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We introduce a new implicit iterative scheme with perturbation for finding the approximate solutions of a hierarchical variational inequality, that is, a variational inequality over the common fixed point set of a finite family of nonexpansive mappings. We establish some convergence theorems for the sequence generated by the proposed implicit iterative scheme. In particular, necessary and sufficient conditions for the strong convergence of the sequence are obtained.

Simple-homotopy for cell complexes is a special type of topological homotopy constructed by elementary collapses and elementary expansions. In this paper, we introduce graph homotopy for graphs and Graham homotopy for hypergraphs and study the relation between the two homotopies and the simple-homotopy for cell complexes. The graph homotopy is useful to describe topological properties of discretized geometric figures, while the Graham homotopy is essential to characterize acyclic hypergraphs and acyclic relational database schemes.

For any 2⩽r⩽∞,n⩾2, we prove the existence of an open set U of C^r‑self‑mappings of any n‑manifold so that a generic map f in U displays a fast growth of the number of periodic points: the number of its n‑periodic points grows as fast as asked. This complements the works of Martens–de Melo–van Strien, Kaloshin, Bonatti–Díaz–Fisher and Turaev, to give a full answer to questions asked by Smale in 1967, Bowen in 1978 and Arnold in 1989, for any manifold of any dimension and for any smoothness. Furthermore, for any 1⩽r<∞ and any k⩾0, we prove the existence of an open set \hat U of k-parameter families in U such that for a generic (f_p)^p ∈ \hat U, for every ∥p∥⩽1, the map f_p displays a fast growth of the number of periodic points. This gives a negative answer to a problem asked by Arnold in 1992 in the finitely smooth case.