The Relevance of a Metric Condition on a Pair of Operators in Common Fixed Point Theory

Let (X, d) be a complete metric space and $$f,g:X\rightarrow X$$ f , g : X → X be two operators satisfying some metric conditions on f and g. We denote by $$F_f$$ F f the fixed point set of f. In this paper, we will study the following problems. I. What are the metric conditions on f and g which imply that all the following conclusions hold? 1. $$F_{f^n}=F_{g^n}=\{ x^* \}$$ F f n = F g n = { x ∗ } for each $$n\in \mathbb {N}^*$$ n ∈ N ∗ ; 2. for each $$x_0\in X$$ x 0 ∈ X , the sequence $$(x_n)_{n\in \mathbb {N}}$$ ( x n ) n ∈ N defined by $$x_{2n}=(gf)^n(x_0), \;\;\; x_{2n+1}=f(x_{2n}),\;\;\forall n\ge 0,$$ x 2 n = ( g f ) n ( x 0 ) , x 2 n + 1 = f ( x 2 n ) , ∀ n ≥ 0 , converges to $$x^*\in X$$ x ∗ ∈ X ; 3. for each $$y_0\in X$$ y 0 ∈ X , the sequence $$(y_n)_{n\in \mathbb {N}}$$ ( y n ) n ∈ N defined by $$y_{2n}=(fg)^n(y_0), \;\;\; y_{2n+1}=g(y_{2n}),\;\;\forall n\ge 0,$$ y 2 n = ( f g ) n ( y 0 ) , y 2 n + 1 = g ( y 2 n ) , ∀ n ≥ 0 , converges to $$x^*\in X$$ x ∗ ∈ X ; 4. for each $$x_0\in X$$ x 0 ∈ X , the sequence $$(f^n(x_0))_{n\in \mathbb {N}}$$ ( f n ( x 0 ) ) n ∈ N converges to $$x^*\in X$$ x ∗ ∈ X ; 5. for each $$x_0\in X$$ x 0 ∈ X , the sequence $$(g^n(x_0))_{n\in \mathbb {N}}$$ ( g n ( x 0 ) ) n ∈ N converges to $$x^*\in X$$ x ∗ ∈ X . II. Under which assumptions does the data dependence phenomenon for the common fixed point problem hold? Other problems, such as well-posedness, Ostrowski property and Ulam-Hyers stability for the common fixed point problem are also considered.