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For a class of multivalued contractions with nonclosed, nonconvex values, the set of all fixed points is proved to be nonempty and arcwise connected. Two applications are then developed. In particular, one of them is concerned with some properties of the set of all classical trajectories corresponding to continuous controls for a given nonlinear control system.

In this paper, two deformation lemmas concerning a family of indefinite, non necessarily continuously differentiable functionals are proved. A critical point theorem, which extends the classical result of Benci-Rabinowitz [14, Theorem 5.29] to the above-mentioned setting, is then deduced.

The aim of this short note is to present a theorem that characterizes the existence of solutions to a class of higher order boundary value problems. This result completely answers a question previously set by the authors in [Differential Integral Equations (1993), 1119–1123].

Let $f:[a,b]\times {ℝ}^{n+1}\to ℝ$ be a Carath’eodory’s function. Let $\left\{t_h\right\}$, with $t_h\in [a,b]$, and $\left\{x_h\right\}$ be two real sequences. In this paper, the family of boundary value problems $$\left\{\begin{array}{c}{x}^{\left(k\right)}=f\left(t,x,x^{\text{'}},...,x^{\left(n\right)}\right)x^{\left(i\right)}\left(t_i\right)=x_i,i=0,1,...,k-1\hfill \end{array}\right.(k=n+1,n+2,n+3,...)$$ is considered. It is proved that these boundary value problems admit at least a solution for each $k\ge \nu$, where $\nu \ge n+1$ is a suitable integer. Some particular cases, obtained by specializing the sequence $\left\{t_h\right\}$, are pointed out. Similar results are also proved for the Picard problem.