Fixed points of functions with max-weighted quasi-arithmetic mean operator

Let λ∈(0,1)λ∈(0,1) and f be a continuous, strictly monotone real-valued function. The weighted quasi-arithmetic mean of two numbers a, b   is defined by a⊗b=f−1(λf(a)+(1−λ)f(b))a⊗b=f−1(λf(a)+(1−λ)f(b)). Let A=[aij]A=[aij] be an n×nn×n real matrix and x=(x1,x2,…,xn)T∈Rnx=(x1,x2,…,xn)T∈Rn. We construct a function ψ(A)=(ψ1(A),ψ2(A),…,ψn(A)):Rn→Rn by ψj(A)(x)=max1⩽l⩽n{xl⊗alj} for all j=1,2,…,nj=1,2,…,n. In this paper we show that ψ(A)ψ(A) has a unique fixed point xˆ(A). Moreover, it can be shown that for each x∈Rnx∈Rn the sequence {x(A,k)}{x(A,k)}, generated by the following iterative scheme: x(A,0)=xx(A,0)=x and x(A,k)=ψ(A)(x(A,k−1))x(A,k)=ψ(A)(x(A,k−1)) for all k⩾1k⩾1, converges to the unique fixed point xˆ(A). Besides, some properties of the fixed point are derived. As an application, our results imply that the max-weighted quasi-arithmetic mean powers of any matrix are always convergent. The continuity of the function η:Mn×n(R)→Rnη:Mn×n(R)→Rn defined by η(A)=xˆ(A) is proposed as well.