CC BY
7URAL MATHEMATICAL JOURNAL, Vol. 8, No. 2, 2022, pp. 71-80
DOI: 10.15826/umj.2022.2.006
PERIODIC SOLUTIONS OF A CLASS OF SECOND ORDER NEUTRAL DIFFERENTIAL EQUATIONS WITH MULTIPLE DIFFERENT DELAYS
Rabah Khemis
Laboratory of Applied Mathematics and History and Didactics of Mathematics (LAMAHIS),
University of 20 August 1955, B.P.26 route d'El-Hadaiek, Skikda, 21000 Algeria
Abdelouaheb Ardjouni
Department of Mathematics and Informatics, Souk Ahras University, Souk Ahras, 41000 Algeria
Ahieme Bouakkaz
Laboratory of Applied Mathematics and History and Didactics of Mathematics (LAMAHIS),
University of 20 August 1955, B.P.26 route d'El-Hadaiek, Skikda, 21000 Algeria
Abstract: The present work mainly probes into the existence and uniqueness of periodic solutions for a class of second-order neutral differential equations with multiple delays. Our approach is based on using Banach and Krasnoselskii's fixed point theorems as well as the Green's function method. Besides, two examples are exhibited to validate the effectiveness of our findings which complement and extend some relevant ones in the literature.
Keywords: Fixed point theorem, Green's function, Neutral differential equation, Periodic solutions.
1. Introduction
We frequently encounter neutral delay differential equations in the modeling of many phenomena in various domains such as physics, biology, population dynamics, medicine, epidemiology, economics, etc.
The investigation on such equations has been one of the most attracting topics in the literature. Recently, these equations have received a considerable attention and many researchers have sought to study them. For some related works, we refer the interested reader to some of them [1, 2, 4, 6, 8-10, 12, 13] and the references cited therein.
Stimulated by the aforementioned publications, we propose the following class of second order neutral differential equations
d2 d d2 r n i
-—x (t) + p(t) -x (t) + q (t) x (t) + -— k (t) x (t) (*) x " (*)) = (1-1)
1=1
where p,q € C(R, (0,oo)), k,c£,r£ € C2 (R, (0, oo)), £ = 1 ,n and e € C(K,[0,oo)) are T-periodic
In the current work, the authors aim is to establish sufficient conditions under which Banach and Krasnoselskii's fixed point theorems are guaranteed to work and hence the existence and uniqueness of periodic solutions of the equation (1.1) are proved. The general idea of our technique is to convert the equation (1.1) into an equivalent integral one in order to pave the way for the application of Banach and Krasnoselskii's fixed point theorems. Indeed, this last one with the help of Arzela-Ascoli theorem and some properties of the obtained Green's kernel, is a proper means for achieving our desired goals.
The key contributions of this work can be summarized as follows.
(i) New sufficient conditions that ensure the existence of periodic solutions of the equation (1.1) are established.
(ii) The studied problems in [1, 3-5, 7, 9, 12] are with globally Lipschitz source terms while this condition is not required here.
The basic frame of this paper is as follows. Section 2, provides some preliminary results and prerequisites that will be used in the sequel. Section 3 is dedicated to the statements and the proofs of our main results. In Section 4, we present two examples to which our main findings can be applied. The conclusion is included in the last section.
functions.
2. Preliminaries
Let
Pt = {x ec(R,Rt), x(t + T) = x(t)}, T> 0,
endowed with the supremum norm
x|| = sup |x (t)| = sup |x (t)| , teR te[o,T]
be a Banach space.
Throughout this paper we will assume that the following hypothesis are fulfilled. Here p, q, k, e, c£ and t£ are T-periodic real-valued functions such that
p(t + T) = p(t), q(t + T) = q(t), k(t + T) = k(t),
e(t + T) = e(t), ct{t + T) = c£{t), t£ {t + T) = r£{t), i = l^n,
(2.1)
p(s)cl,s > 0, / q(s)cl,s> 0, t£(t) > t*£ > 0, £ = l,n.
(2.2)
Lemma 1 [10]. If (2.1) and (2.2) hold and
(2.3)
where
and
Qi + J p (u) d-^^j Rf,
0
then there are continuous and T-periodic functions a and b such that
rT 0
and
b (t) > 0, f a(u)du> 0, a (t) + b(t) = p(t), Jo
d_
dt
b(t) + a(t)b(t) = q(t),
for all t € R. Furthermore, if $ € PT then the equation
x''(t) + p(t)x'(t) + q(t)x(t) = $ (t) has a T-periodic solution. Moreover, the periodic solution can be expressed as
rt+T
/t+1
G(t,s)<(s)ds,
where
G(t, s) =
+■
iexp ru rs / b (v) dv + a (v) dv .Jt Ju du
exp ( r r t+T exp J s a (u) du — 1 ' ru / b (v) dv + J exp o b ( u rs+T a (v) dv u ) du — 1 du
exp ( J a (u) d^j — 1 exp ( J b (u) d^j — 1
Corollary 1 [12]. If G is the Green's function given by (2.4), then G satisfies
G(t, t + T) = G(t, t), G(t + T,s + T) = G(t, s),
exp I / b(v)dv -^G(t,s) =a,(s)G(t,s) - ^Jt
d
—G(t,s) = -b(t)G(t,s) +
exp ^ J b (v) dv^ — 1 exp ^ J^ a (v) dv
ds2
G(t, s) = (a(s) + a'(s)) G(t, s) — (a(s) + b(s))
exp o b (v) dv — 1
exp ^ J^ b (v) dv
exp o b (v) dv — 1
(2.4)
Furthermore, by putting
I (1 I \
A = p(u)du, £>=T2exp(— / In (q(u))du\,
J0 Jo J
Ah = ^ A2 - 4 b) , M2 = \{a + VA2 + 4ft)
T T exp ( / p (u) du ai = ~—77-—■> «2 = - °
T
(eM2 — 1)2' exp I / b (v) dv
(eMl - 1)
2
H (t,s) =
T
exp b (v) dv - 1
-, P =
T
exp b (v) dv
0
T
H* (t,s) =
exp ^ J a (v) dv exp 0 b (v) dv - 1
-, P * =
exp 0 b (v) dv - 1
exp a (v) dv * \./o
T
exp 0 b (v) dv - 1
and if A2 > 4B, then we have
0 <ai < G(t,s) < a2, |H(t,s)|< P, |H*(t,s)|< P*
3. Existence and uniqueness of periodic solutions
Lemma 2. Suppose that (2.1)-(2.3) hold. If x € PT n C2 (R, R), then x is a solution of (1.1) if and only if x is a solution of the following equation
1 1 rt+T
X ^ = l + k (t) ^ C£ ^ X ^ ~ T£ ^ + l + k (t) Jt 6 ^ G ^ dS
t+T a(s) + 6(s) h (s) X (s) - C£ (s) x {s - n (s)) 1H {t, s) ds
1=1 n
k (s) x (s) — ^ cl (s) x (s — ti (s)) G(t, s)ds.
+
A 1 + k (t)
rt+T a(s) + a/(s)
It l + k(t)
(3.1)
i=i
Proof. Let x € Pt n C2 (R, R). From Lemma 1, we get
x (t) =
ft+T
A
9s
d
t+T
k (s) x (s) — ^ ce (s) x (s — T£ (s)) G (t, s)ds+ j e(s)G(t,s)ds
i=i
k (s) X (s) - ce (s) X (s - Te (s)) —G (t, s)
d
i=i
t+T
rt+T r n . d2 rt+T
k (s) x (s) —c-e (s) x (s — Te (s)) G(t,s)ds + / e(s) G (t, s) ds.
i=i
t
0
s
n
t
n
t
Since
d
k (s) x (s) - ce (s) x (s - t£ (s)) —G (t, s)
i=i s
n
= — k (t) x (t) + ^ ce (t) x (t — re (t)),
t+T
i=i
and
then
&
—,G(t, s) = (a(s) + a'(s)) G(t, s) - (a(s) + b(s)) H (t, s), ds2
n ,-t+T
(1 + k (t)) x (t) = ^2 ce (t) x (t — re (t)) + / e (s) G (t, s) ds
ft+T
+ J (a(s) + b(s))
rt+T
(a(s) + a' (s))
k (s) x (s) — ce (s) x (s — re (s))
i=i n
k (s) x (s) — ^ ce (s) x (s — re (s))
i=i
H (t, s) ds G(t, s)ds.
Dividing both sides of the above equation by 1 + k (t), we obtain (3.1). The converse implication can be obtained by the derivation of (3.1). □
Fore ease of exposition, we will use the following notations
A1 = max |a (t)|, At = max la' (t)| , a = max |e (t)| , te[0,T] 1 ie[0,T] 1 1 te[0,T]
ai = max |6(i)| , 5e = max |cf(i)|, I = 1,??.,
p0 = min |k (t)| , p1 = max |k (t)| , pi = max |k' (t)1 . ie[0,T] ie[0,T] te[0,T] 1
Furthermore, we suppose that
1 n
Ti = 7—<
1 + p01=1
and there exists L > 0 which satisfies the following estimate
F2 = ^ + r3L<L,
1 + Po
(3.2)
(3.3)
where
1
1 + Po
f n n \
T(pi + E (ß (Ai + ^i) + «2 (Ai + AÎ) ) + h
1=1 1=1
For employing Krasnoselskii's fixed point theorem, we need to define an operator that can be expressed as a sum of two operators, one of which is continuous and compact and the other is a contraction.
Indeed, from Lemma 2, we can define an operator S : Pt —> Pt as follows
(S<p) (t) = (Sip) (t) + (S2<p) (t),
n
t
n
t
where
and
(S1P) (t) =
1 n 1
J2cAt)<p(t-T£(t)) +
ft+T
1 + k (t)
i=i
1 + k (t) ,/t
e (s) G(t, s)ds,
(S2P) (t) =
rt+T a(s) + b(s)
t 1 + k (t) t+T a(s) + a'(s)
k (s) P(s) - X) ci(s) P(s - Ti(s))
i=i
H (t, s) ds
It 1 + k (t)
k (s) p(s) - Y^ ci(s) p(s - ti (s))
i=i
G(t, s)ds.
Clearly, (Sjp) (t + T) = (Sjp) (t), i = 1,2 which shows that operators Sj are well defined.
To reach our target, it suffices to prove the existence of at least one fixed point of the operator S1 + S2. This is due to the fact that the sought solution of equation (1.1) is just a fixed point of S1 + S2 and vice versa.
n
n
Theorem 1. Suppose that conditions (2.1)-(2.3), (3.2) and (3.3) hold. Then equation (1.1) admits at least one periodic solution x € PT which satisfies ||x|| < L.
Proof. For establishing the existence of periodic solutions, we use Krasnoselskii's fixed point theorem ([11]). The proof will be made in three steps.
Step 1. We show that S1 is a contraction mapping. Let p1, p2 € PT, we have
n (t)
|(sm) (t) - (Sm) (i)| < \<pi (t -U(t)) -<p2(t- n(i))| < ri 11^1 -<P2\\.
1=1 + ( )
From (3.2), we deduce that S1 is a contraction mapping.
Step 2. We show that S2 is continuous and compact mapping. Let p1, p2 € PT. For e > 0 and n = Ae, where
A = - 1 + p0
n
T(p1 + £ Si) (P (A1 + №)+ a (A1 + A^))
1=1
we obtain
||P1 - P2I < n IIS2P1 - S2P2I < e,
which shows the continuity of S2.
On the other hand, let h > 0, K = {p € PT, ||p|| < h} and {pn}neN be a sequence from K. We have
(n
P1 + £ Si
\\02VnU ^-1 /=1 7 (P(Ai+Ati) + a2(Ai + Ai)), (3.4)
1 + Po
and
Tt{S2(pn) {t) = l + m
b(t)(l + k(t))+k> (t) rt+T
(i + k(t))2 Jt
(a(s) + b(s))
k (t) Pu (t) - ^ Cl (t) yn (t - Ti (t))
l=1
n
k (s) Pu (s) - ^ Cl (s) Pu (s - Tl (s))
l=1 n
k (s) Pu (s) - ^ Cl (s) ^ (s - Tl (s))
l=1
H (t, s) ds G(t, s)ds
1
rt+T
1 + k (t)Jt
(a(s) + a' (s))
k (s) Pu (s) - ^ Cl (s) Pu (s - Tl (s))
Hence
where
dt
(S2Pu) (t)
l=1
< r4,
H * (t, s) ds.
(3.5)
r4 = h( p1 + Y, Si
l=1
(Ai+pi) + T/j*(Ai + AÎ) 1 + Po
+T
(P1 (1 + P1) + PÎ) (ß (A1 + P1) + a (A1 + A*))
(1 + Po)2
It follows from (3.4), (3.5) and the Arzela-Ascoli theorem [14] that S2 is a compact operator. Step 3. If L is defined as in (3.3), let
M = € Pt, |M| < L} .
In view of (3.3), if € M, then
||Sm + S2P2II < r2 < L,
which proves that
Si^i + S2^2 € M, € M.
From these three steps, we conclude that the operator S2 + S2 has at least one fixed point x € PT with ||x|| < L. Consequently, the equation (1.1) has at least one periodic solution in M. □
u
d
Theorem 2. Suppose that conditions (2.1)-(2.3) and (3.2) hold. If r3 < 1, then the equation (1.1) has a unique periodic solution x € PT.
Proof. Let <pi,(f2 € PT, we have
|(SPi )(t) - (S^2) (t)| < ra - WW .
Since r3 < 1, the Banach fixed point theorem [11] guarantees that the operator S has a unique fixed point which is the unique periodic solution of the equation (1.1). □
4. Examples
Example 1. Let L = 3n. We consider the following equation
x" (t) + y2x' (t) + ^x (t) + (^.r (t) - (jL sin2 27ri) X (t 7T sin2 2vrt)
V 150
(4.1)
cos2
Here
which implies
C2 (t) = — COS2 2vri, Tl (i) = 7T Sin2 2vri, T'2 (t) = 27r cos4 2vri, 150
100 4 e(i) = Ioiosin 2vrt' T = 1'
12' 24' 144 6' 10'
1 ^ R1
<* -15» +0 '
exp ^ J p (u) du^ - 1
Q1T
~ 22.367 > 1,
1 1 3
Mi = -, M2 = -, a2~ 46.118, 6.5139, = — < 1,
6 4 202
r2 ~ 8.0746 < L = 3n, r ~ 0.36742 < 1.
It follows from Theorem 2 that the equation (4.1) has a unique solution x € PT which satisfies ||x|| < 3n.
The following example shows the usefulness of Theorem 1 when the Banach fixed point theorem cannot be applied.
Example 2. We consider the following equation
5 1 ii (e1/6 - 1)2
(i) + ^ it) + 24® (i) + ((65ei/3_5ei/6+3e5/i2^) ,1/6 ^ ■
-( 2——^——^-—sin2 27ri ) x (i-vrsin2
(25ei/3(_e5e6i/6+3e5/i2 sin2 2* (* " * sin2 2^
j x (t - 2n cos4 2nt) ^ =0.
1/6 - 1 2
- I 4——^——^-— sin2 2vrt I x
Here
5e1/3 - 5e1/6 + 3e5/12
5 1 (e1/6 - 1)2 P(t) = ^, P(t) = 777, Ht)= 6 1 J
12' L y> 24' 5e1/3 - 5e1/6 + 3e5/12'
= "t,5ei/3 _ 5ei/6 + 3e5/i2 cus
24
(e1/6 - 1)2 2 (e1/6 -1)2 2
Cl {t) = ~■)( 1 ' "k 1 '• • '■><" 12 Sin ^ C2 (t) = 45eV3-5eV6 + 3e5A2 C°S ^
T1 (t) = n sin2 2nt, T2 (t) = 2n cos4 2nt, e (t) = 0, T = 1,
which implies
A = —, B = -, A2 = — >4 B2 = -, R1 = -, 12' 24' 144 6' 10'
* = I5Ö +
Ri
exp ^ J p (u) du^ - 1
Q1T
~ 22.367 > 1,
M1 =
1
1
M2 = -, a2~ 46.118, 6.5139, Li = 0.03391 < 1,
r2 = L < L, VL > 0, r3 = 1.
Since r3 = 1, we can not use Theorem 2, but r2 = L < L, so we can apply Theorem 1 to prove that the equation (4.2) has at least one periodic solution x € PT which satisfies ||x|| < L.
6
5. Conclusion
In this paper, by utilizing both the Banach and Krasnoselskii's fixed point theorems and the Green's functions method, a class of second-order neutral differential equations with multiple delays has been investigated. To be more precise, we have discussed the existence and uniqueness of periodic solutions by transforming the equation (1.1) into an equivalent integral one and then by using the Banach and Krasnoselskii's fixed point theorems.
Acknowledgements
The authors are grateful to the referees for their valuable comments which have led to improvement of the presentation.
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