Abstract
New results as regards the existence of positive solutions for first order impulsive differential equations are provided. The method of proof relies on the fixed point theorem and degree theory. Some examples are presented to illustrate the main results.
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1 Introduction
In this paper, we study the existence of positive solutions for the following boundary value problem with impulsive effects:
where \(J=[0,T]\), \(0=t_{0}< t_{1}< t_{2}<\cdots<t_{p}<1=t_{p+1}\), \(\Delta x(t_{k})=x(t_{k}^{+})-x(t_{k}^{-})\), \(x(t_{k}^{+})\), and \(x(t_{k}^{-})\) represent the right limit and left limits of \(u(t)\) at \(t_{k}\), respectively.
Impulsive differential equations serve as basic models to study the dynamics of processes that are subject to sudden changes in their states and its theory has developed fast during the past few years. There has been increasing interest in the investigation for boundary value problems of nonlinear impulsive differential equations and much literature has been published about the existence of solutions for impulsive differential equations, see [1–3] and the references therein. There are some common techniques to approach those problems: the fixed point theorems [4–8], the method of upper and lower solutions [9–12], the topological degree theory [13, 14], the variational method [15–17] and so on. Recently, using the fixed point theorem, Zhang et al. [8] obtained the existence of a positive solution of (1.1), where they required that the function a is of definite sign. In this paper, we continue to discuss (1.1). By using the fixed point theorem in a cone different from the one in [8] and degree theory, we obtain some new conditions which guarantee the existence of single and multiple positive solutions for (1.1). Our results are different from the results in [8] and are new even if \(I_{k}\equiv0\).
2 Main results
Let \(J^{*}=J\backslash\{t_{1},t_{2},\ldots,t_{p}\}\), \(PC(J)=\{u:J\to R| u\in C(J^{*}), u(t_{i}^{+}), u(t_{i}^{-})\mbox{ exist and }u(t_{i}^{-})=u(t_{i}), i=1,2,\ldots,p\}\). \(PC(J)\) is a Banach space with the norm \(\|u\|=\sup\{|u(t)|: t\in J\}\).
Lemma 2.1
[8]
The function \(x\in PC(J)\) is a solution of (1.1) if and only if x is a solution of the following impulsive integral equation:
where
Set
The Green’s function G is of definite sign if \(a\in \Lambda^{+}\cup\Lambda^{-}\). Moreover, \(M>m>0\) if \(a\in\Lambda^{+}\); \(m< M<0\) if \(a\in\Lambda^{-}\).
Define the operator A and cone K on \(PC(J)\) by
where \(\delta=\frac{m}{M} \) if \(a\in\Lambda^{+}\); \(\delta=\frac{M}{m} \) if \(a\in\Lambda^{-}\).
Lemma 2.2
[18]
Let X be a Banach space and K be a cone in X. Suppose \(\Omega_{1}\) and \(\Omega_{2}\) are open subsets of X such that \(0 \in\Omega_{1} \subset\bar{\Omega}_{1} \subset\Omega_{2}\) and suppose that
is a completely continuous operator such that:
-
(i)
\(\inf\| \Phi u\|>0\), \(u\neq\mu\Phi u\) for \(u \in K\cap\partial\Omega_{1}\) and \(\mu\geq1\), and \(u\neq\mu\Phi u\) for \(u \in K\cap\partial\Omega_{2}\) and \(0<\mu \leq1\), or
-
(ii)
\(\inf\| \Phi u\|>0\), \(u\neq\mu\Phi u\) for \(u \in K\cap\partial\Omega_{2}\) and \(\mu\geq1\), and \(u\neq\mu\Phi u\) for \(u \in K\cap\partial\Omega_{1}\) and \(0<\mu \leq 1\).
Then Φ has a fixed point in \(K \cap(\bar{\Omega}_{2} \backslash \Omega_{1})\).
Set
The following theorems are the main results of this paper.
Theorem 2.1
Assume that \(a\in\Lambda^{+}\) and there exist two positive constants \(r< R\) such that
Then (1.1) has at least one solution x with \(r\leq \|x\|\leq R\) if one of the following conditions is satisfied:
- (H1):
-
\(\varphi(r)<\int_{0}^{T}a(s)\,ds\) and \(\psi(R)>\int_{0}^{T}a(s)\,ds\);
- (H2):
-
\(\varphi(R)<\int_{0}^{T}a(s)\,ds\) and \(\psi(r)>\int_{0}^{T}a(s)\,ds\).
Proof
Here, we only prove the case in which (H1) is satisfied. Let \(\Omega_{R}=\{x\in K: \|x\|< R\}\), \(\Omega_{r}=\{x\in K: \|x\|< r\}\). At first, we show that \(A: \bar{\Omega}_{R}\setminus\Omega_{r}\rightarrow K\). For any \(x\in \bar{\Omega}_{R}\setminus\Omega_{r}\), \(\delta r\leq x(t)\leq R\), \(t\in J\). From \(0< m\leq G(t,s)\leq M\) and (2.1), we obtain, for \(x\in \bar{\Omega}_{R}\setminus\Omega_{r}\),
Hence, \(A: \bar{\Omega}_{R}\setminus\Omega_{r}\rightarrow K\). In addition, one easily checks that A is completely continuous.
Next, we show that:
-
(a)
\(x\neq\mu Ax\) for \(x \in K\cap\partial\Omega_{r}\) and \(0<\mu\leq1\),
-
(b)
\(\inf\| Ax\|>0\), \(x\neq\mu A u\) for \(x \in K\cap\partial\Omega_{R}\) and \(\mu\geq1\).
If (a) is not true, there exist \(x \in K\cap\partial\Omega_{r}\) and \(0<\mu\leq1\) with \(x=\mu Ax\). Hence,
Since \(x(t)\geq\delta r>0\), we rewrite (2.2) as
Integrating the first equality in (2.3) from 0 to T, we obtain
which is a contradiction.
Suppose that \(\inf_{x\in K\cap\partial\Omega_{R}}\|Ax\|=0\). There exists the sequence \(x_{n}\in K\cap\partial\Omega_{R}\) such that \(\|Ax_{n}\|\rightarrow0\) as \(n \to\infty\). Noting \(f(t,x_{n}(t))\geq0\), \(I_{k}(x_{n}(t_{k}))\geq0\) and
we obtain \(f(s,x_{n}(s))\rightarrow0\), \(I_{k}(x_{n}(t_{k}))\rightarrow 0\) as \(n\rightarrow\infty\). Hence,
which implies that \(\psi(R)=0\), a contradiction.
Suppose that there exist \(u \in K\cap \partial\Omega_{R}\) and \(\mu\geq1\) with \(u=\mu Au\). Then
Integrating the first equality in (2.4) from 0 to T, we obtain
which is a contradiction.
By Lemma 2.2, there exists \(x\in\bar{\Omega}_{R} \backslash\Omega_{r}\) with \(Ax=x\), which is the positive solution of (1.1). The proof is complete. □
Corollary 2.1
Assume that \(a\in\Lambda^{+}\) and \(f\in C(J\times[0 ,+\infty],[0,+\infty))\), \(I_{k}\in C([0,+\infty), [0,+\infty))\) (\(1\leq k\leq p\)). Then (1.1) has at least one positive solution if one of the following conditions is satisfied:
-
(1)
\(\varphi_{0}<\int_{0}^{T}a(s)\,ds\) and \(\psi_{\infty}>\int_{0}^{T}a(s)\,ds\);
-
(2)
\(\varphi_{\infty}<\int_{0}^{T}a(s)\,ds\) and \(\psi_{0}>\int_{0}^{T}a(s)\,ds\).
Theorem 2.2
Assume that \(a\in\Lambda^{+}\) and there exist \(N+1\) positive constants \(p_{1}< p_{2}<\cdots<p_{N}<p_{N+1}\) such that
Further suppose that one of the following conditions is satisfied:
-
(1)
\(\varphi(p_{2k-1}) <\int_{0}^{T}a(s)\,ds\), \(k=1,2,\ldots,[(N+2)/2]\), \(\psi(p_{2k})> \int_{0}^{T}a(s)\,ds\), \(k=1,2,\ldots,[(N+1)/2]\), or
-
(2)
\(\psi(p_{2k-1}) >\int_{0}^{T}a(s)\,ds\), \(k=1,2,\ldots,[(N+2)/2]\), \(\varphi(p_{2k})<\int_{0}^{T}a(s)\,ds\), \(k=1,2,\ldots,[(N+1)/2]\),
where \([d]\) denotes the integer part of d. Then (1.1) has at least N positive solutions \(x_{k}\in X\), \(k=1,2,\ldots, N\) with \(p_{k}< \| x_{k}\|< p_{k+1}\).
Proof
Assume that (1) holds. The case in which (2) holds is similar. Since φ, ψ are continuous functions, for any \(1\leq j\leq N\), there exist \(r_{j}\), \(R_{j}\) such that \(p_{j}< r_{j}< R_{j}< p_{j+1}\) and
By Theorem 2.1, (1.1) has at least one positive solution \(x_{j}\) with \(r_{j}\leq\|x_{j}\|\leq R_{j}\). This ends the proof. □
Theorem 2.3
Assume that \(a\in\Lambda^{-}\) and there exist two positive constants \(r< R\) such that
Further suppose that (H1) or (H2) is satisfied, where \(\ln \alpha=-\infty\) if \(\alpha\leq0\). Then (1.1) has at least one solution x with \(r\leq\|x\|\leq R\).
The proof of Theorem 2.3 is similar to that of Theorem 2.1 and we omit it.
Theorem 2.4
Assume that \(a\in\Lambda^{+}\) and the following conditions are satisfied:
- (D1):
-
There exist constants \(0<\alpha<\beta\) such that \(f(t,u)\), \(I_{k}(u)\) are nondecreasing in \(u\in[\alpha,\beta]\) and
$$\int_{0}^{T}f(t,\alpha)\,dt+\sum _{k=1}^{p}I_{k}(\alpha)>\frac{\alpha}{m}, \qquad \int_{0}^{T}f(t,\beta)\,dt+\sum _{k=1}^{p}I_{k}(\beta)< \frac{\beta}{M}. $$ - (D2):
-
There exists \(\gamma>\beta\) such that
$$\begin{aligned}& f\in C\bigl(J\times[0,\gamma],[0,+\infty)\bigr), \qquad I_{k}\in C \bigl([0,\gamma ],[0,+\infty)\bigr), \\& \int_{0}^{T}\sup_{0\leq u\leq\gamma}f(t,u)\,dt+ \sum_{k=1}^{p}\sup_{0\leq u\leq\gamma}I_{k}(u) \leq\frac{\gamma}{M}. \end{aligned}$$ - (D3):
-
$$\lim_{x\rightarrow0^{+}} \frac{f(t,x)}{x}=0,\qquad \lim _{x\rightarrow0^{+} }\frac{I_{k}(u)}{u}=0. $$
Then (1.1) has at least two positive solutions in \(\overline{U}_{\gamma}=\{x\in PC(J):\|x\|\leq\gamma\}\).
Proof
Set
where I denotes the identity map. It is easy to check that \(T_{\lambda}:[0,1]\times PC(J)\rightarrow PC(J)\) is compact. We show:
-
(1)
\(T_{1}(\overline{V})\subset V\).
-
(2)
\(\Phi_{\lambda}(u)=0\) implies that \(\|u\|\leq\gamma\).
-
(3)
There exists \(c\in(0,\alpha)\) such that \(\Phi_{\lambda}(u)=0\) admits only the trivial solution in \(\overline{U}_{c}\).
The proof of (1). From (D1), we obtain, for \(\forall u\in V\),
Thus, for all \(u\in V\),
That is, \(T_{1}(\overline{V})\subset V\).
The proof of (2). If \(\varphi_{\lambda}(u)=0\), then
The proof of (3). From (D3), there exists \(0< c<\alpha\) such that
If \(u\in\overline{U}_{c}\) with \(u=T_{\lambda}u\), we have
which implies that \(u\equiv0\).
Let \(U_{c}^{\gamma}=U_{\gamma}/\overline{U}_{c}\) and the degree of \(\Phi_{\lambda}\) at 0 relative to open set D be \(d(\Phi_{\lambda},D,0)\). Since V̅ is a closed, convex set in \(PC(J)\) and \(T_{1}(\partial V)\subset V\), by Schauder’s fixed point theorem, there exists \(u_{1}\in V\) such that \(T_{1}(u_{1})=u_{1}\) and
For \(\lambda\in[0,1]\), \(u\in\partial U_{c}^{\gamma}\), \(\Phi_{\lambda}(u)\neq0\), by the homotopy property of the degree, we obtain
By the additivity property of the degree, one obtains
Hence,
which implies that \(T_{1}\) has at least a fixed point \(u_{2}\in U_{c}^{\gamma}/\overline{V}\). Clearly, \(c\leq\|u_{2}\|\leq\gamma\). In addition,
If there is \(t^{*}\in J\) such that \(u_{2}(t^{*})=0\), by the above inequality, we obtain \(f(s,u_{2}(s))\equiv 0\), \(I_{k}(u_{2}(t_{k}))\equiv0\), which implies that \(u_{2}(t)\equiv0\), a contradiction. Hence, (1.1) has at least two positive solutions in \(\overline{U}_{\gamma}\). The proof is complete. □
Example 2.1
Consider the differential equation with singularity
where κ, a, λ are positive constants, \(c_{k}>0\) and \(h\in C(J,R)\).
Set \(h_{*}=\min\{h(t):t\in J\}\) and \(h^{*}=\max\{h(t):t\in J\}\). We claim that (2.6) has at least one positive solution provided that the following conditions hold:
In fact, \(f(t,u)=au^{-\lambda}+h(t)\), \(\psi_{0}=+\infty\). Moreover, \(\varphi_{\infty}=\gamma_{1}\) if \(h_{*}\geq0\). If \(h_{*}<0\), choose \(b=\sqrt[\lambda]{a/(-h_{*})}\), \(0\leq f(t,u)\leq au^{-\lambda}+h^{*}\leq\gamma_{2}u\) for \((t,u)\in J\times[\delta b,b]\) and \(\varphi(b)=\gamma_{2}T+\gamma_{1}<\kappa T\). By Theorem 2.1, (2.6) has one positive solution.
Remark 2.1
The conditions (2.7) and (2.8) are different from those of Corollary 4.7 in [8].
Example 2.2
Consider the differential equation
Let \(f(t)=t^{3}e^{-0.1t}\), \(I_{1}(t)=10t^{2}/(1+t^{2})\), \(\alpha=0.1\), \(\beta=30\), \(\gamma=10^{4}\). It is easy to check that the conditions (D1)-(D3) hold. Hence, (2.9) has at least two positive solutions.
3 Application
In this section, we consider the differential equation
where \(\lambda\geq0\).
Set \(y=\exp(x^{\lambda+1}(t)/(\lambda+1))\) and \(F(u)=u^{\lambda}f(u)\), then
If (3.2) has a solution \(y(t)>1\), \(t\in J\), then \(x(t)=((\lambda+1)\ln y(t))^{\frac{1}{\lambda+1}}\) is the positive solution of (3.1). By Theorem 2.1, we have the following result.
Theorem 3.1
Assume that \(a\in\Lambda^{+}\), \(f\in C((0,+\infty),[0,+\infty))\), \(c_{k}>0\). If there exist constants \(\delta^{-1}< r< R\) such that
or
then (3.1) has at least one positive solution.
Example 3.1
Consider the differential equation
where \(\lambda\geq0\), \(a\in\Lambda^{+}\), \(f\in C((0 ,+\infty),[0,+\infty))\), μ is a positive real parameter.
Corollary 3.1
There exists \(\mu^{*}>0\) such that (3.3) has at least a positive solution for any \(\mu\in(0,\mu^{*})\).
Proof
Choosing \(r=\delta^{-1}+1\). Since \(u^{\lambda}f(u)\) is continuous in \((0,+\infty)\) and \(\ln u>0\) for \(u\in[\delta r,r]\), we obtain
for some \(C>0\). On the other hand, since \([(1+\mu)^{\lambda+1}-1]/\mu\rightarrow1+\lambda\) as \(\mu\rightarrow0^{+}\), there is \(\sigma>0\) such that
Set \(\mu^{*}=\min \{\sigma, \int_{0}^{T}a(s)\,ds/[CT+2p(\lambda+1)\ln r] \}\), for \(\mu<\mu^{*}\), we have
Taking
one easily checks that
By Theorem 3.1, (3.3) has one positive solution. □
Remark 3.1
Corollary 3.1 admits the case that the function a changes sign. As far as we know, no paper discussed (3.3) when a changes sign.
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Acknowledgements
The authors wish to express their thanks to the referee for his/her very valuable suggestions and careful corrections. The work is supported by the NNSF of China (11171085), Hunan Provincial Natural Science Foundation of China (2015JJ2068).
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Wang, W., Guo, L. New existence results for periodic boundary value problems with impulsive effects. Adv Differ Equ 2015, 275 (2015). https://doi.org/10.1186/s13662-015-0601-9
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DOI: https://doi.org/10.1186/s13662-015-0601-9