Let \((X,d)\) be a metric space. A mapping \(W: X \times X\times[0,1]\to X\) is said to be a convex structure on \(X\) if for each \((x,y,\lambda)\in X\times X\times[0,1]\) and \(u\in X\),
\[ d(u,W(x,y,\lambda))\leq \lambda d(u,x)+(1-\lambda)d(u,y). \]
A metric space \(X\) together with the convex structure \(W\) is called a convex metric space. A nonempty subset \(F\) of \(X\) is said to be \(q\)-starshaped if there exists \(q\in F\) such that \(W(q,x,\lambda)\in F\) whenever \((x,\lambda)\in F\times [0,1]\). Let \(F\) be a \(q\)-starshaped subset of \(X\), \(S\) and \(T\) be self mappings of \(X\) and \(q\) be a fixed point of \(S\). The pair \((S,T)\) is called
(i) \(R\)-subweakly commuting if there exists a real number \(R>0\) such that
\[ d(TSx, STx) \leq R\,d(Sx,Y_q^{T(x)}) \]
for all \(x\in F\),
(ii) uniformly \(R\)-subweakly commuting if then exists a real number \(R>0\) such that
\[ d(T^nSx,ST^nx)\leq R\,d(Sx,Y_q^{T^n(x)}) \]
for all \(x\in F\), where \(Y^{T(x)}_q\equiv\{y_\lambda:y_\lambda=W(q,T(x),\lambda),\) \(\lambda\in[0,1]\}\) and \(d(Sx,y_q^{T(x)})\equiv\inf_{\lambda\in[0,1]}d(S(x),y_\lambda)\).
This paper deals with the study of common fixed points for \(R\)-subweakly and uniformly \(R\)-subweakly commuting mappings in the setting of a convex metric space. The authors also establish results on invariant approximation for these mappings. The results proved in the paper generalize various known results in the literature.