Let \((X,d)\) be a metric space. A map \(W:X^2\times [0,1]\rightarrow X\) is called a convex structure on \(X\) if \[ d(u,W(x,y,\lambda))\leq \lambda d(u,x)+(1-\lambda) d(u,y), \,\forall x,y,u\in X,\,\forall \lambda \in [0,1]. \] A metric space \((X,d)\) together with a convex structure \(W\) on \(X\) is called a convex metric space and is denoted by \((X,d,W)\). A convex metric space \((X,d,W)\) is said to satisfy condition \((W_1)\) if for all \(x,y,z \in X\) and \(t\in (0,1)\), \[ d(W(x,y,t),W(z,y,t))\leq t d(x,z). \] A nonempty subset of a convex metric space \((X,d,W)\) is said to be convex if for all \(x,y\in C\) and \(\lambda \in [0,1]\), \[ W(x,y,\lambda)\in C. \] The main result of the paper is stated as follows.
Theorem 2.4. Let \(C\) be a nonempty, closed, convex and bounded subset of a complete and uniformly convex metric space \((X,d,W)\) satisfying \((W_1)\). If \(T:C\rightarrow C\) is a continuous map satisfying \[ d(Tx,Ty)\leq a_1d(x,y)+a_2 d(Tx,x)+a_3 d(Ty,y)+a_4 d(Tx,y)+a_5 d(Ty,x), \] for all \(x,y \in C\), where \(a_i\geq 0\) and \(\sum_{i=1}^{5}a_i\leq 1\), then \(T\) has a fixed point in \(C\).
A convergence result (Theorem 3.1) for a Krasnoselskii type iteration used to approximate fixed points of a generalized nonexpansive mapping, is also given.