Properties of positive solutions for the operator equation \(Ax=\lambda x\) and applications to fractional differential equations with integral boundary conditions
 Chengbo Zhai^{1}Email author and
 Fang Wang^{1}
https://doi.org/10.1186/s1366201507043
© Zhai and Wang 2015
Received: 14 July 2015
Accepted: 18 November 2015
Published: 1 December 2015
Abstract
In this article we present a new fixed point theorem for a class of generalized concave operators and we establish some properties of positive solutions for the operator equation \(Ax=\lambda x\). Based on them, the existence and uniqueness of positive solutions for a class of fractional differential equations with integral boundary conditions is proved. Moreover, we present some properties of positive solutions to the boundary value problem dependent on the parameter.
Keywords
MSC
1 Introduction
2 Properties of positive solutions for the operator equation \(Ax=\lambda x\)
For the discussion of this section, we first list some basic notations, concepts in ordered Banach spaces. For the convenience of the reader, we refer to [13, 14, 16] for details.
Let \((E,\\cdot\)\) be a real Banach space which is partially ordered by a cone \(P \subset E\), i.e., \(x \leq y\) if and only if \(yx \in P\). If \(x \leq y\) and \(x \neq y\), then we denote \(x < y\) or \(y > x\). By θ we denote the zero element of E. A nonempty closed convex set \(P \subset E\) is a cone if it satisfies (i) \(x \in P\), \(r \geq0 \Rightarrow rx \in P\); (ii) \(x \in P\), \(x \in P \Rightarrow x=\theta\).
P is called normal if there is a constant \(N > 0\) such that, for all \(x,y \in E\), \(\theta\leq x \leq y\) implies \(\ x \\leq N \ y \\); in this case N is the infimum of such constants, it is called the normality constant of P. We say that an operator \(A:E \rightarrow E\) is increasing if \(x \leq y\) implies \(Ax \leq Ay\).
For all \(x,y \in E\), the notation \(x\sim y\) means that there exist \(\lambda> 0\) and \(\mu> 0\) such that \(\lambda x \leq y \leq\mu x\). Clearly, ∼ is an equivalence relation. Given \(h > \theta\) (i.e., \(h \geq\theta\) and \(h \neq\theta\)), we denote by \(P_{h}\) the set \(P_{h}=\{ x \in E \mid x \sim h \}\). It is easy to see that \(P_{h} \subset P\).
Lemma 2.1
(see Theorem 2.1 in [16])
 (D_{1}):

\(A:P\rightarrow P\) is increasing and \(Ah+x_{0}\in P_{h}\) with \(x_{0}\in P\);
 (D_{2}):

for \(x\in P\) and \(t\in(0,1)\), there exists \(\varphi(t)\in(t,1)\) such that \(A(tx)\geq\varphi(t)Ax\).
Theorem 2.1
 (i)
there is \(h_{0}\in P_{h}\) such that \(Ah_{0}\in P_{h}\);
 (ii)
for any \(x\in P\) and \(t\in(0,1)\), there exists \(\varphi(t)\in (t,1)\) such that \(A(tx)\geq\varphi(t)Ax\).
 (1)
the operator equation (2.1) has a unique solution \(x^{*}\) in \(P_{h}\);
 (2)
for any initial value \(x_{0}\in P_{h}\), constructing successively the sequence \(x_{n}=Ax_{n1}\), \(n=1,2,\ldots\) , we have \(x_{n}\rightarrow x^{*}\) as \(n\rightarrow\infty\).
Remark 2.1
We say an operator A is generalized concave if it satisfies the condition (ii) in Theorem 2.1; see [16].
Proof of Theorem 2.1
Next we state and prove some properties of positive solutions for the operator equation (1.1).
Theorem 2.2
 (i)
\(x_{\lambda}\) is strictly decreasing in λ, that is, \(0<\lambda_{1}<\lambda_{2}\) implies \(x_{\lambda_{1}}>x_{\lambda_{2}}\);
 (ii)
if there exists \(\gamma\in(0,1)\) such that \(\varphi(t)\geq t^{\gamma}\) for \(t\in(0,1)\), then \(x_{\lambda}\) is continuous in λ, that is, \(\lambda\rightarrow \lambda_{0}\) (\(\lambda_{0}>0\)) implies \(\x_{\lambda}x_{\lambda_{0}}\\rightarrow0\);
 (iii)
\(\lim_{\lambda\rightarrow\infty}\x_{\lambda}\=0\), \(\lim_{\lambda\rightarrow0^{+}}\x_{\lambda}\=\infty\).
Proof
(iii) Let \(\lambda_{1}=1\), \(\lambda_{2}=\lambda\) in (2.3), we have \(x_{1}\geq \lambda x_{\lambda}\), \(\forall \lambda>1\). Thus, we can easily obtain \(\x_{\lambda}\\leq\frac{N}{\lambda}\x_{1}\\), where N is the normal constant. Let \(\lambda\rightarrow \infty\), then \(\x_{\lambda}\\rightarrow0\). Similarly, let \(\lambda _{1}=\lambda\), \(\lambda_{2}=1 \) in (2.3), then \(x_{\lambda}\geq\frac{1}{\lambda}x_{1}\), \(\forall 0<\lambda<1\). Thus \(\ x_{\lambda}\\geq\frac{1}{N\lambda}\x_{1}\\), where N is the normality constant. Let \(\lambda\rightarrow0+0\), we have \(\ x_{\lambda}\\rightarrow\infty\). □
Remark 2.2
(1) We do not suppose the condition of upperlower solutions which is common in many known results and is difficult to verify. Moreover, we give the iterative forms. The existence of a unique positive solution is proved only in the case where the cone P is normal and the operators A is generalized concave.
(2) The eigenvalue problem for generalized concave operators has not been studied in the literature, so Theorem 2.2 complements the eigenvalue results for generalized concave operators.
3 Properties of positive solutions for problem (1.2)
When \(\lambda=1\) in (1.2), Cabada and Hamdi [10] established the existence of one positive solution for problem (1.2) under sublinear case or superlinear case. The method used there is by GuoKrasnosel’skii fixed point theorem. Different from the main result and the method, we will apply Theorem 2.2 and present some properties of positive solutions for problem (1.2) dependent on the parameter, and the method is also different from those in previous works.
Lemma 3.1
(see [10])
Lemma 3.2
Proof
Theorem 3.1
 (H_{1}):

\(f:[0,1]\times[0,+\infty)\rightarrow[0,+\infty)\) is continuous with \(f(t,0)\not\equiv0\);
 (H_{2}):

\(f(t,x)\) is increasing in x for each \(t\in[0,1]\);
 (H_{3}):

for any \(r\in(0,1)\), there exists \(\varphi(r)\in(r,1)\) such that$$f(t,r x)\geq\varphi(r)f(t,x), \quad \forall t\in[0,1], x\in[0,+\infty). $$
 (1)For any given \(\lambda>0\), problem (1.2) has a unique positive solution \(u^{*}_{\lambda}\) in \(P_{h}\), where \(h(t)=t^{\alpha1}\), \(t\in[0,1]\). Moreover, for any initial value \(u_{0}\in P_{h}\), constructing successively the sequencewe have \(u_{n}(t)\rightarrow u^{*}_{\lambda}(t)\) as \(n\rightarrow+\infty\), where \(G(t,s)\) is given as in Lemma 3.1.$$u_{n}(t)=\lambda \int^{1}_{0}G(t,s)f\bigl(s,u_{n1}(s)\bigr) \, ds,\quad n=1,2,\ldots, $$
 (2)
\(u^{*}_{\lambda}\) is strictly increasing in λ, that is, \(0<\lambda_{1}<\lambda_{2}\) implies \(u^{*}_{\lambda_{1}}< u^{*}_{\lambda_{2}}\).
 (3)
If there exists \(\gamma\in(0,1)\) such that \(\varphi(t)\geq t^{\gamma}\) for \(t\in(0,1)\), then \(u^{*}_{\lambda}\) is continuous in λ, that is, \(\lambda\rightarrow\lambda_{0}\) (\(\lambda_{0}>0\)) implies \(\u^{*}_{\lambda }u^{*}_{\lambda_{0}}\\rightarrow0\).
 (4)
\(\lim_{\lambda\rightarrow0^{+}}\u^{*}_{\lambda}\=0\), \(\lim_{\lambda\rightarrow+\infty}\u^{*}_{\lambda}\=+\infty\).
Proof
In Theorem 3.1, let \(\lambda=1\), we can easily obtain the following conclusions.
Corollary 3.2
Remark 3.1
Comparing Theorem 3.1 and Corollary 3.2 with many main results in the literature, here we present an alternative approach to the study of similar problems under different conditions. Our results cannot only guarantee the existence of a unique positive solution for any given parameter, but they also help to construct an iterative scheme for approximating it. Moreover, we can show that the positive solution with respect to the parameter has some definite properties. So our results are seldom seen in the literature.
Remark 3.2
(1) We can see that there is a large number of functions which satisfy the conditions of Theorem 3.1 or Corollary 3.2. For example, let \(f(t,x)=a(t)[x^{\frac{1}{3}}+b]\), where \(a:[0,1]\rightarrow[0,+\infty)\) is continuous with \(a(t)\not\equiv0\), \(b>0\). Take \(\varphi(r)=r^{\frac{1}{3}}\). Also let \(f(t,x)=g(t)+x^{\frac{1}{2}}+x^{\frac{1}{3}}+\cdots+x^{\frac{1}{n}}+b\), where \(g:[0,1]\rightarrow [0,+\infty)\) is continuous, \(n\geq2\), \(b>0\). Take \(\varphi(r)=r^{\frac{1}{2}}\). We can easily prove that (H_{1})(H_{3}) in Theorem 3.1 hold.
(2) If the Green functions satisfy some properties similar to Lemma 3.2, then Theorems 2.1 and 2.2 can be applied to many fractional differential equations boundary value problems.
Declarations
Acknowledgements
The research was supported by the Youth Science Foundation of China (11201272) and Shanxi Province Science Foundation (2015011005), 131 Talents Project of Shanxi Province (2015).
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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