Positive solutions for a Riemann-Liouville-type impulsive fractional integral boundary value problem

1.   Introduction

In this work, we study the following Riemann-Liouville-type impulsive fractional integral boundary value problem

where $2 < \beta \leq 3$ is a real number, ${}_{t_k} D_t^\beta$ is the Riemann-Liouville fractional derivative, $0 = t_0 < t_1 < \cdots < t_m < t_{m+1} = 1, \ { }_{t_k} D_t^{\beta-1} z\left(t_k^{+}\right) = \lim _{h \rightarrow 0^{+}} { }_{t_k} D_t^{\beta-1} z\left(t_k+h\right)$ and ${ }_{t_k} D_t^{\beta-1} z\left(t_k^{-}\right) = \lim _{h \rightarrow 0^{-}} { }_{t_k} D_t^{\beta-1} z\left(t_k+h\right)$ represent the right and left limits of ${ }_{t_k} D_t^{\beta-1} z(t)$ at $t = t_k$, respectively, ${ }_{t_k} D_t^{\beta-1} z\left(t_k^{-}\right) = { }_{t_k} D_t^{\beta-1} z\left(t_k\right)$, and $\Delta D^{\beta-1} z\left(t_k\right) = { }_{t_k} D_t^{\beta-1} z\left(t_k^{+}\right)-{ }_{t_{k-1}} D_t^{\beta-1} z\left(t_k^{-}\right)$. In addition, the functions $f, g, \alpha, I_k$ satisfy the conditions:

(H0) $f, g\in C([0, 1]\times \mathbb R^+, \mathbb R^+)$, $I_k\in C(\mathbb R^+, \mathbb R^+), \ k = 1, 2, ..., m, \ \mathbb R^+: = [0, +\infty)$,

(H1) $\alpha$ is a function of bounded variation with $\alpha(t)\ge 0$, and $\alpha(t)\not \equiv 0, \ t\in [0, 1]$.

In comparison to integer calculus when describing natural phenomena and objective laws, fractional calculus is more accurate and applicable in physics, chemistry, and engineering. Many scholars have applied the methods of nonlinear analysis to study fractional boundary value problems, and a large number of results have been obtained; 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]} and the references therein. In ^[1], the authors used some fixed-point techniques to study the existence, uniqueness, and multiplicity of positive solutions for the fractional integral boundary value problem

where ${}_0D_t^\alpha, \ {}_0D_t^\beta$ are Riemann-Liouville fractional derivatives. In ^[2], the authors studied the following $p$-Laplacian fractional boundary value problem involving the Riemann-Stieltjes integral:

where ${}_0D_t^\alpha, \ {}_0D_t^\beta, \ {}_0D_t^\gamma$ are Riemann-Liouville fractional derivatives. The authors used fixed point theorems on a sum operator in partial ordering Banach spaces to investigate the existence and uniqueness of positive solutions for their problem.

In ^[3], the authors studied the impulsive fractional integral boundary value problem

and they adopted the contraction mapping principle and the fixed point theorem to establish the existence and uniqueness of nontrivial solutions when the nonlinearities $f, g, I_k$ satisfy some Lipschitz conditions. In ^[4], the authors studied positive solutions for the fractional integral boundary value problem

where $f\in C([0, 1] \times\mathbb R^+, \mathbb R^+)$ satisfies the conditions

(HZ1) $\liminf _{\chi \rightarrow 0^+} \frac{f(t, \chi)}{\chi} > \lambda_1$, $\lim \sup _{\chi \rightarrow +\infty} \frac{f(t, \chi)}{\chi} < \lambda_1$ uniformly with respect to $t \in[0, 1]$,

(HZ2) $\lim \sup _{\chi \rightarrow 0^+} \frac{f(t, \chi)}{\chi} < \lambda_1$, $\lim \inf _{\chi \rightarrow +\infty} \frac{f(t, \chi)}{\chi} > \lambda_1$ uniformly with respect to $t \in[0, 1]$,

where $\lambda_1$ is the first eigenvalue of the operator $(L_{Z1} \chi)(t) = \int_0^1 G_Z(t, s) h(s) \chi(s) d s$ and $G_Z$ is the Green's function.

Motivated by the aforementioned works, in this paper we use the fixed point index to study positive solutions for (1.1) under some conditions concerning the spectral radius of the relevant linear operator. Note that the considered linear operator can include the Riemann-Stieltjes integral condition in (1.1) and the approach is quite different from previous works in the literature. Moreover, we also consider the effect of the impulsive term and our conditions are more general than (HZ1)–(HZ2).

2.   Preliminaries

In this section, we first present the definitions of the Riemann-Liouville-type fractional integral and derivative. For the other necessary definitions and notations, we refer the reader to the books ^[8,13,17].

Definition 2.1. The Riemann-Liouville fractional integral of order $\beta > 0$ of a function $z$ : $(a, +\infty) \rightarrow \mathbb{R}$ is given by

provided that the right-hand side is point-wise defined on $(a, +\infty)$.

Definition 2.2. The Riemann-Liouville fractional derivative of order $\beta > 0$ of a continuous function $z:(a, +\infty) \rightarrow \mathbb{R}$ is given by

where $a > 0, n-1 < \beta \leq n$, provided that the right-hand side is point-wise defined on $(a, +\infty)$.

Let $C([0, 1], \mathbb{R})$ be the Banach space of continuous functions from $[0, 1]$ to $\mathbb{R}$ with the norm $\|z\| =$ $\sup _{0 \leq t \leq 1}|z(t)|$. Define the Banach space $P C^1([0, 1], \mathbb{R})$ as follows

with the norm $\|z\|_{P C^1} = \max \left\{\|z\|, \ \left\|_{t_k} D_t^{\beta-1} z\right\|\right\}$. Let $P = \{z\in C([0, 1], \mathbb{R}): z(t)\ge 0, t\in [0, 1]\}$ and $P_0 = \{z\in P: z(t)\ge t^{\beta-1}\|z\|, t\in [0, 1]\}$. Then $P, P_0$ are cones on $C([0, 1], \mathbb{R})$.

Lemma 2.3. (see [3, Lemma 2.4]) Let $h, V\in C([0, 1], \mathbb{R})$ and $v_k\in \mathbb R, \ k = 1, 2, ..., m$. Then, the boundary value problem

has a solution of the form

where

and

Lemma 2.4. (see ^[9]) The function $G$ has the following properties:

(C1) $G(t, s)\ge 0$ for $t, s\in [0, 1]$;

(C2) $t^{\beta-1}G(1, s)\le G(t, s)\le G(1, s)$ for $t, s\in [0, 1]$.

From Lemma 2.3 and (H0)–(H1), we define an operator $\mathcal T: P\to P$ as follows:

From Lemma 2.3 we see that if there exists $z^*\in P\backslash \{0\}$ such that $\mathcal Tz^* = z^*$, then this $z^*$ is the positive solution for (1.1). Hence, in what follows we study the existence of positive fixed points of the operator $\mathcal T$.

Lemma 2.5. Suppose that (H0)–(H1) hold. Then, $\mathcal T(P)\subset P_0$.

By Lemma 2.4 and the method of [21, Lemma 2.6], we obtain the conclusion, so, we omit its proof.

Lemma 2.6. Let

with $\mu, \nu\ge 0$ and $\mu^2+\nu^2 \not = 0$. Then $\mathcal L_{\mu, \nu}(P)\subset P_0$ and the spectral radius of $\mathcal L_{\mu, \nu}$, denoted by $r(\mathcal L_{\mu, \nu})$, which satisfies the inequality

Proof. If $z\in P$, then from Lemma 2.4(C2) we have

and

Hence, $\mathcal L_{\mu, \nu}(P)\in P_0$, as required.

Let $(L_\mu z)(t) = \mu\int_0^1 G(t, s)z(s)ds$ and $(L_\nu z)(t) = \nu \frac{t^{\beta-1}}{\beta-1} \int_0^1 z(s) d\alpha(s), \ t\in [0, 1].$ Then for all $n\in \mathbb N^+$ we have

and

Consequently, we have

and

where $\textbf{1}(t)\equiv 1, \ t\in [0, 1]$. Therefore, Gelfand's theorem implies that

and

Combining the two inequalities, we get

On the other hand, we note that

and then

Therefore, we obtain (2.3). This completes the proof. □

From Lemma 2.6, we find $r(\mathcal L_{\mu, \nu}) > 0$. Consequently, the Krein-Rutman theorem ^[32] implies that there exists $\zeta_{\mu, \nu} \in P\backslash \{0\}$ such that

From ^[19,33], the conjugate space of $C([0, 1], \mathbb{R})$ is $E^*: = \{\gamma:\gamma \text{ has bounded variation on }[0, 1]\}$. Moreover, the dual cone of $P$ and the bounded linear functional on $C([0, 1], \mathbb{R})$ can be expressed by

Note that $r(\mathcal L_{\mu, \nu}) > 0$ in Lemma 2.6, and there exists $\psi_{\mu, \nu} \in P^* \backslash\{0\}$ such that

where $\mathcal L_{\mu, \nu}^*: E^* \rightarrow E^*$ is the conjugate operator of $\mathcal L_{\mu, \nu}$, denoted by

Lemma 2.7. (see ^[34]) Let $E$ be a Banach space, $\Omega \subset E$ a bounded open set, and $A:\overline{\Omega}\cap P\to P$ a completely continuous operator. If there exists $z_0\in P\setminus\{0\}$ such that $z-Az\neq \lambda z_0, \text{ for all } z\in \partial\Omega\cap P, \lambda \geq 0$, then the fixed point index $i(A, \Omega\cap P, P) = 0$.

Lemma 2.8. (see ^[34]) Let $E$ be a Banach space, $\Omega \subset E$ a bounded open set with $0\in \Omega$, and $A:\overline{\Omega}\cap P\to P$ a completely continuous operator. If $z\neq \lambda Az, \text{ for all } z\in \partial\Omega\cap P, 0\leq \lambda \leq 1$, then the fixed point index $i(A, \Omega\cap P, P) = 1$.

3.   Main results

Consider the coefficients $\mu_i, \nu_i\ge 0$ with $\mu_i^2+\nu_i^2 \not = 0, \ i = 1, 2, 3, 4$. From Lemma 2.6, $r(\mathcal L_{\mu_i, \nu_i}) > 0$. Then there exists $\psi_{\mu_i, \nu_i} \in P^* \backslash\{0\}$ such that

Remark 3.1. Let $z\in P$. Then we have

To see this note that $\psi_{\mu_i, \nu_i}\in P^* \backslash\{0\}$, and from the definition of the Riemann-Stieltjes integral we have

and

for all divisions $t_j$: $0 = t_0 < t_1 < \cdots < t_{n-1} < t_n < t_{n+1} = 1, \ \rho = \max _{1 \leq j \leq n}\left(t_j-t_{j-1}\right), \ \xi_j \in\left[t_{j-1}, t_j\right], \ j =$ $1, 2, \cdots, n$. The other two inequalities can be similarly proven.

Now, we list our assumptions for the nonlinearities $f, g, I_k(k = 1, 2, ..., m)$:

(H2) There exist $\mu_1, \nu_1\ge 0\ (\mu_1^2+\nu_1^2 \not = 0)$ and $l_k\ge 0 \ (\sum\limits_{k = 1}^m l_k^2 \not = 0), k = 1, 2, ..., m$ such that

(H3) There exist $\mu_2, \nu_2\ge 0\ (\mu_2^2+\nu_2^2 \not = 0)$ and $\widetilde{l}_k\ge 0 \ (\sum\limits_{k = 1}^m \widetilde{l}_k^2 \not = 0), k = 1, 2, ..., m$ such that

(H4) There exist $\mu_3, \nu_3\ge 0\ (\mu_3^2+\nu_3^2 \not = 0)$ and $\overline{l}_k\ge 0 \ (\sum\limits_{k = 1}^m \overline{l}_k^2 \not = 0), k = 1, 2, ..., m$ such that

(H5) There exist $\mu_4, \nu_4\ge 0\ (\mu_4^2+\nu_4^2 \not = 0)$ and $\widehat{l}_k\ge 0 \ (\sum\limits_{k = 1}^m \widehat{l}_k^2 \not = 0), k = 1, 2, ..., m$ such that

Theorem 3.2. Suppose that (H0)–(H3) hold. Then, (1.1) has at least one positive solution.

Proof. Let $S_1 = \{z\in P: z-\mathcal Tz = \lambda \widetilde{z}, \ \lambda\ge 0\}$, where $\widetilde{z}\in P_0$ is a fixed element. We first prove that $S_1$ is a bounded set in $P$. Note that $z\in S_1$, and from Lemma 2.5 we have

By (H2) there exist $\widetilde{c}, \widetilde{c}_k > 0(k = 1, 2, ..., m)$ such that

Consequently, if $z\in S_1$, we have

Multiplying by $d\psi_{\mu_1, \nu_1}(t)$ on both sides of (3.3) and integrating over $[0, 1]$, from (3.1) we have

Thus,

There are two cases to consider.

Case 1. $r\left(\mathcal L_{\mu_1, \nu_1}\right)\ge 1$. From (3.2) and (3.4) we have

and thus

Case 2. Now $r\left(\mathcal L_{\mu_1, \nu_1}\right) < 1$. (H2), (3.2), and (3.4) imply that

and then

Combining the two cases, we have proved that $S_1$ is a bounded set, as required. Now, we choose a sufficiently large $R_1 > \sup S_1$ such that

where $B_{R_1} = \{z\in P: \|z\| < R_1\}$. Therefore, Lemma 2.7 implies that

By (H3) there exists $r_1 > 0$ such that

Now, we prove that

where $B_{r_1} = \{z\in P: \|z\| < r_1\}$. If the claim is false, then there exist a $z_1\in \partial B_{r_1}\cap P, \ \lambda_1\in [0, 1]$ such that

By Lemma 2.5, $z_1$ satisfies (3.2), and from (3.7) we have

Multiplying by $d\psi_{\mu_2, \nu_2}(t)$ on both sides of (3.9) and integrating over $[0, 1]$, from (3.1) we obtain

This, together with (3.2), implies that

This contradicts (H3) unless $\|z_1\| = 0$. Note that $\|z_1\| = 0$ also contradicts $z_1\in \partial B_{r_1}\cap P, \ r_1 > 0$. Therefore, we obtain that (3.8) holds, as required. From Lemma 2.8 we have

Note that $R_1$ can be chosen large enough such that $R_1 > \sup S_1$ and $R_1 > r_1$. Therefore, from (3.6) and (3.10) we have

Therefore, the operator $\mathcal T$ has at least one fixed point in $(B_{R_1}\backslash \overline{B}_{r_1})\cap P$. Thus, (1.1) has at least one positive solution. This completes the proof. □

Theorem 3.3. Suppose that (H0) and (H4)–(H5) hold. Then, (1.1) has at least one positive solution.

Proof. By (H4) there exists a sufficiently small $r_2 > 0$ such that

For this $r_2$, we prove that

where $B_{r_2} = \{z\in P: \|z\| < r_2\}$, and $\overline{z}$ is a fixed element in $P_0$. If (3.12) is false, then there exist a $z_2\in \partial B_{r_2}\cap P$, $\lambda_2\ge 0$ such that

Lemma 2.5 implies that $z_2$ satisfies (3.2). Moreover, from (3.11) we have

Multiplying by $d\psi_{\mu_3, \nu_3}(t)$ on both sides of (3.13) and integrating over $[0, 1]$, from (3.1) we obtain

There are two cases to consider.

Cases 1. $r\left(\mathcal L_{\mu_3, \nu_3}\right)\ge 1$. From (3.2) we obtain

which contradicts $z_2\in \partial B_{r_2}\cap P, \ r_2 > 0$.

Cases 2. $r\left(\mathcal L_{\mu_3, \nu_3}\right) < 1$. By (3.2) we have

and it contradicts (H4) unless $\|z_2\| = 0$. We also have a contradiction to $z_2\in \partial B_{r_2}\cap P, \ r_2 > 0$ if $\|z_2\| = 0$.

Therefore, we obtain that (3.12) holds, and Lemma 2.7 implies that

By (H5) there exist $\overline{c}, \overline{c}_k > 0(k = 1, 2, ..., m)$ such that

Let $S_2 = \{z\in P: z = \lambda \mathcal Tz, \lambda\in [0, 1] \}$. We now prove that $S_2$ is bounded in $P$. If $z\in S_2$, then by Lemma 2.5, (3.2) holds, and from (3.15) we have

Multiplying by $d\psi_{\mu_4, \nu_4}(t)$ on both sides of (3.16) and integrating over $[0, 1]$, from (3.1) we obtain

Note that (3.2) and $r\left(\mathcal L_{\mu_4, \nu_4}\right) < 1$, and we have

and (H5) implies that

This implies that $S_2$ is a bounded set in $P$, as required. Therefore, we can choose a large number $R_2 > \max\{\sup S_2, r_2\}$ such that

where $B_{R_2} = \{z\in P: \|z\| < R_2\}$. From Lemma 2.8 we have

As a result, from (3.14) and (3.17) we have

Therefore, the operator $\mathcal T$ has at least one fixed point in $(B_{R_2}\backslash \overline{B}_{r_2})\cap P$. Thus, (1.1) has at least one positive solution. This completes the proof. □

4.   Conclusions

In this paper, we study the existence of positive solutions for the Riemann-Liouville-type impulsive fractional integral boundary value problem (1.1). We first use the Gelfand theorem and the Krein-Rutman theorem to investigate a related positive linear operator, which can include the Riemann-Stieltjes integral condition. Then, the impulsive term is regarded as a perturbation, and we use some conditions concerning the spectral radius of the linear operator to obtain our main results. In this paper we provided a quite different method to study such problems.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

This research was supported by the Talent Introduction Project of Ludong University (grant No. LY2015004). The authors would like to express their heartfelt gratitude to the editors and reviewers for their constructive comments.

Conflict of interest

The authors declare no conflicts of interest.