Nontrivial solutions for a Hadamard fractional integral boundary value problem

1.   Introduction

In this work, we are devoted to studying the existence of nontrivial solutions for the Hadamard-type fractional Riemann-Stieltjes integral boundary value problem

where $D^\beta$ is the Hadamard-type fractional derivative with $\beta \in(2, 3]$, $\delta z(x) = x\frac{dz}{dx}$, and the functions $f, g, \eta$ satisfy the following conditions:

(H1) $f, g\in C([1, e]\times \mathbb R, \mathbb R)$.

(H2) There exist $\gamma_i, \sigma_i\in C([1, e], \mathbb R^+)$ and $M_i\in C(\mathbb R, \mathbb R^+) (i = 1, 2)$ such that

(H3) $\lim_{|z|\to +\infty} \frac{M_i(z)}{|z|} = 0$, $i = 1, 2$.

(H4) $\eta$ is a nondecreasing function in $[1, e]$ with $\eta(1) = 0$.

As an important branch of mathematical analysis, fractional calculus can more accurately describe some dynamic processes with memory and heredity characteristics. In view of outstanding advantages of fractional calculus, it has attracted the great attention of many researchers and developed rapidly; we refer the reader to ^{[1,2,3,4,5,6,7,8,9,10]}. Moreover, we note that originating from the work of Hadamard in 1892, Hadamard fractional calculus is now successfully applied to describe ultra-slow phenomena in the objective world, such as fracture of materials, creep of rocks, etc. Therefore, it is of great significance to study Hadamard-type fractional problems, see ^{[11,12,13,14,15,16,17,18,19]} and the references therein. For example, in ^[11], the authors used the Guo-Krasnosel'skii fixed point theorem to study the existence and nonexistence of positive solutions for the system of Hadamard fractional differential equations

where $f_i (i = 1, 2)$ are nonnegative continuous functions on $[1, e]\times \mathbb R^+\times \mathbb R^+$ and satisfy some $(p_i-1)-$superlinear and $(p_i-1)-$suberlinear growth conditions.

In ^[12], the authors studied the following system of Hadamard fractional differential equations with multipoint Hadamard fractional derivative boundary conditions

where $D^{\vartheta}$ are Hadamard-type fractional derivatives of order $\vartheta \in\left\{p, q, r_1, r_2\right\}, r_1 \in[0, p-1], r_2 \in[0, q-1]$. Using the monotone iterative method, they obtained the existence of monotone positive solutions for their considered problems. In ^[13], the authors used the fixed-point techniques to study the existence and uniqueness results for the following Riemann-Stieltjes integral boundary value problem involving a Hadamard-type fractional differential equation

where $f\in ([1, T] \times \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R})$ satisfies the Lipschitz condition.

Inspired by the aforesaid works, we use the fixed-point methods to study the nontrivial solutions for the Hadamard-type fractional Riemann-Stieltjes integral boundary value problem (1.1). We consider the two-folds: When the nonlinearities $f, g$ are superlinear and suberlinear, we use some conditions concerning the spectral radius of a new linear operator to obtain our existence theorems. When the nonlinearities $f, g$ are asymptotic linear, we use a fixed-point theorem to obtain a nontrivial solution.

2.   Preliminaries

We first briefly provide the definition of the Hadamard-type fractional derivative, which can be founded in ^[11,14,15].

Definition 2.1. Let $g:[1, \infty) \rightarrow \mathbb{R}$, then the Hadamard-type fractional $q$-order derivative is defined as

where $n = [q]+1, [q]$ means the integer part of $q$, and $\ln (\cdot) = \log _e(\cdot)$.

Now, we calculate the Green's function for (1.1).

Lemma 2.2. Let $h, V$ be functions on $[1, e]$, then

has a solution

where

Proof. We first consider the problem

By the methods of ^[15], we can obtain

where $c_i\in \mathbb R, i = 1, 2, 3$. The condition $z(1) = \delta z(1) = 0$ implies that $c_2 = c_3 = 0$. Furthermore, from $\delta z(e) = 0$, we find

then

Next, we consider the problem

which yields that

where $\widetilde{c}_i\in\mathbb R$, $i = 1, 2, 3$. Similarly, $\widetilde{c}_2 = \widetilde{c}_3 = 0$. Consequently, we get

and

Combining (2.1)–(2.4), we can obtain the conclusion of this lemma. This completes the proof.

Lemma 2.3 (see ^[1]). The function $G_\beta$ satisfies the following properties:

(I1) $G_\beta(x, y)\ge 0$ for $x, y\in [1, e]$;

(I2) $(\ln x)^{\beta-1}G_\beta(e, y)\le G_\beta (x, y)\le G_\beta (e, y)$ for $x, y\in [1, e]$.

Let $E: = C[1, e], \|z\|: = \max _{x \in[1, e]}|z(x)|, P: = \{z \in E: z(x) \geq 0, \forall x \in[1, e]\}$, then $(E, \|\cdot\|)$ is a real Banach space and $P$ a cone on $E$. By Lemma 2.2, we can define an operator as follows:

Moreover, it is easy to find that if there is a $z^*\in E\backslash \{0\}$ such that $Tz^* = z^*$, then this $z^*$ is a nontrivial solution for (1.1). Hence, we only need to study the existence of nontrivial fixed points of $T$. For $\xi_i > 0\; (i = 1, 2)$, let $L_{\xi_1, \xi_2}: E\to E$ be defined as

We then see that $L_{\xi_1, \xi_2}$ is a linear operator, and we also obtain the following lemma.

Lemma 2.4. Let $P_0 = \{z\in E: z(x)\ge (\ln x)^{\beta-1} \|z\|, x\in [1, e]\}$, then $L_{\xi_1, \xi_2}(P)\subset P_0$.

This can be easily obtained from Lemma 2.3 (I2), so we don't need to offer its proof.

Lemma 2.5. $r(L_{\xi_1, \xi_2}) > 0$, where $r(L_{\xi_1, \xi_2})$ stands for the spectral radius of $L_{\xi_1, \xi_2}$.

Proof. Let $(L_{\xi_1}z)(x) = \xi_1 \int_1^e G_\beta (x, y) z(y) \frac{dy}{y}, x\in [1, e], z\in E$, then for all $n\in \mathbb N_+$, we have

By Lemma 2.3 (I2), we have

where $\textbf{1}(x)\equiv 1, x\in [1, e]$. Consequently, the Gelfand's theorem implies that

Note that $r(L_{\xi_1, \xi_2})\ge r(L_{\xi_1}) > 0.$ This completes the proof.

Lemma 2.5 and the Krein-Rutman theorem ^[20] imply that there exists $\zeta_{\xi_1, \xi_2} \in P\backslash \{0\}$ such that

Moreover, by Lemma 2.4, we have

Lemma 2.6 (see ^[21]). Let $E$ be a Banach space, $W\subset E$ a bounded open set, and $T: W \rightarrow E$ a continuous compact operator. If there exists $z_0 \in E \backslash\{0\}$ such that

then the topological degree $\operatorname{deg}(I-T, W, 0) = 0$.

Lemma 2.7 (see ^[21]). Let $E$ be a Banach space, $W\subset E$ a bounded open set with $0 \in W$, and $T: W \rightarrow E$ a continuous compact operator. If

then the topological degree $\operatorname{deg}(I-T, W, 0) = 1$.

Lemma 2.8 (see ^[22]). Let $T: E \rightarrow E$ be a completely continuous operator, and $L: E \rightarrow E$ a bounded linear operator. Suppose that $1$ isn't an eigenvalue of $L$ and

then there exists $z^{**}\in E$ such that $Tz^{**} = z^{**}$.

3.   Main results

Now, we list some assumptions on $f$ and $g$, which we need in this section.

(H5) There exist $\xi_1, \xi_2 > 0$ with $r(L_{\xi_1, \xi_2})\ge 1$ such that

(H6) There exist $\xi_3, \xi_4 > 0$ with $r(L_{\xi_3, \xi_4}) < 1$ such that

(H7) There exist $\xi_5, \xi_6 > 0$ with $r(L_{\xi_5, \xi_6}) > 1$ such that

(H8) There exist $\xi_7, \xi_8 > 0$ with $r(L_{\xi_7, \xi_8}) < 1$ such that

(H9) $f, g(x, 0)\not \equiv 0, x\in [1, e]$, and there exist $\xi_9, \xi_{10}$ with $\frac{|\xi_9|}{\beta(\beta-1)\Gamma(\beta)}+|\xi_{10}|\frac{\int_1^e \frac{d\eta(t)}{t}}{\beta-1} < 1$ such that

Theorem 3.1. Let (H1)–(H6) hold, then (1.1) has a nontrivial solution.

Proof. From (H5), there exist $\epsilon_i > 0\; (i = 1, 2)$ and $Z_0 > 0$ such that

From (H3), for any $\varepsilon_i > 0\; (i = 1, 2)$, there exists $Z_1 > Z_0$ such that

Let $M_1^* = \max_{|z|\le Z_1} M_1(z), M_2^* = \max_{|z|\le Z_1} M_2(z)$, then we have

Note that by (H2), for $|z| > Z_1, x\in [1, e]$, and we obtain

Note that $f, g(x, z)$ are bounded on $[1, e]\times [-Z_1, Z_1]$, then let $C_f = (\xi_1+\epsilon_1-\varepsilon_1\|\gamma_2\|)Z_1+\max_{x\in [1, e], |z|\le Z_0}|f(x, z)|$, $C_g = (\xi_2+\epsilon_2-\varepsilon_2\|\sigma_2\|)Z_1+\max_{x\in [1, e], |z|\le Z_0}|g(x, z)|$, so we have

Note that $\varepsilon_i\; (i = 1, 2)$ can be chosen arbitrarily small, and let a sufficiently large $R_1$ satisfy:

where

In the following, we shall prove

where $\zeta_{\xi_1, \xi_2}$ is defined by (2.6) and $B_{R_1}$ is an open ball: $\{z\in E: \|z\| < R_1\}$. If the claim (3.4) isn't satisfied, then there exist $z_1\in \partial B_{R_1}$ and $\mu_1\ge 0$ such that

Note that $\mu_1\not = 0$. If not, $z_1$ is a fixed point of $T$ and the theorem is proved. In order to prove (3.5), let

By $\gamma_1+\gamma_2M_1(z_1)+C_f\in P$ and $\sigma_1+\sigma_2M_2(z_1)+C_g\in P$, Lemma 2.4 implies that

Moreover, we calculate $z_1+\widetilde{z}_1$ as follows:

From (H2), (2.7), and Lemma 2.4 we know

By (3.1), (3.3) and (3.6), note that $\|z_1\| = R_1$, and we have

Note that $\widetilde{z}_1, z_1+\widetilde{z}_1\in P_0$. From (3.2) and (H2), we have

using the fact that

In what follows, we prove that (3.9) holds. Note that $\|z_1\| = R_1$, and by (3.7), we have

Consequently, note that $G_\beta(x, y)\le \frac{(\ln x)^{\beta-1}(1-\log y)^{\beta-2}}{\Gamma(\beta)}, x, y\in [1, e]$, and (3.9) is greater than

(3.3) enables us to obtain

and

As a result, (3.9) holds, as required. From (3.5) and (3.8), we have

Define a set $W = \{\mu: z_1+\widetilde{z}_1 \ge \mu \zeta_{\xi_1, \xi_2}\}$ and $\mu^* = \sup W$, then $\mu_1\in W$ and $\mu^*\ge \mu_1$. Hence, note that $L_{\xi_1, \xi_2}:P\to P$, and we have

which contradicts the definition of $\mu^*$. As a result, (3.4) holds, and Lemma 2.6 implies that

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

In what follows, we prove that

where $B_{r_1} = \{z\in E: \|z\| < r_1\}$. If (3.12) doesn't hold, and there exist $z_2\in \partial B_{r_1}, \; \mu_2\ge 1$, such that

then this, along with (3.11), implies that

Note that $r(L_{\xi_3, \xi_4}) < 1$. This means that $\left(I-L_{\xi_3, \xi_4}\right)^{-1}$ exists and

Consequently, note that $\left(I-L_{\xi_3, \xi_4}\right)^{-1}: P\to P$, and we have

which implies that $z_2(x)\equiv 0, x\in [1, e]$ and contradicts $z_2\in \partial B_{r_1}$. Therefore, Lemma 2.7 implies that

Using this with (3.10), we have

Hence, $T$ has at least one fixed point in $B_{ R_1} \backslash \overline{B}_{r_1}$, i.e., (1.1) has at least one nontrivial solution. This completes the proof.

Now, we consider the case that our nonlinearities are suberlinear. From ^[8], we know that $E$'s conjugate space $E^*: = \{\rho:\rho \text{ has bounded variation on }[1, e]\}$. Moreover, the dual cone of $P$ and the bounded linear functional on $E$ are

Note that $r(L_{\xi_5, \xi_6}) > 1$ in (H7), and from the similar method in ^[23], there exists $\psi_{\xi_5, \xi_6} \in P^* \backslash\{0\}$ such that

where $L_{\xi_5, \xi_6}^*: E^* \rightarrow E^*$ is the conjugate operator of $L_{\xi_5, \xi_6}$, denoted by

Theorem 3.2. Let (H1)–(H4), (H7), and (H8) hold, then (1.1) has a nontrivial solution.

Proof. From (H7), there exists $r_2 > 0$ such that

For this $r_2$, we prove that

where $\omega_0\in P$ is a fixed element and $B_{r_2} = \{z\in E: \|z\| < r_2\}$. We use an argument by indirection, then there exist $z_3\in \partial B_{r_2}, \mu_3\ge 0$ such that

Note that by (3.14), we have

Hence, $\mu_3\ge 0, \omega_0\in P$ enable us to find

and

Multiplying $\frac{d\psi_{\xi_5, \xi_6}(x)}{x}$ on both sides of the above inequalities and integrating over $[1, e]$, from (3.13), we obtain

Note that $\psi_{\xi_5, \xi_6} \in P^* \backslash\{0\}$, then from the definition of the Riemann-Stieltjes integral, we have

where $0 = x_0 < x_1 < \cdots < x_{n-1} < x_n < x_{n+1} = 1, \lambda = \max _{1 \leq i \leq n}\left(x_i-x_{i-1}\right), \forall \xi_i \in\left[x_{i-1}, x_i\right], i =$ $1, 2, \cdots, n$. Therefore, $r(L_{\xi_5, \xi_6}) > 1$ and (3.16) enable us to obtain

Note that for all divisions $x_i$, (3.18) holds, and we only obtain $z_3(x)\equiv 0, x\in [1, e]$. Therefore, this contradicts $z_3\in \partial B_{r_2}$ and (3.15) holds. Lemma 2.6 implies that

By (H8), there exist $d_1, d_2 > 0$ such that

In what follows, we prove that $S = \{z\in E: Tz = \mu z, \mu\ge 1\}$ is a bounded set. If there exist $z_4\in S, \mu_4\ge 1$ such that

then (3.20) implies that

Note that $r(L_{\xi_7, \xi_8}) < 1$. Thus $\left(I-L_{\xi_7, \xi_8}\right)^{-1}$ exists, and

Consequently, note that $\left(I-L_{\xi_7, \xi_8}\right)^{-1}: P\to P$, and we have

and, thus,

which implies that $S$ is a bounded set, as required. Let $R_2 > \sup S$ and $R_2 > r_2$, then we have

Therefore, Lemma 2.7 implies that

Using this with (3.19), we have

Hence, $T$ has at least one fixed point in $B_{ R_2} \backslash \overline{B}_{r_2}$, i.e., (1.1) has at least one nontrivial solution. This completes the proof.

Theorem 3.3. Let (H1), (H4), and (H9) hold, then (1.1) has a nontrivial solution.

Proof. Define a linear operator $L_{\xi_9, \xi_{10}}: E\to E$ as follows:

Now, we prove that $1$ isn't an eigenvalue for $L_{\xi_9, \xi_{10}}$. On the contrary, then there exists $z^*\in E\backslash \{0\}$ such that

By Lemma 2.2, we obtain

Next, we will consider the following two cases.

Case 1. $\xi_9^2 +\xi_{10}^2 = 0$. By Lemma 2.2, we have

where $\widetilde{d}_i\in\mathbb R$, $i = 1, 2, 3$. Note that $\beta\in (2, 3]$, the boundary condition $z^*(1) = \delta z^*(1) = \delta z^*(e) = 0$ implies that $\widetilde{d}_i = 0$, and $z^*(x)\equiv 0, \ x\in [1, e]$. This contradicts $z^*\in E\backslash \{0\}$.

Case 2. $\xi_9^2 +\xi_{10}^2 \not = 0$. From (3.21), we have

By (H9), we obtain that

This is impossible.

As a result, we claim that $1$ isn't an eigenvalue for $L_{\xi_9, \xi_{10}}$, as required.

By (H9), for all $\varepsilon > 0$, there exists $\Lambda > 0$ such that

Note that when $|z|\le \Lambda$ and $x\in [1, e]$, $|f(x, z)-\xi_9 z|$ and $|g(x, z)-\xi_{10} z|$ are bounded. Therefore, there exist $\varrho_i > 0\; (i = 1, 2)$ such that

Hence, (3.22) enables us to obtain

This implies that

Note that from the arbitrariness of $\varepsilon$, we know that

Lemma 2.8 indicates that $T$ has at least one fixed point, and note that $T0\not = 0$ by $f, g(t, 0)\not \equiv 0, t\in [1, e]$. Therefore, (1.1) has at least one nontrivial solution. This completes the proof.

4.   Conclusions

We note that the spectral theory of linear operators can be used to study differential equations, see for example, ^[2,6,9,10]. In ^[9], the authors studied the following fractional boundary value problem

where $\alpha \geq 2, D_{0+}^\alpha$ is the Riemann-Liouville derivative and $f\in C([0, 1]\times (0, +\infty), \mathbb R^+)$. They obtained two existence theorems of positive solutions for their problem when the nonlinearity $f$ satisfies one of the following growth conditions

sublinear condition: $\limsup\limits_{z \rightarrow +\infty} \sup\limits_{t \in[0, 1]} \frac{f(t, z)}{z} < \lambda_1$, and $\liminf\limits_{z \rightarrow 0^+} \inf\limits_{t \in[0, 1]} \frac{f(t, z)}{z} > \lambda_1$;

superlinear condition: $\limsup\limits_{z \rightarrow 0^+} \sup\limits_{t \in[0, 1]} \frac{f(t, z)}{z} < \lambda_1$, and $\liminf\limits_{z \rightarrow +\infty} \inf\limits_{t \in[0, 1]} \frac{f(t, z)}{z} > \lambda_1$, where $\lambda_1 = (r(L))^{-1}$, $r(L)$ is the spectral radius of the linear operator $(L z)(t) = \int_0^1 G_{\text{Wang}}(t, s) q(s) z(s) d s$, $G_{\text{Wang}}$ is their Green's function.

Compared with our problem, we don't incorporate the integral boundary conditions into the Green's function, and consider a new linear operator $(2.5)$. Moreover, we obtain several existence theorems of nontrivial solutions under some conditions regarding the spectral radius of the linear operator. On the other hand, our nonlinearities $f, g$ can be sign-changing, in contrast to the nonlinearities in ^{[1,2,3,4,5,6,7,8,9,10]}, which are assumed to be nonnegative. These imply that our main results generalize and improve the corresponding ones in the works cited above.

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 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 there is no conflict of interest.