Common Fixed Point and Invariant Approximation Results in Certain Metrizable Topological Vector Spaces

Let X be a linear space. A p-norm on X is a real-valued function on X with 0 < p ≤ 1, satisfying the following conditions: (i) ‖x‖p ≥ 0 and ‖x‖p = 0⇔ x = 0, (ii) ‖αx‖p = |α|p‖x‖p, (iii) ‖x+ y‖p ≤ ‖x‖p +‖y‖p for all x, y ∈ X and all scalars α. The pair (X ,‖,‖p) is called a p-normed space. It is a metric linear space with a translation invariant metric dp defined by dp(x, y)= ‖x− y‖p for all x, y ∈ X . If p = 1, we obtain the concept of the usual normed space. It is wellknown that the topology of every Hausdorff locally bounded topological linear space is given by some p-norm, 0 < p ≤ 1 (see [9] and references therein). The spaces lp and Lp, 0 < p ≤ 1 are p-normed spaces. A p-normed space is not necessarily a locally convex space. Recall that dual space X∗ (the dual of X) separates points of X if for each nonzero x ∈ X , there exists f ∈ X∗ such that f (x) = 0. In this case the weak topology on X is well-defined and is Hausdorff. Notice that if X is not locally convex space, then X∗ need not separate the points of X . For example, if X = Lp[0,1], 0 < p < 1, then X∗ = {0} ([12, pages 36 and 37]). However, there are some non-locally convex spaces X (such as the p-normed spaces lp, 0 < p < 1) whose dual X∗ separates the points of X . Let X be a metric linear space and M a nonempty subset of X . The set PM(u)= {x ∈ M : d(x,u)= dist(u,M)} is called the set of best approximants to u∈ X out of M, where dist(u,M) = inf{d(y,u) : y ∈M}. Let f :M →M be a mapping. A mapping T :M →M