- Research
- Open access
- Published:
Positive periodic solutions for a second-order functional differential equation
Boundary Value Problems volume 2012, Article number: 140 (2012)
Abstract
In this paper, the existence results of positive ω-periodic solutions are obtained for the second-order functional differential equation
where is a continuous function which is ω-periodic in t, is a ω-periodic function, . Our discussion is based on the fixed point index theory in cones.
MSC:34C25, 47H10.
1 Introduction
In this paper, we discuss the existence of positive ω-periodic solutions of the second-order functional differential equation with the delay terms of first-order derivative in nonlinearity,
where is a continuous function which is ω-periodic in t and is a ω-periodic delay function, .
For the second-order differential equation without delay and the first-order derivative term in nonlinearity,
the existence problems of periodic solutions have attracted many authors’ attention and concern. Many theorems and methods of nonlinear functional analysis have been applied to research the periodic problems of Equation (2), such as the upper and lower solutions method and monotone iterative technique [1–4], the continuation method of topological degree [5–7], variational method and critical point theory [8–10], the theory of the fixed point index in cones [11–16], etc.
In recent years, the existence of periodic solutions for the second-order delayed differential equations have also been researched by many authors; see [17–24] and the references therein. In some practice models, only positive periodic solutions are significant. In [20, 21, 23], the authors obtained the existence of positive periodic solutions for some delayed second-order differential equations as a special form of the following equation:
by using Krasnoselskii’s fixed point theorem of cone mapping or the theory of the fixed point index in cones. In these works, the positivity of Green’s function of the corresponding linear second-order periodic problems plays an important role. The positivity guarantees that the integral operators of the second-order periodic problems are cone-preserving in the cone
in the Banach space , where is a constant. Hence, the fixed point theorems of cone mapping can be applied to periodic problems of the second-order delay equation (3) as well as Equation (2) (for Equation (2), see [11–16]). However, few people consider the existence of positive periodic solutions of Equation (1). Since the nonlinearity of Equation (1) explicitly contains the delayed first-order derivative term, the corresponding integral operator has no definition on the cone P. Thus, the argument methods used in [20, 21, 23] are not applicable to Equation (1).
The purpose of this paper is to discuss the existence of positive periodic solutions of Equation (1). We will use a different method to treat Equation (1). Our main results will be given in Section 3. Some preliminaries to discuss Equation (1) are presented in Section 2.
2 Preliminaries
Let denote the Banach space of all continuous ω-periodic function with the norm . Let be the Banach space of all continuous differentiable ω-periodic function with the norm
Generally, denotes the n th-order continuous differentiable ω-periodic function space for . Let be the cone of all nonnegative functions in .
Let be a constant. For , we consider the linear second-order differential equation
The ω-periodic solutions of Equation (5) are closely related to the linear second-order boundary value problem
see [14]. It is easy to see that problem (6) has a unique solution which is explicitly given by
where . By [[14], Lemma 1], we have
Lemma 2.1 Let . Then, for every , the linear equation (5) has a unique ω-periodic solution which is given by
Moreover, is a completely continuous linear operator.
Since , for every , by (8), if and , then the ω-periodic solution of Equation (5) for every , and we term it the positive ω-periodic solution. Let
Define a set K in by
It is easy to verify that K is a closed convex cone in .
Lemma 2.2 Let . Then, for every , the positive ω-periodic solution of Equation (5) . Namely, .
Proof Let , . For every , from (8) it follows that
and therefore,
Using (8), we obtain that
For every , since
we have
Hence, . □
Now we consider the nonlinear delay equation (1). Hereafter, we assume that the nonlinearity f satisfies the condition
(F0) There exists such that
Let , then for , , , and Equation (1) is rewritten to
For every , set
Then is continuous. We define an integral operator by
By the definition of the operator S, the positive ω-periodic solution of Equation (1) is equivalent to the nontrivial fixed point of A. From assumption (F0), Lemma 2.1 and Lemma 2.2, we easily see that
Lemma 2.3 and is completely continuous.
We will find the non-zero fixed point of A by using the fixed point index theory in cones. We recall some concepts and conclusions on the fixed point index in [25, 26]. Let E be a Banach space and be a closed convex cone in E. Assume Ω is a bounded open subset of E with the boundary ∂ Ω, and . Let be a completely continuous mapping. If for any , then the fixed point index has a definition. One important fact is that if , then A has a fixed point in . The following two lemmas are needed in our argument.
Lemma 2.4 ([26])
Let Ω be a bounded open subset of E with and be a completely continuous mapping. If for every and , then .
Lemma 2.5 ([26])
Let Ω be a bounded open subset of E and be a completely continuous mapping. If there exists an such that for every and , then .
In the next section, we will use Lemma 2.4 and Lemma 2.5 to discuss the existence of positive ω-periodic solutions of Equation (1).
3 Main results
We consider the existence of positive ω-periodic solutions of the functional differential equation (1). Let satisfy assumption (F0) and be ω-periodic in t. Let be the constant defined by (9) and . For convenience, we introduce the notations
Our main results are as follows.
Theorem 3.1 Let and be ω-periodic in t, . If f satisfies assumption (F0) and the condition
(F1) , ,
then Equation (1) has at least one positive ω-periodic solution.
Theorem 3.2 Let and be ω-periodic in t, . If f satisfies assumption (F0) and the conditions
(F2) ,
then Equation (1) has at least one positive ω-periodic solution.
In Theorem 3.1, the condition (F1) allows to be superlinear growth on x and . For example,
satisfies (F0) with and (F1) with and .
In Theorem 3.2, the condition (F2) allows to be sublinear growth on x and . For example,
satisfies (F0) with and (F2) with and .
Proof of Theorem 3.1 Choose the working space . Let be the closed convex cone in defined by (10) and be the operator defined by (13). Then the positive ω-periodic solution of Equation (1) is equivalent to the nontrivial fixed point of A. Let and set
We show that the operator A has a fixed point in when r is small enough and R is large enough.
By and the definition of , there exist and such that
Let . We now prove that A satisfies the condition of Lemma 2.4 in , namely for every and . In fact, if there exist and such that , then by the definition of A and Lemma 2.1, satisfies the delay differential equation
Since , by the definitions of K and , we have
Hence, from (15) it follows that
By this, (16) and the definition of , we have
Integrating both sides of this inequality from 0 to ω and using the periodicity of , we obtain that
Since , it follows that , which is a contradiction. Hence, A satisfies the condition of Lemma 2.4 in . By Lemma 2.4, we have
On the other hand, since , by the definition of , there exist and such that
Choose and let . Clearly, . We show that A satisfies the condition of Lemma 2.5 in , namely for every and . In fact, if there exist and such that , since , by the definition of A and Lemma 2.1, satisfies the differential equation
Since , by the definition of K, we have
By the latter inequality of (21), we have that . This implies that . Consequently,
By (22) and the former inequality of (21), we have
From this, the latter inequality of (21) and (19), it follows that
By this inequality, (20) and the definition of , we have
Integrating this inequality on I and using the periodicity of , we get that
Since , from this inequality it follows that , which is a contradiction. This means that A satisfies the condition of Lemma 2.5 in . By Lemma 2.5,
Now, by the additivity of fixed point index, (18) and (23), we have
Hence, A has a fixed-point in , which is a positive ω-periodic solution of Equation (1). □
Proof of Theorem 3.2 Let be defined by (14). We prove that the operator A defined by (13) has a fixed point in if r is small enough and R is large enough.
By and the definition of , there exist and such that
Let and . We prove that A satisfies the condition of Lemma 2.5 in , namely for every and . In fact, if there exist and such that , since , by the definition of A and Lemma 2.1, satisfies the delay differential equation
Since , by the definitions of K and , satisfies (17). From (17) and (24) it follows that
By this, (25) and the definition of , we have
Integrating this inequality on and using the periodicity of , we obtain that
Since , from this inequality it follows that , which is a contradiction. Hence, A satisfies the condition of Lemma 2.5 in . By Lemma 2.5, we have
Since , by the definition of , there exist and such that
Choosing , we show that A satisfies the condition of Lemma 2.4 in , namely for every and . In fact, if there exist and such that , then by the definition of A and Lemma 2.1, satisfies the differential equation
Since , by the definition of K, satisfies (21). From the second inequality of (21), it follows that (22) holds. By (22) and the first inequality of (21), we have
From this, the second inequality of (21) and (27), it follows that
By this and (28), we have
Integrating this inequality on and using the periodicity of , we obtain that
Since , from this inequality it follows that , which is a contradiction. This means that A satisfies the condition of Lemma 2.4 in . By Lemma 2.4,
Now, from (26) and (29), it follows that
Hence, A has a fixed-point in , which is a positive ω-periodic solution of Equation (1). □
Example 1 Consider the following second-order differential equation with delay:
where , . If and for , we can verify that
satisfies the conditions (F0) and (F1) for . By Theorem 3.1, the delay equation (30) has at least one positive ω-periodic solution.
Example 2 Consider the functional differential equation
where , , and . If and for . We easily see that
satisfies the conditions (F0) and (F2) for . By Theorem 3.2, the functional differential equation (31) has a positive ω-periodic solution.
References
Leela S: Monotone method for second order periodic boundary value problems. Nonlinear Anal. 1983, 7: 349-355. 10.1016/0362-546X(83)90088-3
Nieto JJ: Nonlinear second-order periodic boundary value problems. J. Math. Anal. Appl. 1988, 130: 22-29. 10.1016/0022-247X(88)90383-6
Cabada A, Nieto JJ: A generation of the monotone iterative technique for nonlinear second-order periodic boundary value problems. J. Math. Anal. Appl. 1990, 151: 181-189. 10.1016/0022-247X(90)90249-F
Cabada A: The method of lower and upper solutions for second, third, forth, and higher order boundary value problems. J. Math. Anal. Appl. 1994, 185: 302-320. 10.1006/jmaa.1994.1250
Gossez JP, Omari P: Periodic solutions of a second order ordinary differential equation: a necessary and sufficient condition for nonresonance. J. Differ. Equ. 1991, 94: 67-82. 10.1016/0022-0396(91)90103-G
Omari P, Villari G, Zanolin F: Periodic solutions of Lienard equation with one-sided growth restrictions. J. Differ. Equ. 1987, 67: 278-293. 10.1016/0022-0396(87)90151-3
Ge W: On the existence of harmonic solutions of Lienard system. Nonlinear Anal. 1991, 16: 183-190. 10.1016/0362-546X(91)90167-Y
Mawhin J, Willem M: Multiple solutions of the periodic boundary value problem for some forced pendulum-type equations. J. Differ. Equ. 1984, 52: 264-287. 10.1016/0022-0396(84)90180-3
Zelati VC: Periodic solutions of dynamical systems with bounded potential. J. Differ. Equ. 1987, 67: 400-413. 10.1016/0022-0396(87)90134-3
Lassoued L: Periodic solutions of a second order superquadratic system with a change of sign in potential. J. Differ. Equ. 1991, 93: 1-18. 10.1016/0022-0396(91)90020-A
Atici FM, Guseinov GS: On the existence of positive solutions for nonlinear differential equations with periodic boundary conditions. J. Comput. Appl. Math. 2001, 132: 341-356. 10.1016/S0377-0427(00)00438-6
Torres PJ: Existence of one-signed periodic solutions of some second-order differential equations via a Krasnoselskii fixed point theorem. J. Differ. Equ. 2003, 190: 643-662. 10.1016/S0022-0396(02)00152-3
Li Y: Positive periodic solutions of nonlinear second order ordinary differential equations. Acta Math. Sin. 2002, 45: 481-488. (in Chinese)
Li Y: Positive periodic solutions of first and second order ordinary differential equations. Chin. Ann. Math., Ser. B 2004, 25: 413-420. 10.1142/S025295990400038X
Li F, Liang Z: Existence of positive periodic solutions to nonlinear second order differential equations. Appl. Math. Lett. 2005, 18: 1256-1264. 10.1016/j.aml.2005.02.014
Graef JR, Kong L, Wang H: Existence, multiplicity, and dependence on a parameter for a periodic boundary value problem. J. Differ. Equ. 2008, 245: 1185-1197. 10.1016/j.jde.2008.06.012
Liu B: Periodic solutions of a nonlinear second-order differential equation with deviating argument. J. Math. Anal. Appl. 2005, 309: 313-321. 10.1016/j.jmaa.2005.01.045
Li JW, Cheng SS: Periodic solutions of a second order forced sublinear differential equation with delay. Appl. Math. Lett. 2005, 18: 1373-1380. 10.1016/j.aml.2005.02.031
Wang Y, Lian H, Ge W: Periodic solutions for a second order nonlinear functional differential equation. Appl. Math. Lett. 2007, 20: 110-115. 10.1016/j.aml.2006.02.028
Wu J, Wang Z: Two periodic solutions of second-order neutral functional differential equations. J. Math. Anal. Appl. 2007, 329: 677-689. 10.1016/j.jmaa.2006.06.084
Wu YX: Existence nonexistence and multiplicity of periodic solutions for a kind of functional differential equation with parameter. Nonlinear Anal. 2009, 70: 433-443. 10.1016/j.na.2007.12.011
Guo CJ, Guo ZM: Existence of multiple periodic solutions for a class of second-order delay differential equations. Nonlinear Anal., Real World Appl. 2009, 10: 3285-3297. 10.1016/j.nonrwa.2008.10.023
Cheung WS, Ren JL, Han W: Positive periodic solution of second-order neutral functional differential equations. Nonlinear Anal. 2009, 71: 3948-3955. 10.1016/j.na.2009.02.064
Lv X, Yan P, Liu D: Anti-periodic solutions for a class of nonlinear second-order Rayleigh equations with delays. Commun. Nonlinear Sci. Numer. Simul. 2010, 15: 3593-3598. 10.1016/j.cnsns.2010.01.002
Deimling K: Nonlinear Functional Analysis. Springer, New York; 1985.
Guo D, Lakshmikantham V: Nonlinear Problems in Abstract Cones. Academic Press, New York; 1988.
Acknowledgements
The research was supported by the NNSFs of China (11261053, 11061031).
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
YL carried out the main part of this article. All authors read and approved the final manuscript.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Li, Y., Li, Q. Positive periodic solutions for a second-order functional differential equation. Bound Value Probl 2012, 140 (2012). https://doi.org/10.1186/1687-2770-2012-140
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1687-2770-2012-140