It is well known that deformation theorems are the basic tools in critical point theory. They can be derived under a condition of Palais-Smale type ((PS), for short). In the classical setting of a C1 functional f defined on a Banach space X (or a C2 Finsler manifold), we refer to [15]; for a continuous functional f defined on a complete metric space X, we refer to [8], the results of which incl...

This is an expanded version of the talk given in the meeting to celebrate the 70th birthday of Prof. Giovanni Castellani. We shortly survey some results in Critical Point Theory.

Since the development of the calculus of variations there has been interest in finding critical points of functionals. This was intensified by the fact that for many equations arising in practice the solutions are critical points of functionals. If a functional G is semibounded, one can find a Palais-Smale (PS) sequence G(uk) → a, G(uk) → 0. These sequences produce critical points if they have ...

Fractional differential equations have been of great interest recently. This is because of both the intensive development of the theory of fractional calculus itself and the applications of such constructions in various scientific fields such as physics, mechanics, chemistry, engineering, etc. Differential equations with impulsive effects arising from the real world describe the dyn...

1. Generalized Sphere Theorem As a first application of the critical point theory of distance funciton, we shall prove Theorem 1.1 (Grove-Shiohama). Let (M, g) be a complete simply connected Riemann-ian manifold with K > 1 4 and diam(M, g) ≥ π, then M is homeomorphic to S m. So Grove-Shiohama's theorem implies the sphere theorem. Proof. Let p, q ∈ M so that dist(p, q) = diam(M, g). In what foll...