Remarks on Some Fixed Point Theorems

Remarks on Lipschitzian Mappings and Some Fixed Point Theorems

Let X, Y be the normed spaces, C ⊂ X a convex set, and T : C → Y a continuous mapping. Some weak conditions implying the Lipschitz continuity of T are presented. Applications to the fixed point theory and theory of composition operators are presented.

Remarks on Recent Fixed Point Theorems

Remarks on Certain Selected Fixed Point Theorems

Theorem 1. Let A, S, I, and J be self-mappings of a complete metric space (X,d) such that the pairs (A,I) and (S,J) are commuting and A(X)⊂ J(X) and S(X)⊂ I(X) such [ 1+pd(Ax,Sy)]d(Ix,Jy) ≤ pmax{d(Ix,Ax)·d(Sy,Jy),d(Ix,Sy)·d(Jy,Ax)} +φ(d(Ax,Sy),d(Ix,Ax),d(Sy,Jy),d(Ix,Sy),d(Jy,Ax)), (1) for all x,y ∈ X where p ≥ 0 and φ ∈ Ψ . Then A, S, I, and J have a unique common fixed point provided one of th...

Some Remarks on Commuting Fixed Point Free Automorphisms of Groups

In this a‌r‌t‌i‌c‌l‌e we will find n‌e‌c‌e‌s‌s‌a‌r‌y and s‌u‌f‌f‌i‌c‌i‌e‌n‌t c‌o‌n‌d‌i‌t‌i‌o‌n‌s for a f‌i‌x‌e‌d  p‌o‌i‌n‌t f‌r‌e‌e autom‌o‌r‌p‌h‌i‌s‌m (f‌p‌f a‌u‌t‌o‌m‌o‌r‌p‌h‌i‌s‌m) o‌f a g‌r‌o‌u‌p t‌o b‌e a c‌o‌m‌m‌u‌t‌i‌n‌g a‌u‌t‌o‌m‌o‌r‌p‌h‌i‌s‌m. F‌o‌r a given prim‌e ‌ we  f‌i‌n‌d t‌h‌e s‌m‌a‌l‌l‌e‌s‌t o‌r‌d‌e‌r o‌f a n‌o‌n a‌b‌e‌l‌i‌a‌n p-g‌r‌o‌u‌p a‌d‌m‌i‌t‌t‌i‌n‌g a c‌o‌m‌m‌u‌t‌i‌n‌g f...

Some Fixed Point Theorems

Introduction. We wish to summarize here some new asymptotic fixed point theorems. By an asymptotic fixed point theorem we mean roughly a theorem in functional analysis in which the existence of fixed points of a map ƒ is proved with the aid of assumptions on the iterates f of ƒ. Such theorems have proved of use in the theory of ordinary and functional differential equations (see [7], [8], [9] a...