This paper presents an extension of the Poincaré-Hopf theorem to generalized critical points of a function on a compact region with nonsmooth boundary, M , defined by a finite number of smooth inequality constraints. Given a function F M → , we define the generalized critical points of F over M , define the index for the critical point, and show that the sum of the indices of the critical point...

A position p in a word w is critical if the minimal local period at equal to global of . According Critical Factorisation Theorem all words length least two have point. We study number η ( ) points square-free ternary , i.e. over three letter alphabet. show that sufficiently long satisfy ≤| |− 5 where | denotes Moreover, bound reached by infinitely many words. On other hand, every has |∕4 point...

The main purpose of this paper is to establish a three critical points result without assuming the coercivity of the involved functional. To this end, a mountain-pass theorem, where the usual Palais-Smale condition is not requested, is presented. These results are then applied to prove the existence of three solutions for a two-point boundary value problem with no asymptotic conditions.

In this paper we study Neumann-type p(x)-Laplacian equation with nonsmooth potential. Firstly, applying a version of the non-smooth three-critical-points theorem we obtain the existence of three solutions of the problem in W (Ω). Finally, we obtain the existence of at least two nontrivial solutions, when α− > p.

This paper deals with the study of a class perturbed nonlinear fractional p-Laplacian differential systems, where by using variational method, two control parameters together recent three critical points theorem Bonanno and Candito for differentiable functionals existence weak solutions has been proved.

the main aim of this paper is to introduce three classes $h^0_{p,q}$, $h^1_{p,q}$ and $th^*_p$ of $p$-harmonic mappings and discuss the properties of mappings in these classes. first, we discuss the starlikeness and convexity of mappings in $h^0_{p,q}$ and $h^1_{p,q}$. then establish the covering theorem for mappings in $h^1_{p,q}$. finally, we determine the extreme points of the class $th^*_{p}$.

‎we study the existence of soliton solutions for a class of‎ ‎quasilinear elliptic equation in $mathbb{textbf{r}}^2$ with critical exponential growth‎. ‎this model has been proposed in the self-channeling of a‎ ‎high-power ultra short laser in matter‎.