AbstractThis article investigates a nonlinear
fractional Caputo–Langevin equationD^{\beta}(D^{\alpha}+\lambda)x(t)=f(t,x(t)),\quad 0<t<1,\,0<\alpha\leq 1,\,1<%
\beta\leq 2,subject to the multi-point boundary conditionsx(0)=0,\qquad\mathcal{D}^{2\alpha}x(1)+\lambda\mathcal{D^{\alpha}}x(1)=0,%
\qquad x(1)=\int_{0}^{\eta}x(\tau)\,d\tau\quad\text{for some }0<\eta<1,where {D^{\alpha}} is the Caputo fractional derivative
of order α, {f:[0,1]\times\mathbb{R}\to\mathbb{R}} is a given continuous
function, and λ is a real number.
Some new existence and uniqueness
results are obtained by applying an interesting fixed point theorem.
A study of a nonlinear coupled system of three fractional differential equations with nonlocal coupled boundary conditions