Steady-state solutions to the equations of motion of second-grade fluids with general Navier-type slip boundary conditions in Hölder spaces

We consider a boundary-value problem for the stationary flow of an incompressible second-grade fluid in a bounded domain. The boundary condition allows for no-slip, Navier-type slip and free slip on different parts of the boundary.
We first establish the well-posedness of a linear auxiliary problem by means of a fixed-point argument in which it is decomposed into a Stokes-type problem and two transport equations. Then we use the method of successive approximations to prove the unique solvability in Hölder spaces of the nonlinear problem with a sufficiently small body force.