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Existence of positive periodic solutions to nonlinear integro-differential equations
Applied Mathematics and Computation 253 (2015) 287–293
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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Existence of positive periodic solutions to nonlinear integro-differential equations Bozˇena Dorociaková ⇑, Rudolf Olach Department of Mathematics, University of Zˇilina, 010 26 Zˇilina, Slovak Republic
a r t i c l e
i n f o
a b s t r a c t This paper deals with the existence of positive periodic solutions for a class of nonlinear integro-differential equations. Such equations arise in the theory of a circulating fuel nuclear reactor. The existence of positive solutions and exponential stability is also treated. The main results are illustrated with several examples. Ó 2014 Elsevier Inc. All rights reserved.
Keywords: Positive periodic solution Existence Nonlinear Integro-differential equation Exponential stability Banach space
1. Introduction In this paper we investigate the existence of positive x-periodic solutions for the nonlinear integro-differential equations of the form
_ þ xðtÞ
Z
t
pðt sÞgðxðsÞÞ ds ¼ 0;
t P T:
ð1Þ
ts
With respect to Eq. (1) in the next sections we will assume the following conditions: (i) p 2 Cð½0; sÞ; RÞ, (ii) g 2 Cðð0; 1Þ; ð0; 1ÞÞ. The Eq. (1) was encountered by [1] in the theory of a circulating fuel nuclear reactor. In this model, x is the neutron density. It is also a good model in one dimensional viscoelasticity in which x is the strain and p is the relaxation function. Such equations have been also proposed to model some biological problems. The stability of zero solution of Eq. (1) was later studied in [2–6] and the references cited therein. To the best of our knowledge there are only a few results on the existence of periodic solutions of Eq. (1). According to applications it is reasonable to consider the positive solutions of (1). In this paper we will obtain existence criteria for the positive x-periodic solutions of Eq. (1). For such equations the existence results in the literature are largely based on the several assumptions for the function pðtÞ and the authors usually do not consider the existence of positive periodic solutions. For example the _ €ðtÞ 0; authors in [3,5] assume that the function pðtÞ satisfies the next conditions: pðsÞ ¼ 0; pðtÞ P 0; pðtÞ 6 0; p 0 6 t 6 s. Authors in [7] studied oscillatory solutions of linear integro-differential equations. Motivated by the discussion above, we will focus on the existence of positive periodic solutions for the Eq. (1) without assumptions above on the function pðtÞ. For related results see also [8,9] and the references therein. In the third section the existence of positive solutions is also considered. In Section 4 we will establish the conditions for the exponential stability of positive solution of Eq. (1). To the ⇑ Corresponding author. E-mail addresses: [email protected] (B. Dorociaková), [email protected] (R. Olach). http://dx.doi.org/10.1016/j.amc.2014.09.086 0096-3003/Ó 2014 Elsevier Inc. All rights reserved.
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B. Dorociaková, R. Olach / Applied Mathematics and Computation 253 (2015) 287–293
best of our knowledge this problem has not been solved yet. For related results we refer readers to [10,11] and the references cited therein. The following fixed point theorem will be used to prove the main results in the next sections. Theorem 1.1 (Schauder’s Fixed Point Theorem [12,13]). Let X be a closed, convex and nonempty subset of a Banach space X. Let S : X ! X be a continuous mapping such that SX is a relatively compact subset of X. Then S has at least one fixed point in X. That is there exists an x 2 X such that Sx ¼ x. 2. Existence of periodic solutions In this section we will study the existence of positive x-periodic solutions of Eq. (1). In the next lemma and theorems we choose T > s > 0. Lemma 2.1. Suppose that there exists a positive and continuous function kðt; sÞ; t s 6 s 6 t, such that
Z
tþx
Z
u
us
t
pðu v Þkðu; v Þ dv du ¼ 0;
Then the function
Z f ðtÞ ¼ exp
t
Z
u
us
T
t P T:
pðu v Þkðu; v Þ dv du ;
ð2Þ
t P T;
is x-periodic. Proof. For t P T we obtain
Z f ðt þ xÞ ¼ exp
tþx
T
Z ¼ exp
t
Z
Z
u us
u us
T
pðu v Þkðu; v Þ dv du
Z pðu v Þkðu; v Þ dv du exp
tþx
t
Z
u
us
pðu v Þkðu; v Þ dv du ¼ f ðtÞ:
Thus the function f ðtÞ is x-periodic. Theorem 2.1. Suppose that there exists a positive and continuous function kðt; sÞ; t s 6 s 6 t, such that (2) holds and
exp
Z
t
Z
u
us
T
Z pðu v Þkðu; v Þ dv du g exp
s
Z
T
u
us
¼ kðt; sÞ; pðu v Þkðu; v Þ dv du
t P T:
ð3Þ
Then Eq. (1) has a positive x-periodic solution. Proof. Let X ¼ fx 2 Cð½T s; 1Þ; RÞg be the Banach space with the norm kxk ¼ suptPTs jxðtÞj. We set
Z f ðtÞ ¼ exp
T
t
Z
u
us
pðu v Þkðu; v Þ dv du ;
t P T:
With regard to Lemma 2.1 we have m 6 f ðtÞ 6 M, where
Z t Z u m ¼ min exp pðu v Þkðu; v Þ dv du ; t2½T;1Þ T us Z t Z u pðu v Þkðu; v Þ dv du : M ¼ max exp t2½T;1Þ
T
us
We now define a closed, bounded and convex subset X of X as follows
X ¼ fx 2 X : xðt þ xÞ ¼ xðtÞ; m 6 xðtÞ 6 M;
t P T;
t P T;
kðt; sÞxðtÞ ¼ gðxðsÞÞ; t P T; t s 6 s 6 t; xðtÞ ¼ 1; T s 6 t 6 Tg: Define the operator S : X ! X as follows
( ðSxÞðtÞ ¼
R R t u v ÞÞ exp T us pðu v Þ gðxð dv du ; xðuÞ 1; T s 6 t 6 T:
t P T;
ð4Þ
B. Dorociaková, R. Olach / Applied Mathematics and Computation 253 (2015) 287–293
289
We will show that for any x 2 X we have Sx 2 X. For every x 2 X and t P T we get
Z ðSxÞðtÞ ¼ exp
t T
Z
Z t Z u gðxðv ÞÞ dv du ¼ exp pðu v Þkðu; v Þ dv du 6 M; xðuÞ T uv
u us
pðu v Þ
and ðSxÞðtÞ P m. For t 2 ½T s; T we have ðSxÞðtÞ ¼ 1, that is ðSxÞðtÞ 2 X. Further for every x 2 X and t P T; t s 6 s 6 t, according to (3) it follows
Z sZ u Z sZ u gðxðv ÞÞ ¼ g exp gððSxÞðsÞÞ ¼ g exp pðu v Þ dv du pðu v Þkðu; v Þ dv du xðuÞ T us T us Z t Z u Z t Z u gðxðv ÞÞ gðxðv ÞÞ pðu v Þ dv du exp pðu v Þ dv du exp xðuÞ xðuÞ T us T us Z sZ u Z t Z u exp ¼ g exp pðu v Þkðu; v Þ dv du pðu v Þkðu; v Þ dv du ðSxÞðtÞ ¼ kðt; sÞðSxÞðtÞ: T
us
T
us
Finally we will show that for x 2 X; t P T the function ðSxÞðtÞ is x-periodic. For x 2 X; t P T and with regard to (2) we get
Z ðSxÞðt þ xÞ ¼ exp
gðxðv ÞÞ dv du xðuÞ us T Z t Z u Z tþx Z u gðxðv ÞÞ gðxðv ÞÞ ¼ exp pðu v Þ dv du exp pðu v Þ dv du xðuÞ xðuÞ t T us us Z tþx Z u ¼ ðSxÞðtÞ exp pðu v Þkðu; v Þ dv du ¼ ðSxÞðtÞ: tþx
Z
u
pðu v Þ
us
t
So ðSxÞðtÞ is x-periodic on ½T; 1Þ. Thus we have proved that Sx 2 X for any x 2 X. We now show that S is completely continuous. First we will show that S is continuous. Let xi ¼ xi ðtÞ 2 X be such that xi ðtÞ ! xðtÞ 2 X as i ! 1. For t P T we have
Z jðSxi ÞðtÞ ðSxÞðtÞj ¼ exp
T
t
Z
u
us
pðu v Þ
Z t Z u gðxi ðv ÞÞ gðxðv ÞÞ dv du exp pðu v Þ dv du : xi ðuÞ xðuÞ T us
By applying the Lebesgue dominated convergence theorem we obtain that
limkðSxi ÞðtÞ ðSxÞðtÞk ¼ 0: i!1
For t 2 ½T s; T the relation above is also valid. This means that S is continuous. We now show that SX is relatively compact. It is sufficient to show by the Arzela–Ascoli theorem that the family of functions fSx : x 2 Xg is uniformly bounded and equicontinuous on ½T s; 1Þ. The uniform boundedness follows from the definition of X. According to (4) for t P T; x 2 X we get
Z t Z t Z u d gðxðv ÞÞ gðxðv ÞÞ ðSxÞðtÞ ¼ pðt v Þ exp pðu v Þ d v du dt xðtÞ xðuÞ ts T us Z t Z t Z u pðt v Þkðt; v Þ dv exp pðu v Þkðu; v Þ dv du 6 M 1 ; ¼ ts
T
M1 > 0:
us
For t 2 ½T s; T; x 2 X we have
d ðSxÞðtÞ ¼ 0: dt This shows the equicontinuity of the family SX, (cf. [14, p. 265]). Hence SX is relatively compact and therefore S is completely continuous. By Theorem 1.1 there is an x0 2 X such that Sx0 ¼ x0 . We see that x0 ðtÞ is a positive x-periodic solution of Eq. (1). The proof is complete. Corollary 2.1. Suppose that there exists a positive and continuous function kðt; sÞ; t s 6 s 6 t, such that (2) holds and
Z s
t
Z
u
us
pðu v Þkðu; v Þ dv du ¼ ln kðt; sÞ;
t P T:
ð5Þ
Then the equation
_ þ xðtÞ
Z
t
ts
pðt sÞxðsÞ ds ¼ 0;
t P T;
ð6Þ
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B. Dorociaková, R. Olach / Applied Mathematics and Computation 253 (2015) 287–293
has a positive x-periodic solution
Z xðtÞ ¼ exp
Z
t
u
us
T
pðu v Þkðu; v Þ dv du ;
t P T:
3. The existence of positive solutions In this section we will investigate the existence of positive solutions of nonlinear integro-differential equation (1). Theorem 3.1. Suppose that there exists a positive and continuous function kðt; sÞ; t s 6 s 6 t, such that (3) holds and
Z
t
pðt sÞkðt; sÞ ds > 0;
t P T:
ð7Þ
ts
Then Eq. (1) has a positive solution
Z xðtÞ ¼ exp
Z
t
u
us
T
pðu v Þkðu; v Þ dv du ;
t P T:
Proof. Let X 1 ¼ fx 2 Cð½T s; 1Þ; RÞg be the set of all continuous bounded functions. Then X 1 is a Banach space with the norm kxk ¼ suptPTs jxðtÞj. We set
Z wðtÞ ¼ exp
t
Z
pðu v Þkðu; v Þ dv du ;
u
us
T
t P T:
We define a closed, bounded and convex subset X1 of X 1 as follows
X1 ¼ fx 2 X 1 : wðtÞ 6 xðtÞ 6 1;
t P T;
t P T; t s 6 s 6 t;
kðt; sÞxðtÞ ¼ gðxðsÞÞ;
T s 6 t 6 T:g
xðtÞ ¼ 1;
Define the operator S1 : X1 ! X 1 as follows
( ðS1 xÞðtÞ ¼
R R t u v ÞÞ exp T us pðu v Þ gðxð dv du ; xðuÞ 1; T s 6 t 6 T:
t P T;
For every x 2 X1 and t P T we obtain
Z ðS1 xÞðtÞ ¼ exp
t
Z
u
us
T
pðu v Þkðu; v Þ dv du 6 1
and ðS1 xÞðtÞ P wðtÞ. For t 2 ½T s; T we get ðS1 xÞðtÞ ¼ 1, that is ðS1 xÞðtÞ 2 X1 . Now we can proceed by the similar way as in the proof of Theorem 2.1. We omit the rest of the proof. Corollary 3.1. Assume that there exists a positive and continuous function kðt; sÞ; t s 6 s 6 t, such that (5) and (7) hold. Then Eq. (6) has a positive solution
Z xðtÞ ¼ exp T
t
Z
u
us
pðu v Þkðu; v Þ dv du ;
t P T:
Corollary 3.2. Assume that there exists a positive and continuous function kðt; sÞ; t s 6 s 6 t, such that (3) and (7) hold and
lim
t!1
Z
t
Z
u
us
T
pðu v Þkðu; v Þ dv du ¼ 1:
Then Eq. (1) has a positive solution which tends to zero. Corollary 3.3. Assume that there exists a positive and continuous function kðt; sÞ; t s 6 s 6 t, such that (3) and (7) hold and
lim
t!1
Z T
t
Z
u
us
pðu v Þkðu; v Þ dv du ¼ a:
Then Eq. (1) has a positive solution which tends to constant ea .
B. Dorociaková, R. Olach / Applied Mathematics and Computation 253 (2015) 287–293
291
4. Stability of positive solution In this section we consider the exponential stability of positive solution of Eq. (1). We denote xðt; T; uÞ; t P T s, for a solution of Eq. (1) satisfying the initial condition xðs; T; uÞ ¼ uðsÞ > 0 for s 2 ½T s; T. Let xðtÞ ¼ xðt; T; uÞ; x1 ðtÞ ¼ xðt; T; u1 Þ and yðtÞ ¼ xðtÞ x1 ðtÞ; t 2 ½T s; 1Þ. Then we get
_ yðtÞ ¼
Z
t
pðt sÞ½gðxðsÞÞ gðx1 ðsÞÞ ds;
t P T:
ts
By the mean value theorem we obtain
_ yðtÞ ¼
Z
t
pðt sÞg 0 ðx Þ½xðsÞ x1 ðsÞ ds;
g 0 ðxÞ ¼
ts
_ yðtÞ ¼
Z
dgðxÞ ; dx
t
pðt sÞg 0 ðx ÞyðsÞ ds;
t P T:
ð8Þ
ts
Definition. Let x1 ðtÞ be a positive solution of Eq. (1) and there exist constants T u;x1 ; K u;x1 and k > 0 such that for every solution xðt; T; uÞ of (1)
jxðt; T; uÞ x1 ðtÞj 6 K u;x1 ekt ;
t P T u;x1 :
Then x1 ðtÞ is said to be exponentially stable. We assume that the function
Fðt; xÞ ¼
Z
t
pðt sÞgðxðsÞÞ ds;
t P T;
ts
is Lipschitzian in second argument. In the next theorem we establish sufficient conditions for the exponential stability of positive solution x1 ðtÞ ¼ xðt; T; 1Þ of Eq. (1).
Theorem 4.1. Suppose that (3) and (7) hold and
p 2 Cð½0; s; ð0; 1ÞÞ;
g 2 C 1 ðð0; 1Þ; ð0; 1ÞÞ;
g 0 ðxÞ P c > 0:
Then Eq. (1) has a positive solution which is exponentially stable. Proof. We will show that there exists a positive k such that
jxðt; T; uÞ x1 ðtÞj 6 K u;x1 ekt ; kT 1
where K u;x1 ¼ maxt2½Ts;T 1 jyðtÞje kt
LðtÞ ¼ jyðtÞje ;
t P T 1 ¼ T þ 2s;
þ 1. Consider the Lyapunov function
t P T1:
We claim that LðtÞ 6 K u;x1 for t P T 1 . On the other hand there exists t > T 1 such that Lðt Þ ¼ K u;x1 . Calculating the upper left derivative of LðtÞ along the solution yðtÞ of (8) we obtain
D ðLðtÞÞ ¼ ekt
Z
t
pðt sÞg 0 ðx ÞjyðsÞj ds þ kjyðtÞjekt ;
t P T1:
ts
For t ¼ t we get
0 6 D ðLðt ÞÞ ¼ c ekt
Z
t
pðt sÞjyðsÞj ds þ kjyðt Þjekt : t s
If yðtÞ > 0; t P T then from (8) it follows that for t P T þ s the function yðtÞ is decreasing and if yðtÞ < 0; t P T then yðtÞ is increasing for t P T þ s. We conclude that jyðtÞj; t P T þ s has decreasing character. Then we obtain
0 6 D ðLðt ÞÞ 6 cjyðt Þjekt
Z
t
t s
Z pðt sÞ ds þ kjyðt Þjekt ¼ c
Z s ¼ c pðuÞ du þ k jyðt Þjekt :
t
pðt sÞ ds þ k jyðt Þjekt
t s
0
For 0 < k < c
Rs 0
pðuÞ du we have a contradiction. Thus jyðtÞjekt 6 K u;x1 for t P T 1 and 0 < k < c
Rs 0
pðtÞ dt. The proof is complete.
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B. Dorociaková, R. Olach / Applied Mathematics and Computation 253 (2015) 287–293
5. Examples The next examples illustrate some results above. Example 1. Consider the integro-differential equation
_ þ xðtÞ
Z
t
pðt sÞxðsÞ ds ¼ 0;
t P T;
ð9Þ
t2p
where pðtÞ ¼ p1 sin t. We choose
kðt; sÞ ¼
c þ sin s ; c þ sin t
c > 1; t 2p 6 s 6 t:
Then for the condition (2), and x ¼ 2p we have
Z
tþ2p
t
¼
p
Z
1
u
1
u2p tþ2p
Z
1
Z
p
sinðu v Þ
1 c þ sin u
t tþ2p
1 c þ sin u
Z
c þ sin v dv du c þ sin u
u
u2p u
sinðu v Þðc þ sin v Þ dv du
Z
ðsin u cos v cos u sin v Þðc þ sin v Þ dv du u2p Z u Z u 1 tþ2p 1 cðsin u cos v dv cos u sin v dv Þ ¼ c þ sin u p t u2p u2p Z u Z u 2 sin v cos v dv cos u sin v dv du þ sin u
¼
p
Z
t
u2p tþ2p
Z
¼
u2p
cos u du ¼ 0; c þ sin u
t
t P T:
For the condition (5) we get
Z
t
Z
u
u2p
s
pðu v Þkðu; v Þ dv du ¼
1
p
Z
t
1 c þ sin u
s
¼ ln kðt; sÞ;
Z
u
u2p
sinðu v Þðc þ sin v Þ dv du ¼
Z
t
s
cos u c þ sin s du ¼ ln c þ sin u c þ sin t
t P T:
All conditions of Corollary 2.1 are satisfied and Eq. (9) has a positive x ¼ 2p-periodic solution
Z xðtÞ ¼ exp ¼ exp
Z
t
Z
u
u2p
T
Z 1 pðu v Þkðu; v Þ dv du ¼ exp
cos u c þ sin t ; du ¼ c þ sin u c þ sin T
t
T
p
t
Z
u
u2p
T
sinðu v Þ
c þ sin v dv du c þ sin u
t P T:
Example 2. Consider the integro-differential equation
_ þ xðtÞ
Z
t
pðt sÞxðsÞ ds ¼ 0;
t P T;
ð10Þ
ts
where pðtÞ ¼ 1s et . We choose kðt; sÞ ¼ ets ; t s 6 s 6 t. Then for the condition (5) we obtain
Z
s
t
Z
u
us
pðu v Þkðu; v Þ dv du ¼
Z
s
t
1
Z
s
u
us
euþv euv dv du ¼
1
s
Z
s
t
Z
u
us
dv du ¼
Z
t
du ¼ t s;
t P T:
s
Thus all conditions of Corollary 3.1 are satisfied and Eq. (10) has the positive solution
Z xðtÞ ¼ exp T
t
1
s
Z
u
us
Z t euþv euv dv du ¼ exp du ¼ etþT ;
t P T;
T
which tends to zero. Example 3. Consider the integro-differential equation
_ þ xðtÞ
Z
t
t2p
pðt sÞxðsÞ ds ¼ 0;
t P T;
ð11Þ
293
B. Dorociaková, R. Olach / Applied Mathematics and Computation 253 (2015) 287–293 s
where pðtÞ ¼ e2p21 cos t. We choose kðt; sÞ ¼ 1þe ; t 2p 6 s 6 t. Then for the condition (5) we have 1þet
Z
Z
2 1 þ ev dv du cosðu v Þ 2p 1 e 1 þ eu s u2p Z t Z u 2 1 ¼ 2p ðcos u cos v þ sin u sin v Þð1 þ ev Þ dv du e 1 s 1 þ eu u2p Z t Z u Z u 2 1 ¼ 2p cos u cos v d v þ sin u sin v dv e 1 s 1 þ eu u2p u2p Z u Z u ev cos v dv þ sin u ev sin v dv du þ cos u u2p u2p Z t Z u Z u 2 1 v v du cos u e cos v d v þ sin u e sin v d v ¼ 2p e 1 s 1 þ eu u2p u2p 2p Z t 2 1 e 1 u e cos uðcos u sin uÞ ¼ 2p e 1 s 1 þ eu 2 e2p 1 u e sin uðsin u þ cos uÞ du þ 2 Z t eu ðcos uðcos u sin uÞ þ sin uðsin u þ cos uÞÞ du ¼ 1 þ eu s Z t eu 1 þ es du ¼ ln ; t P T: ¼ u 1 þ e 1 þ et s t
u
All conditions of Corollary 3.1 are satisfied and Eq. (11) has a positive solution
xðtÞ ¼ exp
2 1 e2p
Z T
which tends to the constant
t
Z
u
u2p
cosðu v Þ
Z t 1 þ e v eu 1 þ et dv du ¼ exp du ¼ ; u u 1þe 1 þ eT T 1þe
t P T; :
1 . 1þeT
Acknowledgements The research was supported by the grant 1/0234/13 of the Scientific Grant Agency of the Ministry of Education of the Slovak Republic. The authors have made the same contribution. All authors read and approved the final manuscript. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]
W.K. Ergen, Kinetics of the circulating fuel nuclear reactor, J. Appl. Phys. 25 (1954) 702–711. T.A. Burton, Stability by Fixed Point Theory for Functional Differential Equations, Dover Publications Inc., New York, 2006. J. Hale, Functional Differential Equations, Springer-Verlag, New York-Heidelberg-Berlin, 1971. J.J. Levin, The asymptotic behavior of a Volterra equation, Proc. Amer. Math. Soc. 14 (1963) 434–451. J.J. Levin, J.A. Nohel, On a nonlinear delay equation, J. Math. Anal. Appl. 8 (1964) 31–44. D. Zhao, S. Yuan, 3/2-stability conditions for a class of Volterra–Levin equations, Nonlinear Anal. 94 (2014) 1–11. R. Olach, H. Šamajová, Oscillations of linear integro-differential equations, Cent. Eur. J. Math. 3 (2005) 98–104. J. Diblík, B. Iricˇanin, S. Stevic´, Z. Šmarda, Note on the existence of periodic solutions of a class of systems of differential-difference equations, Appl. Math. Comput. 232 (2014) 922–928. Z. Jin, H. Wang, A note on positive periodic solutions of delayed differential equations, Appl. Math. Lett. 23 (2010) 581–584. H. Zhang, L. Wang, M. Yang, Existence and exponential convergence of the positive almost periodic solution for a model of hematopoiesis, Appl. Math. Lett. 26 (2013) 38–42. B. Lin, New results on the positive almost periodic solutions for a model of hematopoiesis, Nonlinear Anal. RWA 17 (2014) 252–264. € men, Studia Math. 2 (1930) 171–180. J. Schauder, Der Fixpunktsatz in Funktionalrau A. Granas, J. Dugundji, Fixed Point Theory, Springer, New York, 2003. L.H. Erbe, Q.K. Kong, B.G. Zhang, Oscillation Theory for Functional Differential Equations, Marcel Dekker, New York, 1995.
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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Existence of positive periodic solutions to nonlinear integro-differential equations Bozˇena Dorociaková ⇑, Rudolf Olach Department of Mathematics, University of Zˇilina, 010 26 Zˇilina, Slovak Republic
a r t i c l e
i n f o
a b s t r a c t This paper deals with the existence of positive periodic solutions for a class of nonlinear integro-differential equations. Such equations arise in the theory of a circulating fuel nuclear reactor. The existence of positive solutions and exponential stability is also treated. The main results are illustrated with several examples. Ó 2014 Elsevier Inc. All rights reserved.
Keywords: Positive periodic solution Existence Nonlinear Integro-differential equation Exponential stability Banach space
1. Introduction In this paper we investigate the existence of positive x-periodic solutions for the nonlinear integro-differential equations of the form
_ þ xðtÞ
Z
t
pðt sÞgðxðsÞÞ ds ¼ 0;
t P T:
ð1Þ
ts
With respect to Eq. (1) in the next sections we will assume the following conditions: (i) p 2 Cð½0; sÞ; RÞ, (ii) g 2 Cðð0; 1Þ; ð0; 1ÞÞ. The Eq. (1) was encountered by [1] in the theory of a circulating fuel nuclear reactor. In this model, x is the neutron density. It is also a good model in one dimensional viscoelasticity in which x is the strain and p is the relaxation function. Such equations have been also proposed to model some biological problems. The stability of zero solution of Eq. (1) was later studied in [2–6] and the references cited therein. To the best of our knowledge there are only a few results on the existence of periodic solutions of Eq. (1). According to applications it is reasonable to consider the positive solutions of (1). In this paper we will obtain existence criteria for the positive x-periodic solutions of Eq. (1). For such equations the existence results in the literature are largely based on the several assumptions for the function pðtÞ and the authors usually do not consider the existence of positive periodic solutions. For example the _ €ðtÞ 0; authors in [3,5] assume that the function pðtÞ satisfies the next conditions: pðsÞ ¼ 0; pðtÞ P 0; pðtÞ 6 0; p 0 6 t 6 s. Authors in [7] studied oscillatory solutions of linear integro-differential equations. Motivated by the discussion above, we will focus on the existence of positive periodic solutions for the Eq. (1) without assumptions above on the function pðtÞ. For related results see also [8,9] and the references therein. In the third section the existence of positive solutions is also considered. In Section 4 we will establish the conditions for the exponential stability of positive solution of Eq. (1). To the ⇑ Corresponding author. E-mail addresses: [email protected] (B. Dorociaková), [email protected] (R. Olach). http://dx.doi.org/10.1016/j.amc.2014.09.086 0096-3003/Ó 2014 Elsevier Inc. All rights reserved.
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best of our knowledge this problem has not been solved yet. For related results we refer readers to [10,11] and the references cited therein. The following fixed point theorem will be used to prove the main results in the next sections. Theorem 1.1 (Schauder’s Fixed Point Theorem [12,13]). Let X be a closed, convex and nonempty subset of a Banach space X. Let S : X ! X be a continuous mapping such that SX is a relatively compact subset of X. Then S has at least one fixed point in X. That is there exists an x 2 X such that Sx ¼ x. 2. Existence of periodic solutions In this section we will study the existence of positive x-periodic solutions of Eq. (1). In the next lemma and theorems we choose T > s > 0. Lemma 2.1. Suppose that there exists a positive and continuous function kðt; sÞ; t s 6 s 6 t, such that
Z
tþx
Z
u
us
t
pðu v Þkðu; v Þ dv du ¼ 0;
Then the function
Z f ðtÞ ¼ exp
t
Z
u
us
T
t P T:
pðu v Þkðu; v Þ dv du ;
ð2Þ
t P T;
is x-periodic. Proof. For t P T we obtain
Z f ðt þ xÞ ¼ exp
tþx
T
Z ¼ exp
t
Z
Z
u us
u us
T
pðu v Þkðu; v Þ dv du
Z pðu v Þkðu; v Þ dv du exp
tþx
t
Z
u
us
pðu v Þkðu; v Þ dv du ¼ f ðtÞ:
Thus the function f ðtÞ is x-periodic. Theorem 2.1. Suppose that there exists a positive and continuous function kðt; sÞ; t s 6 s 6 t, such that (2) holds and
exp
Z
t
Z
u
us
T
Z pðu v Þkðu; v Þ dv du g exp
s
Z
T
u
us
¼ kðt; sÞ; pðu v Þkðu; v Þ dv du
t P T:
ð3Þ
Then Eq. (1) has a positive x-periodic solution. Proof. Let X ¼ fx 2 Cð½T s; 1Þ; RÞg be the Banach space with the norm kxk ¼ suptPTs jxðtÞj. We set
Z f ðtÞ ¼ exp
T
t
Z
u
us
pðu v Þkðu; v Þ dv du ;
t P T:
With regard to Lemma 2.1 we have m 6 f ðtÞ 6 M, where
Z t Z u m ¼ min exp pðu v Þkðu; v Þ dv du ; t2½T;1Þ T us Z t Z u pðu v Þkðu; v Þ dv du : M ¼ max exp t2½T;1Þ
T
us
We now define a closed, bounded and convex subset X of X as follows
X ¼ fx 2 X : xðt þ xÞ ¼ xðtÞ; m 6 xðtÞ 6 M;
t P T;
t P T;
kðt; sÞxðtÞ ¼ gðxðsÞÞ; t P T; t s 6 s 6 t; xðtÞ ¼ 1; T s 6 t 6 Tg: Define the operator S : X ! X as follows
( ðSxÞðtÞ ¼
R R t u v ÞÞ exp T us pðu v Þ gðxð dv du ; xðuÞ 1; T s 6 t 6 T:
t P T;
ð4Þ
B. Dorociaková, R. Olach / Applied Mathematics and Computation 253 (2015) 287–293
289
We will show that for any x 2 X we have Sx 2 X. For every x 2 X and t P T we get
Z ðSxÞðtÞ ¼ exp
t T
Z
Z t Z u gðxðv ÞÞ dv du ¼ exp pðu v Þkðu; v Þ dv du 6 M; xðuÞ T uv
u us
pðu v Þ
and ðSxÞðtÞ P m. For t 2 ½T s; T we have ðSxÞðtÞ ¼ 1, that is ðSxÞðtÞ 2 X. Further for every x 2 X and t P T; t s 6 s 6 t, according to (3) it follows
Z sZ u Z sZ u gðxðv ÞÞ ¼ g exp gððSxÞðsÞÞ ¼ g exp pðu v Þ dv du pðu v Þkðu; v Þ dv du xðuÞ T us T us Z t Z u Z t Z u gðxðv ÞÞ gðxðv ÞÞ pðu v Þ dv du exp pðu v Þ dv du exp xðuÞ xðuÞ T us T us Z sZ u Z t Z u exp ¼ g exp pðu v Þkðu; v Þ dv du pðu v Þkðu; v Þ dv du ðSxÞðtÞ ¼ kðt; sÞðSxÞðtÞ: T
us
T
us
Finally we will show that for x 2 X; t P T the function ðSxÞðtÞ is x-periodic. For x 2 X; t P T and with regard to (2) we get
Z ðSxÞðt þ xÞ ¼ exp
gðxðv ÞÞ dv du xðuÞ us T Z t Z u Z tþx Z u gðxðv ÞÞ gðxðv ÞÞ ¼ exp pðu v Þ dv du exp pðu v Þ dv du xðuÞ xðuÞ t T us us Z tþx Z u ¼ ðSxÞðtÞ exp pðu v Þkðu; v Þ dv du ¼ ðSxÞðtÞ: tþx
Z
u
pðu v Þ
us
t
So ðSxÞðtÞ is x-periodic on ½T; 1Þ. Thus we have proved that Sx 2 X for any x 2 X. We now show that S is completely continuous. First we will show that S is continuous. Let xi ¼ xi ðtÞ 2 X be such that xi ðtÞ ! xðtÞ 2 X as i ! 1. For t P T we have
Z jðSxi ÞðtÞ ðSxÞðtÞj ¼ exp
T
t
Z
u
us
pðu v Þ
Z t Z u gðxi ðv ÞÞ gðxðv ÞÞ dv du exp pðu v Þ dv du : xi ðuÞ xðuÞ T us
By applying the Lebesgue dominated convergence theorem we obtain that
limkðSxi ÞðtÞ ðSxÞðtÞk ¼ 0: i!1
For t 2 ½T s; T the relation above is also valid. This means that S is continuous. We now show that SX is relatively compact. It is sufficient to show by the Arzela–Ascoli theorem that the family of functions fSx : x 2 Xg is uniformly bounded and equicontinuous on ½T s; 1Þ. The uniform boundedness follows from the definition of X. According to (4) for t P T; x 2 X we get
Z t Z t Z u d gðxðv ÞÞ gðxðv ÞÞ ðSxÞðtÞ ¼ pðt v Þ exp pðu v Þ d v du dt xðtÞ xðuÞ ts T us Z t Z t Z u pðt v Þkðt; v Þ dv exp pðu v Þkðu; v Þ dv du 6 M 1 ; ¼ ts
T
M1 > 0:
us
For t 2 ½T s; T; x 2 X we have
d ðSxÞðtÞ ¼ 0: dt This shows the equicontinuity of the family SX, (cf. [14, p. 265]). Hence SX is relatively compact and therefore S is completely continuous. By Theorem 1.1 there is an x0 2 X such that Sx0 ¼ x0 . We see that x0 ðtÞ is a positive x-periodic solution of Eq. (1). The proof is complete. Corollary 2.1. Suppose that there exists a positive and continuous function kðt; sÞ; t s 6 s 6 t, such that (2) holds and
Z s
t
Z
u
us
pðu v Þkðu; v Þ dv du ¼ ln kðt; sÞ;
t P T:
ð5Þ
Then the equation
_ þ xðtÞ
Z
t
ts
pðt sÞxðsÞ ds ¼ 0;
t P T;
ð6Þ
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B. Dorociaková, R. Olach / Applied Mathematics and Computation 253 (2015) 287–293
has a positive x-periodic solution
Z xðtÞ ¼ exp
Z
t
u
us
T
pðu v Þkðu; v Þ dv du ;
t P T:
3. The existence of positive solutions In this section we will investigate the existence of positive solutions of nonlinear integro-differential equation (1). Theorem 3.1. Suppose that there exists a positive and continuous function kðt; sÞ; t s 6 s 6 t, such that (3) holds and
Z
t
pðt sÞkðt; sÞ ds > 0;
t P T:
ð7Þ
ts
Then Eq. (1) has a positive solution
Z xðtÞ ¼ exp
Z
t
u
us
T
pðu v Þkðu; v Þ dv du ;
t P T:
Proof. Let X 1 ¼ fx 2 Cð½T s; 1Þ; RÞg be the set of all continuous bounded functions. Then X 1 is a Banach space with the norm kxk ¼ suptPTs jxðtÞj. We set
Z wðtÞ ¼ exp
t
Z
pðu v Þkðu; v Þ dv du ;
u
us
T
t P T:
We define a closed, bounded and convex subset X1 of X 1 as follows
X1 ¼ fx 2 X 1 : wðtÞ 6 xðtÞ 6 1;
t P T;
t P T; t s 6 s 6 t;
kðt; sÞxðtÞ ¼ gðxðsÞÞ;
T s 6 t 6 T:g
xðtÞ ¼ 1;
Define the operator S1 : X1 ! X 1 as follows
( ðS1 xÞðtÞ ¼
R R t u v ÞÞ exp T us pðu v Þ gðxð dv du ; xðuÞ 1; T s 6 t 6 T:
t P T;
For every x 2 X1 and t P T we obtain
Z ðS1 xÞðtÞ ¼ exp
t
Z
u
us
T
pðu v Þkðu; v Þ dv du 6 1
and ðS1 xÞðtÞ P wðtÞ. For t 2 ½T s; T we get ðS1 xÞðtÞ ¼ 1, that is ðS1 xÞðtÞ 2 X1 . Now we can proceed by the similar way as in the proof of Theorem 2.1. We omit the rest of the proof. Corollary 3.1. Assume that there exists a positive and continuous function kðt; sÞ; t s 6 s 6 t, such that (5) and (7) hold. Then Eq. (6) has a positive solution
Z xðtÞ ¼ exp T
t
Z
u
us
pðu v Þkðu; v Þ dv du ;
t P T:
Corollary 3.2. Assume that there exists a positive and continuous function kðt; sÞ; t s 6 s 6 t, such that (3) and (7) hold and
lim
t!1
Z
t
Z
u
us
T
pðu v Þkðu; v Þ dv du ¼ 1:
Then Eq. (1) has a positive solution which tends to zero. Corollary 3.3. Assume that there exists a positive and continuous function kðt; sÞ; t s 6 s 6 t, such that (3) and (7) hold and
lim
t!1
Z T
t
Z
u
us
pðu v Þkðu; v Þ dv du ¼ a:
Then Eq. (1) has a positive solution which tends to constant ea .
B. Dorociaková, R. Olach / Applied Mathematics and Computation 253 (2015) 287–293
291
4. Stability of positive solution In this section we consider the exponential stability of positive solution of Eq. (1). We denote xðt; T; uÞ; t P T s, for a solution of Eq. (1) satisfying the initial condition xðs; T; uÞ ¼ uðsÞ > 0 for s 2 ½T s; T. Let xðtÞ ¼ xðt; T; uÞ; x1 ðtÞ ¼ xðt; T; u1 Þ and yðtÞ ¼ xðtÞ x1 ðtÞ; t 2 ½T s; 1Þ. Then we get
_ yðtÞ ¼
Z
t
pðt sÞ½gðxðsÞÞ gðx1 ðsÞÞ ds;
t P T:
ts
By the mean value theorem we obtain
_ yðtÞ ¼
Z
t
pðt sÞg 0 ðx Þ½xðsÞ x1 ðsÞ ds;
g 0 ðxÞ ¼
ts
_ yðtÞ ¼
Z
dgðxÞ ; dx
t
pðt sÞg 0 ðx ÞyðsÞ ds;
t P T:
ð8Þ
ts
Definition. Let x1 ðtÞ be a positive solution of Eq. (1) and there exist constants T u;x1 ; K u;x1 and k > 0 such that for every solution xðt; T; uÞ of (1)
jxðt; T; uÞ x1 ðtÞj 6 K u;x1 ekt ;
t P T u;x1 :
Then x1 ðtÞ is said to be exponentially stable. We assume that the function
Fðt; xÞ ¼
Z
t
pðt sÞgðxðsÞÞ ds;
t P T;
ts
is Lipschitzian in second argument. In the next theorem we establish sufficient conditions for the exponential stability of positive solution x1 ðtÞ ¼ xðt; T; 1Þ of Eq. (1).
Theorem 4.1. Suppose that (3) and (7) hold and
p 2 Cð½0; s; ð0; 1ÞÞ;
g 2 C 1 ðð0; 1Þ; ð0; 1ÞÞ;
g 0 ðxÞ P c > 0:
Then Eq. (1) has a positive solution which is exponentially stable. Proof. We will show that there exists a positive k such that
jxðt; T; uÞ x1 ðtÞj 6 K u;x1 ekt ; kT 1
where K u;x1 ¼ maxt2½Ts;T 1 jyðtÞje kt
LðtÞ ¼ jyðtÞje ;
t P T 1 ¼ T þ 2s;
þ 1. Consider the Lyapunov function
t P T1:
We claim that LðtÞ 6 K u;x1 for t P T 1 . On the other hand there exists t > T 1 such that Lðt Þ ¼ K u;x1 . Calculating the upper left derivative of LðtÞ along the solution yðtÞ of (8) we obtain
D ðLðtÞÞ ¼ ekt
Z
t
pðt sÞg 0 ðx ÞjyðsÞj ds þ kjyðtÞjekt ;
t P T1:
ts
For t ¼ t we get
0 6 D ðLðt ÞÞ ¼ c ekt
Z
t
pðt sÞjyðsÞj ds þ kjyðt Þjekt : t s
If yðtÞ > 0; t P T then from (8) it follows that for t P T þ s the function yðtÞ is decreasing and if yðtÞ < 0; t P T then yðtÞ is increasing for t P T þ s. We conclude that jyðtÞj; t P T þ s has decreasing character. Then we obtain
0 6 D ðLðt ÞÞ 6 cjyðt Þjekt
Z
t
t s
Z pðt sÞ ds þ kjyðt Þjekt ¼ c
Z s ¼ c pðuÞ du þ k jyðt Þjekt :
t
pðt sÞ ds þ k jyðt Þjekt
t s
0
For 0 < k < c
Rs 0
pðuÞ du we have a contradiction. Thus jyðtÞjekt 6 K u;x1 for t P T 1 and 0 < k < c
Rs 0
pðtÞ dt. The proof is complete.
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B. Dorociaková, R. Olach / Applied Mathematics and Computation 253 (2015) 287–293
5. Examples The next examples illustrate some results above. Example 1. Consider the integro-differential equation
_ þ xðtÞ
Z
t
pðt sÞxðsÞ ds ¼ 0;
t P T;
ð9Þ
t2p
where pðtÞ ¼ p1 sin t. We choose
kðt; sÞ ¼
c þ sin s ; c þ sin t
c > 1; t 2p 6 s 6 t:
Then for the condition (2), and x ¼ 2p we have
Z
tþ2p
t
¼
p
Z
1
u
1
u2p tþ2p
Z
1
Z
p
sinðu v Þ
1 c þ sin u
t tþ2p
1 c þ sin u
Z
c þ sin v dv du c þ sin u
u
u2p u
sinðu v Þðc þ sin v Þ dv du
Z
ðsin u cos v cos u sin v Þðc þ sin v Þ dv du u2p Z u Z u 1 tþ2p 1 cðsin u cos v dv cos u sin v dv Þ ¼ c þ sin u p t u2p u2p Z u Z u 2 sin v cos v dv cos u sin v dv du þ sin u
¼
p
Z
t
u2p tþ2p
Z
¼
u2p
cos u du ¼ 0; c þ sin u
t
t P T:
For the condition (5) we get
Z
t
Z
u
u2p
s
pðu v Þkðu; v Þ dv du ¼
1
p
Z
t
1 c þ sin u
s
¼ ln kðt; sÞ;
Z
u
u2p
sinðu v Þðc þ sin v Þ dv du ¼
Z
t
s
cos u c þ sin s du ¼ ln c þ sin u c þ sin t
t P T:
All conditions of Corollary 2.1 are satisfied and Eq. (9) has a positive x ¼ 2p-periodic solution
Z xðtÞ ¼ exp ¼ exp
Z
t
Z
u
u2p
T
Z 1 pðu v Þkðu; v Þ dv du ¼ exp
cos u c þ sin t ; du ¼ c þ sin u c þ sin T
t
T
p
t
Z
u
u2p
T
sinðu v Þ
c þ sin v dv du c þ sin u
t P T:
Example 2. Consider the integro-differential equation
_ þ xðtÞ
Z
t
pðt sÞxðsÞ ds ¼ 0;
t P T;
ð10Þ
ts
where pðtÞ ¼ 1s et . We choose kðt; sÞ ¼ ets ; t s 6 s 6 t. Then for the condition (5) we obtain
Z
s
t
Z
u
us
pðu v Þkðu; v Þ dv du ¼
Z
s
t
1
Z
s
u
us
euþv euv dv du ¼
1
s
Z
s
t
Z
u
us
dv du ¼
Z
t
du ¼ t s;
t P T:
s
Thus all conditions of Corollary 3.1 are satisfied and Eq. (10) has the positive solution
Z xðtÞ ¼ exp T
t
1
s
Z
u
us
Z t euþv euv dv du ¼ exp du ¼ etþT ;
t P T;
T
which tends to zero. Example 3. Consider the integro-differential equation
_ þ xðtÞ
Z
t
t2p
pðt sÞxðsÞ ds ¼ 0;
t P T;
ð11Þ
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B. Dorociaková, R. Olach / Applied Mathematics and Computation 253 (2015) 287–293 s
where pðtÞ ¼ e2p21 cos t. We choose kðt; sÞ ¼ 1þe ; t 2p 6 s 6 t. Then for the condition (5) we have 1þet
Z
Z
2 1 þ ev dv du cosðu v Þ 2p 1 e 1 þ eu s u2p Z t Z u 2 1 ¼ 2p ðcos u cos v þ sin u sin v Þð1 þ ev Þ dv du e 1 s 1 þ eu u2p Z t Z u Z u 2 1 ¼ 2p cos u cos v d v þ sin u sin v dv e 1 s 1 þ eu u2p u2p Z u Z u ev cos v dv þ sin u ev sin v dv du þ cos u u2p u2p Z t Z u Z u 2 1 v v du cos u e cos v d v þ sin u e sin v d v ¼ 2p e 1 s 1 þ eu u2p u2p 2p Z t 2 1 e 1 u e cos uðcos u sin uÞ ¼ 2p e 1 s 1 þ eu 2 e2p 1 u e sin uðsin u þ cos uÞ du þ 2 Z t eu ðcos uðcos u sin uÞ þ sin uðsin u þ cos uÞÞ du ¼ 1 þ eu s Z t eu 1 þ es du ¼ ln ; t P T: ¼ u 1 þ e 1 þ et s t
u
All conditions of Corollary 3.1 are satisfied and Eq. (11) has a positive solution
xðtÞ ¼ exp
2 1 e2p
Z T
which tends to the constant
t
Z
u
u2p
cosðu v Þ
Z t 1 þ e v eu 1 þ et dv du ¼ exp du ¼ ; u u 1þe 1 þ eT T 1þe
t P T; :
1 . 1þeT
Acknowledgements The research was supported by the grant 1/0234/13 of the Scientific Grant Agency of the Ministry of Education of the Slovak Republic. The authors have made the same contribution. All authors read and approved the final manuscript. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]
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