### Existence of Common Fixed Points for Weakly Compatible and Cq-Commuting Maps and Invariant Approximations

#### Abstract

We prove the existence of common fixed points for three selfmaps $A$, $S$ and $T$ defined on a nonempty subset $E$ of a metric space $(X,d)$ under the assumptions that (i) $A$, $S$ and $T$ satisfy a contractive condition given by $(2.1.1)$; (ii) $S(E)\subseteq A(E)$ and $T(E)\subseteq A(E)$; and (iii) the pairs $(A,S)$ and $(A,T)$ are weakly compatible. We use this result to find the common fixed points of three $C_{q}$-commuting continuous selfmaps defined on $q$-starshaped subset $E$ of a normed space $X$ satisfying certain nonexpansive inequality involving rational expressions.We apply these results to prove the existence of common fixed pointsfrom the set of best approximations.

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