Non-self multivariate contraction mapping principle in Banach spaces

The purpose of this article is to prove the non-self multivariate contraction mapping principle in a Banach space. The main result is the following: let C be a nonempty closed convex subset of a Banach space (X, || . ||). Let T : C → X be a weakly inward N-variables non-self contraction mapping. Then T has a unique multivariate fixed point p ∈ C. That is, there exists a unique element p ∈ C such that T(p, p, . . ., p) = p. In order to get the non-self multivariate contraction mapping principle, the inward and weakly inward N-variables non-self mappings are defined. In addition, the meaning of N-variables non-self contraction mapping T : C → X is the following: ||T x − T y|| ≤ h∇( ||x1 − y1|| , ||x2 − y2|| , · · · , ||xN − yN || ) for all x = (x1, x2, · · · , xN), y = (y1, y2, · · · , yN) ∈ CN, where h ∈ (0, 1) is a constant, and ∇ is an N-variables real function satisfying some suitable conditions. The results of this article improve and extend the previous results given in the literature.