Let $C$ be a closed convex curve of class $C^2$ in the plane. We consider the domain bounded by $C$ a billiard table. Assume that the convex billiard of $C$ is integrable and satisfies a certain property. The property is that the limiting leaves are either closed curves or discrete points in the phase space. Then the set of points with irrational slopes make invariant circles of class $C^1$. If the sets of points with rational slopes do not make invariant circles, then they contains two invariant circles such that they are of class $C^1$ except at finitely many points in $C$.

This paper deals with the controllability of impulsive second order integrodifferential systems in Banach spaces. Sufficient conditions for the controllability are derived with the help of the fixed point theorem due to Sadovskii and the theory of strongly continuous cosine family of operators. An example is provided to show the effectiveness of the proposed results. Further, we study the controllability of second order integrodifferential evolution systems with impulses by using the Schaefer fixed-point theorem.

We prove that the Ricci tensor of a real hypersurface of a complex space form $M^n(c)$, $c\neq 0$, $n\geq 3$, satisfies Ricci pseudo $\eta$-parallel condition if and only if $M$ is pseudo-Einstein.