The purpose of this paper is to present the concept of multivariate contraction mapping in a locally convex topological vector spaces and to prove the multivariate contraction mapping principle in such spaces. The neighborhood-type error estimate formulas are also established. The results of this paper improve and extend Banach contraction mapping principle in the new idea.

The purpose of this paper is to present the concept of contraction mapping in a locally convex topological vector spaces and to prove the generalized contraction mapping principle in such spaces. The neighborhoodtype error estimate formula was also established. The results of this paper improve and extend Banach contraction mapping principle in new idea.

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:

The purpose of this paper is to prove the multivariate contraction mapping principle of N-variables mappings in Menger probabilistic metric spaces. In order to get the multivariate contraction mapping principle, the product spaces of Menger probabilistic metric spaces are subtly defined which is used as an important method for the expected results. Meanwhile, the relative iterative algorithm of the multivariate fixed point is established. The results of this paper improve and extend the contraction mapping principle of single variable mappings in the probabilistic metric spaces.

The purpose of this article is to introduce a new multidirectional hybrid shrinking projection iterative algorithm (or called cloud hybrid shrinking projection iterative algorithm) for solving the common element problems which consist of a generalized split equilibrium problems and fixed point problems for a family of countable quasi-Lipschitz mappings in the framework of Hilbert spaces. It is proved that under appropriate conditions, the sequence generated by the multidirectional hybrid shrinking projection method, converges strongly to some point which is the common fixed point of a family of countable quasi-Lipschitz mappings and the solution of the generalized split equilibrium problems. This iteration algorithm can accelerate the convergence speed of iterative sequence. The main results were also applied to solve split variational inequality problem and split optimization problems. Meanwhile, the main results were also used for solving common problems which consist of a generalized split equilibrium problems and fixed point problems for asymptotically nonexpansive mappings. The results of this paper improve and extend the previous results given in the literature.