In this paper, we consider a countable family of surjective mappings {Tn}n∈N satisfying certain quasi-contractive conditions. We also construct a convergent sequence { Xn } n c∈Nby the quasi-contractive conditions of { Tn } n ∈N and the boundary condition of a given complete and closed subset of a cone metric space X with convex structure, and then prove that the unique limit x＂ of {xn}n∈N is the unique common fixed point of {Tn}n∈N. Finally, we will give more generalized common fixed point theorem for mappings {Ti,j}i,j∈N. The main theorems in this paper generalize and improve many known common fixed point theorems for a finite or countable family of mappings with quasi-contractive conditions.
This paper deals with homology groups induced by the exterior algebra generated by the simplicial compliment of a simplicial complex K. By using ech homology and Alexander duality, the authors prove that there is a duality between these homology groups and the simplicial homology groups of K.
In 1981, Cohen constructed an infinite family of homotopy elements ζk∈ π＊（S） represented by h0bk ∈ ExtA3,2（p-1）（pk＋1＋1）（z/p,Z/p） in the Adams spectral sequence, where p 〉 2 and k ≥ 1. In this paper, we make use of the Adams spectral sequence and the May spectral sequence to prove that the composite map ζn-1β2γs＋3 is nontrivial in the stable homotopy groups of spheres πt（s,n）-s-8（S）, where p ≥7, n 〉 3, 0≤s 〈p-5 andt（s,n） =2（p-1）[pn＋（s＋3）p2＋（s＋4）p＋（s＋3）]＋s.
