Let \(\mathbb{R}^{m}\) be an m-dimensional Euclidean space. We prove our main results for the system
$$ \textstyle\begin{cases} ({{}_{0}^{C}D}^{\alpha ,\rho }x )(t)=e^{\frac{\rho -1}{ \rho }t} [A_{0}x(t)+A_{1}x(t-\tau )+f(t,x(t),x(t-\tau )) ],\quad t \in [0,T], \\ x(t)=\varphi (t), \quad t\in [-\tau ,0], \end{cases} $$
(24)
where \({{}_{0}^{C}D}^{\alpha ,\rho }\) denotes the GPF derivative of Caputo type of order \(\alpha \in (0,1)\), the state vector \(x: [-\tau ,T] \to \mathbb{R}^{m}\), the constant matrices \(A_{0}\) and \(A_{1}\) are of appropriate dimensions, the nonlinearity \(f:[0,T]\times \mathbb{R} ^{m} \times \mathbb{R}^{m} \to \mathbb{R}^{m}\), and the initial function \(\varphi :[-\tau ,0] \to \mathbb{R}^{m}\). By virtue of the results obtained in the previous sections, we prove the uniqueness and obtain an estimate for the solutions of system (24). Moreover, a numerical example is presented to demonstrate the applicability of the main results.
Let \(\vert \cdot \vert \) be any Euclidean norm and \(\Vert \cdot \Vert \) be the matrix norm induced by this vector. Denote by \(\mathcal{C}:=C([-\tau ,0],\mathbb{R}^{m})\) the set of all continuous functions. Clearly, the space \(\mathcal{C}\) is a Banach space induced by the norm \(\Vert z \Vert _{\mathcal{C}}:= \sup_{ t \in [-\tau ,0] }|z(t)|\).
Throughout the remaining part of the paper, we make use of the following assumptions:
-
(H.1)
The nonlinearity \(f \in C([0,T]\times \mathbb{R}^{m} \times \mathbb{R}^{m},\mathbb{R}^{m})\) satisfies the Lipschitz condition. That is, there exists a positive constant \(L_{1}>0\) such that
$$\begin{aligned}& \bigl\Vert f\bigl(t,x(t),x(t-\tau )\bigr)-f\bigl(t,y(t),y(t-\tau )\bigr) \bigr\Vert \\& \quad \le L_{1} \bigl( \bigl\Vert x(t)-y(t) \bigr\Vert + \bigl\Vert x(t-\tau )-y(t-\tau ) \bigr\Vert \bigr) \end{aligned}$$
for \(t \in [0,T]\).
-
(H.2)
There exists a positive constant \(L_{2}\) such that \(\| f(t,x(t),x(t-\tau ))\| \le L_{2}\).
In what follows, we provide a representation for the solutions of system (24) that will be useful in the subsequent analysis.
Lemma 7
The function
\(x: [-\tau ,0] \to \mathbb{R}^{m}\)
is a solution of system (24) if and only if
$$ \textstyle\begin{cases} x(t) = \varphi (0)e^{\frac{\rho -1}{\rho }t} \\ \hphantom{x(t) ={}}{}+ ({{}_{0}I}^{\alpha , \rho }e^{\frac{\rho -1}{\rho }s} [A_{0}x(s)+A_{1}x(s-\tau )+f(s,x(s),x(s- \tau )) ] )(t),\quad t \in [0,T], \\ x(t)=\varphi (t),\quad t \in [-\tau ,0]. \end{cases} $$
(25)
Proof
For \(t \in [-\tau ,0]\), it is clear that \(x(t)=\varphi (t)\) is the solution of (24). We apply the operator \({{}_{0}D}^{\alpha , \rho }\) on both sides of equation (25) with Proposition 1 and Lemma 3 to obtain, for \(t \in [0,T]\),
$$ \bigl({{}_{0}D}^{\alpha ,\rho }x \bigr) (t) = \varphi (0) \frac{\rho ^{\alpha }e^{\frac{\rho -1}{\rho }t}t^{-\alpha }}{\varGamma (1-\alpha )} + e^{\frac{ \rho -1}{\rho }t} \bigl[A_{0}x(t)+A_{1}x(t- \tau )+f\bigl(t,x(t),x(t-\tau )\bigr) \bigr]. $$
By using the relation of the Caputo and Riemann–Liouville GPF derivatives in Proposition 3, it follows that
$$ \bigl({{}_{0}^{C}D}^{\alpha ,\rho }x \bigr) (t) = e^{\frac{\rho -1}{\rho }t} \bigl[A_{0}x(t)+A_{1}x(t-\tau )+f \bigl(t,x(t),x(t-\tau )\bigr) \bigr]. $$
For system (24), we can see that \(x(t)=\varphi (t)\), \(t \in [-\tau ,0]\). For \(t \in [0,T]\), we apply the operator \({{}_{0}I}^{\alpha ,\rho }\) on both sides of equation (24) to get
$$ \bigl({{}_{0}I}^{\alpha ,\rho }{{}_{0}^{C}D}^{\alpha ,\rho }x \bigr) (t) = \bigl({{}_{0}I}^{\alpha ,\rho }e^{\frac{\rho -1}{\rho }s} \bigl[A_{0}x(s)+A _{1}x(s-\tau )+f\bigl(s,x(s),x(s-\tau ) \bigr) \bigr] \bigr) (t). $$
In view of Lemma 5, one can easily see that
$$ x(t)= \varphi (0)e^{\frac{\rho -1}{\rho }t}+ \bigl({{}_{0}I}^{\alpha , \rho }e^{\frac{\rho -1}{\rho }s} \bigl[A_{0}x(s)+A_{1}x(s-\tau )+f\bigl(s,x(s),x(s- \tau ) \bigr) \bigr] \bigr) (t). $$
□
Uniqueness of solutions
The first main application in this paper is provided in the following theorem.
Theorem 1
Let condition (H.1) hold. If
x
and
y
are two solutions for system (24), then
\(x=y\).
Proof
Let x and y be two solutions of system (24). Denote \(z=x-y\). Then, one can easily figure out that \(z(t)=0\) for \(t \in [- \tau ,0]\). This implies that system (24) has a unique solution for \(t \in [-\tau ,0]\).
For \(t \in [0,T]\), however, we have
$$ z(t)= \bigl({{}_{0}I}^{\alpha ,\rho }e^{\frac{\rho -1}{\rho }s} \bigl[A_{0}z(s)+A _{1}z(s-\tau )+f\bigl(s,x(s),x(s-\tau ) \bigr)-f\bigl(s,y(s),y(s-\tau )\bigr) \bigr] \bigr) (t). $$
If \(t \in [0,\tau ]\), then \(z(t-\tau )=0\). Therefore,
$$ z(t)= \bigl({{}_{0}I}^{\alpha ,\rho }e^{\frac{\rho -1}{\rho }s} \bigl[A_{0}z(s)+f\bigl(s,x(s),x(s- \tau )\bigr)-f\bigl(s,y(s),y(s-\tau )\bigr) \bigr] \bigr) (t). $$
(26)
This implies
$$\begin{aligned} \bigl\Vert z(t) \bigr\Vert \le & \bigl({{}_{0}I}^{\alpha ,\rho }e^{\frac{\rho -1}{ \rho }s} \bigl[ \Vert A_{0} \Vert \bigl\Vert z(s) \bigr\Vert + \bigl\Vert f\bigl(s,x(s),x(s- \tau )\bigr)-f\bigl(s,y(s),y(s-\tau )\bigr) \bigr\Vert \bigr] \bigr) (t) \\ \le & ({{}_{0}I}^{\alpha ,\rho }e^{\frac{\rho -1}{\rho }s} \bigl[ \Vert A_{0} \Vert \bigl\Vert z(s) \bigr\Vert + L_{1} \bigl( \bigl\Vert x(s) - y(s) \bigr\Vert + \bigl\Vert x(s-\tau ) - y(s-\tau ) \bigr\Vert \bigr] \bigr) (t) \\ =& \bigl({{}_{0}I}^{\alpha ,\rho }e^{\frac{\rho -1}{\rho }s} \bigl[ \bigl( \Vert A_{0} \Vert +L_{1}\bigr) \bigl\Vert z(s) \bigr\Vert + L_{1} \bigl\Vert z(s-\tau ) \bigr\Vert \bigr] \bigr) (t) \\ =& \bigl( \Vert A_{0} \Vert +L_{1}\bigr) \bigl({{}_{0}I}^{\alpha ,\rho }e^{\frac{ \rho -1}{\rho }s} \bigl\Vert z(s) \bigr\Vert \bigr) (t). \end{aligned}$$
(27)
By applying the result of Corollary 3, we have
$$ \bigl\Vert z(t) \bigr\Vert \le (0)\cdot E_{\alpha } \bigl( \Vert A_{0} \Vert +L_{1}, t \bigr), $$
(28)
which implies that \(x(t)=y(t)\) for \(t \in I_{\tau }\).
For \(t \in [\tau ,T]\), we get
$$\begin{aligned} z(t) =& \bigl({{}_{0}I}^{\alpha ,\rho }e^{\frac{\rho -1}{\rho }s} \bigl[A _{0}z(s)+f\bigl(s,x(s),x(s-\tau )\bigr)-f\bigl(s,y(s),y(s-\tau )\bigr) \bigr] \bigr) (t) \\ &{}+ \bigl({{}_{0}I}^{\alpha ,\rho }e^{\frac{\rho -1}{\rho }s} \bigl[A_{1}z(s- \tau ) \bigr] \bigr) (t). \end{aligned}$$
(29)
It follows that
$$\begin{aligned} \bigl\Vert z(t) \bigr\Vert \le & \bigl({{}_{0}I}^{\alpha ,\rho }e^{\frac{\rho -1}{\rho }s} \bigl[ \Vert A_{0} \Vert \bigl\Vert z(s) \bigr\Vert + \bigl\Vert f\bigl(s,x(s),x(s-\tau )\bigr)-f\bigl(s,y(s),y(s- \tau )\bigr) \bigr\Vert \bigr] \bigr) (t) \\ &{}+ \bigl({{}_{0}I}^{\alpha ,\rho }e^{\frac{\rho -1}{\rho }s} \bigl[ \Vert A _{1} \Vert \bigl\Vert z(s-\tau ) \bigr\Vert \bigr] \bigr) (t) \\ \le & \bigl( \Vert A_{0} \Vert +L_{1}\bigr) \bigl({{}_{0}I}^{\alpha ,\rho }e^{\frac{ \rho -1}{\rho }s} \bigl\Vert z(s) \bigr\Vert \bigr) (t) +\bigl( \Vert A_{1} \Vert +L_{1} \bigr) \bigl({{}_{0}I}^{\alpha ,\rho }e^{\frac{\rho -1}{\rho }s} \bigl\Vert z(s- \tau ) \bigr\Vert \bigr) (t). \end{aligned}$$
Let \(\bar{z}(t)=\sup_{\theta \in [-\tau ,0]}\| z(t+\theta ) \|\), then we get
$$\begin{aligned} \bar{z}(t) \le & \bigl( \Vert A_{0} \Vert +L_{1} \bigr) \bigl({{}_{0}I}^{\alpha , \rho }e^{\frac{\rho -1}{\rho }s}\bar{z}(s) \bigr) (t) +\bigl( \Vert A_{1} \Vert +L_{1}\bigr) \bigl({{}_{0}I}^{\alpha ,\rho }e^{\frac{\rho -1}{\rho }s} \bar{z}(s) \bigr) (t) \\ \le & \bigl( \Vert A_{0} \Vert + \Vert A_{1} \Vert +2L_{1}\bigr) \bigl({{}_{0}I}^{ \alpha ,\rho }e^{\frac{\rho -1}{\rho }s} \bar{z}(s) \bigr) (t). \end{aligned}$$
(30)
By applying the result of Corollary 3, we obtain
$$ \bigl\Vert z(t) \bigr\Vert \le \bar{z}(t) \le (0)\cdot E_{\alpha } \bigl( \Vert A_{0} \Vert + \Vert A_{1} \Vert +2L_{1} , t \bigr). $$
(31)
Hence, we end up with \(x(t)=y(t)\) for \(t \in [-\tau ,T]\). □
Bound for solutions
In this subsection, we provide a bound for the solution of system (24).
Theorem 2
Let condition (H.2) hold. Then the following estimate for the solution
\(x(t)\)
of system (24) is valid:
$$ \bigl\Vert x(t) \bigr\Vert \le \biggl[ \Vert \varphi \Vert + \bigl(L_{2}+\bigl(\Vert A_{0} \Vert +\Vert A_{1}\Vert \bigr)\Vert \varphi \Vert \bigr) \frac{t^{\alpha }}{ \rho ^{\alpha }\varGamma (\alpha +1)} \biggr]E_{\alpha } \bigl(\bigl( \|A_{0}\|+\|A _{1}\|\bigr)\varGamma (\alpha ), t \bigr). $$
(32)
Proof
For \(t \in [0,T]\), the solution of system (24) has the form
$$ x(t)= \varphi (0)e^{\frac{\rho -1}{\rho }t}+ \bigl({{}_{0}I}^{\alpha , \rho }e^{\frac{\rho -1}{\rho }s} \bigl[A_{0}x(s)+A_{1}x(s-\tau )+f\bigl(s,x(s),x(s- \tau ) \bigr) \bigr] \bigr) (t). $$
(33)
Using the fact \(e^{\frac{\rho -1}{\rho }t}\leq 1\) for all \(t\in [0,T]\), it follows that
$$\begin{aligned} \bigl\Vert x(t) \bigr\Vert \le & \bigl\Vert \varphi (0) \bigr\Vert + \bigl({{}_{0}I}^{\alpha ,\rho }e^{\frac{\rho -1}{\rho }s} \bigl\Vert A_{0}x(s)+A_{1}x(s-\tau )+f\bigl(s,x(s),x(s- \tau )\bigr) \bigr\Vert \bigr) (t) \\ \le & \Vert \varphi \Vert + \Vert A_{0} \Vert \bigl({{}_{0}I}^{\alpha , \rho }e^{\frac{\rho -1}{\rho }s} \bigl\Vert x(s) \bigr\Vert \bigr) (t) + \Vert A _{1} \Vert \bigl({{}_{0}I}^{\alpha ,\rho }e^{\frac{\rho -1}{\rho }s} \bigl\Vert x(s-\tau ) \bigr\Vert \bigr) (t) \\ &{}+ \bigl({{}_{0}I}^{\alpha ,\rho }e^{\frac{\rho -1}{\rho }s} \bigl\Vert f \bigl(s,x(s),x(s- \tau )\bigr) \bigr\Vert \bigr) (t). \end{aligned}$$
By assumption (H.2) and Proposition 1, the above inequality can be rewritten as follows:
$$\begin{aligned} \bigl\Vert x(t) \bigr\Vert \le & \Vert \varphi \Vert +\bigl( \Vert A_{0} \Vert + \Vert A_{1} \Vert \bigr) \Bigl({{}_{0}I}^{\alpha ,\rho }e^{\frac{\rho -1}{\rho }s} \Bigl[ \sup _{\theta \in [-\tau ,0]} \bigl\Vert x(s+\theta ) \bigr\Vert + \Vert \varphi \Vert \Bigr] \Bigr) (t) \\ &{}+L_{2} \bigl({{}_{0}I}^{\alpha ,\rho }e^{\frac{\rho -1}{\rho }s}(1) \bigr) (t) \\ =& \Vert \varphi \Vert +\bigl(L_{2}+\bigl( \Vert A_{0} \Vert + \Vert A_{1} \Vert \bigr) \Vert \varphi \Vert \bigr) \frac{t^{\alpha }}{\rho ^{\alpha }\varGamma (\alpha +1)} \\ &{}+\bigl( \Vert A_{0} \Vert + \Vert A_{1} \Vert \bigr) \Bigl({{}_{0}I}^{\alpha ,\rho }e ^{\frac{\rho -1}{\rho }s} \Bigl[\sup _{\theta \in [-\tau ,0]} \bigl\Vert x(s+ \theta ) \bigr\Vert \Bigr] \Bigr) (t). \end{aligned}$$
Let \(v(t)=\Vert \varphi \Vert +(L_{2}+(\Vert A_{0}\Vert +\Vert A_{1} \Vert )\Vert \varphi \Vert ) \frac{T^{\alpha }}{\rho ^{\alpha }\varGamma (\alpha +1)}\), then v is a nondecreasing function. Therefore, Corollary 3 with \(w(t)= \|A_{0}\|+\|A_{1}\|\) implies that
$$ \bigl\Vert x(t) \bigr\Vert \le \sup_{\theta \in [-\tau ,0]} \bigl\Vert x(t+\theta ) \bigr\Vert \le v(t) E_{\alpha } \bigl(\bigl( \Vert A_{0} \Vert + \Vert A_{1} \Vert \bigr)\varGamma ( \alpha ), t \bigr). $$
(34)
Hence, the solution x of (24) satisfies the estimate
$$ \bigl\Vert x(t) \bigr\Vert \le \biggl[ \Vert \varphi \Vert +\bigl(L_{2}+\bigl( \Vert A_{0} \Vert + \Vert A_{1} \Vert \bigr) \Vert \varphi \Vert \bigr) \frac{t^{\alpha }}{ \rho ^{\alpha }\varGamma (\alpha +1)} \biggr]E_{\alpha } \bigl(\bigl( \Vert A_{0} \Vert + \Vert A _{1} \Vert \bigr)\varGamma (\alpha ), t \bigr). $$
(35)
The proof is complete. □
Example 1
Consider the nonlinear delay proportional fractional system of the form
$$ \textstyle\begin{cases} ({{}_{0}^{C}D}^{\frac{1}{2},\frac{1}{3}}x )(t)=e^{-2t} [3x(t)+x(t-2)+2 \cos x(t)-\cos x(t-2) ], \quad t \in [0,1], \\ x(t)=\sin 2t,\quad t\in [-2,0]. \end{cases} $$
(36)
This corresponds to equation (24) with \(\alpha = 1/2\), \(\rho = 1/3\), \(A_{0}=3\), \(A_{1}=1\), \(T=1\), and \(\tau = 2\). The nonlinearity has the form \(f(t,x(t),x(t-\tau )) = 2\cos x(t)-\cos x(t-2)\). Therefore, we have
$$\begin{aligned}& \bigl\Vert f\bigl(t,x(t),x(t-\tau )\bigr) - f\bigl(t,y(t),y(t-\tau )\bigr) \bigr\Vert \\& \quad = \bigl\Vert 2\cos x(t)-\cos x(t-2)-2\cos y(t)+\cos y(t-2) \bigr\Vert \\& \quad \le 2 \bigl( \bigl\Vert \cos x(t) -\cos y(t) \bigr\Vert + \bigl\Vert \cos x(t-2) -\cos y(t-2) \bigr\Vert \bigr). \end{aligned}$$
Then assumption (H.1) holds with \(L_{1}=2\). By the consequence of Lemma 7, system (36) has a unique solution. Moreover,
$$ \bigl\Vert f\bigl(t,x(t),x(t-\tau )\bigr) \bigr\Vert = \bigl\Vert 2\cos x(t)-\cos x(t-2) \bigr\Vert \le 3, $$
which implies that assumption (H.2) is satisfied with \(L_{2}=3\). By Theorem 2, the solution x of system (36) has the estimate
$$ \bigl\Vert x(t) \bigr\Vert \le \biggl[1+\frac{24\sqrt{3}}{\sqrt{\pi }}t ^{\frac{1}{2}} \biggr] \sum_{k=0}^{\infty }\frac{(4\sqrt{\pi })^{k}t ^{\frac{k}{2}}}{\varGamma (\frac{k}{2}+1)}. $$