New common fixed point theorems and invariant approximation in convex metric spaces

As relevant definitions and results are already discussed in preceding chapters (i.e. Chapters 1 and 2), so there is no reason to revisit those again. As far as results of this chapter are concerned, these are proved using newly introduced concepts of subcompatibility and subsequential continuity contained in [33] (Bouhadjera, GodetThobie, Common fixed theorems for pairs of subcompatible maps, 17 June 2009. [math.FA]). Also, we prove common fixed point theorems for a pair of maps in convex metric spaces which are essentially patterned after a theorem of Huang and Li (Fixed point theorems of compatible mappings in convex metric spaces, Soochow J. Math. 22(3) (1996), 439—447). In process, we also derive some related fixed point theorems and utilize a few of these results to prove theorems on best approximation. Essentially, the following theorem due to Huang and Li [93] has inspired our studies in this chapter. Theorem 3.1.1.([93]) Let (X , d) be a convex metric space and K be a nonempty closed convex subset of X . If (T ,S) is a compatible pair of self-mapping defined on K such that for all x, y ∈ K, d(T x,T y) ≤ ad(Sx,Sy) + bmax{d(Sx, T x), d(Sy, T y)} +cmax{d(Sx,Sy), d(Sx, T x), d(Sy, T y), 1 2 (d(Sx, T y) + d(Sy, T x))} (3.1.1.1)