A uniqueness method to a new Hadamard fractional differential system with four-point boundary conditions

Abstract

Abstract In this article, we discuss a new Hadamard fractional differential system with four-point boundary conditions {DαHu(t)+f(t,v(t))=lf,t∈(1,e),DβHv(t)+g(t,u(t))=lg,t∈(1,e),u(j)(1)=v(j)(1)=0,0≤j≤n−2,u(e)=av(ξ),v(e)=bu(η),ξ,η∈(1,e), $$\textstyle\begin{cases} {}^{H} D^{\alpha}u(t)+f(t,v(t))=l_{f},\quad t\in(1,e),\\ {}^{H} D^{\beta}v(t)+g(t,u(t))=l_{g},\quad t\in(1,e),\\ u^{(j)}(1)=v^{(j)}(1)=0, \quad 0\leq j\leq n-2,\\ u(e)=av(\xi),\qquad v(e)=bu(\eta),\quad \xi, \eta\in(1,e), \end{cases} $$ where a,b $a,b$ are two parameters with 00 $l_{f}, l_{g}>0$ are constants, and DαH,HDβ ${}^{H} D^{\alpha}, {}^{H} D^{\beta}$ are the Hadamard fractional derivatives of fractional order. Based upon a fixed point theorem of increasing φ- (h,r) $(h,r)$-concave operators, we establish the existence and uniqueness of solutions for the problem dependent on two constants lf,lg $l_{f}, l_{g}$.

Keywords

• Hadamard fractional derivative
• Existence and uniqueness
• φ- ( h , r ) $(h,r)$ -concave operator