New results for nonlinear fractional jerk equations with resonant boundary value conditions

1.   Introduction

It is well known that a three-dimensional dynamical system:

can be transformed into a third-order differential equation of the form:

The above equation involves a third derivative of the variable $x$, which in a mechanical system is a rate of change of acceleration, it is called a jerk equation. Jerk equations are widely used in applied science and engineering, such as it can describe a three-order dynamic vibration model. In recent years, the discussion of jerk equations have attracted the attention of scholars and some results have been obtained. For recent publication on jerk equations, we refer the reader to see ^{[1,2,3,4,5,6,7]}.

In 2017, by using classical contraction mapping theorem, Elsonbaty and El-Sayed ^[1] obtained the existence and uniqueness of the solution of the system:

where the dot denotes differentiation with respect to time, $a$ and $c$ are positive parameters, and $c\in \mathbb{R}$.

In ^[2], a second approximate solution of nonlinear jerk equation:

was obtained by using modified harmonic balance method.

Very recently, Liu and Chang ^[3] developed two iterative algorithms to determine the periodic solutions of the following nonlinear jerk equation:

where the initial values $x_0, \dot{x}_0$ and $\ddot{x}_0$ for the periodic solution of the above are unknown and the period is denoted by $T > 0$.

What is interesting to us is the appearance of the article ^[4]. Prakash and Singh presented a new fractional order jerk equations which do not have equilibrium point:

where $f = -y(t)+3y^2(t)-x^2(t)-x(t)z(t)+\beta-W(x(t))y(t)$, $\alpha, \beta, \gamma$ are fractional order and $0 < \alpha, \beta, \gamma\leq1$. In addition, the authors successfully designed a fractional-order backstepping controller to stabilise the chaos in the system. This proposed system is novel in the sense that the jerk systems involving fractional order. It enlightens us to think the problem: how to observe the existence result of fractional jerk differential equation.

Differential equations with fractional order arise from a variety of applications in various fields of science and engineering, such as electromagnetic, mechanics, biology, chemistry etc. (see ^[8]). Over last two decades, there has been significant progress in the area of fractional differential equations. Though many works have been done dealing with the existence of solutions to fractional differential equation (see ^{[9,10,11,12,13,14,15,16,17,18,19,20,21]}), no results on boundary value problem of fractional jerk equation are reported in the literature. To the best of the authors' knowledge, this is the first paper dealing with resonant boundary value problem of fractional jerk equations.

In this paper, we are interested in the solutions to the following fractional jerk equations:

with boundary value conditions given as

where $t\in(0, 1)$, $0 < \alpha, \beta, \gamma \leq1$, $0 < \xi\leq1$, $D_{0+}^{\alpha}, D_{0+}^{\beta}, D_{0+}^{\gamma}$ denote the Caputo fractional derivatives and $f:[0, 1]\times \mathbb{R}^{3 }\rightarrow \mathbb{R}$ is continuous.

It is easy to check that the following fractional jerk equation:

is equivalent to (1.1). Due to conditions (1.2), Eq (1.3) happens to be at resonance in the sense that the associated linear homogeneous equation:

has $u(t) = c, c\in \mathbb{R}$ as nontrivial solutions.

In addition, we can see if $\alpha = \beta = \gamma = 1$, the Eq (1.3) can be rewritten as

which is a standard nonlinear jerk equation. Furthermore, if $\xi\neq1$, then the above equation is a nonlocal initial value problems of nonlinear jerk equation; if $\xi = 1$, then it is a nonlinear jerk equations with periodic boundary conditions. If $u(t), t\in [0, 1]$ satisfies $u(0) = u(1)$, then boundary value condition $u(0) = u(1)$ is called periodic boundary conditions. According to different values of $\xi, \alpha, \beta, \gamma$, we will get diversiform types of boundary value conditions. Thus, this paper enriches and extends the existing literatures, such as ^[4,5,6,7].

The remaining part of this article is organized as follows. In Section 2, we will state some necessary notations, definitions and lemmas. In Section 3, by applying the coincidence degree theory due to Mawhin ^[22], we obtain the existence of solutions of (1.3). In the last section, an example is given to illustrate our results.

2.   Preliminaries

We present the necessary definitions and lemmas from fractional calculus theory that will be used to prove our main theorems.

Definition 2.1 (^[8]). The Riemann-Liouville fractional integral of order $\alpha > 0$ of a function $f:(0, \infty)\to \mathbb{R}$ is given by

provided that the right-hand side is pointwise defined on $(0, \infty)$.

Definition 2.2 (^[8]). The Caputo fractional derivative of order $\alpha > 0$ of a continuous function $f:(0, \infty)\to \mathbb{R}$ is given by

where $n-1 < \alpha\leq n$, provided that the right-hand side is pointwise defined on $(0, \infty)$.

Lemma 2.3 (^[8]). Let $n-1 < \alpha\leq n$, $u\in C(0, 1)\cap L^1(0, 1)$, then

where $c_{i}\in \mathbb{R}$, $i = 0, 1, \cdots, n-1$.

Lemma 2.4. If $\beta > 0, \; \alpha+\beta > 0$, then the equation

is satisfied for continuous function $f$.

Firstly, we briefly recall some definitions on the coincidence degree theory. For more details, see ^[22].

Let $Y, Z$ be real Banach spaces, $L:\operatorname{dom} L \subset Y \to Z$ be a Fredholm map of index zero and $P:Y\to Y$, $Q:Z\to Z$ be continuous projectors such that

It follows that

is invertible. We denote the inverse of this map by $K_P$. If ${\Omega}$ is an open bounded subset of $Y$, the map $N$ will be called $L$-compact on $\overline{\Omega}$ if $QN(\overline{\Omega})$ is bounded and $K_{P, Q}N = K_P(I-Q)N:\overline{\Omega}\to Y$ is compact.

Theorem 2.5 (^[22]). Let $L$ be a Fredholm operator of index zero and $N$ be $L$-compact on $\overline{\Omega}$. Suppose that the following conditions are satisfied:

$(1) \; Lx \neq \lambda Nx$ for each $(x, \lambda) \in \left[(\operatorname{dom}L\backslash {\rm Ker } L) \cap \partial\Omega\right]\times (0, 1)$;

$(2) \; Nx \not\in \operatorname{Im} L$ for each $x \in {\rm Ker } L \cap \partial\Omega$;

$(3) \; \deg (JQN|_{{\rm Ker } L}, \, \Omega\cap{\rm Ker } L, \, 0)\neq0$, where $Q:Z\to Z$ is a continuous projection as above with $\operatorname{Im}L = {\rm Ker } Q$ and $J:\operatorname{Im}Q\to {\rm Ker } L$ is any isomorphism.

Then the equation $Lx = Nx$ has at least one solution in $\operatorname{dom}L\cap \overline{\Omega}$.

3.   Main results

In this section, we will discuss the existence of solutions to Eq (1.3).

To obtain our main results, we impose the following condition:

$\rm {(H_0)} \; \alpha+\beta+\gamma > 2$.

We define $E = C[0, 1]$ with the norm $\|u\|_\infty = \max\limits_{0 \le t \le 1}|u(t)|$ and $X = \{u(t): u(t), u'(t), u''(t)\in E \}$ with the norm ${{\| u \|}_{X}} = \| u \|_\infty+\| u' \|_\infty+\|u''(t)\|_\infty$. It is clear that $(E, \|\cdot\|_\infty)$ and $(X, \|\cdot\|_X)$ are Banach spaces.

Define

where

Then the three-point boundary value problem (1.3) can be written as

Lemma 3.1. $L$ is defined as (3.1), then

Proof. It is obvious that the operator $L$ is linear. By $Lu = 0$ and Lemmas 2.3, we have $\big(D_{0+}^{\beta}(D_{0^+}^{\alpha}u)\big)(t) = c_{0}, c_{0}\in \mathbb{R}.$ In view of $\big(D_{0+}^{\beta}(D_{0^+}^{\alpha}u)\big)(0) = 0$, we get $c_{0} = 0$. Then, $\big(D_{0+}^{\beta}(D_{0^+}^{\alpha}u)\big)(t) = 0$. Thus, by Lemmas 2.3 again, we obtain $D_{0^+}^{\alpha}u(t) = c_{1}, c_{1}\in\mathbb{R}$. In view of $D_{0^+}^{\alpha}u(0) = 0$, we get $c_{1} = 0$. So, $D_{0^+}^{\alpha}u(t) = 0$. Then, according to $u(0) = u(\xi)$, we have $u(t) = c, c\in\mathbb{R}.$ Hence, we obtain (3.3).

Next, we prove (3.4) hold. Let $x\in {\rm Im}L$, so there exists $u\in {\rm dom}L$ such that $x(t) = \big(D_{0+}^{\gamma}(D_{0+}^{\beta}(D_{0^+}^{\alpha}u))\big)(t).$ By Lemma 2.3 and the definition of ${\rm dom}L$, we have

In view of $D_{0+}^{\beta}u(0) = \big(D_{0+}^{\beta}(D_{0^+}^{\alpha}u)\big)(0) = 0$, we get $c_0 = c_1 = 0$. Hence,

According to $u(0) = u(\xi)$, we have $I_{0+}^{\alpha+\beta+\gamma }x(\xi) = 0.$ On the other hand, suppose $x$ satisfies the above equations. Let $u(t) = I_{0+}^{\alpha+\beta+\gamma }x(t)$, we can prove $u(t)\in {\rm dom} L$ and $L u(t) = x$. Then, (3.4) holds.

For simplicity of notation, we write $p = \frac{ 1+(\alpha+\beta+ \gamma)^2 }{\Gamma(1+\alpha+\beta+ \gamma)}$.

Lemma 3.2. The mapping $L:\mathit{\text{dom}} L \subset Y\rightarrow Z$ is a Fredholm operator of index zero.

Proof. The linear continuous projector operator $P$ can be defined as

Obviously, $P^2 = P$. It is clear that

It follows from $u = u-Pu+Pu$ that $Y = {\rm Ker}P+{\rm Ker }L$. For $u\in {\rm Ker}L\cap {\rm Ker }P$, then $u = c, c\in \mathbb{R}$. Furthermore, by the definition of ${\rm Ker}P$, we have $c = 0$. Thus,

The linear operator $Q$ can be defined as

Obviously, $Qx \cong \mathbb{R}.$ For $x(t)\in E$, we have

So, the operator $Q$ is idempotent. It follows from $x = x-Qx+Qx$ that $E = {\rm Im}L+{\rm Im}Q.$ Moreover, by ${\rm Ker}Q = {\rm Im}L$ and $Q ^2 = Q$, we get ${\rm Im}L\cap {\rm Im}Q = \{0\}$. Hence,

Now, $\mathop{\rm Ind}L = \mathop{\rm dim\, {\rm Ker }}L - \mathop{\rm codim\, Im}L = 0,$ and so $L$ is a Fredholm mapping of index zero.

For every $u\in X$, we have

Furthermore, the operator $K_P:\operatorname{Im}L\to \operatorname{dom}L \cap {\rm Ker } P$ can be defined by

For $x(t)\in {\rm Im} L$, we have

On the other hand, for $u\in {\rm dom} L \cap {\rm Ker } P$, according to Lemma 2.3 and the definitions of ${\rm dom} L$ and ${\rm Ker } P$, we have

Combining (3.6) and (3.7), we have $K_P = (L_{{\rm dom} L \cap{\rm Ker } P })^{-1}$.

For $x\in {\rm Im} L$ and $\alpha+\beta+\gamma > 2$, we have

Again for $u\in \Omega_1$, $u\in {\rm dom}(L)\setminus{\rm Ker}(L)$, then $(I-P)u\in {\rm dom}L\cap {\rm Ker }P$ and $LPu = 0$, thus from (3.8), we have

With the similar proof showed in ^[13], we have the following lemma.

Lemma 3.3. $K_P(I-Q)N:Y\rightarrow Y$ is completely continuous.

Theorem 3.4. Assume the following conditions hold:

$\rm {(H_1)}$ There exist nonnegative functions $\psi(t), \varphi_0(t), \varphi_1(t), \varphi_2(t) \in E$, such that for all $t\in[0, 1]$, $(u_1, u_2, u_{3}) \in \mathbb{R}^{3}$, one has

$\rm {(H_2)}$ There exists $A > 0$ such that for $\big(t, u, u', u''\big)$, if $|u | > A$ for all $t\in [0, 1]$, one has

$\rm {(H_3)}$ There exists $B > 0$ such that if $|c | > B, c\in \mathbb{R}$, one has either

or

Then, BVP (1.3) has at least a solution in $X$ provided that

where $\varphi = \max\{\|\varphi_0(t)\| _\infty, \|\varphi_1(t)\| _\infty, \|\varphi_2(t)\| _\infty \}$.

Proof. Let

For $Lu = \lambda Nu\in {\rm Im}L = {\rm Ker }Q$, by (3.2) and the definition of ${\rm Ker }Q$, we have

According to $\rm {(H_2)}$, there exists $t_0 \in (0, 1)$ such that $|u(t_0)|\leq A$. By $L u = \lambda N u, \; u\in {\rm dom}L \setminus{\rm Ker }L$, that is $\big(D_{0+}^{\gamma}(D_{0+}^{\beta}(D_{0^+}^{\alpha}u))\big)(t) = \lambda Nu$, we have

Substituting $t = t_{0}$ into the above equation, we get

So, we have

Substituting $t = 0$ into the above equation, together with $|u (t_0)|\leq A$ and $\rm {(H_1)}$, we have

So

By (3.10) and $\rm {(H_1)}$, we have

According to (3.9), we can derive

Thus, we have $\Omega_1$ is bounded.

Let

For $u\in {\rm Ker }L$, then $u(t) = c, c\in \mathbb{R}$. In view of $Nu\in {\rm Im}L = {\rm Ker }Q$, we have $Q (Nu) = 0$, that is

By $\rm {(H_2)}$, there exist constants $t_0 \in [0, 1]$ such that

Therefore, $\Omega_2$ is bounded.

Let

For $u\in {\rm Ker }L$, then $u(t) = c_0$. By the definition of the set $\Omega_3$, we have

Next, the proof is divided into three cases.

Case 1. If $\lambda = 1$, then $c_0 = 0.$

Case 2. If $\lambda = 0$, similar to the proof of the boundness of $\Omega_2$, we have $|c_0|\leq A$.

Case 3. If $\lambda \in (0, 1)$, we also have $|c_0|\leq B$. Otherwise, if $|c_0| > B$, in view of the first part of $\rm {(H_3)}$, we obtain

which contradicts (3.13). Thus, $\Omega_3$ is bounded.

If the second part of $\rm {(H_3)}$ holds, we can prove the set

is bounded.

Let $\Omega = \{x\in X: \|x\|_X < \max\{M, A, B\}+1\}$. It follows from Lemma 3.2 and Lemma 3.3 that $L$ is a Fredholm operator of index zero and $N$ is $L-$compact on $\bar{\Omega}$. By (3.11) and (3.12), we get that the following two conditions of Theorem 2.5 are satisfied

(1). $Lu\neq \lambda Nu$, for every $u\in [({\rm dom}L\setminus {Ker}L)\cap \partial \Omega]\times(0, 1)$;

(2). $Nu\notin {\rm Im}L$ for every $u\in{\rm Ker }L\cap \partial \Omega$.

Let $H(u, \lambda) = \pm \lambda Iu+(1-\lambda)JQNu$, where $I$ is the identical operator. By the proof of the boundness of $\Omega_3$, we have that $H(u, \lambda)\neq 0$ for $u\in {\rm Ker }L\cap \partial\Omega$. Therefore, via the homotopy property of degree, we obtain

So, the condition (3) of Theorem 2.5 is satisfied. Applying Theorem 2.5, we conclude that $Lu = Nu$ has at least one solution in $\operatorname{dom}L\cap \overline{\Omega}$. Then fractional jerk equation (1.1) has at least one solution and the proof is complete.

4.   Example

Example 4.1. Let us consider the following fractional jerk equation:

where

Corresponding to BVP (1.3), we have that $\alpha = 0.8$, $\beta = 0.7$, $\gamma = 0.6$, $\alpha+\beta+\gamma = 2.1 > 2$. By calculation, we have

By a simple proof, we have

Choose $\varphi_0(t) = \frac{3}{20}$, $\varphi_1(t) = \frac{1}{10}$, $\varphi_2 = \varphi_3 = 0$, then (H$_1$) is satisfied. Furthermore,

Hence, (3.9) holds. In addition, by a simple calculation, by choosing $A = B = \frac{\sqrt{2}}{2}$, then condition (H$_2$) and (H$_3$) are satisfied. Consequently, By Theorem 3.4, BVP (4.1) has at least one solution.

5.   Conclusions

In this article, we considered a class of nonlinear fractional jerk equations with boundary value conditions at resonance. By using the theory of Mawhin's continuation theorem, we obtained the existence of solutions of boundary value problems (1.1). An example is presented to illustrate the main results. To the authors' knowledge, the existence of solutions for fractional jerk BVPs at resonance has not been reported. So, it is novel in term of fractional jerk equations and our result enriches and extends some related results in the literature. There is still more work to be done in the future on this interesting problem. For example, seeking the existence of positive solutions for fractional jerk equations with resonant boundary conditions.

Acknowledgments

This research was supported by the Research Fund for the Doctoral Program of Shandong Jiaotong University, Shandong Jiaotong University “Climbing” Research Innovation Team Program (No.SDJTUC1803), a project of Shandong Province Higher Educational Science and Technology Program (No.J17KA073, J18KA330), the key research and development project of Shandong Province (No.2018GGX105011, No.2019GGX101055) and University outstanding youth innovation team development plan of Shandong Province (No.2019KJN023). This research was also funded by the National Natural Science Foundation of China (11671181, 61803230) and the Fundamental Research Funds for the Central Universities (2020YJSLX01).

Conflict of interest

The authors declare no conflicts of interest in this paper.