Uniqueness and stability of coupled sequential fractional differential equations with boundary conditions
In this paper, the authors study the following coupled system of nonlinear Caputo sequential fractional differential equations of the form \[ \begin{cases} ^{C}D^{(\alpha+1)}(D+k)u(t)=f(t,u(t),v(t)), \quad t\in [0,T], 1<a\leq 2, k>0,\\ {}^{C}D^{(\beta-1)}(D+k)u(t)=g(t,u(t),v(t)), \quad t\in [0,T], 1<\beta\leq 2, k>0,\\ u(0)=\lambda v'(T), \quad u'(0)=0,\\ v(0)=\mu u'(T), \quad v'(0)=0, \end{cases} \] where \({}^{C}D^{\xi}\) denote the Caputo fractional derivatives of order \(\xi\in\{\alpha,\beta\}\), \(f,g: [0,T]\times \mathbb{R}\to \mathbb{R}\) are given continuous functions and \(\lambda, \mu\) are real constants. Existence and uniqueness results are obtained via Banach fixed point theorem and Leray-Schauder alternative. Also Ulam-Hyers stability for the given problem is studied. The main results are well-illustrated with the aid of a numerical example.
scientific article; zbMATH DE number 7598556