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26URAL MATHEMATICAL JOURNAL, Vol. 3, No. 2, 2017
ON THE OSCILLATION OF A THIRD ORDER NONLINEAR DIFFERENTIAL EQUATIONS WITH NEUTRAL TYPE
V. Ganesan
Department of Mathematics, Aringar Anna Government Arts College, Namakkal, Tamilnadu, India ganesan_vgp@rediffmail.com
M. Sathish Kumar
Department of Mathematics, Paavai Engineering College (Autonomous), Namakkal, Tamilnadu, India msksjv@gmail.com
Abstract: In this article, we investigate the oscillation behavior of the solutions of the third-order nonlinear differential equation with neural type of the form
(ai(í)(«2(t)Z'(t))')' + q(t)f (x(a(t))) =0, t > to > 0,
where Z(t) := x(t) + p(t)xa(r(t)). Some new oscillation results are presented that extend those results given in the literature.
AMS Subject Classification: 34C10, 34K11.
Key words: Oscillation, Non-linear, Neutral differential equation, Third order.
1. Introduction
Consider the third order non-linear neutral delay differential equation
(ai(í)(a2(í)Z'(í))')' + q(t)f(x(a(t))) = 0, t > to > 0, (E)
where Z(t) := x(t) + p(t)xa(r(t)) and 0 < a < 1 is a ratio of odd positive integers. Throughout this paper, without further mention, let
(Ai) ai(t) € C([to, +to)), a(t) > 0 for i = 1, 2 and p(t), q(t) € C([to, +to)), q(t) > 0; (A2) t(t) € C([to, +to)), t(t) < t, ff(t) € C([to, +ro)), a(t) < t;
(A3) f is nondecreasing and uf (u) > k > 0 for u = 0 and lim t(t) = lim a(t) = to.
t—t—
By a solution of equation (E) we mean a nontrivial real valued function x(t) € C([Tx, to)), Tx > to, which has the property Z'(t) € C 1([Tx, to)), a2(t)Z'(t) € C 1([Tx, to)), a1(t)(a2(t)Z'(t))' € C 1([Tx, to)) and satisfies (E) on [Tx, to). We consider only those solutions x(t) of (E) which satisfy sup{|x(t)| : t > T} > 0 for all T > Tx. A solution of (E) is called oscillatory if it has arbitrarily large zeros on [Tx, to) and otherwise, it is said to be non-oscillatory. Equation (E) is called almost oscillatory if all its solutions are oscillatory or convergent to zero asymptotically.
In the last years, a great deal of interest in oscillatory properties of neutral functional differential equations has been shown, we refer the reader to [1-8] and the references cited therein. A number
of authors including B. Baculikova and J. DZurina [10], T. Candan and Dahiya [11, 12], Graef et al. [7], and E. Thandapani and Li [8] have studied the oscillatory behavior of solutions of third order neutral differential equations in the form of equation (E) when a = 1.
Recently, Lin and Tang [13] explored the oscillation of first-order neutral differential equation with a super-linear neutral term
m
[x(t) - p(xa(t - t)]' + q(t) JJ |x(t - (Tjsgn[x(t - gj)] = 0,
j=i
where a > 1. Ravi P. Agarwal et al. [14] concerned with oscillation of a certain class of second-order differential equations with a sub-linear neutral term
(a(t) [x(t) + p(t)xa(t(t))]' )' + q(t)x(((t)) =0, t > to > 0,
where 0 < a < 1 is a ratio of odd positive integers and E. Thandapani et al. [9] established sufficient conditions for the oscillation of all solutions of a nonlinear differential equation
(a(t) [x(t) + p(t)xa(T(t))]' )' + q(t)xfi(((t)) = 0, t > to > 0,
where a and fi are ratio of odd positive integers. The above observation shows that this paper extend the results in third order.
This article presents the further investigation of the oscillations of (E). The following two cases:
i'O 1 i'O 1
-dt = oo, / —— dt = oo, (1.1)
'to
al(t) ' J to a2(t)
1 f™ 1
-dt < oo, / —--dt = oo, (1.2)
J to ai(t) ' J to a2(t)
are studied.
The paper is organized as follows. In Section 2, we present sufficient conditions for the oscillation of all solutions of (E) and in Section 3, we provide some examples to illustrate the main results.
In the following, all functional inequalities considered in this paper are assumed to hold eventually, that is, they are satisfied for all t large enough. Without loss of generality, we can deal only with the positive solutions of (E).
2. Main result
In this section, we state and prove our main results for the equation (E). For convenience, we use the notations
___ai{u) J
Ml~a ) ' - f* du
Mt) = ( i - . m -J" (2.i)
'ti
a1 (u)
Theorem 1. Let 0 < p(t) < pi < 1. If (1.1) holds and it there exists a positive function $ € Ci([to, to), R), such that for all sufficiently large t3 > t2 > ti > to we have
limsup r (<f>(s)kq(s)p*(s)Q(s) - ds = oo (2.2)
t^™ JtA 4^(s)
OO
and
q(s)ds
dudv = to (2.3)
J to a2(v) Jv ai(u) holds for all constants M > 0, then (E) is almost oscillatory.
Proof. Suppose that x(t) is a positive solution of (E). By condition (1.1), there exist two possible cases:
(1) Z(t) > 0, Z'(t) > 0, Mt)Z'(t))' > 0, (ai(t)(a2(t)Z'(t))')' < 0,
(2) Z(t) > 0, Z'(t) < 0, (a2(t)Z'(t))' > 0, (ai(t)(a2(t)Z'(t))')' < 0, for t > ti, ti is large enough. Assume Z (t) satisfying property (1), then
'ai(t)(a2(t)Z'(t))')' = -q(t)f (x(a(t))) < -fcq(t)x(a(t)) < 0.
If there exists t > ii such that Z(t) > 0, Z(a(t)) > 0, Z'(t) > 0, then Z(t) is monotonically increasing, there exists a constant M > 0 such that Z(t) > M and by the definition of Z we have
x(t) = Z{t) -V{t)xa{a{t)) > Z(t) — p(t)Za(a(t)) > (l - Z(t) = Mt)Z(t), (2.4)
where p*(t) is defined in (2.1). Let
w(t) > 0 for t > ti. Differentiating (2.5), we obtain
w/(i) = ^(t)ai(t)(a2(t)Z'(t)y + ^t)(a1(t)(a2(t)Zf(t)),y
-0(t)
a2(t)Z' (t) ' a2(t)Z' (t)
ai (t)(ai (t) (a2 (t) Z' (t))')(a2 (t) Z' (t))'
(a2 (t)Z'(t))2
Since (ai(t)(a2(t)Z'(t))')' < 0, then ai(t)(a2(t)Z'(t))' is decreasing, so
a2(i)Z'(i)> rai(s)(a2(;f(s))/^>al(t)(a2(t)Z-(t))- * ./ti ai(s) ./ti ai(s)
which implies that
a2{t)Z'{t) /tti ds/ai(s)/
< 0. (2.6)
Thus,
du /"s du
w v ; 4 r du a2(s) - [* du Jt2 a2(s) V ;
/t2 du a2(s) f* jmj_ jt2
ai(u) Jti ai(u)
DO
OO
DO
1
1
s
for t > t2 > ti.
It follows from (E), (2.4), and (2.5) that
a/(f) < - - <Kt)kq(t)Mt). Z(a(t))
Ht) 0(t)ai(t) rw ^w^*wa2(t)Z'(t)'
that is,
'. , H (t) . . w2(t) . . . . Z(((t)) a2(((t))Z' (((t))
W (i) < -rrvw(i) - ,, v , v - é(t)kq(t)pJt) , , , ,, V ■
From (2.6) and (2.7) follows
Mt) /-a du \ Mt) du
>(+\ ^ fltt) u\ w2(i) AU\U u\ u\J*2 V«2(s) Jtl ai(u)/ Jtl ai(u)
w (t) - " » - -p^r--
Jtl cn(u) Jtl ai(u)
Mt) du \
Jt1
ai(t)(H' (t))2
<-
- <l>(t)q(t)kp*(t)e(t) +
4H(t)
which implies
W'(t) < -mmkMtMt) + ai(^))2.
Integrating the last inequality from t3 (> t2) to t we obtain
jf ($(s)q(s)kp*(s)@(s) - aiif^j}S))2) ds < W(ts).
Letting t ^ to, it contradicts to (2.2).
Assume the case (2) holds. Using the similar proof of [10, Lemma 2], we can get limt^^ x(t) = 0 due to condition (2.3). □
Theorem 2. Let 0 < p(t) < pi < 1. If (1.2) holds and there exists a positive function ^ € Ci([t0, to), R), such that for all sufficiently large t3 > t2 > ti > t0, one has (2.2) and (2.3). If
where
c™ 1
r
i(i):= I
-ds,
ai (s)
holds for all constants M > 0, then (E) is almost oscillatory.
Proof. Suppose that x(t) is a positive solution of (E). By condition (1.2), there exist three possible cases:
(1) Z(t) > 0, Z'(t) > 0, Mt)Z'(t))' > 0, (ai(t)(a2(t)Z'(t))')' < 0,
(2) Z(t) > 0, Z'(t) < 0, (a2(t)Z'(t))' > 0, (ai(t)(a2(t)Z'(t))')' < 0, and
(3) Z(t) > 0, Z'(t) > 0, (a2(t)Z'(t))' < 0, (ai(t)(a2(t)Z'(t))')' < 0, for t > ti, ti is large enough.
In cases (1) and (2) we can obtain the conclusion of Theorem 2 by applying the proof of Theorem 1. Assume that case (3) holds, (ai(t)(a2(t)Z'(t))')' < 0 and ai(t)(a2(t)Z'(t))' is nonincreasing. Thus,
we get
ai(s)(a2(s)Z'(s))' < ai(t)(a2(t)Z'(t))', s > t > ti.
Dividing the above inequality by a1(s) and integrating from t to l, we obtain
Letting l ^ to, we have
that is,
ai(t)(a2(i)Z/(i))/ ds
a2(t)Z'(t) I ai(s) - L
t
(2.9)
Now define ^ as
t > ti.
(2.10)
Then <p(t) < 0 for t > ti. Therefore, by (2.9) and (2.10), we obtain
■¿(t)p(t) < 1.
(2.11)
(2.12)
In view of case (3), we see that
(2.13)
Thus,
Z(t)
/ ds/a2(s)
< 0 implies
ra(t> ds
Z{a{t)) Jtl a2(s) m -
r* ds Iti a2(s)
From (2.10) and (2.12)-(2.14) we have
p'(t) < —q(t)fcp*(t)
!tl a2(s) ai(t)'
Multiplying the above inequality by ¿(t) and integrating it from t2 (> t1) to t, we have
ft , \ V(t)5(t)-V(t2)5(t2) + j 5(s)kq(s)p*(s)[ J ^j)ds
7t2 ai(s) 7t2 ai (s)
/>t / Ms) dv \ /'t
<p(t)S(t) - <p(t2)S(t2) + jf S(s)kq(s)p*(s)( jf -
ds
t2 4^(s)ai(s)
+
t
t2
5(s) . , 1 1
from which it follows that t
t2
J U(s)q(s)fcp*(s)
rCT(s) dv \
2
ds < 0,
1
Iti a2(vW 4^(s)ai (s)
ds < 1 + p(t2)5(t2)
(2.14)
due to (2.11). Letting t ^ to, we come to the contradiction (2.8). Then the result of the Theorem follows. □
3. Examples
In this section we will present some examples to illustrate the main results. Example 1. Consider a third-order neutral differential equation
t-i/2 ti/2
x{t) + ^x3/5(t - 1)
+ -^x(t- 2) = 0, i>l,
(3.1)
t(t) = t — 1, and a(t) = t — 2. We obtain p*(t) = 1 —
1/4
M2/5 ;
where A > 0 is a constant. Let a = 3/5, ai(t) = t 1/2, a2(t) = t1/2, p(t) = 1/4, g(t) = —
ti/2
1
1
'to
a2(v) ai(u)
?i/2
ds
dudv = to.
A
and
r ( 1 is du ,
as
®(f) =
ft2 \a2(s) Jtl ai(u)J _(t- 2)3 + 6(t - 2)1/2ii/2 _ 3c
f* ¿3/2 _ ¿3/2
Jti
where c = t2/3 + 2tf2t2/2. Pick 0(t) = 1, then
A/ 1/4 ^ (S - + 6(s-2)1/H1l/2-3c
I si1-Mm)h -jz^-=
if 1/4 < M2/5. Hence, by Theorem 1, every solution of equation (3.1) is either oscillatory or converges to zero as t ^ to when 1/4 < M2/5.
Example 2. Consider a third-order neutral differential equation
(V [«(t) + l-xl'\t/8)] + \ (l + ^)x{t/2) =0, t > 1. (3.2)
Let a = 1/3, ai(t) = t2, «2(t) = 1, p(t) = 1/2, q(t) = 1/t( 1 + 2/27t2/3), t(t) = t/8 and a(t) = t/2. We obtain
and
then
1/2 x r ds 1 Jti a2(s) 2
'Î2
jt ^(s)g(s)p*(s) 1 - 0.5M-2/3)
rW dv \ 1 ds
Jti a2(v) J 4ô(s)ai(s)y
1 „-1/3 h 27 s2 2tl „-4/3" 27S - 4s
if 0.5 < M2/3. Hence, by Theorem 2, every solution of equation (3.2) is either oscillatory or converges to zero as t ^ to when 0.5 < M2/3 and x(t) = t-1 is such a solution of (3.2).
Acknowledgement
The authors would like to thank very much the anonymous referees for valuable suggestions, corrections and comments, which result in a great improvement of the original manuscript.
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