Let \(X\) be a linear space. A \(p\)-norm on \(X\) is a real-valued function \(\|\cdot\|_p\) on \(X\) with \(0 0\) such that \(\| f^h gx- gf^h x\|_p\leq R\,d_p(gx, Y^{f^hx}_q)\) for all \(x\in E\), \item[(b)] \(C_q\)-commuting if \(gfx= fgx\) for all \(x\in C_q(g,f)\), where \(C_q(g, f)= \bigcup\{C(g,f_\lambda): 0\leq \lambda\leq 1\}\), \(f_\lambda x= (1-\lambda)q+\lambda fx\), \(C(g,f_\lambda)\) being the set of common fixed points of \(g\) and \(f_\lambda\). \end{itemize}} This paper deals with the study of common fixed points for \(C_q\)-commuting and uniformly \(R\)-subweakly commuting mappings in \(p\)-normed spaces. As applications, results on invariant approximation for these mappings are derived. These results extend, generalize and unify various known results. It is remarked that all results of the paper remain valid in the setup of metrizable locally convex topological vector spaces \((X,d)\), where \(d\) is translation invariant and \(d(\lambda x,\lambda y)\leq\lambda d(x,y)\) for each \(\lambda\in ]0,1[\) and every \(x,y\in X\).