Approximate fixed point theorems of cyclical contraction mapping

S.A.M. Mohsenialhosseini

Let $X_{i}$, $i=1,\dots,m$, are subsets of a metric space $X$ and also $T: \cup_{i=1}^{m}X \to \cup_{i=1}^{m}X_{i}$, and $T(X_{1})\subseteq X_{2},\ldots,T(X_{m-1})\subseteq X_{m},T(X_{m})\subseteq X_{1}$. We are going to consider element $X\in \cup_{i=1}^{m}X_{i}$ such that $d(x,Tx)\leq\epsilon$ for some $\epsilon>0$. The existence results regarding approximate fixed points proved for the several operators such as Chatterjea and Zamfirescu on metric space (not necessarily complete). These results can be exploited to establish new approximate fixed point theorems for cyclical contraction maps. In addition, there is a new class of cyclical operators and contraction mapping on metric space (not necessarily complete) which do not need to be continuous. Finally, some examples are presented to illustrate our results for new approximate fixed point theorems on cyclical contraction maps.