  This paper is devoted to study the existence and uniqueness of solutions of a
one parameter family of nonlinear fractional differential equation with mixed
boundary value conditions. Riemann-Liouville fractional derivative is
considered. An exhaustive study of the sign of the related Green's function is
done.
  Under suitable assumptions on the asymptotic behavior of the nonlinear part
of the equation at zero and at infinity, and by application of the fixed theory
of compact operators defined in suitable cones, it is proved the existence of
at least one solution of the considered problem. Moreover it is developed the
method of lower and upper solutions and it is deduced the existence of
solutions by a combination of both techniques. In some particular situations,
the Banach contraction principle is used to ensure the uniqueness of solutions.
