Abstract
We consider the second-order nonlocal impulsive differential system
where the weight functions \(a(t)\), \(b(t)\), and \(\omega (t)\) change sign on \([0,1]\), and \(g(t)\not \equiv 0\) and \(h(t)\not \equiv 0\) on \([0,1]\). By constructing a cone \(K_{1}\times K_{2}\), which is the Cartesian product of two cones in space \(PC[0,1]\), and applying the well-known fixed point theorem of cone expansion and compression in \(K_{1}\times K_{2}\), we obtain conditions for the existence and multiplicity of positive solutions of a nonlocal indefinite impulsive differential system. An example is given to illustrate the main results.
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1 Introduction
It is generally accepted that the theory and applications of differential equations with impulsive effects are an important area of investigation, since it is far richer than the corresponding theory of differential equations without impulsive effects. Various population models, biological system models, ecology models, biotechnology models, pharmacokinetics models, and optimal control models, which are characterized by the fact that per sudden changing of their state, can be expressed by impulsive differential equations. For an introduction of general theory of impulsive differential equations, we refer the reader to the references [1] and [2], whereas the applications of impulsive differential equations can be found in [3–5]. Some classical methods have been widely used to study impulsive differential equations: the theory of critical point theory and variational methods [6–8], fixed point theorems in cones [9–25], and bifurcation theory [26, 27]. In particular, we would like to mention some results of Lin and Jiang [28] and Feng and Xie [29]. Lin and Jiang [28] considered the following Dirichlet boundary value problem with impulse effects:
and by means of the fixed point index theory in cones the authors obtained some sufficient conditions for the existence of multiple positive solutions for problem (1.1).
Recently, using fixed point theorems in a cone, Feng and Xie [29] studied the existence of positive solutions for the following problem:
In comparison with numerous results on the impulsive differential equations, it is less known about the impulsive differential systems, even for the nonlocal impulsive differential systems.
Moreover, we see that increasing attention has been paid to the study of boundary value problems with integral boundary conditions; for example, see Liu, Sun, Zhang, and Wu [30], Zhang, Feng, and Ge [31], Zhang and Ge [32], Hao et al. [33–35], Yan, Zuo, and Hao [36], Zhang et al. [37, 38], Sun, Liu, and Wu [39], Lin and Zhao [40], and Ahmad, Alsaedi, and Alghamdi [41]. This problem contains two-, three-, and multipoint boundary value problems as particular cases; for instance, see Karakostas and Tsamatos [42], Feng and Ge [43], Jiang, Liu, and Wu [44], Lan [45], Zhang et al. [46–50], Feng, Du, and Ge [51], Ahmad and Alsaedi [52], Mao and Zhao [53], Liu, Hao, and Wu [54], and the references therein. Specifically, Boucherif [55] exploited the fixed point theorem in cones to study the following problem:
The author obtained several excellent results on the existence of positive solutions to problem (1.3).
Feng, Ji, and Ge [56] began to study the following boundary value problem with integral boundary conditions in abstract spaces:
Applying the fixed point theory in a cone for strict set contraction operators, the authors investigated the existence, nonexistence, and multiplicity of positive solutions for problem (1.4).
Recently, Kong [57] considered the existence and uniqueness of positive solutions for the second-order singular boundary value problem:
The author examined the uniqueness of the solution and its dependence on the parameter λ for problem (1.5) by using the mixed monotone operator theory.
Simultaneously, an indefinite problem has attracted the attention of Ma and Han [58], López-Gómez and Tellini [59], Boscaggin and Zanolin [60], Feltrin and Zanolin [61], Boscaggin et al. [62, 63], Sovrano and Zanolin [64], Bravo and Torres [65], Wang and An [66], and Yao [67]. Ma and Han [58] considered the following boundary value problem:
where \(a\in C[0,1]\) may change sign, and λ is a parameter. They proved the existence, multiplicity, and stability of positive solutions for problem (1.6) by applying bifurcation techniques.
Aapplying the shooting method, Sovrano and Zanolin [60] presented a multiplicity result for positive solutions for the Neumann problem
where the weight function \(a\in C[0,1]\) has indefinite sign.
Recently, Wang and An [66] dealt with the existence and multiplicity of positive solutions for the second-order differential system
where \(a(t)\), \(b(t)\), and \(g(t)\) are allowed to change sign on \([0,1]\). For the latest results of indefinite problems, please refer to Jiao and Zhang [68], Feltrin and Sovrano [69], and Zhang [70].
For all we know, in the literature there are no papers on multiple positive solutions for analogous indefinite impulsive differential systems with nonlocal boundary value conditions. More precisely, the study of \(a(t)\), \(b(t)\), and \(\omega (t)\) changing sign on \([0,1]\) is still open for the second-order nonlocal impulsive differential system
where \(a(t)\), \(\omega (t)\), \(b(t)\) change sign on \([0,1]\), \(t_{k}\) (\(k=1,2,\ldots,n\); where n is a fixed positive integer) are fixed points such that \(0< t_{1}< t_{2}<\cdots <t_{k}<\cdots <t_{n}<1\), \(\Delta x|_{t=t_{k}}\) denotes the jump of \(x(t)\) at \(t=t_{k}\), that is, \(\Delta x|_{t=t_{k}}=x(t_{k}^{+})-x(t_{k}^{-})\), where \(x(t_{k}^{+})\) and \(x(t_{k}^{-})\) represent the right- and left-hand limits of \(x(t)\) at \(t=t_{k}\), respectively; \(\Delta y|_{t=t_{k}}\) has a similar meaning for \(y(t)\). In addition, a, ω, b, f, \(I_{k}\), and \(J_{k}\) (\(k=1,2,\ldots,n\)) satisfy (\(H_{1}\)) \(a, \omega, b: [0,1]\rightarrow (-\infty,+\infty)\) are continuous, and there exists a constant \(\xi \in (0, 1)\) such that
Moreover, \(a(t)\), \(\omega (t)\), \(b(t)\) do not vanish identically on any subinterval of \([0,1]\).
- (\(H_{2}\)):
-
\(f:[0,+\infty)\rightarrow [0,+\infty)\) is continuous.
- (\(H_{3}\)):
-
\(I_{k}:[0,+\infty)\rightarrow [0,+\infty)\) is continuous.
- (\(H_{4}\)):
-
\(J_{k}:[0,+\infty)\rightarrow [0,+\infty)\) is continuous.
- (\(H_{5}\)):
-
\(h,g\in L^{1}[0,1]\) are nonnegative, and \(\nu,\nu_{1} \in [0,1)\), where
$$ \nu = \int_{0}^{1}g(s)\,ds, \qquad \nu_{1}= \int_{0}^{1}h(s)\,ds. $$(1.10)
Let \(J=[0,1]\) and \(J'=J\backslash \{t_{1},t_{2},\ldots,t_{n}\}\). The basic space used in this paper is \(PC[0,1]= \{u| u:[0,1]\rightarrow R\mbox{ is continuous at }t\neq t_{k}, u(t_{k}^{-})=u(t_{k}), \mbox{and } u(t_{k}^{+})\mbox{ exists}, k=1,2,\ldots, n \}\). Then \(PC[0,1]\) is a real Banach space with the norm
For convenience, consider \(PC_{1}[0,1]= \{x: x\mbox{ is continuous at } t\neq t_{k},x(t_{k}^{-})=x(t_{k}), \mbox{and }x(t_{k}^{+})\ \mbox{exists}, k=1,2,\ldots, n \}\), which is a real Banach space with norm
and \(PC_{2}[0,1]= \{y: y\mbox{ is continuous at }t\neq t_{k},x(t_{k} ^{-})=x(t_{k}), \mbox{and }y(t_{k}^{+})\mbox{ exists}, k=1,2,\ldots, n \}\), which is a real Banach space with norm
Clearly, \(PC_{1}[0,1]\times PC_{2}[0,1]\) is also a real Banach space with norm
By a positive solution of system (1.9) we mean a pair of functions \((x,y)\) with \(x\in C^{2}(J')\cap PC_{1}[0,1]\) and \(y\in C^{2}(J') \cap PC_{2}[0,1]\) such that \((x,y)\) satisfies system (1.9) and \(x, y\geq 0\), \(t\in J'\), \(x,y\not \equiv 0\).
Remark 1.1
The technique to deal with the impulsive term is completely different from that of [6–27].
Remark 1.2
When we consider nonlocal differential systems with indefinite weights, another difficulty is to prove \(T: K_{1}\times K _{2}\rightarrow K_{1}\times K_{2}\); for details, see Lemma 2.3.
Remark 1.3
It is not difficult to see that Proposition 2.3 of [67] plays key roles in the proofs of main results of [66] and [67]. However, it is invalid for nonlocal problems; for details, see Corollary 4.1.
Remark 1.4
In comparison with other related indefinite problems [58–66], the main features of this paper are as follows.
-
(i)
\(I_{k}, J_{k}\neq 0\) (\(k=1,2,\ldots,n\)) are introduced.
-
(ii)
Nonlocal boundary conditions are introduced.
-
(iii)
\(K_{1}\times K_{2}\) is the Cartesian product of two cones in the space \(PC[0,1]\).
We define \(a^{\pm }(t)\), \(\omega^{\pm }(t)\), and \(b^{\pm }(t)\) as
so that
Inspired by the references mentioned, in this paper, we investigate the existence and multiplicity of positive solutions for system (1.9). By constructing a cone \(K_{1}\times K_{2}\), which is the Cartesian product of two cones in the space \(PC[0,1]\), and using the well-known fixed point theorem of cone expansion and compression, we obtain conditions for the existence and multiplicity of positive solutions of system (1.9). We remark that this is probably the first time that the existence and multiplicity of positive solutions of impulsive differential systems with indefinite weight and integral boundary conditions have been studied.
The rest of this paper is organized as follows. In Sect. 2, we give some preliminary results. Section 3 is devoted to state and prove the main results. Finally, an example is given in Sect. 4.
2 Preliminaries
In this section, we give some preliminary results for the convenience of later use and reference. It is clear that system (1.9) is equivalent to the following two boundary value problems:
and
Lemma 2.1
Assume that (\(H_{1}\)), (\(H_{2}\)), (\(H_{4}\)), and (\(H_{5}\)) hold. Then problem (2.1) has a unique solution y, which can be expressed in the form
where
Proof
First, suppose that u is a solution of problem (2.1). It is easy to see by integration of problem (2.1) that
Integrating again, we get
Letting \(t=1\) in (2.8), we find
Substituting the boundary condition \(y(0)=\int_{0}^{1}g(t)y(t)\,dt\) and (2.10) into (2.9), we obtain
where
Therefore we have
and
Let
Then
The proof of sufficiency is complete.
Conversely, let u be a solution of (2.1). Direct differentiation of (2.4) and (2.5) implies, for \(t\neq t_{k}\),
Evidently,
The lemma is proved. □
Proposition 2.1
Let \(G(t,s)\), \(G'_{s}(t,s)\), \(H(t,s)\), and \(H'_{s}(t,s)\) be given as in Lemma 2.1. If \(\nu \in [0,1)\), then we get
where
Proof
By the definition of \(G(t,s)\) and \(H(t,s)\), relations (2.11) and (2.12) are simple to prove.
Next, we consider (2.13). In fact, from (1.10) and (2.12) we get
and
This shows that (2.13) holds.
Similarly, by the the definition of \(G'_{s}(t,s)\) and \(H'_{s}(t,s)\), we can prove that (2.14) holds. □
Remark 2.1
From (2.5) we can prove that
Proof
It follows from (2.5) and (2.7) that
□
Lemma 2.2
Assume that (\(H_{1}\))–(\(H_{3}\)) and (\(H_{5}\)) hold. Then problem (2.2) has a unique solution x given by
where
Proof
The proof of Lemma 2.2 is similar to that of Lemma 2.1. □
Proposition 2.2
Let \(H_{1}\) and \(H'_{1s}\) be given as in Lemma 2.1. If \(\nu_{1}\in [0,1)\), then we get
where
Remark 2.2
From (2.18) we can prove that
Remark 2.3
Let \((x,y)\) be a solution of system (1.9). Then from Lemma 2.1 and Lemma 2.2 we have
where
To obtain the existence and multiplicity of a positive solution of system (1.9), we make the following hypotheses:
- (\(H_{6}\)):
-
There exists a constant \(\sigma_{1}\) satisfying \(0<\sigma _{1}<\xi \) such that
$$ \sigma_{1} \int_{\sigma_{1}}^{\xi }H(t,s)b^{+}(s)\,ds\geq \xi \int_{ \xi }^{1}H(t,s)b^{-}(s)\,ds; $$ - (\(H_{7}\)):
-
There exists a constant \(\sigma_{2}\) satisfying \(0<\sigma _{2}<\xi \) such that
$$ \rho \sigma_{2} \int_{\sigma_{2}}^{\xi }H_{1}(t,s)G(s,s)a^{+}(s) \,ds \geq \gamma \xi \int_{\xi }^{1}H_{1}(t,s)a^{-}(s) \,ds; $$ - (\(H_{8}\)):
-
There exists a constant μ satisfying \(0<\mu \leq 1\) such that
$$ f(\omega)\geq \mu \varphi (\omega),\quad \omega \in [0,+\infty), $$where \(\varphi (\omega)=\max \{f(\rho):0\leq \rho \leq \omega \}\);
- (\(H_{9}\)):
-
There exist constants \(0<\alpha <+\infty \) with \(\alpha \neq 1\) and \(k_{1},k_{2},l_{1},l_{2},m_{1},m_{2}>0\) such that
$$\begin{aligned}& k_{1}x^{\alpha }\leq f(x)\leq k_{2}x^{\alpha }, \qquad l_{1}x^{\alpha } \leq I_{k}(x)\leq l_{2}x^{\alpha }, \\& m_{1}y^{\alpha }\leq J_{k}(y) \leq m_{2}y^{\alpha }, \quad x, y\in [0,+\infty); \end{aligned}$$ - (\(H_{10}\)):
-
There exists \(0<\sigma_{3}<\xi \) satisfying \(\frac{\sigma _{3}}{2}< t_{1}<\sigma_{3}\) such that
$$ \sigma^{\alpha }_{3}\mu^{2}k_{1} \int_{\sigma_{3}}^{\xi }H_{1}(t,s) \omega^{+}(s)\,ds\geq k_{2}\xi^{\alpha } \int_{\xi }^{1}H_{1}(t,s)\omega ^{-}(s)\,ds. $$
Obviously, \(\varphi:[0,+\infty)\rightarrow [0,+\infty)\) is nondecreasing. Moreover, if f is nondecreasing, then \(f=\varphi \) and \(\mu =1\).
We denote
If \(x\in K_{1}\), then it is easy to see that \(\|x\|_{PC_{1}}=\max_{0\leq t\leq \xi }|x(t)|\). Similarly, we have \(\|y\|_{PC_{2}}= \max_{0\leq t\leq \xi }|y(t)|\). Also, for a positive number r, we define
and then we get
For any \((x,y)\in K_{1}\times K_{2}\), define the mappings \(T_{1}:K _{1}\rightarrow PC_{1}[0,1]\), \(T_{2}:K_{2}\rightarrow PC_{2}[0,1]\), and \(T:K_{1}\times K_{2}\rightarrow PC_{1}[0,1]\times PC_{2}[0,1]\) as follows:
Remark 2.4
It follows from Lemmas 2.1–2.2 and Remark 2.3 that \((x,y)\) is a solution of system (1.9) if and only if \((x,y)\) is a fixed point of operator T.
Lemma 2.3
Assume that (\(H_{1}\))–(\(H_{10}\)) hold. Then \(T(K_{1}\times K_{2})\subset K_{1}\times K_{2}\), and \(T:K_{1}\times K _{2}\rightarrow K_{1}\times K_{2}\) is completely continuous.
Proof
For any \((x,y)\in K_{1}\times K_{2}\), we prove that \(T(x,y)\in K_{1}\times K_{2}\), that is, \(T_{1}x\in K_{1}\) and \(T_{2}y\in K_{2}\). In view of (2.20), we know that
where
and then we have \((T_{1}x)^{\prime}(1)=0\). From (2.14) and (2.20) we get
and
Similarly, we have \((T_{2}y)^{\prime}(1)=0\), \((T_{2}y)(0)=\int_{0}^{1}g(t)(T _{2}y)(t)\,dt\).
Define the function \(q: [0,1]\rightarrow [0,1]\) as follows:
-
if \(x(1)=0\), then
$$ q(t)=\min \biggl\{ \frac{t}{\xi },\frac{1-t}{1-\xi } \biggr\} ,\quad \forall t \in J; $$ -
if \(x(1)>0\), then
$$ q(t)=\min \biggl\{ \frac{t}{\xi },1 \biggr\} , \quad \forall t\in J. $$
Since \(\sigma_{1}<\xi \), \(\max_{0\leq t\leq 1}q(t)=1\) and \(\min_{\sigma_{i}\leq t\leq \xi }q(t)=\frac{\sigma_{i}}{\xi }\), \(i=1,2,3\).
Let \(x\in K_{1}\). Then x is concave on \([0,\xi ]\) and convex on \([\xi,1]\). Noticing that \(x(0)=\int_{0}^{1}h(t)x(t)\,dt\) and \(x'(1)=0\), we get
First of all, for any \(x\in K_{1}\), we show that
Indeed, for \(x\in K_{1}\), we obtain
Then by (\(H_{6}\)) we have
Secondly, for any \(x\in K_{1}\), we prove that
For \(t\in J\), since \(\int_{0}^{1}H(t,s)b(s)x(s)\,ds\geq 0\), we have
and then it follows from (\(H_{7}\)) that
Similarly, for any \(y\in K_{2}\), since \(\sum_{k=1}^{n}H'_{ \tau }(s,t_{k})I_{k} (y(t_{k}) )\geq 0\), we get
Thirdly, for any \(x\in K_{1}\), we prove that
Since φ is nondecreasing, we also obtain
Therefore, for any \(x\in K_{1}\), it follows from (\(H_{8}\))–(\(H_{10}\)) that
which shows that
Thus, for \((x,y)\in K_{1}\times K_{2}\),
Moreover, by direct calculation we derive
which shows that \(T_{1}x\) and \(T_{2}y\) are concave on \([0,\xi ]\) and convex on \([\xi,1]\). It follows that \(T_{1}x\in K_{1}\) and \(T_{2}y\in K_{2}\). Thus \(T(K_{1}\times K_{2})\subset K_{1}\times K _{2}\).
Finally, by standard methods and the Arzelà–Ascoli theorem we can prove that T is completely continuous. □
Remark 2.5
In [66] and [67], it is not difficult to see that the function \(q(t)\) plays an important role in the proof of completely continuous operator. If \(x(0)=x(1)=0\), then we can define \(q(t)= \min \{\frac{t}{\xi },\frac{1-t}{1-\xi } \}\). However, if \(x(0)=\int_{0}^{1}h(t)x(t)\,dt\) and \(x'(1)=0\), then the definition of \(q(t)\) is invalid. This shows that when \(x(0)=\int_{0}^{1}h(t)x(t)\,dt\) and \(x'(1)=0\), we require a special technique to give a fine definition of \(q(t)\).
In fact, a fine definition of \(q(t)\) is very difficult to give when \(x(0)=\int_{0}^{1}h(t)x(t)\,dt\) and \(x'(1)=0\). This is probably the main reason why there is almost no paper studying the existence of positive solutions for the class of second-order nonlocal differential systems with indefinite weights and even for second-order nonlocal impulsive differential systems with indefinite weights.
Remark 2.6
The idea of the proof of Lemma 2.3 comes from Theorem 3.1 of [67].
The following lemma is very crucial in our argument.
Lemma 2.4
(Theorem 2.3.4 of [71], Fixed point theorem of cone expansion and compression of norm type)
Let \(\varOmega_{1}\) and \(\varOmega_{2}\) be two bounded open sets in a real Banach space E such that \(0\in \varOmega_{1}\) and \(\bar{\varOmega }_{1}\subset \varOmega_{2}\). Let an operator \(T: K\cap (\bar{\varOmega }_{2}\backslash \varOmega_{1})\rightarrow K\) be completely continuous, where K is a cone in E. Suppose that one of the following two conditions is satisfied:
-
(a)
\(\|Tx\|\leq \|x\|\), \(\forall x\in K\cap \partial \varOmega_{1}\), and \(\|Tx\|\geq \|x\|\), \(\forall x\in K\cap \partial \varOmega_{2}\),
and
-
(b)
\(\|Tx\|\geq \|x\|\), \(\forall x\in K\cap \partial \varOmega_{1}\), and \(\|Tx\|\leq \|x\|\), \(\forall x\in K\cap \partial \varOmega_{2}\),
is satisfied. Then T has at least one fixed point in \(K\cap (\bar{ \varOmega }_{2}\backslash \varOmega_{1})\).
3 Main results
In this part, applying Lemma 2.4, we obtain the following three existence theorems.
Theorem 3.1
Assume that (\(H_{1}\))–(\(H_{10}\)) hold. If \(\alpha >1\), then system (1.9) admits at least one positive solution.
Proof
On one hand, considering the case \(\alpha >1\), by (\(H_{9}\)) we get
Furthermore, there exist \(r',r''>0\) such that
where \(\varepsilon_{1}\), \(\varepsilon_{2}\), \(\varepsilon_{3}\) satisfy
Let
and choose \(r=\min \{(4A)^{-1},(4\varepsilon_{3}A')^{-1},r',r'' \}\). Then for any \((x,y)\in (K_{1}\times K_{2})\cap \partial \varOmega_{r}\), we have \(\|(x,y)\|=r\), and by (2.20) we get
Consequently,
On the other hand, since \(\alpha >1\), it follows from (\(H_{9}\)) that
which shows that there exist \(R',R''>0\) such that
where \(\varepsilon_{4}\), \(\varepsilon_{5}\), \(\varepsilon_{6}\) satisfy
where
If \((x,y)\in K_{1}\times K_{2}\), then x, y are two nonnegative concave functions on \([0,\xi ]\). So we get
It follows that \(\min_{\frac{\sigma_{i}}{2}\leq t\leq \sigma _{i}}x(t)\geq \theta_{i}\|x\|_{PC_{1}}\), \(\min_{\frac{\sigma_{i}}{2}\leq t\leq \sigma_{i}}y(t)\geq \theta _{i}\|y\|_{PC_{2}}\), \(i=1,2,3\), where
Let
\(R_{1}>\max \{(3B)^{-1},\frac{R^{\prime}}{\theta_{3}},r \}\), \(R_{2}>\max \{\frac{R^{\prime\prime }}{\theta_{3}},r \}\), and \(R=\max \{R_{1},R_{2} \}\). Then for any \((x,y)\in (K_{1} \times K_{2})\cap \partial \varOmega_{R}\), we have
and
Consequently,
Therefore, applying Lemma 2.4 to (3.4) and (3.11), we can show that T has at least one fixed point
The proof of Theorem 3.1 is completed. □
The following theorem deals with the multiplicity of system (1.9). For convenience, we introduce the following notations:
Theorem 3.2
Assume that (\(H_{1}\))–(\(H_{10}\)) hold. Suppose that \(0<\alpha <1\) and there exist constants d and r satisfying \(0< d<\min \{(4A)^{-1},(4A'\varGamma)^{-1},r\}\) such that
where A and \(A'\) are defined in (3.1). Then system (1.9) admits at least two positive solutions.
Proof
If \(0<\alpha <1\), then by (\(H_{9}\)) we know that
-
(i)
\(\lim_{x\rightarrow 0}\frac{f(x)}{x}\geq \lim_{x\rightarrow 0}\frac{k_{1}x^{\alpha }}{x}=\infty\), \(\lim_{x\rightarrow 0}\frac{I_{k}(x)}{x}\geq \lim_{x\rightarrow 0}\frac{l_{1}x^{\alpha }}{x}=\infty\), \(\lim_{y\rightarrow 0}\frac{J_{k}(y)}{y}\geq \lim_{y\rightarrow 0}\frac{m_{1}y^{\alpha }}{y}=\infty\);
-
(ii)
\(\lim_{x\rightarrow \infty }\frac{f(x)}{x}\leq \lim_{x\rightarrow \infty }\frac{k_{2}x^{\alpha }}{x}=0\), \(\lim_{x\rightarrow \infty }\frac{I_{k}(x)}{x}\leq \lim_{x\rightarrow \infty }\frac{l_{2}x^{\alpha }}{x}=0\), \(\lim_{y\rightarrow \infty }\frac{J_{k}(y)}{y}\leq \lim_{y\rightarrow \infty }\frac{m_{2}y^{\alpha }}{y}=0\).
From (i) it follows that there exists a sufficiently small positive constant r such that
where \(\varepsilon_{7}\), \(\varepsilon_{8}\), \(\varepsilon_{9}\) satisfy
with \(\delta (t)\) defined in (3.5).
Therefore, for any \((x,y)\in (K_{1}\times K_{2})\cap \partial \varOmega _{r}\), we get from (3.6) that
Consequently,
Next, let us turn to (ii), which shows that there exist \(R',R''>r\) such that
where \(\varepsilon_{1}\), \(\varepsilon_{2}\), \(\varepsilon_{3}\) satisfy
Let
Then
Let
Choosing \(\max \{6M,6M^{*},r\}< R<(6A)^{-1}\), for any \((x,y)\in (K_{1} \times K_{2})\cap \partial \varOmega_{R}\), similarly to the proof of (3.2) and (3.3), we get
which shows that
Finally, since \(0< d<\min \{(4A)^{-1},(4A'\varGamma)^{-1},r\}\), for \((x,y)\in (K_{1}\times K_{2})\cap \partial \varOmega_{d}\), it follows from (3.12) that
which shows that
Therefore, applying Lemma 2.4 to (3.15), (3.19), and (3.22), we can show that T has at least two fixed points
The proof of Theorem 3.2 is completed. □
Corollary 3.1
Assume that (\(H_{1}\))–(\(H_{10}\)) hold. If \(0<\alpha <1\), then system (1.9) admits at least one positive solution.
Proof
It follows from the proof of Theorem 3.2 that Corollary 3.1 holds. □
Corollary 3.2
Assume that (\(H_{1}\))–(\(H_{10}\)) hold. Suppose that \(\alpha >1\) and there exist two constants \(d_{1}\) and r satisfying \(0< d_{1}< r=\min \{(4A)^{-1},(4\varepsilon_{3}A')^{-1},r',r'' \}\) such that
where A, \(A'\), \(\varepsilon_{3}\), \(r'\) and \(r''\) are defined in Theorem 3.1. Then system (1.9) admits at least two positive solutions.
Proof
Similarly to the proof of Theorem 3.1, we can obtain that (3.4) and (3.11) hold. Then, similarly to the proof (3.22), we get
This finishes the proof of Corollary 3.2. □
Finally, in the case \(0<\alpha <1\), we consider the existence of three positive solutions for system (1.9).
Theorem 3.3
Assume that (\(H_{1}\))–(\(H_{10}\)) hold and there exist four positive numbers η, \(\eta_{1}\), \(\eta_{2}\), and γ such that one of the following conditions is satisfied:
- (\(H_{11}\)):
-
\(0<\alpha <1\), \(0<\eta =\max \{\eta_{1}, \eta_{2}\}< \min \{r,(4A)^{-1},(4A'\varGamma)^{-1}\}\leq r\leq \max \{6M,6M^{*},r\} <R<(6A)^{-1}<\gamma \), and
$$\begin{aligned}& f(x)< (D)^{-1}\eta, \qquad I_{k}(x)< (\varLambda)^{-1} \eta, \\& J_{k}(y)< \varGamma \eta, \quad \forall x\in [\theta_{3} \eta_{1}, \eta_{1}], y \in [\theta_{3} \eta_{2}, \eta_{2}], \\& f(x)>\bigl(D^{*}\bigr)^{-1}\gamma, \qquad I_{k}(x)>\bigl(\varLambda^{*}\bigr)^{-1} \gamma, \\& J _{k}(y)>\varGamma^{*} \gamma,\quad \forall x, y\in [0, \gamma ], \end{aligned}$$
where r, R, A, M, \(M^{*}\), D, \(D^{*}\), γ, \(\gamma^{*}\), \(\theta_{3}\), Λ, and \(\varLambda^{*}\) are defined in Theorems 3.1 and 3.2, respectively. Then system (1.9) admits at least three positive solutions.
Proof
Since \(0<\alpha <1\), from the proof of Theorem 3.2 we know that
By the first part of (\(H_{11}\)), for any \((x,y)\in (K_{1}\times K_{2}) \cap \partial \varOmega_{\eta }\), we obtain
and similarly to the proof of (3.22), we get
Considering the second part of (\(H_{11}\)), for any \((x,y)\in (K_{1} \times K_{2})\cap \partial \varOmega_{\gamma }\), we have
This shows that
Therefore, applying Lemma 2.4 to (3.24), (3.25), (3.26), and (3.29) respectively, we can show that T has at least three fixed points \((x_{i},y_{i})\) (\(i=1,2,3\)) satisfying
This gives the proof of Theorem 3.3. □
4 An example
Example 4.1
Let \(n=1\) and \(t_{1}=\frac{1}{10}\). Consider the following system:
where \(I_{1}(x)=\frac{x^{3}}{2}\), \(J_{1}(y)=\frac{y^{3}}{4}\), \(h(t)\equiv 1\), \(g(t)=t\), and
Conclusion 4.1
System (4.1) admits at least one positive solution.
For convenience, we give a corollary of Proposition 2.3 in [67].
Corollary 4.1
Consider the following system:
where \(\alpha >0\), and \(k(t)\) satisfies the changing-sign condition
and
If there exists \(0<\sigma <\xi \) such that
then the following inequalities hold:
Proof
Similarly to the proof of Proposition 2.3 in [67], we can prove that
Hence it follows from (2.4) that
Next, letting \(s=\xi -\frac{\xi -\sigma }{1-\xi }\eta \), \(\eta \in [0,1- \xi ]\), we get
letting \(s=\xi +\eta \), \(\eta \in [0,1-\xi ]\), we have
Now, from assumption (4.4), for all \((t,\eta)\in [0,1]\times [0,1- \xi ]\), we have
Finally, by integrating in η both sides of (4.7) from 0 to \(1-\xi \) it follows that inequality (4.5) holds. □
Similarly, we can show that inequality (4.6) holds.
Proof of Example 4.1
From the definitions of \(a(t)\), \(b(t)\), and \(g(t)\) we know that \(\xi =\frac{1}{3}\).
Step 1. We show that (\(H_{6}\)) holds. For fixed \(c_{1}=c_{2}=1\), \(\sigma_{1}=\frac{1}{6}\), \(\mu =1\), and \(\alpha =1\), (4.4) is equivalent to the inequality
Letting \(\frac{1}{3}-\frac{1}{4}\tau =\varrho \), this inequality is equivalent to
By the definition of \(b(t)\) the last inequality holds obviously. It is clear that by (4.5) (\(H_{6}\)) is reasonable.
Step 2. We show (\(H_{7}\)) holds. Similarly to Step 1, letting \(c_{1}=1\), \(c_{2}=\frac{36}{5}\), \(\sigma_{2}=\frac{1}{6}\), \(\mu =1\), and \(\alpha =1\), by (4.6) we get
It is easy to see by calculating that
Furthermore, from inequality (4.8) it follows that
So, (\(H_{7}\)) holds.
Step 3. Similarly to Step 1, letting \(c_{1}=c_{2}=1\), \(\sigma_{3}=\frac{1}{6}\), \(\mu =1\), and \(\alpha =3\), we get that (\(H_{10}\)) holds.
Hence it follows from Theorem 3.1 that system (4.1) admits at least one positive solution for \(\alpha >1\). □
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Jiao, L., Zhang, X. A class of second-order nonlocal indefinite impulsive differential systems. Bound Value Probl 2018, 163 (2018). https://doi.org/10.1186/s13661-018-1082-z
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DOI: https://doi.org/10.1186/s13661-018-1082-z