Using functional arguments, some existence results for the infinite boundary value problem $\dot{x}=F\left(t,x\right),x\left(-\infty \right)=x\left(+\infty \right)$ are given. A solution of this problem is frequently called, from Poincaré, homoclinic.

We determine the class of functions, the divided difference of which, at $n$ distinct numbers, is a continuous function of the product of these numbers.

We consider the nonlinear second order conjugate eigenvalue problem on a time scale: $y{}^{\Delta \Delta }\left(t\right)+\lambda a\left(t\right)f\left(y\left(\sigma \left(t\right)\right)\right)=0,t\in \left[0,1\right],y\left(0\right)=0=y\left(\sigma \left(1\right)\right)$ . Values of the parameter $\lambda$ (eigenvalues) are determined for which this problem has a positive solution. The methods used here extend recent results by allowing for a broader class of functions for $a\left(t\right)$.

We prove the existence of vortex local minimizers to Ginzburg-Landau functional with a global magnetic effect. A domain perturbating method is developed, which allows us to extend a local minimizer on a nonsimply connected superconducting material to the local minimizer with vortex on a simply connected material.

We prove a Morse type lemma for, possibly degenerate, critical points of a $C^1$ function twice strongly differentiable at those points, which allows us to recover, for Finsler metrics, the theorem of Gromoll and Meyer on the existence of infinitely many closed geodesics.

We consider quasilinear strongly resonant problems with discontinuous right-hand side. To develop an existence theory we pass to a multivalued problem by, roughly speaking, filling in the gaps at the discontinuity points. We prove the existence of at least three nontrivial solutions. Our approach uses the nonsmooth critical point theory for locally Lipschitz functionals due to Chang (1981) and a generalized version of the Ekeland variational principle. At the end of the paper we show that the nonsmooth Palais-Smale (PS)-condition implies the coercivity of the functional, extending this way a well-known result of the “smooth” case.