On four-point nonlocal boundary value problems of nonlinear integro-differential equations of fractional order (Q711274)

The authors consider the four-point nonlocal boundary value problem in a Banach space \(X\), i.e. \[ ^cD^qx(t)=f(t,x(t),(\phi x)(t),(\psi x)(t)),\;\;0<t<1, \;\;1<q<2, \] \[ x'(0)+ax(\eta_1)=0, \;\;bx'(1)+x(\eta_2)=0,\;\;0<\eta_1\leq \eta_2<1, \] where \(^cD\) is the Caputo's fractional derivative, \(f:[0,1]\times X\times X\times X\times X\to X\) is continuous, \(\phi\), \(\psi\) are Volterra integral operators and \(a,c\in (0,1)\). By using fixed point arguments they prove an existence and uniqueness result of solutions for the problem above.