Using the Krasnoselskii’s fixed point theorem and the contraction mapping principle we give sufficient conditions for the existence and uniqueness of solutions for initial value problems for delay fractional differential equations with the mixed Riemann-Liouville and Caputo fractional derivatives. At the end, an example is given to illustrate our main results.
The fractional differential equations is a hot topic of research due to its various applications in many scientific disciplines such as physics, chemistry, biology, engineering, viscoelasticity, signal processing, electrotechnical, electrochemistry and controllability, see [1,2,3,4,5,6] and the references therein. The neutral fractional differential equations have been studied extensively in the last decades and different technics have been used to solve it, for example, fixed point theorems, upper and lower solution method, spectral theory, etc. For some recent contributions in fractional boundary value problems, we refer [7,8,9] and the references therein. To the best of our knowledge, the use of mixed fractional derivative in neutral fractional differential equations which is an important type of fractional differential equations is still not sufficiently generalized. Our main aim is to solve mixed fractional differential equations.
Benchohra et al. [7], investigated the existence of solutions for the following Riemann-Liouville fractional order functional differential equation with infinite delay \begin{equation*} \begin{cases} ^{RL}D^{\alpha }[u(t)-g(t,u_{t})]=f(t,u_{t}),& \ t\in \lbrack 0,T],\ 0 < \alpha < 1, \\[4pt] u\left( t\right) =\phi \left( t\right) ,\text{ } & t\in \left( -\infty ,0\right] . \end{cases} \end{equation*}
Agarwal et al. [8], studied the initial value problem of fractional neutral Caputo fractional derivative
\begin{equation*} \begin{cases} ^{C}D^{\alpha }[u(t)-g(t,u_{t})]=f(t,u_{t}),\ t\in \left( t_{0},\infty \right) ,\ t_{0}\geq 0,\text{ }0< \alpha < 1, \\ u_{t_{0}}=\phi , \end{cases} \end{equation*} and established the existence results of solutions of this problem by using Krasnoselskii's fixed point theorem. In [9], Ahmad et al. studied the existence and uniqueness of solutions to the following boundary value problem \begin{equation*} \begin{cases} D^{\alpha }\left( D^{\beta }u(t)-g\left( t,u_{t}\right) \right) =f(t,u_{t}),\ & t\in \lbrack 1,b], \\[4pt] u(t)=\phi (t),\ & t\in \lbrack 1-\tau ,1], \\ D^{\beta }u(1)=\eta \in \mathbb{R},% \end{cases}% \end{equation*} where \(D^{\alpha }\) and \(D^{\beta }\) are the Caputo-Hadamard fractional derivatives, \(0< \alpha ,\beta < 1\).Motivated and inspired by above mentioned works, in this paper we investigate the existence and uniqueness of solutions for the following initial value problem of the mixed Riemann-Liouville and Caputo fractional functional differential equation with delay
The organization of this paper is as follows: In Section 2 we recall some useful preliminaries and present the equivalent fixed point problem corresponding to (1). In Section 3, we discuss the existence and uniqueness of solutions for (1) via fixed point theory. An example is constructed for illustrating the obtained results.
Definition 1 ([2,4,6]). The Riemann-Liouville fractional integral of the function \(u\) of order \(% \alpha >0\) is defined by \begin{equation*} I^{\alpha }u\left( t\right) =\frac{1}{\Gamma \left( \alpha \right) }% \int_{0}^{t}\frac{u\left( s\right) }{\left( t-s\right) ^{1-\alpha }}ds, \end{equation*} where \(\Gamma \) is the Euler gamma function defined by \(\Gamma \left( \alpha \right) =\int_{0}^{\infty }e^{-t}t^{\alpha -1}dt\).
Definition 2 ([2,4,6]). The Riemann-Liouville fractional derivative of the function \(u\) of order \(% \alpha \in \left( n-1,n\right] \) is defined by \begin{equation*} ^{RL}D^{\alpha }u\left( t\right) =\frac{1}{\Gamma \left( n-\alpha \right) }% \frac{d^{n}}{dt^{n}}\int_{0}^{t}\frac{u\left( s\right) }{\left( t-s\right) ^{\alpha -n+1}}ds. \end{equation*}
Definition 3 ([2,4,6]). The Caputo fractional derivative of the function \(u\) of order \(\alpha \in \left( n-1,n\right] \) is defined by \begin{equation*} ^{C}D^{\alpha }u\left( t\right) =\frac{1}{\Gamma \left( n-\alpha \right) }% \int_{0}^{t}\frac{u^{\left( n\right) }\left( s\right) }{\left( t-s\right) ^{\alpha -n+1}}ds. \end{equation*} Let \(\alpha >0\) be a real number, we have following results:
Lemma 1 ([4]). The unique solution of the linear fractional differential equation \begin{equation*} ^{RL}D^{\alpha }u(t)=0, \end{equation*} is given by \begin{equation*} u(t)=c_{1}t^{\alpha -1}+c_{2}t^{\alpha -2}+c_{3}t^{\alpha -3}+…+c_{n}t^{\alpha -n},\text{ }c_{i}\in \mathbb{R},\text{ }i=1,2,…,n. \end{equation*} where \(n=\left[ \alpha \right] +1,\) \(\left[ \alpha \right] \) denotes the integer part of \(\alpha \).
Lemma 2 ([4]). The unique solution of the linear fractional differential equation \begin{equation*} ^{C}D^{\alpha }u(t)=0, \end{equation*} is given by \begin{equation*} u(t)=c_{1}+c_{2}t+…+c_{n}t^{n-1},\text{ }c_{i}\in \mathbb{R},\ i=1,2,…,n, \end{equation*} where \(n=\left[ \alpha \right] +1\).
Lemma 3.
(1) is equivalent to the following integral equation
Proof. Using Lemma 1, equation one of (1) can be written as \begin{equation*} ^{C}D^{\beta }u(t)=I^{\alpha }f(t,u\left( t-\tau \right) )+g(t,u\left( t-\tau \right) )+c_{0}t^{\alpha -1}. \end{equation*} Using the condition \(\underset{t\rightarrow 0}{\lim }t^{1-\alpha \text{ }% C}D^{\beta }u(t)=0\), we get \(c_{0}=0\). On the other hand, from Lemma 2, one gets \begin{equation*} u(t)=I^{\alpha +\beta }f(t,u\left( t-\tau \right) )+I^{\beta }g(t,u\left( t-\tau \right) )+c_{1}+c_{2}t. \end{equation*} Clearly \(u\left( 0\right) =\phi \left( 0\right)\), so we obtain \(c_{1}=\phi \left( 0\right) \) and because \(u^{\prime }(0)=0\), we find \(c_{2}=0\), then we get the integral equation \begin{eqnarray*} u(t) &=&\frac{1}{\Gamma (\alpha +\beta )}\int_{0}^{t}\left( t-s\right) ^{\alpha +\beta -1}f(s,u\left( s-\tau \right) )ds \\ &&+\frac{1}{\Gamma (\beta )}\int_{0}^{t}\left( t-s\right) ^{\beta -1}g(s,u\left( s-\tau \right) )ds+\phi \left( 0\right) . \end{eqnarray*} Our main results are based on the following Krasnoselskii fixed point theorem and the contraction mapping principle.
Theorem 1 (Krasnoselskii fixed point theorem[10,11])
If \(\mathcal{M}\) is a nonempty bounded, closed and convex subset
of a Banach space \(E\) and \(\mathcal{A}\) and \(\mathcal{B}\) two operators defined
on \(\mathcal{M}\) with values in \(E\) such that
Theorem 2 (Contraction mapping principle[10,11]) Let \(E\) be a Banach space. If \(\mathcal{H}:E\rightarrow E\) is a contraction, then \(\mathcal{H}\) has a unique fixed point in \(E\).
Theorem 3. Assume that \(\left( H1\right) -\left( H3\right) \) hold. Then (1) has at least one solution on \([-\tau ,T]\), provided
Proof. We show that the operators \(\mathcal{A}\) and \(\mathcal{B}\) defined by (4) and (5) satisfied all hypothesis of Theorem 1. Choosing \(% R\geq \frac{\Vert \phi \Vert _{C}+\frac{T^{\alpha +\beta }}{\Gamma (\alpha +\beta +1)}\Vert \zeta \Vert _{\infty }}{1-\frac{T^{\beta }\Vert \eta \Vert _{\infty }}{\Gamma (\beta +1)}}\) and define \(B_{R}=\{u\in C([-\tau ,T],% \mathbb{R}):\Vert u\Vert _{\lbrack -\tau ,T]}\leq R\}\), then for any \(u,v\in B_{R}\), we have
This means that \(\mathcal{A}u+\mathcal{B}v\in B_{R}\) for any \(u,v\in B_{R}\). On the other hand, operator \(\mathcal{A}\) is continuous from the continuity of \(f\).
It remains to prove the compactness of the operator \(\mathcal{A}\). Clearly, (7) show that \(\mathcal{A}\) is uniformly bounded on \(B_{R}\) and \begin{equation*} \left\Vert \mathcal{A}v\right\Vert \leq \frac{T^{\alpha +\beta }}{\Gamma (\alpha +\beta +1)}\Vert \zeta \Vert _{\infty }. \end{equation*}
Also, for \(t_{1},t_{2}\in \left[ 0,T\right] \), \(t_{1}< t_{2}\) we have \begin{align*} |\left( \mathcal{A}u\right) (t_{2})-\left( \mathcal{A}u\right) (t_{1})|& \leq \frac{1}{\Gamma (\alpha +\beta )}\int_{0}^{t_{1}}\left[ \left( t_{2}-s\right) ^{\alpha +\beta -1}-\left( t_{1}-s\right) ^{\alpha +\beta -1}% \right] \left\vert f(s,u\left( s-\tau \right) )\right\vert ds \\ &\;\; +\frac{1}{\Gamma (\alpha +\beta )}\int_{t_{1}}^{t_{2}}\left( t_{2}-s\right) ^{\alpha +\beta -1}\left\vert f(s,u\left( s-\tau \right) )\right\vert ds \\ & =\frac{\Vert \zeta \Vert _{\infty }}{\Gamma (\alpha +\beta +1)}\left( t_{2}^{\alpha +\beta }-t_{1}^{\alpha +\beta }\right) , \end{align*} which is independent of \(u\) and tends to zero as \(t_{2}\rightarrow t_{1}\). Thus, the set \(\left\{ \mathcal{A}u,\text{ }u\in B_{R}\right\} \) is equicontinuous and hence it is relatively compact. So, by Ascoli-Arzela theorem, \(\mathcal{A}\) is compact on \(B_{R}\).
It remains to show that \(\mathcal{B}\) is a contraction. Let \(u,v\in C([-\tau ,T],\mathbb{R})\), then for all \(t\in \lbrack -\tau ,T]\), we have \begin{align*} \left\vert \left( \mathcal{B}u\right) (t)-\left( \mathcal{B}v\right) (t)\right\vert &\leq \frac{1}{\Gamma (\beta )}\int_{0}^{t}\left( t-s\right) ^{\beta -1}\left\vert g(s,u\left( s-\tau \right) )-g(s,v\left( s-\tau \right) )\right\vert ds \\ & \leq \frac{T^{\beta }\Vert \eta \Vert _{\infty }}{\Gamma (\beta +1)}\Vert u-v\Vert _{\lbrack -\tau ,T]}, \end{align*} from (6) \(\mathcal{A}\) is a contraction operator.
Thus all the assumptions of Theorem 1 are satisfied. So the conclusion of Theorem 3 implies that (1) has at least one continuous solution on \([-\tau ,T]\).
Now, we use the contraction principle mapping to investigate uniqueness results for (1).
Theorem 4.
Assume that \(\left( H1\right) -\left( H3\right) \) hold. Then (1) has a unique solution on \(\left[ -\tau ,T\right] \), provided
Proof. We claim that \(\mathcal{H}\) is contraction mapping, this show that \(\mathcal{ H}\) has a unique fixed point which is the unique solution of (1). To this end, let \(u,v\in C([-\tau ,T],\mathbb{R})\), then for all \(t\in \lbrack -\tau ,T]\), we have \begin{align*} \left\vert \left( \mathcal{H}u\right) (t)-\left( \mathcal{H}v\right) (t)\right\vert &\leq \frac{1}{\Gamma (\alpha +\beta )}\int_{0}^{t}\left( t-s\right) ^{\alpha +\beta -1}\left\vert f(s,u\left( s-\tau \right) )-f(s,v\left( s-\tau \right) )\right\vert ds \\ & \;\;+\frac{1}{\Gamma (\beta )}\int_{0}^{t}\left( t-s\right) ^{\beta -1}\left\vert g(s,u\left( s-\tau \right) )-g(s,v\left( s-\tau \right) )\right\vert ds \\ & \leq \left[ \frac{T^{\alpha +\beta }\Vert \zeta \Vert _{\infty }}{\Gamma (\alpha +\beta +1)}+\frac{T^{\beta }\Vert \eta \Vert _{\infty }}{\Gamma (\beta +1)}\right] \Vert u-v\Vert _{\lbrack -\tau ,T]}. \end{align*} Therefore \(\mathcal{H}\) is a contraction. Thus, the conclusion of Theorem 7 follows by the contraction mapping principle.
Now, we give an example to illustrate the usefulness of our main results.
Example 1. Consider (1) with \(\alpha =0.5\), \(\beta =1.5\), \(T=1\), \(f(t,x)=\frac{% \sin (t^{2}\arctan x^{3})}{1+t^{2}}\), \(g(t,x)=\cos t\left( t^{2}e^{-e^{-x}}-% \frac{t^{2}}{e}+\frac{\sin tx}{e}\right) \). Clearly \(f\) and \(g\) are continuous functions and \begin{equation*} \left\vert f(t,x)\right\vert \leq \frac{\pi }{2}\frac{t^{2}}{1+t^{2}}=\zeta \left( t\right) , \end{equation*} and \begin{equation*} g\left( t,0\right) =0,\text{ }\left\vert g(t,x)-g(t,y)\right\vert \leq \frac{% t^{2}}{e}\cos t\Vert x-y\Vert _{\lbrack -\tau ,T]}=\eta \left( t\right) \Vert x-y\Vert _{\lbrack -\tau ,T]}, \end{equation*} (since \(z+e^{-z}\geq 1\) for all real \(z\) then one gets \(\left\vert e^{-e^{-x}}-e^{-e^{-y}}\right\vert \leq \frac{\left\vert x-y\right\vert }{e}\) ). Also, \(\frac{T^{\beta }\Vert \eta \Vert _{\infty }}{\Gamma (\beta +1)}% \simeq 0.27674< 1\). Thus, by Theorem 3, (1) with above data has at least one solution on \(\left[ -\tau ,1\right] \).