Existence and Ulam stability results for a class of boundary value problem of neutral pantograph equations with complex order (Q827615)

Let the fractional BVPs for the nonlinear neutral pantograph equations with complex order be given by \[ \left\{ \begin{array} [c]{c} D_{0^{+}}^{\theta}x\left( t\right) =f\left( t,x\left( t\right) ,x\left( \lambda t\right) ,D_{0^{+}}^{\theta}x\left( \lambda t\right) \right) ,\text{ }\theta=m+i\alpha\text{, }\lambda\in\left( 0,1\right) \text{, }t\in J\text{:=}\left[ 0,T\right] ,\\ ax\left( 0\right) +bx\left( T\right) =c, \end{array} \right. \] where \(D_{0^{+}}^{\theta}\) is the Caputo fractional derivative of order \(\theta\in \mathbb{C} \). Let \(\alpha\in \mathbb{R} ^{+}\), \(0<\alpha<1\), \(m\in\left( 0,1\right] \), and \(f:J\times \mathbb{R} ^{3}\rightarrow \mathbb{R} \) \ is a given continuous function. Here \(a,b,c\) are real constants with \(a+b\neq0.\) The following lemma is used: Let \(\theta=m+i\alpha\), \ \(0<m\leq1\), \(\alpha\in \mathbb{R} ^{+}\) and \(f:J\times \mathbb{R} ^{3}\rightarrow \mathbb{R} \) be continous. A function \(x\) is a solution of the fractional integral equation \[ x\left( t\right) =x_{0}+\frac{1}{\Gamma\left( \theta\right) }\int_{0} ^{t}\left( t-s\right) ^{\theta-1}f\left( s,x\left( s\right) ,x\left( \lambda s\right) ,D_{0^{+}}^{\theta}x\left( \lambda s\right) \right) ds \] if and only if \(x\) is a solution of the initial value problem for the following fractional neutral pantograph differential equation with complex order \begin{align*} D_{0^{+}}^{\theta}x\left( t\right) & =f\left( t,x\left( t\right) ,x\left( \lambda t\right) ,D_{0^{+}}^{\theta}x\left( \lambda t\right) \right) ,\text{ }\lambda\in\left( 0,1\right) \text{, }t\in J:=\left[ 0,T\right] \\ x\left( 0\right) & =x_{0}. \end{align*} By means of Banach's contraction principle, and Schaefer's fixed point theorem, they obtain the existence and uniqueness, Ulam's-Hyers and Ulam's-Hyers-Rassias stability result for the above problem.