A Picard-S Iterative Method for Approximating Fixed Point of Weak-Contraction Mappings

Abstract

We study the convergence analysis of a Picard-S iterative method for a particular class of weak-contraction mappings and give a data dependence result for fixed points of these mappings. Also, we show that the Picard-S iterative method can be used to approximate the unique solution of mixed type Volterra-Fredholm functional nonlinear integral equation
\begin{equation*}
x\left( t\right) =F\left( t,x\left( t\right),\int\nolimits_{a_{1}}^{t_{1}}\cdots \int\nolimits_{a_{m}}^{t_{m}}K\left(
t,s,x\left( s\right) \right) ds,\int\nolimits_{a_{1}}^{b_{1}}\cdots
\int\nolimits_{a_{m}}^{b_{m}}H\left( t,s,x\left( s\right) \right) ds\right).
\end{equation*}
Furthermore, with the help of the Picard-S iterative method, we establish a data dependence result for the solution of integral equation mentioned above.