Exact Solutions and Hyers-Ulam Stability of the Nonhomogeneous Riemann-Liouville Fractional Oscillatory Differential Equations with Pure Delay

In this paper, to overcome certain difficulties encountered in previous studies, we first investigate a formula for the Laplace transform of the Riemann-Liouville derivative with a lower limit $a$ . We then obtain the exact solutions of nonhomogeneous Riemann-Liouville fractional differential equations with pure delay. In addition, this paper explores the Hyers-Ulam stability of nonhomogeneous fractional oscillatory differential equations with pure delay.

• Riemann-Liouville Fractional Derivative
• Hyers-Ulam Stability
• Laplace Transform

1. Introduction and Preliminaries

There have been many studies on the well-posedness and Hyers-Ulam stability of PDE [1] [2] . In recent decades, the solutions of the Cauchy problem with delay have been studied by many authors [3] [4] . Many researchers have focused on the stability and controllability study of solutions of fractional Caputo differential equations with delay [5] - [11] and fewer researchers have studied fractional Riemann-Liouville differential equations with delay [12] - [15] . The stability theory of fractional order differential equations allows the solution to remain accurate to $\delta$ ( $\delta =C\epsilon$) in the presence of a perturbation in the right end term of the equation. For time-lagged systems, this stability avoids the amplifying effect of errors in the time-lag loop. For example, in control system design, it ensures that sensor noise does not trigger violent oscillations in the control commands.

Fan and Pan [16] gave the solutions and the finite time stability of Equation (1) by using the Mittag-Leffler matrix-type functions and the constant variational method in 2023. The authors stated that it was difficult to compute the solutions of system (1) directly by using the Laplace transform, because they didn’t find a suitable formula for the Laplace transform of the Riemann-Liouville fractional derivative with the lower limit $a$. Motivated to the difficulty left over from the literature [16] , we further study the nonhomogeneous Riemann-Liouville fractional oscillatory differential equations with pure delay of the following form

$\{\begin{array}{ll}{}^R{D}_{-{\tau }^{+}}^{\alpha }y\left(t\right)=\beta y\left(t-\tau \right)+f\left(t\right),\hfill & t\in J:=\left(0,T\right],\tau >0,\hfill \\ y\left(t\right)=\phi \left(t\right),\hfill & -\tau <t\le 0,\hfill \\ J_{-{\tau }^{+}}^{2-\alpha }y\left(-{\tau }^{+}\right)=m,{}^R{D}_{-{\tau }^{+}}^{\alpha -1}y\left(-{\tau }^{+}\right)=n,\hfill & \hfill \end{array}$ (1)

where $\alpha \in \left(1,2\right)$, $f:J\to {ℝ}^n$ is a continuous function, $T=C\tau$ for a fixed $C\in \left\{1,2,\cdots \right\}$, $\beta$ denotes a constant matrix, ${}^R{D}_{-{\tau }^{+}}^{\alpha }$ and ${}^R{J}_{-{\tau }^{+}}^{\alpha }$ are, respectively, the Riemann-Liouville fractional derivativeand integral of $\alpha \in \left(1,2\right)$ with the lower limit $-{\tau }^{+}$ defined by

${}^R{D}_{-{\tau }^{+}}^{\alpha }f\left(t\right)=\frac{1}{\text{Γ}\left(2-\alpha \right)}\frac{{\text{d}}^2}{\text{d}{t}^2}{\displaystyle {\int }_{-\tau }^t{\left(t-s\right)}^{1-\alpha }f\left(s\right)\text{d}s},t>-\tau .$

${}^R{J}_{-{\tau }^{+}}^{\alpha }f\left(t\right)=\frac{1}{\text{Γ}\left(\alpha \right)}{\displaystyle {\int }_{-\tau }^t{\left(t-s\right)}^{\alpha -1}f\left(s\right)\text{d}s},t>-\tau .$

To obtain the exact solutions of system (1), we firstly study the exact solutions of the homogeneous fractional delay differential equation

$\{\begin{array}{ll}{}^R{D}_{-{\tau }^{+}}^{\alpha }y\left(t\right)=\beta y(t-\tau ),\hfill & t\in J:=\left(0,T\right],\tau >0,\hfill \\ y\left(t\right)=\phi \left(t\right),\hfill & -\tau <t\le 0,\hfill \\ J_{-{\tau }^{+}}^{2-\alpha }y\left(-{\tau }^{+}\right)=m,{}^R{D}_{-{\tau }^{+}}^{\alpha -1}y\left(-{\tau }^{+}\right)=n.\hfill & \hfill \end{array}$ (2)

Then, by means of the generalized Mittag-Leffler type matrix functions and the Laplace transform, we will solve the solutions of Equation (1) and derive its Hyers-Ulam stability.

Currently, we introduce two delayed Mittag-Leffler type matrix functions. We use the norm $\Vert y\Vert ={\displaystyle {\sum }_{i=1}^n\left|y_i\right|}$, $y\in {ℝ}^{n\times 1}$ and the matrix norm $\Vert \beta \Vert ={\text{max}}_{1\le j\le n}{\displaystyle {\sum }_{i=1}^n\left|{\beta }_{ij}\right|}$, $\beta \in {ℝ}^{n\times n}$.

Definition 1. (See [16] ) The delayed Mittag-Leffler type matrix function ${\mathcal{Q}}_{\alpha }^{\tau }\left(t\right):ℝ\to {ℝ}^{n\times n}$ is defined as

${\mathcal{Q}}_{\alpha }^{\tau }\left(t\right)=\{\begin{array}{ll}\theta ,\hfill & -\infty <t\le -\tau ,\hfill \\ I\frac{{\left(t+\tau \right)}^{\alpha -1}}{\text{Γ}\left(\alpha \right)},\hfill & -\tau <t\le 0,\hfill \\ I\frac{{\left(t+\tau \right)}^{\alpha -1}}{\text{Γ}\left(\alpha \right)}+\beta \frac{t^{2\alpha -1}}{\text{Γ}\left(2\alpha \right)},\hfill & 0<t\le \tau ,\hfill \\ ⋮\hfill & ⋮\hfill \\ I\frac{{\left(t+\tau \right)}^{\alpha -1}}{\text{Γ}\left(\alpha \right)}+\beta \frac{t^{2\alpha -1}}{\text{Γ}\left(2\alpha \right)}+{\beta }^2\frac{{\left(t-\tau \right)}^{3\alpha -1}}{\text{Γ}\left(3\alpha \right)}+\cdots +{\beta }^k\frac{{\left(t-\left(k-1\right)\tau \right)}^{\left(k+1\right)\alpha -1}}{\text{Γ}\left(\left(k+1\right)\alpha \right)},\hfill & \left(k-1\right)\tau <t\le k\tau .\hfill \end{array}$

Definition 2. (See [16] ) The delayed Mittag-Leffler type matrix function ${\mathcal{W}}_{\alpha }^{\tau }\left(t\right):ℝ\to {ℝ}^{n\times n}$ is defined as

${\mathcal{W}}_{\alpha }^{\tau }\left(t\right)=\{\begin{array}{ll}\theta ,\hfill & -\infty <t\le -\tau ,\hfill \\ I\frac{{\left(t+\tau \right)}^{\alpha -2}}{\text{Γ}\left(\alpha -1\right)},\hfill & -\tau <t\le 0,\hfill \\ I\frac{{\left(t+\tau \right)}^{\alpha -2}}{\text{Γ}\left(\alpha -1\right)}+\beta \frac{t^{2\alpha -2}}{\text{Γ}\left(2\alpha -1\right)},\hfill & 0<t\le \tau ,\hfill \\ ⋮\hfill & ⋮\hfill \\ I\frac{{\left(t+\tau \right)}^{\alpha -2}}{\text{Γ}\left(\alpha -1\right)}+\beta \frac{t^{2\alpha -2}}{\text{Γ}\left(2\alpha -1\right)}+{\beta }^2\frac{{\left(t-\tau \right)}^{3\alpha -2}}{\text{Γ}\left(3\alpha -1\right)}+\cdots +{\beta }^k\frac{{\left(t-\left(k-1\right)\tau \right)}^{\left(k+1\right)\alpha -2}}{\text{Γ}\left(\left(k+1\right)\alpha -1\right)},\hfill & \left(k-1\right)\tau <t\le k\tau .\hfill \end{array}$

Lemma 1. (See [16] ) For ${\mathcal{Q}}_{\alpha }^{\tau }\left(t\right)$ and ${\mathcal{W}}_{\alpha }^{\tau }\left(t\right)$, we have

i) $\frac{\text{d}}{\text{d}t}{\mathcal{Q}}_{\alpha }^{\tau }\left(t\right)={\mathcal{W}}_{\alpha }^{\tau }\left(t\right)$ for all $t\in ℝ\backslash \left\{-\tau \right\}$.

ii) ${\mathcal{Q}}_{\alpha }^{\tau }\left(t\right)$ is a solution of Eq.(2), which meets starting conditions ${\mathcal{Q}}_{\alpha }^{\tau }\left(t\right)=I\frac{{\left(t+\tau \right)}^{\alpha -1}}{\text{Γ}\left(\alpha \right)}$, $-\tau <t\le 0$, ${\mathcal{J}}_{-{\tau }^{+}}^{2-\alpha }{\mathcal{Q}}_{\alpha }^{\tau }\left(-{\tau }^{+}\right)=0$. and ${}^R{\mathcal{D}}_{-{\tau }^{+}}^{\alpha -1}{\mathcal{Q}}_{\alpha }^{\tau }\left(-{\tau }^{+}\right)=1$.

iii) ${\mathcal{W}}_{\alpha }^{\tau }\left(t\right)$ is a solution of Eq. (2), which meets starting conditions ${\mathcal{W}}_{\alpha }^{\tau }\left(t\right)=I\frac{{\left(t+\tau \right)}^{\alpha -2}}{\text{Γ}\left(\alpha -1\right)}$, $-\tau <t\le 0$, ${\mathcal{J}}_{-{\tau }^{+}}^{2-\alpha }{\mathcal{W}}_{\alpha }^{\tau }\left(-{\tau }^{+}\right)=1$ and ${}^R{\mathcal{D}}_{-{\tau }^{+}}^{\alpha -1}{\mathcal{W}}_{\alpha }^{\tau }\left(-{\tau }^{+}\right)=0$.

Lemma 2. (See [13] ) If the Laplace transform of $f_1\left(t\right)=f\left(t-a\right)$ exists for $t\ge a$. Then we find

$ℒ\left(f_1\left(t\right)\right)\left(s\right)={\text{e}}^{-as}{\displaystyle {\int }_{-a}^0{\text{e}}^{-ts}f\left(t\right)\text{d}t}+{\text{e}}^{-as}ℒ\left(f\left(t\right)\right)\left(s\right).$

Lemma 3. When $f\left(t\right)$ is integrable on $\left[a,+\infty \right)$, the Laplace transform of $f\left(t\right)$ is defined by $F\left(s\right)=ℒ\left(f\left(t\right)\right)\left(s\right)={\displaystyle {\int }_a^{\infty }{\text{e}}^{-st}f\left(t\right)\text{d}t}$, we obtain

$ℒ\left(f^{\left(n\right)}\left(t\right)\right)\left(s\right)=s^nℒ\left(f\left(t\right)\right)\left(s\right)-\underset{k=0}{\overset{n-1}{{\displaystyle \sum }}}{s}^k{\text{e}}^{-sa}{f}^{\left(n-k-1\right)}\left(a\right).$ (3)

Proof. We prove it with the help of mathematical induction method. When $n=1$, we get

$ℒ\left(f^{\prime }\left(t\right)\right)\left(s\right)={\displaystyle {\int }_a^{\infty }{\text{e}}^{-st}{f}^{\prime }\left(t\right)\text{d}t}=sℒ\left(f\left(t\right)\right)\left(s\right)-{\text{e}}^{-sa}f\left(a\right).$

Assuming Equation (1.3) holds when $n=k$, that is $ℒ\left(f^{\left(k\right)}\left(t\right)\right)\left(s\right)=s^kℒ\left(f\left(t\right)\right)\left(s\right)-s^{k-1}{\text{e}}^{-sa}f\left(a\right)-\cdots -s{\text{e}}^{-sa}{f}^{\left(k-2\right)}\left(a\right)-{\text{e}}^{-sa}{f}^{\left(k-1\right)}\left(a\right)$.

Let $n=k+1$, we have

$\begin{array}{c}ℒ\left(f^{\left(k+1\right)}\left(t\right)\right)\left(s\right)=ℒ\left(\frac{\text{d}}{\text{d}t}{f}^{\left(k\right)}\left(t\right)\right)\left(s\right)=sℒ\left(f^k\left(t\right)\right)\left(s\right)-{\text{e}}^{-sa}f\left(a\right)\\ =s^{k+1}ℒ\left(f\left(t\right)\right)\left(s\right)-s^k{\text{e}}^{-sa}f\left(a\right)-\cdots -s{\text{e}}^{-sa}{f}^{\left(k-1\right)}\left(a\right)-{\text{e}}^{-sa}f\left(a\right).\end{array}$

Thus, Lemma 3 is proved.

Utilizing the Lemma 3 and the method of proof in Section 2.8.2 of literature [17] , we get

Lemma 4. The Laplace integral transform of Riemann-Liouville fractional derivative with the lower limit $a$ is given by

$ℒ\left({}^R{D}_a^{\alpha }f\left(t\right)\right)\left(s\right)=s^{\alpha }ℒ\left(f\left(t\right)\right)\left(s\right)-\underset{k=0}{\overset{n-1}{{\displaystyle \sum }}}{s}^k{\text{e}}^{-sa}\left[{}^R{D}_a^{\alpha -k-1}f\left(a\right)\right].$ (4)

Proof. The Laplace transform of the function $t^{\alpha -1}$ is

$G\left(s\right)=ℒ\left(t^{\alpha -1}\right)\left(s\right)=\text{Γ}\left(\alpha \right)s^{-\alpha },$ (5)

then, by applying the Laplace transform formula of the convolution equation, we derive the Laplace transform of the Riemann-Liouville fractional integral

$ℒ\left(J_a^{\alpha }\right)\left(s\right)=s^{-\alpha }F\left(s\right).$ (6)

Next we explore for the Laplace transform of the Riemannian-Liouville fractional order derivative, and for this purpose we write the following form

$J_a^n{D}_a^{\alpha }f\left(t\right)=J_a^n{g}^{\left(\alpha \right)}\left(t\right),$

$g\left(t\right)=J_a^{n-\alpha }f\left(t\right)=\frac{1}{n-\alpha }{\displaystyle {\int }_a^t{\left(t-\tau \right)}^{n-\alpha -1}f\left(\tau \right)\text{d}\tau },n-1\le \alpha <n.$

According to the Lemma 3, we obtain

$ℒ\left(D_a^{\alpha }f\left(t\right)\right)\left(s\right)=s^{n}G\left(s\right)-\underset{k=0}{\overset{n-1}{{\displaystyle \sum }}}{s}^{k}g\left(n-\alpha -1\right)\left(a\right),$ (7)

the Laplace transform of $g\left(t\right)$ follows from (6)

$G\left(s\right)=s^{-\left(n-\alpha \right)}F\left(s\right).$ (8)

Furthermore, based on the definition of Riemann-Liouville fractional derivative, we derive the following result

$g^{\left(n-k-1\right)}\left(t\right)=\frac{{\text{d}}^{n-k-1}}{\text{d}{t}^{n-k-1}}{J}_a^{n-\alpha }f\left(t\right)=D_a^{\alpha -k-1}f\left(t\right),$ (9)

then, substituting (8) and (9) into (7), we have

$ℒ\left({}^R{D}_a^{\alpha }f\left(t\right)\right)\left(s\right)=s^{\alpha }ℒ\left(f\left(t\right)\right)\left(s\right)-\underset{k=0}{\overset{n-1}{{\displaystyle \sum }}}{s}^k{\text{e}}^{-sa}{\left[{}^R{D}_a^{\alpha -k-1}f\left(t\right)\right]}_{t=a}.$

Lemma 5. (See [16] ) Assuming $1<\alpha <2,\tau >0$, the solution of system (2) can be expressed as

$u\left(t\right)={\mathcal{Q}}_{\alpha }^{\tau }\left(t\right)n+{\mathcal{W}}_{\alpha }^{\tau }\left(t\right)m+{\displaystyle {\int }_{-\tau }^0{\mathcal{Q}}_{\alpha }^{\tau }}\left(t-\tau -\xi \right)\left({}^R{D}_{-{\tau }^{+}}^{\alpha }\phi \right)\left(\xi \right)\text{d}\xi .$ (10)

Lemma 6. (Gronwall’s inequality) (See [18] ) Let $y\left(t\right)$ be a nonnegetive summable function on $\left(0,Q\right]$, and $y\left(t\right)\le C_1{\displaystyle {\int }_0^{t}y\left(s\right)\text{d}s}+C_2$ for a.e. $t\in \left(0,Q\right]$ and constants $C_1,C_2\ge 0$. Then $y\left(t\right)\le C_2\left(1+C_{1}t{\text{e}}^{C_{1}t}\right)$, for a.e. $0<t\le Q$.

Definition 3. The system (1) is Hyers-Ulam stable on $\left(0,T\right]$ if there is, for a given constant $\epsilon >0$, a function $\bar{y}\left(t\right)$ satisfying the inequality

$\Vert {}^R{D}_{-{\tau }^{+}}^{\alpha }\bar{y}\left(t\right)-\beta \bar{y}\left(t-\tau \right)-f\left(t\right)\Vert \le \epsilon ,t\in \left(0,T\right],$ (11)

and satisfying conditions of system (1), there exists a solution $y\left(t\right)$ of problem (1) and a number $C>0$ such that $\Vert \bar{y}\left(t\right)-y\left(t\right)\Vert \le C\epsilon$ for every $t\in \left(0,T\right]$.

2. Exact solutions

Lemma 7. The solution of the problem (1) satisfies

$ℒ\left(y\left(t\right)\right)\left(s\right)={\left(s^{\alpha }-\beta {\text{e}}^{-\tau s}\right)}^{-1}\left({\mathcal{A}}_{\tau }\left(s\right)+ℒ\left(f\left(t\right)\right)\left(s\right)\right),$

where ${\mathcal{A}}_{\tau }\left(s\right)=\beta {\text{e}}^{-\tau s}{\displaystyle {\int }_{-\tau }^0{\text{e}}^{-sz}\phi \left(z\right)\text{d}z}$.

Proof. Performing the Laplace transform on both sides of the system (1), according to (4) and Lemma 2, we get

$ℒ\left({}^R{D}_{-{\tau }^{+}}^{\alpha }y\left(t\right)\right)\left(s\right)=\beta ℒ\left(y\left(t-\tau \right)\right)\left(s\right)+ℒ\left(f\left(t\right)\right)\left(s\right),$

then

$s^{\alpha }ℒ\left(y\left(t\right)\right)\left(s\right)=\beta \left({\text{e}}^{-\tau s}{\displaystyle {\int }_{-\tau }^0{\text{e}}^{-sz}y\left(z\right)\text{d}z}+{\text{e}}^{-s\tau }ℒ\left(y\left(t\right)\right)\left(s\right)\right)+ℒ\left(f\left(t\right)\right)\left(s\right),$

hence

$ℒ\left(y\left(t\right)\right)\left(s\right)={\left(s^{\alpha }-\beta {\text{e}}^{-\tau s}\right)}^{-1}\left({\mathcal{A}}_{\tau }\left(s\right)+ℒ\left(f\left(t\right)\right)\left(s\right)\right).$

The proof is completed.

Corollary 1. The solution of problem (2) satisfies

$ℒ\left(u\left(t\right)\right)\left(s\right)={\left(s^{\alpha }-\beta {\text{e}}^{-\tau s}\right)}^{-1}\left({\mathcal{A}}_{\tau }\left(s\right)\right).$

Lemma 8. Let $1<\alpha <2,\tau >0$, we have $ℒ\left({\mathcal{Q}}_{\alpha }^{\tau }\left(t-\tau \right)\right)\left(s\right)={\left(s^{\alpha }I-\beta {\text{e}}^{-\tau s}\right)}^{-1}I$.

Proof. According to the Lemma 1 (ii) and Corollary 1, we find

$\begin{array}{c}ℒ\left({\mathcal{Q}}_{\alpha }^{\tau }\left(t\right)\right)\left(s\right)={\left(s^{\alpha }I-\beta {\text{e}}^{-s\tau }\right)}^{-1}\cdot \beta {\text{e}}^{-s\tau }{\displaystyle {\int }_{-\tau }^0{\text{e}}^{-sz}\frac{{\left(z+\tau \right)}^{\alpha -1}}{\text{Γ}\left(\alpha \right)}\text{d}z}\\ ={\left(s^{\alpha }I-\beta {\text{e}}^{-s\tau }\right)}^{-1}\cdot \beta {\displaystyle {\int }_{-\tau }^0{\text{e}}^{-s\left(z+\tau \right)}\frac{{\left(z+\tau \right)}^{\alpha -1}}{\text{Γ}\left(\alpha \right)}\text{d}z}\\ ={\left(s^{\alpha }I-\beta {\text{e}}^{-s\tau }\right)}^{-1}\cdot \frac{\beta }{\text{Γ}\left(\alpha \right)}{\displaystyle {\int }_{-\tau }^0{\text{e}}^{-s\left(z+\tau \right)}{\left(z+\tau \right)}^{\alpha -1}\text{d}z}\\ ={\left(s^{\alpha }I-\beta {\text{e}}^{-s\tau }\right)}^{-1}\cdot \frac{\beta }{s^{\alpha }}I.\end{array}$

Combining this with the Lemma 2, we have

$\begin{array}{c}ℒ\left({\mathcal{Q}}_{\alpha }^{\tau }\left(t-\tau \right)\right)\left(s\right)={\text{e}}^{-s\tau }{\displaystyle {\int }_{-\tau }^0{\text{e}}^{-sz}\frac{{\left(z+\tau \right)}^{\alpha +1}}{\text{Γ}\left(\alpha \right)}\text{d}z}+{\text{e}}^{-s\tau }ℒ\left({\mathcal{Q}}_{\alpha }^{\tau }\left(t\right)\right)\left(s\right)\\ =\frac{I}{\text{Γ}\left(\alpha \right)}{\text{e}}^{-s\tau }{\displaystyle {\int }_{-\tau }^0{\text{e}}^{-sz}{\left(z+\tau \right)}^{\alpha -1}\text{d}z}+{\text{e}}^{-s\tau }\cdot {\left(s^{\alpha }I-\beta {\text{e}}^{-s\tau }\right)}^{-1}\cdot \frac{\beta }{s^{\alpha }}\\ ={\left(s^{\alpha }I-\beta {\text{e}}^{-s\tau }\right)}^{-1}I.\end{array}$

This proof is completed.

Theorem 1. The fractional delay differential equation (1) has a unique solution $y\left(t\right)$, and $y\left(t\right)$ can be written in this form

$y\left(t\right)={\mathcal{Q}}_{\alpha }^{\tau }\left(t\right)n+{\mathcal{W}}_{\alpha }^{\tau }\left(t\right)m+{\displaystyle {\int }_{-\tau }^0{\mathcal{Q}}_{\alpha }^{\tau }}\left(t-\tau -\xi \right)\left({}^R{D}_{-\tau +}^{\alpha }\phi \right)\left(\xi \right)\text{d}\xi +{\mathcal{Q}}_{\alpha }^{\tau }\left(t-\tau \right)\ast f\left(t\right).$ (12)

Proof. The implicit form of this solution is presented by the lemma 6 by using the Laplace transformation. Now all that is needed is to solve the Laplace inverse transformation of this

${ℒ}^{-1}\left({\left(s^{\alpha }I-\beta {\text{e}}^{-sz}\right)}^{-1}\left(\beta {\text{e}}^{-s\tau }{\displaystyle {\int }_{-\tau }^0{\text{e}}^{-sz}\phi \left(z\right)\text{d}z}+ℒ\left(f\left(t\right)\right)\left(s\right)\right)\right)\left(t\right).$

By Corollary 1, we need only discuss ${ℒ}^{-1}\left({\left(s^{\alpha }I-\beta {\text{e}}^{-s\tau }\right)}^{-1}ℒ\left(f\left(t\right)\right)\left(s\right)\right)\left(t\right)$

$\begin{array}{c}{ℒ}^{-1}\left({\left(s^{\alpha }I-\beta {\text{e}}^{-s\tau }\right)}^{-1}ℒ\left(f\left(t\right)\right)\left(s\right)\right)\left(t\right)={ℒ}^{-1}\left(ℒ\left({\mathcal{Q}}_{\tau }^{\alpha }\left(t-\tau \right)\right)\left(s\right)ℒ\left(f\left(t\right)\right)\left(s\right)\right)\left(t\right)\\ ={ℒ}^{-1}\left(ℒ\left({\mathcal{Q}}_{\tau }^{\alpha }\left(t-\tau \right)\ast f\left(t\right)\right)\left(s\right)\right)\left(t\right)\\ ={\mathcal{Q}}_{\alpha }^{\tau }\left(t-\tau \right)\ast f\left(t\right).\end{array}$ (13)

Hence

$y\left(t\right)={\mathcal{Q}}_{\alpha }^{\tau }\left(t\right)n+{\mathcal{W}}_{\alpha }^{\tau }\left(t\right)m+{\displaystyle {\int }_{-\tau }^0{\mathcal{Q}}_{\alpha }^{\tau }}\left(t-\tau -\xi \right)\left({}^R{D}_{-\tau +}^{\alpha }\phi \right)\left(\xi \right)\text{d}\xi +{\mathcal{Q}}_{\alpha }^{\tau }\left(t-\tau \right)\ast f\left(t\right).$

This proof is completed.

3. Hyers-Ulam stability

Theorem 2. Let $\beta \in {ℝ}^{n\times n}$, $\alpha \in \left(1,2\right)$ and $f\left(t\right)$ is a function defined on ${ℝ}^n$, then the system (1) is Hyers-Ulam stable on $\left(0,T\right]$.

Proof. Let $\bar{y}\left(t\right)$ satisfy the inequality (11) and the conditions of system (1). Putting

$\mathcal{K}\left(t\right)={}^R{D}_{-{\tau }^{+}}^{\alpha }\bar{y}\left(t\right)-\beta \bar{y}\left(t-\tau \right)-f\left(t\right),t\in \left(0,T\right].$ (14)

From the Definition 3, it follows that $\Vert \mathcal{K}\left(t\right)\Vert \le \epsilon$. Using the Laplace transform for the left and right sides of (14), we have

$ℒ\left(\mathcal{K}\left(t\right)\right)\left(s\right)=\left(s^{\alpha }I-\beta {\text{e}}^{-\tau s}\right)ℒ\left(\bar{y}\left(t\right)\right)\left(s\right)-\beta {\text{e}}^{-\tau s}{\displaystyle {\int }_{-\tau }^0{\text{e}}^{-sz}\phi \left(z\right)\text{d}z}-ℒ\left(f\left(t\right)\right)\left(s\right).$

Thus

$ℒ\left(\bar{y}\left(t\right)\right)\left(s\right)=\frac{\beta {\text{e}}^{-s\tau }{\displaystyle {\int }_{-\tau }^0{\text{e}}^{-sz}\phi \left(z\right)\text{d}z}+ℒ\left(f\left(t\right)\right)\left(s\right)}{s^{\alpha }I-\beta {\text{e}}^{-\tau s}}+\frac{ℒ\left(\mathcal{K}\left(t\right)\right)\left(s\right)}{s^{\alpha }I-\beta {\text{e}}^{-\tau s}}.$

According to Theorem 1, we find a solution $y\left(t\right)$ of equation (1). By the Lemma 6, we have

$ℒ\left(\bar{y}\left(t\right)-y\left(t\right)\right)\left(s\right)=\frac{ℒ\left(\mathcal{K}\left(t\right)\right)\left(s\right)}{s^{\alpha }I-\beta {\text{e}}^{-s\tau }}.$

Using the convolution theorem and formula (13), we obtain

$\frac{ℒ\left(\mathcal{K}\left(t\right)\right)\left(s\right)}{s^{\alpha }I-\beta {\text{e}}^{-s\tau }}=ℒ\left(Q_{\alpha }^{\tau }\left(t-\tau \right)\right)\left(s\right)\cdot ℒ\left(\mathcal{K}\left(t\right)\right)\left(s\right)=ℒ\left(Q_{\alpha }^{\tau }\left(t-\tau \right)\ast \mathcal{K}\left(t\right)\right)\left(s\right).$

Hence, we have $\bar{y}\left(t\right)-y\left(t\right)=Q_{\alpha }^{\tau }\left(t-\tau \right)\ast \mathcal{K}\left(t\right)$. For any $t\in \left(k\tau ,\left(k+1\right)\tau \right]$, $k=0,1,2,\cdots ,N-1$, we get

$\begin{array}{c}\Vert \bar{y}\left(t\right)-y\left(t\right)\Vert =\Vert {\displaystyle {\int }_0^t\mathcal{K}}\left(t-\xi \right)Q_{\alpha }^{\tau }\left(\xi -\tau \right)\text{d}\xi \Vert \\ \le \epsilon {\displaystyle {\int }_0^t\Vert Q_{\alpha }^{\tau }\left(\xi -\tau \right)\Vert \text{d}\xi }\\ =\epsilon {\displaystyle {\int }_0^t\Vert \underset{j=1}{\overset{k}{{\displaystyle \sum }}}\frac{{\beta }^j{\left(\xi -k\tau \right)}^{\left(j+1\right)\alpha -1}}{\text{Γ}\left(\left(j+1\right)\alpha \right)}\Vert \text{d}\xi }.\end{array}$

Let $C={\max }_{0<t\le T}{\displaystyle {\int }_0^t\Vert Q_{\alpha }^{\tau }\left(\xi -\tau \right)\Vert \text{d}\xi }$, then $\Vert \bar{y}\left(t\right)-y\left(t\right)\Vert \le C\epsilon$. Thus the system (1) has Hyers-Ulam stability.

Hypothesis 1. Assume that $g\left(t\right)$ is a non-negative function and belongs to $L\left(0,T\right]$. Then there exists $W_g>0$, such that

$\frac{1}{\text{Γ}\left(\alpha \right)}{\displaystyle {\int }_0^t\frac{g\left(s\right)}{{\left(t-s\right)}^{1-\alpha }}\text{d}s}\le W_{g}g\left(t\right),t\in J.$

Theorem 3. Let $\Vert {}^R{D}_{-{\tau }^{+}}^{\alpha }\bar{y}\left(t\right)-\beta \bar{y}\left(t-\tau \right)-f\left(t\right)\Vert \le \epsilon g\left(t\right)$, $t\in \left(0,T\right]$, where $\epsilon >0$. Assuming that there exists a number $M>0$ such that $\beta {\left(t-s\right)}^{\alpha -1}\le M$, then $\exists$ a solution $z\left(t\right)\in J$ of (1) with

$\left|y\left(t\right)-z\left(t\right)\right|\le \epsilon K\left(t\right),$

then the system (1) is the generalized Ulam-Hyers stable.

Proof. Since

$-\epsilon g\left(t\right)\le {}^R{D}_{-{\tau }^{+}}^{\alpha }y\left(t\right)-\beta y\left(t-\tau \right)-f\left(t\right)\le \epsilon g\left(t\right),$

then, we get

$-\epsilon \frac{1}{\text{Γ}\left(\alpha \right)}{\displaystyle {\int }_0^t\frac{g\left(s\right)}{{\left(t-s\right)}^{1-\alpha }}\text{d}s}\le y\left(t\right)-\beta J^{\alpha }y\left(t-\tau \right)-J^{\alpha }f\left(t\right)\le \epsilon \frac{1}{\text{Γ}\left(\alpha \right)}{\displaystyle {\int }_0^t\frac{g\left(s\right)}{{\left(t-s\right)}^{1-\alpha }}\text{d}s}.$

For the following equation ${}^R{D}_{-{\tau }^{+}}^{\alpha }z\left(t\right)=\beta z\left(t-\tau \right)+f\left(t\right)$, we obtain $z\left(t\right)=\beta J^{\alpha }z\left(t-\tau \right)+J^{\alpha }f\left(t\right)$. Therefore,

$\begin{array}{c}\left|y\left(t\right)-z\left(t\right)\right|=\left|y\left(t\right)-\beta J^{\alpha }y\left(t-\tau \right)-J^{\alpha }f\left(t\right)+\beta J^{\alpha }\left(y-z\right)\left(t-\tau \right)\right|\\ \le \epsilon W_{g}g\left(t\right)+\left|\beta J^{\alpha }\left(y-z\right)\left(t-\tau \right)\right|\\ \le \epsilon W_{g}g\left(t\right)+\frac{1}{\text{Γ}\left(\alpha \right)}{\displaystyle {\int }_0^t\left|\beta {\left(t-s\right)}^{\alpha -1}\left[y\left(s\right)-z\left(s\right)\right]\right|\text{d}s}\\ \le \epsilon W_{g}g\left(t\right)+\frac{M}{\text{Γ}\left(\alpha \right)}{\displaystyle {\int }_0^t\left|y\left(s\right)-z\left(s\right)\right|\text{d}s}.\end{array}$

Based on the Gronwall’s inequality, we have the following estimates $\left|y\left(t\right)-z\left(t\right)\right|\le \epsilon C_2\left(1+C_1{\text{e}}^{C_{1}t}\right)$, where $C_2=W_{g}g\left(t\right)$, $C_1=\frac{M}{\text{Γ}\left(\alpha \right)}>0$. Thus, the system (1) is the generalized Ulam-Hyers stable.

4. Example

This section will confirm the soundness of the theoretical results with a concrete example.

Example. Assume that $\alpha =1.8,\tau =0.15,k=4$ and $T=0.6$. Consider

$\{\begin{array}{ll}{}^R{D}_{-{0.15}^{+}}^{1.8}y\left(t\right)=\beta y\left(t-0.15\right)+f\left(t\right),\hfill & 0<t\le 0.6,\tau >0,\hfill \\ \phi \left(t\right)={\left({\left(t+0.15\right)}^2,\frac{{\left(t+0.15\right)}^3}{2}\right)}^{\text{T}},\hfill & 0.15<t\le 0,\hfill \\ J_{-{0.15}^{+}}^{0.2}Y\left(-{0.15}^{+}\right)=m=0,\hfill & \hfill \\ {}^R{D}_{-{0.15}^{+}}^{0.8}Y\left(-{0.15}^{+}\right)=n=0,\hfill & \hfill \end{array}$ (15)

where $y\left(t\right)={\left(y_1\left(t\right),y_2\left(t\right)\right)}^{\text{T}}$, $\beta =\left[\begin{array}{cc}0.45& 0\\ 0& 0.5\end{array}\right]$, $f\left(t\right)={\left(\frac{t^2}{2},t^3\right)}^{\text{T}}$.

According to Theorem 3, the solution of system (7) will exhibit a particular form when the time parameter $t$ is in the interval $\left(0,0.6\right]$.

$\begin{array}{c}y\left(t\right)=Q_{1.8}^{0.15}\left(t\right)\cdot n+W_{1.8}^{0.15}\left(t\right)\cdot m+{\displaystyle {\int }_{-0.15}^0{Q}_{1.8}^{0.15}}\left(t-0.15-\xi \right)\left({}^R{D}_{-0.15+}^{1.8}\phi \right)\left(\xi \right)\text{d}\xi \\ +Q_{1.8}^{0.15}\left(t-0.15\right)\ast f\left(t\right),\end{array}$

where

$\begin{array}{l}{\displaystyle {\int }_{-0.15}^0{Q}_{1.8}^{0.15}}\left(t-0.15-\xi \right)\left({}^R{D}_{-0.15+}^{1.8}\phi \right)\left(\xi \right)\text{d}\xi \\ ={\displaystyle {\int }_{-0.15}^0{Q}_{1.8}^{0.15}}\left(t-0.15-\xi \right)\cdot \left(\begin{array}{l}\frac{2.64}{\Gamma \left(0.2\right)}{\left(t+0.15\right)}^{0.2}B\left[3,0.2\right]\hfill \\ \frac{3.52}{\Gamma \left(0.2\right)}{\left(t+0.15\right)}^{1.2}B\left[4,0.2\right]\hfill \end{array}\right)\text{d}\xi .\end{array}$

Let $C={\max }_{0<t\le 0.6}{\displaystyle {\int }_0^t\Vert Q_{1.8}^{0.15}\left(\xi -0.15\right)\Vert \text{d}\xi }=0.002207$, then

$\left|y\left(t\right)-z\left(t\right)\right|\le \epsilon {\displaystyle {\int }_0^t\Vert Q_{\alpha }^{\tau }\left(\xi -\tau \right)\Vert \text{d}\xi }\le 0.002207\epsilon .$

That is to say, the system (1) is Hyers-Ulam stable on $\left(0,0.6\right]$.

5. Conclusion

In this paper, we discuss the Laplace transform of the higher-order derivative that is integrable on $\left[a,+\infty \right)$ and the Riemann-Liouville derivatives with a lower limit $a$. Based on these transforms and the generalized delayed Mittag-Leffler type matrix functions, we study the exact solutions of the nonhomogeneous Riemann-Liouville fractional oscillatory differential equation of order $\alpha \in \left(1,2\right)$ by using the Laplace transform technique. And explores the Hyers-Ulam stability of the nonhomogeneous fractional oscillatory differential equation with pure delay. We successfully solved the difficulty left over from the literature [16] .

Acknowledgement

The authors would like to thank the anonymous referees for their valuable comments and suggestions which help substantially improve the quality of the paper. Wang F was partially supported by the Natural Science Foundation of Hunan Province (No. 2023JJ0007), the National Natural Science Foundation of China (No. 12001064), the Hunan Provincial Research Project on Teaching Reform in Colleges and Universities (No. HNJG-2021-0462), the National First-class Offfine Undergraduate Course Complex Variable Functions and Integral Transformations and the Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering (Grant No.2017TP1017).

References

1. Wang, F., Xue, L., Zhao, K. and Zheng, X. (2021) Global Stabilization and Boundary Control of Generalized Fisher/KPP Equation and Application to Diffusive SIS Model. Journal of Differential Equations, 275, 391-417. >https://doi.org/10.1016/j.jde.2020.11.031

2. Wang, F. and Gao, Y. (2022) The Analysis of Hyers-Ulam Stability for Heat Equations with Time-Dependent Coefficient. Mathematics, 10, Article 4355. >https://doi.org/10.3390/math10224355

3. Khusainov, D.Y., Diblík, J., Růžičková, M. and Lukáčová, J. (2008) Representation of a Solution of the Cauchy Problem for an Oscillating System with Pure Delay. Nonlinear Oscillations, 11, 276-285. >https://doi.org/10.1007/s11072-008-0030-8

4. Diblík, J., Fečkan, M. and Pospíšil, M. (2013) Representation of a Solution of the Cauchy Problem for an Oscillating System with Two Delays and Permutable Matrices. Ukrainian Mathematical Journal, 65, 64-76. >https://doi.org/10.1007/s11253-013-0765-y

5. Wang, F. and Yao, Z. (2016) Approximate Controllability of Fractional Neutral Differential Systems with Bounded Delay. Fixed Point Theory, 17, 495-508.

6. Wang, F., Liu, Z. and Li, J. (2013) Complete Controllability of Fractional Neutral Differential Systems in Abstract Space. Abstract and Applied Analysis, 2013, Article ID: 529025. >https://doi.org/10.1155/2013/529025

7. Kilbas, A.A.A., Srivastava, H.M. and Trujillo, J.J. (2006) Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, 204.

8. Elshenhab, A.M. and Wang, X.T. (2021) Representation of Solutions for Linear Fractional Systems with Pure Delay and Multiple Delays. Mathematical Methods in the Applied Sciences, 44, 12835-12850. >https://doi.org/10.1002/mma.7585

9. Elshenhab, A.M. and Wang, X. (2022) Controllability and Hyers-Ulam Stability of Differential Systems with Pure Delay. Mathematics, 10, Article 1248. >https://doi.org/10.3390/math10081248

10. Liang, Y., Shi, Y. and Fan, Z. (2022) Exact Solutions and Hyers-Ulam Stability of Fractional Equations with Double Delays. Fractional Calculus and Applied Analysis, 26, 439-460. >https://doi.org/10.1007/s13540-022-00122-3

11. Hatime, N., Melliani, S., Mfadel, A.E. and Elomari, M. (2023) Existence, Uniqueness, and Finite-Time Stability of Solutions for Caputo Fractional Differential Equations with Time Delay. Computational Methods for Differential Equations, 11, 785-802.

12. Agarwal, R., Hristova, S. and O’Regan, D. (2020) Existence and Integral Representation of Scalar Riemann-Liouville Fractional Differential Equations with Delays and Impulses. Mathematics, 8, 607. >https://doi.org/10.3390/math8040607

13. Liu, L., Dong, Q. and Li, G. (2021) Exact Solutions and Hyers-Ulam Stability for Fractional Oscillation Equations with Pure Delay. Applied Mathematics Letters, 112, Article ID: 106666. >https://doi.org/10.1016/j.aml.2020.106666

14. Li, M. and Wang, J. (2019) Finite Time Stability and Relative Controllability of Riemann-Liouville Fractional Delay Differential Equations. Mathematical Methods in the Applied Sciences, 42, 6607-6623. >https://doi.org/10.1002/mma.5765

15. Liu, S., Zhou, X., Li, X. and Jiang, W. (2017) Asymptotical Stability of Riemann-Liouville Fractional Singular Systems with Multiple Time-Varying Delays. Applied Mathematics Letters, 65, 32-39. >https://doi.org/10.1016/j.aml.2016.10.002

16. Pan, R. and Fan, Z. (2023) Analyses of Solutions of Riemann-Liouville Fractional Oscillatory Differential Equations with Pure Delay. Mathematical Methods in the Applied Sciences, 46, 10450-10464. >https://doi.org/10.1002/mma.9132

17. Podlubny, I. (1999) Fractional Differential Equations. Academic Press.

18. Evans, L.C. (1998) Partial Differential Equations (Graduate Studies in Mathematics). American Mathematical Society, Providence.