Solvability of a Riemann–Liouville-Type Fractional-Impulsive Differential Equation with a Riemann–Stieltjes Integral Boundary Condition

Abstract

In this work, we address the solvability of a Riemann–Liouville-type fractional-impulsive integral boundary value problem. Under some conditions on the spectral radius corresponding to the related linear operator, we use fixed-point methods to obtain several existence theorems for our problem. In particular, we obtain the existence of multiple positive solutions via the Avery–Peterson fixed-point theorem. Note that our linear operator depends on the impulsive term and the integral boundary condition.

1. Introduction

In this work, we address the Riemann–Liouville-type fractional integral boundary value problem with impulses

$$\left\{\begin{array}{c}-D_{0+}^{\alpha }\chi \left(t\right)=h_1(t,\chi \left(t\right)),t\in (0,1)\backslash {\left\{t_k\right\}}_{k=1}^m,\hfill \\ \Delta \chi \left(t_k\right)=J_k\left(\chi \left(t_k\right)\right),k=1,2,\cdots ,m,\hfill \\ \chi \left(0\right)={\chi }^{\prime }\left(0\right)=\cdots ={\chi }^{(n-2)}\left(0\right),{\chi }^{\prime }\left(1\right)={\int }_0^1{h}_2(s,\chi \left(s\right))d\eta \left(s\right),\hfill \end{array}\right.$$

(1)

where $D_{0+}^{\alpha }$ is an $\alpha$-order Riemann–Liouville-type fractional derivative with $\alpha \in (n-1,n]$, $n\ge 3$, $n\in {\mathbb{N}}^{+}$. Further, $h_i(i=1,2),\eta ,J_k,t_k$ satisfy the conditions

(H1) The sequence ${\left\{t_k\right\}}_{k=1}^m$ satisfies $0=t_0<t_1<\cdots <t_m<t_{m+1}=1,\Delta \chi \left(t_k\right)=\chi \left(t_k^{+}\right)-\chi \left(t_k^{-}\right)$, where $\chi \left(t_k^{-}\right)=\chi \left(t_k\right)$, and $\chi \left(t_k^{+}\right)={lim}_{h\to 0}\chi \left(t_k+h\right)$ and $\chi \left(t_k^{-}\right)={lim}_{h\to 0}\chi \left(t_k-h\right)$ are the right- and left-hand limits of $\chi \left(t\right)$ at $t=t_k$, respectively,

(H2) $h_i\in C([0,1]\times {\mathbb{R}}^{+},{\mathbb{R}}^{+})$ ($i=1,2$);

(H3) $J_k\in C\left({\mathbb{R}}^{+},{\mathbb{R}}^{+}\right),k=1,2,\cdots ,m$;

(H4) $\eta \left(t\right)$ is a non-decreasing and nonconstant function for $t\in [0,1]$.

Fractional-order impulsive differential systems have witnessed extensive applications across a diverse range of fields, including physics, chemistry, aerodynamics, and the electrodynamics of complex media. These systems play a crucial role in accurately modeling various real-world phenomena that traditional integer-order models may fail to capture effectively. For example, in epidemiology, a Riemann–Liouville-type fractional-impulsive model characterizes the spread of viral mutant strains by integrating memory effects (via fractional derivatives) and abrupt transmission changes (via impulses). The model for infected individuals $I\left(t\right)$ is defined as

$$D_{0+}^{\alpha }I\left(t\right)=\beta I\left(t\right){\left({\displaystyle \frac{I\left(t\right)}{N}}\right)}^{\gamma }-\mu I\left(t\right),t\notin \left\{t_1,t_2,\dots ,t_k\right\},$$

where $D_{0+}^{\alpha }$ denotes the Riemann–Liouville-type fractional derivative of order $\alpha \in (0,1)$, capturing slow immune decay or long-range contact patterns. At mutation times $t_i$, the impulsive jumps occur

$$\beta \left(t_i^{+}\right)=\beta \left(t_i^{-}\right)·\left(1+{\delta }_i\right),\mu \left(t_i^{+}\right)=\mu \left(t_i^{-}\right)·\left(1-{ϵ}_i\right)$$

with ${\delta }_i>0$ (transmission enhancement) and ${ϵ}_i>0$ (reduced recovery rate) quantifying mutant impacts.

In recent years, a significant amount of research has been conducted on the initial-boundary value problems of fractional-order impulsive differential equations. As a result, numerous valuable findings and theoretical results have emerged in the literature; see for example [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] and the references cited therein. In [1], the authors studied the following Caputo-type fractional impulsive boundary value problem

$$\left\{\begin{array}{c}\lambda {}^{c}D_q^{\alpha }\zeta \left(t\right)+(1-\lambda ){}^{c}D_q^{\beta }\zeta \left(t\right)=f(t,\zeta \left(t\right)),t\in \mathcal{J}=[0,T],t\ne t_{\sigma },\hfill \\ \Delta \zeta \left(t_{\sigma }\right)={\mathcal{I}}_{\sigma }\left(\zeta \left(t_{\sigma }\right)\right),\Delta {\zeta }^{\prime }\left(t_{\sigma }\right)={\widehat{\mathcal{I}}}_{\sigma }\left(\zeta \left(t_{\sigma }\right)\right),\hfill \\ \zeta \left(T\right)=\mu \zeta \left(0\right)+\eta T{\zeta }^{\prime }\left(0\right),T{\zeta }^{\prime }\left(T\right)=\gamma \zeta \left(0\right)+\delta T{\zeta }^{\prime }\left(0\right),\hfill \end{array}\right.$$

where ${}^{c}D_q^{\alpha },{}^{c}D_q^{\beta }$ denote Caputo-type fractional q-derivatives, and the nonlinearities $f,{\mathcal{I}}_{\sigma },{\widehat{\mathcal{I}}}_{\sigma }(\sigma =1,2,3,\cdots ,k)$ satisfy some Lipschitz conditions and bounded conditions. They established their existence and uniqueness theorems of solutions by means of the fixed-point theorems of Banach and Schaefer.

In [2], the authors used the Guo–Krasnosel’skii fixed-point theorem to investigate the multiplicity of positive solutions of the fractional-impulsive integral boundary value problem

$$\left\{\begin{array}{c}{\left(\varphi \left({}^{c}D_{0+}^{\alpha }\zeta \left(t\right)\right)\right)}^{\prime }+\gamma \left(t\right)f(t,\zeta \left(\tau \left(t\right)\right))=0,t\in \mathcal{J}=[0,1],t\ne t_k,\hfill \\ \zeta \left(0\right)=a\zeta \left(1\right),{\zeta }^{\prime }\left(1\right)=b{\zeta }^{\prime }\left(0\right)+\lambda \left[\zeta \right],\hfill \\ \Delta \zeta \left(t_k\right)=I_k\left(\zeta \left(t_k\right)\right),\Delta {\zeta }^{\prime }\left(t_k\right)=J_k\left(\zeta \left(t_k\right)\right),\hfill \end{array}\right.$$

where ${}^{c}D_{0+}^{\alpha }$ denotes the Caputo-type fractional derivative, and $f,I_k,J_k$ satisfy some sublinear growth conditions. In [3], the authors researched the fractional-order boundary value problem

$$\left\{\begin{array}{c}{}^{c}D_{s_i^{+}}^{\beta }\left({}^{c}D_{s_i^{+}}^{\alpha }+\mathcal{A}\right)\zeta \left(t\right)=f\left(t,\zeta \left(t\right),{}^{c}D_{s_i^{+}}^{\alpha }\zeta \left(t\right)\right),t\in \left(s_i,t_{i+1}\right]\subset J,\hfill \\ \zeta \left(t\right)=g_i\left(t,\zeta \left(t_i^{-}\right)\right),t\in \left(t_i,s_i\right]\subset J,\hfill \\ \zeta \left(0\right)=\zeta \left(1\right)=\zeta \left({\eta }_{i+1}\right)=\zeta \left({\delta }_{i+1}\right)=0,{\eta }_{i+1},{\delta }_{i+1}\in \left(s_i,t_{i+1}\right)\subset J,\hfill \end{array}\right.$$

where $J=[0,1],\alpha \in (0,1),\beta \in (1,2),{}^{c}D_{s_i^{+}}^{\alpha },{}^{c}D_{s_i^{+}}^{\beta }(i=0,1,\cdots ,m)$ are the Caputo-type fractional derivatives, and $\mathcal{A}\in C^2(\mathbb{R},\mathbb{R})$ is a bounded operator. When the nonlinearities satisfy some Lipschitz conditions, the authors investigated the solvability and the Ulam–Hyers stability for their problem.

In [4], the authors researched the solvability of the following fractional-order triple-point integral boundary value problems:

$$\left\{\begin{array}{c}{D}_{0+}^{\alpha }\zeta \left(t\right)=f\left(t,\zeta ,{\zeta }^{\prime },D_{0+}^{\alpha -1}\zeta \right),t\ne t_k,\hfill \\ \Delta D_{0+}^{\alpha -1}\zeta \left(t_k\right)=I_k\left(\zeta \left(t_k\right)\right),k=1,\dots ,m,\hfill \\ \zeta \left(0\right)={\zeta }^{\prime }\left(0\right)=0,{\zeta }^{\prime }\left(1\right)={\int }_0^{\eta }g(s,\zeta \left(s\right))ds,\hfill \end{array}\right.$$

where $D_{0+}^{\alpha },D_{0+}^{\alpha -1}$ are the Riemann–Liouville fractional derivatives. When $f,g,I_k$ satisfy some Lipschitz conditions and sublinear growth conditions, the authors obtained some existence theorems for their system. In [33], the author considered impulsive fractional boundary value problems

$$\left\{\begin{array}{c}{}^{c}D^q\chi \left(t\right)=f(t,\chi \left(t\right)),1<q\le 2,t\in [0,1]\backslash \left\{t_1,t_2,\dots ,t_p\right\},\hfill \\ \Delta \chi \left(t_k\right)=Q_k\left(\chi \left(t_k\right)\right),\Delta {\chi }^{\prime }\left(t_k\right)=I_k\left(\chi \left(t_k\right)\right),k=1,2,\dots ,p,\hfill \\ a\chi \left(0\right)-b{\chi }^{\prime }\left(0\right)={\chi }_0,c\chi \left(1\right)+d{\chi }^{\prime }\left(1\right)={\chi }_1,\hfill \end{array}\right.$$

where ${}^{c}D^q$ is the Caputo fractional derivative, $a\ge 0, b>0, c\ge 0, d>0 (\delta =ac+ad+bc\ne 0)$ and ${\chi }_0,{\chi }_1\in \mathbb{R}$. The author in [33] used the Guo–Krasnosel’skii fixed-point theorem to prove that their system has a nontrivial solution under the conditions

$${\left(\mathrm{H}\right)}_{\mathrm{Wang}}\underset{\chi \to 0}{lim}{\displaystyle \frac{f(t,\chi )}{\chi }}=0,\underset{\chi \to 0}{lim}{\displaystyle \frac{Q_k\left(\chi \right)}{\chi }}=0\mathrm{and}\underset{\chi \to 0}{lim}{\displaystyle \frac{I_k\left(\chi \right)}{\chi }}=0,$$

uniformly on $t\in [0,1]$ and $k=1,2,\dots ,p$.

Inspired by the above related works, in this paper, we use fixed-point methods to study the solvability of (1). We first establish a new linear positive operator that includes the impulsive term and the integral boundary condition and study its spectral radius. Then, under some conditions involving the spectral radius of this operator, we obtain some existence theorems for our considered problem. Also, we use some bounded conditions to obtain the existence of multiple positive solutions via the Avery–Peterson fixed-point theorem. We note that our growth conditions for the nonlinearities $h_i(i=1,2)$ and the impulsive functions $J_k(k=1,2,\cdots ,m)$ are much more general and applicable than ${\left(\mathrm{H}\right)}_{\mathrm{Wang}}$ (see (H5)–(H8) in Section 3).

2. Preliminaries

In this section, we provide the definition of the Riemann–Liouville-type fractional-order derivative. For more details about fractional calculus, we refer the reader to [5,6].

Definition 1.

The $\alpha (>0)$-order Riemann–Liouville-type fractional-order derivative of a continuous function $\chi :(0,\infty )\to \mathbb{R}$ is defined by

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

where $n-1<\alpha \le n$.

In what follows, we transform (1) into its equivalent integral expression and obtain the following lemma.

Lemma 1

(see [7,8]). Let (H1) and (H4) hold and $V_i(i=1,2),{\overline{V}}_k(k=1,2,\cdots ,m)$ be given functions. Then,

$$\left\{\begin{array}{c}-D_{0+}^{\alpha }\chi \left(t\right)=V_1\left(t\right),t\in (0,1)\backslash {\left\{t_k\right\}}_{k=1}^m,\hfill \\ \Delta \chi \left(t_k\right)={\overline{V}}_k\left(t_k\right),k=1,2,\cdots ,m,\hfill \\ \chi \left(0\right)={\chi }^{\prime }\left(0\right)=\cdots ={\chi }^{(n-2)}\left(0\right),{\chi }^{\prime }\left(1\right)={\int }_0^1{V}_2\left(s\right)d\eta \left(s\right)\hfill \end{array}\right.$$

has a solution

$$\chi \left(t\right)={\int }_0^1\mathcal{G}(t,s)V_1\left(s\right)ds+{\displaystyle \frac{t^{\alpha -1}}{\alpha -1}}{\int }_0^1{V}_2\left(s\right)d\eta \left(s\right)+t^{\alpha -1}\sum _{t\le t_k<1}{t}_k^{1-\alpha }{\overline{V}}_k\left(t_k\right),t\in [0,1],$$

where

$$\mathcal{G}(t,s)={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}\left\{\begin{array}{cc}{t}^{\alpha -1}{(1-s)}^{\alpha -2}-{(t-s)}^{\alpha -1},\hfill & 0\le s\le t\le 1,\hfill \\ t^{\alpha -1}{(1-s)}^{\alpha -2},\hfill & 0\le t\le s\le 1.\hfill \end{array}\right.$$

(2)

Lemma 2

(see [7]). $\mathcal{G}$ has the following properties:

(i) 

$\mathcal{G}\in C([0,1]\times [0,1],{\mathbb{R}}^{+})$;

(ii) 

$\mathcal{G}(t,s)>0,t,s\in (0,1)$;

(iii) 

$t^{\alpha -1}s{(1-s)}^{\alpha -2}\le \Gamma \left(\alpha \right)\mathcal{G}(t,s)\le s{(1-s)}^{\alpha -2},t,s\in [0,1]$.

Note that $\alpha \in (n-1,n]$, $n\ge 3$, and $t^{\alpha -1}{(1-s)}^{\alpha -2}$ is continuous on $t,s\in [0,1]$, and ${(t-s)}^{\alpha -1}$ is continuous on $0\le s\le t\le 1$. This means that $\mathcal{G}$ is continuous on $[0,1]\times [0,1]$.

Let $PC[0,1]=\{\chi \in [0,1]\to {\mathbb{R}}^{+}:$$\chi \mathrm{is}\mathrm{continuous}\mathrm{in}[0,1]\backslash {\left\{t_k\right\}}_{k=1}^m,$$\mathrm{and}\chi \left(t_k^{-}\right),$$\chi \left(t_k^{+}\right)$ exist with $\chi \left(t_k^{-}\right)=\chi \left(t_k\right),k=1,2,\cdots ,m\}$, and $\parallel \chi \parallel ={sup}_{t\in [0,1]}\left|\chi \left(t\right)\right|$. Then, $PC[0,1]$ is a Banach space, and P is a cone with $P=\{\chi \in PC[0,1]:\chi (t)\ge 0,t\in [0,1\left]\right\}$. From Lemma 1, we know that (1) is equivalent to the following integral expression:

$$\begin{array}{cc}\hfill \chi \left(t\right)& ={\int }_0^1\mathcal{G}(t,s)h_1(s,\chi \left(s\right))ds+{\displaystyle \frac{t^{\alpha -1}}{\alpha -1}}{\int }_0^1{h}_2(s,\chi \left(s\right))d\eta \left(s\right)+t^{\alpha -1}{\displaystyle \sum _{t\le t_k<1}}{t}_k^{1-\alpha }{J}_k\left(\chi \left(t_k\right)\right)\hfill \\ & =\left(\mathsf{\Theta }\chi \right)\left(t\right),t\in [0,1].\hfill \end{array}$$

Note that, if there is a ${\chi }^{\ast }\in P\backslash \left\{0\right\}$ such that $\mathsf{\Theta }{\chi }^{\ast }={\chi }^{\ast }$, i.e., ${\chi }^{\ast }$ is the fixed point of $\mathsf{\Theta }$, then this ${\chi }^{\ast }$ is a positive solution for (1). Moreover, note that, regarding Lemma 2(i), by Lemma 2.9 in [8] and the Ascoli–Arzela theorem, we can prove that $\mathsf{\Theta }:P\to P$ is a completely continuous operator.

Define a set

$$P_0=\{\chi \in P:\chi \left(t\right)\ge t^{\alpha -1}\parallel \chi \parallel ,t\in [0,1]\}.$$

Then, $P_0$ is a cone on $PC[0,1]$, and we obtain the following lemma.

Lemma 3.

Let (H1)–(H4) hold. Then, $\mathsf{\Theta }\left(P\right)\subset P_0$.

Proof. 

For $\chi \in P$, from Lemma 2(iii), we have

$$\left(\mathsf{\Theta }\chi \right)\left(t\right)\le {\int }_0^1{\displaystyle \frac{s{(1-s)}^{\alpha -2}}{\Gamma \left(\alpha \right)}}{h}_1(s,\chi \left(s\right))ds+{\displaystyle \frac{1}{\alpha -1}}{\int }_0^1{h}_2(s,\chi \left(s\right))d\eta \left(s\right)+\sum _{t\le t_k<1}{t}_k^{1-\alpha }{J}_k\left(\chi \left(t_k\right)\right),$$

and thus we can find

$$\begin{array}{cc}\hfill \left(\mathsf{\Theta }\chi \right)\left(t\right)& \ge {\int }_0^1{\displaystyle \frac{t^{\alpha -1}s{(1-s)}^{\alpha -2}}{\Gamma \left(\alpha \right)}}{h}_1(s,\chi \left(s\right))ds+{\displaystyle \frac{t^{\alpha -1}}{\alpha -1}}{\int }_0^1{h}_2(s,\chi \left(s\right))d\eta \left(s\right)+t^{\alpha -1}{\displaystyle \sum _{t\le t_k<1}}{t}_k^{1-\alpha }{J}_k\left(\chi \left(t_k\right)\right)\hfill \\ & \ge t^{\alpha -1}\parallel \mathsf{\Theta }\chi \parallel ,t\in [0,1].\hfill \end{array}$$

The proof is completed. □

For positive constants ${\xi }_i(i=1,2,3)$, we define a linear operator as follows:

$$\begin{array}{cc}\hfill \left(L_{{\xi }_1,{\xi }_2,{\xi }_3}\chi \right)\left(t\right)=& {\xi }_1{\int }_0^1\mathcal{G}(t,s)\chi \left(s\right)ds+{\xi }_2{\displaystyle \frac{t^{\alpha -1}}{\alpha -1}}{\int }_0^1\chi \left(s\right)d\eta \left(s\right)\hfill \\ & +{\xi }_3{t}^{\alpha -1}{\displaystyle \sum _{t\le t_k<1}}{t}_k^{1-\alpha }\chi \left(t_k\right),\chi \in P,t\in [0,1].\hfill \end{array}$$

(3)

From (H1), (H4), and Lemma 2(i)–(ii), we have $L_{{\xi }_1,{\xi }_2,{\xi }_3}:P\to P$ is a positive linear operator, and now we will prove that its spectral radius, denoted by $r\left(L_{{\xi }_1,{\xi }_2,{\xi }_3}\right)$, is positive.

Lemma 4

(see [24]). The spectral radius $r\left(L_{{\xi }_1,{\xi }_2,{\xi }_3}\right)>0$.

Proof. 

We define linear operators as follows:

$$\begin{array}{l}\left(L_{{\xi }_1}\chi \right)\left(t\right)={\xi }_1{\int }_0^1\mathcal{G}(t,s)\chi \left(s\right)ds,\left(L_{{\xi }_2}\chi \right)\left(t\right)={\xi }_2{\displaystyle \frac{t^{\alpha -1}}{\alpha -1}}{\int }_0^1\chi \left(s\right)d\eta \left(s\right),\\ \left(L_{{\xi }_3}\chi \right)\left(t\right)={\xi }_3{t}^{\alpha -1}{\displaystyle \sum _{t\le t_k<1}}{t}_k^{1-\alpha }\chi \left(t_k\right)={\xi }_3{\displaystyle \sum _{k=1}^m}H(t,t_k)\chi \left(t_k\right),\chi \in P,t\in [0,1],\end{array}$$

where

$$H(t,t_k)=\left\{\begin{array}{c}{t}^{\alpha -1}{t}_k^{1-\alpha },0\le t\le t_k<1,\hfill \\ 0,0<t_k<t\le 1.\hfill \end{array}\right.$$

Therefore, we can find

$$\left(L_{{\xi }_1,{\xi }_2,{\xi }_3}\chi \right)\left(t\right)=\left(L_{{\xi }_1}\chi \right)\left(t\right)+\left(L_{{\xi }_2}\chi \right)\left(t\right)+\left(L_{{\xi }_3}\chi \right)\left(t\right),\chi \in P,t\in [0,1].$$

For all $n\in {\mathbb{N}}^{+}$, we have

$$(L_{{\xi }_1}^n\chi )\left(t\right)={\xi }_1^n\underset{n}{\underbrace{{\int }_0^1\cdots {\int }_0^1}}\mathcal{G}(t,s_1)\cdots \mathcal{G}(s_{n-1},s_n)\chi \left(s_n\right)d{s}_1\cdots d{s}_n,$$

and Lemma 2(iii) implies that

$$\begin{array}{l}\parallel L_{{\xi }_3}^n\parallel \\ \ge {\xi }_1^n{\displaystyle \underset{t\in [0,1]}{sup}}{t}^{\alpha -1}\underset{n}{\underbrace{{\int }_0^1\cdots {\int }_0^1}}{\displaystyle \frac{s_1{(1-s_1)}^{\alpha -2}}{\Gamma \left(\alpha \right)}}\cdots {\displaystyle \frac{s_{n-1}^{\alpha -1}{s}_n{(1-s_n)}^{\alpha -2}}{\Gamma \left(\alpha \right)}}d{s}_1\cdots d{s}_n\\ ={\xi }_1^n{\left({\int }_0^1{\displaystyle \frac{s^{\alpha }{(1-s)}^{\alpha -2}}{\Gamma \left(\alpha \right)}}ds\right)}^{n-1}{\int }_0^1{\displaystyle \frac{s{(1-s)}^{\alpha -2}}{\Gamma \left(\alpha \right)}}ds.\end{array}$$

Hence, from Gelfand’s theorem, we have

$$\begin{array}{cc}\hfill r\left(L_{{\xi }_1}\right)& ={\displaystyle \underset{n\to \infty }{lim\; inf}}\sqrt[n]{\parallel L_{{\xi }_3}^n\parallel }\ge {\displaystyle \underset{n\to \infty }{lim\; inf}}\sqrt[n]{{\xi }_1^n{\left({\int }_0^1{\displaystyle \frac{s^{\alpha }{(1-s)}^{\alpha -2}}{\Gamma \left(\alpha \right)}}ds\right)}^{n-1}{\int }_0^1{\displaystyle \frac{s{(1-s)}^{\alpha -2}}{\Gamma \left(\alpha \right)}}ds}\hfill \\ & ={\xi }_1{\int }_0^1{\displaystyle \frac{s^{\alpha }{(1-s)}^{\alpha -2}}{\Gamma \left(\alpha \right)}}ds.\hfill \end{array}$$

Note that $L_{{\xi }_2},L_{{\xi }_3}$ are positive linear operators, and thus we have

$$r\left(L_{{\xi }_1,{\xi }_2,{\xi }_3}\right)\ge r\left(L_{{\xi }_1}\right)\ge {\displaystyle \frac{\alpha {\xi }_1\Gamma (\alpha -1)}{\Gamma \left(2\alpha \right)}}>0.$$

(4)

This completes the proof. □

Lemma 5

(Krein–Rutman; see [34]). Let P be a reproducing cone in a real Banach space E and $L:E\to E$ be a compact linear operator with $L\left(P\right)\subset P.$ If L’s spectral radius is positive, then there exists $\xi \in P\backslash \left\{0\right\}$ such that $L\xi =r\left(L\right)\xi$.

Therefore, Lemmas 4 and 5 imply that there exists ${\phi }_{{\xi }_1,{\xi }_2,{\xi }_3}\in P\backslash \left\{0\right\}$ such that

$$L_{{\xi }_1,{\xi }_2,{\xi }_3}{\phi }_{{\xi }_1,{\xi }_2,{\xi }_3}=r\left(L_{{\xi }_1,{\xi }_2,{\xi }_3}\right){\phi }_{{\xi }_1,{\xi }_2,{\xi }_3}.$$

(5)

Note that, by Lemma 3, we obtain $L_{{\xi }_1,{\xi }_2,{\xi }_3}$ maps from P into $P_0$, and (5) implies that

$${\phi }_{{\xi }_1,{\xi }_2,{\xi }_3}\in P_0.$$

(6)

Remark 1.

Note that

$$r\left(L_{{\xi }_1}\right)\le \parallel L_{{\xi }_1}\parallel =\underset{t\in [0,1]}{sup}{\xi }_1{\int }_0^1\mathcal{G}(t,s)ds\le {\xi }_1{\int }_0^1{\displaystyle \frac{s{(1-s)}^{\alpha -2}}{\Gamma \left(\alpha \right)}}ds={\displaystyle \frac{{\xi }_1\Gamma (\alpha -1)}{\alpha {\Gamma }^2\left(\alpha \right)}},$$

and

$$r\left(L_{{\xi }_2}\right)\le \parallel L_{{\xi }_2}\parallel \le {\displaystyle \frac{{\xi }_2}{\alpha -1}}{\int }_0^{1}d\eta \left(s\right),r\left(L_{{\xi }_3}\right)\le \parallel L_{{\xi }_3}\parallel \le {\xi }_3\sum _{k=1}^m{t}_k^{1-\alpha }.$$

Therefore, we have

$$\begin{array}{cc}\hfill r\left(L_{{\xi }_1,{\xi }_2,{\xi }_3}\right)& \le r\left(L_{{\xi }_1}\right)+r\left(L_{{\xi }_2}\right)+r\left(L_{{\xi }_3}\right)\hfill \\ & \le {\displaystyle \frac{{\xi }_1\Gamma (\alpha -1)}{\alpha {\Gamma }^2\left(\alpha \right)}}+{\displaystyle \frac{{\xi }_2}{\alpha -1}}{\int }_0^{1}d\eta \left(s\right)+{\xi }_3{\displaystyle \sum _{k=1}^m}{t}_k^{1-\alpha }.\hfill \end{array}$$

(7)

Lemma 6

(see [35]). Let $\mathsf{\Omega }\subset E$ be a bounded open set and $A:\overline{\mathsf{\Omega }}\cap P\to P$ be a continuous compact operator. If there exists ${\zeta }_0\in P\backslash \left\{0\right\}$ such that $\zeta -A\zeta \ne \lambda {\zeta }_0,\forall \lambda \ge 0,\zeta \in 𝜕\mathsf{\Omega }\cap P$, then the fixed point-index $i(A,\mathsf{\Omega }\cap P,P)=0$.

Lemma 7

(see [35]). Let $\mathsf{\Omega }\subset E$ be a bounded open set with $0\in \mathsf{\Omega }$, and $A:\overline{\mathsf{\Omega }}\cap P\to P$ be a continuous compact operator. If $\zeta \ne \lambda A\zeta ,\forall \zeta \in 𝜕\mathsf{\Omega }\cap P,0\le \lambda \le 1$, then the fixed-point index $i(A,\mathsf{\Omega }\cap P,P)=1$.

The map $\tilde{\varphi }$ is called a non-negative continuous convex functional on cone P in a real Banach space E if $\tilde{\varphi }:P\to {\mathbb{R}}^{+}$ is continuous and

$$\tilde{\varphi }(tx+(1-t)y)\le t\tilde{\varphi }\left(x\right)+(1-t)\tilde{\varphi }\left(y\right),x,y\in P,t\in [0,1].$$

The map $\tilde{\beta }$ is called a non-negative continuous concave functional on cone P in a real Banach space E if $\tilde{\beta }:P\to {\mathbb{R}}^{+}$ is continuous and

$$\tilde{\beta }(tx+(1-t)y)\ge t\tilde{\beta }\left(x\right)+(1-t)\tilde{\beta }\left(y\right),x,y\in P,t\in [0,1].$$

Let $\tilde{\gamma },\tilde{\theta }$ be non-negative continuous convex functionals on $P,\tilde{\varphi }$ be a non-negative continuous concave functional on P, and $\tilde{\psi }$ be a non-negative continuous functional on P. Then, for $\tilde{a},\tilde{b},\tilde{c},\tilde{d}>0$, we define the following convex sets:

$$\begin{array}{cc}& P(\tilde{\gamma },\tilde{d})=\{x\in P\mid \tilde{\gamma }\left(x\right)<\tilde{d}\}\hfill \\ & P(\tilde{\gamma },\tilde{\varphi },\tilde{b},\tilde{d})=\{x\in P\mid \tilde{b}\le \tilde{\varphi }\left(x\right),\tilde{\gamma }\left(x\right)\le \tilde{d}\}\hfill \\ & P(\tilde{\gamma },\tilde{\theta },\tilde{\varphi },\tilde{b},\tilde{c},\tilde{d})=\{x\in P\mid \tilde{b}\le \tilde{\varphi }\left(x\right),\tilde{\theta }\left(x\right)\le \tilde{c},\tilde{\gamma }\left(x\right)\le \tilde{d}\}\hfill \end{array}$$

and a closed set

$$R(\tilde{\gamma },\tilde{\psi },\tilde{a},\tilde{d})=\{x\in P\mid \tilde{a}\le \tilde{\psi }\left(x\right),\tilde{\gamma }\left(x\right)\le \tilde{d}\}.$$

Lemma 8

(see [36]). Let P be a cone in Banach space E, $\tilde{\gamma },\tilde{\theta }$ be non-negative continuous convex functionals on P, $\tilde{\varphi }$ be a non-negative continuous concave functional on P, and $\tilde{\psi }$ be a non-negative continuous functional on P. Then, for $\tilde{l},\tilde{d}>0$ such that

$$\begin{array}{cc}& \tilde{\psi }\left(\lambda x\right)\le \lambda \tilde{\psi }\left(x\right),0\le \lambda \le 1,\hfill \\ & \tilde{\varphi }\left(x\right)\le \tilde{\psi }\left(x\right),\parallel x\parallel \le \tilde{l}\tilde{\gamma }\left(x\right),x\in \overline{P(\tilde{\gamma },\tilde{d})},\hfill \end{array}$$

where $\overline{P(\tilde{\gamma },\tilde{d})}$ is the closure of $P(\tilde{\gamma },\tilde{d})$. Let $\mathsf{\Theta }:\overline{P(\tilde{\gamma },\tilde{d})}\to \overline{P(\tilde{\gamma },\tilde{d})}$ be completely continuous and there exist $\tilde{a},\tilde{b},\tilde{c}>0$ with $\tilde{a}<\tilde{b}$ such that

(i) 

$\{x\in P(\tilde{\gamma },\tilde{\theta },\tilde{\varphi },\tilde{b},\tilde{c},\tilde{d})\mid \tilde{\varphi }\left(x\right)>\tilde{b}\}\ne \varnothing$ and $\tilde{\varphi }\left(\mathsf{\Theta }x\right)>\tilde{b}$ for $x\in P(\tilde{\gamma },\tilde{\theta },\tilde{\varphi },\tilde{b},\tilde{c},\tilde{d})$;

(ii) 

$\tilde{\varphi }\left(\mathsf{\Theta }x\right)>\tilde{b}$ for $x\in P(\tilde{\gamma },\tilde{\varphi },\tilde{b},\tilde{d})$ with $\tilde{\theta }\left(\mathsf{\Theta }x\right)>\tilde{c}$;

(iii) 

$0\notin R(\tilde{\gamma },\tilde{\psi },\tilde{a},\tilde{d})$ and $\tilde{\psi }\left(\mathsf{\Theta }x\right)<\tilde{a}$ for $x\in R(\tilde{\gamma },\tilde{\psi },\tilde{a},\tilde{d})$ with $\tilde{\psi }\left(x\right)=\tilde{a}$.

Then, Θ has at least three fixed points $x_1,x_2,x_3\in \overline{P(\tilde{\gamma },\tilde{d})}$ such that

$$\tilde{\gamma }\left(x_i\right)\le \tilde{d},i=1,2,3;\tilde{b}<\tilde{\varphi }\left(x_1\right);\tilde{a}<\tilde{\psi }\left(x_2\right),\tilde{\varphi }\left(x_2\right)<\tilde{b};\tilde{\psi }\left(x_3\right)<\tilde{a}.$$

3. Main Results

Now, we list our assumptions for $h_i(i=1,2)$ and $J_k(k=1,2,\cdots ,m)$ in this section.

(H5) There exist ${\xi }_{11},{\xi }_{12},{\xi }_{13}>0$ such that $r\left(L_{{\xi }_{11},{\xi }_{12},{\xi }_{13}}\right)\ge 1$ and

$$\underset{\chi \to +\infty }{lim\; inf}{\displaystyle \frac{h_1(t,\chi )}{\chi }}>{\xi }_{11},\underset{\chi \to +\infty }{lim\; inf}{\displaystyle \frac{h_2(t,\chi )}{\chi }}\ge {\xi }_{12},\mathrm{uniformly}\mathrm{on}t\in [0,1],$$

and

$$\underset{\chi \to +\infty }{lim\; inf}{\displaystyle \frac{J_k\left(\chi \right)}{\chi }}\ge {\xi }_{13},\mathrm{uniformly}\mathrm{on}k=1,2,\cdots ,m.$$

(H6) There exist ${\xi }_{21},{\xi }_{22},{\xi }_{23}>0$ such that $r\left(L_{{\xi }_{21},{\xi }_{22},{\xi }_{23}}\right)<1$ and

$$\underset{\chi \to 0^{+}}{lim\; sup}{\displaystyle \frac{h_1(t,\chi )}{\chi }}\le {\xi }_{21},\underset{\chi \to 0^{+}}{lim\; sup}{\displaystyle \frac{h_2(t,\chi )}{\chi }}\le {\xi }_{22},\mathrm{uniformly}\mathrm{on}t\in [0,1],$$

and

$$\underset{\chi \to 0^{+}}{lim\; sup}{\displaystyle \frac{J_k\left(\chi \right)}{\chi }}\le {\xi }_{23},\mathrm{uniformly}\mathrm{on}k=1,2,\cdots ,m.$$

(H7) There exist ${\xi }_{31},{\xi }_{32},{\xi }_{33}>0$ such that $r\left(L_{{\xi }_{31},{\xi }_{32},{\xi }_{33}}\right)\ge 1$ and

$$\underset{\chi \to 0^{+}}{lim\; inf}{\displaystyle \frac{h_1(t,\chi )}{\chi }}\ge {\xi }_{31},\underset{\chi \to 0^{+}}{lim\; inf}{\displaystyle \frac{h_2(t,\chi )}{\chi }}\ge {\xi }_{32},\mathrm{uniformly}\mathrm{on}t\in [0,1],$$

and

$$\underset{\chi \to 0^{+}}{lim\; inf}{\displaystyle \frac{J_k\left(\chi \right)}{\chi }}\ge {\xi }_{33},\mathrm{uniformly}\mathrm{on}k=1,2,\cdots ,m.$$

(H8) There exist ${\xi }_{41},{\xi }_{42},{\xi }_{43}>0$ such that $r\left(L_{{\xi }_{41},{\xi }_{42},{\xi }_{43}}\right)<1$ and

$$\underset{\chi \to +\infty }{lim\; sup}{\displaystyle \frac{h_1(t,\chi )}{\chi }}\le {\xi }_{41},\underset{\chi \to +\infty }{lim\; sup}{\displaystyle \frac{h_2(t,\chi )}{\chi }}\le {\xi }_{42},\mathrm{uniformly}\mathrm{on}t\in [0,1],$$

and

$$\underset{\chi \to +\infty }{lim\; sup}{\displaystyle \frac{J_k\left(\chi \right)}{\chi }}\le {\xi }_{43},\mathrm{uniformly}\mathrm{on}k=1,2,\cdots ,m.$$

Theorem 1.

Let (H1)–(H6) hold. Then, (1) has at least one positive solution.

Proof. 

By (5), there exists ${\phi }_{{\xi }_{11},{\xi }_{12},{\xi }_{13}}\in P\backslash \left\{0\right\}$ such that

$$L_{{\xi }_{11},{\xi }_{12},{\xi }_{13}}{\phi }_{{\xi }_{11},{\xi }_{12},{\xi }_{13}}=r\left(L_{{\xi }_{11},{\xi }_{12},{\xi }_{13}}\right){\phi }_{{\xi }_{11},{\xi }_{12},{\xi }_{13}},$$

(8)

and

$${\phi }_{{\xi }_{11},{\xi }_{12},{\xi }_{13}}\in P_0.$$

(9)

From (H5), there exist ${\epsilon }_1,c_1>0$ such that

$$h_1(t,\chi )\ge ({\xi }_{11}+{\epsilon }_1)\chi -c_1,h_2(t,\chi )\ge {\xi }_{12}\chi -c_1,\chi \ge 0,t\in [0,1],$$

(10)

and

$$J_k\left(\chi \right)\ge {\xi }_{13}\chi -c_1,\chi \ge 0,k=1,2,\cdots ,m.$$

(11)

Next, we claim that there exists an adequately large

$$R_1\ge {\displaystyle \frac{c_1\Gamma \left(2\alpha \right)\left[\frac{1+\Gamma \left(\alpha \right){\int }_0^{1}d\eta \left(s\right)}{(\alpha -1)\Gamma \left(\alpha \right)}+{\displaystyle \sum _{k=1}^m}{t}_k^{1-\alpha }\right]}{{\epsilon }_1\alpha \Gamma (\alpha -1)}}$$

(12)

such that

$$\chi -\mathsf{\Theta }\chi \ne \lambda {\phi }_{{\xi }_{11},{\xi }_{12},{\xi }_{13}},\chi \in 𝜕B_{R_1}\cap P,\lambda \ge 0,$$

(13)

where ${\phi }_{{\xi }_{11},{\xi }_{12},{\xi }_{13}}$ is defined by (8), and $B_{R_1}=\{\chi \in PC[0,1]:\parallel \chi \parallel <R_1\}$. We prove it by contradiction. Suppose there exist ${\chi }_1\in 𝜕B_{R_1}\cap P,{\lambda }_1\ge 0$ such that

$${\chi }_1=\mathsf{\Theta }{\chi }_1+{\lambda }_1{\phi }_{{\xi }_{11},{\xi }_{12},{\xi }_{13}}.$$

(14)

Note that ${\lambda }_1\ne 0$ (otherwise, $\mathsf{\Theta }{\chi }_1={\chi }_1$ implies that this ${\chi }_1$ is a positive solution for (1), and our theorem is proved). This, together with (9) and Lemma 3, imply that

$${\chi }_1\in P_0.$$

(15)

Note that ${\displaystyle \sum _{k=1}^m}{t}_k^{1-\alpha }\ge {\displaystyle \sum _{t\le t_k<1}}{t}_k^{1-\alpha }$ and $\parallel {\chi }_1\parallel = R_1$, and, by (15), (12), and (2), we have

$$\begin{array}{l}{\epsilon }_1{\int }_0^1\mathcal{G}(t,s){\chi }_1\left(s\right)ds-c_1\left[{\int }_0^1\mathcal{G}(t,s)ds+{\displaystyle \frac{t^{\alpha -1}}{\alpha -1}}{\int }_0^{1}d\eta \left(s\right)+t^{\alpha -1}{\displaystyle \sum _{t\le t_k<1}}{t}_k^{1-\alpha }\right]\\ \ge {\epsilon }_1{\int }_0^1{\displaystyle \frac{t^{\alpha -1}s{(1-s)}^{\alpha -2}}{\Gamma \left(\alpha \right)}}{s}^{\alpha -1}\parallel {\chi }_1\parallel ds-c_1[{\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_0^1{t}^{\alpha -1}{(1-s)}^{\alpha -2}ds\\ \hspace{1em}+{\displaystyle \frac{t^{\alpha -1}}{\alpha -1}}{\int }_0^{1}d\eta \left(s\right)+t^{\alpha -1}{\displaystyle \sum _{t\le t_k<1}}{t}_k^{1-\alpha }]\\ =t^{\alpha -1}\left[{\displaystyle \frac{{\epsilon }_1\alpha \Gamma (\alpha -1)R_1}{\Gamma \left(2\alpha \right)}}-c_1\left({\displaystyle \frac{1+\Gamma \left(\alpha \right){\int }_0^{1}d\eta \left(s\right)}{(\alpha -1)\Gamma \left(\alpha \right)}}+{\displaystyle \sum _{t\le t_k<1}}{t}_k^{1-\alpha }\right)\right]\\ \ge 0,t\in [0,1].\end{array}$$

This, together with (10) and (11), imply that

$$\begin{array}{cc}\hfill \left(\mathsf{\Theta }{\chi }_1\right)\left(t\right)& \ge {\int }_0^1\mathcal{G}(t,s)[({\xi }_{11}+{\epsilon }_1){\chi }_1\left(s\right)-c_1]ds+{\displaystyle \frac{t^{\alpha -1}}{\alpha -1}}{\int }_0^1[{\xi }_{12}{\chi }_1\left(s\right)-c_1]d\eta \left(s\right)\hfill \\ & \hspace{1em}+t^{\alpha -1}{\displaystyle \sum _{t\le t_k<1}}{t}_k^{1-\alpha }[{\xi }_{13}{\chi }_1\left(t_k\right)-c_1]\hfill \\ & \ge \left(L_{{\xi }_{11},{\xi }_{12},{\xi }_{13}}{\chi }_1\right)\left(t\right)+{\epsilon }_1{\int }_0^1\mathcal{G}(t,s){\chi }_1\left(s\right)ds\hfill \\ & \hspace{1em}-c_1{\int }_0^1\mathcal{G}(t,s)ds-{\displaystyle \frac{c_1{t}^{\alpha -1}}{\alpha -1}}{\int }_0^{1}d\eta \left(s\right)-c_1{t}^{\alpha -1}{\displaystyle \sum _{t\le t_k<1}}{t}_k^{1-\alpha }\hfill \\ & \ge \left(L_{{\xi }_{11},{\xi }_{12},{\xi }_{13}}{\chi }_1\right)\left(t\right).\hfill \end{array}$$

(16)

Let ${\lambda }^{\ast }=supS$ with $S:=\{\lambda :{\chi }_1\ge \lambda {\phi }_{{\xi }_{11},{\xi }_{12},{\xi }_{13}}\}$. Then, from (14), we have ${\lambda }_1\in S$ and ${\lambda }^{\ast }\ge {\lambda }_1$. By (5), (14), and (16), we obtain

$$\begin{array}{cc}\hfill {\chi }_1& \ge L_{{\xi }_{11},{\xi }_{12},{\xi }_{13}}{\chi }_1+{\lambda }_1{\phi }_{{\xi }_{11},{\xi }_{12},{\xi }_{13}}\hfill \\ & \ge L_{{\xi }_{11},{\xi }_{12},{\xi }_{13}}{\lambda }^{\ast }{\phi }_{{\xi }_{11},{\xi }_{12},{\xi }_{13}}+{\lambda }_1{\phi }_{{\xi }_{11},{\xi }_{12},{\xi }_{13}}\hfill \\ & =[{\lambda }^{\ast }r\left(L_{{\xi }_{11},{\xi }_{12},{\xi }_{13}}\right)+{\lambda }_1]{\phi }_{{\xi }_{11},{\xi }_{12},{\xi }_{13}}.\hfill \end{array}$$

Since ${\lambda }^{\ast }r\left(L_{{\xi }_{11},{\xi }_{12},{\xi }_{13}}\right)+{\lambda }_1>{\lambda }^{\ast }$ with $r\left(L_{{\xi }_{11},{\xi }_{12},{\xi }_{13}}\right)\ge 1,{\lambda }_1>0$, and thus this contradicts the definition of ${\lambda }^{\ast }$. Therefore, (13) holds. Now, Lemma 6 implies that

$$i(\mathsf{\Theta },B_{R_1}\cap P,P)=0.$$

(17)

By (H6), there exists an adequately small $r_1\in (0,R_1)$ such that

$$h_1(t,\chi )\le {\xi }_{21}\chi ,h_2(t,\chi )\le {\xi }_{22}\chi ,\chi \in [0,r_1],t\in [0,1],$$

(18)

and

$$J_k\left(\chi \right)\le {\xi }_{23}\chi ,\chi \in [0,r_1],k=1,2,\cdots ,m.$$

(19)

Now, we prove that

$$\mathsf{\Theta }\chi \ne \lambda \chi ,\chi \in 𝜕B_{r_1}\cap P,\lambda \ge 1,$$

(20)

where $B_{r_1}=\{\chi \in PC[0,1]:\parallel \chi \parallel <r_1\}$. We argue by contradiction. Suppose there exist ${\chi }_2\in 𝜕B_{r_1}\cap P,{\lambda }_2\ge 1$ such that

$$\mathsf{\Theta }{\chi }_2={\lambda }_2{\chi }_2.$$

(21)

Note that ${\lambda }_2>1$ (otherwise, $\mathsf{\Theta }{\chi }_2={\chi }_2$ implies that this ${\chi }_2$ is a positive solution for (1), and our theorem is proved). Consequently, by (18), (19), and (21), we have

$$\begin{array}{cc}\hfill {\chi }_2\left(t\right)& ={\lambda }_2^{-1}\left(\mathsf{\Theta }{\chi }_2\right)\left(t\right)\hfill \\ & \le {\lambda }_2^{-1}\left[{\xi }_{21}{\int }_0^1\mathcal{G}(t,s){\chi }_2\left(s\right)ds+{\xi }_{22}{\displaystyle \frac{t^{\alpha -1}}{\alpha -1}}{\int }_0^1{\chi }_2\left(s\right)d\eta \left(s\right)+{\xi }_{23}{t}^{\alpha -1}{\displaystyle \sum _{t\le t_k<1}}{t}_k^{1-\alpha }{\chi }_2\left(t_k\right)\right]\hfill \\ & ={\lambda }_2^{-1}\left(L_{{\xi }_{21},{\xi }_{22},{\xi }_{23}}{\chi }_2\right)\left(t\right).\hfill \end{array}$$

This implies that ${\chi }_2\le {\lambda }_2^{-1}{L}_{{\xi }_{21},{\xi }_{22},{\xi }_{23}}{\chi }_2$, and thus we can obtain a sequence ${\left\{{\lambda }_2^{-n}{L}_{{\xi }_{21},{\xi }_{22},{\xi }_{23}}^n{\chi }_2\right\}}_{n=1}^{\infty }$ as follows:

$$\begin{array}{cc}\hfill {\chi }_2& \le {\lambda }_2^{-1}{L}_{{\xi }_{21},{\xi }_{22},{\xi }_{23}}{\chi }_2\le {\lambda }_2^{-1}{L}_{{\xi }_{21},{\xi }_{22},{\xi }_{23}}\left({\lambda }_2^{-1}{L}_{{\xi }_{21},{\xi }_{22},{\xi }_{23}}{\chi }_2\right)\hfill \\ & ={\lambda }_2^{-2}{L}_{{\xi }_{21},{\xi }_{22},{\xi }_{23}}^2{\chi }_2\le \cdots \le {\lambda }_2^{-n}{L}_{{\xi }_{21},{\xi }_{22},{\xi }_{23}}^n{\chi }_2\le \cdots .\hfill \end{array}$$

Note that $L_{{\xi }_{21},{\xi }_{22},{\xi }_{23}}$ is a compact operator, and hence we have

$$\parallel L_{{\xi }_{21},{\xi }_{22},{\xi }_{23}}^n\parallel \ge {\displaystyle \frac{\parallel L_{{\xi }_{21},{\xi }_{22},{\xi }_{23}}^n{\chi }_2\parallel }{\parallel {\chi }_2\parallel }}\ge {\displaystyle \frac{{\lambda }_2^n\parallel {\chi }_2\parallel }{\parallel {\chi }_2\parallel }},$$

and Gelfand’s theorem implies that

$$r\left(L_{{\xi }_{21},{\xi }_{22},{\xi }_{23}}\right)=\underset{n\to \infty }{lim\; inf}\sqrt[n]{\parallel L_{{\xi }_{21},{\xi }_{22},{\xi }_{23}}^n\parallel }\ge {\lambda }_2>1.$$

This contradicts (H6), and thus (20) holds. Now, Lemma 7 implies that

$$i(\mathsf{\Theta },B_{r_1}\cap P,P)=1.$$

(22)

From (17) and (22), we have

$$i(\mathsf{\Theta },(B_{R_1}\backslash {\overline{B}}_{r_1})\cap P,P)=i(\mathsf{\Theta },B_{R_1}\cap P,P)-i(\mathsf{\Theta },B_{r_1}\cap P,P)=-1,$$

and this indicates that $\mathsf{\Theta }$ has at least one fixed point in $(B_{R_1}\backslash {\overline{B}}_{r_1})\cap P$; i.e., (1) has at least one positive solution. The proof is completed. □

Theorem 2.

Let (H1)–(H4) and (H7)–(H8) hold. Then, (1) has at least one positive solution.

Proof. 

By (5), there exists ${\phi }_{{\xi }_{31},{\xi }_{32},{\xi }_{33}}\in P\backslash \left\{0\right\}$ such that

$$L_{{\xi }_{31},{\xi }_{32},{\xi }_{33}}{\phi }_{{\xi }_{31},{\xi }_{32},{\xi }_{33}}=r\left(L_{{\xi }_{31},{\xi }_{32},{\xi }_{33}}\right){\phi }_{{\xi }_{31},{\xi }_{32},{\xi }_{33}}.$$

(23)

From (H7), there exists an adequately small $r_2>0$ such that

$$h_1(t,\chi )\ge {\xi }_{31}\chi ,h_2(t,\chi )\ge {\xi }_{32}\chi ,\chi \in [0,r_2],t\in [0,1],$$

(24)

and

$$J_k\left(\chi \right)\ge {\xi }_{33}\chi ,\chi \in [0,r_2],k=1,2,\cdots ,m.$$

(25)

Now, we will show that

$$\chi -\mathsf{\Theta }\chi \ne \lambda {\phi }_{{\xi }_{31},{\xi }_{32},{\xi }_{33}},\chi \in 𝜕B_{r_2}\cap P,\lambda \ge 0,$$

(26)

where ${\phi }_{{\xi }_{31},{\xi }_{32},{\xi }_{33}}$ is defined by (23), and $B_{r_2}=\{\chi \in PC[0,1]:\parallel \chi \parallel <r_2\}$. Suppose the contrary. Then, there exist ${\chi }_3\in 𝜕B_{r_2}\cap P,{\lambda }_3\ge 0$ such that

$${\chi }_3=\mathsf{\Theta }{\chi }_3+{\lambda }_3{\phi }_{{\xi }_{31},{\xi }_{32},{\xi }_{33}}.$$

(27)

Note that ${\lambda }_3>0$ (otherwise, $\mathsf{\Theta }{\chi }_3={\chi }_3$ implies that this ${\chi }_3$ is a positive solution for (1), and our theorem is proved). Now, (24) and (25) enable us to obtain

$$\begin{array}{cc}\hfill \left(\mathsf{\Theta }{\chi }_3\right)\left(t\right)& \ge {\xi }_{31}{\int }_0^1\mathcal{G}(t,s){\chi }_3\left(s\right)ds+{\xi }_{32}{\displaystyle \frac{t^{\alpha -1}}{\alpha -1}}{\int }_0^1{\chi }_3\left(s\right)d\eta \left(s\right)+{\xi }_{33}{t}^{\alpha -1}{\displaystyle \sum _{t\le t_k<1}}{t}_k^{1-\alpha }{\chi }_3\left(t_k\right)\hfill \\ & =\left(L_{{\xi }_{31},{\xi }_{32},{\xi }_{33}}{\chi }_3\right)\left(t\right).\hfill \end{array}$$

This, combined with (27), implies that

$${\chi }_3\ge L_{{\xi }_{31},{\xi }_{32},{\xi }_{33}}{\chi }_3+{\lambda }_3{\phi }_{{\xi }_{31},{\xi }_{32},{\xi }_{33}}.$$

(28)

Let ${\lambda }^{\ast \ast }=supW$ with $W=\{\lambda :{\chi }_3\ge \lambda {\phi }_{{\xi }_{31},{\xi }_{32},{\xi }_{33}}\}$. Then, ${\lambda }_3\in W$ and ${\lambda }^{\ast \ast }\ge {\lambda }_3$. From (28), we have

$${\chi }_3\ge L_{{\xi }_{31},{\xi }_{32},{\xi }_{33}}\left({\lambda }^{\ast \ast }{\phi }_{{\xi }_{31},{\xi }_{32},{\xi }_{33}}\right)+{\lambda }_3{\phi }_{{\xi }_{31},{\xi }_{32},{\xi }_{33}}=({\lambda }^{\ast \ast }r\left(L_{{\xi }_{31},{\xi }_{32},{\xi }_{33}}\right)+{\lambda }_3){\phi }_{{\xi }_{31},{\xi }_{32},{\xi }_{33}}.$$

Note that ${\lambda }^{\ast \ast }r\left(L_{{\xi }_{31},{\xi }_{32},{\xi }_{33}}\right)+{\lambda }_3>{\lambda }^{\ast \ast }$ with $r\left(L_{{\xi }_{31},{\xi }_{32},{\xi }_{33}}\right)\ge 1,{\lambda }_3>0$; this contradicts the definition of ${\lambda }^{\ast \ast }$. Consequently, (26) holds. Now, Lemma 6 implies that

$$i(\mathsf{\Theta },B_{r_2}\cap P,P)=0.$$

(29)

From (H8), there exists $c_2>0$ such that

$$h_1(t,\chi )\le {\xi }_{41}\chi +c_2,h_2(t,\chi )\le {\xi }_{42}\chi +c_2,\chi \ge 0,t\in [0,1],$$

(30)

and

$$J_k\left(\chi \right)\le {\xi }_{43}\chi +c_2,\chi \ge 0,k=1,2,\cdots ,m.$$

(31)

Define a set $T=\{\chi \in P:\mathsf{\Theta }\chi =\lambda \chi ,\lambda \ge 1\}$, and now we claim that T is bounded in P. Suppose there exist ${\chi }_4\in T,{\lambda }_4\ge 1$ such that

$$\mathsf{\Theta }{\chi }_4={\lambda }_4{\chi }_4.$$

(32)

Note that ${\lambda }_4>1$ (otherwise, $\mathsf{\Theta }{\chi }_4={\chi }_4$ implies that this ${\chi }_4$ is a positive solution for (1), and our theorem is proved), and, by (30)–(32), we have

$$\begin{array}{cc}\hfill {\chi }_4\left(t\right)& \le \left(\mathsf{\Theta }{\chi }_4\right)\left(t\right)\le {\int }_0^1\mathcal{G}(t,s)[{\xi }_{41}{\chi }_4\left(s\right)+c_2]ds\hfill \\ & \hspace{1em}+{\displaystyle \frac{t^{\alpha -1}}{\alpha -1}}{\int }_0^1[{\xi }_{42}{\chi }_4\left(s\right)+c_2]d\eta \left(s\right)+t^{\alpha -1}{\displaystyle \sum _{t\le t_k<1}}{t}_k^{1-\alpha }[{\xi }_{43}{\chi }_4\left(t_k\right)+c_2]\hfill \\ & ={\xi }_{41}{\int }_0^1\mathcal{G}(t,s){\chi }_4\left(s\right)ds+{\xi }_{42}{\displaystyle \frac{t^{\alpha -1}}{\alpha -1}}{\int }_0^1{\chi }_4\left(s\right)d\eta \left(s\right)+{\xi }_{43}{t}^{\alpha -1}{\displaystyle \sum _{t\le t_k<1}}{t}_k^{1-\alpha }{\chi }_4\left(t_k\right)\hfill \\ & \hspace{1em}+c_2\left[{\int }_0^1\mathcal{G}(t,s)ds+{\displaystyle \frac{t^{\alpha -1}}{\alpha -1}}{\int }_0^{1}d\eta \left(s\right)+t^{\alpha -1}{\displaystyle \sum _{t\le t_k<1}}{t}_k^{1-\alpha }\right]\hfill \\ & \le \left(L_{{\xi }_{41},{\xi }_{42},{\xi }_{43}}{\chi }_4\right)\left(t\right)+c_3,\hfill \end{array}$$

where

$$c_3=c_2{\int }_0^1{\displaystyle \frac{s{(1-s)}^{\alpha -2}}{\Gamma \left(\alpha \right)}}ds+{\displaystyle \frac{c_2}{\alpha -1}}{\int }_0^{1}d\eta \left(s\right)+c_2{\displaystyle \sum _{k=1}^m}{t}_k^{1-\alpha }.$$

On iterating this inequality, we find

$${\chi }_4\le L_{{\xi }_{41},{\xi }_{42},{\xi }_{43}}^{n+1}{\chi }_4+L_{{\xi }_{41},{\xi }_{42},{\xi }_{43}}^n{c}_3+\cdots +L_{{\xi }_{41},{\xi }_{42},{\xi }_{43}}{c}_3+c_3.$$

Note that $r\left(L_{{\xi }_{41},{\xi }_{42},{\xi }_{43}}\right)<1$, and we have

$$\underset{n\to \infty }{lim}{L}_{{\xi }_{41},{\xi }_{42},{\xi }_{43}}^{n+1}{\chi }_4=0,\underset{n\to \infty }{lim}[L_{{\xi }_{41},{\xi }_{42},{\xi }_{43}}^n{c}_3+\cdots +L_{{\xi }_{41},{\xi }_{42},{\xi }_{43}}{c}_3+c_3]={(I-L_{{\xi }_{41},{\xi }_{42},{\xi }_{43}})}^{-1}{c}_3.$$

This enables us to obtain

$$\parallel {\chi }_4\parallel \le \parallel {(I-L_{{\xi }_{41},{\xi }_{42},{\xi }_{43}})}^{-1}{c}_3\parallel .$$

Thus, T is a bounded set in P. Then, we can take an adequately large $R_2>max\{r_2,supT\}$ such that

$$\mathsf{\Theta }\chi \ne \lambda \chi ,\chi \in 𝜕B_{R_2}\cap P,\lambda \ge 1,$$

where $B_{R_2}=\{\chi \in PC[0,1]:\parallel \chi \parallel <R_2\}$. Now, Lemma 2.9 implies that

$$i(\mathsf{\Theta },B_{R_2}\cap P,P)=1.$$

(33)

From (29) and (33), we have

$$i(\mathsf{\Theta },(B_{R_2}\backslash {\overline{B}}_{r_2})\cap P,P)=i(\mathsf{\Theta },B_{R_2}\cap P,P)-i(\mathsf{\Theta },B_{r_2}\cap P,P)=1,$$

and this indicates that $\mathsf{\Theta }$ has at least one fixed point in $(B_{R_2}\backslash {\overline{B}}_{r_2})\cap P$; i.e., (1) has at least one positive solution. The proof is completed. □

For any given positive constant ${\overline{t}}_0\in (0,1)$, and, from Lemma 3, we have

$$\mathsf{\Theta }\left(P\right)\subset P_{01},$$

(34)

where

$$P_{01}=\left\{\chi \in P:\underset{t\in [t_0,1]}{min}\chi \left(t\right)\ge {\overline{t}}_0^{\alpha -1}\parallel \chi \parallel \right\}.$$

Theorem 3.

Let (H1)–(H4) hold and there exist $0<\tilde{a}<\tilde{b}{\overline{t}}_0^{1-\alpha }<\tilde{c}=\tilde{d}$ such that

(H9) there exist $M_1,M_2,N_k(k=1,2,\cdots ,m)$ such that

$${\displaystyle \frac{M_1\Gamma (\alpha -1)}{\alpha {\Gamma }^2\left(\alpha \right)}}+{\displaystyle \frac{M_2}{\alpha -1}}{\int }_0^{1}d\eta \left(t\right)+\sum _{k=1}^m{t}_k^{1-\alpha }{N}_k\le 1$$

with

$$h_1(t,\chi )\le \tilde{d}{M}_1,h_2(t,\chi )\le \tilde{d}{M}_2,J_k\left(\chi \right)\le \tilde{d}{N}_k,(t,\chi )\in [0,1]\times [0,\tilde{d}],$$

(H10) there exists $M_3$ such that

$$M_3{\int }_{{\overline{t}}_0}^1{\displaystyle \frac{{\overline{t}}_0^{\alpha -1}t{(1-t)}^{\alpha -2}}{\Gamma \left(\alpha \right)}}dt>1,{\displaystyle \frac{M_3}{M_1}}<{\displaystyle \frac{\tilde{d}}{\tilde{b}}}$$

with

$$h_1(t,\chi )\ge \tilde{b}{M}_3,(t,\chi )\in [{\overline{t}}_0,1]\times [\tilde{b},\tilde{c}],$$

(H11) $h_1(t,\chi )\le \tilde{a}{M}_1,h_2(t,\chi )\le \tilde{a}{M}_2,J_k\left(\chi \right)\le \tilde{a}{N}_k,(t,\chi )\in [0,1]\times [0,\tilde{a}]$.

Then, (1) has at least three positive solutions.

Proof. 

On the cone P, we define several functionals as follows:

$$\tilde{\gamma }\left(\chi \right)=\tilde{\psi }\left(\chi \right)=\tilde{\theta }\left(\chi \right)=\underset{t\in [0,1]}{sup}\chi \left(t\right),\tilde{\alpha }\left(\chi \right)=\underset{t\in \left[t_0,1\right]}{inf}\chi \left(t\right).$$

Then, we easily know that $\tilde{\alpha }$ is the non-negative continuous concave functional, $\tilde{\gamma }$ and $\tilde{\theta }$ are the non-negative continuous convex functionals, and $\tilde{\psi }$ is the non-negative continuous functional. From the above definitions, we see that

$$\tilde{\alpha }\left(\chi \right)\le \tilde{\psi }\left(\chi \right)\mathrm{and}\parallel \chi \parallel \le \tilde{\gamma }\left(\chi \right)\mathrm{for}\chi \in \overline{P(\tilde{\gamma },\tilde{d})}.$$

Next, we will prove that

$$\mathsf{\Theta }:\overline{P(\tilde{\gamma },\tilde{d})}\to \overline{P(\tilde{\gamma },\tilde{d})}.$$

(35)

Indeed, if $\chi \in \overline{P(\tilde{\gamma },\tilde{d})}$, i.e., $\chi \in [0,\tilde{d}]$, by (H9), we have

$$\begin{array}{cc}\hfill \tilde{\gamma }\left(\mathsf{\Theta }\chi \right)& ={\displaystyle \underset{t\in [0,1]}{sup}}\left[{\int }_0^1\mathcal{G}(t,s)h_1(s,\chi \left(s\right))ds+{\displaystyle \frac{t^{\alpha -1}}{\alpha -1}}{\int }_0^1{h}_2(s,\chi \left(s\right))d\eta \left(s\right)+t^{\alpha -1}{\displaystyle \sum _{t\le t_k<1}}{t}_k^{1-\alpha }{J}_k\left(\chi \left(t_k\right)\right)\right]\hfill \\ & \le {\int }_0^1{\displaystyle \frac{s{(1-s)}^{\alpha -2}}{\Gamma \left(\alpha \right)}}\tilde{d}{M}_{1}ds+{\displaystyle \frac{1}{\alpha -1}}{\int }_0^1\tilde{d}{M}_{2}d\eta \left(s\right)+{\displaystyle \sum _{k=1}^m}{t}_k^{1-\alpha }\tilde{d}{N}_k\hfill \\ & \le \tilde{d}.\hfill \end{array}$$

Thus, (35) holds.

Let $\chi \left(t\right)={\displaystyle \frac{\tilde{b}+\tilde{d}}{2}}$, and this $\chi \in P(\tilde{\gamma },\tilde{\theta },\tilde{\alpha },\tilde{b},\tilde{c},\tilde{d})$ with $\tilde{\alpha }\left(\chi \right)>\tilde{b}$, which means that $\{\chi \in P(\tilde{\gamma },\tilde{\theta },\tilde{\alpha },\tilde{b},\tilde{c},\tilde{d})\mid \tilde{\alpha }\left(z\right)>\tilde{b}\}\ne ⌀$. Thus, for $\chi \in P(\tilde{\gamma },\tilde{\theta },\tilde{\alpha },\tilde{b},\tilde{c},\tilde{d})$, we have

$$\tilde{b}\le \chi \left(t\right)\le \tilde{c}\mathrm{for}t\in \left[{\overline{t}}_0,1\right].$$

Consequently, by (H10), we obtain

$$\begin{array}{cc}\hfill \tilde{\alpha }\left(\mathsf{\Theta }\chi \right)& ={\displaystyle \underset{t\in \left[{\overline{t}}_0,1\right]}{inf}}\left[{\int }_0^1\mathcal{G}(t,s)h_1(s,\chi \left(s\right))ds+{\displaystyle \frac{t^{\alpha -1}}{\alpha -1}}{\int }_0^1{h}_2(s,\chi \left(s\right))d\eta \left(s\right)+t^{\alpha -1}{\displaystyle \sum _{t\le t_k<1}}{t}_k^{1-\alpha }{J}_k\left(\chi \left(t_k\right)\right)\right]\hfill \\ & \ge {\displaystyle \underset{t\in \left[{\overline{t}}_0,1\right]}{inf}}{\int }_0^{1}G(t,s)h_1(s,\chi \left(s\right))ds\hfill \\ & \ge {\int }_{{\overline{t}}_0}^1{\displaystyle \frac{{\overline{t}}_0^{\alpha -1}s{(1-s)}^{\alpha -2}}{\Gamma \left(\alpha \right)}}\tilde{b}{M}_{3}ds\hfill \\ & >\tilde{b}.\hfill \end{array}$$

Hence, we have

$$\tilde{\alpha }\left(\mathsf{\Theta }\chi \right)>\tilde{b},\mathrm{for}\mathrm{all}\chi \in P(\tilde{\gamma },\tilde{\theta },\tilde{\alpha },\tilde{b},\tilde{c},\tilde{d}).$$

For $\chi \in P(\tilde{\gamma },\tilde{\alpha },\tilde{b},\tilde{d})$ with $\tilde{\theta }\left(\mathsf{\Theta }\chi \right)>\tilde{c}$, by (34), we have

$$\tilde{\alpha }\left(\mathsf{\Theta }\chi \right)=\underset{t\in \left[{\overline{t}}_0,1\right]}{inf}\left(\mathsf{\Theta }\chi \right)\left(t\right)\ge {\overline{t}}_0^{\alpha -1}\parallel \mathsf{\Theta }\chi \parallel ={\overline{t}}_0^{\alpha -1}\tilde{\theta }\left(\mathsf{\Theta }\chi \right)>{\overline{t}}_0^{\alpha -1}\tilde{c}>\tilde{b}.$$

Since $\tilde{\psi }\left(0\right)=0<\tilde{a}$, then $0\notin R(\tilde{\gamma },\tilde{\psi },\tilde{a},\tilde{d})$. Suppose that $\chi \in R(\tilde{\gamma },\tilde{\theta },\tilde{a},\tilde{d})$ with $\tilde{\psi }\left(\chi \right)=\tilde{a}$. Then,

$$0\le \chi \left(t\right)\le \tilde{a},t\in [0,1].$$

Consequently, from (H11), we have

$$\begin{array}{cc}\hfill \tilde{\psi }\left(\mathsf{\Theta }\chi \right)& ={\displaystyle \underset{t\in [0,1]}{sup}}\left[{\int }_0^1\mathcal{G}(t,s)h_1(s,\chi \left(s\right))ds+{\displaystyle \frac{t^{\alpha -1}}{\alpha -1}}{\int }_0^1{h}_2(s,\chi \left(s\right))d\eta \left(s\right)+t^{\alpha -1}{\displaystyle \sum _{t\le t_k<1}}{t}_k^{1-\alpha }{J}_k\left(\chi \left(t_k\right)\right)\right]\hfill \\ & \le {\int }_0^1{\displaystyle \frac{s{(1-s)}^{\alpha -2}}{\Gamma \left(\alpha \right)}}\tilde{a}{M}_{1}ds+{\displaystyle \frac{1}{\alpha -1}}{\int }_0^1\tilde{a}{M}_{2}d\eta \left(s\right)+{\displaystyle \sum _{k=1}^m}{t}_k^{1-\alpha }\tilde{a}{N}_k\hfill \\ & \le \tilde{a}.\hfill \end{array}$$

Now, all the conditions in Lemma 8 hold, and this implies that (1) has at least three positive solutions regarding ${\chi }_1,{\chi }_2$, and ${\chi }_3$ such that

$$\tilde{\gamma }\left({\chi }_i\right)\le \tilde{d},\mathrm{for}i=1,2,3,\tilde{\psi }\left({\chi }_1\right)<\tilde{a}<\tilde{\psi }\left({\chi }_2\right),\tilde{\alpha }\left({\chi }_2\right)<\tilde{b}<\tilde{\alpha }\left({\chi }_3\right).$$

The proof is completed. □

Remark 2.

(i) 

Our hypotheses (H5)–(H8) are inspired by [37], and (H9)–(H11) are inspired by [25]. (H1) is usually used to set the basic condition for the impulsive workspace $PC[0,1]$, and (H2)–(H4) are used to ensure that the solutions obtained for our problem are positive solutions.

(ii) 

We now compare the results in this paper with those in [24]. First, we note that, in [24], the impulsive term is merely regarded as a perturbation. When constructing a linear operator similar to $L_{{\xi }_1,{\xi }_2,{\xi }_3}$ in (3), they only take into account the influence brought by the integral boundary condition, and there is no impulsive term (i.e., ${\xi }_3=0$). Moreover, they also discuss the spectral radius of the conjugate operator.

In [24], the authors constructed a positive linear operator, namely

$$\left(L_{{\xi }_1,{\xi }_2}\chi \right)\left(t\right)={\xi }_1{\int }_0^1\mathcal{G}(t,s)\chi \left(s\right)ds+{\xi }_2{\displaystyle \frac{t^{\alpha -1}}{\alpha -1}}{\int }_0^1\chi \left(s\right)d\eta \left(s\right),\chi \in P,t\in [0,1],$$

(36)

and its conjugate operator is

$$\left(L_{{\xi }_1,{\xi }_2}^{\ast }\gamma \right)\left(t\right):={\xi }_1{\int }_0^{t}ds{\int }_0^1\mathcal{G}(\tau ,s)d\gamma \left(\tau \right)+{\xi }_2\eta \left(t\right){\int }_0^1{\displaystyle \frac{{\tau }^{\alpha -1}}{\alpha -1}}d\gamma \left(\tau \right),\gamma \in E^{\ast },$$

(37)

where $E^{\ast }:=\{\gamma :\gamma \mathrm{has}\mathrm{bounded}\mathrm{variation}\mathrm{on}[0,1]\}$, and ${\xi }_1,{\xi }_2$ are non-negative constants. By the Krein–Rutman theorem, there exist ${\zeta }_{{\xi }_1,{\xi }_2}\in P\backslash \left\{0\right\}$ and ${\psi }_{{\xi }_1,{\xi }_2}\in P^{\ast }\backslash \left\{0\right\}$ such that

$$L_{{\xi }_1,{\xi }_2}{\zeta }_{{\xi }_1,{\xi }_2}=r\left(L_{{\xi }_1,{\xi }_2}\right){\zeta }_{{\xi }_1,{\xi }_2},L_{{\xi }_1,{\xi }_2}^{\ast }{\psi }_{{\xi }_1,{\xi }_2}=r\left(L_{{\xi }_1,{\xi }_2}\right){\psi }_{{\xi }_1,{\xi }_2},$$

where $P^{\ast }=\left\{\gamma \in E^{\ast }:\gamma \mathrm{is}\text{non-decreasing}\mathrm{on}[0,1]\right\}$. Now, we list some conditions in [24]:

(H2)_[24] There exist ${\mu }_1,v_1\ge 0\left({\mu }_1^2+v_1^2\ne 0\right)$ and $l_k\ge 0\left({\displaystyle \sum _{k=1}^m}{l}_k^2\ne 0\right)$ such that

$$\mathrm{if}r\left(L_{{\mu }_1,v_1}\right)<1\Rightarrow \sum _{k=1}^m{l}_k{t}_k^{\alpha -1}{\int }_0^{1}H\left(t,t_k\right)d{\psi }_{{\mu }_1,v_1}\left(t\right)>\left[1-r\left(L_{{\mu }_1,v_1}\right)\right]{\int }_0^{1}d{\psi }_{{\mu }_1,v_1}\left(t\right),$$

$$\underset{\chi \to +\infty }{lim\; inf}{\displaystyle \frac{f(t,\chi )}{\chi }}\ge {\mu }_1,\underset{\chi \to +\infty }{lim\; inf}{\displaystyle \frac{g(t,\chi )}{\chi }}\ge v_1\mathrm{uniformly}\mathrm{on}t\in [0,1],$$

and

$$\underset{\chi \to +\infty }{lim\; inf}{\displaystyle \frac{I_k\left(\chi \right)}{\chi }}\ge l_k,k=1,2,\cdots ,m.$$

(H3)_[24] There exist ${\mu }_2,v_2\ge 0\left({\mu }_2^2+v_2^2\ne 0\right)$ and $\widetilde{l_k}\ge 0\left({\displaystyle \sum _{k=1}^m}\widetilde{l_k}\ne 0\right)$ such that

$$\mathrm{if}r\left(L_{{\mu }_2,v_2}\right)<1\Rightarrow \left[1-r\left(L_{{\mu }_2,v_2}\right)\right]{\int }_0^1{t}^{\alpha -1}d{\psi }_{{\mu }_2,v_2}\left(t\right)>\sum _{k=1}^m{\tilde{l}}_k{\int }_0^{1}H\left(t,t_k\right)d{\psi }_{{\mu }_2,v_2}\left(t\right),$$

$$\underset{\chi \to 0^{+}}{lim\; sup}{\displaystyle \frac{f(t,\chi )}{\chi }}\le {\mu }_2,\underset{\chi \to 0^{+}}{lim\; sup}{\displaystyle \frac{g(t,\chi )}{\chi }}\le v_2\mathrm{uniformly}\mathrm{on}t\in [0,1],$$

and

$$\underset{\chi \to 0^{+}}{lim\; inf}{\displaystyle \frac{I_k\left(\chi \right)}{\chi }}\le {\tilde{l}}_k,k=1,2,\cdots ,m.$$

(H4)_[24] There exist ${\mu }_3,v_3\ge 0\left({\mu }_3^2+v_3^2\ne 0\right)$ and ${\overline{l}}_k\ge 0\left({\displaystyle \sum _{k=1}^m}{\overline{l}}_k^2\ne 0\right)$ such that

$$\mathrm{if}r\left(L_{{\mu }_3,{\nu }_3}\right)<1\Rightarrow \sum _{k=1}^m{\overline{l}}_k{l}_k^{\alpha -1}{\int }_0^{1}H\left(t,t_k\right)d{\psi }_{{\mu }_3,{\nu }_3}\left(t\right)>\left[1-r\left(L_{{\mu }_3,{\nu }_3}\right)\right]{\int }_0^{1}d{\psi }_{{\mu }_3,{\nu }_3}\left(t\right),$$

$$\underset{\chi \to 0^{+}}{lim\; inf}{\displaystyle \frac{f(t,\chi )}{\chi }}\ge {\mu }_3,\underset{\chi \to 0^{+}}{lim\; inf}{\displaystyle \frac{g(t,\chi )}{\chi }}\ge v_3\mathrm{uniformly}\mathrm{on}t\in [0,1],$$

and

$$\underset{\chi \to 0^{+}}{lim\; inf}{\displaystyle \frac{I_k\left(\chi \right)}{\chi }}\ge {\overline{l}}_k,k=1,2,\cdots ,m.$$

(H5)_[24] There exist ${\mu }_4,v_4\ge 0\left({\mu }_4^2+v_4^2\ne 0\right)$ and $\widehat{l_k}\ge 0\left({\displaystyle \sum _{k=1}^m}\widehat{l_k^2}\ne 0\right)$ such that

$$\mathrm{if}r\left(L_{{\mu }_4,v_4}\right)<1\Rightarrow \left(1-r\left(L_{{\mu }_4,v_4}\right)\right){\int }_0^1{t}^{\alpha -1}d{\psi }_{{\mu }_4,v_4}\left(t\right)>\sum _{k=1}^m{\widehat{l}}_k{\int }_0^{1}H\left(t,t_k\right)d{\psi }_{{\mu }_4,v_4}\left(t\right),$$

$$\underset{\chi \to +\infty }{lim\; sup}{\displaystyle \frac{f(t,\chi )}{\chi }}\le {\mu }_4,\underset{z\to +\infty }{lim\; sup}{\displaystyle \frac{g(t,\chi )}{\chi }}\le v_4\mathrm{uniformly}\mathrm{on}t\in [0,1],$$

and

$$\underset{\chi \to +\infty }{lim\; inf}{\displaystyle \frac{I_k\left(\chi \right)}{\chi }}\le {\widehat{l}}_k,k=1,2,\cdots ,m.$$

From these conditions, it can be seen that not only are there restrictions on the spectral radius but also on the growth conditions for the impulsive function $I_k$ when $r\left(L_{{\mu }_i,v_i}\right)<1,i=1,2,3,4$. However, our conditions (H5)–(H8) only have restrictions on the spectral radius, which is a major improvement from an application viewpoint.

In the current paper, we construct our linear operator $L_{{\xi }_1,{\xi }_2,{\xi }_3}$, which contains the impulsive term and the integral boundary condition, and there is no need to discuss its conjugate operator. Under some conditions involving the spectral radius of this linear operator, we can still obtain the existence of positive solutions for our impulsive system when the nonlinearities grow sublinearly and superlinearly. This is the first study of impulsive differential equations by using this kind of operator.

4. Examples

In this section, we will provide several examples to verify our main theorems (the idea is from [25,37]). Let $n=m=3,\alpha =2.5,{\overline{t}}_0=0.25,t_1=0.2,t_2=0.3,t_3=0.5,\eta \left(t\right)=t,t\in [0,1]$. From (4) and (7), we have

$$0.09{\xi }_1\left(0.27{\xi }_2\right)\le r\left(L_{{\xi }_1,{\xi }_2,{\xi }_3}\right)\le 0.2{\xi }_1+0.67{\xi }_2+20.1{\xi }_3,$$

In view of the above inequalities, we can adjust the constants ${\xi }_{i1},{\xi }_{i2},{\xi }_{i3}(i=1,2,3,4)$ so that they satisfy the conditions of the spectral radius in (H5)–(H8).

Example 1.

Let $h_1(t,\chi )=e^t{\chi }^{{\kappa }_1},h_2(t,\chi )=(t+{\kappa }_3){\chi }^{{\kappa }_2},J_k\left(\chi \right)={\chi }^{k+1}$ with ${\kappa }_1,{\kappa }_2>1,{\kappa }_3>0$ for $\chi \ge 0,t\in [0,1],k=1,2,3$. Then, we have

$$\underset{\chi \to +\infty }{lim\; inf}{\displaystyle \frac{h_1(t,\chi )}{\chi }}=\underset{\chi \to +\infty }{lim\; inf}{\displaystyle \frac{e^t{\chi }^{{\kappa }_1}}{\chi }}=+\infty ,\underset{\chi \to +\infty }{lim\; inf}{\displaystyle \frac{h_2(t,\chi )}{\chi }}=\underset{\chi \to +\infty }{lim\; inf}{\displaystyle \frac{(t+{\kappa }_3){\chi }^{{\kappa }_2}}{\chi }}=+\infty ,$$

uniformly on $t\in [0,1]$, and

$$\underset{\chi \to +\infty }{lim\; inf}{\displaystyle \frac{J_k\left(\chi \right)}{\chi }}=\underset{\chi \to +\infty }{lim\; inf}{\displaystyle \frac{{\chi }^{k+1}}{\chi }}=+\infty ,uniformlyonk=1,2,3.$$

On the other hand, by direct computation, we obtain

$$\underset{\chi \to 0^{+}}{lim\; sup}{\displaystyle \frac{h_1(t,\chi )}{\chi }}=\underset{\chi \to 0^{+}}{lim\; sup}{\displaystyle \frac{e^t{\chi }^{{\kappa }_1}}{\chi }}=0,\underset{\chi \to 0^{+}}{lim\; sup}{\displaystyle \frac{h_2(t,\chi )}{\chi }}=\underset{\chi \to 0^{+}}{lim\; sup}{\displaystyle \frac{(t+{\kappa }_3){\chi }^{{\kappa }_2}}{\chi }}=0,$$

uniformly on $t\in [0,1]$, and

$$\underset{\chi \to 0^{+}}{lim\; sup}{\displaystyle \frac{J_k\left(\chi \right)}{\chi }}=\underset{\chi \to 0^{+}}{lim\; sup}{\displaystyle \frac{{\chi }^{k+1}}{\chi }}=0,uniformlyonk=1,2,3.$$

Therefore, (H1)–(H6) hold, and, from Theorem 1, (1) has at least one positive solution.

Example 2.

Let $h_1(t,\chi )={\omega }_1\left(t\right){\chi }^{{\kappa }_4},h_2(t,\chi )={\omega }_2\left(t\right){\chi }^{{\kappa }_5},J_k\left(\chi \right)=\sqrt[k+1]{\chi }$ with ${\kappa }_4,{\kappa }_5\in (0,1)$, ${\omega }_i\left(t\right)>0$ for $\chi \ge 0,t\in [0,1],i=1,2,k=1,2,3$. Then, we have

$$\underset{\chi \to 0^{+}}{lim\; inf}{\displaystyle \frac{h_1(t,\chi )}{\chi }}=\underset{\chi \to 0^{+}}{lim\; inf}{\displaystyle \frac{{\omega }_1\left(t\right){\chi }^{{\kappa }_4}}{\chi }}=+\infty ,\underset{\chi \to 0^{+}}{lim\; inf}{\displaystyle \frac{h_2(t,\chi )}{\chi }}=\underset{\chi \to 0^{+}}{lim\; inf}{\displaystyle \frac{{\omega }_2\left(t\right){\chi }^{{\kappa }_5}}{\chi }}=+\infty ,$$

uniformly on $t\in [0,1]$, and

$$\underset{\chi \to 0^{+}}{lim\; inf}{\displaystyle \frac{J_k\left(\chi \right)}{\chi }}=\underset{\chi \to 0^{+}}{lim\; inf}{\displaystyle \frac{\sqrt[k+1]{\chi }}{\chi }}=+\infty ,uniformlyonk=1,2,3.$$

On the other hand, we find

$$\underset{\chi \to +\infty }{lim\; sup}{\displaystyle \frac{h_1(t,\chi )}{\chi }}=\underset{\chi \to +\infty }{lim\; sup}{\displaystyle \frac{{\omega }_1\left(t\right){\chi }^{{\kappa }_4}}{\chi }}=0,\underset{\chi \to +\infty }{lim\; sup}{\displaystyle \frac{h_2(t,\chi )}{\chi }}=\underset{\chi \to +\infty }{lim\; sup}{\displaystyle \frac{{\omega }_2\left(t\right){\chi }^{{\kappa }_5}}{z}}=0,$$

uniformly on $t\in [0,1]$, and

$$\underset{\chi \to +\infty }{lim\; sup}{\displaystyle \frac{J_k\left(\chi \right)}{\chi }}=\underset{\chi \to +\infty }{lim\; sup}{\displaystyle \frac{\sqrt[k+1]{\chi }}{\chi }}=0,uniformlyonk=1,2,3.$$

As a result, (H1)–(H4) and (H7)–(H8) hold, and, from Theorem 2, (1) has at least one positive solution.

Example 3.

Let $\tilde{a}=1,\tilde{b}=10,\tilde{c}=\tilde{d}=650$, $M_1=0.8,M_2=0.2,M_3=50,N_1=0.02,N_2=0.03,N_3=0.1$. Now, we choose

$$h_1(t,\chi )=\left\{\begin{array}{c}{\displaystyle \frac{t}{4}}+0.519{\chi }^3,(t,\chi )\in [0,1]\times [0,10],\hfill \\ {\displaystyle \frac{t}{4}}-0.014\chi +519.14,(t,\chi )\in [0,1]\times [10,650],\hfill \\ {\displaystyle \frac{t}{4}}+510.04e^{650}{e}^{-\chi },(t,\chi )\in [0,1]\times [650,+\infty ),\hfill \end{array}\right.$$

and

$$h_2(t,\chi )={\displaystyle \frac{1}{11}}(t+\chi ),J_1\left(\chi \right)={\displaystyle \frac{\chi }{50}},J_2\left(\chi \right)={\displaystyle \frac{3\chi }{100}},J_3\left(\chi \right)={\displaystyle \frac{\chi }{10}},\chi \ge 0,t\in [0,1].$$

Note, (H1)–(H4) and (H9)–(H11) hold, and, from Theorem 3, (1) has at least three positive solutions.

5. Conclusions

In this paper, we addressed the existence and multiplicity of positive solutions for (1). We first established a new linear positive operator, which includes the impulsive term and the integral boundary condition, and then we used the Krein–Rutman theorem to study its spectral radius. Then, under some conditions involving the spectral radius of this linear operator, we obtained some existence theorems when $h_i(i=1,2)$ and $J_k(k=1,2,\cdots ,m)$ grow sublinearly and superlinearly. Finally, we used some bounded conditions to obtain the existence of multiple positive solutions via the Avery–Peterson fixed-point theorem.

In a future study, we hope to discuss the existence of nontrivial solutions when the nonlinearities $h_i,J_k$ can be sign-changing. In this situation, we will look to find some similar growth conditions related to the linear operator (3), which is a key issue to be considered.