Suppose $C$ is any nonempty subset of a Banach space $X$. A mapping $T:Crightarrow C$ is said to satisfy condition $(B_{gamma,mu})$ if there exists $gammain[0,1]$ and $muin[0,frac{1}{2}]$ with $2muleqgamma$ such that for each two elements $x,yin C$,

begin{equation*}

gamma||x-Tx||leq||x-y||+mu||y-Ty||

end{equation*}

begin{equation*}

text{implies} ||Tx-Ty||leq(1-gamma)||x-y||+mu(||x-Ty||+||y-Tx||).

end{equation*}

In this research, we suggest some convergence results for these mappings under a up-to-date iterative process in a Banach space setting. Our results are new and improve some known results of the literature.