Fixed point theorem of generalized quasi-contractive mapping in cone metric space

In this paper, we introduce the generalized quasi-contractive mapping ff in a cone metric space (X,d)(X,d). ff is called a generalized quasi-contractive if there is a real λ∈[0,1)λ∈[0,1) such that for all x,y∈Xx,y∈X, d(fx,fy)≤λsd(fx,fy)≤λs for some s∈co{0,d(fx,fy),d(x,y),d(x,fx),d(y,fy),d(x,fy),d(y,fx)}.s∈co{0,d(fx,fy),d(x,y),d(x,fx),d(y,fy),d(x,fy),d(y,fx)}. It is proved that if XX is a complete cone metric space with normal cone then ff has a unique fixed point. A example is given, which shows that our result is a genuine generalization of quasi-contractive mapping.