 Research
 Open Access
Fixed point theorems for a class of nonlinear sumtype operators and application in a fractional differential equation
 Hui Wang^{1},
 Lingling Zhang^{1}Email author and
 Xiaoqiang Wang^{2}
 Received: 2 May 2018
 Accepted: 5 September 2018
 Published: 18 September 2018
Abstract
In this paper, we consider the fixed point for a class of nonlinear sumtype operators ‘\(A+B+C\)’ on an ordered Banach space, where A, B are two mixed monotone operators, C is an increasing operator. Without assuming the existence of upperlower solutions or compactness or continuity conditions, we prove the unique existence of a positive fixed point and also construct two iterative schemes to approximate it. As applications, we research a nonlinear fractional differential equation with multipoint fractional boundary conditions. By using the obtained fixed point theorems of sumtype operator, we get the sufficient conditions which guarantee the existence and uniqueness of positive solutions. At last, a specific example is provided to illustrate our result.
Keywords
 Sumtype operator
 Existence and uniqueness
 Positive solution
 Fractional differential equation
 Fractional boundary condition
MSC
 47H10
 47H07
 34B10
 34B18
1 Introduction
With a significant development and extensive applications in various differential and integral equations, nonlinear operators theory has been an active area of research in nonlinear functional analysis. Over the past several decades, much attention has been paid to various fixed point theorems for the single nonlinear operator, and a lot of important results have been obtained, see for example [1–10]. Thereinto, without requiring the operators to be continuous or compact or having the upperlower solutions, the authors present some important and interesting fixed point theorems (see [1, 4–10]).
In [11], the operator A stands for a mixed monotone operator, B represents a sublinear operator. By using the partial ordering theory and monotone iterative technique, Sang gets the existence and uniqueness of solutions. Note that, in this study, the author does not require the operators to have upperlower solutions.
In [12], \(A:P\times P\rightarrow P\) is a mixed monotone operator verifying a more general concave property, \(B:E\times E\rightarrow E\) is a sublinear operator. Amor proves that the operator equation has exactly one fixed point in \([u_{0},v_{0}]\).
In [13], \(A:P\times P\rightarrow P\) is a mixed monotone operator, \(B:P\rightarrow P\) is an increasing subhomogeneous operator or αconcave operator. By applying a fixed point theorem for mixed monotone operators, Zhai and Hao get some existence and uniqueness results of positive solutions.
As applications, the above results on the sumtype nonlinear operators have been widely applied to study nonlinear differential and integral equations, see [11–23] and the references therein. In [12], the new fixed point theorems are used to prove positive solutions to a second order Neumann boundary value problem, a Sturm–Liouville boundary value problem, and a nonlinear elliptic boundary value problem for the Lane–Emden–Fowler equation. In [13, 19, 20], authors investigate the existence and uniqueness results for some kinds of fractional differential equations and nonlinear elastic beam equations via the fixed point theorems in [13]. Also, based on a method originally due to Zhai and Anderson [14], Feng et al. present the existence and uniqueness of positive solutions for nonlinear elastic beam equations, Lane–Emden–Fowler equations, and a class of fractional differential equation with integral boundary conditions in [14, 21]. Besides, these fixed point theorems can also be used to deal with some singular problems. In [22, 23], the unique existence of positive solutions for singular fractional differential systems with coupled integral boundary conditions and with integral boundary conditions is studied by fixed point theorem in [15].
 (1)
\(A(tx,t^{1}y)\geq t^{\alpha}A(x,y)\), \(\alpha\in(0,1)\), \(\forall x, y\in P\), \(t\in(0,1)\).
 (2)
\(B(\cdot,y):P\rightarrow P\) is concave for fixed y; \(B(x,\cdot):P\rightarrow P\) is convex for any fixed x.
 (3)
C is a subhomogeneous operator.
 (1)
\(A(tx,t^{1}y)\geq t A(x,y)\), \(\forall x, y\in P\), \(t\in(0,1)\).
 (2)
\(B(\cdot,y):P\rightarrow P\) is concave for fixed y; \(B(x,\cdot) :P\rightarrow P\) is convex for any fixed x.
 (3)
C is a αconcave operator.
After obtaining the unique existence results for operator Eq. (1.1), we also construct two iterative sequences for uniformly approximating the positive solution. Then, we utilize the obtained fixed point theorems to study the existence and uniqueness of positive solutions for a nonlinear fractional differential equation with multipoint fractional boundary conditions. Also, we give a specific example to demonstrate our abstract result. The characteristic features presented in this paper are as follows. Firstly, to our knowledge, in the existing literature, there is almost no research on the fixed point for sumtype operator \('A+B+C'\) with operators satisfying the conditions showed in case one and case two. Hence, our research presents new methods to study nonlinear equation problems. Secondly, the work presented in this paper is the generalization and improvement of sumtype operator equation studied in [13], where B is a null operator. Other particular cases of our research were investigated in [15], where C is a null operator. Finally, we also discuss the solution of the nonlinear eigenvalue equation \(A(x, x) + B(x, x) +Cx= \lambda x\) and give dependency to the parameter. Hence, it is worthwhile to investigate the operator Eq. (1.1).
The content of this paper is organized as follows. In Sect. 2, we present some definitions, lemmas that will be used in the proofs of our theorems. In Sect. 3, we consider the unique existence of positive solutions for the operator Eq. (1.1) without assuming operators to be continuous and compact. In Sect. 4, we utilize the results obtained in Sect. 3 to study the existence and uniqueness of positive solutions for nonlinear fractional differential equation with multipoint fractional boundary conditions, and we also give a concrete example to illustrate our result.
2 Preliminaries
For convenience of the reader, we present some definitions, lemmas, and basic results that will be used in the proofs of our theorems. For more details, we refer the reader to [1–3, 24, 25].
Definition 2.1
([24])
 (i)
If \(x\in P\), \(\lambda\geq0\), then \(\lambda x\in P\);
 (ii)
If \(x\in P\) and \(x\in P\), then \(x=\theta\),
About the cone P, we also have the following definitions.
A cone P is said to be solid if P̊ is nonempty, where \(\mathring{P}=\{x\in P \mid x\mbox{ is an interior point of } P\}\).
A cone 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 \(\x\\leq N\y\ \), where N is called the normality constant of P.
Given \(h>\theta\) (i.e., \(h\geq\theta\) and \(h\neq\theta\)), we denote by \(P_{h}\) the set \(P_{h}=\{x\in E\mid x\sim h\}\), in which ∼ is an equivalence relation, i.e., \(x\sim y\) means that there exist \(\lambda >0\) and \(\mu>0\) such that \(\lambda x\leq y\leq\mu x\) for all \(x,y\in E\). It is easy to see that \(P_{h}\subset P\) is convex and \(\lambda P_{h}=P_{h}\) for all \(\lambda>0\). If \(\mathring{P}\neq\theta\) and \(h\in\mathring{P}\), it is clear that \(P_{h}=\mathring{P}\).
Definition 2.2
([2])
An operator \(A:P\times P\rightarrow P\) is said to be a mixed monotone operator if \(A(x,y)\) is increasing in x and decreasing in y, i.e., \(\forall u_{i}, v_{i}(i=1,2)\in P\), \(u_{1}\leq u_{2}\), \(v_{1}\geq v_{2}\) 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 \(A(x,x)=x\).
Definition 2.3
([25])
An operator \(A: E\rightarrow E\) is said to be increasing if for any \(x, y\in E\), \(x\leq y\) implies \(Ax\leq Ay\).
Definition 2.4
([25])
Definition 2.5
([25])
Definition 2.6
([3])
Lemma 2.1
([9])
 \((\mathrm{A}_{1})\) :

There exists \(h\in P\) with \(h\neq\theta\) such that \(T(h,h)\in P_{h}\);
 \((\mathrm{A}_{2})\) :

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 study some fixed point theorems for a class of sumtype operators. This problem is equivalent to researching the existence and uniqueness of positive solutions for the operator equation \(A(x,x)+B(x,x)+Cx=x\), where A, B are the mixed monotone operators with different properties, C is an increasing operator. 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
 (\(\mathrm{L}_{1}\)):

Let \(\alpha\in(0,1)\) for any \(t\in(0,1)\),$$ A\bigl(tx,t^{1}y\bigr)\geq t^{\alpha}A(x,y),\quad \forall x,y\in P. $$(3.1)
 (\(\mathrm{L}_{2}\)):

For any fixed \(y\in P\), \(B(\cdot, y):P\rightarrow P\) is concave, that is, for any \(x_{1}, x_{2}\in P\) with \(x_{2}\leq x_{1}\) and every \(t\in(0,1)\), we haveFor any fixed \(x\in P\), \(B(x,\cdot):P\rightarrow P\) is convex, that is, for any \(y_{1}, y_{2}\in P\) with \(y_{2}\leq y_{1}\) and every \(t\in (0,1)\), we have$$ B\bigl(tx_{1}+(1t)x_{2},y\bigr)\geq tB(x_{1},y)+(1t)B(x_{2},y); $$(3.2)$$ B\bigl(x,ty_{1}+(1t)y_{2}\bigr)\leq tB(x,y_{1})+(1t)B(x,y_{2}). $$(3.3)
 (\(\mathrm{L}_{3}\)):

There exists a constant \(b\in[\frac {1}{2},1]\) such that$$ B(\theta,lh)\geq b B(lh,\theta), \quad\forall l\geq1. $$
 (\(\mathrm{L}_{4}\)):

There is \(h\in P\), \(h> \theta\), such that \(A(h,h)\in P_{h}\), \(B(h,h)\in P_{h}\), and \(Ch\in P_{h}\).
 (\(\mathrm{L}_{5}\)):

There exists a constant \(\delta_{0}>0\) such that$$ A(x,y)\geq\delta_{0}\bigl[B(x,y)+Cx\bigr], \quad\forall x, y\in P. $$(3.4)
 (1)
\(A: P_{h}\times P_{h}\rightarrow P_{h}\), \(B: P_{h}\times P_{h}\rightarrow P_{h}\), \(C: P_{h}\rightarrow P_{h}\);
 (2)There exist \(u_{0},v_{0}\in P_{h}\) and \(r\in(0,1)\) such that$$rv_{0}\leq u_{0}< v_{0},\qquad u_{0} \leq A(u_{0},v_{0})+B(u_{0},v_{0})+Cu_{0} \leq A(v_{0},u_{0})+B(v_{0},u_{0})+Cv_{0} \leq v_{0}; $$
 (3)
The operator equation \(A(x,x)+B(x,x)+Cx=x\) 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 \(y_{n}\rightarrow x^{\ast}\) as \(n\rightarrow\infty\).$$\begin{aligned} &x_{n}=A(x_{n1},y_{n1})+B(x_{n1},y_{n1})+Cx_{n1}, \quad n=1,2,\ldots , \\ &y_{n}=A(y_{n1},x_{n1})+B(y_{n1},x_{n1})+Cy_{n1}, \quad n=1,2,\ldots, \end{aligned}$$
Proof
The proof will be divided into two steps.
Step one: We demonstrate \(A: P_{h}\times P_{h}\rightarrow P_{h}\), \(B: P_{h}\times P_{h}\rightarrow P_{h}\), \(C: P_{h}\rightarrow P_{h}\).
By the proof of Theorem 3.1, we can obtain the following corollary.
Corollary 3.1
Corollary 3.2
Proof
If we set operator \(T=\lambda^{1}(A+B+C)\) (\(\lambda>0\)), similar to the proof of Theorem 3.1, we can easily obtain that the operator T satisfies all the conditions of Lemma 2.1. □
Corollary 3.3
 (\(\mathrm{L}'_{4}\)):

There is \(h\in P\), \(h> \theta\), such that \(A(h,h)\in P_{h}\) and \(B(h,h)\in P_{h}\).
 (\(\mathrm{L}'_{5}\)):

There exists a constant \(\delta_{0}>0\) such that$$ A(x,y)\geq\delta_{0}B(x,y), \quad \forall x, y\in P. $$
 (1)
\(A: P_{h}\times P_{h}\rightarrow P_{h}\), \(B: P_{h}\times P_{h}\rightarrow P_{h}\);
 (2)There exist \(u_{0},v_{0}\in P_{h}\) and \(r\in(0,1)\) such that$$rv_{0}\leq u_{0}< v_{0},\qquad u_{0} \leq A(u_{0},v_{0})+B(u_{0},v_{0})\leq A(v_{0},u_{0})+B(v_{0},u_{0})\leq v_{0}; $$
 (3)
The operator equation \(A(x,x)+B(x,x)=x\) 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 \(y_{n}\rightarrow x^{\ast}\) as \(n\rightarrow\infty\).$$\begin{aligned} &x_{n}=A(x_{n1},y_{n1})+B(x_{n1},y_{n1}), \quad n=1,2,\ldots, \\ &y_{n}=A(y_{n1},x_{n1})+B(y_{n1},x_{n1}), \quad n=1,2,\ldots, \end{aligned}$$
Theorem 3.2
 (\(\mathrm{L}_{6}\)):

For any \(t\in(0,1)\),$$ A\bigl(tx,t^{1}y\bigr)\geq t A(x,y),\quad \forall x, y \in P. $$(3.13)
 (\(\mathrm{L}_{7}\)):

There exists a constant \(\delta_{0}>0\) such thatThen conclusions \((1)\)–\((4)\) in Theorem 3.1 also hold true.$$ A(x,y)+B(x,y)\leq\delta_{0}Cx, \quad \forall x, y\in P. $$(3.14)
Proof
Corollary 3.4
 (\(\mathrm{L}_{6}'\)):

There is \(h\in P\), \(h> \theta\), such that \(B(h,h)\in P_{h}\) and \(Ch\in P_{h}\).
 (\(\mathrm{L}_{7}'\)):

There exists a constant \(\delta_{0}>0\) such that$$ B(x,y)\leq\delta_{0}Cx, \quad \forall x, y\in P. $$
 (1)
\(B: P_{h}\times P_{h}\rightarrow P_{h}\), \(C: P_{h}\rightarrow P_{h}\);
 (2)There exist \(u_{0}, v_{0}\in P_{h}\) and \(r\in(0,1)\) such that$$rv_{0}\leq u_{0}< v_{0},\qquad u_{0} \leq B(u_{0},v_{0})+Cu_{0}\leq B(v_{0},u_{0})+Cv_{0}\leq v_{0}; $$
 (3)
The operator equation \(B(x,x)+Cx=x\) 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 \(y_{n}\rightarrow x^{\ast}\) as \(n\rightarrow\infty\).$$\begin{aligned} &x_{n}=B(x_{n1},y_{n1})+Cx_{n1},\quad n=1,2,\ldots, \\ &y_{n}=B(y_{n1},x_{n1})+Cy_{n1},\quad n=1,2,\ldots, \end{aligned}$$
Corollary 3.5
In view of the fact that \(P_{h}=\mathring{P}\) with \(h\neq\theta\) and \(h\in\mathring{P}\), we suppose that operators \(A, B:P_{h}\times P_{h}\rightarrow P_{h}\), \(C:P_{h}\rightarrow P_{h}\) or \(A, B:\mathring {P}\times\mathring{P}\rightarrow\mathring{P}\), \(C:\mathring {P}\rightarrow\mathring{P}\) with P is a solid cone, then \(A(h,h)\in P_{h}\), \(B(h,h)\in P_{h}\) and \(Ch\in P_{h}\) are automatically satisfied. So if we let \(D=\mathring{P}\) or \(P_{h}\), we have the following results.
Corollary 3.6
Let \(C: D\rightarrow D\) be an increasing subhomogeneous operator, \(A,B : D\times D\rightarrow D\) be two mixed monotone operators. Assume that
(\(\mathrm{M}_{2}\)) For any fixed \(y\in D\), \(B(\cdot, y):D\rightarrow D\) is concave, for any fixed \(x\in D\), \(B(x,\cdot):D\rightarrow D\) is convex.
Corollary 3.7
Assume that \(C:D\rightarrow D\) is an increasing αconcave operator, \(A,B:D\times D\rightarrow D\) are two mixed monotone operators satisfying conditions (\(\mathrm{M}_{2}\))–(\(\mathrm{M}_{4}\)) and also meeting the following hypotheses:
4 Applications
Definition 4.1
([28])
Lemma 4.1
(see Lemmas 1 and 2 in [33])
By Lemmas 3 and 4 in [33], we can easily obtain that the Green function \(G(t,s)\) defined by (4.2) has the following properties.
Lemma 4.2
 (1)
\(G(t,s)\) is continuous on the unit square \([0,1]\times[0,1]\);
 (2)
\(G(t,s)\geq0\) for each \((t,s)\in[0,1]\times[0,1]\);
 (3)
\(t^{\nu1}J(s)\leq G(t,s) \leq\sigma t^{\nu1},\quad \forall t, s\in[0,1]\),
Theorem 4.1
 \((H_{1})\) :

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

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

For any \(\lambda\in(0,1)\), there exists a constant \(\alpha\in (0,1)\) such that \(\forall t\in[0,1]\), \(u,v\in[0,+\infty)\), \(f(t,\lambda u,\lambda^{1} v)\geq\lambda^{\alpha}f(t,u,v)\); for fixed \(t\in[0,1]\), \(v\in[0,+\infty)\), \(g(t,\cdot, v)\) is concave, for fixed \(t\in[0,1]\), \(u\in[0,+\infty)\), \(g(t,u,\cdot)\) is convex; and for all \(\lambda\in(0,1)\), \(\forall t\in[0,1]\), \(u\in[0,+\infty)\), \(h(t,\lambda u)\geq\lambda h(t,u)\);
 \((H_{4})\) :

There exists \(\frac{1}{2}\leq b\leq1\) such that \(g(t,0,lh(t))\geq bg(t,lh(t),0)\), \(\forall l\geq1\);
 \((H_{5})\) :

There exists a constant \(\delta_{0}>0\) such that \(f(t,u,v)\geq\delta_{0}(g(t,u,v)+h(t,u))\), \(t\in[0,1]\), \(u, v\in [0,+\infty)\).
 (1)There exist \(u_{0},v_{0}\in P_{h}\) and \(r\in(0,1)\) such that \(rv_{0}\leq u_{0}< v_{0}\) andwhere \(h(t)=t^{\alpha1}\), \(t\in[0,1]\);$$ \textstyle\begin{cases} u_{0}(t)\leq\int _{0}^{1}G(t,s)[f(s,u_{0}(s),v_{0}(s))+g(s,u_{0}(s),v_{0}(s))+h(s,u_{0}(s))]\,ds,&t\in [0,1],\\ v_{0}(t)\geq\int _{0}^{1}G(t,s)[f(s,v_{0}(s),u_{0}(s))+g(s,v_{0}(s),u_{0}(s))+h(s,v_{0}(s))]\,ds,&t\in [0,1], \end{cases} $$
 (2)
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 both \({x_{n}(t)}\) and \({y_{n}(t)}\) converge uniformly to \(u^{\ast}(t)\) for all \(t\in[0,1]\).$$ \textstyle\begin{cases} x_{n+1}(t)= \int _{0}^{1}G(t,s)[f(s,x_{n}(s),y_{n}(s))+g(s,x_{n}(s),y_{n}(s))+h(s,x_{n}(s))]\,ds,&n=0,1,2,\ldots ,\\ y_{n+1}(t)= \int _{0}^{1}G(t,s)[f(s,y_{n}(s),x_{n}(s))+g(s,y_{n}(s),x_{n}(s))+h(s,y_{n}(s))]\,ds,&n=0,1,2,\ldots , \end{cases} $$
Proof
Firstly, it follows from \((H_{1})\) and the fact \(G(t,s)\geq0\), \(\forall t, s\in[0,1]\) that \(A, B : P\times P\rightarrow P\) and \(C : P\rightarrow P\). Further, by \((H_{2})\), we can easily obtain A, B are two monotone operators, and C is an increasing operator.
By using Theorem 3.2, we can easily prove the following result.
Theorem 4.2
Assume that \((H_{1})\), \((H_{2})\), and \((H_{4})\) hold, and the following conditions are also satisfied.
\((H_{6})\) \(f(t,\lambda u,\lambda^{1} v)\geq\lambda f(t,u,v)\), \(\forall t\in[0,1]\), \(\lambda\in(0,1)\), \(u,v\in[0,+\infty)\); for fixed \(t\in[0,1]\), \(v\in[0,+\infty)\), \(g(t,\cdot, v)\) is concave, for fixed \(t\in[0,1]\), \(u\in[0,+\infty)\), \(g(t,u,\cdot)\) is convex; and there exists a constant \(\alpha\in(0,1)\) such that \(h(t,\lambda u)\geq \lambda^{\alpha} h(t,u)\) for \(\lambda\in(0,1)\), \(t\in[0,1]\), \(u\in [0,+\infty)\);
\((H_{7})\) There exists a constant \(\delta_{0}>0\) such that \(f(t,u,v)+g(t,u,v)\leq\delta_{0} h(t,u)\), \(\forall t\in[0,1]\), \(u,v\in[0,+\infty)\).
Then, conclusions (1)–(3) of Theorem 4.1 still hold.
In what follows, we give a concrete example to illustrate our main result.
Example 4.1
 (1)
\(f,g:[0,1] \times[0,+\infty)\times[0,+\infty)\rightarrow [0,+\infty)\) and \(h:[0,1]\times[0,+\infty)\rightarrow[0,+\infty)\) are continuous with \(f(t,0,1)=4+2^{\frac{1}{2}}t^{2}\not\equiv0\), \(g(t,0,1)=\frac{5}{3}+\frac{1}{3e}t^{2}\not\equiv0\), \(h(t,0)=1\neq0\).
 (2)
Obviously, \(f(t,x,y)\), \(g(t,x,y)\) are increasing in \(x\in[0,+\infty )\) for fixed \(t\in[0,1]\) and \(y\in[0,+\infty)\), decreasing in \(y\in [0,+\infty)\) for fixed \(t\in[0,1]\) and \(x\in[0,+\infty)\), and \(h(t,x)\) is increasing in \(x\in[0,+\infty)\) for fixed \(t\in[0,1]\).
 (3)For \(\lambda\in(0,1)\), \(t\in[0,1]\), \(x,y\in[0,+\infty)\), taking \(\alpha=\frac{1}{2}\), we have$$ \begin{aligned} f\bigl(t,\lambda x,\lambda^{1} y\bigr)&=( \lambda x+1)^{\frac{1}{2}}+\bigl(\lambda ^{1} y+1\bigr)^{\frac{1}{2}}+3t^{2} \\ &\geq\lambda^{\frac{1}{2}}\bigl((x+1)^{\frac{1}{2}}+(y+1)^{\frac {1}{2}}+3t^{2} \bigr)=\lambda^{\alpha}f(t,x,y). \end{aligned} $$
5 Conclusions
In this paper, we investigate a class of nonlinear sumtype operators without considering the existence of upperlower solutions or compactness or continuity. The sufficient conditions have been established for such sumtype operators to have a unique positive fixed point in \(P_{h}\), and two iterative sequences are also constructed to converge to the fixed point. Further, we apply the obtained results to prove the existence and uniqueness of positive solutions for a nonlinear fractional differential equation with multipoint fractional boundary conditions. The main contribution is that we provide a new method to deal with the unique positive solution of the nonlinear differential equations. Our study enriches the fixed point theorems of nonlinear sumtype operators.
Declarations
Acknowledgements
We are thankful to the editor and the anonymous reviewers for many valuable suggestions to improve this paper.
Availability of data and materials
Data sharing not applicable to this article as no datasets were generated or analysed during the current study.
Funding
This project is supported by the Key Research and Development Plan of Shanxi Province (No. 201703D2210311) and Innovation Project of Shanxi Postgraduate Education (No. 2017BY039).
Authors’ contributions
HW participated in the design of the study and drafted the manuscript. LZ and XW carried out the theoretical studies and helped to draft the manuscript. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
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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