Positive solutions to boundary value problems of a high-order fractional differential equation in a Banach space

Abstract

In this paper, by using the fixed-point theorem in the cone of strict-set-contraction operators, we study a class of higher-order boundary value problems of nonlinear fractional differential equation in a Banach space. The sufficient conditions for the existence of at least two positive solutions is obtained. In addition, an example to illustrate the main results is given.

1 Introduction

Fractional calculus is a generalization of ordinary differentiation and integration to arbitrary noninteger order. The fractional differential equations have been of great interest recently. This is because of both the intensive development of the theory of fractional calculus itself and its numerous applications in various fields of science and engineering including fluid flow, rheology, control, electrochemistry, electromagnetic, porous media and probability, etc. (see [1–4]).

In recent years, the existence and uniqueness of solutions of the initial and boundary value problems for fractional equations have been extensively studied (see [4–16] and the references therein). But there are few works that deal with the existence of solutions of nonlinear fractional differential equations in Banach spaces; see [17–24]. In [19], Hussein investigated the existence of pseudo solutions for the following nonlinear m-point boundary value problem of fractional type:

$$\{\begin{array}{c}{D}_{0^{+}}^{q}u(t)+a(t)f(t,u(t))=0,0<t<1,\hfill \\ u(0)=u^{\prime }(0)=\cdots =u^{(n-2)}(0)=0,u(1)={\sum }_{i=1}^{m-2}{\zeta }_{i}u({\eta }_i)\hfill \end{array}$$

in a reflexive Banach space E, where $D_{0^{+}}^q$ is the pseudo fractional differential operator of order $n-1<q\le n$, $n\ge 2$.

In [22], by the monotone iterative technique and the Mönch fixed point theorem, Lv et al. investigated the existence of a solution to the following Cauchy problem for the differential equation with fractional order in a real Banach space E:

$${}^{C}D^{q}u(t)=f(t,u(t)),u(0)=u_0,$$

where ${}^{C}D^{q}u(t)$ is the Caputo derivative of order $0<q<1$.

By means of Darbo’s fixed point theorem, Su [23] established the existence result of solutions to the following boundary value problem of fractional differential equation on unbounded domain $[0,+\mathrm{\infty })$:

$$\{\begin{array}{c}{D}_{0^{+}}^{q}u(t)=f(t,u(t)),t\in [0,+\mathrm{\infty }),1<q\le 2,\hfill \\ u(0)=\theta ,D_{0^{+}}^{q-1}u(\mathrm{\infty })=u_{\mathrm{\infty }}\hfill \end{array}$$

in a Banach space E. $D_{0^{+}}^q$ is the Riemann-Liouville fractional derivative.

Being directly inspired by [9, 15, 19, 22] but taking quite a different method from that in [15, 19–23], we discuss in this paper the following high-order boundary value problem (BVP for short) in a Banach space E:

$$\{\begin{array}{c}{D}_{0+}^{q}u(t)+f(t,u,u^{\prime },\dots ,u^{(n-2)})=\theta ,\mathrm{\forall }t\in (0,1),q\in (n-1,n],\hfill \\ u^{(i)}(0)=0,0\le i\le n-3,\hfill \\ \alpha u^{(n-2)}(0)-\beta u^{(n-1)}(0)=\theta ,\gamma u^{(n-2)}(1)+\delta u^{(n-1)}(1)=\theta ,\hfill \end{array}$$

(1.1)

where θ is the zero element of E, $n\ge 2$, α, β, γ and δ are nonnegative constants satisfying ${\rho }^{-1}=\alpha \gamma +\alpha \delta +\beta \gamma >0$, and $D_{0+}^q$ is the Caputo fractional derivative. Note that the nonlinear term f depends on u and its derivatives $u^{\prime },u^{″},\dots ,u^{(n-2)}$.

The paper is organized as follows. In Section 2 we give some basic definitions in Riemann-Liouville fractional calculus and the Kuratowski noncompactness. In Section 3 we present the expression and properties of Green’s function associated with BVP (1.1), and by using the fixed-point theorem for strict-set-contraction operators and introducing a new cone Ω, we obtain the existence of at least two positive solutions for BVP (1.1) under certain conditions on the nonlinearity. Moreover, an example illustrating our main result is given in Section 4.

2 Preliminaries and lemmas

For convenience of the reader, we present here some definitions and preliminaries which are used throughout the paper. These definitions and lemmas can be found in the recent literature such as [1, 5].

Definition 2.1 ([1])

The Riemann-Liouville fractional integral of order $q>0$ of a function $y(t)$ is given by

$$I_{0+}^{q}y(t)=\frac{1}{\mathrm{\Gamma }(q)}{\int }_0^t{(t-s)}^{q-1}y(s)ds,$$

provided that the right-hand side is defined pointwise.

Definition 2.2 ([1])

The fractional derivative of order $q>0$ of a function $y:{\mathbf{R}}_0^{+}\to \mathbf{R}$ is given by

$$D_{0+}^{q}y(t)=\frac{1}{\mathrm{\Gamma }(n-q)}{\left(\frac{d}{dt}\right)}^n{\int }_0^t{(t-s)}^{n-q-1}y(s)ds,$$

where $n=[q]+1$, $[q]$ denotes the integer part of number q, provided that the right-hand side is defined pointwise. In particular, for $q=n$, $D_{0+}^{q}y(t)=y^{(n)}(t)$.

Lemma 2.3 ([1])

Let $q>0$. Then the fractional differential equation

$$D_{0+}^{q}y(t)=0$$

has the unique solution $y(t)=c_1{t}^{q-1}+c_2{t}^{q-2}+\cdots +c_n{t}^{q-n}$, $c_i\in \mathbf{R}$, $i=1,2,\dots ,n$, here $n-1<q\le n$.

Lemma 2.4 ([5])

Let $q>0$. Then the following equality holds for $y\in L(0,1)$, $D_{0+}^{q}y\in L(0,1)$:

$$I_{0+}^q{D}_{0+}^{q}y(t)=y(t)+c_1{t}^{q-1}+c_2{t}^{q-2}+\cdots +c_N{t}^{q-N}$$

for some $c_i\in \mathbf{R}$, $i=1,2,\dots ,N$, here N is the smallest integer greater than or equal to q.

Let the real Banach space E with the norm $\parallel \cdot \parallel$ be partially ordered by a cone P of E, i.e., $u\le v$ if and only if $v-u\in P$, and P is said to be normal if there exists a positive constant N such that $\theta \le u\le v$ implies $\parallel u\parallel \le N\parallel v\parallel$, where the smallest N is called the normal constant of P. For details on cone theory, see [25].

The basic space used in this paper is $C[I,E]$. For any $u\in C[I,E]$, evidently, $(C[I,E],{\parallel \cdot \parallel }_C)$ is a Banach space with the norm ${\parallel u\parallel }_C={sup}_{t\in I}|u(t)|$, and $P=\{u\in C[I,E]:u(t)\ge \theta \text{ for }t\in I\}$ is a cone of the Banach space $C[I,E]$. We use α, ${\alpha }_C$ to denote the Kuratowski noncompactness measure of bounded sets in the spaces E, $C(I,E)$, respectively. As for the definition of the Kuratowski noncompactness measure, we refer to Ref. [25].

Definition 2.5 ([25], Strict-set contraction operator)

Let $E_1$, $E_2$ be real Banach spaces, $S\subset E_1$. $T:S\to E_2$ is a continuous and bounded operator. If there exists a constant k such that $\alpha (T(S))\le k\alpha (S)$, then T is called a k-set contraction operator. When $k<1$, T is called a strict-set contraction operator.

Lemma 2.6 ([25])

If $D\subset C[I,E]$ is bounded and equicontinuous, then $\alpha (D(t))$ is continuous on I and

$${\alpha }_C(D)=\underset{t\in I}{max}\alpha (D(t)),\alpha \left(\{{\int }_{I}u(t)dt:u\in D\}\right)\le {\int }_I\alpha (D(t))dt,$$

where $D(t)=\{u(t):u\in D,t\in I\}$.

Lemma 2.7 ([25])

Let K be a cone in a Banach space E. Assume that ${\mathrm{\Omega }}_1$, ${\mathrm{\Omega }}_2$ are open subsets of E with $\theta \in {\mathrm{\Omega }}_1,{\overline{\mathrm{\Omega }}}_1\subset {\mathrm{\Omega }}_2$. If $T:K\cap ({\overline{\mathrm{\Omega }}}_2\setminus {\mathrm{\Omega }}_1)\to K$ is a strict-set contraction operator such that either

• (i) $\parallel Tx\parallel \le \parallel x\parallel$, $x\in K\cap \partial {\mathrm{\Omega }}_1$ and $\parallel Tx\parallel \ge \parallel x\parallel$, $x\in K\cap \partial {\mathrm{\Omega }}_2$, or

• (ii) $\parallel Tx\parallel \ge \parallel x\parallel$, $x\in K\cap \partial {\mathrm{\Omega }}_1$ and $\parallel Tx\parallel \le \parallel x\parallel$, $x\in K\cap \partial {\mathrm{\Omega }}_2$,

then T has a fixed point in $K\cap ({\overline{\mathrm{\Omega }}}_2\setminus {\mathrm{\Omega }}_1)$.

In the paper, we always assume that the following three assumptions hold:

• (H0) $0<{\rho }^{\ast }:=\rho {\int }_0^1\phi (s)ds<+\mathrm{\infty }$, where $\phi (s)$ is defined as

  $$\phi (s)=\frac{1}{\mathrm{\Gamma }(q-n+2)}(\beta +\alpha s)[\gamma (1-s)+\delta (q-n+1)]{(1-s)}^{q-n}.$$

• (H1) There exist $a\in C[I,R_{+}]$ and $h\in C[R_{+}^{n-1},R_{+}]$ such that

  $$\begin{array}{r}\parallel f(t,u_1,\dots ,u_{n-1})\parallel \le a(t)h(\parallel u_1\parallel ,\dots ,\parallel u_{n-1}\parallel ),\\ \mathrm{\forall }t\in I,u_k\in P,k=1,\dots ,n-1.\end{array}$$

  (2.1)

• (H2) $f:I\times P_r^{n-1}\to P$ for any $r>0$, f is uniformly continuous on $I\times P_r^{n-1}$ and there exist nonnegative constants $L_k$, $k=1,\dots ,n-1$, with

  $$\frac{2}{{\rho }^{\ast }}(\sum _{k=1}^{n-2}\frac{L_k}{(n-2-k)!}+L_{n-1})<1$$

  (2.2)

  such that

  $$\alpha (f(t,D_1,D_2,\dots ,D_{n-1}))\le \sum _{k=1}^{n-1}{L}_k\alpha (D_k),\mathrm{\forall }t\in I,\text{ bounded }{D}_k\in P_r,$$

  (2.3)

  where $P_r=\{u\in P:\parallel u\parallel \le r\}$.

3 Main results

Lemma 3.1 Given $y\in C[I,E]$, then the unique solution of

$$\{\begin{array}{c}{D}_{0^{+}}^{q-n+2}x(t)+y(t)=0,0<t<1,n-1<q\le n,n>2,\hfill \\ \alpha x(0)-\beta x^{\prime }(0)=0,\gamma x(1)+\delta x^{\prime }(1)=0\hfill \end{array}$$

(3.1)

is

$$x(t)={\int }_0^{1}G(t,s)y(s)ds,$$

(3.2)

where

$$G(t,s)=\{\begin{array}{cc}{G}_1(t,s)=-\frac{{(t-s)}^{q-n+1}}{\mathrm{\Gamma }(q-n+2)}+G_0(t,s),\hfill & 0\le s\le t\le 1,\hfill \\ G_0(t,s),\hfill & 0\le t\le s\le 1,\hfill \end{array}$$

(3.3)

and

$$G_0(t,s)=\frac{\rho (\beta +\alpha t)}{\mathrm{\Gamma }(q-n+2)}[\gamma (1-s)+\delta (q-n+1)]{(1-s)}^{q-n}.$$

Proof Deduced from Lemma 2.4, we have

$$x(t)=-I_{0+}^{q-n+2}y(t)+c_1+c_{2}t$$

for some $c_1,c_2\in \mathbf{\text{R}}$. Then we get

$$\begin{array}{c}x(t)=-\frac{1}{\mathrm{\Gamma }(q-n+2)}{\int }_0^t{(t-s)}^{q-n+1}y(s)ds+c_1+c_{2}t,\hfill \\ x^{\prime }(t)=-\frac{1}{\mathrm{\Gamma }(q-n+1)}{\int }_0^t{(t-s)}^{q-n}y(s)ds+c_2.\hfill \end{array}$$

By boundary conditions $\alpha x(0)-\beta x^{\prime }(0)=0$, $\gamma x(1)+\delta x^{\prime }(1)=0$, and noting that $\mathrm{\Gamma }(q-n+2)=(q-n+1)\mathrm{\Gamma }(q-n+1)$, we have

$$c_1=\frac{\rho \beta }{\mathrm{\Gamma }(q-n+2)}{\int }_0^1[\gamma (1-s)+\delta (q-n+1)]{(1-s)}^{q-n}y(s)ds$$

and

$$c_2=\frac{\rho \alpha }{\mathrm{\Gamma }(q-n+2)}{\int }_0^1[\gamma (1-s)+\delta (q-n+1)]{(1-s)}^{q-n}y(s)ds.$$

Thus

$$\begin{array}{rcl}x(t)& =& -{\int }_0^t\frac{{(t-s)}^{q-n+1}}{\mathrm{\Gamma }(q-n+2)}y(s)ds\\ +\frac{\rho \beta }{\mathrm{\Gamma }(q-n+2)}{\int }_0^1[\gamma (1-s)+\delta (q-n+1)]{(1-s)}^{q-n}y(s)ds\\ +\frac{\rho \alpha t}{\mathrm{\Gamma }(q-n+2)}{\int }_0^1[\gamma (1-s)+\delta (q-n+1)]{(1-s)}^{q-n}y(s)ds\\ =& -{\int }_0^t\frac{{(t-s)}^{q-n+1}}{\mathrm{\Gamma }(q-n+2)}y(s)ds+\rho (\beta +\alpha t){\int }_0^1\frac{[\gamma (1-s)+\delta (q-n+1)]{(1-s)}^{q-n}}{\mathrm{\Gamma }(q-n+2)}y(s)ds\\ =& {\int }_0^{1}G(t,s)y(s)ds,\end{array}$$

where $G(t,s)$ is Green’s function defined by (3.3). This completes the proof. □

Moreover, there is one paper [10] in which the following statement was shown.

Lemma 3.2 ([10])

The function $G(t,s)$ defined by Lemma  2.3 has the following properties:

• (i) $G(t,s)$ is continuous on $[0,1]\times [0,1]$;

• (ii) if $\beta >\frac{n-q}{q-n+1}\alpha$, then $0<G(t,s)\le G(s,s)$ for any $t,s\in [0,1]$.

Lemma 3.3 If $\beta >\frac{n-q}{q-n+1}\alpha$, then the function $G(t,s)$ satisfies:

$$\lambda \phi (s)\le G(t,s)\le \rho \phi (s)\mathit{\text{for}}t,s\in [0,1],$$

where

$$\begin{array}{c}\lambda :=min\{\frac{4\rho \alpha \gamma \delta [\beta (q-n+1)+\alpha (q-n)]}{\zeta },\frac{4\rho \alpha \beta \gamma \delta (q-n+1)}{\zeta }\},\hfill \\ \zeta :={[\alpha \gamma -\beta \gamma +\alpha \delta (q-n+1)]}^2+4\alpha \beta \gamma [\gamma +\delta (q-n+1)].\hfill \end{array}$$

Proof According to (3.3) and Lemma 3.2, we have

$$G^{\ast }(s)\le G(t,s)\le \rho \phi (s),$$

where

$$G^{\ast }(s)=\{\begin{array}{cc}{G}_1(1,s),\hfill & 0\le s<\frac{\beta \gamma -\alpha \delta (q-n)}{\alpha \delta +\beta \gamma },\hfill \\ G_0(0,s),\hfill & \frac{\beta \gamma -\alpha \delta (q-n)}{\alpha \delta +\beta \gamma }\le s<1.\hfill \end{array}$$

Since

$$\begin{array}{rcl}\underset{0<s<1}{inf}\frac{G_1(1,s)}{\phi (s)}& =& \underset{0<s<1}{inf}\frac{-{\rho }^{-1}(1-s)+(\beta +\alpha )[\gamma (1-s)+\delta (q-n+1)]}{{\rho }^{-1}(\beta +\alpha s)[\gamma (1-s)+\delta (q-n+1)]}\\ \ge & \frac{\rho \delta [\beta (q-n+1)+\alpha (q-n)]}{(\beta +\alpha s)[\gamma (1-s)+\delta (q-n+1)]}\\ \ge & \frac{4\rho \alpha \gamma \delta [\beta (q-n+1)+\alpha (q-n)]}{{[\alpha \gamma -\beta \gamma +\alpha \delta (q-n+1)]}^2+4\alpha \beta \gamma [\gamma +\delta (q-n+1)]}\\ =& \frac{4\rho \alpha \gamma \delta [\beta (q-n+1)+\alpha (q-n)]}{\zeta },\end{array}$$

and

$$\begin{array}{rcl}\underset{0<s<1}{inf}\frac{G_0(0,s)}{\phi (s)}& =& \underset{0<s<1}{inf}\frac{\rho [\beta \gamma (1-s)+\beta \delta (q-n+1)]}{(\beta +\alpha s)[\gamma (1-s)+\delta (q-n+1)]}\\ \ge & \frac{4{\rho }^{-1}\alpha \beta \gamma \delta (q-n+1)}{{[\alpha \gamma -\beta \gamma +\alpha \delta (q-n+1)]}^2+4\alpha \beta \gamma [\gamma +\delta (q-n+1)]}\\ =& \frac{4\rho \alpha \beta \gamma \delta (q-n+1)}{\zeta },\end{array}$$

then we get

$$\lambda \phi (s)\le G(t,s)\le G(s,s)\le \rho \phi (s).$$

 □

Lemma 3.4 Let $u(t)=I_{0^{+}}^{n-2}x(t)$, $x\in C[I,E]$. Then problem (1.1) can be transformed into the following modified problem:

$$\{\begin{array}{c}{D}_{0^{+}}^{q-n+2}x(t)+f(s,I_{0^{+}}^{n-2}x(s),\dots ,I_{0^{+}}^{1}x(s),x(s))=\theta ,\hfill \\ \alpha x(0)-\beta x^{\prime }(0)=\theta ,\gamma x(1)+\delta x^{\prime }(1)=\theta ,\hfill \end{array}$$

(3.4)

where $0<t<1$, $n-1<q\le n$, $n\ge 2$. Moreover, if $x\in C[I,E]$ is a solution of problem (3.3) and $x\ge \theta$, $x\not\equiv \theta$, then the function $u(t)=I_{0^{+}}^{n-2}x(s)$ is a positive solution of (1.1).

To obtain a positive solution, we construct a cone Ω by

$$\mathrm{\Omega }=\{x(t)\in P:x(t)\ge \frac{\lambda }{\rho }x(s),\mathrm{\forall }t,s\in I\},$$

(3.5)

where $P=\{x\in C[I,E]:x(t)\ge \theta ,t\in I\}$.

Define an integral operator $T:\mathrm{\Omega }\to E$ by

$$(Tx)(t)={\int }_0^{1}G(t,s)f(s,I_{0^{+}}^{n-2}x(s),I_{0^{+}}^{n-3}x(s),\dots ,I_{0^{+}}^{1}x(s),x(s))ds,0\le t\le 1.$$

(3.6)

Lemma 3.5 Assume that (H0)-(H2) hold. Then $T:\mathrm{\Omega }\to \mathrm{\Omega }$ is a strict-set contraction operator.

Proof From Lemma 3.3 and (3.6), we obtain

$$(Tx)(t)\ge \lambda {\int }_0^1\phi (s)f(s,I_{0^{+}}^{n-2}{x}_j(s),\dots ,I_{0^{+}}^1{x}_j(s),x_j(s))ds.$$

On the other hand,

$$\begin{array}{rcl}(Tx)(s)& =& {\int }_0^{1}G(s,s)f(s,I_{0^{+}}^{n-2}{x}_j(s),\dots ,I_{0^{+}}^1{x}_j(s),x_j(s))ds\\ \le & \rho {\int }_0^1\phi (s)f(s,I_{0^{+}}^{n-2}{x}_j(s),\dots ,I_{0^{+}}^1{x}_j(s),x_j(s))ds\\ =& \frac{\rho }{\lambda }{\int }_0^1\lambda \phi (s)f(s,I_{0^{+}}^{n-2}{x}_j(s),\dots ,I_{0^{+}}^1{x}_j(s),x_j(s))ds\le \frac{\rho }{\lambda }(Tx)(t).\end{array}$$

Then $(Tx)(t)\ge \frac{\lambda }{\rho }(Tx)(s)$, which implies $(Tx)\in \mathrm{\Omega }$, i.e., $T(\mathrm{\Omega })\subset \mathrm{\Omega }$.

Next we prove that T is continuous on Ω. Let $\{x_j\},\{x\}\subset \mathrm{\Omega }$ and ${\parallel x_j-x\parallel }_{\mathrm{\Omega }}\to 0$ ($j\to \mathrm{\infty }$). Hence $\{x_j\}$ is a bounded subset of Ω. Thus, there exists $r>0$ such that $r={sup}_j{\parallel x_j\parallel }_{\mathrm{\Omega }}<\mathrm{\infty }$ and ${\parallel x\parallel }_{\mathrm{\Omega }}\le r$.

According to the properties of f, for $\mathrm{\forall }\epsilon >0$, there exists $J>0$ such that

$$\parallel f(s,I_{0^{+}}^{n-2}{x}_j(s),\dots ,I_{0^{+}}^1{x}_j(s),x_j(s))-f(s,I_{0^{+}}^{n-2}x(s),\dots ,I_{0^{+}}^{1}x(s),x(s))\parallel \le \frac{\epsilon }{{\rho }^{\ast }}$$

for $j\ge J$, $\mathrm{\forall }t\in I$.

Then

$$\begin{array}{rcl}\parallel (T{x}_j)(t)-(Tx)(t)\parallel & \le & {\int }_0^{1}G(t,s)\parallel f(s,I_{0^{+}}^{n-2}{x}_j(s),\dots ,I_{0^{+}}^1{x}_j(s),x_j(s))\\ -f(s,I_{0^{+}}^{n-2}x(s),\dots ,I_{0^{+}}^{1}x(s),x(s))\parallel ds\\ \le & \rho {\int }_0^1\phi (s)ds\cdot \parallel f(s,I_{0^{+}}^{n-2}{x}_j(s),\dots ,I_{0^{+}}^1{x}_j(s),x_j(s))\\ -f(s,I_{0^{+}}^{n-2}x(s),\dots ,I_{0^{+}}^{1}x(s),x(s))\parallel <\epsilon .\end{array}$$

Therefore, $\mathrm{\forall }\epsilon >0$ for any $t\in I$ and $j\ge J$. We get

$$\parallel (T{x}_j)(t)-(Tx)(t)\parallel \to 0.$$

This implies T is continuous on Ω.

By the properties of continuity of $G(t,s)$, it is easy to see that T is equicontinuous on I.

Finally, we are going to show that T is a strict-set contraction operator. Let $D\subset \mathrm{\Omega }$ be bounded. Then, by condition (H1), Lemma 3.1 implies ${\alpha }_C(TD)={max}_{t\in I}\alpha ((TD)(t))$. It follows from (3.6) that

$$\begin{array}{rcl}\alpha ((TD)(t))& =& \alpha \left(\{{\int }_0^{1}G(t,s)f(s,(T_{1}x)(s),\dots ,(T_{n-2}x)(s),x(s))ds\in D\}\right)\\ \le & {\int }_0^1\alpha \left(\{G(t,s)f(s,(T_{1}x)(s),\dots ,(T_{n-2}x)(s),x(s))ds\in D\}\right)\\ \le & \rho {\int }_0^1\phi (s)\alpha \left(f(I\times \left(I_{0^{+}}^{n-2}D\right)(I)\times \cdots \times \left(I_{0^{+}}^{1}D\right)(I)\times D(I))\right)ds\\ \le & {\rho }^{\ast }(\sum _{k=1}^{n-2}{L}_k\alpha (\left(I_{0^{+}}^{n-1-k}D\right)(I))+L_{n-1}\alpha (D(I))),\end{array}$$

which implies

$${\alpha }_C(TD)\le {\rho }^{\ast }(\sum _{k=1}^{n-2}{L}_k\alpha (\left(I_{0^{+}}^{n-1-k}D\right)(I))+L_{n-1}\alpha (D(I))).$$

(3.7)

Obviously,

$$\begin{array}{rl}\alpha \left(I_{0^{+}}^{n-1-k}D\right)(I)& =\alpha \left(\{{\int }_0^s\frac{{(s-\nu )}^{n-2-k}}{(n-2-k)!}x(s)ds:\nu \in [0,s],s\in I,k=1,\dots ,n-2\}\right)\\ \le \frac{1}{(n-2-k)!}\alpha (D(I)).\end{array}$$

(3.8)

Using a similar method as in the proof of Theorem 2.1.1 in [25], we have

$$\alpha (D(I))\le 2{\alpha }_C(D).$$

(3.9)

Therefore, it follows from (3.7), (3.8) and (3.9) that

$${\alpha }_C(TD)\le \frac{2}{{\rho }^{\ast }}(\sum _{k=1}^{n-2}\frac{L_k}{(n-2-k)!}+L_{n-1}){\alpha }_C(D).$$

Noticing that (2.3), we obtain that T is a strict-set contraction operator. This completes the proof. □

Now we are in a position to give the main result of this work.

Theorem 3.6 Let the cone P be normal and conditions (H0)-(H2) hold. In addition, assume that the following conditions are satisfied:

• (H3) There exist $u^{\ast }\in P\mathrm{\setminus }\{\theta \}$, $c_1\in C[I,R_{+}]$ and $h_1\in C[P^{n-1},R_{+}]$ such that

  $$f(t,u_1,\dots ,u_{n-1})\ge c_1(t)h_1(u_1,\dots ,u_{n-1})u^{\ast },\mathrm{\forall }t\in I,u_k\in P,$$

  and

  $$h_1^{\mathrm{\infty }}=\underset{{\sum }_{k=1}^{n-1}\parallel u_k\parallel \to \mathrm{\infty }}{lim}\frac{h_1(u_1,\dots ,u_{n-1})}{{\sum }_{k=1}^{n-1}\parallel u_k\parallel }>{\mathrm{△}}^{-1},\mathit{\text{as}}{u}_k\in P,$$

  where $\mathrm{△}:=N^{-2}{\rho }^{-1}({\lambda }^2{\int }_0^1\phi (s)c_1(s)ds\parallel u^{\ast }\parallel )$.

• (H4) There exist $u_{\ast }\in P\mathrm{\setminus }\{\theta \}$, $c_2\in C[I,R_{+}]$ and $h_2\in C[P^{n-1},R_{+}]$ such that

  $$f(t,u_1,\dots ,u_{n-2})\ge c_2(t)h_2(u_1,\dots ,u_{n-1})u_{\ast },\mathrm{\forall }t\in I,u_k\in P,$$

  and

  $$h_1^0=\underset{{\sum }_{k=1}^{n-1}\parallel u_k\parallel \to 0}{lim}\frac{h_2(u_1,\dots ,u_{n-1})}{{\sum }_{k=1}^{n-1}\parallel u_i\parallel }>{\mathrm{\nabla }}^{-1},\mathit{\text{as}}{u}_k\in P,$$

  where $\mathrm{\nabla }:=N^{-2}{\rho }^{-1}({\lambda }^2{\int }_0^1\phi (s)c_2(s)ds\parallel u_{\ast }\parallel )$.

• (H5) There exists $\xi >0$ such that

  $$\rho N{M}_{\xi }{\int }_0^1\phi (s)a(s)ds<\xi ,$$

  where $M_{\xi }={max}_{u_k\in P_{\xi }}\{h(\parallel u_1\parallel ,\dots ,\parallel u_{n-1}\parallel )\}$.

Then problem (1.1) has at least two different positive solutions.

Proof Consider condition (H3), there exists $r_1>0$ such that

$$h_1(u_1,\dots ,u_{n-1})\ge (h_1^{\mathrm{\infty }}-{\epsilon }_1)\sum _{k=1}^{n-1}\parallel u_k\parallel ,\mathrm{\forall }{u}_k\in P,\sum _{k=1}^{n-1}\parallel u_k\parallel \ge r_1,$$

where ${\epsilon }_1>0$ satisfies $(h_1^{\mathrm{\infty }}-{\epsilon }_1)\mathrm{△}\ge 1$.

Therefore,

$$f(t,u_1,\dots ,u_{n-1})\ge (h_1^{\mathrm{\infty }}-{\epsilon }_1)\sum _{k=1}^{n-1}\parallel u_k\parallel \cdot c_1(t)u^{\ast },\mathrm{\forall }{u}_k\in P,\sum _{k=1}^{n-1}\parallel u_k\parallel \ge r_1.$$

Take

$$R>max\{N\rho {\lambda }^{-1}{r}_1,\xi \}.$$

Then, for $t\in I$, $x\in \mathrm{\Omega }$, ${\parallel x\parallel }_{\mathrm{\Omega }}=R$, we have by (3.5)

$$\parallel x(t)\parallel \ge \lambda {(\rho N)}^{-1}{\parallel x\parallel }_{\mathrm{\Omega }}\ge \lambda {(\rho N)}^{-1}R>r_1.$$

Thus

$$\begin{array}{rcl}(Tx)(t)& =& {\int }_0^{1}G(t,s)f(s,I_{0^{+}}^{n-2}x(s),\dots ,I_{0^{+}}^{1}x(s),x(s))ds\\ \ge & {\int }_0^1\lambda \phi (s)f(s,I_{0^{+}}^{n-2}x(s),\dots ,I_{0^{+}}^{1}x(s),x(s))ds\\ \ge & (h_1^{\mathrm{\infty }}-{\epsilon }_1)\lambda {\int }_0{u}^1\phi (s)(\sum _{k=1}^{n-2}\parallel I_{0^{+}}^{n-1-k}x(s)\parallel +\parallel x(s)\parallel )c_1(s)ds\cdot u^{\ast }\\ \ge & (h_1^{\mathrm{\infty }}-{\epsilon }_1)\lambda {\int }_0^1\phi (s)c_1(s)\parallel x(s)\parallel ds\cdot u^{\ast }\\ \ge & (h_1^{\mathrm{\infty }}-{\epsilon }_1){\lambda }^2{(\rho N)}^{-1}{\parallel x\parallel }_{\mathrm{\Omega }}({\int }_0^1\phi (s)c_1(s)ds)\cdot u^{\ast }\\ =& (h_1^{\mathrm{\infty }}-{\epsilon }_1)\frac{{\lambda }^2}{\rho N^2}({\int }_0^1\phi (s)c_1(s)ds\parallel u^{\ast }\parallel )\cdot \frac{N{\parallel x\parallel }_{\mathrm{\Omega }}}{\parallel u^{\ast }\parallel }{u}^{\ast }\\ \ge & \frac{N{\parallel x\parallel }_{\mathrm{\Omega }}}{\parallel u^{\ast }\parallel }\cdot u^{\ast },\end{array}$$

(3.10)

and consequently,

$${\parallel Tx\parallel }_{\mathrm{\Omega }}\ge {\parallel x\parallel }_{\mathrm{\Omega }},\mathrm{\forall }x\in \mathrm{\Omega },{\parallel x\parallel }_{\mathrm{\Omega }}=R.$$

(3.11)

Similarly, by condition (H4), there exists $r_2>0$ such that

$$h_2(u_1,\dots ,u_{n-1})\ge (h_1^0-{\epsilon }_2)\sum _{k=1}^{n-1}\parallel u_k\parallel ,\mathrm{\forall }{u}_k\in P,0<\sum _{k=1}^{n-1}\parallel u_k\parallel \le r_2,$$

where ${\epsilon }_2>0$ satisfies $(h_1^0-{\epsilon }_2)\mathrm{\nabla }\ge 1$.

Therefore,

$$f(t,u_1,\dots ,u_{n-1})\ge (h_1^0-{\epsilon }_2)\sum _{k=1}^{n-1}\parallel u_k\parallel \cdot c_2(t)u_{\ast },\mathrm{\forall }{u}_k\in P,0<\sum _{k=1}^{n-1}\parallel u_k\parallel \le r_2.$$

Choose

$$0<r<min\{{\left(\sum _{k=0}^{n-2}\frac{1}{k!}\right)}^{-1}{r}_2,\xi \}.$$

Then, for $t\in I_{\tau }$, $x\in \mathrm{\Omega }$, ${\parallel x\parallel }_{\mathrm{\Omega }}=r$, similar to (3.10), we have

$$\begin{array}{rl}(Tx)(t)& \ge (h_1^0-{\epsilon }_2)\lambda {\int }_0^1\phi (s)c_2(s)\parallel x(s)\parallel ds\cdot u_{\ast }\\ \ge (h_1^0-{\epsilon }_2){\lambda }^2{(\rho N)}^{-1}{\parallel x\parallel }_{\mathrm{\Omega }}({\int }_0^1\phi (s)c_2(s)ds)\cdot u_{\ast }\\ =(h_1^0-{\epsilon }_2)\frac{{\lambda }^2}{\rho N^2}({\int }_0^1\phi (s)c_2(s)ds\parallel u_{\ast }\parallel )\cdot \frac{N{\parallel x\parallel }_{\mathrm{\Omega }}}{\parallel u_{\ast }\parallel }{u}_{\ast }\\ \ge \frac{N{\parallel x\parallel }_{\mathrm{\Omega }}}{\parallel u_{\ast }\parallel }\cdot u_{\ast },\end{array}$$

which implies

$${\parallel Tx\parallel }_{\mathrm{\Omega }}\ge {\parallel x(s)\parallel }_{\mathrm{\Omega }},\mathrm{\forall }x\in \mathrm{\Omega },{\parallel x\parallel }_{\mathrm{\Omega }}=r.$$

(3.12)

On the other hand, according to Lemma 3.3 and (3.6), we get

$$(Tx)(t)\le \rho {\int }_0^1\phi (s)f(s,I_{0^{+}}^{n-2}x(s),\dots ,I_{0^{+}}^{1}x(s),x(s))ds.$$

(3.13)

By condition (H1), for $t\in I$, $x\in \mathrm{\Omega }$, ${\parallel x\parallel }_{\mathrm{\Omega }}=\xi$, we have

$$\parallel f(t,u_1,\dots ,u_{n-1})\parallel \le a(t)h(\parallel u_1\parallel ,\dots ,\parallel u_{n-1}\parallel )\le M_{\xi }a(t).$$

Therefore,

$${\parallel (Tx)(t)\parallel }_{\mathrm{\Omega }}\le \rho N{M}_{\xi }\cdot {\int }_0^1\phi (s)a(s)ds<\xi ={\parallel x\parallel }_{\mathrm{\Omega }}.$$

(3.14)

Applying Lemma 2.7 to (3.12), (3.13) and (3.14) yields that T has a fixed point $x_1\in {\overline{\mathrm{\Omega }}}_{r,\xi }$, $r\le \parallel x_1\parallel \le \xi$, and a fixed point $x_2\in {\overline{\mathrm{\Omega }}}_{\xi ,R}$, $\xi \le \parallel x_2\parallel \le R$. Noticing (3.13), we get $\parallel x_1\parallel \ne \xi$ and $\parallel x_2\parallel \ne \xi$. This and Lemma 3.4 complete the proof. □

Theorem 3.7 Let the cone P be normal and conditions (H0) ∼ (H3) hold. In addition, assume that the following condition is satisfied:

(H6)

$$\frac{h(\parallel u_1\parallel ,\dots ,\parallel u_{n-1}\parallel )}{{\sum }_{k=1}^{n-1}\parallel u_k\parallel }\to 0,\mathit{\text{as}}{u}_k\in P,\sum _{k=1}^{n-1}\parallel u_k\parallel \to 0^{+}.$$

(3.15)

Then problem (1.1) has at least one positive solution.

Proof By (H3), we can choose $R>N\rho {\lambda }^{-1}{r}_1$. As in the proof of Theorem 3.6, it is easy to see that (3.11) holds. On the other hand, considering (3.15), there exists $r_3>0$ such that

$$h(\parallel u_1\parallel ,\dots ,\parallel u_{n-1}\parallel )\le {\epsilon }_3\sum _{k=1}^{n-1}\parallel u_k\parallel ,\text{for }t\in I,u_k\in P,0<\sum _{k=1}^{n-1}\parallel u_k\parallel \le r_3,$$

where ${\epsilon }_3>0$ satisfies

$${\epsilon }_3={(N\rho \sum _{k=1}^{n-1}\frac{1}{k!}{\int }_0^1\phi (s)a(s)ds)}^{-1}.$$

Choose $0<r^{\ast }<min\{{({\sum }_{k=0}^{n-2}\frac{1}{k!})}^{-1}{r}_3,R\}$. For $t\in I$, $x\in \mathrm{\Omega }$, ${\parallel x\parallel }_{\mathrm{\Omega }}=r^{\ast }$, it follows from (3.5) that

$$\begin{array}{r}0<{(\rho N)}^{-1}\lambda r^{\ast }\le \parallel x\parallel <r_3,\\ 0<{(\rho N)}^{-1}\lambda r^{\ast }\le \sum _{k=1}^{n-1}\parallel I_{0^{+}}^{n-1-k}x(t)\parallel \le \sum _{k=1}^{n-1}\frac{1}{k!}\parallel x\parallel <r_3.\end{array}$$

(3.16)

Then, for $t\in I$, $x\in \mathrm{\Omega }$, ${\parallel x\parallel }_{\mathrm{\Omega }}=r^{\ast }$, we have

$$\begin{array}{rcl}\parallel (Tx)(t)\parallel & \le & \rho {\int }_0^1\phi (s)\parallel f(s,I_{0^{+}}^{n-2}x(s),\dots ,I_{0^{+}}^{1}x(s),x(s))\parallel ds\\ \le & \rho {\int }_0^1\phi (s)a(s)h(\parallel I_{0^{+}}^{n-2}x(s)\parallel ,\dots ,\parallel I_{0^{+}}^{1}x(s)\parallel ,\parallel x(s)\parallel )ds\\ \le & {\epsilon }_3\rho {\int }_0^1\phi (s)a(s)(\sum _{k=1}^{n-2}\parallel I_{0^{+}}^{n-1-k}x(s)\parallel +\parallel x(s)\parallel )ds\\ \le & N{\epsilon }_3\rho \sum _{k=0}^{n-1}\frac{1}{k!}{r}^{\ast }{\int }_0^1\phi (s)a(s)ds=r^{\ast },\end{array}$$

and consequently,

$${\parallel (Tx)(t)\parallel }_{\mathrm{\Omega }}\le {\parallel x\parallel }_{\mathrm{\Omega }},\mathrm{\forall }x\in \mathrm{\Omega },\parallel x\parallel \le r^{\ast }.$$

(3.17)

Since $0<r^{\ast }<R$, applying Lemma 2.7 to (3.11) and (3.17) yields that T has a fixed point $x_0\in {\overline{\mathrm{\Omega }}}_{r^{\ast },R}$, $r^{\ast }\le \parallel x_0\parallel \le R$. This and Lemma 3.5 complete the proof. □

4 An example

Consider the following system of scalar differential equations of fractional order:

$$\{\begin{array}{c}-D^{\frac{5}{2}}{u}_k(t)=\frac{1}{24}{u}_k(t)+\frac{1+t}{324k^2}\{{[2u_{k+2}(t)+2u_{k+1}^{\prime }(t)+{\sum }_{j=1}^{\mathrm{\infty }}{u}_j(t)+{\sum }_{j=1}^{\mathrm{\infty }}{u}_{2j}^{\prime }(t)]}^3\hfill \\ \phantom{-D^{\frac{5}{2}}{u}_k(t)=}+\sqrt{3u_{2k}(t)+3u_{k+1}^{\prime }(t)+{\sum }_{j=1}^{\mathrm{\infty }}{u}_j(t)+{\sum }_{j=1}^{\mathrm{\infty }}{u}_j^{\prime }(t)}\},t\in I,\hfill \\ u_k(0)=u_k^{\prime }(0)=0,\frac{1}{4}{u}_k(0)-\frac{1}{2}{u}_k^{\prime }(0)=0,\hfill \\ u_k(1)+\frac{1}{2}{u}_k^{\prime }(1)=0,k=1,2,3,\dots .\hfill \end{array}$$

(4.1)

Conclusion Problem (4.1) has at least two positive solutions.

Proof Let $E=l^1=\{u=(u_1,u_2,\dots ,u_k,\dots ):{\sum }_{k=1}^{\mathrm{\infty }}|u_k|<\mathrm{\infty }\}$ with the norm $\parallel u\parallel ={\sum }_{k=1}^{\mathrm{\infty }}|u_k|$, and $P=\{(u_1,\dots ,u_k,\dots ):u_k\ge 0,k=1,2,3,\dots \}$. Then P is a normal cone in E with a normal constant $N=1$, and system (4.1) can be regarded as a boundary value problem of the form (1.1). In this situation, $q=\frac{5}{2}$, $n=3$, $\alpha =\frac{1}{4}$, $\beta =\frac{1}{2}$, $\gamma =1$, $\delta =\frac{1}{2}$, $u=(u_1,\dots ,u_n,\dots )$, $f=(f_1,f_2,\dots ,f_n,\dots )$, in which

$$\begin{array}{rl}{f}_k(t,u_k,u_k^{\prime })=& \frac{1}{24}{u}_k(t)+\frac{1+t}{324k^2}\{{[2u_{k+2}(t)+2u_{k+1}^{\prime }(t)+\sum _{j=1}^{\mathrm{\infty }}{u}_j(t)+\sum _{j=1}^{\mathrm{\infty }}{u}_{2j}^{\prime }(t)]}^3\\ +\sqrt{3u_{2k}(t)+3u_{k+1}^{\prime }(t)+\sum _{j=1}^{\mathrm{\infty }}{u}_j(t)+\sum _{j=1}^{\mathrm{\infty }}{u}_j^{\prime }(t)}\}.\end{array}$$

(4.2)

By calculating, we have $\rho =\frac{4}{5}$, $\lambda \approx 0.1515$, and

$$\frac{1}{2}=\beta >\frac{n-q}{q-n+1}\alpha =\frac{1}{4},$$

and

$${\rho }^{\ast }=\rho {\int }_0^1\phi (s)ds=\frac{1}{5\sqrt{\pi }}{\int }_0^1(2+s)(\sqrt{1-s}+\frac{1}{4\sqrt{1-s}})ds=\frac{44}{75\sqrt{\pi }}\approx 0.3310,$$

which implies that condition (H0) is satisfied. Observing the inequality ${\sum }_{k=1}^{\mathrm{\infty }}\frac{1}{k^2}<2$, we get, by (4.2),

$$\parallel f(t,u,v)\parallel =\sum _{k=1}^{\mathrm{\infty }}|f_k(t,u_k,v_k)|\le \frac{1+t}{3}(\frac{1}{8}\parallel u\parallel +\frac{1}{2}{(\parallel u\parallel +\parallel v\parallel )}^3+\frac{1}{27}\sqrt{\parallel u\parallel +\parallel v\parallel }).$$

(4.3)

Hence (H1) is satisfied for $a(t)=\frac{1+t}{3}$, and

$$h(x,y)=\frac{1}{8}x+\frac{1}{2}{(x+y)}^3+\frac{1}{27}\sqrt{x+y}.$$

Now, we check condition (H2). Obviously, $f:I\times P_r^2\to P$ for any $r>0$, and f is uniformly continuous on $I\times P_r^2$. Let $f=f^{(1)}+f^{(2)}$, where $f^{(1)}=(f_1^{(1)},\dots ,f_k^{(1)},\dots )$ and $f^{(2)}=(f_1^{(2)},\dots ,f_k^{(2)},\dots )$, in which

$$\begin{array}{rl}{f}_k^{(1)}(t,u,v)=& \frac{1+t}{324k^2}\{{[2u_{k+2}(t)+2u_{k+1}^{\prime }(t)+\sum _{j=1}^{\mathrm{\infty }}{u}_j(t)+\sum _{j=1}^{\mathrm{\infty }}{u}_{2j}^{\prime }(t)]}^3\\ +\sqrt{3u_{2k}(t)+3u_{k+1}^{\prime }(t)+\sum _{j=1}^{\mathrm{\infty }}{u}_j(t)+\sum _{j=1}^{\mathrm{\infty }}{u}_j^{\prime }(t)}\}(k=1,2,3,\dots ),\end{array}$$

(4.4)

and

$$f_k^{(2)}(t,u,v)=\frac{1}{24}{u}_k(t)(k=1,2,3,\dots ).$$

(4.5)

For any $t\in I$ and bounded subsets $D_1,D_2\subset E$, by (4.4), (4.5), we know

$$\alpha (f^{(2)}(I,D_1,D_2))\le \frac{1}{24}\alpha (D_1),\mathrm{\forall }t\in I,D_1,D_2\subset E,$$

(4.6)

and

$$\begin{array}{rcl}0& \le & \parallel f^{(1)}(t,u,v)\parallel =\sum _{k=1}^{\mathrm{\infty }}|f_k^{(1)}(t,u_k,v_k)|\\ \le & {(\parallel u\parallel +\parallel v\parallel )}^3+\frac{2}{27}\sqrt{\parallel u\parallel +\parallel v\parallel },\mathrm{\forall }t\in I,u,v\in E.\end{array}$$

Similar to the proof of [[25], Example 2.12], we have

$$\alpha (f^{(1)}(t,D_1,D_2))=0,\mathrm{\forall }t\in I,\text{ bounded sets }{D}_1,D_2\subset E.$$

(4.7)

It follows from (4.6) and (4.7) that

$$\alpha (f(I,D_1,D_2))\le \frac{1}{24}\alpha (D_1),\mathrm{\forall }t\in I,D_1,D_2\subset E,$$

and

$$\frac{2}{{\rho }^{\ast }}(\sum _{k=1}^{n-2}\frac{L_k}{(n-2-k)!}+L_{n-1})=\frac{24}{29\sqrt{\pi }}\approx 0.2518<1,$$

i.e., condition (H2) holds for $L_1=\frac{1}{24}$, $L_2=0$.

On the other hand, by (4.2), we have

$$f_k(t,u,v)\ge \frac{1+t}{324k^2}{(\parallel u\parallel +\parallel v\parallel )}^3,\mathrm{\forall }t\in I_{\tau },u,v\in P(k=1,2,3,\dots ),$$

and

$$f_k(t,u,v)\ge \frac{1+t}{324k^2}\sqrt{\parallel u\parallel +\parallel v\parallel },\mathrm{\forall }t\in I_{\tau },u,v\in P(k=1,2,3,\dots ).$$

Hence condition (H3) is satisfied for

$$c_1(t)=\frac{1+t}{324k^2},h_{1,k}(u,v)={(\parallel u\parallel +\parallel v\parallel )}^3,\text{and}{u}^{\ast }=(1,\dots ,\frac{1}{k^2},\dots ),$$

in this situation,

$$h_{1,k}^{\mathrm{\infty }}=\underset{\parallel u\parallel +\parallel v\parallel \to \mathrm{\infty }}{lim}\frac{{(\parallel u\parallel +\parallel v\parallel )}^3}{\parallel u\parallel +\parallel v\parallel }=\mathrm{\infty }>{\mathrm{△}}^{-1}.$$

And condition (H4) is also satisfied for

$$c_2(t)=\frac{1+t}{324k^2},h_{2,k}(u,v)=\sqrt{\parallel u\parallel +\parallel v\parallel },\text{and}{u}_{\ast }=(1,\dots ,\frac{1}{k^2},\dots ),$$

in this situation,

$$h_{2,k}^{\mathrm{\infty }}=\underset{\parallel u\parallel +\parallel v\parallel \to 0}{lim}\frac{\sqrt{\parallel u\parallel +\parallel v\parallel }}{\parallel u\parallel +\parallel v\parallel }=\mathrm{\infty }>{\mathrm{\nabla }}^{-1}.$$

Finally, choose $\xi =1$. It is easy to check that condition (H5) is satisfied. In this case, $M_{\xi }\approx 4.1774$, and so

$$\rho N{M}_{\xi }{\int }_0^1\phi (s)a(s)ds\approx 0.9604<\xi =1.$$

From Theorem 3.6, the conclusion follows and the proof is complete. □