Definition Let $M$ be a subset of a metric space $X$. The set $M$ is called $q$- starshaped if $ kx+(1-k)q \in M$ for all $x\in M$ and $k\in[0,1]$, where $q$ is an element of $M$.
Definition Let $M$ be subset of metric space $X$ and $T, I :M\to M$ be self map. Then the pair $(T,I)$ is called compatible if $$\lim_{n\to\infty}d(ITx_n, TIx_n)=0$$ whenever ${x_n}$ is a sequence such that $\lim_{n\to\infty}Ix_n=\lim_{n\to\infty}Tx_n=t$ for some $t\in M$.
Definition Let $M$ be subset of metric space $X$ and $T :M\to M$ be self map. A point $x\in M$ is said to be fixed point of $T$ if $Tx=x$.
Definition Let $M$ be a $q$ starshaped subset of a metric space $X$ and Let $M$ be subset of metric space $X$ and $T, I :M\to M$ be self map with $q\in F(I)$, where $F(I) $ is the set of fixed point of $I$. Define $S_{q}(I,T)=\cup_{k\in [0,1]}S(I,T_{k})$ where $T_{k}(x)=(1-k)q+kT(x)$ and $$S(I,T_{k})=\{\{x_{n}\}\in M: \lim_{n\to\infty}I(x_n)=\lim_{n\to\infty}T_{k}(x_n)=t\in M \implies\lim_{n\to\infty}d(T_{k}I(x_n), IT_{k}(x_n)=0\}.$$ Then $I$ and $T$ are sub-compatible if $$\lim_{n\to\infty}d(TIx_n, ITx_n)=0$$
for all {$x_n$}$\in S_{q}(I,T)$.
Question : Is it true that every sub-compatible maps are compatible?
Question Explain the definition of $S(I,T_k)$. What type of sequences lies in $S(I,T_k)$.
