Nontrivial solutions for a fourth-order Riemann-Stieltjes integral boundary value problem

1.   Introduction

In this paper we study the existence of solutions for the following integral boundary value problem of the fourth-order differential equation

where $f$ is a continuous function on $[0, 1]\times \mathbb R$, $\int_0^1 u(t)d \alpha(t)$ denotes the Riemann-Stieltjes integral, $\alpha$ is a function of bounded variation and satisfies the condition

(H1) $\alpha(t)\ge 0, t\in [0, 1]$ with $\int_0^1 t d\alpha(t)\in [0, 1)$.

Boundary value problems can describe many phenomena in the applied sciences such as nonlinear diffusion, thermal ignition of gases and concentration in chemical or biological problems. There are many papers in the literature considering the existence of solutions using Leray-Schauder degree, the method of upper-lower solutions and the Guo-Krasnoselskii fixed point theorem in cones; we refer the reader to ^{[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32]} and the references cited therein. In ^[4] the authors used the Guo-Krasnoselskii fixed point theorem to study the existence of positive solutions of the fourth-order integral boundary value problem

and in ^[13] the authors investigated monotone positive solutions for the nonlinear fourth-order boundary value problem with integral and multi-point boundary conditions

where $f \in C([0, 1] \times \mathbb{R}^{+} \times \mathbb{R}^{+}, \mathbb{R}^{+})$ satisfies some superlinear and sublinear growth conditions. In ^[14] the authors studied the existence and uniqueness of positive solutions for the fourth-order $m$-point boundary value problem

where $f \in C\left([0, 1] \times \mathbb{R}^{+}, \mathbb{R}^{+}\right)$ satisfies the following conditions:

$\text{(H)}_\text{Hao1} \ \ \lim _{u \rightarrow \infty} \inf \min _{t \in[0, 1]} \frac{f(t, u)}{u} > \lambda^*, \ \lim _{u \rightarrow 0^{+}} \sup \max _{t \in[0, 1]} \frac{f(t, u)}{u} < \lambda^*,$ and

$\text{(H)}_\text{Hao2} \ \ \lim _{u \rightarrow 0^{+}} \inf \min _{t \in[0, 1]} \frac{f(t, u)}{u} > \lambda^*, \ \lim _{u \rightarrow \infty} \sup \max _{t \in[0, 1]} \frac{f(t, u)}{u} < \lambda^*,$ where $\lambda^*$ is the first eigenvalue of the eigenvalue problem

with the boundary conditions in (1.2).

Note all integral boundary conditions include the two-point, three-point and multi-point boundary conditions as special cases and naturally this kind of problem has interested researchers; see for example ^{[1,2,4,8,9,11,13,19,22,24,25,26,27,28,30,31]} and the references cited therein. In ^[11] the author studied the following nonlocal fractional boundary value problem with a Riemann-Stieltjes integral boundary condition

where $D^{\alpha}$ is the standard Caputo derivative, $f:[0, 1] \times \mathbb R^+ \rightarrow \mathbb R^+$ is continuous, $\gamma[u] = \int_{0}^{1} u(s) d A(s)$ and in ^[31] the authors studied the eigenvalue problem for a class of singular $p$-Laplacian fractional differential equations involving the Riemann-Stieltjes integral boundary condition

where ${D}_{t}^{\beta}$ and ${D}_{t}^{\alpha}$ are the standard Riemann-Liouville derivatives, and $f(t, x):(0, 1) \times(0, +\infty) \rightarrow \mathbb R^+$ is continuous.

As is well known, due to the non-locality of fractional calculus, more and more problems in physics, electromagnetism, electrochemistry, diffusion and general transport theory can be described by the fractional calculus approach. As a new modeling tool, it has a wide range of applications in many fields. However, in the process of research, more and more scholars have found that a variety of important dynamical problems exhibit fractional-order behavior that may vary with time, space or other conditions. This phenomenon indicates that variable-order fractional calculus is a natural choice, which provides an effective mathematical framework for the description of complex mathematics. For more definitions of fractional derivatives and physical understandings, we refer the reader to ^[3,33,34,35].

Motivated by the aforementioned works, in this paper we use topological degree and the method of upper-lower solutions to study the fourth-order Riemann-Stieltjes integral boundary value problem (1.1), and obtain existence theorems for nontrivial solutions and extremal solutions. Moreover, we note that the conditions in this paper are more general than $\text{(H)}_\text{Hao1}$ and $\text{(H)}_\text{Hao2}$. Finally, some appropriate examples to illustrate our main results are given.

2.   Preliminaries

In this section motivated by the variational iteration method (see [13,Lemma 1]), we first obtain an equivalent integral equation for our problem (1.1). Let

Then we have

and

By using $u''(0) = u''(1) = 0$ we obtain

and

Note that

and hence

Therefore, we obtain

where

We multiply both sides of (2.1) by $d\alpha(t)$ and integrate over $[0, 1]$, then (note (H1))

and

Consequently, we have

where

Lemma 2.1. $K(t, s)$ has the following properties:

(ⅰ) $K(t, s) = \int_0^1 H(t, \tau)H(\tau, s)d\tau$, where

(ⅱ) $K(t, s) > 0$ for $t, s\in (0, 1)$;

(ⅲ) $\frac{1}{30}t(1-t)s(1-s)\le K(t, s)\le \frac{1}{6}s(1-s)$ for $t, s\in [0, 1]$;

(ⅳ) $K(t, s)\le \frac{1}{6}t(1-t)$ for $t, s\in [0, 1]$.

By simple calculations we obtain Lemma 2.1(ⅰ). Moreover, note that $H$ satisfies $t(1-t)s(1-s)\le H(t, s)\le s(1-s)$ and $H(t, s)\le t(1-t)$ for $t, s\in [0, 1]$, so we can easily obtain Lemma 2.1 (ⅲ)–(ⅳ), so we here omit their proofs.

Lemma 2.2. $\Theta(t, s)$ has the following properties:

(ⅰ) $\Theta(t, s) > 0$ for $t, s\in (0, 1)$;

(ⅱ) $\Theta(t, s)\ge \frac{\int_0^1 t(1-t)d\alpha(t)}{30[1-\int_0^1 td\alpha(t)]}ts(1-s)$ for $t, s\in [0, 1]$;

(ⅲ) $\Theta(t, s)\le \frac{1}{6}\left[1+\frac{\alpha(1)}{1-\int_0^1 td\alpha(t)}\right] s(1-s)$ for $t, s\in [0, 1]$;

(ⅳ) $\Theta(t, s)\le \frac{1}{6}t\left[1+\frac{\int_0^1 t(1-t)d\alpha(t)}{1-\int_0^1 td\alpha(t)}\right]$ for $t, s\in [0, 1]$.

These conclusions can be obtained from Lemma 2.1.

Let $E: = C[0, 1], \|u\|: = \max _{t \in[0, 1]}|u(t)|, P: = \{u \in E: u(t) \geq 0, \forall t \in[0, 1]\}$. Then $(E, \|\cdot\|)$ is a real Banach space and $P$ a cone on $E$.

Define a linear operator:

Then $B: E \rightarrow E$ is a completely continuous, positive, linear operator, and its spectral radius, denoted by $r(B)$, is $\frac{1}{\pi^4}$. Let an operator $L_{\xi}(\xi > 0)$ be given by

Now $L_{\xi}: P \rightarrow P$ is a completely continuous, linear, positive operator. Note that the spectral radius $r\left(L_{\xi}\right) \geq \xi r(B) > 0$. Then the Krein-Rutman theorem ^[17] implies that there exists $\varphi_{\xi} \in P \backslash\{0\}$ such that

Define an operator $A: C[0, 1] \rightarrow C[0, 1]$ as

It is clear that $u^*$ is a solution of (1.1) if and only if $Au^* = u^*$, i.e.,

and (H1) implies that

Therefore, the operator $A$ can also be expressed as

Lemma 2.3. Let $(L_\Theta u)(t) = \int_0^1 \Theta(t, s)u(s)ds$. Then $L_\Theta(P)\subset P_{01}$, where

Proof. If $u\in P$, from Lemma 2.2(ⅲ)–(ⅳ) we have

and

This completes the proof.

Lemma 2.4. (see ^[10]) Let $\Omega$ be a bounded open set in a Banach space $E$, and $T: \Omega \rightarrow E$ a continuous compact operator. If there exists $x_0 \in E \backslash\{0\}$ such that

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

Lemma 2.5. (see ^[10]) Let $\Omega$ be a bounded open set in a Banach space $E$ with $0 \in \Omega$, and $T: \Omega \rightarrow E$ a continuous compact operator. If

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

3.   Nontrivial solutions for (1.1)

In this section, we assume that the nonlinearity $f$ satisfies the conditions:

(H2) $f \in C([0, 1] \times \mathbb{R}, \mathbb{R})$. Moreover, there exist three functions $\gamma_i\in C([0, 1], \mathbb R^+), i = 1, 2,$ and $\mathcal M\in C(\mathbb R, \mathbb R^+)$ with $\gamma_2(t)\not \equiv 0, t\in [0, 1]$ such that

(H3) $\lim _{|x| \rightarrow +\infty} \frac{\mathcal M(x)}{|x|} = 0$.

(H4) There exists $\xi_1 > 0$ such that $r(L_{\xi_1})\ge 1$ and

(H5) There exists $\xi_2 > 0$ such that $r(L_{\xi_2}) < 1$ and

Theorem 3.1. Suppose that (H1)–(H5) hold. Then (1.1) has at least one nontrivial solution.

Proof. From (2.2) there exists $\varphi_{\xi_1} \in P \backslash\{0\}$ such that $L_{\xi_1}\varphi_{\xi_1} = r(L_{\xi_1})\varphi_{\xi_1}$, i.e.,

Note that $r(L_{\xi_1})\ge 1$. We multiply both sides of the above equation by $d\alpha(t)$ and integrate over $[0, 1]$ (note (H1)) so we obtain

and

Consequently, we have

where

Let

Now

Indeed, from Lemma 2.1(ⅲ) we have

and

Note that

Therefore, by Lemma 2.3 we obtain

By (H4), there exist $\varepsilon_0 > 0$ and $X_0 > 0$ such that

For any fixed $\varepsilon$ with $\varepsilon_0-\|\gamma_2\| \varepsilon > 0$, from (H3) there exists $X_1 > X_0$ such that

Note from (H2), we also obtain

Let $C_{X_1} = (\xi_1+\varepsilon_0-\varepsilon\|\gamma_2\|)X_1+ \max_{t\in [0, 1], |x|\le X_1} |f(t, x)|, \mathcal M^* = \max_{ |x|\le X_1} \mathcal M(x)$, and we have

Note that $\varepsilon$ can be chosen arbitrarily small, and we let

where

In what follows, we prove that

where $\varphi_{\xi_1}$ is defined in (3.1), and $B_{\mathcal{R}_1} = \{u \in E:\|u\| < \mathcal{R}_1\}$. Suppose the contrary. Then there exist $u_1\in \partial B_{\mathcal R_1}$ and $\mu_1\ge 0$ such that

Note that $\mu_1\not = 0$ (otherwise, $u_1$ is a solution for (1.1) and the theorem is proved). Let

Then (H1) implies that

Note that $\gamma_1(s)+\gamma_2(s)\mathcal M(u_1(s))+C_{X_1}\ge 0, s\in [0, 1]$, and by (3.3) we have

Moreover, from (3.7) we have

i.e.,

From (H1) we get

Hence, we have

Note that $f(s, u_1(s))+\gamma_1(s)+\gamma_2(s)\mathcal M(u_1(s))+C_{X_1}\ge 0, s\in [0, 1]$. Then (3.2) and (3.3) imply that

implies that

Now, we estimate the norm of $\widetilde{u}_1$. Note that (3.5) and $\|u_1\| = \mathcal R_1$, from Lemma 2.2 (ⅲ) and (3.4) we have

From (3.8) we have $u_1(t)+\widetilde{u}_1(t)\ge t \frac{\int_0^1 t(1-t)d\alpha(t)}{5[r(L_{\xi_1})+\int_0^1(1-t) d\alpha(t)]} \|u_1+\widetilde{u}_1\|\ge t \frac{\int_0^1 t(1-t)d\alpha(t)}{5[r(L_{\xi_1})+\int_0^1(1-t) d\alpha(t)]} (\|u_1\|-\|\widetilde{u}_1\|), t\in [0, 1].$ Note (3.5), and

Then Lemma 2.2 (ⅳ) implies that

Therefore, from (3.4) we have

Together with (3.7), we have

Define

Clearly, $\mu^*\ge \mu_1$, and $u_1+\widetilde{u}_1\ge \mu^* \varphi_{\xi_1}$. Note that $L_{\xi_1}\varphi_{\xi_1} = r(L_{\xi_1})\varphi_{\xi_1}$ and we have

which contradicts the definition of $\mu^*$($r(L_{\xi_1})\ge 1$). Therefore, (3.6) holds, and from Lemma $2.4$ we obtain

From (H5) there exists $r_1\in (0, \mathcal R_1)$ such that

Now for this $r_1$, we prove that

Suppose the contrary. Then there exist $u_2 \in \partial B_{r_1}$ and $\mu_2 \geq 1$ such that

where $B_{r_1} = \{u \in E:\|u\| < r_1\}$. Consequently, we have

Let $v_2(t) = |u_2(t)|\in P, t\in [0, 1]$. Then we have

Note that $r(L_{\xi_2}) < 1$, which implies that $\left(I-L_{\xi_2}\right)^{-1}$ exists, and

Consequently, note that $\left(I-L_{\xi_2}\right)^{-1}: P\to P$, and we have

Hence, $\|v_2\| = 0 \Rightarrow \|u_2\| = 0$, and this contradicts $u_2\in \partial B_{r_1}$. Thus, (3.12) holds, and Lemma $2.5$ implies that

Combining this with (3.11) we have

Therefore the operator $A$ has at least one fixed point in $B_{\mathcal R_1} \backslash \overline{B}_{r_1}$. Equivalently, (1.1) has at least one nontrivial solution. This completes the proof.

4.   Extremal solutions for (1.1)

In this section we use the method of upper-lower solutions to study the existence of extremal solutions for (1.1). We first provide the definitions of upper and lower solutions.

Definition 4.1. We say that $u \in E$ is an upper solution of (1.1) if

Definition 4.2. We say that $u \in E$ is a lower solution of (1.1) if

Lemma 4.3. Suppose that (H1) holds. Let $u\in E$ satisfy

Then $u(t)\ge 0, t\in [0, 1]$; here $c(t)$ satisfies the condition

(H6) $-\pi^4 < c(t) < c_0$, $t\in [0, 1]$, and $c_0: = 4 k_0^4$ with $k_0$ being the smallest positive solution of the equation $\tan k = \tanh k$ (i.e., $k_0 \approx 3.9266$ and $\left.c_0 \approx 950.8843\right)$.

Proof. From ^[6,7,32] we introduce a result. Let $L_c: W \rightarrow C[0, 1]$ be defined by $L_c u = u^{(4)}+c(t) u.$ Then by (H6), $L_c$ has a positive inverse, where $W = \left\{u \in C^4([0, 1]): u(0) = u(1) = u^{\prime \prime}(0) = u^{\prime \prime}(1) = 0\right\}$. In (4.1) let $u^{(4)}(t)+c(t) u(t) = z(t)\ge 0$ and $\chi_1 = u(1)-\int_0^1 u(t)d\alpha (t)\ge 0$, then we have

is equivalent to

where $G$ is defined in [32,Lemma 2.1].

We multiply both sides of (4.3) by $d\alpha(t)$ and integrate over $[0, 1]$, then (H1) enables us to obtain

and

Therefore, we have

where

Note that $G(t, s)\ge 0, t, s\in [0, 1]$. Then, (H1) implies that

This completes the proof.

For $v_{0}, w_{0} \in E$ with $v_{0}(t) \leq w_{0}(t)$ for $t \in[0, 1]$, we denote an ordered interval:

Also, we list our other assumptions in this section.

(H7) There exist $w_{0}, v_{0} \in E$ which are the upper and lower solutions of problem (1.1), respectively, and $v_{0}(t) \leq w_{0}(t), t \in[0, 1].$

(H8) $f \in C([0, 1] \times \mathbb{R}, \mathbb{R})$ and

Theorem 4.4. Suppose that (H1) and (H6)–(H8) hold. Then there exist monotone iterative sequences $\left\{v_{n}\right\}, \left\{w_{n}\right\} \subset\left[v_{0}, w_{0}\right]$ such that $v_{n} \to v^{*}, w_{n} \to w^{*}$ as $n \to \infty$ uniformly in $\left[v_{0}, w_{0}\right]$, and $v^{*}, w^{*}$ are the minimal and the maximal solution of (1.1) in $\left[v_{0}, w_{0}\right]$, respectively.

Proof. We define two sequences $\{w_n\}, \{v_n\}\subset E$ satisfying the following boundary value problems

and

Step 1. We prove

Let $x(t) = v_1(t)-v_0(t)$. Then we have

From Lemma 4.3, $x(t)\ge 0$, i.e., $v_1(t)\ge v_0(t), t\in [0, 1]$.

Let $y(t) = w_0(t)-w_1(t)$. Then we obtain

Lemma 4.3 implies that $y(t)\ge 0$, i.e., $w_0(t)\ge w_1(t), t\in [0, 1]$.

Let $h(t) = w_{1}(t)-v_{1}(t)$. Then we have

and thus

Lemma 4.3 enable us to obtain $h(t)\ge 0$, i.e., $w_1(t)\ge v_1(t), t\in [0, 1]$.

As a result, (4.6) holds.

Step 2. We prove that $w_{1}, v_{1}$ are upper and lower solutions of problem (1.1), respectively.

From (H8) and (4.4) we have

and note that

From Definition 4.2, $v_1$ is a lower solution for (1.1).

From (H8) and (4.5) we have

and

From Definition 4.1, $w_1$ is an upper solution for (1.1).

Therefore, for $v_{n-1}, v_n, w_{n-1}, w_n$ we can use the method in Steps 1 and 2 to obtain

and $w_{n}, v_{n} \in E$ are upper and lower solutions of problem (1.1), respectively.

Using mathematical induction, it is easy to verify that

It is easy to conclude that $\left\{v_{n}\right\}_{n = 0}^{\infty}$ and $\left\{w_{n}\right\}_{n = 0}^{\infty}$ are uniformly bounded in $E$, and from the monotone bounded theorem we have

Step 3. We prove (1.1) has solutions.

Note that (4.4) and (4.5) are respectively equivalent to the following integral equations

and

Let $n\to \infty$ and we have

and

These two integral equations can be transformed into the following boundary value problems

and

i.e., $v^*, w^*$ are solutions for (1.1).

Step 4. We prove that $v^{*}$ and $w^{*}$ are extremal solutions for (1.1) in $\left[v_{0}, w_{0}\right]$.

Let $u \in\left[v_{0}, w_{0}\right]$ be any solution for (1.1). We assume that $v_{m}(t) \leq u(t) \leq w_{m}(t), t \in[0, 1]$ for some $m$. Let $p(t) =$ $u(t)-v_{m+1}(t), q(t) = w_{m+1}(t)-u(t)$. Then from (1.1), (4.4) and (H8) we have

and this leads to the following boundary value problem

Lemma 4.3 implies that $p(t)\ge 0$, i.e., $u(t)\ge v_{m+1}(t), t\in [0, 1]$.

By (1.1), (4.5) and (H8) we have

and this leads to the following boundary value problem

Lemma 4.3 implies that $q(t)\ge 0$, i.e., $w_{m+1}(t) \ge u(t), t\in [0, 1]$.

Combining the above two cases, we have

Applying mathematical induction, we obtain $v_{n}(t) \leq u(t) \leq w_{n}(t)$ on $[0, 1]$ for any $n$. Taking the limit, we conclude $v^{*}(t) \leq u(t) \leq w^{*}(t), t \in[0, 1]$. This completes the proof.

Remark 4.1. As noted in ^[32], the Green's function $G$ in (4.2) has no explicit expression, but this does not affect our result. In our study we only use its positiveness and continuity.

5.   Examples

Now, we provide some examples to illustrate our main results.

Example 5.1. From (2.2) we have

Let $\alpha(t) = \frac{1}{2}t, t\in [0, 1]$. Then we can choose $\xi_1\ge \pi^4, \xi_2\in (0, 18)$ such that

Let $\gamma_1(t)\equiv \zeta_1\in (\xi_1, +\infty), \gamma_2(t)\equiv \zeta_2\in (0, \zeta_1+\xi_2]$, and $f(t, x) = \zeta_1|x|-\zeta_2 \mathcal M(x), \mathcal M(x) = \ln (|x|+1), x \in \mathbb{R}, t\in [0, 1]$. Then $\lim _{|x| \rightarrow +\infty} \frac{\mathcal M(x)}{|x|} = 0$, and $\lim _{|x| \rightarrow +\infty} \frac{\zeta_1|x|-\zeta_2 \mathcal M(x)}{|x|} = \zeta_1 >$ $\xi_1, \lim _{|x| \rightarrow 0} \frac{|\zeta_1|x|-\zeta_2 \mathcal M(x)|}{|x|} = |\zeta_1-\zeta_2|\le\xi_2$. Therefore, (H1)–(H5) hold. From Theorem 3.1, (1.1) has a nontrivial solution.

Example 5.2. Let $\alpha(t) = \frac{1}{2}t,$ and $v_0(t) = -t^4+2t^3-5t, w_0(t) = t^4-2t^3+5t, f(t, u) = 5tu(t), t\in [0, 1]$. Then we have

and

Moreover,

Then (H1) and (H6)–(H8) hold. From Theorem 4.4, (1.1) has two extremal solutions.

6.   Conclusions

In this paper we use topological degree and the method of upper-lower solutions to study the existence of solutions for (1.1). When the nonlinearity satisfies some superlinear growth conditions involving the first eigenvalue corresponding to the relevant linear operator we obtain nontrivial solutions. Also, when the nonlinearity satisfies a one-sided Lipschitz condition, we use the method of upper-lower solutions to obtain extremal solutions. We also provide two iterative sequences for these solutions.

Acknowledgments

This research was supported by the Nature Science Foundation of Anhui Provincial Education Department (Grant Nos. KJ2020A0735, KJ2021ZD0136), the Foundation of Suzhou University (Grant Nos. 2019XJZY02, szxy2020xxkc03).

Conflict of interest

The authors declare no conflict of interest.