Iterative unique positive solutions for a new class of nonlinear singular higher order fractional differential equations with mixed-type boundary value conditions

In this paper, we consider a new class of singular nonlinear higher order fractional boundary value problems supplemented with sum of Riemann–Stieltjes integral type and nonlocal infinite-point discrete type boundary conditions. The fractional derivative of different orders is involved in the nonlinear terms and boundary conditions, and the nonlinear terms are allowed to be singular in regard to not only time variable but also space variables. A unique positive solution is established by using the fixed point theorem of mixed monotone operator. In addition, some significant properties of the unique solution depending on the parameter λ are stated. In the end, two examples are worked out to illustrate our main results.

In this paper, we are investigating the following singular nonlinear higher order fractional boundary value problem (BVP for short):

where \(D_{0+}^{\gamma }\) is the Riemann–Liouville fractional derivative of γ order, \(\lambda >0\) is a parameter, \(n-1<\gamma \leq n\) (\(n\geq 3\)), \(k-1<\nu _{k}\), \(q_{k}\leq k\) (\(k=1, 2,\ldots ,n-2\)), \(\nu _{n-2}-q_{k}\leq n-2-k\) (\(k=1, 2,\ldots ,n-2\)), \(1<\gamma -\nu _{n-2}\leq 2\), \(\nu _{n-2}\leq \gamma _{0}\leq n-1\), \(\gamma -\gamma _{0}\geq 1\), \(a_{i}\geq 0\) (\(i=1,2,\ldots ,p\)), \(\nu _{n-2}\leq \alpha _{i}\leq \gamma _{0}\) (\(i=1, 2, \ldots , p\)), \(b_{j}\geq 0\) (\(j=1,2, \ldots \)), \(\nu _{n-2}\leq \beta _{j}\leq \gamma _{0}\) (\(j=1, 2, \ldots \)), \(0<\xi _{1}<\xi _{2}<\cdots <\xi _{j}<\cdots <1\); \(I_{i}\subseteq [0, 1]\) (\(i=1,2,\ldots ,p\)) is measurable; \(w_{i}: (0, 1)\rightarrow \mathbb{R}_{+}=[0, +\infty )\) is continuous with \(w_{i}\in L^{1}(0, 1)\), and \(\int _{0}^{1}w_{i}(s)z(s)\,dA_{i}(s)\) denotes the Riemann–Stieltjes integral, in which \(A_{i}: I_{i} \rightarrow \mathbb{R}\) (\(i=1,2, \ldots ,p\)) is a function of bounded variation. \(f :(0,1)\times (0, + \infty )^{n-1}\rightarrow \mathbb{R}_{+}\) (\(\mathbb{R}_{+}=[0,+\infty )\)) is continuous. A function \(z\in C[0,1]\) is called a positive solution of BVP (1.1) if it satisfies (1.1) and \(z(t)>0\) for \(t\in (0, 1)\).

In recent years, the fractional differential equations have drawn the attention of many famous researchers, readers can refer to [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41] and the references therein. It is caused by the applications of fractional differential equations in a proposed framework for describing significant phenomena, for example, the deflection of an elastic beam, the non-Newtonian fluid theory, the degrading of polymer materials, etc. Some interesting results can be found in [1,2,3,4,5].

In [7], the authors considered the following nonlinear fractional differential equation:

where \(D_{0+}^{\alpha }\) is the Riemann–Liouville fractional derivative, \(n-1<\alpha \leq n\) (\(n\geq 3\)), \(1\leq \beta \leq n-2\), \(0\leq \gamma \leq \beta \), \(0<\xi _{1}<\xi _{2}<\cdots <\xi _{m}<1\), \(Tz(t)=\int _{0}^{t}K(t, s)z(s)\,ds\), and \(Sz(t)=\int _{0}^{1}H(t, s)z(s)\,ds\). By using the Banach contraction mapping principle and the Krasnosel’skii fixed point theorem, they obtained the existence of nonnegative solutions for this problem.

By using the Banach contraction map principle and the theory of \(u_{0}\)-positive linear operator, Zhang and Zhong in [8] studied the following fractional differential equation:

where \(D_{0^{+}}^{\alpha }\) is the Riemann–Liouville derivative, \(n-1<\alpha \leq n\) (\(n\geq 3\)), \(\beta \geq 1\), \(\alpha -\beta >1\), \(0<\eta \leq 1\), \(\lambda >0\) is a parameter, \(h\in L^{1}[0, 1]\), \(f:[0,1]\times \mathbb{R}\rightarrow \mathbb{R}\) is continuous. They got the existence and uniqueness of solutions for this problem.

Based on the reducing method of fractional orders, the Schauder fixed point theorem, and the upper and lower solutions method, Zhang, Liu, and Wu in [9] obtained an eigenvalue interval for the existence of positive solutions of the following fractional differential equation:

where \(D_{0+}^{\alpha }\) is the Riemann–Liouville fractional derivative, \(n-1<\alpha \leq n\) (\(n\geq 3\)), \(n-i-1\leq \alpha -\mu _{i}\leq n-i\) (\(i=1, 2,\ldots , n-2\)), \(\mu -\mu _{n-1}>0\), \(\alpha - \mu _{n-1}\leq 2\), \(\alpha -\mu >1\), \(a_{j}\geq 0\) (\(j=1,2,\ldots , p-2\)), \(0<\xi _{1}<\xi _{2}<\cdots <\xi _{p-2}<1\); \(f:(0, +\infty )^{n}\rightarrow \mathbb{R}_{+}\) is continuous and is nonincreasing in \(x_{i}>0\) for \(i=1,2,\ldots ,n\).

Inspired by the above-mentioned papers, we investigate BVP (1.1). As far as we know, BVP (1.1) has seldom been researched up to now, and the novelty of this paper lies in three aspects. Firstly, the boundary conditions are the combination of sum of Riemann–Stieltjes integral type boundary conditions and nonlocal infinite-point discrete type boundary conditions, which involves fractional derivative of different orders. This fact suggests BVP (1.1) is more general than the above-mentioned literature. For instance, let \(I_{1}=(0, 1)\), \(I_{i}=0\) (\(i= 2, 3,\ldots , p\)), and \(b_{j}= 0\) (\(j=1,2,\ldots \)), then the boundary condition of BVP (1.1) reduces to the boundary condition in [16]; if \(b_{j}=0\) (\(j=1,2,\ldots \)), \(\alpha _{i}=0\) \(w_{i}=1\) (\(i= 2, 3,\ldots , p\)), then the boundary condition is equal to [37]; and if \(I_{1}=(0, \eta )\) (\(\eta \in (0, 1)\)), \(I_{i}=0\) (\(i= 2, 3,\ldots , p\)), the boundary condition of BVP (1.1) is the same as [18]. Meanwhile the work to check the properties of the corresponding Green’s function is too hard. Secondly, the nonlinearity f contains different orders of fractional derivative of the unknown function. In general, many papers consider these kinds of boundary value problem in the space \(E=\{u\in C[0, 1]: D_{0+} ^{\nu _{i}}u\in C[0, 1], i=1,2,\ldots ,n-2\}\), which makes the study extremely difficult. In this paper, we use the reducing method to transform BVP (1.1) into a relatively low-order equivalent problem, which could be considered in the space \(C[0, 1]\), and is a good way to do this. Some interesting results of the reducing method can be found in [9, 17, 23, 25, 27, 29, 31] and the references therein. Thirdly, there is much to be learned about the theory and applications of mixed monotone operator, recently. Especially, many papers have taken it into the research for fractional boundary value problems. Some interesting results can be found in [10, 11, 15, 21,22,23, 25, 27] and the references therein. Thus, in this paper, by using the fixed point theorem of mixed monotone operator, we obtain the uniqueness of positive solution under the assumption that f may be singular with respect to both the time and space variables. It is worth mentioning that some important properties of the unique solution rely on the parameter λ.

The paper is organized as follows. In Sect. 2, we present some preliminary setting, derive the corresponding Green’s function, and transform BVP (1.1) into a relatively low-order equivalent problem, in which the nonlinear term has no fractional derivatives. In Sect. 3, we pay particular attention to establishing the uniqueness of positive solutions and consider some relative properties of the unique positive solution. In Sect. 4, two examples are devoted to our main results.

Let E be a Banach space and P be a cone in E. P is said to be normal if there exists a constant \(N>0\) such that, for any \(u, v \in E\), \(\theta \leq u\leq v\) implies \(\|u\|\leq N\|v\|\), the smallest constant, which satisfies this inequality, is called the normality constant of P. Then E is partially ordered by P, i.e., \(u\leq v\) if and only if \(v-u\in P\). For any \(u, v\in E\), the notation \(u\sim v\) means that there exist constants \(\lambda >0\) and \(\mu >0\) such that \(\lambda u\leq v\leq \mu u\). Obviously, ∼ is an equivalence relation. For fixed \(e\in P_{e}\) and \(e>\theta \), we denote \(P_{e}=\{u\in E : u\sim e\}=\{u\in E : \omega e\leq u \leq \frac{1}{\omega }e, 0<\omega <1\}\). It is easy to see that \(P_{e}\subset P\) is a component of P.

Definition 2.1

([11])

Let E be a Banach space and \(D\subset E\). The operator \(A :D\times D \rightarrow E\) is called a mixed monotone operator if \(A(u, v)\) is increasing in \(u\in D\) and decreasing in \(v\in D\), i.e., \(u_{i}\), \(v_{i} \in D\) (\(i=1,2\)), \(u_{1}\leq u_{2}\), \(v_{1}\geq v_{2}\) imply \(A(u_{1}, v_{1})\leq A(u _{2}, v_{2})\). An element \(u\in D\) is called a fixed point of A if \(A(u, u)=u\).

Lemma 2.2

([10, 12])

Let P be a normal cone in the Banach space E, and \(A, B: P_{e}\times P_{e} \rightarrow P_{e}\) be two mixed monotone operators which satisfy the following conditions:

1. (i)

   For any \(\mu \in (0, 1)\), there exists \(\varphi (\mu )\in ( \mu ,1]\) such that

   $$ A\bigl(\mu u,\mu ^{-1}v\bigr)\geq \varphi (\mu )A(u, v), \quad \forall u, v \in P_{e}. $$
2. (ii)

   For any \(\mu \in (0,1)\), \(u, v\in P_{e}\),

   $$ B\bigl(\mu u, \mu ^{-1}v\bigr)\geq \mu B(u, v). $$
3. (iii)

   There exists a constant \(\kappa >0\) such that \(A(u, v)\geq \kappa B(u, v)\), \(\forall u, v\in P_{e}\).

Then there exists a unique fixed point \(u^{*}\in P_{e}\) such that \(A(u^{*}, u^{*})+B(u^{*}, u^{*})=u^{*}\). And for any initial values \(u_{0}\), \(v_{0}\in P_{e}\), by constructing successively the sequences as follows:

we have \(u_{n}\rightarrow u^{*} \) and \(v_{n}\rightarrow u^{*} \) in E, as \(n\rightarrow \infty \).

Lemma 2.3

([10, 12])

Suppose that operators A and B satisfy all the conditions of Lemma 2.2. Then the equation

has a unique solution \(u_{\lambda }\) in \(P_{e}\) for all \(\lambda >0\), which satisfies:

1. (i)

   If there exists \(r\in (0, 1)\) such that

   $$ \varphi (\mu )\geq \frac{\mu ^{r}-\mu }{\kappa }+\mu ^{r},\quad \forall \mu \in (0, 1), $$

   then \(u_{\lambda }\) is continuous with respect to \(\lambda \in (0, + \infty )\). That is, for any \(\lambda _{0}\in (0,+\infty )\),

   $$ \Vert u_{\lambda }-u_{\lambda _{0}} \Vert \rightarrow 0, \quad \textit{as } \lambda \rightarrow \lambda _{0}. $$
2. (ii)

   If

   $$ \varphi (\mu )\geq \frac{\mu ^{\frac{1}{2}}-\mu }{\kappa }+ \mu ^{\frac{1}{2}},\quad \forall \mu \in (0, 1), $$

   then \(0<\lambda _{1}< \lambda _{2}\) implies \(u_{\lambda _{1}}< u_{\lambda _{2}}\).

3. (iii)

   If there exists \(r\in (0, \frac{1}{2})\) such that

   $$ \varphi (\mu )\geq \frac{\mu ^{r}-\mu }{\kappa }+\mu ^{r},\quad \forall \mu \in (0, 1), $$

   then

   $$ \lim_{\lambda \rightarrow 0^{+}} \Vert u_{\lambda } \Vert =0, \qquad \lim_{\lambda \rightarrow +\infty } \Vert u_{\lambda } \Vert =+\infty . $$

Definition 2.4

([5])

Let \(\alpha >0\). The Riemann–Liouville fractional integral of order α of a function \(z: (0, \infty ) \rightarrow \mathbb{R} \) is given by

provided that the right-hand side is pointwise defined on \((0, \infty )\).

Definition 2.5

([5])

Let \(\alpha >0\). The Riemann–Liouville fractional derivative of order α of a continuous function \(z: (0, \infty ) \rightarrow \mathbb{R}\) is given by

where \(n = [\alpha ]+1\), \([\alpha ]\) denotes the integer part of α, provided that the right-hand side is pointwise defined on \((0, \infty )\).

Lemma 2.6

([6])

Let \(z\in C(0, 1) \cap L^{1}(0, 1)\). Then the fractional differential equation

has a unique solution

where n is the smallest integer greater than or equal to α.

Lemma 2.7

([6])

Let \(z \in C(0, 1) \cap L^{1}(0, 1)\) and \(D_{0^{+}}^{\alpha }z\in C(0, 1) \cap L^{1}(0,1)\). Then

where n is the smallest integer greater than or equal to α.

Lemma 2.8

([5])

Suppose that \(z\in C(0, 1) \cap L^{1}(0, 1)\), then

1. (i)

   \(I_{0^{+}}^{\alpha }I_{0^{+}}^{\beta }z(t)=I_{0^{+}} ^{\alpha +\beta }z(t)\) for α, \(\beta >0\);

2. (ii)

   \(D_{0^{+}}^{\beta }I_{0^{+}}^{\alpha }z(t)=I_{0^{+}} ^{\alpha -\beta }z(t)\) for \(\alpha \geq \beta >0\).

Lemma 2.9

Suppose that BVP (1.1) has a solution \(z\in C[0, 1]\), then \(u=D_{0^{+}}^{\nu _{n-2}}z\) is a solution of the following boundary value problem:

where \(1<\gamma -\nu _{n-2}\leq 2\). On the other hand, if we assume BVP (2.1) has a solution \(u\in C[0,1]\), then BVP (1.1) has a solution \(z=I_{0^{+}}^{\nu _{n-2}}u\).

Proof

Suppose that \(z\in C[0, 1]\) and satisfies BVP (1.1). Let

It follows from Lemma 2.7 that

where \(c_{i}\in \mathbb{R}\) (\(i=1,2,\ldots , n-2\)). Hence,

Since \(z(0)=0,\ldots \) , \(D_{0^{+}}^{q_{n-3}}z(0)=0\), we immediately obtain that \(c_{1}=\cdots =c_{n-2}=0\). Thus,

By using (ii) in Lemma 2.8, we have

and

Similar to (2.5), we have

It follows from (2.2)–(2.5) that

On the basis of (2.6), (2.8), and (2.9), we have

From (2.7), we have

Combining (2.10)–(2.12), we deduce that \(u =D_{0^{+}}^{\nu _{n-2}}z \) is a solution of BVP (2.1).

On the other hand, we consider the case that BVP (2.1) has a solution \(u\in C[0,1]\). Let \(z(t)=I_{0^{+}}^{\nu _{n-2}}u(t)\), \(t\in [0,1]\). Then \(z=I_{0^{+}}^{\nu _{n-2}}u\) is a solution of BVP (1.1). The proof is similar to Lemma 3 in [31]. So, we omit details. □

Remark 2.10

With the analysis of Lemma 2.9, it is enough to show that the work on searching solutions of BVP (1.1) is equivalent to finding solutions of BVP (2.1). Accordingly, we will focus on seeking the solutions of BVP (2.1) in the rest of this paper.

Lemma 2.11

Let \(x\in C(0,1)\cap L^{1}(0,1)\). Then the boundary value problem

is equivalent to

where

in which

Obviously, \(K(t, s)\) is continuous on \([0, 1]\times [0, 1]\).

Proof

By using Lemma 2.7, we may express (2.13) as

where \(c_{1}\), \(c_{2}\in \mathbb{R}\). Since \(D_{0^{+}}^{q_{n-2}-\nu _{n-2}}u(0)=0\), we get \(c_{2}=0 \) and rewrite (2.16) as

With the help of (ii) in Lemma 2.8, we have

and

which combined with the boundary condition

yields

where

Applying (2.18) into (2.17), we can obtain

The proof is complete. □

Lemma 2.12

Let \(\sigma >0\) (defined in (2.19) of Lemma 2.11), \(\int _{I_{i}}s^{\gamma -\alpha _{i}-1}w_{i}(s)\,dA_{i}(s)\geq 0\) (\(i=1, 2,\ldots , p\)), and \(0<\sum_{j=1}^{\infty }\frac{b_{j}}{ \varGamma (\gamma -\beta _{j})}\xi _{j}^{\gamma -\beta _{j}-1}<\infty \). Then the functions \(K_{0}(t, s)\), \(K_{i}(t, s)\) (\(i=1, 2,\ldots , p\)) and \(H_{j}(t, s)\) (\(j=1, 2,\ldots \)) given in Lemma 2.11 have the following properties:

1. (i)

   \(t^{\gamma -\nu _{n-2}-1}k_{0}(s)\leq K_{0}(t, s)\leq \frac{1}{ \varGamma (\gamma -\nu _{n-2})}t^{\gamma -\nu _{n-2}-1}\), where

   $$ k_{0}(s)=\frac{1}{\varGamma (\gamma -\nu _{n-2})}(1-s)^{\gamma -\gamma _{0}-1} \bigl(1-(1-s)^{\gamma _{0}-\nu _{n-2}}\bigr). $$
2. (ii)

   \(t^{\gamma -\alpha _{i}-1}k_{i}(s)\leq K_{i}(t, s)\leq \frac{a _{i}}{\sigma \varGamma (\gamma -\nu _{n-2})\varGamma (\gamma -\alpha _{i})}t ^{\gamma -\alpha _{i}-1}\) (\(i=1, 2,\ldots , p\)), where

   $$ k_{i}(s)=\frac{a_{i}}{\sigma \varGamma (\gamma -\nu _{n-2})\varGamma (\gamma -\alpha _{i})}(1-s)^{\gamma -\gamma _{0}-1} \bigl(1-(1-s)^{\gamma _{0}-\alpha _{i}}\bigr). $$
3. (iii)

   \(t^{\gamma -\beta _{j}-1}h_{j}(s)\leq H_{j}(t, s)\leq \frac{b _{j}}{\sigma \varGamma (\gamma -\nu _{n-2})\varGamma (\gamma -\beta _{j})}t ^{\gamma -\beta _{j}-1}\) (\(j=1, 2,\ldots \)), where

   $$ h_{j}(s)=\frac{b_{j}}{\sigma \varGamma (\gamma -\nu _{n-2})\varGamma (\gamma -\beta _{j})}(1-s)^{\gamma -\gamma _{0}-1} \bigl(1-(1-s)^{\gamma _{0}-\beta _{j}}\bigr). $$

Proof

(i) For \(s\leq t\),

For \(t\leq s\),

Using the same argument again, it is straightforward to infer (ii) and (iii). The proof is complete. □

Lemma 2.13

Let \(\sigma >0\) (defined in (2.19) of Lemma 2.11), \(\int _{I_{i}}s^{\gamma -\alpha _{i}-1}w_{i}(s)\,dA_{i}(s)\geq 0\) (\(i=1, 2,\ldots , p\)), and \(0<\sum_{j=1}^{\infty }\frac{b_{j}}{ \varGamma (\gamma -\beta _{j})}\xi _{j}^{\gamma -\beta _{j}-1}<\infty \). Then the Green’s function \(K(t, s)\) defined in Lemma 2.11 satisfies:

1. (i)

   \(K(t, s)\leq Q_{1}e(t)\) for \(t, s\in [0, 1]\), where \(e(t)=t ^{\gamma -\nu _{n-2}-1}\),

   $$ \begin{aligned} Q_{1}&= \frac{1}{\varGamma (\gamma -\nu _{n-2})}+\sum _{i=1}^{p}\frac{a_{i}}{ \sigma \varGamma (\gamma -\nu _{n-2})\varGamma (\gamma - \alpha _{i})} \int _{I_{i}}\tau ^{\gamma -\alpha _{i}-1}w_{i}(\tau ) \,dA_{i}(\tau ) \\ &\quad{} + \sum_{j=1}^{\infty } \frac{b_{j}}{\sigma \varGamma (\gamma -\nu _{n-2}) \varGamma (\gamma -\beta _{j})}\xi _{j}^{\gamma -\beta _{j}-1}. \end{aligned} $$
2. (ii)

   \(K(t, s)\geq Q_{2}(s)e(t)\) for \(t, s\in [0, 1]\), where

   $$ Q_{2}(s)=k_{0}(s)+\sum_{i=1}^{p}k_{i}(s) \int _{I_{i}} \tau ^{\gamma -\alpha _{i}-1}w_{i}(\tau ) \,dA_{i}(\tau )+\sum_{j=1}^{ \infty }H_{j}( \xi _{j},s). $$
3. (iii)

   \(K(t, s)>0\) for \(t, s\in (0, 1)\).

Proof

The conclusion can be easily given by Lemma 2.12. So we omit it. □

In this paper, we equip \(E=C[0, 1]\) with the norm \(\|u\|=\sup_{0\leq t\leq 1}|u(t)|\). Then \((E, \|\cdot \|)\) is a Banach space. Let \(P=\{u\in E:u(t)\geq 0, t\in [0, 1]\}\) be a cone in E. Let us define a nonlinear operator \(T_{\lambda }:P\rightarrow P\) by

It is easy to check that BVP (2.1) has a solution if and only if the operator \(T_{\lambda }\) has a fixed point.

Theorem 3.1

Suppose that \(f\in C((0,1)\times (0,+ \infty )^{n-1}, \mathbb{R}_{+})\) satisfies:

\((\mathrm{H}_{1})\) :

There exist two functions \(f_{1}\), \(f_{2}\in C((0,1)\times (0,+\infty )^{2(n-1)},\mathbb{R}_{+})\) such that

$$ \begin{aligned} f(t,x_{1},x_{2},\ldots ,x_{n-1})&= f_{1}(t,x_{1},x_{2}, \ldots ,x_{n-1},x _{1},x_{2},\ldots ,x_{n-1}) \\ &\quad{} +f_{2}(t,x_{1},x_{2},\ldots ,x_{n-1},x _{1},x_{2},\ldots ,x_{n-1}). \end{aligned} $$

\((\mathrm{H}_{2})\) :

For all \(t\in (0,1)\) and \((y_{1}, y_{2}, \ldots , y_{n-1})\in (0,+\infty )^{n-1}\), \(f_{1}(t, x _{1}, x_{2}, \ldots , x_{n-1}, y_{1}, y_{2}, [4] \ldots , y_{n-1})\), \(f_{2}(t, x_{1}, x_{2}, \ldots , x_{n-1}, y_{1}, y_{2},\ldots , y_{n-1})\) are increasing in \((x_{1}, x_{2}, \ldots , x_{n-1})\in (0,+\infty )^{n-1}\); for all \(t\in (0,1)\) and \((x_{1}, x_{2}, \ldots , x_{n-1})\in (0,+\infty )^{n-1}\), \(f_{1}(t, x_{1}, x_{2}, \ldots , x _{n-1}, y_{1}, y_{2},\ldots , y_{n-1})\), \(f_{2}(t, x_{1}, x_{2}, \ldots , x_{n-1}, y_{1}, y_{2},\ldots , y_{n-1})\) are decreasing in \((y_{1}, y_{2}, \ldots , y_{n-1})\in (0,+\infty )^{n-1}\).

\((\mathrm{H}_{3})\) :

For all \(\mu \in (0,1)\), there exists \(\varphi (\mu )\in (\mu , 1]\) such that, for all \(t\in (0,1)\) and \((x_{1}, x_{2}, \ldots , x_{n-1})\), \((y_{1}, y_{2}, \ldots , y_{n-1})\in (0,+\infty )^{n-1}\),

$$\begin{aligned}& \begin{aligned} &f_{1}\bigl(t, \mu x_{1}, \mu x_{2}, \ldots , \mu x_{n-1}, \mu ^{-1} y_{1}, \mu ^{-1} y_{2},\ldots , \mu ^{-1} y_{n-1}\bigr) \\ &\quad \geq \varphi (\mu ) f _{1}(t, x_{1}, x_{2}, \ldots , x_{n-1}, y_{1}, y_{2},\ldots , y_{n-1}), \end{aligned} \\& \begin{aligned} &f_{2}\bigl(t, \mu x_{1}, \mu x_{2}, \ldots , \mu x_{n-1}, \mu ^{-1} y_{1}, \mu ^{-1} y_{2},\ldots , \mu ^{-1} y_{n-1}\bigr) \\ &\quad \geq \mu f_{2}(t, x_{1}, x_{2}, \ldots , x_{n-1}, y_{1}, y_{2},\ldots , y_{n-1}). \end{aligned} \end{aligned}$$

\((\mathrm{H}_{4})\) :

There exists a constant \(\kappa >0\) such that, for all \(t\in (0,1)\) and \((x_{1}, x_{2}, \ldots , x_{n-1})\), \((y_{1}, y_{2}, \ldots , y_{n-1})\in (0,+\infty )^{n-1}\),

$$ f_{1}(t, x_{1}, x_{2}, \ldots , x_{n-1}, y_{1}, y_{2},\ldots , y_{n-1}) \geq \kappa f_{2}(t, x_{1}, x_{2}, \ldots , x_{n-1}, y_{1}, y_{2}, \ldots , y_{n-1}). $$

\((\mathrm{H}_{5})\) :

The functions \(f_{1}\) and \(f_{2}\) satisfy

$$\begin{aligned}& 0< \int _{0}^{1}f_{1}\bigl(s,1,1, \ldots ,1,s^{\gamma -1},s^{\gamma -1}, \ldots ,s^{\gamma -1}\bigr) \,ds< +\infty , \\& 0< \int _{0}^{1}f_{2}\bigl(s,1,1, \ldots ,1,s^{\gamma -1},s^{\gamma -1}, \ldots ,s^{\gamma -1}\bigr) \,ds< +\infty . \end{aligned}$$

And at the same time, \(z_{\lambda }^{*}\) satisfies:

1. (i)

   If there exists \(r\in (0, 1)\) such that

   $$ \varphi (\mu )\geq \frac{\mu ^{r}-\mu }{\kappa }+\mu ^{r},\quad \forall \mu \in (0, 1), $$

   then \(z_{\lambda }^{*}\) is continuous with respect to \(\lambda \in (0,+ \infty )\), i.e., for ∀ \(\lambda _{0}\in (0,+\infty )\),

   $$ \bigl\Vert z_{\lambda }^{*}-z_{\lambda _{0}}^{*} \bigr\Vert \rightarrow 0, \quad \textit{as } \lambda \rightarrow \lambda _{0}. $$
2. (ii)

   If

   $$ \varphi (\mu )\geq \frac{\mu ^{\frac{1}{2}}-\mu }{\kappa }+ \mu ^{\frac{1}{2}},\quad \forall \mu \in (0, 1), $$

   then \(0<\lambda _{1}< \lambda _{2}\) implies \(z_{\lambda _{1}}^{*}< z_{ \lambda _{2}}^{*}\).

3. (iii)

   If there exists \(r\in (0, \frac{1}{2})\) such that

   $$ \varphi (\mu )\geq \frac{\mu ^{r}-\mu }{\kappa }+\mu ^{r},\quad \forall \mu \in (0, 1), $$

   then

   $$ \lim_{\lambda \rightarrow 0^{+}} \bigl\Vert z_{\lambda }^{*} \bigr\Vert =0, \qquad \lim_{\lambda \rightarrow +\infty } \bigl\Vert z_{\lambda }^{*} \bigr\Vert =+\infty . $$

Moreover, for any initial values \(z_{0}\), \(\tilde{z}_{0}\in P_{e}\), by constructing successively the sequences as follows:

we have \(z_{n}\rightarrow z_{\lambda }^{*}\) and \(\tilde{z}_{n}\rightarrow z_{\lambda }^{*}\) in E, as \(n\rightarrow \infty \).

Proof

Let \(P_{e}=\{u\in E: u\sim e\}\), where \(e(t)=t^{ \gamma -\nu _{n-2}-1}\). Then \(P_{e}\) is a component of P. Now, we define two operators \(A_{\lambda }, B_{\lambda }: P_{e}\times P_{e} \rightarrow P\) by

Combining the definition of \(T_{\lambda }\) in (2.20) and \((H_{1})\), we have

Then we can conclude that u is the solution of BVP (2.1) if u satisfies \(u=A_{\lambda }(u, u)+B_{\lambda }(u, u)\).

We prove that \(A_{\lambda }, B_{\lambda }: P_{e}\times P_{e}\rightarrow P \) are well defined at first. For any \(u, v\in P_{e}\), there exists a constant \(\omega \in (0,1)\) such that \(\omega e(t)\leq u(t)\leq \frac{1}{ \omega }e(t)\), \(\omega e(t)\leq v(t)\leq \frac{1}{\omega }e(t)\), \(t\in [0, 1]\). Moreover, by the definition of fractional integral and \(e(t)\leq 1\), for all \(t\in [0,1]\),

and

Thus, by \(\mathrm{(H_{1})}\), \(\mathrm{(H_{2})}\), \(\mathrm{(H_{3})}\), \(\mathrm{(H_{5})}\), (3.2), and (3.3), we know that, for all \(t\in [0,1]\),

where

Similarly, for all \(t\in [0,1]\),

So, \(A_{\lambda }, B_{\lambda }: P_{e}\times P_{e}\rightarrow P\) are well defined.

Now, we prove that \(A_{\lambda }, B_{\lambda }: P_{e}\times P_{e} \rightarrow P_{e}\). Taking a constant \(W>1\) such that

Then from \(\mathrm{(H_{1})}\), \(\mathrm{(H_{2}),}\) and \(\mathrm{(H_{3})}\), for all \(t\in [0,1]\),

and

On the other hand, from (3.4) and (3.5), we know, for all u, \(v\in P_{e}\), \(t\in [0, 1]\),

and

So, \(A_{\lambda }, B_{\lambda }: P_{e}\times P_{e}\rightarrow P_{e}\).

In fact, from \(\mathrm{(H_{2})}\), it is easy to check that \(A_{\lambda }\), \(B_{\lambda }\) are mixed monotone operators. Furthermore, it follows from \(\mathrm{(H_{3})}\) that, for all \(\mu \in (0,1)\), there exists \(\varphi (\mu )\in (\mu , 1]\) such that, for any \(u, v\in P_{e}\), \(t\in [0, 1]\),

and

By \(\mathrm{(H_{4})}\), we infer that there exists \(\kappa >0\) such that, for all u, \(v\in P_{e}\), \(t\in [0, 1]\),

Combining (3.11)–(3.13) and using Lemma 2.2, we infer that there exists a unique fixed point \(u^{*}_{\lambda }\in P_{e}\) such that

That is, BVP (2.1) has a unique solution \(u^{*}_{\lambda }\in P_{e}\). Since \(u^{*}_{\lambda }\in P_{e}\), there exists a constant \(\eta _{\lambda }\in (0, 1)\) such that

Moreover, for any \(\lambda >0\), let \(\tilde{A}= (\lambda ^{-1} A_{ \lambda })\) and \(\tilde{B}= (\lambda ^{-1} B_{\lambda })\). Obviously, Ã and B̃ satisfy all the conditions of Lemma 2.3. With the preceding proof, we can infer that \(u^{*}_{\lambda }\) is the unique positive solution of the following equation:

By means of Lemma 2.3, we know that \(u^{*}_{\lambda }\) satisfies:

1. (1)

   If there exists \(r\in (0, 1)\) such that

   $$ \varphi (\mu )\geq \frac{\mu ^{r}-\mu }{\kappa }+\mu ^{r},\quad \forall \mu \in (0, 1), $$

   then \(u^{*}_{\lambda }\) is continuous with respect to \(\lambda \in (0, +\infty )\). That is, for ∀ \(\lambda _{0}\in (0,+\infty )\),

   $$ \bigl\Vert u^{*}_{\lambda }-u^{*}_{\lambda _{0}} \bigr\Vert \rightarrow 0, \quad \text{as } \lambda \rightarrow \lambda _{0}. $$
2. (2)

   If

   $$ \varphi (\mu )\geq \frac{\mu ^{\frac{1}{2}}-\mu }{\kappa }+ \mu ^{\frac{1}{2}},\quad \forall \mu \in (0, 1), $$

   then \(0<\lambda _{1}< \lambda _{2}\) implies \(u^{*}_{\lambda _{1}}< u^{*} _{\lambda _{2}}\).

3. (3)

   If there exists \(r\in (0, \frac{1}{2})\) such that

   $$ \varphi (\mu )\geq \frac{\mu ^{r}-\mu }{\kappa }+\mu ^{r},\quad \forall \mu \in (0, 1), $$

   then

   $$ \lim_{\lambda \rightarrow 0^{+}} \bigl\Vert u^{*}_{\lambda } \bigr\Vert =0, \qquad \lim_{\lambda \rightarrow +\infty } \bigl\Vert u^{*}_{\lambda } \bigr\Vert =+\infty . $$

Furthermore, by Lemma 2.2, we can infer that, for any initial values \(u_{0}, v_{0}\in P_{e}\), by constructing successively the sequences as follows:

we have

Finally, by what we have proved in Lemma 2.9, we know \(z^{*}_{\lambda }=I^{\nu _{n-2}}u^{*}_{\lambda }\) is the unique positive solution of BVP (1.1). From (3.14), we know that \(z^{*}\) satisfies

And from the monotonicity and continuity of fractional integral, we get:

1. (i)

   If there exists \(r\in (0, 1)\) such that

   $$ \varphi (\mu )\geq \frac{\mu ^{r}-\mu }{\kappa }+\mu ^{r},\quad \forall \mu \in (0, 1), $$

   then for ∀ \(\lambda _{0}\in (0,+\infty )\),

   $$ \bigl\Vert z^{*}_{\lambda }-z^{*}_{\lambda _{0}} \bigr\Vert \rightarrow 0, \quad \text{as } \lambda \rightarrow \lambda _{0}. $$
2. (ii)

   If

   $$ \varphi (\mu )\geq \frac{\mu ^{\frac{1}{2}}-\mu }{\kappa }+ \mu ^{\frac{1}{2}},\quad \forall \mu \in (0, 1), $$

   then \(0<\lambda _{1}< \lambda _{2}\) implies \(z^{*}_{\lambda _{1}}< z^{*} _{\lambda _{2}}\).

3. (iii)

   If there exists \(r\in (0, \frac{1}{2})\) such that

   $$ \varphi (\mu )\geq \frac{\mu ^{r}-\mu }{\kappa }+\mu ^{r},\quad \forall \mu \in (0, 1), $$

   then

   $$ \lim_{\lambda \rightarrow 0^{+}} \bigl\Vert z^{*}_{\lambda } \bigr\Vert =0, \qquad \lim_{\lambda \rightarrow +\infty } \bigl\Vert z^{*}_{\lambda } \bigr\Vert =+\infty . $$

Furthermore, for any initial values \(z_{0}, \tilde{z}_{0}\in P_{e}\), by constructing successively the sequences as follows:

we have

The proof of Theorem 3.1 is completed. □

Example 4.1

We consider the following problem:

where \(\lambda >0\) is a parameter, and

Let

\(n=3\), \(\gamma =\frac{5}{2}\), \(\gamma _{0}=\frac{3}{2}\), \(\nu _{1}= \frac{3}{5}\), \(q_{1}=\frac{2}{3}\), \(a_{1}=2\), \(a_{2}=\frac{1}{2}\), \(\alpha _{1}=\frac{5}{4}\), \(\alpha _{2}=\frac{11}{8}\), \(I_{1}=[0, 1]\), \(I_{2}=[0, \frac{2}{3}]\), \(b_{j}=(5j-4)^{-1}(5j+1)^{-1}\) (\(j=1,2, \ldots \)), \(\beta _{j}= \frac{3}{2}-\frac{1}{2^{(7+j)}}\) (\(j=1,2, \ldots \)), \(\xi _{j}=(28+2j)^{-1}\) (\(j=1,2,\ldots \)), \(w_{1}(t)=t^{ \frac{3}{4}}(1-t)^{2}\), \(w_{2}(t)=t^{\frac{7}{8}}(1+t^{2})^{-1}\). Then problem (4.1) can be transformed into BVP (1.1) for \(\lambda >0\).

By simple computation, we have a rough estimate

and

which means the properties of Green’s function in Lemma 2.13 are achieved. Let

and

Then

It is easy to check the following conditions:

1. (1)

   For all \(t\in (0,1)\) and \((w, z)\in (0,+\infty )^{2}\), \(f_{1}(t, u, v, w, z)\), \(f_{2}(t, u, v, w, z)\) are increasing in \((u, v)\in (0,+\infty )^{2}\); for all \(t\in (0,1)\) and \((u, v)\in (0,+ \infty )^{2}\), \(f_{1}(t, u, v, w, z)\), \(f_{2}(t, u, v, w, z)\) are decreasing in \((w, z)\in (0,+\infty )^{2}\).

2. (2)

   Let \(\varphi (\mu )=\mu ^{\frac{1}{3}}\). Then, for \(\mu \in (0,1)\), \(t\in (0,1)\), and \((u, v, w, z)\in (0,+\infty )^{4}\),

   $$\begin{aligned}& \begin{aligned} f_{1}\bigl(t, \mu u, \mu v, \mu ^{-1} w, \mu ^{-1} z\bigr)&= 5\mu ^{\frac{1}{4}} t ^{-\frac{1}{3}}u^{\frac{1}{8}}z^{-\frac{1}{8}}+4\mu ^{\frac{1}{6}}u ^{\frac{1}{6}}\bigl(1-t^{2}\bigr)^{-\frac{1}{2}} \\ &\quad{} +\bigl(6\mu ^{\frac{1}{3}}v^{ \frac{1}{3}}+1\bigr) \bigl(1+t^{2}\bigr)^{-1}+ 7\mu ^{\frac{1}{5}} (tw)^{-\frac{1}{5}} \\ &\geq \mu ^{\frac{1}{3}} f_{1}(t, u, v, w, z), \end{aligned} \\& \begin{aligned} f_{2}\bigl(t, \mu u, \mu v, \mu ^{-1} w, \mu ^{-1} z\bigr)&= \mu ^{\frac{1}{4}}t^{-\frac{1}{3}}u^{\frac{1}{8}}z^{-\frac{1}{8}}+ \mu ^{\frac{1}{6}}u ^{\frac{1}{6}} \bigl(1-t^{2}\bigr)^{-\frac{1}{2}} \\ &\quad{} + \mu ^{\frac{1}{4}}v^{ \frac{1}{4}}\bigl(1+t^{2} \bigr)^{-1}+ \mu ^{\frac{4}{15}} (tw)^{-\frac{1}{5}} (w+ \mu )^{-\frac{4}{5}} \\ &\geq \mu f_{2}(t, u, v, w, z). \end{aligned} \end{aligned}$$
3. (3)

   Let \(\kappa =4\). Then, for all \((u, v, w, z)\in (0,+\infty )^{4}\),

   $$ f_{1}(t, u, v, w, z)\geq \kappa f_{2}(t, u, v, w, z). $$
4. (4)

   The functions \(f_{1}\) and \(f_{2}\) satisfy

   $$\begin{aligned}& \begin{aligned} 0&< \int _{0}^{1}f_{1}\bigl(s, 1, 1, s^{\gamma -1}, s^{\gamma -1}\bigr)\,ds\\ &= \int _{0}^{1} \bigl(5s^{-\frac{25}{48}}+4 \bigl(1-s^{2}\bigr)^{-\frac{1}{2}}+7\bigl(1+s ^{2} \bigr)^{-1}+7s^{-\frac{1}{2}} \bigr)< +\infty , \end{aligned} \\& 0< \int _{0}^{1}f_{2}\bigl(s, 1, 1, s^{\gamma -1}, s^{\gamma -1}\bigr)\,ds\leq \int _{0}^{1} \bigl(s^{-\frac{25}{48}}+ \bigl(1-s^{2}\bigr)^{-\frac{1}{2}}+\bigl(1+s^{2} \bigr)^{-1}+s ^{-\frac{1}{2}} \bigr)< +\infty . \end{aligned}$$

Let \(r=\frac{7}{15}<\frac{1}{2}\). It is easy to check that

1. (i)
   $$ \varphi (\mu )=\mu ^{\frac{1}{3}}\geq \frac{\mu ^{r}-\mu }{\kappa }+\mu ^{r}=\frac{5}{4}\mu ^{\frac{7}{15}}- \frac{1}{4}\mu ,\quad \forall \mu \in (0, 1). $$
2. (ii)
   $$ \varphi (\mu )=\mu ^{\frac{1}{3}}\geq \frac{\mu ^{\frac{1}{2}}-\mu }{ \kappa }+\mu ^{\frac{1}{2}}=\frac{5}{4}\mu ^{\frac{1}{2}}- \frac{1}{4} \mu ,\quad \forall \mu \in (0, 1). $$

Therefore, the assumptions of Theorem 3.1 are satisfied. Then BVP (4.1) has a unique solution \(z_{\lambda }^{*}\) in P, and there exists a constant \(\eta _{\lambda }\in (0, 1)\) such that

And at the same time, \(z_{\lambda }^{*}\) satisfies:

1. (i)

   \(z_{\lambda }^{*}\) is continuous with respect to \(\lambda \in (0,+\infty )\), i.e., for ∀ \(\lambda _{0}\in (0,+\infty )\),

   $$ \bigl\Vert z_{\lambda }^{*}-z_{\lambda _{0}}^{*} \bigr\Vert \rightarrow 0, \quad \text{as } \lambda \rightarrow \lambda _{0}. $$
2. (ii)

   \(0<\lambda _{1}< \lambda _{2}\) implies \(z_{\lambda _{1}}^{*}< z _{\lambda _{2}}^{*}\).

3. (iii)
   $$ \lim_{\lambda \rightarrow 0^{+}} \bigl\Vert z_{\lambda }^{*} \bigr\Vert =0, \qquad \lim_{\lambda \rightarrow +\infty } \bigl\Vert z_{\lambda }^{*} \bigr\Vert =+\infty . $$

Moreover, for any initial values \(z_{0}\), \(\tilde{z}_{0}\in P_{e}\), by constructing successively the sequences as follows:

we have \(z_{n}\rightarrow z_{\lambda }^{*}\) and \(\tilde{z}_{n}\rightarrow z_{\lambda }^{*}\) in E, as \(n\rightarrow \infty \).

Example 4.2

We consider the following problem:

where

Let

\(\gamma =\frac{8}{3}\) (\(n=3\)), \(\gamma _{0}=\frac{3}{2}\), \(\nu _{1}= \frac{2}{3}\), \(q_{1}=\frac{3}{4}\), \(a_{1}=2\), \(a_{2}=\frac{1}{2}\), \(\alpha _{1}=\frac{5}{4}\), \(\alpha _{2}=\frac{11}{8}\), \(I_{1}=[0, 1]\), \(I_{2}=[0, \frac{2}{3}]\), \(b_{j}= (5j-4)^{-1}(5j+1)^{-1}\) (\(j=1,2, \ldots \)), \(\beta _{j}=\frac{5}{3}- 2^{-(7+j)}\) (\(j=1,2,\ldots \)), \(\xi _{j}=(28+2j)^{-1}\) (\(j=1,2,\ldots \)), \(w_{1}(t)=t^{\frac{3}{4}}(1-t)^{2}\), \(w_{2}(t)=t^{\frac{7}{8}}(1+t ^{2})^{-1}\). Then problem (4.2) can be transformed into BVP (1.1) for \(\lambda =1\). By simple computation, we have a rough estimate:

and

which means the properties of Green’s function in Lemma 2.13 are achieved. Let

and

Then

It is easy to check the following conditions:

1. (1)

   For all \(t\in (0,1)\) and \((w, z)\in (0,+\infty )^{2}\), \(f_{1}(t, u, v, w, z)\), \(f_{2}(t, u, v, w, z)\) are increasing in \((u, v)\in (0,+\infty )^{2}\); for all \(t\in (0,1)\) and \((u, v)\in (0,+ \infty )^{2}\), \(f_{1}(t, u, v, w, z)\), \(f_{2}(t, u, v, w, z)\) are decreasing in \((w, z)\in (0,+\infty )^{2}\).

2. (2)

   Let \(\varphi (\mu )=\mu ^{\frac{1}{2}}\). Then, for \(\mu \in (0,1)\), \(t\in (0,1)\), and \((u, v, w, z)\in (0,+\infty )^{4}\),

   $$\begin{aligned}& \begin{aligned} &f_{1}\bigl(t, \mu u, \mu v, \mu ^{-1} w, \mu ^{-1} z\bigr)\\ &\quad = t^{-\frac{1}{3}}( \mu u)^{\frac{1}{4}}\bigl(\mu ^{-1}z\bigr)^{-\frac{1}{4 }}+ (\mu v)^{\frac{1}{3}} \bigl(1-t^{2}\bigr)^{-\frac{1}{2} } + t^{-\frac{1}{4} }\bigl(\mu ^{-1}w\bigr)^{- \frac{1}{4} } \\ &\quad \geq \mu ^{\frac{1}{2}} f_{1}(t, u, v, w, z), \end{aligned} \\& \begin{aligned}& f_{2}\bigl(t, \mu u, \mu v, \mu ^{-1} w, \mu ^{-1} z\bigr)\\ &\quad = t^{-\frac{1}{4}} ( \mu u)^{\frac{1}{4}} \bigl(\mu ^{-1}z\bigr)^{-\frac{1}{4} }+ (\mu v)^{ \frac{1}{3}} \bigl(1-t^{2}\bigr)^{-\frac{1}{2} } + \mu (tw)^{-\frac{1}{4}}(w+ \mu )^{-\frac{3}{4}} \\ &\quad \geq \mu f_{2}(t, u, v, w, z). \end{aligned} \end{aligned}$$
3. (3)

   Let \(\kappa =1\). Then, for all \((u, v, w, z)\in (0,+\infty )^{4}\),

   $$ f_{1}(t, u, v, w, z)\geq f_{2}(t, u, v, w, z). $$
4. (4)

   The functions \(f_{1}\) and \(f_{2}\) satisfy

   $$\begin{aligned}& 0< \int _{0}^{1}f_{1}\bigl(s, 1, 1, s^{\gamma -1}, s^{\gamma -1}\bigr)\,ds= \int _{0}^{1} \bigl(s^{-\frac{3}{4}}+ \bigl(1-s^{2}\bigr)^{-\frac{1}{2}}+s^{- \frac{2}{3}} \bigr)< + \infty , \\& 0< \int _{0}^{1}f_{2}\bigl(s, 1, 1, s^{\gamma -1}, s^{\gamma -1}\bigr)\,ds\leq \int _{0}^{1} \bigl(s^{-\frac{2}{3}}+ \bigl(1-s^{2}\bigr)^{-\frac{1}{2}}+s^{- \frac{2}{3}} \bigr)< + \infty . \end{aligned}$$

Thus BVP (4.2) has a unique positive solution \(z_{1}^{*}\). Then BVP (1.1) has a unique solution \(z_{1}^{*}\) in P, and there exists a constant \(\eta _{1}\in (0, 1)\) such that

Moreover, for any initial values \(z_{0}\), \(\tilde{z}_{0}\in P_{e}\), by constructing successively the sequences as follows:

we have \(z_{n}\rightarrow z_{1}^{*}\) and \(\tilde{z}_{n}\rightarrow z _{1}^{*}\) in E, as \(n\rightarrow \infty \).

Acknowledgements

The authors would like to thank the editors for their helpful suggestions.

Availability of data and materials

Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

Funding

The authors are supported financially by the National Natural Science Foundation of China (11871302), the Natural Science Foundation of Shandong Province of China (ZR2017MA036), and the support from the Australian Research Council for the research is also acknowledged.

Affiliations

1. School of Mathematical Sciences, Qufu Normal University, Qufu, People’s Republic of China

   • Fang Wang
   •  & Lishan Liu

2. Department of Mathematics and Statistics, Curtin University, Perth, Australia

   • Lishan Liu
   •  & Yonghong Wu

Contributions

All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.

Corresponding author

Correspondence to Lishan Liu.

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