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 Open Access
Eigenvalue problem for fractional differential equations with nonlinear integral and disturbance parameter in boundary conditions
 Wenxia Wang^{1}Email author and
 Xiaotong Guo^{2}
 Received: 7 December 2015
 Accepted: 27 January 2016
 Published: 10 February 2016
Abstract
This paper is concerned with the existence, nonexistence, uniqueness, and multiplicity of positive solutions for a class of eigenvalue problems of nonlinear fractional differential equations with a nonlinear integral term and a disturbance parameter in the boundary conditions. By using fixed point index theory we give the critical curve of eigenvalue λ and disturbance parameter μ that divides the range of λ and μ for the existence of at least two, one, and no positive solutions for the eigenvalue problem. Furthermore, by using fixed point theorem for a sum operator with a parameter we establish the maximum eigenvalue interval for the existence of the unique positive solution for the eigenvalue problem and show that such a positive solution depends continuously on the parameter λ for given μ. In particular, we give estimates for the critical value of parameters. Two examples are given to illustrate our main results.
Keywords
 integral boundary value problem
 eigenvalue problem
 disturbance parameter
 Caputo fractional derivative
 positive solution
 cone
MSC
 34B18
 34B15
1 Introduction and preliminaries
Fractional differential equations have been extensively investigated in recent years, due to a wide range of applications in various fields of sciences and engineering such as control, porous media, electromagnetic, and so forth; see [1–18] and the references therein. Since the integral boundary value problems can better describe the actual phenomenon, the existence of positive solutions for fractional integral boundary value problems has attracted considerable attention, and fruits from research into it emerge continuously. For a small sample of such a work, we refer the reader to [19–27] and the references therein.
The FEP (1) seems to be studied for the first time. The purpose of this paper is to find the critical curve of parameters λ and μ dividing the range of λ and μ for the existence of at least two, one, and no positive solutions and to establish the maximum eigenvalue interval for the existence of the unique positive solution for the eigenvalue problem. The main tools used in this paper are fixed point index theory, the fixed point theorem of a sum operator with a parameter, and a sufficient and necessary condition for the existence of a fixed point for a concave operator. In particular, the positivity of a solution \(x(t)\) of FEP (1) means that \(x(t)\geq0\) for \(t\in[0,1]\) and \(x(t)>0\) for \(t\in(0,1)\).
The paper is organized as follows. In Section 2, we establish an operator equation that is equivalent to FEP (1) and present properties of solutions of FEP (1). In Section 3, we obtain the critical curve of λ and μ and establish an eigenvalue interval for the existence of at least two, one, and no positive solutions for FEP (1) according to the range of the disturbance parameter μ. These results show that the impact of the parameter μ on the eigenvalue interval for the existence of positive solutions and on their number. In Section 4, under some mild assumption, we establish the maximum eigenvalue interval for the existence of the unique positive solution for FEP (1) and show that such a positive solution depends continuously on the parameter λ. In particular, we give estimates for the critical values of parameters. Finally, two examples are given to illustrate our main results.
For convenience of the readers, we first present some basic notation and results that will be used in the proofs of our theorems. We refer to [1–4] for details.
Definition 1.1
Lemma 1.2
If \(x\in AC^{n}[0,1]\), then the Caputo fractional derivative \({}^{C}D_{0^{+}}^{\alpha}x(t)\) exists almost everywhere on \([0,1]\), where \(AC^{n}[0,1]=\{x\in C^{n1}[0,1] \mid x^{(n1)}\textit{ is absolutely continuous} \}\), and n is the smallest integer greater than or equal to α.
Lemma 1.3
In the rest of this section, we present some notation and some known results on cone theory. We refer to [35, 36], and [37] for details.
Let E be a real Banach space partially ordered by a cone \(P\subset E\), that is, \(x\leq y\) iff \(yx\in P\). If \(x\leq y\) and \(x\neq y\), then we write \(x< y\) or \(y>x\). By θ we denote the zero element of E. A cone P is said to be normal if there exists a positive number N, called the normal constant of P, such that \(\theta\leq x\leq y\) implies \(\Vert x\Vert\leq N\Vert y\Vert\). For \(u,v\in E\), \(u\leq v\), denote \([u,v]=\{x\in E \mid u\leq x\leq v\}\).
Let \(D\subset E\). An operator \(T:D\rightarrow E\) is said to be increasing if for \(x,y\in D\), \(x\leq y\Rightarrow Tx\leq Ty\). An element \(x^{*}\in D\) is called a fixed point of T if \(Tx^{*}=x^{*}\).
Lemma 1.4
 (i)
If \(Tx\neq\rho x\) for \(x\in P\cap\partial\Omega\) and \(\rho\geq1\), then \(i(T,P\cap\Omega,P)=1\).
 (ii)
If \(Tx\neq x\) and \(\Tx\\geq\x\\) for \(x\in P\cap \partial\Omega\), then \(i(T,P\cap\Omega,P)=0\).
Lemma 1.5
([37])
 (G1)
\(A(P_{e})\subset P_{e}^{*}\) and \(A(rx)\geq rAx\) for \(x\in P_{e}\) and \(r\in(0,1)\);
 (G2)
\(T(P_{e})\subset P_{e}\), and there exists \(\tau\in(0,1)\) such that \(T(rx)\geq r^{\tau}Tx\) for \(x\in P_{e}\), \(r\in(0,1)\).
 (i)
the recurrent sequence \(u_{n}=\lambda Au_{n1}+ Tu_{n1}\) (\(n=1,2,\ldots\)) for any \(u_{0}\in P_{e}\) converges to \(x_{\lambda}\), that is, \(\lim_{n\rightarrow+\infty}\u_{n}x_{\lambda}\=0\);
 (ii)
\(x_{\lambda}\) is increasing in λ for \(\lambda\in[0,\lambda^{*})\);
 (iii)
\(x_{\lambda}\) is continuous with respect to λ for \(\lambda\in[0,\lambda^{*})\).
Lemma 1.6
([37])
2 Properties of positive solutions and equivalent operator equation
In this section, we will apply Lemma 1.3 to present the existence and uniqueness results for a solution of a linear fractional boundary value problem; moreover, we present the operator equation equivalent to FEP (1). This is important for our research.
We set \(E=C[0,1]\), the Banach space of all continuous functions on \([0,1]\) with the norm \(\x\=\max\{x(t) \mid t\in[0,1]\} \). Let \(P=\{x\in C[0,1]\mid x(t)\geq0, t\in[0,1]\}\). It is clear that P is a normal cone with normal constant 1.
Lemma 2.1
Proof
Lemma 2.2
Proof
It is clear by (4) that also \(0\leq G(t,s)\leq G(s,s)\) for \(0\leq t\leq s\leq1\). This ends the proof. □
Lemma 2.3
Proof
Lemma 2.4
Proof
According to Lemma 2.1, it is evident that the solution of FEP (1) is the solution of the integral equation (10).
By Lemmas 2.3 and 2.4 we can get the following result.
Lemma 2.5
If \(x\in P\backslash\{\theta\}\) is a solution of FEP (1), then \(x(t)\geq0\) for \(t\in[0,1]\) and \(x(t)>0\) for \(t\in(0,1)\), that is, \(x(t)\) is a positive solution of FEP (1).
Lemma 2.6
The operator \(C_{(\lambda,\mu)}:P\to K\) is completely continuous.
Proof
According to Lemma 2.3, it is easy to verify that \(C_{(\lambda,\mu)}(P)\subset K\).
3 Existence and nonexistence results
 (H1)
\(f(t,x)\) is nondecreasing in \(x\in[0,+\infty)\) for fixed \(t \in[0,1]\);
 (H2)
\(g(x)\) is nondecreasing in \(x\in[0,+\infty)\);
 (H3)
there exists \(r_{0}>0\) such that \(g(r_{0})\int_{0}^{1}k(s)\,ds< r_{0}\);
 (H4)
\(g_{\infty}:=\liminf_{x\to+\infty}\frac{g(x)}{x}>\frac {a+b}{\gamma_{0}(a\sigma_{1}+b)\int_{\sigma_{1}}^{\sigma_{2}}k(s)\,ds}\) and \(\int_{\sigma_{1}}^{\sigma_{2}}k(s)\,ds>0\);
 (H5)
\(f_{\infty}:=\limsup_{x\to+\infty}\min_{t\in[\sigma _{1},\sigma_{2}]}\frac{f(t,x)}{x} =+\infty\).
In this section, we assume that \(f(t,0)\not\equiv0\), \(t\in[0,1]\).
Lemma 3.1
Suppose that (H1) and (H2) hold. If \(x^{*}\in P\) is a fixed point of \(C_{(\lambda,\mu)}\) and \(\lambda+\mu \neq0\), then \(x^{*}\neq\theta\); moreover, \(x^{*}\) is a positive solution of FEP (1).
Proof
From Lemma 2.6 we have \(C_{(\lambda,\mu)}(P)\subset K\). If (H1) and (H2) hold and \(\lambda+\mu\neq0\), then by Lemma 3.1, \(x^{*}\in K\) is not a fixed point of \(C_{(\lambda,\mu )}\), which implies that \(x^{*}\) is not a positive solution of FEP (1).
Lemma 3.2
 (i)
If there exist \(\lambda_{0}, \mu_{0}>0\) such that \(C_{(\lambda _{0},\mu_{0})}\) has a fixed point \(x_{0}\in K\), then \(C_{(\lambda,\mu)}\) has a fixed point \(x\in K\) for any \(0\leq\lambda\leq \lambda_{0}\), \(0\leq\mu\leq\mu_{0}\), \(\lambda+\mu\neq0\).
 (ii)
If there exist \(\lambda_{0}', \mu_{0}'>0\) such that \(C_{(\lambda_{0}', \mu_{0}')}\) has no fixed points in K, then \(C_{(\lambda,\mu)}\) has no fixed points in K for \(\lambda\geq\lambda _{0}'\), \(\mu\geq\mu_{0}'\).
Proof
It is evident that conclusion (i) implies conclusion (ii). This completes the proof. □
Denote \(f_{m}(x)=\max_{t\in[0,1]}f(t,x)\) for \(x\geq0\). If (H1) holds, then \(f_{m}(x)>0\) for \(x\geq0\).
Theorem 3.3
Suppose that (H1), (H2), and (H3) hold. Then there exist \(\lambda_{0}, \mu_{0}>0\) such that \(C_{(\lambda,\mu)}\) has at least one fixed point \(x\in K\) for \(0\leq \lambda\leq\lambda_{0}\) and \(0\leq\mu\leq\mu_{0}\) with \(\lambda+\mu\neq0\).
Proof
Theorem 3.4
 (i)
FEP (1) has at least two positive solutions for \(0\leq\mu<\mu^{*}\) and \(0<\lambda<\lambda^{*}(\mu)\);
 (ii)
FEP (1) has at least one positive solution for \(0\leq\mu\leq\mu^{*}\), \(\lambda=\lambda^{*}(\mu)\) or \(\mu=\mu^{*}\), \(0\leq \lambda\leq\lambda^{*}(\mu^{*})\) or \(0<\mu\leq\mu^{*}\), \(\lambda=0\);
 (iii)
FEP (1) has no positive solutions for \(0\leq \mu\leq\mu^{*}\), \(\lambda>\lambda^{*}(\mu)\) or \(\mu>\mu^{*}\), \(\lambda\geq0\).
Proof
We prove all statements by five steps.
Case 2. \(\{x_{n}\}_{1}^{+\infty}\) is unbounded, that is, there exists a subsequence of \(\{x_{n}\}_{1}^{+\infty}\), still denoted by \(\{x_{n}\} _{1}^{+\infty}\), such that \(\lim_{n\to+\infty}\x_{n}\=+\infty\).
Step 2. When \(\mu=\mu^{*}\), we prove that there exist \(\bar{\lambda}\geq0\) and \(\bar{x}\in K\) such that \(C_{(\bar{\lambda},\mu^{*})}\bar{x}=\bar{x}\).
By (13) there exists an increasing sequence \(\{\mu_{n}\}_{1}^{+\infty }\subset\Lambda\) such that \(\lim_{n\to+\infty}\mu_{n}=\mu^{*}\). By (12) there exist \(\lambda_{n}>0\) and \(x_{n}\in K\) such that \(C_{(\lambda_{n},\mu_{n})}x_{n}=x_{n}\). Arguing similarly to Case 2, we can show that \(\{x_{n}\}_{1}^{+\infty}\) is bounded, that is, there exists a constant \(M>0\) such that \(\x_{n}\\leq M\), \(n=1,2,\ldots \) . Arguing similarly to Case 1, we obtain that \(\{\lambda_{n}\}_{1}^{+\infty}\) is bounded and denote \(\lambda'=\sup\{\lambda_{n} \mid n=1,2,\ldots\}\).
By the preceding discussion we obtain that \(\mu^{*}>0\) and that the function \(\lambda^{*}:[0,\mu^{*}]\to[0,+\infty)\) defined by (17) and (20) satisfies (11).
Step 4. It is evident by (14), (15), (18), (21), and the discussion of Step 2, together with Lemma 3.2. that conclusions (ii) and (iii) hold.
Step 5. We prove conclusion (i), that is, that FEP (1) has at least two positive solutions for \(0\leq\mu<\mu^{*}\) and \(0<\lambda <\lambda^{*}(\mu)\).

the left side of \(\mbox{(23)} \geq(\lambda^{*}(\mu)\lambda)\int_{0}^{1}G(t_{0},s)f(s,\bar{x}^{*}(s))\,ds >0\), and

the right side of \(\mbox{(23)}=e(t_{0})\int_{0}^{1}k(s) (g(x_{1}^{*}(s))g(\bar{x}^{*}(s)) )\,ds \leq0\),
Remark 3.5
Similarly to the proof of Theorem 3.4, we obtain the following result.
Theorem 3.6
 (i)
FEP (1) has at least two positive solutions for \(0<\lambda<\lambda^{*}\) and \(0\leq\mu<\mu^{*}(\lambda)\);
 (ii)
FEP (1) has at least one positive solution for \(0\leq \lambda<\lambda^{*}\), \(\mu=\mu^{*}(\lambda)\), or \(\lambda=\lambda^{*}\), \(0\leq\mu\leq\mu^{*}(\lambda)\), or \(0<\lambda\leq\lambda^{*}\), \(\mu=0\);
 (iii)
FEP (1) has no positive solutions for \(0\leq\lambda \leq\lambda^{*}\), \(\mu>\mu^{*}(\lambda)\) or \(\lambda>\lambda^{*}\), \(\mu\geq0\).
Remark 3.7
4 Uniqueness and dependence on parameter
 (H6)
for any \(r\in(0,1)\) and \(x\in[0,+\infty)\), \(f(t,rx)\geq r f(t,x)\) for \(t\in[0,1]\);
 (H7)
for any \(r\in(0,1)\) and \(x\in[0,+\infty)\), \(g(rx)\geq r g(x)\);
 (H8)
\(\lim_{x\to+\infty}g(x)<+\infty\);
 (H9)there exists a constant \(\kappa\in(0,1)\) such that$$f(t,rx)\geq r^{\kappa}f(t,x), \quad\forall t\in[0,1], r\in(0,1), x\in [0, \infty). $$
Let \(e(t)=\frac{at+b}{a+b}\). Define \(P_{e}\) and \(P_{e}^{*}\) as in Section 1 and the operators A and \(T_{\mu}\) as in Section 2.
Lemma 4.1
 (i)
\(A:P\to P\) is an increasing operator, and \(A(P)\subset P_{e}^{*}\);
 (ii)
\(T_{\mu}: P\to P\) is an increasing operator, and \(T_{\mu}(P)\subset P_{e}\) for \(\mu>0\);
Proof
Lemma 4.2
Proof
Lemma 4.3
Proof
Remark 4.4
By Lemma 4.1 we have \(C_{(\lambda,\mu)}(P)\subset P_{e}\). This means that if x is not a fixed point of \(C_{(\lambda,\mu)}\) in \(P_{e}\), then x is not a fixed point of \(C_{(\lambda,\mu)}\) in P. So, in this section, we study the existence and nonexistence of the unique fixed point of \(C_{(\lambda,\mu)}\) in \(P_{e}\).
Theorem 4.5
 (i)
for any \(u_{0}\in P\), setting \(u_{n}(t)=\lambda\int_{0}^{1}G(t,s)f(s,u_{n1}(s))\,ds+ e(t) (\int _{0}^{1}k(s)g(u_{n1}(s))\,ds+ \mu )\), \(n=1,2,\ldots\) , we have \(\lim_{n\rightarrow\infty}\u_{n}x_{\lambda}\=0\);
 (ii)
\(x_{\lambda}\) is increasing in λ for \(\lambda\in [0,\lambda^{*}(\mu))\);
 (iii)
\(x_{\lambda}\) is continuous with respect to λ for \(\lambda\in[0,\lambda^{*}(\mu))\).
Proof
Given \(\mu>0\), Lemma 4.1 and (H6) imply that A satisfies condition (G1). It is easy to see by Lemmas 4.1 and 4.2 that \(T_{\mu}\) satisfies condition (G2). Consequently, Theorem 4.5 follows from Lemma 1.5. The proof is complete. □
Theorem 4.6
Proof
Corollary 4.7
 (i)
for any \(u_{0}\in P\), setting \(u_{n}=\lambda Au_{n1}+T_{\mu}u_{n1 }\) (\(n=1,2,\ldots\)), we have \(\lim_{n\rightarrow\infty}\u_{n}x_{\lambda}\=0\);
 (ii)
\(x_{\lambda}\) is nondecreasing in λ for \(\lambda\in [0,+\infty)\);
 (iii)
\(x_{\lambda}\) is continuous with respect to λ for \(\lambda\in[0,+\infty)\);
 (iv)
if \(f(t,1)\not\equiv0\), then \(\lim_{\lambda\to0+}\ x_{\lambda}x_{\mu}\=0\) and \(\lim_{\lambda\to+\infty}\x_{\lambda}\=+\infty\), where \(x_{\mu}\) is the unique fixed point of \(T_{\mu}\) in \(P_{e}\).
Proof
When \(g(x)\) is a constant function c (≥0), it is evident that g satisfies (H2), (H7), and (H8). For given \(\mu>0\), \(T_{\mu}x(t)=(c\int_{0}^{1}k(s)\,ds+\mu)e(t):=x_{\mu}(t)\). We can obtain the following results.
Corollary 4.8
 (i)
there exists \(\lambda^{*}(\mu)\geq\frac{\Gamma(\alpha +1)}{2F_{\infty}}\) such that FEP (1) with \(g\equiv c\) has a unique positive solution \(x_{\lambda}\) for \(\lambda\in[0,\lambda^{*}(\mu))\) and has no solution for \(\lambda\geq\lambda^{*}(\mu)\). Moreover, for any \(u_{0}\in P\), setting \(u_{n}= x_{\mu}+\lambda Au_{n1} \) (\(n=1,2,\ldots\)), we have \(\lim_{n\rightarrow\infty}\u_{n}x_{\lambda}\=0\);
 (ii)
\(x_{\lambda}\) is nondecreasing in λ for \(\lambda\in[0,\lambda^{*}(\mu))\);
 (iii)
\(x_{\lambda}\) is continuous with respect to λ for \(\lambda\in[0,\lambda^{*}(\mu))\); moreover, \(\lim_{\lambda\to0+}\ x_{\lambda}x_{\mu}\=0\).
Corollary 4.9
 (i)
FEP (1) with \(g\equiv c\) has a unique positive solution \(x_{\lambda}\) for any \(\lambda\in[0,+\infty)\); moreover, for any \(u_{0}\in P\), setting \(u_{n}=x_{\mu}+\lambda Au_{n1} \) (\(n=1,2,\ldots\)), we have \(\lim_{n\rightarrow\infty}\u_{n}x_{\lambda}\=0\);
 (ii)
\(x_{\lambda}\) is nondecreasing in λ for \(\lambda\in [0,+\infty)\);
 (iii)
\(x_{\lambda}\) is continuous with respect to λ for \(\lambda\in[0,+\infty)\);
 (iv)
\(\lim_{\lambda\to0+}\x_{\lambda}x_{\mu}\=0\) and \(\lim_{\lambda\to+\infty}\x_{\lambda}\=+\infty\).
Corollary 4.10
Assume that (H1) and (H9) hold. Then conclusions (i), (ii), (iii), and (iv) in Corollary 4.9 hold.
Remark 4.11
5 Examples
In the section, we give two concrete examples to illustrate our main results.
Example 5.1
 (i)
FEP (32) has at least two positive solutions for \(0<\mu<\mu ^{*}\) and \(0\leq\lambda<\lambda^{*}(\mu)\);
 (ii)
FEP (32) has at least one positive solution for \(0\leq\mu<\mu ^{*}\) and \(\lambda=\lambda^{*}(\mu)\);
 (iii)
FEP (32) has no positive solutions for \(0\leq\mu\leq\mu ^{*}\), \(\lambda>\lambda^{*}(\mu)\) or \(\mu>\mu^{*}\), \(\lambda\geq0\).
Example 5.2

for \(0\leq x \leq1\), we have \(f(t,rx)=\frac{tr}{3}x=rf(t,x)\);

if \(x >1\) and \(0< rx\leq1\), then we have \(f(t,rx)=\frac{tr}{3}x\geq \frac{r t}{3}(\frac{1}{2}x+\frac{1}{2}\sqrt{x})\geq \frac{tr}{6}(x+\sqrt{x})= rf(t,x)\);

if \(x >1\) and \(rx>1\), then we have \(f(t,rx)=\frac{t}{6}(rx+\sqrt {rx})\geq\frac{tr}{6}(x+\frac{1}{\sqrt{r}}\sqrt{x})\geq rf(t,x)\),
 (i)
for any \(u_{0}\in P\), setting \(u_{n}(t)=\lambda\int_{0}^{1}G(t,s)f(s,u_{n1}(s))\,ds+e(t) (\int _{0}^{1}k(s)g(u_{n1}(s))\,ds+ \mu )\), \(n=1,2,\ldots\) , we have \(\lim_{n\rightarrow\infty}\ u_{n}x_{\lambda}\=0\);
 (ii)
\(x_{\lambda}\) is increasing in λ for \(\lambda\in[0,\lambda ^{*}(\mu))\);
 (iii)
\(x_{\lambda}\) is continuous with respect to λ for \(\lambda \in[0,\lambda^{*}(\mu))\).
Declarations
Acknowledgements
The authors sincerely thank the reviewers for their valuable suggestions and useful comments. This research was supported by the NNSF of China (11361047), the University Natural Science Research Develop Foundation of Shanxi Province of China (20111021, 2013156), and Research Project Supported by Shanxi Scholarship Council of China (2013102).
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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