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 Open Access
Positive solutions for a secondorder differential equation with integral boundary conditions and deviating arguments
 Xuemei Zhang^{1}Email author and
 Meiqiang Feng^{2}
Received: 3 June 2015
Accepted: 19 November 2015
Published: 2 December 2015
Abstract
Using a wellknown fixed point theorem on cones, we study the number of positive solutions for a secondorder differential equation with integral boundary conditions and deviating arguments. We discuss our problems under two cases when the deviating arguments are delayed and advanced. Our results extend and improve those of Boucherif (Nonlinear Anal. 70:364371, 2009) and Kong (Nonlinear Anal. 72:26282638, 2010) by generalizing the nonlinearity \(f(t,u(t))\) to \(f(t,u(\alpha(t)))\) with general \(\alpha(t)\not\equiv t\). The dependence of solutions on the parameter λ is also studied.
Keywords
 differential equations with advanced or delayed arguments
 integral boundary conditions
 number of positive solutions
 parameter dependence of positive solutions
1 Introduction
 (H)\(f:[0,\infty)\rightarrow(0,\infty)\) is nondecreasing, and there exists \(\nu\in(0,1)\) such that$$f(kx)\geq k^{\nu}f(x),\quad \mbox{for }k\in(0,1) \mbox{ and } x\in[0,+ \infty). $$
 (H^{∗}):

\(\frac{f(s)}{\varphi_{p}(s)}\) is strictly decreasing in \((0,\infty)\).
But to the best of our knowledge, there are no results for the dependence of positive solution \(x_{\lambda}(t)\) on the parameter λ of secondorder boundary value problems with deviating arguments without a condition similar to (H) or (H^{∗}). The objective of the present paper is to fill this gap.
 (H_{1}):

\(g\in C^{1}(J,(0,+\infty))\), \(\alpha\in C(J,J)\);
 (H_{2}):

\(\omega\in C((0,1), [0,+\infty))\) withand ω does not vanish on any subinterval of \((0,1)\);$$0< \int_{0}^{1}\omega(s)\,ds< \infty $$
 (H_{3}):

\(f\in C([0,1]\times[0,+\infty), [0,+\infty))\) with \(f(t,x)>0\) for all t and \(x>0\);
 (H_{4}):

\(h\in C[0,1]\) is nonnegative with \(\nu\in[0,a)\), where$$ \nu= \int_{0}^{1}h(t)\,dt. $$(1.2)
Some special cases of (1.1) have been investigated. For example, Boucherif [1] considered problem (1.1) under the case that \(\lambda=1\), \(g(t)\equiv1\), \(\omega (t)\equiv1\), and \(\alpha(t)\equiv t\) on J. By using Krasnoselskii’s fixed point theorem in a cone, the author proved the existence results of positive solution for problem (1.1).
Kong [2] considered problem (1.1) under the case that \(g(t)\equiv1\), \(\omega(t)\equiv1\), and \(\alpha(t)\equiv t\) on J. By using the mixed monotone operator theory, the author obtained the existence and uniqueness of positive solutions for problem (1.1).
It is clear that \(i_{0}, i_{\infty}=0,1\mbox{ or }2\). Then we shall show that problem (1.1) has \(i_{0}\) or \(i_{\infty}\) positive solution(s) for sufficiently large or small λ, respectively.
The paper is organized in the following fashion. In Section 2, we provide some necessary background. In particular, we state some properties of the Green’s function associated with problem (1.1). In Section 3, we use a wellknown fixed point theorem to study the existence, multiplicity and nonexistence of positive solutions for problem (1.1) with advanced argument α. In Section 4 we discuss the dependence of solution \(x_{\lambda}(t)\) on the parameter λ for problem (1.1) with advanced argument α and we formulate sufficient conditions under which delayed problem (1.1) has positive solutions in Section 5. Finally, an example is given to illustrate the main results in Section 6.
2 Preliminaries
Let \(E=C[0,1]\). It is well known that E is a real Banach space with the norm \(\ \cdot\\) defined by \(\x\=\max_{t\in J}x(t)\).
In our main results, we will make use of the following definitions and lemmas.
Definition 2.1
(see [39])
 (i)
\(cu+dv \in P\) for all \(u, v \in P\) and all \(c\geq0\), \(d\geq 0\) and
 (ii)
\(u, u \in P \) implies \(u=0\).
Every cone \(P \subset E\) induces an ordering in E given by \(x\leq y\) if and only if \(yx \in P\).
Definition 2.2
A function \(x\in E\cap C^{2}(0,1)\) is called a solution of problem (1.1) if it satisfies (1.1). If \(x(t)\geq0\) and \(x(t)\not\equiv0\) on J, then x is called a positive solution of problem (1.1).
Lemma 2.1
Proof
The proof is similar to that of Lemma 2.1 in [11]. □
Lemma 2.2
Proof
It follows from the definition of \(G(t,s)\) and \(H(t,s)\) that (2.5) holds. Now, we show that (2.6) also holds.
Remark 2.1
From (2.11), we know that a function \(x\in K\) is a solution of problem (1.1) if and only if x is a fixed point of operator T, and we obtain Lemma 2.3.
Lemma 2.3
Assume that (H_{1})(H_{4}) hold. If x is a fixed point of operator T, then \(x\in E\cap C^{2}(0,1)\), and x is a solution of problem (1.1).
Lemma 2.4
Suppose that (H_{1})(H_{4}) hold. Then \(T(K) \subset K\) and \(T: K\rightarrow K\) is completely continuous.
Proof
Finally, similar to the proof of Theorem 4.1 in [40], one can prove that \(T: K \rightarrow K\) is completely continuous. This gives the proof of Lemma 2.4. □
In the rest of this section, we state a wellknown fixed point theorem which we need later.
Lemma 2.5
(see [39])
 (i)
\(\Ax\\leq\x\\), \(\forall x\in P\cap\partial \Omega_{1}\) and \(\Ax\\geq\x\\), \(\forall x\in P\cap\partial\Omega_{2}\), or
 (ii)
\(\Ax\\geq\x\\), \(\forall x\in P\cap\partial \Omega_{1}\) and \(\Ax\\leq\x\\), \(\forall x\in P\cap\partial\Omega_{2}\),
The fixed point theorem in a cone has often been used to study the existence and multiplicity of positive solutions of boundary value problems over the last several years. As recent example, we mention the paper of Baleanu et al. [41].
3 Existence of positive solutions for problem (1.1) under \(\alpha(t)\geq t\) on J
In this section, we show that problem (1.1) has \(i_{0}\) or \(i_{\infty}\) positive solution(s) for sufficiently large or small λ under \(\alpha(t)\geq t\) on J.
Theorem 3.1
 (i)
If \(i_{0}=1\textit{ or }2\), then there exists \(\lambda_{0}>0\) such that problem (1.1) has \(i_{0}\) positive solution(s) for \(\lambda>\lambda _{0}\).
 (ii)
If \(i_{\infty}=1\textit{ or }2\), then there exists \(\lambda_{0}>0\) such that problem (1.1) has \(i_{\infty}\) positive solution(s) for \(0<\lambda <\lambda_{0}\).
 (iii)
If \(i_{0}=0\) or \(i_{\infty}=0\), then problem (1.1) has no positive solution for sufficiently large or small λ, respectively.
Proof
Thus by (i) of Lemma 2.5, it follows from (3.1) and (3.2) that T has a fixed point x in \(K\cap(\bar{\Omega}_{r}\backslash\Omega _{r_{1}})\) with \(r_{1}\leq\x\\leq r\). Lemma 2.3 implies that problem (1.1) has at least one positive solution x with \(r_{1}\leq\x\\leq r\).
Thus by (ii) of Lemma 2.5, it follows from (3.1) and (3.3) that T has a fixed point x in \(K\cap(\bar{\Omega}_{r_{2}}\backslash\Omega _{r})\) with \(r\leq\x\\leq r_{2}\). Lemma 2.3 implies that problem (1.1) has at least one positive solution x with \(r\leq\x\\leq r_{2}\).
Consequently, it follows from Lemma 2.3 that problem (1.1) has two positive solutions for \(\lambda>\lambda_{0}\) if \(f^{0}=f^{\infty}=0\).
Thus by (ii) of Lemma 2.5, it follows from (3.6) and (3.7) that T has a fixed point x in \(K\cap(\bar{\Omega}_{r}\backslash\Omega _{r_{1}})\) with \(r_{1}\leq\x\\leq r\). Lemma 2.3 shows that problem (1.1) has at least one positive solution x with \(r_{1}\leq\x\\leq r\).
Thus by (i) of Lemma 2.5, it follows from (3.6) and (3.8) that T has a fixed point x in \(K\cap(\bar{\Omega}_{r_{2}}\backslash\Omega _{r})\) with \(r\leq\x\\leq r_{2}\). Lemma 2.3 shows that problem (1.1) has at least one positive solution x with \(r\leq\x\\leq r_{2}\).
Assume \(y\in K\) is a positive solution of problem (1.1). We will show that this leads to a contradiction for \(\lambda> \lambda_{0}=[ab^{2}\gamma\eta\delta\beta]^{1}\Delta\).
Since \(0\leq t\leq\alpha(t)\leq1\) on J, it follows from \(0\leq x(t)\leq h_{3}\), \(x(t)\geq h_{4}\), and \(h_{3}\leq x(t)\leq h_{4}\) on J that \(0\leq x(\alpha(t))\leq h_{3}\), \(x(\alpha(t))\geq h_{4}\), and \(h_{3}\leq x(\alpha(t))\leq h_{4}\) on J, respectively.
Assume \(y\in K\) is a positive solution of problem (1.1). We will show that this leads to a contradiction for \(0<\lambda< \lambda_{0}=[a\gamma D \eta^{*}\beta]^{1}\Delta\).
Theorem 3.2 is a direct consequence of the proof of Theorem 3.1(iii). Under the conditions of Theorem 3.2 we are able to give explicit intervals of λ such that (1.1) has no positive solution.
Theorem 3.2
 (i)
If there exists \(l>0\) such that \(f(t,x)\geq lx\) for \(t\in J\) and \(x\in[0,\infty)\), then there exists \(\lambda_{0}>0\) such that problem (1.1) has no positive solution for \(\lambda>\lambda_{0}\).
 (ii)
If there exists \(L>0\) such that \(f(t,x)\leq Lx\) for \(t\in J\) and \(x\in[0,\infty)\), then there exists \(\lambda_{0}>0\) such that problem (1.1) has no positive solution for \(0<\lambda<\lambda_{0}\).
Theorem 3.3
Proof
We give the proof under two cases of \(f_{\infty}>f^{0}\) and \(f_{\infty}< f^{0}\).
Now, considering \(f^{0}\) and \(f_{\infty}\), there exists \(r_{1}>0\) such that \(f(t,x)\leq(f^{0}+\varepsilon)x\) for \(t\in J\) and \(0\leq x\leq r_{1}\).
On the other hand, there exists \(L>0\) with \(L>r_{1}\) such that \(f(t,x)\geq(f_{\infty}\varepsilon)x\) for \(t\in J\) and \(x\geq L\).
Since \(0\leq t\leq\alpha(t)\leq1\), it follows from \(0\leq x(t)\leq r_{1}\) on J that \(x(\alpha(t))\geq L\).
It follows from Lemma 2.5 that T has a fixed point in \(K\cap(\bar{\Omega}_{r_{2}}\backslash\Omega_{r_{1}})\). Consequently, problem (1.1) has a positive solution.
Now, turning to \(f^{0}\) and \(f_{\infty}\), there exists \(r_{1}>0\) such that \(f(t,x)\geq(f^{0}\varepsilon)x\) for \(t\in J\) and \(0\leq x\leq r_{1}\).
On the other hand, there exists \(L>0\) with \(L>r_{1}\) such that \(f(t,x)\leq(f_{\infty}+\varepsilon)x\) for \(t\in J\) and \(x\geq L\).
It follows from Lemma 2.5 that T has a fixed point in \(K\cap(\bar{\Omega}_{r_{2}}\backslash\Omega_{r_{1}})\). Consequently, problem (1.1) has a positive solution. □
Corollary 3.1
Proof
The proof is similar to that of Theorem 3.3. □
4 The dependence of solution \(x_{\lambda}(t)\) on the parameter λ for problem (1.1) under \(\alpha(t)\geq t\) on J
 \((\mathrm{H}_{3})^{*}\) :

\(f\in C([0,1]\times[0,+\infty), [0,+\infty))\).
Theorem 4.1
 (i)
If \(f^{0}=0\) and \(f_{\infty }=\infty\), then for every \(\lambda>0\) problem (1.1) has a positive solution \(x_{\lambda}(t)\) satisfying \(\lim_{\lambda\rightarrow0^{+}}\x_{\lambda}\ =\infty\).
 (ii)
If \(f_{0}=\infty\) and \(f^{\infty}=0\), then for every \(\lambda>0\) problem (1.1) has a positive solution \(x_{\lambda}(t)\) satisfying \(\lim_{\lambda\rightarrow 0^{+}}\x_{\lambda}\=0\).
Proof
By definition, \(i_{0}=i_{\infty}=1\) implies that \(f^{0}=0\) and \(f_{\infty}=\infty\) or \(f_{0}=\infty\) and \(f^{\infty}=0\). We need only prove this theorem under the condition \(f^{0}=0\) and \(f_{\infty}=\infty\) since the proof is similar to that of \(f_{0}=\infty\) and \(f^{\infty}=0\).
Applying (i) of Lemma 2.5 to (4.1) and (4.2) shows that the operator T has a fixed point \(x_{\lambda}\in K\cap(\bar{\Omega}_{R}\backslash\Omega_{r})\). Consequently, it follows from Lemma 2.3 that problem (1.1) has a positive solution \(x_{\lambda}\in K\cap(\bar{\Omega}_{R}\backslash\Omega _{r})\) with \(r\leq\x_{\lambda}\\leq R\).
5 Positive solutions of problem (1.1) for the case of \(\alpha (t)\leq t\) on J
Now we deal with problem (1.1) for the case of \(\alpha(t)\leq t\) on J. Let E, K, and T be as defined in Section 2. Similarly as Lemmas 2.22.4, we can prove the following results.
Lemma 5.1
Lemma 5.2
Assume that (H_{1})(H_{4}) hold. If x is a fixed point of the operator T, then \(x\in E\cap C^{2}(0,1)\), and x is a solution of problem (1.1).
Lemma 5.3
Assume that (H_{1})(H_{4}) hold. Then \(T(K)\subset K\) and \(T:K\rightarrow K\) is completely continuous.
By analogous methods, we have the following results.
Theorem 5.1
 (i)
If \(i_{0}=1\textit{ or }2\), then there exists \(\lambda_{0}>0\) such that problem (1.1) has \(i_{0}\) positive solution(s) for \(\lambda>\lambda _{0}\).
 (ii)
If \(i_{\infty}=1\textit{ or }2\), then there exists \(\lambda_{0}>0\) such that problem (1.1) has \(i_{\infty}\) positive solution(s) for \(0<\lambda <\lambda_{0}\).
 (iii)
If \(i_{0}=0\) or \(i_{\infty}=0\), then problem (1.1) has no positive solution for sufficiently large or small λ, respectively.
Theorem 5.2
 (i)
If there exists \(l>0\) such that \(f(t,x)\geq lx\) for \(t\in J\) and \(x\in[0,\infty)\), then there exists \(\lambda_{0}>0\) such that problem (1.1) has no positive solution for \(\lambda>\lambda_{0}\).
 (ii)
If there exists \(L>0\) such that \(f(t,x)\leq lx\) for \(t\in J\) and \(x\in[0,\infty)\), then there exists \(\lambda_{0}>0\) such that problem (1.1) has no positive solution for \(0<\lambda<\lambda_{0}\).
Theorem 5.3
Corollary 5.1
Theorem 5.4
 (i)
If \(f^{0}=0\) and \(f_{\infty }=\infty\), then for every \(\lambda>0\) problem (1.1) has a positive solution \(x_{\lambda}(t)\) satisfying \(\lim_{\lambda\rightarrow0^{+}}\x_{\lambda}\ =\infty\).
 (ii)
If \(f_{0}=\infty\) and \(f^{\infty}=0\), then for every \(\lambda>0\) problem (1.1) has a positive solution \(x_{\lambda}(t)\) satisfying \(\lim_{\lambda\rightarrow 0^{+}}\x_{\lambda}\=0\).
6 An example
To illustrate how our main results can be used in practice we present an example.
Example 6.1
This means that problem (6.1) involves the advanced argument α. For example, we can take \(\alpha(t)=\sqrt[3]{t}\). It is clear that ω is singular at \(t=0\) and f is both nonnegative and continuous.
We claim that problem (6.1) has at least one positive solution for any \(\lambda>\frac{e^{2}(e+1)}{4}\).
Proof
It follows from the definition of g, ω, f, α, and h that (H_{1})(H_{4}) hold, and \(f^{0}=0\).
Therefore, for any \(\lambda>\lambda_{0}=\frac{e^{2}(e+1)}{4}\), it follows from Theorem 3.1(i) that problem (6.1) has a positive solution. □
Declarations
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 anonymous referees for their constructive comments and suggestions, which has greatly improved this paper.
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.
Authors’ Affiliations
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