Fixed point results for generalized convex orbital Lipschitz operators

Abstract Krasnoselskii’s iteration is a classical and important method for approximating the fixed point of an operator that satisfies certain conditions. Many authors have used this approach to obtain several famous fixed point theorems for different types of operators. It is well known that Kirk’s iteration can be seen as a generalization of Krasnoselskii’s iteration, in which the iterates are generated by a certain generalized averaged mapping. This approximation method is of great practical significance because the iterative formula contains more information related to the operator in question. The purpose of this study is to define weak ( α n , β i ) \left({\alpha }_{n},{\beta }_{i}) -convex orbital Lipschitz operators. These concepts not only extend the previously introduced Popescu-type convex orbital ( λ , β ) \left(\lambda ,\beta ) -Lipschitz operators in Fixed-point results for convex orbital operators, (Demonstr. Math. 56 (2023), 20220184), but also encompass many classical contractive operators. Popescu also proved a fixed point result for his proposed operator using the graphic contraction principle and obtained an approximation of the fixed point with Krasnoselskii’s iterates. To extend Popescu’s main results from Krasnoselskii’s iterative scheme to Kirk’s iterative scheme, several fixed point theorems are established, in which an appropriate Kirk’s iterative algorithm can be used to approximate the fixed point of a k k -fold averaged mapping associated with our presented convex orbital Lipschitz operators. These results not only generalize, but also complement the existing results documented in the previous literature.