Existence of solutions for a system of fractional differential equations with coupled nonlocal boundary conditions
The authors consider the following system of nonlinear fractional differential equations \[ \begin{aligned} ^{c}D^{\alpha}_{0+}u(t)&=f(t,v(t),^{c}D^p_{0+}v(t)),\;t\in (0,1),\;\alpha\in (n-1,n],\;p\in (0,1),\\ ^{c}D^{\beta}_{0+}v(t)&=g(t,u(t),^{c}D^q_{0+}u(t)),\;t\in (0,1),\;\beta\in (m-1,m],\;q\in (0,1),\end{aligned} \] with solutions subjected to the coupled nonlocal integral boundary conditions \[ \begin{aligned} u(0)&=\phi(v),\;u'(0)=0, \ldots, u^{(n-2)}(0)=0, u(1)=\lambda I^{\gamma}_{0+}v(\xi),\;\xi\in (0,1),\\ v(0)&=\psi(v),\;v'(0)=0, \ldots, v^{(m-2)}(0)=0, v(1)=\mu I^{\delta}_{0+}u(\eta),\;\eta\in (0,1),\end{aligned} \] where \(^{c}D^{\chi}_{0+}\) denotes the Caputo fractional derivative of order \(\chi\), \(I^{\kappa}_{0+}\) denotes the Riemann-Liouville fractional integral of order \(\kappa,\) \(f,g: [0,1]\times {\mathbb R}\times {\mathbb R}\to {\mathbb R}\), \(\phi, \psi: C([0,1],{\mathbb R})\to \mathbb R\) are continuous functions and \(\lambda, \mu \in \mathbb R\). Existence results are obtained by using the Schauder fixed point theorem and the nonlinear alternative of Leray-Schauder type. Examples illustrating the obtained results are also presented.
Caputo fractional differential equations
scientific article; zbMATH DE number 6958684
scientific article