  Let $M$ be a Riemannian $2$-sphere. A classical theorem of Lyusternik and
Shnirelman asserts the existence of three distinct simple non-trivial periodic
geodesics on $M$. In this paper we prove that there exist three simple periodic
geodesics with lengths that do not exceed $20d$, where $d$ is the diameter of
$M$.
  We also present an upper bound that depends only on the area and diameter for
the lengths of the three simple periodic geodesics with positive indices that
appear as minimax critical values in the classical proofs of the
Lyusternik-Shnirelman theorem.
  Finally, we present better bounds for these three lengths for "thin" spheres,
when the area $A$ is much less than $d^2$, where the bounds for the lengths of
the first two simple periodic geodesics are asymptotically optimal, when
${A\over d^2}\longrightarrow 0$.
