Abstract
Using a fixed point theorem of cone expansion and compression of norm type and a new method to deal with the impulsive term, we prove that the second-order singular impulsive Neumann boundary value problem has denumerably many positive solutions. Noticing that \(M>0\), our main results improve many previous results.
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1 Introduction
We are concerned with the existence of denumerably many positive solutions of the second-order singular impulsive Neumann boundary value problem
where M is a positive constant, \(J=[0,1]\), \(t_{k} \in\mathrm{R}\), \(k =1,\ldots,m\), \(m \in\mathrm{N,}\) satisfy \(0=t_{0}< t_{1}<t_{2}<\cdots <t_{m}<t_{m+1}=1\), \(-\Delta x'|_{t=t_{k}}\) denotes the jump of \(x'(t)\) at \(t=t_{k}\), that is, \(-\Delta x'|_{t=t_{k}}=x'((t_{k})^{+})-x'((t_{k})^{-})\), here \(x'((t_{k})^{+})\) and \(x'((t_{k})^{-})\), respectively, represent the right-hand limit and left-hand limit of \(x'(t)\) at \(t=t_{k}\).
In addition, ω, f and \(I_{k}\) satisfy the following conditions:
- \((H_{1})\) :
-
\(\omega(t)\in L^{p}[0,1]\) for some \(p\in[1,+\infty)\), and there exists \(N>0\) such that \(\omega(t)\geq N\) a.e. on J;
- \((H_{2})\) :
-
\(f\in C(J\times\mathrm{R^{+}}, \mathrm{R^{+}})\), \(I_{k}\in C( \mathrm{R^{+}}, \mathrm{R^{+}})\), where \(\mathrm {R^{+}}=[0,+\infty)\);
- \((H_{3})\) :
-
there exists a sequence \(\{t_{i}'\}_{i=1}^{\infty} \) such that \(t_{1}'<\delta\), where \(\delta=\min\{t_{1},\frac{1}{2}\} \), \(t_{i}' \downarrow t^{*}\geq0 \) and \(\lim_{t\rightarrow t_{i}'} \omega(t) =+\infty\) for all \(i=1, 2,\dots\).
For the case \(M=0\) and \(I_{k}=0\) (\(k=1,2,\ldots,m\)), problem (1.1) reduces to the problem studied by Kaufmann and Kosmatov in [1]. By using Krasnosel’skiĭ’s fixed point theorem and Hölder’s inequality, the authors showed the existence of countably many positive solutions. The other related results can be found in [2–13]. However, there are almost no papers considering second-order impulsive Neumann boundary value problem with infinitely many singularities. To identify a few, we refer the reader to [14–27] and the references therein.
The main reason is that \(M\neq0\) in problem (1.1), which shows that the solution of problem (1.1) has no concave properties. On the other hand, under the case \(M\neq0\) and \(\omega(t)\) with infinitely many singularities, the properties of the corresponding Green’s function for problem (1.1) are more complicated.
Our plan of the paper is as follows: in Section 2, we collect some well-known results to be used in the subsequent sections. In particular, we also present some new properties of Green’s function under the case \(M\neq0\) and \(\omega(t)\) with infinitely many singularities. In Section 3, we obtain some new sufficient conditions for the existence of denumerably many positive solutions for problem (1.1). In Section 4, we give an example of a family functions \(\omega(t)\) such that \((H_{3})\) holds.
2 Preliminaries
In this installment, we list some definitions and lemmas which are needed throughout this paper.
Let \(J'=J\setminus \{t_{1},t_{2},\ldots,t_{m} \}\) and \(E=C[0,1]\). We define \(\mathit {PC}^{1}[0,1]\) in E by
Then \(\mathit {PC}^{1}[0,1]\) is a real Banach space with norm
where \(\Vert x \Vert _{\infty}=\sup_{t\in J} \vert x(t) \vert \), \(\Vert x' \Vert _{\infty}=\sup_{t\in J} \vert x'(t) \vert \).
Suppose that \(G(t,s)\) is the Green’s function of the boundary value problem
then
Lemma 2.1
By the definition of \(G(t,s)\) and the properties of \(sinhx\) and \(coshx\), we have the following results.
-
(a)
For any \(t, s\in J\), there is
$$ A=\frac{1}{\gamma\sinh\gamma}\leq G(t,s)\leq\frac{\cosh\gamma}{\gamma \sinh\gamma}=B. $$(2.4)Then it follows from (2.4) that
$$A\leq G(t,s)\leq G(s,s)\leq B. $$ -
(b)
For any \(\tau\in(0,\delta)\),
$$ \frac{D'_{k}}{\gamma\sinh\gamma}\leq G(t,s)\leq\frac{\cosh\gamma(1-\tau )\cosh\gamma\tau'_{k}}{\gamma\sinh\gamma}, \quad \forall t\in \bigl[ \tau,\tau '_{k} \bigr], s\in J, $$(2.5)where
$$\tau'_{k}=\max\{1-\tau, 1-t_{k}\},\quad\quad D'_{k}=\max \bigl\{ \cosh\gamma\tau, \cosh\gamma \bigl(1- \tau'_{k} \bigr) \bigr\} , \quad k=1,2,3,\ldots, m. $$ -
(c)
$$ G_{t}'(t,s)= \frac{1}{\sinh\gamma} \textstyle\begin{cases} -\sinh\gamma(1-t)\cosh\gamma s,& 0\leq s\leq t\leq1,\\ \sinh\gamma(1-s)\cosh\gamma t,& 0\leq t\leq s\leq1, \end{cases} $$(2.6)
and
$$ \max_{t,s\in J,t\neq s} \bigl\vert G_{t}'(t,s) \bigr\vert \leq \sinh\gamma. $$(2.7)
Proof
We can get equations (2.4)-(2.7) by the definition of \(G(t,s)\), so we omit it here. □
To establish the existence of positive solutions to problem (1.1), for a fixed \(\tau\in(0,\delta)\), we construct the cone \(K_{\tau}\) in \(\mathit {PC}^{1}[0,1]\) by
where
It is easy to see \(K_{\tau}\) is a closed convex cone of \(\mathit {PC}^{1}[0,1]\).
Let \(\{\tau_{i}\}_{i=1}^{\infty}\) be such that \(t_{i+1}'<\tau _{i}<t_{i}'\), \(i=1,2,\dots\). Then for any \(i\in\mathrm{N}\), we define the cone \(K_{\tau_{i}}\) by
where
It is easy to see \(K_{\tau_{i}}\) is a closed convex cone of \(\mathit {PC}^{1}[0,1]\).
Remark 2.1
For any \(i=1,2,\ldots\) , \(k=1,2,\ldots,m\), it follows from the definition of \(\sigma_{k}\) and \(\sigma_{ik}\) that \(0<\sigma _{k},\sigma_{ik} <1\).
Lemma 2.2
If \((H_{1})\)-\((H_{3})\) hold, then problem (1.1) has a unique solution x given by
Proof
The proof is similar to that of Lemma 2.4 in [26]. □
Definition 2.1
A function \(x(t)\) is said to be a solution of problem (1.1) on J if:
-
(i)
\(x(t)\) is absolutely continuous on each interval \((0,t_{1}]\) and \((t_{k},t_{k+1}]\), \(k =1,2,\ldots,n\);
-
(ii)
for any \(k =1,2,\ldots,m\), \(x(t_{k}^{ +})\), \(x(t_{k}^{-})\) exist;
-
(iii)
\(x(t)\) satisfies (1).
Define an operator \(T: K_{\tau} \to \mathit {PC}^{1}[0,1]\) by
From (2.14), we know that \(x(t)\in \mathit {PC}^{1}[0,1]\) is a solution of problem (1.1) if and only if x is a fixed point of the operator T. Also, for a positive number r, define \(\Omega_{r}\) by
Note that \(\partial\Omega_{r}= \{x\in \mathit {PC}^{1}[0,1]: \Vert x \Vert _{\mathit {PC}^{1}}=r \}\) and \(\bar{\Omega}_{r}= \{x\in \mathit {PC}^{1}[0,1]: \Vert x \Vert _{\mathit {PC}^{1}}\leq r \}\).
Definition 2.2
An operator is called completely continuous if it is continuous and maps bounded sets into pre-compact sets.
Lemma 2.3
Assume that \((H_{1})\)-\((H_{3})\) hold. Then \(T(K_{\tau })\subset K_{\tau} \) and \(T: K_{\tau} \to K_{\tau}\) is a completely continuous.
Proof
For \(t\in J\), \(x\in K_{\tau}\), it follows from ((2.5)) and (2.14) that
On the other hand, it follows from (2.6), (2.7) and (2.14) that
For any \(t\in J\), combined with (2.15) and (2.16), we have
Then, by (2.5), (2.8) and (2.17), we have
Evidently, \(T(K_{\tau})\subset K_{\tau}\).
Next, we prove that the operator \(T: K_{\tau}\to K_{\tau}\) is a completely continuous.
It is obvious that T is continuous.
Let \(B_{d}=\{x\in \mathit {PC}^{1}[0,1] \mid \Vert x \Vert _{\mathit {PC}^{1}}\le d\}\) be bounded set. Then, for all \(x\in B_{d}\), by the definition of \(\Vert Tx \Vert _{\infty}\), \(\Vert Tx' \Vert _{\infty}\), \(\Vert Tx \Vert _{\mathit {PC}^{1}}\), we have
and
where
Therefore \(T(B_{d})\) is uniformly bounded.
On the other hand, for all \(t_{1}, t_{2}\in J_{k}\) with \(t_{1}< t_{2}\), we have
Noting (2.7), we know that \(G'(t,s)\) is a constant and
which shows that \(T(B_{d})\) is equicontinuous. The Arzelà-Ascoli theorem implies that T is completely continuous, and the lemma is proved. □
Lemma 2.4
Hölder
Let \(e\in L^{p}[a,b]\) with \(p>1\), \(h\in L^{q}[a,b]\) with \(q>1\) and \(\frac{1}{p}+\frac{1}{q}=1\). Then \(eh\in L^{1}[a,b]\) and
Let \(e\in L^{1}[a,b]\), \(h\in L^{\infty}[a,b]\). Then \(eh\in L^{1}[a,b]\) and
Lemma 2.5
See [28]; fixed point theorem of cone expansion and compression of norm type
Let E be a Banach space, P be a cone in E. Assume that \(\Omega _{1}\), \(\Omega_{2}\) are bounded open subsets in E with \(\theta \in\Omega_{1}\) and \(\bar{\Omega}_{1}\subset\Omega_{2}\), where θ denotes zero operator. Suppose \(A : P \cap(\bar{\Omega}_{2} \setminus \Omega_{1})\rightarrow P \) is completely continuous such that either
-
(i)
\(\Vert Ax \Vert \leq \Vert x \Vert \), \(\forall x\in P\cap\partial\Omega_{1}\); \(\Vert Ax \Vert \geq \Vert x \Vert \), \(\forall x\in P\cap\partial\Omega_{2} \);
-
(ii)
\(\Vert Ax \Vert \leq \Vert x \Vert \), \(\forall x\in P\cap\partial\Omega_{2}\); \(\Vert Ax \Vert \geq \Vert x \Vert \), \(\forall x\in P\cap\partial\Omega_{1} \).
Then A has a fixed point in \(P \cap(\bar{\Omega}_{2} \setminus \Omega_{1})\).
3 Main results
In this section, using Lemmas 2.1-2.5, we give our main results in the case \(\omega\in L^{P}[0,1]\); \(p>1\), \(p=1\) and \(p=\infty\).
For convenience, we write
Firstly, we consider the case \(p>1\).
Theorem 3.1
Assume that \((H_{1})\)-\((H_{3})\) hold. Let \(\{r_{i}\} _{i=1}^{\infty}\) and \(\{R_{i}\}_{i=1}^{\infty}\) be such that
where
For each natural number i, we assume that f and \(I_{k}\) satisfy:
- (\(H_{4}\)):
-
For any \(t\in J\), \(x\in[0,R_{i}]\), \(f(t,x)\leq M_{0}R_{i}\), and for any \(x\in[0,R_{i}]\), \(k\in\{1,2,\dots,m\}\), \(I_{k}(x(t_{k}))\leq M_{0}R_{i}\), where
$$0< M_{0}\leq\frac{\rho_{0}}{D+mB}. $$ - (\(H_{5}\)):
-
For any \(t\in J\), \(x\in[\sigma_{ik}r_{i},r_{i}]\), \(f(t,x)\geq L_{0}r_{i}\), and for any \(x\in[\sigma_{ik}r_{i},r_{i}]\), \(k\in\{1,2,\dots,m\}\), \(I_{k}(x)\geq L_{0}r_{i}\).
Then problem (1.1) has denumerably many positive solutions \(\{x_{i}(t)\}_{i=1}^{\infty}\) such that
Proof
We consider the following open subset sequences \(\{\Omega_{1,i}\} _{i=1}^{\infty}\) and \(\{\Omega_{2,i}\}_{i=1}^{\infty}\) of \(\mathit {PC}^{1}[0,1]\):
Let \(\{\tau_{i}\}_{i=1}^{\infty}\) be as in the hypothesis and note that \(0< t_{i+1}'<\tau_{i}<t_{i}'<\delta\), \(i=1,2,\dots\).
For fixed i, we assume that \(x\in K_{\tau_{i}}\cap\partial\Omega _{2,i}\), then for any \(t\in J\)
Noticing (2.5) and (2.14), for all \(x\in K_{\tau_{i}}\cap\partial\Omega _{2,i}\), by \((H_{1})\) and \((H_{5})\), we have
which shows that
On the other hand, for all \(t\in J\), \(x\in P_{i}\cap\partial\Omega _{1,i}\), we have \(x(t)\leq \Vert x \Vert _{\mathit {PC}^{1}}=R_{i}\).
Noticing (2.4) and (2.14), for all \(t\in J\), \(x\in K_{\tau_{i}}\cap \partial\Omega_{1,i}\), by \((H_{4})\), we have
Moreover, by (2.6), (2.16) and \((H_{4})\), we have
Applying Lemma 2.5 to (3.1) and (3.4) shows that the operator T has a fixed point \(x_{i}\in K_{\tau_{i}}\cap(\bar{\Omega}_{2,i}/ \Omega_{1,i})\) such that \(r_{i}\leq \Vert x_{i} \Vert \leq R_{i} \). Since \(i\in\mathrm{N}\) was arbitrary, the proof is complete. □
The following results deal with the case \(p=\infty\).
Theorem 3.2
Assume that \((H_{1})\)-\((H_{3})\) hold. Let \(\{a_{i}\} _{i=1}^{\infty}\) and \(\{b_{i}\}_{i=1}^{\infty}\) be such that
For each natural number i, we assume that f and \(I_{k}\) satisfy (\(H_{4}\)) and (\(H_{5}\)), then problem (1.1) has denumerably many positive solutions \(\{x_{i}(t)\}_{i=1}^{\infty}\) such that
Proof
Let \(\Vert G \Vert _{1} \Vert \omega \Vert _{\infty}\) replace \(\Vert G \Vert _{q} \Vert \omega \Vert _{p}\) and repeat the previous argument. □
Finally, we consider the case of \(p=1\).
Theorem 3.3
Assume that \((H_{1})\)-\((H_{3})\) hold. Let \(\{a_{i}\} _{i=1}^{\infty}\) and \(\{b_{i}\}_{i=1}^{\infty}\) be such that
For each natural number i, we assume that f and \(I_{k}\) satisfy (\(H_{4}\)) and (\(H_{5}\)), then the problem (1.1) has denumerably many positive solutions \(\{x_{i}(t)\}_{i=1}^{\infty}\) such that
Proof
Similar to the proof of (3.2) and (3.3), for all \(t\in[\tau _{i},\delta-\tau_{i}]\), \(x\in K_{\tau_{i}}\cap\partial\Omega_{1,i}\), then \(x(t)\leq \Vert x \Vert _{\mathit {PC}^{1}}=R_{i}\).
Since (2.4) and (2.14), for all \(x\in K_{\tau_{i}}\cap\partial\Omega _{1,i}\), by \((H_{4})\), we have
and by (2.4), (2.7), (2.16) and \((H_{4})\),
Similarly to the proof of Theorem 3.1, we can finish the proof of Theorem 3.3. □
4 An example
From Section 3, it is not difficult to see that \((H_{3})\) plays an important role in the proof that problem (1.1) has denumerably many positive solutions. As an example, we consider a family of functions \(\omega(t)\) as follows.
Example 4.1
Let \(k=m=1\), \(t_{1}=\frac{1}{3}\), and
It is easy to see that
and
where \(\sum_{i=1}^{\infty}\frac{1}{(i+1)(i+2)(i+3)(i+4)}=\frac{1}{72}\).
Let
where
From \(\sum_{i=1}^{\infty}\frac{1}{n^{4}}=\frac{\pi^{4}}{90}\) and \(\sum_{i=1}^{\infty}\frac{1}{n^{2}}=\frac{\pi^{4}}{6}\), we have
Thus, it is easy to see
Therefore, \(\omega(t)\in L^{1}[0,1]\), which satisfies condition \((H_{3})\).
References
Kaufmann, ER, Kosmatov, N: A multiplicity result for a boundary value problem with infinitely many singularities. J. Math. Anal. Appl. 269, 444-453 (2002)
Kosmatov, N: On a singular conjugate boundary value problem with infinitely many solutions. Math. Sci. Res. Hot-Line 4, 9-17 (2000)
Liang, SH, Zhang, JH: The existence of countably many positive solutions for nonlinear singular m-point boundary value problems. J. Comput. Appl. Math. 214, 78-89 (2008)
Liang, SH, Zhang, JH: The existence of countably many positive solutions for some nonlinear singular three-point impulsive boundary value problems. Nonlinear Anal. 71, 4588-4597 (2009)
Liu, B: Positive solutions three-points boundary value problems for one-dimensional p-Laplacian with infinitely many singularities. Appl. Math. Lett. 17, 655-661 (2004)
Su, H, Wei, ZL, Xu, FY: The existence of countably many positive solutions for a system of nonlinear singular boundary value problems with the p-Laplacian operator. J. Math. Anal. Appl. 325, 319-332 (2007)
Xu, F, Wu, Y, Liu, L, Zhou, Y: Positive solutions of three-point boundary value problems for higher-order p-Laplacian with infinitely many singularities. Discrete Dyn. Nat. Soc. 2006, 69073 (2006)
Liang, SH, Zhang, JH: The existence of countably many positive solutions for nonlinear singular m-point boundary value problems on time scales. J. Comput. Appl. Math. 223, 291-303 (2009)
Liang, SH, Zhang, JH: The existence of countably many positive solutions for some nonlinear three-point boundary problems on the half-line. Nonlinear Anal. 70, 3127-3139 (2009)
Ji, D, Bai, Z, Ge, W: The existence of countably many positive solutions for singular multipoint boundary value problems. Nonlinear Anal. 72, 955-964 (2010)
Liang, SH, Zhang, JH: Positive solutions for singular third-order boundary value problem with dependence on the first order derivative on the half-line. Acta Appl. Math. 111, 27-43 (2010)
Kosmatov, N: Countably many solutions of a fourth order boundary value problem. Electron. J. Qual. Theory Differ. Equ. 2004, 12 (2004)
Davis, JM, Erbe, LH, Henderson, J: Multiplicity of positive solutions for higher order Sturm-Liouville problems. Rocky Mt. J. Math. 31, 169-184 (2001)
Cabada, A, Sanchez, L: A positive operator approach to the Neumann problem for a second order ordinary differential equation. J. Math. Anal. Appl. 204, 774-785 (1996)
Cabada, A, Pouso, RL: Existence result for the problem \((\phi(u'))' = f(t, u, u')\) with periodic and Neumann boundary conditions. Nonlinear Anal. 30, 1733-1742 (1997)
Cabada, A, Habets, P, Lois, S: Monotone method for the Neumann problem with lower and upper solutions in the reverse order. Appl. Math. Comput. 117, 1-14 (2001)
Gao, S, Chen, L, Nieto, JJ, Torres, A: Analysis of a delayed epidemic model with pulse vaccination and saturation incidence. Vaccine 24, 6037-6045 (2006)
Chu, J, Sun, Y, Chen, H: Positive solutions of Neumann problems with singularities. J. Math. Anal. Appl. 337, 1267-1272 (2008)
Dang, H, Oppenheimer, SF: Existence and uniqueness results for some nonlinear boundary value problems. J. Math. Anal. Appl. 198, 35-48 (1996)
Dong, Y: A Neumann problem at resonance with the nonlinearity restricted in one direction. Nonlinear Anal. 51, 739-747 (2002)
Erbe, LH, Wang, H: On the existence of positive solutions of ordinary differential equations. Proc. Am. Math. Soc. 120, 743-748 (1994)
Ma, R: Existence of positive radial solutions for elliptic systems. J. Math. Anal. Appl. 201, 375-386 (1996)
Yazidi, N: Monotone method for singular Neumann problem. Nonlinear Anal. 49, 589-602 (2002)
Sun, J, Li, W: Multiple positive solutions to second order Neumann boundary value problems. Appl. Math. Comput. 146, 187-194 (2003)
Jiang, D, Liu, H: Existence of positive solutions to second order Neumann boundary value problem. J. Math. Res. Expo. 20, 360-364 (2000)
Liu, X, Li, Y: Positive solutions for Neumann boundary value problems of second-order impulsive differential equations in Banach spaces. Abstr. Appl. Anal. 2012, 401923 (2012)
Zhang, X: Parameter dependence of positive solutions for second-order singular Neumann boundary value problems with impulsive effects. Abstr. Appl. Anal. 2014, 968792 (2014)
Guo, D, Lakshmikantham, V: Nonlinear Problems in Abstract Cones. Academic Press, New York (1988)
Acknowledgements
This work is sponsored by the National Natural Science Foundation of China (11301178) and the Beijing Natural Science Foundation (1163007), the Scientific Research Project of Construction for Scientific and Technological Innovation Service Capacity (71E1610973) and the teaching reform project of Beijing Information Science & Technology University (2015JGYB41). The authors are grateful to the anonymous referees for their constructive comments and suggestions, which have greatly improved this paper.
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Wang, M., Feng, M. Infinitely many singularities and denumerably many positive solutions for a second-order impulsive Neumann boundary value problem. Bound Value Probl 2017, 50 (2017). https://doi.org/10.1186/s13661-017-0784-y
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DOI: https://doi.org/10.1186/s13661-017-0784-y