The Fixed Point and Mann Iteration of a Modified Isotonic Operator

Abstract:

      This paper consists of two parts. In the first part, we discuss the H\"{o}lder continuity of Cauchy-type integral operator $T$ of isotonic functions and the relationship between $\|T[f]\|_{\alpha}$ and $\|f\|_{\alpha}$. In the second part, firstly, we introduce a modified Cauchy-type integral operator $T'$ and demonstrate that the operator $T'$ has a unique fixed point by the Contraction Mapping Principle. Then we give the Mann iterative sequence and prove that the Mann iterative sequence strongly converges to the fixed point of the modified Cauchy-type integral operator $T'$.