Abstract
The authors consider the existence of positive periodic solution for the impulsive functional differential equation \(u'(t)=-a(t)g(u(h_{1}(t)))u(t)+\lambda b(t)f (t, u(h_{2}(t)), \int_{-\zeta}^{0}e(s)u(t+s)\, ds )\), \(t\in\mathbf{R}\), \(t\neq t_{k}\), \(u(t_{k}^{+})-u(t_{k})=\mu I_{k}(t_{k},u(t_{k}))\), \(k\in\mathbf{Z}\), where \(\lambda>0 \) and \(\mu>0\) are two parameters. Several new and more general existence and multiplicity results are derived in terms of different values of \(\lambda>0\) and \(\mu>0\). Here we not only consider the case that g is bounded, but the case that g is not necessarily bounded is also considered. Our results improve those in (Ma et al. in J. Math. Anal. Appl. 384:527-535, 2011; Li et al. in Comput. Math. Appl. 56:2556-2560, 2008). Moreover, the parameter dependence of the periodic solution is also studied.
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1 Introduction
Impulsive functional differential equations are emerging as an important area of investigation, since various real-world simulation processes which depend on their prehistory and are subject to short time disturbances, such as population dynamics, biology, biotechnology, industrial robotic, pharmacokinetics, optimal control, etc., can be expressed by functional differential equations with impulses; see [1–5]. Moreover, Cushing [6] pointed out that any biological or environmental parameters are naturally subject to fluctuations in time, due to seasonal effects of weather, food supply, mating habits, hunting or harvesting seasons, etc. Therefore, it is necessary and important to consider models with periodic ecological parameters or perturbations, which may be more natural (see [7–27]). For example, Liu and Li [9] considered the following first-order functional differential equation with a parameter:
By using the eigenvalue theory of operators, the authors established several sufficient conditions for the existence of positive T-periodic solutions.
In [10], Wang investigated the following first-order functional differential equation with a parameter:
and by virtue of the fixed point theorems in a cone, the author obtained the existence, non-existence, and multiplicity of positive ω-periodic solutions under suitable conditions imposed on the nonlinear term f.
Yan [28] pointed out that (A) and (B) include various models. For example, Cooke and Kaplan [29], and Cooke and Yorke [30] studied, respectively, the equation
for epidemic and population growth and gonorrhea epidemics.
Mackey and Glass [31] considered the following equation:
for model of hematopoiesis (blood cell production).
For some other recent excellent results and applications on (A) and (B), we refer the reader to papers by Jin and Wang [32], Ye et al. [33], Li et al. [34], Graef and Kong [35, 36], Franco et al. [37], Wu and Wang [38], Padhi and Srivastava [39], Olach [40], Amster and Idels [41], Chen et al. [42], Ma et al. [43] and Weng and Sun [44].
As regards the ideal of an impulsive perturbation, in [28], a pioneer paper concerning the solvability of periodic problem, Yan considered the following impulsive functional differential equation:
By utilizing a well-known fixed point index theorem in cone, the author got some sufficient conditions of the existence and non-existence of positive periodic solutions of (C).
In this paper, we investigate the existence of positive periodic solutions of the impulsive functional differential equation
where \(\lambda>0 \) and \(\mu>0\) are parameters, \(e\in C([-\zeta ,0],\mathbf{R})\), \(\int_{-\zeta}^{0}e(s)\, ds=1\), and \(\{t_{k}, k\in\mathbf {Z}\}\) is an increasing sequence of real number with \(\lim_{k\rightarrow\pm\infty}t_{k}=\pm\infty\). In addition, let
We make the following hypotheses:
- (H1):
-
\(a, b \in C(\mathbf{R},\mathbf{R}_{+})\) are ω-periodic functions satisfying \(\int_{0}^{\omega}a(t)\, dt>0\), \(\int_{0}^{\omega}b(t)\, dt>0\);
- (H2):
-
\(h_{i}\in C(\mathbf{R},\mathbf{R}_{+})\) (\(i=1,2\)) is ω-periodic function, \(g\in C(\mathbf{R}_{+},\mathbf{R}_{+})\);
- (H3):
-
\(f\in C(\mathbf{R}\times\mathbf{R}_{+}\times\mathbf{R}_{+}, \mathbf{R}_{+})\) is ω-periodic function in t; moreover, \(f(t,x,y)>0\) for all \(t, x>0\) and \(y>0\);
- (H4):
-
\(I_{k}\in C(\mathbf{R}\times\mathbf{R}_{+}, \mathbf{R}_{+})\) is ω-periodic function in t and \(I_{k}(t,x)>0\) for all t and \(x>0\), \(k\in\mathbf{Z}\); moreover, there exists a positive integer p such that
$$I_{k+p}(t_{k+p},x)=I_{k}(t_{k},x),\quad t_{k+p}=t_{k}+\omega, k\in\mathbf{Z}. $$
Without loss of generality, we assume that \([0,\omega]\cap\{t_{k},k\in \mathbf{Z}\}=\{t_{0},t_{1},\ldots,t_{p-1}\}\).
Some special cases of (1.1) have been investigated. For example, Li et al. [34] discussed problem (1.1) in the case that \(\mu\equiv1\) and \(I_{k}(t,u)=I_{k}(u)\), \(k\in\mathbf{Z}^{+}\). By using the well-known Leggett-Williams multiple fixed point theorem, the authors obtained the existence of multiple positive periodic solutions of problem (1.1). However, in [34], Li et al. only dealt with the case that there exist two positive constants l and L such that
It is interesting to know whether there is a positive periodic solution of (1.1) when g does not satisfy (1.2). Very recently, Ma et al. [43] studied the existence of positive periodic solutions of (1.1) under the following two cases:
where l and L are two positive constants. However, Ma et al. obtained the existence of at least one positive periodic solution only for \(I_{k}(t,u)\equiv0\), \(k\in\mathbf{Z}\).
At the same time, we notice that there is almost no paper except [28] studying impulsive functional differential equations with two parameters. By using a well-known fixed point index theorem due to Krasnoselskii, the author obtained the following results in the case g is bounded and \(e(t)\equiv0\), \(t\in[-\zeta,0]\).
Theorem A
(Theorem 3.1 of [28])
Let the following conditions hold.
- (A1):
-
\(\lambda>0\) and \(\mu\geq0\) are parameters.
- (A2):
-
\({t_{k}}\), \(k\in\mathbf{Z}\), is an increasing sequence of real number with \(\lim_{k\rightarrow\pm\infty}t_{k}=\pm\infty\).
- (A3):
-
h and \(f: \mathbf{R}\times\mathbf{R}^{+}\rightarrow \mathbf{R}^{+}\) satisfy Carathéodory conditions, that is, \(h(t,y)\) and \(f (t,y)\) are locally Lebesgue measurable in t for each fixed y and are continuous in y for each fixed t, are ω-periodic functions in t. Moreover, \(f(t,y)>0\) for all t and \(y>0\). \(\tau: \mathbf{R}\rightarrow\mathbf{R}\) is locally bounded Lebesgue measurable ω-periodic function.
- (A4):
-
There exist ω-periodic functions \(a_{1}\) and \(a_{2}: \mathbf{R}\rightarrow\mathbf{R}^{+}\) which are locally bounded Lebesgue measurable so that \(a_{1}(t)y \leq h(t,y) \leq a_{2}(t)y\) for all \(y>0\) and \(\lim_{y\rightarrow0^{+}} h(t,y)y \) exists, \(\int_{0}^{\omega}a_{1}(t)\, dt>0\).
- (A5):
-
\(I_{k}: \mathbf{R}\times[0, \infty) \rightarrow\mathbf{R}\), \(k\in Z\), satisfy Carathéodory conditions and are ω-periodic functions in t and there exists an integer ρ such that \(I_{k+\rho }(t_{k+\rho},y)=I_{k}(t_{k},y)\), \(t_{k+\rho}=t_{k}+\omega\), \(k\in\mathbf{Z}\). Moreover, \(I_{k}(t,0)=0\) for all \(k\in\mathbf{Z}\).
- (A6):
-
p and \(q:\mathbf{R}\rightarrow\mathbf{R}^{+}\) are positive bounded Lebesgue measurable ω-periodic functions and is bounded away from zero.
Suppose further that \(f^{0}\), \(f_{\infty}\), \(I_{\infty}\), and \(I^{0}\) are positive constants, where
If
and
then problem (1.1) has a positive ω-periodic solution.
It is not difficult to see that the conditions of Theorem A are not the optimal conditions which guarantee the existence of at least one positive periodic solution for problem (1.1). In fact, if
or
we can prove that problem (1.1) has at least one positive periodic solution, respectively.
Motivated by the papers mentioned above, we will extend the results of [9, 10, 28, 32, 34, 43] to problem (1.1) in the case that g satisfies (1.2), (1.3), and (1.4), respectively. Moreover, the parameter dependence of the periodic solution is also studied.
The rest of the paper is organized as follows. In Section 2, we study the existence and multiplicity results of positive periodic solutions for problem (1.1) when the function g satisfies (1.2). In Section 3, we establish the dependence results of the positive periodic solution on the parameter for problem (1.1). The final section of the paper studies the existence of positive periodic solutions for problem (1.1) when the function g satisfies (1.3) or (1.4). Some remarks are also given to explain why we do not consider the multiplicity of positive periodic solutions and the dependence results of the positive periodic solution on the parameter when the function g satisfies (1.3) or (1.4).
2 Existence results under (1.2)
2.1 Main results
In this section, we state the main results, including existence and multiplicity results of positive periodic solutions for problem (1.1).
For convenience, we introduce the following notations:
where \(J=[0,\omega]\), \(k=0,1,\ldots,p-1\). Moreover, we choose four numbers r, \(r_{1}\), \(r_{2}\), and R satisfying
where δ is defined in (2.9).
Theorem 2.1
Let (H1)-(H4) hold. Assume that \(0< f_{\infty}\leq f^{\infty}<+\infty\), \(0< I_{\infty}(k)\leq I^{\infty }(k)<+\infty\), \(k=0,1,\ldots,p-1\), and the function g satisfies (1.2). Then
-
(i)
there exist \(\lambda_{0}>0\) and \(\mu_{0}>0\) such that, for any \(\lambda>\lambda_{0}\) and \(\mu>\mu_{0}\), problem (1.1) has a positive periodic solution \(u(t)\) with
$$ \delta r\leq u(t)\leq\frac{1}{\delta}R,\quad t\in J; $$(2.2) -
(ii)
there exist \(\bar{\lambda}_{0}>0\) and \(\bar{\mu}_{0}>0\) such that, for any \(0<\lambda<\bar{\lambda}_{0}\) and \(0<\mu<\bar{\mu}_{0}\), problem (1.1) has a positive periodic solution \(u(t)\), \(t\in J\) with property (2.2).
Theorem 2.2
Let (H1)-(H4) hold. Assume that \(0< f_{0}\leq f^{0}<+\infty\), \(0< I_{0}(k)\leq I^{0}(k)<+\infty\), \(k=0,1,\ldots,p-1\) and the function g satisfies (1.2). Then
-
(i)
there exist \(\lambda_{0}>0\) and \(\mu_{0}>0\) such that, for any \(\lambda>\lambda_{0}\) and \(\mu>\mu_{0}\), problem (1.1) has a positive periodic solution \(u(t)\) with
$$ \delta r\leq u(t)\leq R,\quad t\in J; $$(2.3) -
(ii)
there exist \(\bar{\lambda}_{0}>0\) and \(\bar{\mu}_{0}>0\) such that, for any \(0<\lambda<\bar{\lambda}_{0}\) and \(0<\mu<\bar{\mu}_{0}\), problem (1.1) has a positive periodic solution \(u(t)\), \(t\in J\) with property (2.3).
Remark 2.1
Some ideas of Theorems 2.1 and 2.2 are from Yan [28].
Theorem 2.3
Assume that (H1)-(H4) hold and the function g satisfies (1.2).
-
(i)
If \(f^{\infty}=0\) and \(I^{\infty}(k)=0\), \(k=0,1,\ldots,p-1\), then there exist \(\lambda_{0}>0\) and \(\mu_{0}>0\) such that, for any \(\lambda >\lambda_{0}\) and \(\mu>\mu_{0}\), problem (1.1) has a positive periodic solution \(u(t)\), \(t\in J\) with property (2.2).
-
(ii)
If \(f^{0}=0\) and \(I^{0}(k)=0\), \(k=0,1,\ldots,p-1\), then there exist \(\lambda_{0}>0\) and \(\mu_{0}>0\) such that, for any \(\lambda >\lambda_{0}\) and \(\mu>\mu_{0}\), problem (1.1) has a positive periodic solution \(u(t)\) with property (2.3).
-
(iii)
If \(f^{0}=f^{\infty}=I^{\infty}(k)=I^{0}(k)=0\), \(k=0,1,\ldots,p-1\), then there exist \(\lambda_{0}>0\) and \(\mu_{0}>0\) such that, for any \(\lambda >\lambda_{0}\) and \(\mu>\mu_{0}\), problem (1.1) has at least two positive periodic solutions \(u_{1}(t)\) and \(u_{2}(t)\) with
$$ \delta r\leq u_{1}(t)\leq r_{1}< \delta r_{2} \leq u_{2}(t)\leq R, \quad t\in J. $$(2.4)
Theorem 2.4
Assume that (H1)-(H4) hold and the function g satisfies (1.2).
-
(i)
If \(f_{\infty}=+\infty\) and \(I_{\infty}(k)=+\infty\), \(k=0,1,\ldots,p-1\), then there exist \(\bar{\lambda}_{0}>0\) and \(\bar{\mu}_{0}>0\) such that, for any \(0<\lambda<\bar{\lambda}_{0}\) and \(0<\mu<\bar{\mu}_{0}\), problem (1.1) has a positive periodic solution \(u(t)\), \(t\in J\) with property (2.2).
-
(ii)
If \(f_{0}=+\infty\) and \(I_{0}(k)=+\infty\), \(k=0,1,\ldots,p-1\), then there exist \(\bar{\lambda}_{0}>0\) and \(\bar{\mu}_{0}>0\) such that, for any \(0<\lambda<\bar{\lambda}_{0}\) and \(0<\mu<\bar{\mu}_{0}\), problem (1.1) has a positive periodic solution \(u(t)\), \(t\in J\) with property (2.3).
-
(iii)
If \(f_{0}=f_{\infty}=I^{\infty}(k)=I_{0}(k)=+\infty\), \(k=0,1,\ldots,p-1\), then there exist \(\bar{\lambda}_{0}>0\) and \(\bar{\mu}_{0}>0\) such that, for any \(0<\lambda<\bar{\lambda}_{0}\) and \(0<\mu<\bar{\mu}_{0}\), problem (1.1) has at least two positive periodic solutions \(u_{1}(t)\) and \(u_{2}(t)\) with
$$ \delta r\leq u_{1}(t)\leq r_{1}< \delta r_{2} \leq u_{2}(t)\leq\frac {1}{\delta}R, \quad t\in J. $$(2.5)
Remark 2.2
Some ideas of the proof of Theorem 2.3 and Theorem 2.4 are from Wang [10].
2.2 Preliminaries
In our main results, we will make use of the following definitions and lemmas.
Definition 2.1
A function \(u:\mathbf{R}\rightarrow(0,\infty )\) is called a positive solution of (1.1) if it satisfies:
-
(i)
\(u(t)\) is absolutely continuous on each \((t_{k},t_{k+1})\);
-
(ii)
for each \(k\in\mathbf{Z}\), \(u(t_{k}^{+})\) and \(u(t_{k}^{-})\) exist and \(u(t_{k}^{-})=u(t_{k})\);
-
(iii)
\(u(t)\) satisfies (1.1).
Definition 2.2
A solution \(u(t)\) of (1.1) on R is called ω-periodic if \(u(t+\omega)=u(t)\).
Let E be the Banach space
with \(\|u\|=\sup_{0 \leq t\leq\omega}|u(t)|\). We denote
for all \(r>0\) in the sequel.
Lemma 2.1
(See [34])
Assume that (H1)-(H4) hold and the function g satisfies (1.2). Then \(u\in E\) is a solution of problem (1.1) if and only if u is a solution of the integral equation
where
Further, it follows from (2.7) and (1.2) that
where \(\sigma=e^{\int_{0}^{\omega}a(t)\, dt}\).
Define a cone K in E by
where
It is easy to see K is a closed non-empty subset of E.
Define \(T_{\lambda}^{\mu}:K\rightarrow E\) by
From (2.10) and Lemma 2.1, it is easy to obtain the following result.
Lemma 2.2
(See [16])
Assume that (H1)-(H4) hold. Problem (1.1) is equivalent to the fixed point problem of \(T_{\lambda}^{\mu}\) in K.
Lemma 2.3
(See Lemmas 2.1 and 2.2 in [28])
Assume that (H1)-(H4) hold and the function g satisfies (1.2). Then \(T_{\lambda}^{\mu}:K\rightarrow K\) is completely continuous.
The following well-known result of the fixed point is crucial in our arguments.
Lemma 2.4
(See [45])
Let P be a cone in a real Banach space E. Assume \(\Omega_{1}\), \(\Omega_{2}\) are bounded open sets in E with \(0 \in\Omega_{1}\), \(\bar{\Omega}_{1}\subset \Omega_{2}\). If
is completely continuous such that either
-
(a)
\(\|Ax\|\leq\|x\|\), \(\forall x\in P\cap\partial\Omega_{1}\) and \(\|Ax\|\geq\|x\|\), \(\forall x\in P\cap\partial\Omega_{2}\), or
-
(b)
\(\|Ax\|\geq\|x\|\), \(\forall x\in P\cap\partial\Omega_{1}\) and \(\|Ax\|\leq\|x\|\), \(\forall x\in P\cap\partial\Omega_{2}\),
then A has at least one fixed point in \(P\cap(\bar{\Omega}_{2}\setminus\Omega_{1})\).
2.3 Proofs of the main results
For convenience we write
Proof of Theorem 2.1
Part (i). Noticing that \(f(t,x,y)>0\), \(I_{k}(t,x)>0\) (\(k=0,1,\ldots,p-1\)) for all \(t, x>0\), and \(y>0\), we can define, for \(r>0\),
where
Since \(\int_{-\zeta}^{0}e(s)\, ds=1\), it follows from \(\delta r\leq x\leq r\) that
Let
Then for any \(u\in K\cap\partial\Omega_{r}\) and \(\lambda>\lambda_{0}\), \(\mu>\mu_{0}\), we have
which implies that
If \(0< f^{\infty}<+\infty\), \(0< I^{\infty}(k)<+\infty\), \(k=0,1,\ldots,p-1\), then there exist \(l_{1}>0\), \(l_{2}>0\), and \(R>r>0\) such that
where \(l_{1}\) satisfies
\(l_{2}\) satisfies
Let \(\eta=\frac{R}{\delta}\). Thus, when \(u\in K\cap\partial\Omega_{\eta}\) we have
Since \(\int_{-\zeta}^{0}e(s)\, ds=1\), it follows from \(x\geq R\) that
Therefore, it follows from (2.10) that
This yields
Applying (b) of Lemma 2.4 to (2.11) and (2.14) shows that \(T_{\lambda}^{\mu}\) has a fixed point \(u\in K\cap(\bar{\Omega}_{\eta}\setminus\Omega_{r})\) with \(r \leq\| u\| \leq\eta=\frac{1}{\delta}R\). Hence, since for \(u\in K\) we have \(u(t)\geq\delta\|u\|\), \(t \in J\), it follows that (2.2) holds. This gives the proof of part (i).
Part (ii). Noticing that \(f(t,x,y)>0\), \(I_{k}(t,x)>0\) (\(k=0,1,\ldots,p-1\)) for all \(t, x>0\), and \(y>0\), we can define, for \(r>0\),
where
Since \(\int_{-\zeta}^{0}e(s)\, ds=1\), it follows from \(0\leq x\leq r\) that
Let
Then, for \(u\in K\cap\partial\Omega_{r}\) and \(\lambda<\bar{\lambda} _{0}\), \(\mu<\bar{\mu}_{0}\), we have
This implies
If \(0< f_{\infty}<+\infty\), \(0< I_{\infty}(k)<+\infty\), \(k=0,1,\ldots,p-1\), then there exist \(l_{3}>0\), \(l_{4}>0\), and \(R> r>0\) such that
where \(l_{3}\) satisfies
\(l_{4}\) satisfies
Let \(\bar{\eta}=\frac{R}{\delta}\). Thus, when \(u\in K\cap\partial\Omega_{\bar{\eta}}\) we have
Since \(\int_{-\zeta}^{0}e(s)\, ds=1\), it follows from \(x\geq R\) that
Therefore, for any \(u\in K\cap\partial\Omega_{\bar{\eta}}\), we have
which implies that
Applying (a) of Lemma 2.4 to (2.15) and (2.18) shows that \(T_{\lambda}^{\mu}\) has a fixed point \(u\in K\cap(\bar{\Omega}_{\bar{\eta}}\setminus\Omega_{\bar{r}})\) with \(r \leq\|u\| \leq\bar{\eta}=\frac{1}{\delta}R\). Hence, since for \(u\in K\) we have \(u(t)\geq\delta\|u\|\), \(t \in J\), it follows that (2.3) holds. This finishes the proof of part (ii). □
Proof of Theorem 2.2
Part (i). Noticing that \(f(t,x,y)>0\), \(I_{k}(t,x)>0\) (\(k=0,1,\ldots,p-1\)) for all \(t, x>0\), and \(y>0\), we can define, for \(R>0\),
where
Let
Then, for any \(u\in K\cap\partial\Omega_{R}\) and \(\lambda>\lambda_{0}\), \(\mu>\mu_{0}\), we have
which implies that
If \(0< f^{0}<+\infty\), \(0< I^{0}(k)<+\infty\), \(k=0,1,\ldots,p-1\), then there exist \(l_{1}>0\), \(l_{2}>0\), and \(0< r< R\) such that
where \(l_{1}\) and \(l_{2}\) satisfy (2.12) and (2.13), respectively.
Therefore, for \(u\in K\cap\partial\Omega_{r}\), we have
This yields
Applying (a) of Lemma 2.4 to (2.19) and (2.20) shows that \(T_{\lambda}^{\mu}\) has a fixed point \(u\in K\cap(\bar{\Omega}_{R}\setminus\Omega_{r})\) with \(r \leq\|u\| \leq R\). Hence, since for \(u\in K\) we have \(u(t)\geq\delta\|u\|\), \(t \in J\), it follows that (2.3) holds. This gives the proof of part (i).
Part (ii). Noticing that \(f(t,x,y)>0\), \(I_{k}(t,x)>0\) (\(k=0,1,\ldots,p-1\)) for all \(t, x>0\), and \(y>0\), we can define, for \(R>0\),
where
Let
Then, for \(u\in K\cap\partial\Omega_{R}\) and \(\lambda<\bar{\lambda}_{0}\), \(\mu<\bar{\mu}_{0}\), we have
This implies
If \(0< f_{0}<+\infty\), \(0< I_{0}(k)<+\infty\), \(k=0,1,\ldots,p-1\), then there exist \(l_{3}>0\), \(l_{4}>0\), and \(0< r< R\) such that
where \(l_{3}\) and \(l_{4}\) satisfy (2.16) and (2.17), respectively.
Therefore, for any \(u\in K\cap\partial\Omega_{r}\), we have
which implies that
Applying (a) of Lemma 2.4 to (2.21) and (2.22) shows that \(T_{\lambda}^{\mu}\) has a fixed point \(u\in K\cap(\bar{\Omega}_{R}\setminus\Omega_{r})\) with \(r \leq\|u\| \leq R\). Hence, since for \(u\in K\) we have \(u(t)\geq\delta\|u\|\), \(t \in J\), it follows that (2.3) holds. This finishes the proof of part (ii). □
Proof of Theorem 2.3
Similar to the proof of Theorem 2.1(i) and Theorem 2.2(i), respectively, one can show that Theorem 2.3(i) and (ii) hold.
Consider part (iii). Choose two numbers \(r_{1}\) and \(r_{2}\) satisfying (2.1). By Theorem 2.1(i) and Theorem 2.2(i), there exist \(\lambda_{0}>0\) and \(\mu_{0}>0\) such that
Since \(f^{0}=f^{\infty}=I^{\infty}(k)=I^{0}(k)=0\), \(k=0,1,\ldots,p-1\), from the proof of Theorem 2.1(i), Theorem 2.2(i), and (2.1), it follows that
and
Applying Lemma 2.4 to (2.23)-(2.25) shows that \(T_{\lambda}^{\mu}\) has two fixed points \(u_{1}\) and \(u_{2}\) such that \(u_{1}\in K\cap(\bar{\Omega}_{r_{1}}\setminus\Omega_{r})\) and \(u_{2}\in K\cap(\bar{\Omega}_{R}\setminus\Omega_{r_{2}})\). These are the desired distinct positive periodic solutions of problem (1.1) for \(\lambda_{0}>0\) and \(\mu_{0}>0\) satisfying (2.4). Then the result of part (iii) follows. □
Proof of Theorem 2.4
Similar to the proof of those Theorem 2.1(ii) and Theorem 2.2(ii), respectively, one can show that Theorem 2.4(i) and (ii) hold.
Now consider part (iii). Choose two numbers \(r_{1}\) and \(r_{2}\) satisfying (2.1). By Theorem 2.1(ii) and Theorem 2.2(ii), there exist \(\bar{\lambda}_{0}>0\) and \(\bar{\mu}_{0}>0\) such that
Since \(f_{0}=f_{\infty}=I_{\infty}(k)=I_{0}(k)=+\infty\), \(k=0,1,\ldots ,p-1\), from the proof of Theorem 2.1(ii) and Theorem 2.2(ii) and (2.1), it follows that
and
Applying Lemma 2.4 to (2.26)-(2.28) shows that \(T_{\lambda}^{\mu}\) has two fixed points \(u_{1}\) and \(u_{2}\) such that \(u_{1}\in K\cap(\bar{\Omega}_{r_{1}}\setminus\Omega_{r})\) and \(u_{2}\in K\cap(\bar{\Omega}_{R}\setminus\Omega_{r_{2}})\). These are the desired distinct positive periodic solutions of problem (1.1) for \(0<\lambda<\bar{\lambda}_{0}\) and \(0<\mu<\bar{\mu}_{0}\) satisfying (2.5). Then proof of part (iii) is complete. □
3 The parameter dependence of positive periodic solution under (1.2)
In this section, we consider the parameter dependence of positive periodic solution in the case \(\lambda=\mu\). For convenience, let
where
and d is a positive number.
- \((\mathrm{H}_{3})'\) :
-
Let \(f\in C(\mathbf{R}\times\mathbf{R}_{+}\times\mathbf{R}_{+}, \mathbf{R}_{+})\) be ω-periodic functions in t.
Theorem 3.1
Assume that (H1), (H2), \((\mathrm{H}_{3})'\), and (H4) hold and the function g satisfies (1.2). Then the following two conclusions hold.
- (H5):
-
If \(f^{0}=0\), \(I^{0}(k)=0\), and \(f_{\infty}=\infty\), \(I_{\infty }(k)=\infty\), \(k=0,1,\ldots,p-1\), then for every \(\lambda>0\) problem (1.1) has a positive periodic solution \(u_{\lambda}(t)\) satisfying \(\lim_{\lambda\rightarrow0^{+}}\|u_{\lambda}\|=\infty\).
- (H6):
-
If \(f_{0}=\infty\), \(I_{0}(k)=\infty\), and \(f^{\infty}=0\), \(I^{\infty}(k)=0\), \(k=0,1,\ldots,p-1\), then for every \(\lambda>0\) problem (1.1) has a positive periodic solution \(u_{\lambda}(t)\) satisfying \(\lim_{\lambda\rightarrow 0^{+}}\|u_{\lambda}\|=0\).
Remark 3.1
Assume that \(f_{\infty}=\infty\) or \(I_{\infty }(k)=\infty\), \(k=0,1,\ldots,p-1\) in (H5) or \(f_{0}=\infty\) or \(I_{0}(k)=\infty\), \(k=0,1,\ldots,p-1\) in (H6). Then the conclusions of Theorem 3.1 also hold.
Remark 3.2
Theorem 3.1 improves and develops Theorem 2.3.7 in [45] essentially.
Remark 3.3
Theorem 3.1 only considers the special case \(\mu=\lambda\). However, we do not give any information on the general case \(\mu\neq\lambda\).
Proof of Theorem 3.1
We need only to prove this theorem under condition (H5) since the proof is similar when (H6) holds.
For given \(\lambda>0\), similar to the proof of (2.18) and (2.20), it follows from (H5) that there exists \(0< r< R\) such that
and
Hence, condition (a) of Lemma 2.4 is satisfied for the operator \(T_{\lambda}^{\mu}\), which implies that \(T_{\lambda}^{\mu}\) has a fixed point \(u_{\lambda}\) in \(\bar{\Omega}_{R}\setminus\Omega_{r}\).
It remains to prove \(\|u_{\lambda}\|=+\infty\) as \(\lambda\rightarrow0^{+}\). In fact, if not, there exist a number \(d>0\) and a sequence \(\lambda_{n}\rightarrow0^{+}\) such that
Furthermore, the sequence \(\{\|u_{\lambda_{n}}\|\}\) contains a subsequence that converges to a number η (\(0\leq\eta\leq d\)). For simplicity, suppose that \(\{\|u_{\lambda_{n}}\|\}\) itself converges to η.
If \(\eta>0\), then \(\|u_{\lambda_{n}}\|>\frac{\eta}{2}\) for sufficiently large n (\(n>\mathbf{N}\)), and therefore
which contradicts \(\lambda_{n}\rightarrow0^{+}\).
If \(\eta=0\), then \(\|u_{\lambda_{n}}\|\rightarrow0\) for sufficiently large n (\(n>\mathbf{N}\)), and therefore it follows from (H5) that for any \(\varepsilon>0\) there exists \(r_{3}>0\) such that
and hence it follows from (2.10) that
Since ε is arbitrary, we have \(\lambda_{n}\rightarrow+\infty\) (\(n\rightarrow+\infty\)) in contradiction with \(\lambda_{n}\rightarrow0^{+}\). Therefore, \(\|u_{\lambda}\|\rightarrow+\infty\) as \(\lambda\rightarrow0^{+}\) and the proof is complete. □
4 Existence under (1.3) and (1.4)
Firstly, let us denote that for each fixed constant \(\rho>0\),
Theorem 4.1
Assume that (H1)-(H4) hold and g satisfies (1.3).
- (H7):
-
If \(f^{0}=0\) and \(I^{0}(k)=0\), \(k=0,1,\ldots,p-1\), then there exist \(\lambda_{\rho}>0\) and \(\mu_{\rho}>0\) such that problem (1.1) has a positive periodic solution u with
$$ \sup_{t\in J}u(t)\leq\rho \quad \textit{for } \lambda> \lambda_{\rho} \textit{ and }\mu>\mu_{\rho }. $$(4.3) - (H8):
-
If \(f_{0}=+\infty\) and \(I_{0}(k)=+\infty\), \(k=0,1,\ldots,p-1\), such that problem (1.1) has a positive periodic solution u with
$$ \sup_{t\in J}u(t)\leq\rho\quad \textit{for } 0< \lambda< \lambda_{\rho} \textit{ and } 0< \mu< \mu _{\rho}. $$(4.4)
Corollary 4.1
Assume that (H1)-(H4) hold and the function g satisfies (1.3).
-
(i)
If \(f^{0}=0\) and \(I^{0}(k)=0\), \(k=0,1,\ldots,p-1\), then there exists \(\lambda_{\rho}>0\) or \(\mu_{\rho}>0\) such that problem (1.1) has a positive periodic solution \(u(t)\) with property (4.3).
-
(ii)
If \(f_{0}=+\infty\) or \(I_{0}(k)=+\infty\), \(k=0,1,\ldots,p-1\), then there exist \(\bar{\lambda}_{\rho}>0\) and \(\bar{\mu}_{\rho}>0\) such that problem (1.1) has a positive periodic solution \(u(t)\), \(t\in J\) with property (4.4).
Let ρ be the constant in Theorem 4.1. Define a cone \(K[\rho]\) in E by
For every \(\iota>0\), let
and
Define an operator \(T_{\lambda}:E\rightarrow E\) by
where
It follows from (4.1) and (4.2) that
and hence, for \(u\in\bar{\Omega}_{\rho}\), we have
where σ is defined in (2.8).
Similarly, for \(0< r<\rho\), we can define
Obviously,
and, for \(u\in\bar{\Omega}_{r}\), we have
Lemma 4.1
(See [43])
Assume that (H1)-(H4) hold. Let \(\rho>0\) be fixed. Then for each r: \(0< r\leq\rho\), we have \(T(\bar{\Omega}_{r})\subset K[\rho]\) and \(T:\bar{\Omega}_{r}\rightarrow K[\rho]\) is compact and continuous.
Remark 4.1
(See [43])
Notice that for the cone \(K[\rho]\) defined by (4.3) we cannot obtain
However, Lemma 4.1 shows that, for every \(r\in(0,\rho]\),
which is enough for us to apply the fixed point theorem in cones.
Lemma 4.2
(See [43])
Assume that (H1)-(H4) hold. If u is a fixed point of T in \(K[\rho]\), then u is a periodic solution of problem (1.1).
Proof of Theorem 4.1
Let \(R=\rho\) and \(\alpha(\rho)\) replace α, \(\beta(\rho)\) replace β in the proof of Theorem 2.2(i) and (ii) and repeat the argument of Theorem 2.2(i) and (ii). Then we can prove that Theorem 4.1 holds. □
Remark 4.2
Let the condition (1.4) replace (1.3) in Theorem 4.1, the results of Theorem 4.1 also hold.
Remark 4.3
Since the function \(G_{u}(t,s)\) is related to u and the operator T does not satisfy
we have the following facts.
-
(i)
\(u\rightarrow0\) means there exists a sufficiently small positive number r such that \(u\in[0,r]\). So one can easily choose a positive number ρ to control r.
-
(ii)
\(u\rightarrow\infty\) means there exists a sufficiently large positive number R such that \(u\in[R,\infty]\), which implies that it is very difficult to find a positive number η to control R.
Therefore, we only prove the existence of positive periodic solutions for problem (1.1) under (1.3) and f and \(I_{k}\) (\(k=0,1,\ldots,p-1\)) satisfying one of the following conditions:
-
(i)
\(f^{0}=0\) and \(I^{0}(k)=0\), \(k=0,1,\ldots,p-1\);
-
(ii)
\(f_{0}=\infty\) and \(I_{0}(k)=\infty\), \(k=0,1,\ldots,p-1\).
However, we do not give any information on the existence of positive periodic solutions for problem (1.1) in the four following cases:
-
(iii)
(1.3) holds, \(f^{\infty}=0\) and \(I^{\infty}(k)=0\), \(k=0,1,\ldots,p-1\);
-
(iv)
(1.3) holds, \(f_{\infty}=\infty\) and \(I_{\infty }(k)=\infty\), \(k=0,1,\ldots,p-1\);
-
(v)
(1.4) holds, \(f^{\infty}=0\) and \(I^{\infty}(k)=0\), \(k=0,1,\ldots,p-1\);
-
(vi)
(1.4) holds, \(f_{\infty}=\infty\) and \(I_{\infty }(k)=\infty\), \(k=0,1,\ldots,p-1\).
This and the proof of Theorem 3.1 imply the following.
Remark 4.4
If (1.3) or (1.4) hold, then we cannot obtain the parameter dependence results of positive periodic solution for problem (1.1).
5 An example
To illustrate how our main results can be used in practice, we present an example. Let
Example 5.1
Consider the following equation:
where \(\lambda>0 \) and \(\mu>0\) are parameters, and
Conclusion 5.1
For any \(\lambda>\frac{3e^{\frac {4}{3}}(e^{\frac{4}{3}}-1)^{2}}{14\pi(e^{\frac{1}{3}}-1)}\) and \(\mu>\frac {e^{\frac{4}{3}}(e^{\frac{4}{3}}-1)^{2}}{3(e^{\frac{1}{3}}-1)}\), problem (5.1) has a positive 2π-periodic solution.
Proof
It follows from (5.3) that (H1)-(H4) of Theorem 2.1 hold. Noticing that \(p=1\), we have
Letting \(t_{0}=\pi\), then
So we can take
On the other hand, it follows from (5.1) that
and
Therefore, it follows from Theorem 2.1 that there exist \(\lambda _{0}=\frac{3e^{\frac{4}{3}}(e^{\frac{4}{3}}-1)^{2}}{14\pi(e^{\frac {1}{3}}-1)}>0\) and \(\mu_{0}=\frac{e^{\frac{4}{3}}(e^{\frac {4}{3}}-1)^{2}}{3(e^{\frac{1}{3}}-1)}>0\) such that, for any \(\lambda >\lambda_{0}\) and \(\mu>\mu_{0}\), problem (5.1) has a positive 2π-periodic solution. □
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Acknowledgements
This work is sponsored by the project NSFC (11301178), the Fundamental Research Funds for the Central Universities (2014ZZD10, 2014MS58) and the Scientific Research Common Program of Beijing Municipal Commission of Education (KM201511232018). The authors are grateful to the anonymous referees for their constructive comments and suggestions, which have greatly improved this paper.
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XZ completed the main study and carried out the results of this article. MF checked the proofs and verified the calculation. All authors read and approved the final manuscript.
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Zhang, X., Feng, M. Multi-parameter, impulsive effects and positive periodic solutions of first-order functional differential equations. Bound Value Probl 2015, 137 (2015). https://doi.org/10.1186/s13661-015-0401-x
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DOI: https://doi.org/10.1186/s13661-015-0401-x