatory.

Proof. Let x(t) be a nonoscillatory solution of equation (1.1), say, x(t) > 0 for t [greater than or equal to] [t.sub.0]. By Lemma 1.1, we distinguish the two cases:

Case (I). Let (1.3) hold. Then there exist positive constants [c.sub.1] and [c.sub.2] and a [t.sub.1] [greater than or equal to] [t.sub.0] such that

[x.sup.[DELTA][DELTA][DELTA]](t) [less than or equal to] [c.sub.1] and [x.sub.[DELTA][DELTA]](t) [greater than or equal to] [c.sub.2] for t [greater than or equal to] [t.sub.1].

Thus, it follows that

x(t) [less than or equal to] [[bar.c].sub.1][h.sub.3](t,[t.sub.1]) for t [greater than or equal to] [t.sub.1] (3.3)

and

x(t) [greater than or equal to] [[bar.c].sub.2][h.sub.2](t,[t.sub.1]) for t [greater than or equal to] [t.sub.1], (3.4)

where [[bar.c].sub.1] and [[bar.c].sub.2] are positive constants. As in the proof of Theorem 2.1, we obtain (2.3). Using (1.7), (3.3) and (3.4) in (2.3), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which contradicts condition (3.1).

Case (II). Let (1.4) hold. As in the proof of Theorem 2.1, we obtain (2.4) and (2.5) which takes the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Using (2.4), (3.3) and (1.7) in the above inequality, we obtain a contradiction to (3.2). This completes the proof.

Next, we establish the following results.

Theorem 3.2. Suppose that equation (1.1) is strongly sublinear. If

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.5)

for any constant c [not equal to] 0 and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.6)

for any constant [bar.c] [not equal to] 0, then equation (1.1) is oscillatory.

Proof. Let x(t) be a nonoscillatory solution of equation (1.1), say, x(t) > 0 for t [greater than or equal to] [t.sub.0]. By Lemma 1.1, there are two cases to consider:

Case (I). Let (1.3) hold. As in the proof of Theorem 3.1, we obtain (3.3) and as in Theorem 3.2, we obtain (2.18). Integrating equation (1.1) from [t.sub.1] [greater than or equal to] [t.sub.0] to u [greater than or equal to] t and letting u [right arrow] [infinity], we get (2.12). Using (1.7) and (3.3) in (2.12), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.7)

Using (2.18) in the above inequality, we find

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

from which, denoting the right-hand side by z(t), we get

-[z.sup.-[beta]](t)[z.sup.[DELTA]](t) [greater than or equal to] [([??]/[[bar.c].sub.1]).sup.[beta]]f(t,[[bar.c].sub.1] [h.sub.3](t,[t.sub.1])), t [greater than or equal to] [t.sub.1].

Using Lemma 1.2 (right-hand side), we have

[-[([(x(t).sup.1-[beta]]).sup.[DELTA]]/1-[beta]] [greater than or equal to] [([??]/[[bar.c].sub.1]).sup.[beta]]f(t,[[bar.c].sub.1][h.sub.3](t,[t.sub.1])) for t [greater than or equal to] [t.sub.1]

Integrating this inequality from [t.sub.1] to t [greater than or equal to] [t.sub.1], we obtain a contradiction to condition (3.5).

Case (II). Let (1.4) hold. As in the proof of Theorem 2.2, we obtain (2.14) and

-[x.sup.[DELTA][DELTA]](t) [greater than or equal to] [[integral].sup.[infinity].sub.t] [[integral].sup.[infinity].sub.[tau]] f(s,[x.sup.[sigma]](s))[DELTA]s[DELTA][tau] for t [greater than or equal to] [t.sub.1] (3.8)

Using (1.7) and (3.3) in (3.8), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.9)

Integrating this inequality from t [greater than or equal to] [t.sub.1] to u [greater than or equal to] t, letting u [right arrow] [infinity] and using (2.14), we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

whenever t [greater than or equal to] [t.sub.1]. The rest of the proof is similar to that of Case (I) above and hence omitted. This completes the proof.

Theorem 3.3. Suppose that equation (1.1) is strongly sublinear. If

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.10)

for any constant c [not equal to] 0 and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.11)

for any constant [bar.c] [not equal to] 0, then equation (1.1) is oscillatory.

Proof. Let x(t) be a nonoscillatory solution of equation (1.1), say, x(t) > 0 for t [greater than or equal to] [t.sub.0]. By

Lemma 1.1, we consider the following two cases:

Case (I). Let (1.3) hold. As in the proof of Theorem 2.3, we obtain (2.18) and as in the proof of Theorem 3.2, we get (3.7), i.e.,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Using (2.18) in this inequality one easily finds

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Taking lim sup of both sides of the above inequality as t [right arrow] [infinity], we obtain a contradiction to (3.10).

Case (II). Let (1.4) hold. As in the proof of Theorem 2.2, we obtain (2.14) and as in Theorem 3.2, we get (3.9). Integrating (3.9) from t [greater than or equal to] [t.sub.1] to u [greater than or equal to] t and letting u [right arrow] [infinity], we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Using (2.14) in the above inequality, we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Taking lim sup of both sides of this inequality as t [right arrow] [infinity], we obtain a contradiction to (3.11). This completes the proof.

The following result is concerned with the oscillation of sublinear equation (1.1), i.e., f satisfies (1.7) with [beta] = 1.

Theorem 3.4. Suppose that equation (1.1) is sublinear. If

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for any constants c [not equal to] 0, [b.sub.1] > 0 and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for any constant [bar.c] [not equal to] 0 and [b.sub.2] > 0, then equation (1.1) is oscillatory.

Proof. The proof follows from Theorem 3.3, and hence omitted.

As illustrative examples, we apply Theorem 3.1 to equation (2.23) and Theorem 3.2 to equation (2.26) and obtain

Corollary 3.1. Suppose that equation (2.23) is strongly sublinear. If

[[integral].sup.[infinity]][s.sup.-[beta]][absolute value of f(s,c[s.sup.3])]ds = [infinity]

for any constant c [not equal to] 0 and

[[integral].sup.[infinity]][s.sup.1-3[beta]][absolute value of f(s,[bar.c][s.sup.3])]ds = [infinity]

for any constant [bar.c] [not equal to] 0, then equation (2.23) is oscillatory.

Corollary 3.2. Suppose that equation (2.26) is strongly sublinear. If

[[infinity].summation over (s=[t.sub.0])][absolute value of f(s,c[(s).sup.(3)])] = [infinity]

for any constant c [not equal to] 0 and

[[infinity].summation over (v=[t.sub.0])][v.sup.[beta]]([[infinity].summation over ([tau]=v)][[infinity].summation over (s=[tau])][([(s).sup.(3)]).sup.-[beta]][absolute value of f(s,[bar.c][(s).sup.(3)])]) = [infinity]

for any constant [bar.c] [not equal to] 0, then equation (2.26) is oscillatory.

Remark 3.1. The results of this paper extend some of our earlier results presented in [5,6] to fourth order dynamic equation (1.1). We also note that these results can be extended to delay dynamic equation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where g: T [right arrow] T, g(t) [less than or equal to] t and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Remark 3.2. We have applied some of the obtained results to cases when T = R and T = Z (see Corollaries 2.1, 2.2, 3.1 and 3.2). We can as well employ other type of time scales, e.g., [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] with q > 1, T = [N.sup.2], etc. see [3,4].

Acknowledgments. The authors would like to thank the editors for their helpful suggestions and professional assistance during the preparation of the revised form of this work.

REFERENCES

[1] R. P. Agarwal, M. Bohner and S. R. Grace: Oscillation criteria for first-order forced nonlinear dynamic equations, Can. Appl. Math. Q., 15(2007), No. 3, 223-233.

[2] R. P. Agarwal, M. Bohner, D. O'Regan and A. C. Peterson: Dynamic equations on time scales: a survey, J. Comput. Appl. Math., 141(2002), No. 1-2, 1-26.

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

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

[5] S. R. Grace, M. Bohner and R. P. Agarwal: On the oscillation of second-order half-linear dynamic equations, J. Difference Equ. Appl., 15(2009), No. 5, 451-460.

[6] S. R. Grace, R. P. Agarwal, M. Bohner and D. O'Regan: Oscillation of second order strongly superlinear and strongly sublinear dynamic equations, Commun. Nonlinear Sci. Numer. Simul., 14(2009), 3463-3471.

[7] T. Kusano, A. Ogata and H. Usami: On the oscillation of solutions of second order quasilinear ordinary differential equations, Hiroshima Math. J., 23(1993), No.3, 645-667.

Cairo University

Faculty of Engineering

Department of Engineering Mathematics

Orman, Giza 12221, Egypt

E-mail address: srgrace@eng.cu.edu.eg

Texas A&M University--Kingsville

Department of Mathematics

Kingsville, TX 78363, USA

E-mail address: Agarwal@tamuk.edu

National University of Ireland

Department of Mathematics

Galway, Ireland

E-mail address: donal.oregan@nuigalway.ie

Received: June 6, 2011. Revised: September 1, 2011.

2000 Mathematics Subject Classification: 34C10
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Author:Grace, Said R.; Agarwal, Ravi P.; O'Regan, Donal
Publication:Journal of Advanced Mathematical Studies
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Date:Jan 1, 2012
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