- Research
- Open Access
- Published:
Multiple positive solutions to some second-order integral boundary value problems with singularity on space variable
Boundary Value Problems volume 2017, Article number: 90 (2017)
Abstract
This article deals with integral boundary value problems of the second-order differential equations
where \(a\in C(J)\), \(b\in C(J, R_{-})\), \(f\in C(J_{+}\times R_{+}, R^{+})\) and \(g, h\in L^{1}(J)\) are nonnegative. The result of the existence of two positive solutions is established by virtue of fixed point index theory on cones. Especially, the nonlinearity f permits the singularity on the space variable.
1 Introduction
The aim of this article is to study the existence of two positive solutions to the following nonlinear second-order differential equation involving integral boundary value conditions:
where \(a\in C(J)\), \(b\in C(J, R_{-})\), \(f\in C(J_{+}\times R_{+}, R^{+})\) and \(g, h\in L^{1}(J)\) are nonnegative, \(J=[0,1]\), \(J_{+}=(0,1)\), \(R^{+}=[0,+\infty)\), \(R_{+}=(0,+\infty)\), \(R_{-}=(-\infty,0)\). Singularities of the nonlinearity f are related to both \(t=0,1\) and \(u=0\).
Recently, there has been a considerable increase in the investigation of nonlocal boundary value problems; see [1–11] for integer order and [12–22] for fractional differential equations. Based on a specially constructed cone, the existence as well as nonexistence results on positive solutions for the following second-order integral BVPs are obtained in an abstract space in Feng et al. [1]:
here \(f \in C([0, 1] \times P, P), \theta\) represents the zero element of E, and g is nonnegative and integrable. Also, in an abstract space, by the fixed point theorem of strict set contraction operators, Zhang et al. [2] obtained the existence results of solutions for some second-order integral boundary value problems with the impulsive effect.
For general differential operator, when the nonlinearity f is continuous, Feng and Ge [3] studied multiple positive solutions for the following singular m-point boundary value problems:
here \(\lambda> 0\), \(\xi_{i}\in(0, 1)\), \(\alpha_{i}\in R_{+}\) (\(i = 1, 2, \ldots, m-2\)) are known constants and L represents the linear operator
where \(a \in C(J)\) and \(b \in C(J, R_{+})\), \(f \in C(J \times R^{+}, R^{+})\), \(w \in C(J_{+}, R_{+})\). As is well known, two, three and multi-point BVPs may be looked upon as a special case of integral boundary value problems. For integral conditions, with some so-called first eigenvalue of the related linear operator, Liu et al. [4] formulated the existence results for BVP (1). The whole discussion relied on the fixed point index theorems. With the impulsive effect, under different combinations of super-linear and sub-linear condition on nonlinear term and the impulses, some results of existence of multiple positive solutions as well as nonexistence results for BVP (1) are obtained in Hao et al. [5].
We attempt in this article to study the existence of two positive solutions of BVP (1). The interesting points focus on two aspects. First, singularities of the nonlinearity f are related not only to the time but also to the space variables. Second, compared with [4], the method and conditions used to get result of multiple positive solutions are quite different from that used in [4]. The integral of the nonlinearity on some special bounded set is considered in this paper. The tools used to obtain the main result are fixed point index theorems on cones. Obviously, the result obtained in this paper can be analogously given for the Riemann-Stieltjes integral case after some minor modifications.
2 Preliminaries and several lemmas
Let \(\psi_{1}\) and \(\psi_{2}\) be the unique solution of the BVP
and
respectively. By [8], we know that \(\psi_{1}\), \(\psi_{2}\) are strictly increasing and strictly decreasing on J, respectively.
We adopt the following assumptions throughout this article.
- \(\mathrm{(H_{1})}\) :
-
\(a\in C(J)\), \(b\in C(J, R_{-})\);
- \(\mathrm{(H_{2})}\) :
-
\(g, h: J_{+}\to R^{+}\) are integrable, and \(k_{1}>0\), \(k_{4}>0\), \(k>0\), where
$$\begin{gathered} k_{1}=1- \int_{0}^{1}\psi_{2}(s)g(s)\,\mathrm{d}s, \qquad k_{2}= \int_{0}^{1}\psi_{1}(s)g(s)\,\mathrm{d}s, \\k_{3}= \int_{0}^{1}\psi_{2}(s)h(s)\,\mathrm{d}s, \qquad k_{4}=1- \int_{0}^{1}\psi_{1}(s)h(s)\,\mathrm{d}s , \\k=k_{1}k_{4}-k_{2}k_{3}; \end{gathered} $$ - \(\mathrm{(H_{3})}\) :
-
\(f\in C(J_{+}\times R_{+}, R^{+})\);
- \(\mathrm{(H_{4})}\) :
-
there exist three functions \(\widehat{a}, \widehat{b}\in C(J_{+}, R^{+})\), \(\widehat{g}\in C(R_{+}, R^{+})\) satisfying
$$f(t,u)\leq\widehat{a}(t)\widehat{g}(u)+\widehat{b}(t),\quad \forall t\in J_{+}, u \in R_{+}, $$where, in addition,
$$\widehat{a}_{r}^{*}= \int_{0}^{1}\mathcal{H}(t)\widehat{a}(t)\widehat {g}_{r}(t)\,\mathrm{d}t< +\infty, \qquad \widehat{b}^{*}= \int_{0}^{1}\mathcal {H}(t)\widehat{b}(t) \,\mathrm{d}t< +\infty, $$and
$$\widehat{g}_{r}(t)=\max\bigl\{ \widehat{g}(u): \gamma(t)r \leq u\leq r \bigr\} ,\quad \forall r>0, $$here, \(\gamma(t)\) is defined in (6), \(\mathcal{H}(t)\) is defined in (7);
- \(\mathrm{(H_{5})}\) :
-
there exists a function \(\widehat{c}\in C(J_{+}, R^{+})\) satisfying
$$\frac{f(t,u)}{\widehat{c}(t)u}\to+\infty\quad \mbox{as } u\to +\infty $$uniformly for \(t\in J_{+}\), and in addition,
$$\widehat{c}^{*}= \int_{0}^{1}\widehat{c}(t)\,\mathrm{d}t< +\infty; $$ - \(\mathrm{(H_{6})}\) :
-
there exists a function \(\widehat{d}\in C(J_{+}, R^{+})\) satisfying
$$\frac{f(t,u)}{\widehat{d}(t)}\to+\infty \quad\mbox{as } u\to0^{+} $$uniformly for \(t\in J_{+}\), and in addition,
$$\widehat{d}^{*}= \int_{0}^{1}\widehat{d}(t)\,\mathrm{d}t< +\infty. $$
Lemma 1
[4]
Assume that \((\mathrm{H_{1}})\) and \((\mathrm{H_{2}})\) hold. Then, for any \(y\in C(J_{+})\cap L^{1}(J)\), the BVP
has a unique solution u that can be expressed in the form
where
Moreover, \(u(t)\geq0\) on J provided \(y\geq0\).
By Remark 2.1 in [4], we have
Lemma 2
[4] Suppose that \((\mathrm{H_{1}})\) and \((\mathrm{H_{2}})\) hold, then, for any \(t,s \in J\), we have
where \(\gamma(t)=\min\{\psi_{1}(t), \psi_{2}(t)\}\), \(t\in J\) and
Let \(E=C(J)\) be the standard Banach space with the maximum norm and P be the typical cone of nonnegative continuous functions in the form
Let \(P_{mn}=\{u\in P, m\leq\|u\|\leq n\}\), \(P_{r}=\{u\in P: \|u\|\leq r\}\) for \(n>m>0\), \(r>0\).
First, we give an operator \(T:P\setminus\{0\}\to C(J)\) as follows:
Lemma 3
If conditions \((\mathrm{H_{1}})\) and \((\mathrm{H_{2}})\) are satisfied, then, for any \(n>m>0\), \(T:P_{mn}\to P\) is a completely continuous operator.
Proof
For any given \(u\in P_{mn}\), we have \(m\leq\|u\|\leq n\). From the construction of P, we have
Clearly, for any \(n>m>0\), condition \((\mathrm{H_{4}})\) means that
where
It follows from (8), \((\mathrm{H_{3}})\), \((\mathrm{H_{4}})\) and Lemma 2 that
and
which shows that T makes sense. According to Lemma 2, we have for any \(t\in J\)
Hence,
At the same time, by Lemma 2 and (14), we get
This indicates that T maps \(P_{mn}\) into P.
Next, we shall show the complete continuity of the operator T. Let \(u_{n}, \bar{u}\in P_{mn}\), with \(\|u_{n}-\bar{u}\|\to0\) (\(n\to\infty \)); then \(\lim _{n\to\infty}u_{n}(t)=\bar{u}(t)\), \(t\in J\). Let
By \((\mathrm{H_{1}})\),
Similar to (12), for \(u_{n}, \bar{u}\in P_{mn}\), one has
Then one gets
The Lebesgue dominated convergence theorem together with (16) and (17) generates
That is to say \(T_{1}: P_{mn}\to L^{1}(J)\) is continuous. Furthermore, the complete continuity of the operator \(T_{2}:L^{1}(J)\to C(J)\) can easily be verified by the Arzela-Ascoli theorem and a standard discussion. Hence, by the property of compound operators we see that \(T=T_{2}\circ T_{1}: P_{mn}\to C(J)\) is completely continuous. □
Lemma 4
[23]
Let E be a Banach space, \(P\subset E\) a cone in E. For \(r>0\), define \(P_{r}=\{u\in P: \|u\|\leq r\}\). Assume that \(T:P_{r}\to P\) is a compact map such that \(Tu\neq u\) for \(u\in\partial P_{r}=\{u\in P: \|u\| =r\}\).
-
(i)
If \(\|u\|\leq\|Tu\|\), \(\forall u\in \partial P_{r}\), then
$$i(T, P_{r}, P)=0. $$ -
(ii)
If \(\|u\|\geq\|Tu\|\), \(\forall u\in \partial P_{r}\), then
$$i(T, P_{r}, P)=1. $$
3 Main results
Theorem 1
Let conditions \((\mathrm{H_{1}})\)-\((\mathrm {H_{6}})\) be satisfied. Furthermore, there exists a constant \(\widehat {r}>0\) satisfying
here, \(\widehat{a}_{\widehat{r}}^{*}\) and \(\widehat{b}^{*}\) are given in \((\mathrm{H_{4}})\). Then the BVP (1) admits at least two positive solutions \(x^{*}\) and \(x^{**}\) such that \(0<\|x^{*}\|<\widehat{r}<\|x^{**}\|\).
Proof
From Lemma 3, we know that for any \(n>m>0\), the operator T maps \(P_{mn}\) into P and is completely continuous. Next, we are in a position to show that T has two distinct positive fixed points \(x^{*}\), \(x^{**}\) such that \(0<\|x^{*}\|<\widehat{r}<\|x^{**}\|\).
From \((\mathrm{H_{5}})\) we know there exists a constant \(r_{1}>0\) satisfying
Let \(0<\gamma_{\frac{1}{4}}=\min_{t\in[\frac{1}{4},\frac{3}{4}]}\{ \psi_{1}(t), \psi_{2}(t)\}<1\). Choose
For \(u\in\partial P_{r_{2}}\), considering the definition of cone P, we have
So, we get from (19), (20), (21) that
i.e., \(\|Tu\|>\|u\|\), \(u\in\partial P_{r_{2}}\). Therefore, by Lemma 4,
By condition \((\mathrm{H_{6}})\), there exists a constant \(r_{3}>0\) satisfying
Choose
For \(u\in\partial P_{r_{4}}\), we have
So, we get from (24) and (25) that
i.e., \(\|Tu\|>\|u\|\), \(u\in\partial P_{r_{4}}\). Therefore, by Lemma 4,
In a similar manner, for \(u\in\partial P_{\widehat{r}}\), by \((\mathrm {H_{4}})\), Lemma 2 and (14), we get
i.e., \(\|Tu\|<\|u\|\), \(u\in\partial P_{\widehat{r}}\). Therefore, by Lemma 4, we get
Now, the additivity of the fixed point index together with (23), (27), (29) implies that
and
Hence, T has two fixed points \(x^{*}\) and \(x^{**}\) which belong to \(P_{\widehat{r}}\setminus\mathring{P}_{r_{4}}\) and \(P_{r_{2}}\setminus \mathring{P}_{\widehat{r}}\), respectively, such that \(0< r_{4}<\|x^{*}\|<\widehat{r}<\|x^{**}\|\leq r_{2}\). □
4 An example
Example 1
Consider the following second-order singular integral BVPs:
Conclusion
BVP (30) admits at least two positive solutions \(x^{*}\) and \(x^{**}\) satisfying \(0<\|x^{*}\|<1<\|x^{**}\|\).
Proof
Clearly, BVP (30) has the form (1), in which \(a(t)\equiv1\), \(b(t)\equiv-2\), \(f(t,u)=\frac{1}{560\sqrt [8]{(1-t)}} (u^{2}+\frac{1}{3\sqrt[4]{u}} )+\frac{1}{420\sqrt [3]{t(1-t)}}\), \(g(s)=s\), \(h(s)=s^{2}\). Obviously, \(f(t,u)\) permits singularities at \(t=0, 1\) and \(u=0\).
Let \(\psi_{1}\) and \(\psi_{2}\) satisfy
By a simple computation, we get
It is not difficult to see that \(0< G(t,s)< 2s\) and
For any given \(r>0\), we can see that \((\mathrm{H_{4}})\) is valid for \(\widehat{a}(t)=\frac{1}{560\sqrt[8]{(1-t)}}\), \(g(u)=u^{2}+\frac {1}{3\sqrt[4]{u}}\), \(b(t)=\frac{1}{420\sqrt[3]{t(1-t)}}\), and it follows from \(0\leq\frac{3}{2(e-e^{-2})^{2}}t(1-t)^{2} \leq\gamma (t)\leq1\) that
\(\widehat{b}^{*}=\int_{0}^{1}\mathcal{H}(t)\widehat{b}(t)\,\mathrm {d}t<0.0684\). Clearly, \((\mathrm{H_{5}})\) and \((\mathrm{H_{6}})\) hold for \(\widehat{c}(t)=\widehat{d}(t)=\frac{1}{560\sqrt[8]{(1-t)}}\), and \(c^{*}=d^{*}= 0.0020\). Take \(\widehat{r}=1\); we have by (31)
Consequently, (18) holds and our conclusion can be deduced from Theorem 1. □
References
Feng, M, Ji, D, Ge, W: Positive solutions for a class of boundary value problem with integral boundary conditions in Banach spaces. J. Comput. Appl. Math. 222, 351-363 (2008)
Zhang, X, Feng, M, Ge, W: Existence of solutions of boundary value problems with integral boundary conditions for second-order impulsive integro-differential equations in Banach spaces. J. Comput. Appl. Math. 233, 1915-1926 (2010)
Feng, M, Ge, W: Positive solutions for a class of m-point singular boundary value problems. Math. Comput. Model. 46, 375-383 (2007)
Liu, L, Hao, X, Wu, Y: Positive solutions for singular second order differential equations with integral boundary conditions. Math. Comput. Model. 57, 836-847 (2013)
Hao, X, Liu, L, Wu, Y: Positive solutions for second order impulsive differential equations with integral boundary conditions. Commun. Nonlinear Sci. Numer. Simul. 16, 101-111 (2011)
Zhang, X, Feng, M, Ge, W: Existence result of second-order differential equations with integral boundary conditions at resonance. J. Math. Anal. Appl. 353, 311-319 (2009)
Kong, L: Second order singular boundary value problems with integral boundary conditions. Nonlinear Anal. 72, 2628-2638 (2010)
Ma, R, Wang, H: Positive solutions of nonlinear three-point boundary-value problems. J. Math. Anal. Appl. 279, 216-227 (2003)
Cui, Y, Zou, Y: An existence and uniqueness theorem for a second order nonlinear system with coupled integral boundary value conditions. Appl. Math. Comput. 256, 438-444 (2015)
Kang, P, Wei, Z: Three positive solutions of singular nonlocal boundary value problems for systems of nonlinear second-order ordinary differential equations. Nonlinear Anal. 70, 444-451 (2009)
Jiang, J, Liu, L, Wu, Y: Second-order nonlinear singular Sturm-Liouville problems with integral boundary conditions. Appl. Math. Comput. 215, 1573-1582 (2009)
Zhang, X: Positive solutions for a class of singular fractional differential equation with infinite-point boundary value conditions. Appl. Math. Lett. 39, 22-27 (2015)
Wang, G: Explicit iteration and unbounded solutions for fractional integral boundary value problem on an infinite interval. Appl. Math. Lett. 47, 1-7 (2015)
Ahmad, B, Sivasundaram, S: Existence of solutions for impulsive integral boundary value problems of fractional order. Nonlinear Anal. Hybrid Syst. 4, 134-141 (2010)
Yuan, C: Two positive solutions for \((n-1,1)\)-type semipositone integral boundary value problems for coupled systems of nonlinear fractional differential equations. Commun. Nonlinear Sci. Numer. Simul. 17, 930-942 (2012)
Cabada, A, Hamdi, Z: Nonlinear fractional differential equations with integral boundary value conditions. Appl. Math. Comput. 228, 251-257 (2014)
Zhang, X, Wang, L, Sun, Q: Existence of positive solutions for a class of nonlinear fractional differential equations with integral boundary conditions and a parameter. Appl. Math. Comput. 226, 708-718 (2014)
Zhao, K, Liang, J: Solvability of triple-point integral boundary value problems for a class of impulsive fractional differential equations. Adv. Differ. Equ. 2017, 50 (2017)
Zhao, K, Liu, J: Multiple monotone positive solutions of integral BVPs for a higher-order fractional differential equation with monotone homomorphism. Adv. Differ. Equ. 2016, 20 (2016)
Zhao, K: Impulsive integral boundary value problems of the higher-order fractional differential equation with eigenvalue arguments. Adv. Differ. Equ. 2015, 382 (2015)
Zhao, K: Triple positive solutions for two classes of delayed nonlinear fractional FDEs with nonlinear integral boundary value conditions. Bound. Value Probl. 2015, 181 (2015)
Zhao, K: Multiple positive solutions of integral BVPs for high-order nonlinear fractional differential equations with impulses and distributed delays. Dyn. Syst. 30(2), 208-223 (2015)
Guo, D, Lakshmikantham, V: Nonlinear Problems in Abstract Cones. Academic Press, San Diego (1988)
Acknowledgements
The project is supported financially by a Project of Shandong Province Higher Educational Science and Technology Program (J15LI16), NSFC cultivation project of Jining Medical University, the Natural Science Foundation of Shandong Province of China (ZR2015AL002), the National Natural Science Foundation of China (11571296, 11571197, 11371221, 11071141) and the Foundations for Jining Medical College Natural Science (JY2015KJ019, JY2015BS07, JYQ14KJ06).
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
QZ and XZ obtained the results in an equally joint work. All the authors read and approved the final manuscript.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Zhong, Q., Zhang, X. Multiple positive solutions to some second-order integral boundary value problems with singularity on space variable. Bound Value Probl 2017, 90 (2017). https://doi.org/10.1186/s13661-017-0822-9
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13661-017-0822-9
MSC
- 34B15
- 34B18
Keywords
- integral boundary value
- cone
- two positive solutions
- singularity