R E S E A R C H
Open Access
Classifications of solutions of second-order
nonlinear neutral differential equations of
mixed type
E Thandapani
1, S Padmavathi
1and S Pinelas
2**Correspondence:
2Departamento de Ciências Exactas
e Naturais, Academia Militar, Av. Conde Castro Guimarães, Amadora, 2720-113, Portugal
Full list of author information is available at the end of the article
Abstract
In this paper the authors classified all solutions of the second-order nonlinear neutral differential equations of mixed type,
(a
(t)(x
(t) +bx(t–τ
1) +cx(t+τ
2))
)
+p(t)xα(t–σ
1) +q(t)xβ(t+σ
2) = 0,into four classes and obtained conditions for the existence/non-existence of solutions in these classes. Examples are provided to illustrate the main results.
MSC: 34C15
Keywords: oscillation; second order; neutral differential equation; mixed type
1 Introduction
This paper is concerned with the second-order nonlinear neutral differential equations of mixed type of the form
a(t)x(t) +bx(t–τ) +cx(t+τ)
+p(t)xα(t–σ) +q(t)xβ(t+σ) = , (.)
t≥t> , subject to the following conditions: (c) a∈C([t,∞),R)and is positive for allt≥t;
(c) bandcare constants;τ,τ,σandσare nonnegative constants; (c) αandβare the ratio of odd positive integers;
(c) p,q∈C([t,∞),R).
By a solution of equation (.), we mean a functionx∈C([Tx,∞),R) for someTx≥t,
which has the propertiesx(t) +bx(t–τ) +cx(t+τ)∈C([Tx,∞),R) anda(t)(x(t) +bx(t– τ) +cx(t+τ))∈C((Tx,∞),R) and satisfies equation (.) on [Tx,∞). As is customary,
a solution of equation (.) is oscillatory if it is neither eventually positive nor eventually negative; otherwise, it is called non-oscillatory. A non-oscillatory solutionx(t) of equation (.) is said to be weakly oscillatory ifx(t) is non-oscillatory andx(t) is oscillatory for large value oft.
Second-order neutral delay differential equations have applications in problems dealing with vibrating masses attached to an elastic and also appear, as the Euler equation, in some vibrational problems (see [] and []).
In [] the authors considered equation (.) withq(t) = ,b=h(t),c= andα= for all t≥t, and classified all solutions of (.) into four classes and obtained criteria for
the existence/non-existence of solutions in these classes. In [] the authors considered equation (.) withq≤,b=c(t),c= ,α= ,β= for allt≥t, and classified all solutions
into four classes and obtained the solutions in these classes. Forp(t) = orq(t) = and
c= for allt≥t, the oscillatory and asymptotic behavior of solutions of equation (.) is
discussed in [, ] and [].
In [–] the authors considered equation (.) with a(t)≡ , α =β = or α =β
and obtained conditions for the oscillation of all solutions of equation (.). Motivated by this observation, in this paper we consider the cases p,q ≥ and p, q changes sign for all large t, to give sufficient conditions in order that every solution of equa-tion (.) is either oscillatory or weakly oscillatory and to study the asymptotic na-ture of non-oscillatory solutions of equation (.) with respect to their asymptotic be-havior. All the solutions of equation (.) may be a prioridivided into the following classes:
M+=x=x(t) solution of (.): there existstx≥tsuch thatx(t)x(t)≥,t≥tx
,
M–=x=x(t) solution of (.): there existstx≥tsuch thatx(t)x(t)≤,t≥tx
,
OS=x=x(t) solution of (.): there exists{tn},tn→ ∞such thatx(tn) =
,
WOS=x=x(t) solution of (.)x(t)= for all largetbutx(t) oscillates.
In Section , we obtain sufficient conditions for the existence/non-existence in the above said classes. In Section , we discuss the asymptotic behavior of solutions in the solutions
M+andM–. Examples are provided to illustrate the main results.
2 Existence results
First, we examine the existence of solutions of equation (.) in the classM+.
Theorem . With respect to the differential equation(.),assume that (H) b≥andc≥;
(H) p(t)≥for allt≥t; (H) α>βandσ≤σ.
If
lim sup t→∞
t
t
Lp(s) +q(s)ds=∞, (.)
for every L> ,then M+=∅.
Proof Suppose that equation (.) has a solutionx∈M+. Without loss of generality, we
may assume that there existst≥tsuch thatx(t) > and x(t)≥ for allt≥t. (The
Now,
Integrating the last inequality fromttot, we obtain
a(t)z(t)
From condition (.), we obtain
lim inf
Theorem . Assume that condition(H)holds.Further assume that (H) q(t)≥for allt≥t;
As in the proof of Theorem ., we havexβ–α(t–σ
Integrating the last inequality fromttot, we have
a(t)z(t)
From condition (.), we obtain
lim inf
Theorem . Assume that conditions(H)and(H)hold.Further assume that (H) α=β.
Proof Suppose that equation (.) has a solutionx∈M+. Without loss of generality, we may assume that there existst≥tsuch thatx(t) > and x(t)≥ for allt≥t. (The
Integrating the last inequality fromttot, we have
a(t)z(t)
From condition (.), we obtain
which contradicts the fact thatz(t)≥ for all larget. This completes the proof of the
theorem.
Theorem . Assume that conditions(H)-(H)hold.Further assume that (H) – <b+c≤,withb< andc> .
If
∞
t
a(s)ds=∞ and
∞
t
Lp(s) +q(s)ds=∞, (.)
for every L≥,then M+=∅.
Proof Suppose that equation (.) has a solutionx∈M+. Without loss of generality, we
may assume that there existst≥tsuch thatx(t) > and x(t)≥ for allt≥t. (The
proof ifx(t) < andx(t)≤ is similar for larget.) Letz(t) =x(t) +bx(t–τ) +cx(t+τ)≥
( +b+c)x(t–τ) > , fort≥t. From equation (.), we have
a(t)z(t) = –
p(t)x
α(t–σ
)
xβ(t–σ
)
+q(t)x
β(t+σ
)
xβ(t–σ
)
xβ(t–σ)
≤–Lp(t) +q(t)xβ(t–σ)
≤, t≥t.
Hence,a(t)z(t) is non-increasing fort≥t, and we claim thata(t)z(t)≥ fort≥t. If
a(t)z(t) < fort≥t, thena(t)z(t)≤a(t)z(t) < fort≥t> . Now,
z(t)≤a(t)z
(t
)
a(t) <
and integrating the last inequality fromttot, we obtain
z(t) –z(t)≤
t
t
a(t)z(t)
a(s) ds.
This implies thatz(t)→–∞ast→ ∞, which is a contradiction. Thusa(t)z(t)≥. Now, proceeding as in the proof of Theorem . and using condition (.), we have
lim t→∞
a(t)z(t)
xβ(t–σ
)
= –∞.
This contradicts the fact thatz(t)≥ for all larget. This completes the proof of the
the-orem.
Theorem . Assume that conditions(H), (H), (H)and(H)hold.If
∞
t
a(s)ds=∞ and
∞
t
p(s) +Lq(s)
ds=∞, (.)
Proof The proof is similar to that of Theorem . and hence the details are omitted.
Theorem . Assume that conditions(H), (H), (H)and(H)hold.If
∞
t
a(s)ds=∞ and
∞
t
p(s) +q(s)ds=∞, (.)
then M+=∅.
Proof The proof is similar to that of Theorem . and hence the details are omitted. Next, we examine the problem of the existence of solutions of equation (.) in the class
M–.
Theorem . Assume that conditions(H)-(H)and(H)hold.Further assume that (H) τ≤σ;
(H)
γ
du
uβ <∞and –γ
du
uβ > –∞for someγ > .
If
lim sup t→∞
t
T
a(s)
s
T
Lp(v) +q(v)dv
ds=∞ (.)
for all T≥tand for every L> ,then M–=∅.
Proof Suppose that equation (.) has a solutionx∈M–. Without loss of generality, we
may assume that there existst≥tsuch thatx(t) > and x(t)≤ for allt≥t. (The
proof is similar ifx(t) < andx(t)≥ for larget.) Letz(t) =x(t) +bx(t–τ) +cx(t+τ),
then in view of (H),z(t) > andz(t)≤ for allt≥t. As in the proof of Theorem ., we
obtain
a(t)z(t)
xβ(t–σ
)
– a(t)z
(t
)
xβ(t
–σ)≤
–
t
t
Lp(s) +q(s)ds, t≥t
or
z(t)
xβ(t–σ
)≤
–
a(t)
t
t
Lp(s) +q(s)ds, t≥t. (.)
Sincexis non-increasing and by (H), we see thatz(t)≤( +b+c)x(t–σ)≥ and
zβ(t)≤( +b+c)β
xβ(t–σ). (.)
Combining (.) and (.), we have
z(t)
zβ(t)≤
z(t) ( +b+c)βxβ(t–σ
)
≤ –
a(t)( +b+c)β t
t
Integrating the last inequality fromttot, yields
and by condition (.) we see that
lim sup
which contradicts condition (H). This completes the proof of the theorem.
Theorem . Assume that conditions(H), (H), (H)and(H)hold.Further assume that
Proof Suppose that equation (.) has a solutionx∈M–. Without loss of generality, we
may assume that there existst≥tsuch thatx(t) > and x(t)≤ for allt≥t. (The
Combining (.) and (.), we have
The rest of the proof is similar to that of Theorem . and hence the details are omitted. This completes the proof of the theorem.
Theorem . Assume that conditions(H), (H), (H), (H), (H)and(H)hold.If
Proof The proof is similar to that of Theorem . and hence the details are omitted. Next, we establish sufficient conditions under which any solution of equation (.) is either oscillatory or weakly oscillatory.
Theorem . If conditions(H), (H), (H), (H)and(.)hold,then every solution of
equation(.)is either oscillatory or weakly oscillatory.
Proof From Theorem ., it follows that for equation (.) we haveM+=∅. To complete
the proof, it suffices to show that for equation (.),M–=∅. Letxbe a solution of equation
Integrating the last inequality fromttotand using condition (.), we have
w(t)≤w(t) +
By Gronwall’s inequality, we obtain
or
z(t)≤ M
a(t), t≥t. The integration yields
z(t)≤z(t) +M
t
t
a(s)ds→–∞
ast→ ∞by condition (.). This contradiction completes the proof.
Theorem . If conditions(H), (H), (H)and(.)hold,then every solution of equation
(.)is either oscillatory or weakly oscillatory.
Proof The proof is similar to that of Theorem . and hence the details are omitted.
Theorem . If conditions(H), (H), (H)and(.)hold,then every solution of equation
(.)is either oscillatory or weakly oscillatory.
Proof The proof is similar to that of Theorem . and hence the details are omitted.
3 Behavior of solutions in the classesM+andM–
First, we study the asymptotic behavior of solutions in the classM–.
Theorem . Assume that conditions(H)-(H)and(H)hold.If condition(.)holds,
then for every solution x(t)∈M–,we havelim
t→∞x(t) = .
Proof The argument used in the proof of Theorem . again leads to (.). This implies thatlimt→∞z(t) = . Butz(t)≥x(t) for allt≥timplies thatlimt→∞x(t) = and the proof
is complete.
Theorem . Assume that conditions(H), (H), (H)and(H)hold.If condition(.)
holds,then for every solution x(t)∈M–,we havelimt→∞x(t) = .
Proof The argument used in the proof of Theorem . again leads to (.). This implies thatlimt→∞z(t) = . Butz(t)≥x(t) for allt≥timplies thatlimt→∞x(t) = and the proof
is complete.
Theorem . Assume that conditions(H), (H), (H)and(H)hold.If condition(.)
holds,then for every solution x(t)∈M–,we havelim
t→∞x(t) = .
Proof The argument used in the proof of Theorem . again leads to (.). This implies thatlimt→∞z(t) = . Butz(t)≥x(t) for allt≥timplies thatlimt→∞x(t) = and the proof
is complete.
Theorem . If the assumptions(H)-(H)hold and
Proof Letxbe a solution of equation (.) such thatx∈M+. Without loss of generality,
we assume that there existst≥tsuch thatx(t) > andx(t) > for allt≥t, for some Integrating the last inequality, we obtain
w(t)≥w(t) + ative for all large values oft. Thus
Sincez(t) =x(t) +bx(t–τ) +cx(t+τ) andx(t) is nondecreasing, we havez(t)≤( +b+
c)x(t+τ). In view of (.) we getlimt→∞x(t) =∞. This completes the proof.
Theorem . If the assumptions(H), (H)and(H)hold and
lim sup t→∞
t
T
p(s) +Lq(s)
s
T
a(v)dv
ds=∞ (.)
for all T≥t,and for any L> is satisfied,then every solution in the class M+ is
un-bounded.
Proof The proof is similar to that of Theorem . and hence the details are omitted.
Theorem . If the assumptions(H), (H)and(H)hold and
lim sup t→∞
t
T
p(s) +q(s)
s
T
a(v)dv
ds=∞ (.)
for all T≥tis satisfied,then every solution in the class M+is unbounded.
Proof The proof is similar to that of Theorem . and hence the details are omitted.
4 Examples
In this section we present some examples to illustrate the main results.
Example Consider the differential equation
t
x(t) + x(t– ) + x(t+ )
+
t(t– )x
(t– ) +
t(t+ )x(t+ ) = (.)
fort≥. All the conditions of Theorem . are satisfied except condition (.). We see that equation (.) has a solutionx(t) =t∈M+since it satisfies equation (.).
Example Consider the differential equation
t
x(t) + x(t– ) + x(t+ )
+
t(t– )x(t– ) +
t(t+ )x
(t+ ) = (.)
fort≥. All the conditions of Theorem . are satisfied except condition (.). We see that equation (.) has a solutionx(t) =t∈M+since it satisfies equation (.).
Example Consider the differential equation
t
x(t) +x(t– ) +x(t+ )
+
t(t– )x
(t– ) +
t(t+ )x
(t+ ) = (.)
Example Consider the differential equation
tx(t) – x(t– ) +x(t+ )+ t
(t– )x(t– ) +
t
(t+ )x
(t+ ) = (.)
fort≥. All the conditions of Theorem . are satisfied except conditions (H) and (.).
In fact, it has a solutionx(t) =t∈M+.
Example Consider the differential equation
et
x(t) +
ex(t– ) +ex(t+ )
+et–x(t– ) + et+x(t+ ) = (.)
fort≥. Since the function x(t) =e–t is a solution of equation (.), we haveM–=∅.
Moreover, condition (.) holds in Theorem ., while condition (H) is not satisfied.
Example Consider the differential equation
et
x(t) +
ex(t– ) +e
x(t+ )
+e(t–)x(t– ) + et+x(t+ ) = (.)
fort≥. Since the functionx(t) =e–t is a solution of equation (.), we haveM–=∅.
For this equation condition (.) does not hold in Theorem ., while condition (H) is
satisfied.
Example Consider the differential equation (.). It is easy to see that all the conditions of Theorem . are satisfied. In fact,x(t) =e–t∈M–is a solution of equation (.) such
thatx(t) =e–t→ ast→ ∞.
Example Consider the differential equation (.). It can easily be seen that all the con-ditions of Theorem . are satisfied. Herex(t) =t∈M+is a solution of equation (.) such
thatx(t)→ ∞ast→ ∞.
We conclude this paper with the following remark.
Remark In this paper we obtained conditions for the non-existence of solutions in the classesM+andM–and the existence of solutions in the classes OS and WOS. It would be
interesting to extend the results of this paper to the following equation:
a(t)x(t) +b(t)x(t–τ) +c(t)x(t+τ)
+p(t)xα(t–σ) +q(t)xβ(t+σ) =r(t),
wherer(t) is a real valued continuous function.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
Author details
1Ramanujan Institute for Advanced Study in Mathematics, University of Madras, Chennai, 600 005, India.2Departamento
de Ciências Exactas e Naturais, Academia Militar, Av. Conde Castro Guimarães, Amadora, 2720-113, Portugal.
Received: 23 November 2012 Accepted: 3 December 2012 Published: 28 December 2012
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