Abstract
In this paper, we study the existence of multiple positive solutions of the second-order periodic boundary value problems for functional differential equations with impulse. The proof of our main results is based upon the fixed point index theorem in cones.
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1 Introduction
In this paper, we will consider the existence of positive solutions for boundary value problems of second order impulsive functional differential equations of the form
where , is a continuous function, ( be given in Section 2), , , , , . , , . , () denote the right limit (left limit) of at , and .
Impulsive differential equations describe processes which experience a sudden change of their state at certain moments. The theory of impulse differential equations has been a significant development in recent years and played a very important role in modern applied mathematical models of real processes arising in phenomena studied in physics, population dynamics, chemical technology and biotechnology; see [1–3].
Many papers have been published about the existence analysis of periodic boundary value problems of first and second order for ordinary or functional or integro-differential equations with impulsive. We refer the readers to the papers [4–23]. For instance, in [6], He and Yu investigated the following problem:
By using the coincidence degree, Dong [4] studied the following periodic boundary value problems (PBVP) for first-order functional differential equations with impulse:
It is remarkable that the author required for . The author also obtained the existence of one solution of PBVP (1.2).
To study periodic boundary value problems for first and second order functional differential equations with impulse, the approaches used in [4, 6–8, 13–16, 20–23] are the monotone iterative technique and the method of upper and lower solutions. What they obtained is the existence of at least one solution if there is a pair of upper and lower solutions. However, in some cases it is difficult to find upper and lower solutions for general differential equations.
As we know, the fixed point theorem of cone expression and compression is extensively used to study the existence of multiple solutions of boundary value problems for second-order differential equations. In paper [5], Ma considers the following periodic boundary value problem:
and he obtained some sufficient conditions for the existence of at least one positive solution of the PBVP (1.3).
In [19], by applying the fixed point theorem of cone expression and compression, Liu investigates the existence of multiple positive solutions of the following problem:
Motivated by the results in [5, 19, 24], the aim of this paper is to consider the existence of multiple positive solutions for the PBVP (1.1) by using some properties of the Green function and the fixed point index theorem in cones.
This paper will be divided into three sections. In Section 2, we provide some preliminaries and establish several lemmas which will be used throughout Section 3. In Section 3, we shall give the existence theorems of multiplicity positive solutions of PBVP (1.1).
2 Preliminary and lemmas
Let := {; is continuous everywhere except for a finite number of points at which and exist and }, then is a normed space with the norm
Let , and represent the set of a continuous and continuously differentiable on , respectively. Moreover, for we define . Furthermore, we denote:
= { is a map from into R such that is continuous at , and , exist and for ; for }.
= {, and exist; and for }. .
Clearly, is a Banach space with the norm for . is also a Banach space with the norm .
We need to assume the following conditions:
(A1) ; for , and ;
(A2) for and ;
(A3) () are continuous and for .
Lemma 2.1 (see [25])
Let E be a Banach spaces and be a cone in E. Let and . Assume that is a completely continuous operator such that for .
-
(i)
If for , then .
-
(ii)
If for , then .
Lemma 2.2 For any , and . Then the problem
has a unique solution
where
Proof Integrating the first equation of (2.1) over the interval for , we get
It follows from the boundary conditions , and (2.4) that
Together with (2.4), we obtain (2.2). □
By the standard discussion, we have the following lemma which will be used later.
Lemma 2.3 The Green function defined in (2.3) has the following properties:
-
(i)
, ;
-
(ii)
, ;
-
(iii)
, , , where , , and
Form Lemma 2.2, the problem (1.1) is equivalent to the integral equation:
Definition 2.4 A function is called a positive solution of PBVP (1.1), if it satisfies the PBVP (1.1) and on , and on J.
Define a cone K in as follows:
Lemma 2.5 The operator is completely continuous.
Proof It is easy to show that holds. Let B be a bounded subset in . By virtue of the Ascoli-Arzela theorem, we only show that is bounded in and is equicontinuous. For any and , , we have . Therefore, the set is uniformly bounded with respect to on . Then there exist two constants , such that
Taking
It follows from (2.5), (2.7), and (2.8) that is bounded in .
Let and with . There are three possibilities:
Case I. If , then
Case II. If , then we have .
Case III. If , then
Clearly, in either case, it follows from the continuity of and the uniform continuity of φ in that for any , there exists a positive constant δ, independent of t, and u, whenever , such that holds. Therefore, is equicontinuous. □
3 The main results
In this section, we shall consider the existence of multiple positive solutions for the periodic boundary value problems (1.1).
For convenience sake, we set
For the first theorem we need the following hypotheses:
(C1) There exists a constant such that for ,
where and are two positive constants satisfying:
(C2) There exists a constant satisfying:
such that
for , where , are constants satisfying:
Theorem 3.1 Assume that (C1), (C2), , and hold. Then PBVP (1.1) has at least two positive solutions , with .
Proof For any , we have , . It follows from the definitions of and that
Then for with , it follows from (2.5) and assumption (C1) that
Now if we set , then (3.4) shows that for . Thus, Lemma 2.1 yields
For with , from (2.5) and assumption (C2), we have
Set . Then (3.6) shows that for .
Hence, Lemma 2.1 implies that
According to and , choose a constant such that
and
where , are constants satisfying:
For with , from (2.5), Lemma 2.5 and (3.9), by using the same method to get (3.3), we can get
Now if we set . Then (3.10) shows that for . Thus, an application of Lemma 2.1 again shows that
Since , it follows from (3.5), (3.7), (3.11), and the additivity of the fixed index that
Thus, S has a fixed point in , and a fixed point in . They are positive solutions of the PBVP (1.1) and
The proof is complete. □
For the second theorem we need the following hypotheses:
(C3) There exists a constant satisfying
such that for ,
where , are positive constants satisfying:
(C4) () are continuous nonincreasing functions such that for .
(C5) There exists a constant with such that for any ,
and
where , , and
Theorem 3.2 Assume that (C3)-(C5), and hold. Then PBVP (1.1) has at least two positive solutions , with .
Proof For any with . According to (2.5) and assumption (C3), we have
Set . Then (3.13) shows that for . Thus Lemma 2.1 implies
On the other hand, it follows from (2.5) and assumption (C5) that
Set . Then (3.15) implies that for . Thus an application of Lemma 2.1 again shows that
In view of assumption (C4) and , there are two possibilities:
Case 1. Suppose that f is unbounded, then there exists a constant satisfying:
such that
and
where is a positive constant satisfying
If with , then it follows from (2.5), (3.17), and (3.18) that
Case 2. Suppose that f is bounded. Then there exists a constant N such that
Taking
For and , from (2.5), we have
Choose . Hence, in either case, we always may set
such that for . Thus, Lemma 2.1 yields
Since , it follows from (3.14), (3.16), (3.20), and the additivity of the fixed point index that
Thus, S has a fixed point in , and a fixed point in . They are positive solutions of the PBVP (1.1) and
The proof is complete. □
Example 3.3 Consider the following PBVP:
where
Then PBVP (3.21) has at least two positive solutions , with .
Proof PBVP (3.21) can be regarded as a PBVP of the form (1.1), where
and , , , .
First we have . By choosing , , we get . On the other hand, it follows from (3.21) and (3.22) that and are satisfied. Finally, we show that (C1) and (C2) hold. Choose , , , , , . By calculation, we get
and
Then it is not difficult to see that the conditions (C1) and (C2) hold.
By Theorem 3.1, PBVP (3.21) has at least two positive solutions , with . □
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Acknowledgements
The authors are highly grateful for the referees’ careful reading and comments on this paper. The research is supported by Hunan Provincial Natural Science Foundation of China (Nos. 13JJ3106 and 12JJ2004); it is also supported by the National Natural Science Foundation of China (Nos. 61074067 and 11271372).
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Zhao, Y., Chen, H. & Qin, B. Periodic boundary value problems for second-order functional differential equations with impulse. Adv Differ Equ 2014, 134 (2014). https://doi.org/10.1186/1687-1847-2014-134
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DOI: https://doi.org/10.1186/1687-1847-2014-134