In this paper, we use the Banach fixed point theorem to obtain the existence, interval of existence and uniqueness of solutions for nonlinear hybrid implicit Caputo-Hadamard fractional differential equations. We also use the generalization of Gronwall’s inequality to show the estimate of the solutions.
The concept of fractional calculus is a generalization of the ordinary differentiation and integration to arbitrary non integer order. Fractional differential equations with and without delay arise from a variety of applications including in various fields of science and engineering such as applied sciences, practical problems concerning mechanics, the engineering technique fields, economy, control systems, physics, chemistry, biology, medicine, atomic energy, information theory, harmonic oscillator, nonlinear oscillations, conservative systems, stability and instability of geodesic on Riemannian manifolds, dynamics in Hamiltonian systems, etc. In particular, problems concerning qualitative analysis of linear and nonlinear fractional differential equations with and without delay have received the attention of many authors, see [1,2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14,15, 16, 17] and the references therein.
Recently, Ahmad and Ntouyas [3] discussed the existence of solutions for the hybrid Hadamard differential equation \begin{equation*} \left\{ \begin{array}{l} ^{H}D^{\alpha }\left( \frac{x(t)}{g\left( t,x\left( t\right) \right) } \right) =f\left( t,x\left( t\right) \right) ,\text{ }t\in \left[ 1,T\right] , \\ \left. ^{H}I^{\alpha }x(t)\right\vert _{t=1}=\eta , \end{array} \right. \end{equation*} where \(^{H}D^{\alpha }\) is the Hadamard fractional derivative of order \( 0< \alpha \leq 1\). By employing the Dhage fixed point theorem, the authors obtained existence results.
In [4], Ardjouni and Djoudi studied the existence, interval of existence and uniqueness of solutions for nonlinear implicit Caputo-Hadamard fractional differential equations with nonlocal conditions \begin{equation*} \left\{ \begin{array}{l} \mathfrak{D}_{1}^{\alpha }\left( x\left( t\right) \right) =f\left( t,x\left( t\right) ,\mathfrak{D}_{1}^{\alpha }\left( x\left( t\right) \right) \right) ,\ t\in \left[ 1,T\right] , \\ x\left( 1\right) +g\left( x\right) =x_{0}, \end{array} \right. \end{equation*} where \(f:[1,T]\times\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}\) and \(g:C\left( [1,T],\mathbb{R}\right) \rightarrow\mathbb{R}\) are nonlinear continuous functions and \(\mathfrak{D}_{1}^{\alpha }\) denotes the Caputo-Hadamard fractional derivative of order \(0< \alpha < 1\).
Motivated by these works, we study the existence, interval of existence and uniqueness of solution for the following nonlinear hybrid implicit Caputo-Hadamard fractional differential equation
Definition 1. [12] The Hadamard fractional integral of order \(\alpha >0\) for a continuous function \(x:\left[ 1,+\infty \right) \rightarrow\mathbb{R}\) is defined as \begin{equation*} \mathfrak{I}_{1}^{\alpha }x\left( t\right) =\frac{1}{\Gamma \left( \alpha \right) }\int_{1}^{t}\left( \log \frac{t}{s}\right) ^{\alpha -1}x\left( s\right) \frac{ds}{s},\text{ }\alpha >0. \end{equation*}
Definition 2. [12] The Caputo-Hadamard fractional derivative\ of order \(\alpha \) for a continuous function \(x:\left[ 1,+\infty \right) \rightarrow\mathbb{R}\) is defined as \begin{equation*} \mathfrak{D}_{1}^{\alpha }x\left( t\right) =\frac{1}{\Gamma \left( n-\alpha \right) }\int_{1}^{t}\left( \log \frac{t}{s}\right) ^{n-\alpha -1}\delta ^{n}\left( x\right) \left( s\right) \frac{ds}{s},\text{ }n-1< \alpha < n, \end{equation*} where \(\delta ^{n}=\left( t\frac{d}{dt}\right) ^{n}\), \(n=\left[ \alpha \right] +1\).s
Lemma 3. [12] Let \(\alpha >0\), \(n\in\mathbb{N}\). Suppose \(x\in C^{n-1}\left( \left[ 1,+\infty \right) \right) \) and \( \delta ^{\left( n\right) }x\) exists almost everywhere on any bounded interval of \(\left[ 1,+\infty \right) \). Then \begin{equation*} \mathfrak{I}_{1}^{\alpha }\left[ \mathfrak{D}_{1}^{\alpha }x\right] \left( t\right) =x(t)-\sum\limits_{k=0}^{n-1}\frac{\delta ^{\left( k\right) }x\left( 1\right) }{\Gamma \left( k+1\right) }\left( \log t\right) ^{k}. \end{equation*} In particular, when \(0< \alpha < 1,\ \mathfrak{I}_{1}^{\alpha }\left[ \mathfrak{D}_{1}^{\alpha }x\right] \left( t\right) =x(t)-x(1)\).
Lemma 4.[12] For all \(\mu >0\) and \(v>-1\), then \begin{equation*} \frac{1}{\Gamma \left( \mu \right) }\int_{1}^{t}\left( \log \frac{t}{s} \right) ^{\mu -1}\left( \log s\right) ^{v}ds=\frac{\Gamma \left( v+1\right) }{\Gamma \left( \mu +v+1\right) }\left( \log t\right) ^{\mu +v}. \end{equation*}
The following generalization of Gronwall’s lemma for singular kernels plays an important rsole in obtaining our main results.Lemma 5. [15] Let \(x:\left[ 1,T\right] \rightarrow \left[ 0,\infty \right) \) be a real function and \(w\) is a nonnegative locally integrable function on \( \left[ 1,T\right] \). Assume that there is a constant \(a>0\) such that for \( 0< \alpha < 1\) \begin{equation*} x(t)\leq w(t)+a\int_{1}^{t}\left( \log \frac{t}{s}\right) ^{\alpha -1}x(s) \frac{ds}{s}. \end{equation*} Then, there exists a constant \(k=k(\alpha )\) such that \begin{equation*} x(t)\leq w(t)+ka\int_{1}^{t}\left( \log \frac{t}{s}\right) ^{\alpha -1}w(s) \frac{ds}{s}, \end{equation*} for every \(t\in \left[ 1,T\right] \).
Lemma 6. If the functions \(f:\left[ 1,T\right] \times\mathbb{R}\rightarrow\mathbb{R}\), \(g:\left[ 1,T\right] \times\mathbb{R}\rightarrow\mathbb{R}\backslash \left\{ 0\right\} \) and \(h:\left[ 1,T\right] \times\mathbb{R}^{2}\rightarrow\mathbb{R}\) are continuous, then the initial value problem (1) is equivalent to the nonlinear fractional Volterra integro-differential equation \begin{eqnarray*} x(t) &=&f\left( t,x(t)\right) +\theta g(t,x\left( t\right) ) +\frac{g\left( t,x\left( t\right) \right) }{\Gamma \left( \alpha \right) } \int_{1}^{t}\left( \log \frac{t}{s}\right) ^{\alpha -1}h\left( s,x\left( s\right) ,\mathfrak{D}_{1}^{\alpha }\left( \frac{x\left( s\right) -f\left( s,x(s)\right) }{g(s,x\left( s\right) )}\right) \right) \frac{ds}{s}, \end{eqnarray*} for \(t\in \left[ 1,T\right] \).
Theorem 7.
Let \(T>0\). Assume that the continuous functions \(f:\left[ 1,T
\right] \times\mathbb{R}\rightarrow\mathbb{R}\), \(g:\left[ 1,T\right] \times\mathbb{R}\rightarrow\mathbb{R}\backslash \left\{ 0\right\} \) and \(h:\left[ 1,T\right] \times\mathbb{R}^{2}\rightarrow\mathbb{R}\)
satisfy the following conditions
(H1) There exists \(M_{g}\in\mathbb{R}^{+}\) such that
\begin{equation*}
\left\vert g\left( t,u\right) \right\vert \leq M_{g},
\end{equation*}
for all \(u\in\mathbb{R}\) and \(t\in \left[ 1,T\right] \).
(H2) There exists \(M_{h}\in\mathbb{R}^{+}\) such that
\begin{equation*}
\left\vert h\left( t,u,v\right) \right\vert \leq M_{h},
\end{equation*}
for all \(u,v\in\mathbb{R}\) and \(t\in \left[ 1,T\right] \).
(H3) There exist \(K_{1},K_{2},K_{3}\in\mathbb{R}^{+},K_{4}\in \left( 0,1\right) \) with \(K_{1}+K_{2}\left\vert \theta
\right\vert \in \left( 0,1\right) \) such that
\begin{eqnarray*}
\left\vert f\left( t,u\right) -f\left( t,u^{\ast }\right) \right\vert &\leq
&K_{1}\left\vert u-v\right\vert , \\
\left\vert g\left( t,u\right) -g\left( t,u^{\ast }\right) \right\vert &\leq
&K_{2}\left\vert u-v\right\vert ,
\end{eqnarray*}
and
\begin{equation*}
\left\vert h\left( t,u,v\right) -h\left( t,u^{\ast },v^{\ast }\right)
\right\vert \leq K_{3}\left\vert u-u^{\ast }\right\vert +K_{4}\left\vert
v-v^{\ast }\right\vert ,
\end{equation*}
for all \(u,v,u^{\ast },v^{\ast }\in\mathbb{R}\) and \(t\in \left[ 1,T\right] \).
Let
Proof. Let \begin{equation*} \mathfrak{D}_{1}^{\alpha }\left( \frac{x\left( t\right) -f\left( t,x(t)\right) }{g\left( t,x(t)\right) }\right) =z_{x}\left( t\right) ,\text{ }x\left( 1\right) =\theta g\left( 1,x(1)\right) +f\left( 1,x(1)\right) , \end{equation*} then by Lemma 6, we have \begin{equation*} x(t)=f\left( t,x(t)\right) +\theta g(t,x\left( t\right) )+\frac{g\left( t,x\left( t\right) \right) }{\Gamma \left( \alpha \right) } \int_{1}^{t}\left( \log \frac{t}{s}\right) ^{\alpha -1}z_{x}\left( s\right) \frac{ds}{s}, \end{equation*} where \begin{equation*} z_{x}\left( t\right) =h\left( t,f\left( t,x(t)\right) +\theta g(t,x\left( t\right) )+g\left( t,x\left( t\right) \right) \mathfrak{I}_{1}^{\alpha }z_{x}\left( t\right) ,z_{x}\left( t\right) \right) . \end{equation*} That is \(x\left( t\right) =f\left( t,x(t)\right) +\theta g(t,x\left( t\right) )+g\left( t,x\left( t\right) \right) \mathfrak{I}_{1}^{\alpha }z_{x}\left( t\right) \). Define the mapping \(P:C\left( \left[ 1,b\right] ,\mathbb{R}\right) \rightarrow C\left( \left[ 1,b\right] ,\mathbb{R}\right) \) as follows \begin{equation*} \left( Px\right) \left( t\right) =f\left( t,x(t)\right) +\theta g(t,x\left( t\right) )+\frac{g\left( t,x\left( t\right) \right) }{\Gamma \left( \alpha \right) }\int_{1}^{t}\left( \log \frac{t}{s}\right) ^{\alpha -1}z_{x}\left( s\right) \frac{ds}{s}. \end{equation*} It is clear that the fixed points of \(P\) are solutions of (1). Let \( x,y\in C\left( \left[ 1,b\right] ,\mathbb{R}\right) \), then we have
Theorem 8. Assume that \(f:\left[ 1,T\right] \times \mathbb{R}\rightarrow \mathbb{R}\), \(g:\left[ 1,T\right] \times \mathbb{R}\rightarrow\mathbb{R}\backslash \left\{ 0\right\} \) and \(h:\left[ 1,T\right] \times\mathbb{R}^{2}\rightarrow\mathbb{R}\) satisfy (H1), (H2) and (H3). If \(x\) is a solution of (1), then \begin{eqnarray*} \left\vert x\left( t\right) \right\vert &\leq &\left( \frac{\left( 1-K_{4}\right) \left( 1-\left( K_{1}+K_{2}\left\vert \theta \right\vert \right) \right) \Gamma \left( \alpha +1\right) +M_{g}K_{3}K\left( \log T\right) ^{\alpha }}{\left( 1-K_{4}\right) \left( 1-\left( K_{1}+K_{2}\left\vert \theta \right\vert \right) \right) \Gamma \left( \alpha +1\right) }\right) \left( Q_{1}+\left\vert \theta \right\vert Q_{2}+\frac{ M_{g}Q_{3}\left( \log T\right) ^{\alpha }}{\left( 1-K_{4}\right) \Gamma \left( \alpha +1\right) }\right) , \end{eqnarray*} where \(Q_{1}=\underset{t\in \left[ 1,T\right] }{\sup }\left\vert f\left( t,0\right) \right\vert ,\) \(Q_{2}=\underset{t\in \left[ 1,T\right] }{\sup } \left\vert g\left( t,0\right) \right\vert ,\ Q_{3}=\underset{t\in \left[ 1,T \right] }{\sup }\left\vert h\left( t,0,0\right) \right\vert \) and \(K\in\mathbb{R}^{+}\) is a constant.
Proof. Let \begin{equation*} \mathfrak{D}_{1}^{\alpha }\left( \frac{x\left( t\right) -f\left( t,x(t)\right) }{g\left( t,x(t)\right) }\right) =z_{x}\left( t\right) ,\text{ }x\left( 1\right) =\theta g\left( 1,x(1)\right) +f\left( 1,x(1)\right) , \end{equation*} then by Lemma 6, \(x(t)=f\left( t,x(t)\right) +\theta g(t,x\left( t\right) )+g\left( t,x\left( t\right) \right) \mathfrak{I}_{1}^{\alpha }z_{x}\left( t\right) \). Then by (H1), (H2) and (H3), for any \(t\in \left[ 1,T\right] \) we have \begin{align*} \left\vert x\left( t\right) \right\vert & \leq \left\vert f\left( t,x(t)\right) \right\vert +\left\vert \theta \right\vert \left\vert g(t,x\left( t\right) )\right\vert +\left\vert g\left( t,x\left( t\right) \right) \right\vert \left\vert \mathfrak{I}_{1}^{\alpha }z_{x}\left( t\right) \right\vert \\ & \leq \left\vert f\left( t,x(t)\right) -f\left( t,0\right) \right\vert +\left\vert f\left( t,0\right) \right\vert +\left\vert \theta \right\vert \left( \left\vert g(t,x\left( t\right) )-g(t,0)\right\vert +\left\vert g(t,0)\right\vert \right) +M_{g}\left\vert \mathfrak{I}_{1}^{\alpha }z_{x}\left( t\right) \right\vert \\ & \leq K_{1}\left\vert x\left( t\right) \right\vert +Q_{1}+\left\vert \theta \right\vert \left( K_{2}\left\vert x\left( t\right) \right\vert +Q_{2}\right) +M_{g}\mathfrak{I}_{1}^{\alpha }\left\vert z_{x}\left( t\right) \right\vert . \end{align*} On the other hand, for any \(t\in \left[ 1,T\right] \) we get \begin{eqnarray*} \left\vert z_{x}\left( t\right) \right\vert =\left\vert h\left( t,x\left( t\right) ,z_{x}\left( t\right) \right) \right\vert &\leq& \left\vert h\left( t,x\left( t\right) ,z_{x}\left( t\right) \right) -h\left( t,0,0\right) \right\vert +\left\vert h\left( t,0,0\right) \right\vert \\ &\leq& K_{3}\left\vert x\left( t\right) \right\vert +K_{4}\left\vert z_{x}\left( t\right) \right\vert +\left\vert h\left( t,0,0\right) \right\vert \\ &\leq& \frac{K_{3}}{1-K_{4}}\left\vert x\left( t\right) \right\vert +\frac{ Q_{3}}{1-K_{4}}. \end{eqnarray*} Therefore \begin{equation*} \left\vert x\left( t\right) \right\vert \leq K_{1}\left\vert x\left( t\right) \right\vert +Q_{1}+\left\vert \theta \right\vert \left( K_{2}\left\vert x\left( t\right) \right\vert +Q_{2}\right) +M_{g}\mathfrak{I} _{1}^{\alpha }\left( \frac{K_{3}}{1-K_{4}}\left\vert x\left( t\right) \right\vert +\frac{Q_{3}}{1-K_{4}}\right) . \end{equation*} Thus \begin{eqnarray*}\left( 1-\left( K_{1}+K_{2}\left\vert \theta \right\vert \right) \right) \left\vert x\left( t\right) \right\vert &\leq& Q_{1}+\left\vert \theta \right\vert Q_{2}+\frac{M_{g}Q_{3}\left( \log T\right) ^{\alpha }}{\left( 1-K_{4}\right) \Gamma \left( \alpha +1\right) } +\left( \frac{M_{g}K_{3}}{\left( 1-K_{4}\right) \left( 1-\left( K_{1}+K_{2}\left\vert \theta \right\vert \right) \right) }\right)\\&&\times \left( \mathfrak{I}_{1}^{\alpha }\left\{ \left( 1-\left( K_{1}+K_{2}\left\vert \theta \right\vert \right) \right) \left\vert x\left( t\right) \right\vert \right\} \right) . \end{eqnarray*} By Lemma 5, there is a constant \(K=K\left( \alpha \right) \) such that \begin{eqnarray*} \left( 1-\left( K_{1}+K_{2}\left\vert \theta \right\vert \right) \right) \left\vert x\left( t\right) \right\vert &\leq& Q_{1}+\left\vert \theta \right\vert Q_{2}+\frac{M_{g}Q_{3}\left( \log T\right) ^{\alpha }}{\left( 1-K_{4}\right) \Gamma \left( \alpha +1\right) }+\left( \frac{M_{g}K_{3}K\left( \log T\right) ^{\alpha }}{\left( 1-K_{4}\right) \left( 1-\left( K_{1}+K_{2}\left\vert \theta \right\vert \right) \right) \Gamma \left( \alpha +1\right) }\right) \\ && \times\left( Q_{1}+\left\vert \theta \right\vert Q_{2}+\frac{ M_{g}Q_{3}\left( \log T\right) ^{\alpha }}{\left( 1-K_{4}\right) \Gamma \left( \alpha +1\right) }\right) \\&\leq& \left( \frac{\left( 1-K_{4}\right) \left( 1-\left( K_{1}+K_{2}\left\vert \theta \right\vert \right) \right) \Gamma \left( \alpha +1\right) +M_{g}K_{3}K\left( \log T\right) ^{\alpha }}{\left( 1-K_{4}\right) \left( 1-\left( K_{1}+K_{2}\left\vert \theta \right\vert \right) \right) \Gamma \left( \alpha +1\right) }\right) \\ && \times \left( Q_{1}+\left\vert \theta \right\vert Q_{2}+\frac{ M_{g}Q_{3}\left( \log T\right) ^{\alpha }}{\left( 1-K_{4}\right) \Gamma \left( \alpha +1\right) }\right) . \end{eqnarray*} Hence \begin{eqnarray*} \left\vert x\left( t\right) \right\vert &\leq &\left( \frac{\left( 1-K_{4}\right) \left( 1-\left( K_{1}+K_{2}\left\vert \theta \right\vert \right) \right) \Gamma \left( \alpha +1\right) +M_{g}K_{3}K\left( \log T\right) ^{\alpha }}{\left( 1-K_{4}\right) \left( 1-\left( K_{1}+K_{2}\left\vert \theta \right\vert \right) \right) \Gamma \left( \alpha +1\right) }\right) \left( Q_{1}+\left\vert \theta \right\vert Q_{2}+\frac{ M_{g}Q_{3}\left( \log T\right) ^{\alpha }}{\left( 1-K_{4}\right) \Gamma \left( \alpha +1\right) }\right) . \end{eqnarray*} This completes the proof.