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 Open Access
The solution for a class of sum operator equation and its application to fractional differential equation boundary value problems
 Hui Wang^{1} and
 Lingling Zhang^{1}Email author
 Received: 28 May 2015
 Accepted: 27 October 2015
 Published: 5 November 2015
Abstract
In this paper we study a class of sum operator equation \(Ax+Bx+C(x,x)=x\) on ordered Banach spaces, where A is an increasing operator, B is a decreasing operator, and C is a mixed monotone operator. The existence and uniqueness of its positive solution are obtained by using the properties of cone and fixed point theorems for mixed monotone operators. As an application, we utilize the obtained results to study the existence and uniqueness of positive solutions for nonlinear fractional differential equation boundary value problems.
Keywords
 operator equation
 fixed point theorem
 fractional differential equation
 positive solution
1 Introduction
Over the past several decades, nonlinear functional analysis has been an active area of research in mechanics, elasticity, fluid dynamics, and so on. As an important branch of nonlinear functional analysis, the nonlinear operator theorem and its application in nonlinear differential equations have attracted much attention (see [1–7]). It is well known that the existence and uniqueness of positive solutions to nonlinear operator equations are very important in theory and applications. Many authors have studied this problem; for a small sample of such work, we refer the reader to [8–16].
Reference [14] has successively considered the sum operator equation \(Mx+Qx+Nx=x\) on ordered Banach spaces, where M is an increasing, αconcave operator, Q is an increasing subhomogeneous operator, and N is a homogeneous operator. The existence and uniqueness of its positive solutions are obtained by using the properties of cones and a fixed point theorem for increasing general βconcave operators.
In [15], the sum operator equation \(A(x,x)+Bx=x\) has been considered. A is a mixed monotone operator and B is an increasing αconcave (or subhomogeneous) operator. By using the properties of cones and a fixed point theorem for mixed monotone operators, respectively, the author established the existence and uniqueness of positive solutions for the operator equation.
The content of this paper is organized as follows. In Section 2, we present some definitions, lemmas and basic results that will be used in the proofs of our theorems. In Section 3, we consider the existence and uniqueness of positive solutions for the operator equation (1.1). In Section 4, we utilize the results obtained in Section 3 to study the existence and uniqueness of positive solutions for nonlinear fractional differential equation boundary value problems.
2 Preliminaries
For convenience of the reader, we present here some definitions, lemmas, and basic results that will be used in the proofs of our theorems.
Suppose that \((E,\\cdot\)\) is 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. Recall that a nonempty closed convex set \(P\subset E\) is a cone if it satisfies (i) \(x\in P\), \(\lambda\geq0\Rightarrow\lambda x\in P\); (ii) \(x\in P\), \(x\in P \Rightarrow x=\theta\).
Putting \(\mathring{P}=\{x\in P \mid x \text{ is an interior point of } P\}\), a cone P is said to be solid if P̊ is nonempty. Moreover, P is called normal if there exists 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 called the normality constant of P. If \(x_{1}, x_{2}\in E\), the set \([x_{1},x_{2}]=\{x\in E\mid x_{1}\leq x\leq x_{2}\} \) is called the order interval between \(x_{1}\) and \(x_{2}\). We say that an operator \(A: E\rightarrow E\) is increasing (decreasing) if \(x\leq y\) implies \(Ax\leq Ay\) (\(Ax\geq Ay\)).
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.d., \(h\geq\theta \) 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 \(P_{h}\subset P\).
Definition 2.1
([17])
Definition 2.2
([17])
Definition 2.3
([17])
\(A:P\times P\rightarrow P\) is said to be a mixed monotone operator if \(A(x,y)\) is increasing in x and decreasing in y, i.e., \(u_{i},v_{i}\ (i=1,2)\in P\), \(u_{1}\leq u_{2}\), \(v_{1}\geq v_{2}\) imply \(A(u_{1},v_{1})\leq A(u_{2},v_{2})\). An element \(x\in P\) is called a fixed point of A if \(A(x,x)=x\).
Lemma 2.4
(See Lemma 2.1 and Theorem 2.1 in [12])
 (A_{1}):

there exists \(h\in P\) with \(h\neq\theta\) such that \(T(h,h)\in P_{h}\);
 (A_{2}):

for any \(u,v\in P\) and \(t\in(0,1)\), there exists \(\varphi(t)\in(t,1]\) such that \(T(tu,t^{1}v)\geq\varphi(t)T(u,v)\).
 (1)
\(T:P_{h}\times P_{h}\rightarrow P_{h}\);
 (2)
there exist \(u_{0},v_{0}\in P_{h}\) and \(r\in(0,1)\) such that \(rv_{0}\leq u_{0}< v_{0}\), \(u_{0}\leq T(u_{0},v_{0})\leq T(v_{0},u_{0})\leq v_{0} \);
 (3)
T has a unique fixed point \(x^{\ast}\) in \(P_{h}\);
 (4)for any initial values \(x_{0},y_{0}\in P_{h}\), constructing successively the sequenceswe have \(x_{n}\rightarrow x^{\ast}\) and \(y_{n}\rightarrow x^{\ast}\) as \(n\rightarrow\infty\).$$ x_{n}=T(x_{n1},y_{n1}) , \qquad y_{n}=T(y_{n1},x_{n1}),\quad n=1,2,\ldots, $$
3 Main results
In this section we consider the existence and uniqueness of positive solutions for the operator equation \(Ax+Bx+C(x,x)=x\). We assume that E is a real Banach space with a partial order introduced by a normal cone P of E. Take \(h\in E\), \(h>\theta\), \(P_{h}\) is given as in the preliminaries.
Theorem 3.1
 (H_{1}):

there is \(h_{0}\in P_{h}\) such that \(Ah_{0}\in P_{h}\), \(Bh_{0}\in P_{h}\), \(C(h_{0},h_{0})\in P_{h}\);
 (H_{2}):

there exists a constant \(\delta>0\) such that \(C(x,y)\geq\delta(Ax+By)\), \(\forall x,y\in P\).
 (1)
\(A: P_{h}\rightarrow P_{h}\), \(B: P_{h}\rightarrow P_{h}\), \(C: P_{h}\times P_{h}\rightarrow P_{h}\);
 (2)there exist \(u_{0},v_{0}\in P_{h}\) and \(r\in(0,1)\) such that$$ rv_{0}\leq u_{0}< v_{0} , \qquad u_{0}\leq Au_{0}+Bv_{0}+C(u_{0},v_{0}) \leq Av_{0}+Bu_{0}+C(v_{0},u_{0})\leq v_{0} ; $$
 (3)
the operator equation \(Ax+Bx+C(x,x)=x\) has a unique solution \(x^{\ast}\) in \(P_{h}\);
 (4)for any initial values \(x_{0},y_{0}\in P_{h}\), constructing successively the sequenceswe have \(x_{n}\rightarrow x^{\ast}\) and \(y_{n}\rightarrow y^{\ast}\) as \(n\rightarrow\infty\).$$\begin{aligned}& x_{n}=Ax_{n1}+By_{n1}+C(x_{n1},y_{n1}) , \\& y_{n}=Ay_{n1}+Bx_{n1}+C(y_{n1},x_{n1}) , \quad n=1,2,\ldots , \end{aligned}$$
Proof
First step: we will demonstrate \(A: P_{h}\rightarrow P_{h}\), \(B: P_{h}\rightarrow P_{h}\), \(C: P_{h}\times P_{h}\rightarrow P_{h}\).
The second step is to demonstrate the conclusions (2)(4) are correct.
From the proof of Theorem 3.1, we can easily prove the following conclusion.
Corollary 3.2
Let \(\alpha\in(0,1)\). Suppose that \(A: P_{h}\rightarrow P_{h}\) is an increasing subhomogeneous operator, \(B: P_{h}\rightarrow P_{h}\) is a decreasing operator, \(C: P_{h}\times P_{h}\rightarrow P_{h}\) is a mixed monotone operator, assume that (3.1) and (H_{2}) hold. Then the conclusions (2)(4) of Theorem 3.1 hold.
Corollary 3.3
 (H_{3}):

there is \(h_{0}\in P_{h}\) such that \(Ah_{0}\in P_{h}\), \(C(h_{0},h_{0})\in P_{h}\);
 (H_{4}):

there exists a constant \(\delta>0\) such that \(C(x,y)\geq\delta Ax\), \(\forall x,y\in P\).
 (1)
\(A: P_{h}\rightarrow P_{h}\), \(C: P_{h}\times P_{h}\rightarrow P_{h}\);
 (2)there exist \(u_{0},v_{0}\in P_{h}\) and \(r\in(0,1)\) such that$$ rv_{0}\leq u_{0}< v_{0} , \qquad u_{0}\leq Au_{0}+C(u_{0},v_{0})\leq Av_{0}+C(v_{0},u_{0})\leq v_{0} ; $$
 (3)
the operator equation \(Ax+C(x,x)=x\) has a unique solution \(x^{\ast }\) in \(P_{h}\);
 (4)for any initial values \(x_{0},y_{0}\in P_{h}\), constructing successively the sequenceswe have \(x_{n}\rightarrow x^{\ast}\) and \(y_{n}\rightarrow y^{\ast}\) as \(n\rightarrow\infty\).$$ x_{n}=Ax_{n1}+C(x_{n1},y_{n1}) ,\qquad y_{n}=Ay_{n1}+C(y_{n1},x_{n1}) ,\quad n=1,2,\ldots , $$
Corollary 3.4
 (H_{5}):

there is \(h_{0}\in P_{h}\) such that \(Bh_{0}\in P_{h}\), \(C(h_{0},h_{0})\in P_{h}\);
 (H_{6}):

there exists a constant \(\delta>0\) such that \(C(x,y)\geq\delta By\), \(\forall x,y\in P\).
 (1)
\(B: P_{h}\rightarrow P_{h}\), \(C: P_{h}\times P_{h}\rightarrow P_{h}\);
 (2)there exist \(u_{0},v_{0}\in P_{h}\) and \(r\in(0,1)\) such that$$ rv_{0}\leq u_{0}< v_{0} , \qquad u_{0}\leq Bv_{0}+C(u_{0},v_{0})\leq Bu_{0}+C(v_{0},u_{0})\leq v_{0} ; $$
 (3)
the operator equation \(Bx+C(x,x)=x\) has a unique solution \(x^{\ast }\) in \(P_{h}\);
 (4)for any initial values \(x_{0},y_{0}\in P_{h}\), constructing successively the sequenceswe have \(x_{n}\rightarrow x^{\ast}\) and \(y_{n}\rightarrow y^{\ast}\) as \(n\rightarrow\infty\).$$ x_{n}=By_{n1}+C(x_{n1},y_{n1}) ,\qquad y_{n}=Bx_{n1}+C(y_{n1},x_{n1}) , \quad n=1,2,\ldots , $$
Corollary 3.5
 (1)there exist \(u_{0},v_{0}\in P_{h}\) and \(r\in(0,1)\) such that$$ rv_{0}\leq u_{0}< v_{0} ,\qquad u_{0} \leq C(u_{0},v_{0})\leq C(v_{0},u_{0}) \leq v_{0} ; $$
 (2)
the operator equation \(C(x,x)=x\) has a unique solution \(x^{\ast}\) in \(P_{h}\);
 (3)for any initial values \(x_{0},y_{0}\in P_{h}\), constructing successively the sequenceswe have \(x_{n}\rightarrow x^{\ast}\) and \(y_{n}\rightarrow y^{\ast}\) as \(n\rightarrow\infty\).$$ x_{n}=C(x_{n1},y_{n1}) , \qquad y_{n}=C(y_{n1},x_{n1}) , \quad n=1,2,\ldots, $$
Remark 3.6
Corollaries 3.3, 3.4, 3.5 which have been studied in [12, 15, 16] are special cases of Theorem 3.1. In this sense, our results extend and supplement the results in [12, 15, 16].
Theorem 3.7
 (\(\mathrm{H}_{1}'\)):

there is \(h_{0}\in P_{h}\) such that \(Ah_{0}\in P_{h}\), \(Bh_{0}\in P_{h}\), \(C(h_{0},h_{0})\in P_{h}\);
 (\(\mathrm{H}_{2}'\)):

there exists a constant \(\delta>0\) such that \(Ax+C(x,y)\leq\delta By\), \(\forall x,y\in P\).
Proof
Theorem 3.8
 (\(\mathrm{H}_{1}''\)):

there is \(h_{0}\in P_{h}\) such that \(Ah_{0}\in P_{h}\), \(Bh_{0}\in P_{h}\), \(C(h_{0},h_{0})\in P_{h}\);
 (\(\mathrm{H}_{2}''\)):

there exists a constant \(\delta>0\) such that \(By+C(x,y)\leq\delta Ax\), \(\forall x,y\in P\).
Proof
4 Applications
Definition 4.1
([23])
Lemma 4.2
Lemma 4.3
Lemma 4.4
Proof
Lemma 4.5
 (1)
\(G(t,s)\geq0\), \((t,s)\in[0,1]\times[0,1]\);
 (2)
\(\frac{1}{\Gamma(\nu)}s(2s)(1s)^{\nu3}t^{\nu1}\leq G(t,s)\leq \frac{1}{\Gamma(\nu)}(1s)^{\nu3}t^{\nu1}\) for \(t,s\in[0,1]\).
Proof
Theorem 4.6
 (L_{1}):

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

\(f(t,u,v)\) is increasing in \(u\in[0,+\infty)\) for fixed \(t\in[0,1]\) and \(v\in[0,+\infty)\), decreasing in \(v\in[0,+\infty)\) for fixed \(t\in[0,1]\) and \(u\in[0,+\infty)\), and \(g(t,u)\) is increasing in \(u\in[0,+\infty)\) for fixed \(t\in[0,1]\), and \(q(t,v)\) is decreasing in \(v\in[0,+\infty)\) for fixed \(t\in[0,1]\);
 (L_{3}):

\(g(t,\lambda u)\geq\lambda g(t,u)\) for \(\lambda\in(0,1)\), \(t\in[0,1]\), \(u\in[0,+\infty)\), and \(q(t,\lambda^{1}v)\geq\lambda q(t,v)\) for \(\lambda\in(0,1)\), \(t\in[0,1]\), \(v\in[0,+\infty)\), and there exists a constant \(\alpha\in(0,1)\) such that \(f(t,\lambda u,\lambda ^{1} v)\geq\lambda^{\alpha}f(t,u,v)\), \(\forall t\in[0,1]\), \(\lambda\in (0,1)\), \(u,v\in[0,+\infty)\);
 (L_{4}):

there exists a constant \(\delta>0\) such that \(f(t,u,v)\geq \delta(g(t,u)+q(t,v))\), \(t\in[0,1]\), \(u,v\geq0\).
 (1)there exists \(u_{0},v_{0}\in P_{h}\) and \(r\in(0,1)\) such that \(rv_{0}\leq u_{0}< v_{0}\) andwhere \(h(t)=t^{\nu1}\), \(t\in[0,1]\);$$ \textstyle\begin{cases} u_{0}(t)\leq\int_{0}^{1}G(t,s)[f(s,u_{0}(s),v_{0}(s))+g(s,u_{0}(s))+q(s,v_{0}(s))]\, ds,\quad t\in[0,1], \\ v_{0}(t)\geq\int_{0}^{1}G(t,s)[f(s,v_{0}(s),u_{0}(s))+g(s,v_{0}(s))+q(s,u_{0}(s))]\, ds,\quad t\in[0,1], \end{cases} $$
 (2)
the problem (4.1) has a unique positive solution \(u^{\ast}\) in \(P_{h}\);
 (3)for any \(x_{0},y_{0}\in P_{h}\), constructing successively the sequenceswe have \(\x_{n}u^{\ast}\\rightarrow0\) and \(\y_{n}u^{\ast }\\rightarrow0\) as \(n\rightarrow\infty\).$$ \textstyle\begin{cases} x_{n+1}(t)= \int_{0}^{1}G(t,s)[f(s,x_{n}(s),y_{n}(s))+g(s,x_{n}(s))+q(s,y_{n}(s))]\, ds,\quad n=0,1,2,\ldots, \\ y_{n+1}(t)= \int_{0}^{1}G(t,s)[f(s,y_{n}(s),x_{n}(s))+g(s,y_{n}(s))+q(s,x_{n}(s))]\, ds,\quad n=0,1,2,\ldots, \end{cases} $$
Proof
First, we prove that C is a mixed monotone operator, A is increasing and B is decreasing.
Second, we show that B, C satisfies the condition (3.1) and A is subhomogeneous operator.
Third, we show that \(Ah\in P_{h}\), \(Bh\in P_{h}\), and \(C(h,h)\in P_{h}\).
Lastly, we show the condition (H_{2}) of Theorem 3.1 is satisfied.
By using Theorem 3.7, we can easily prove the following conclusion.
Theorem 4.7
 (L_{5}):

\(f(t,\lambda u,\lambda^{1} v)\geq\lambda f(t,u,v)\), \(\forall t\in[0,1]\), \(\lambda\in(0,1)\), \(u,v\in[0,+\infty)\) and \(g(t,\lambda u)\geq\lambda g(t,u)\) for \(\lambda\in(0,1)\), \(t\in[0,1]\), \(u\in[0,+\infty)\), and there exists a constant \(\alpha\in(0,1)\) such that \(q(t,\lambda^{1}v)\geq\lambda^{\alpha} q(t,v)\) for \(\lambda\in (0,1)\), \(t\in[0,1]\), \(v\in[0,+\infty)\);
 (L_{6}):

there exists a constant \(\delta>0\) such that \(g(t,u)+f(t,u,v)\leq\delta q(t,v)\), \(t\in[0,1]\), \(u,v \geq0\).
By using Theorem 3.8, we can easily prove the following conclusion.
Theorem 4.8
 (L_{7}):

\(f(t,\lambda u,\lambda^{1} v)\geq\lambda f(t,u,v)\), \(\forall t\in[0,1]\), \(\lambda\in(0,1)\), \(u,v\in[0,+\infty)\) and there exists a constant \(\alpha\in(0,1)\) such that \(g(t,\lambda u)\geq \lambda^{\alpha} g(t,u)\) for \(\lambda\in(0,1)\), \(t\in[0,1]\), \(u\in [0,+\infty)\), and \(q(t,\lambda^{1}v)\geq\lambda q(t,v)\) for \(\lambda \in(0,1)\), \(t\in[0,1]\), \(v\in[0,+\infty)\);
 (L_{8}):

there exists a constant \(\delta>0\) such that \(q(t,v)+f(t,u,v)\leq\delta g(t,u)\), \(t\in[0,1]\), \(u,v\geq0\).
Example 4.9
Declarations
Acknowledgements
The research was partially supported by the National Natural Science Foundation of China (No. 61250011) and (No. 61473180) and State Key Laboratory of Explosion Science and Technology (Beijing Institute of Technology) Openend Funds.
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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