Abstract
In this paper, we introduce a new class of mappings and investigate their fixed point property. In the first direction, we prove a fixed point theorem for general higher-order contraction mappings in a given metric space and finally prove an approximate fixed point property for general higher-order nonexpansive mappings in a Banach space.
Highlights
Given a complete metric space (X, d), the most well-studied types of self-maps are referred to as Lipschitz mappings, which are given by the metric inequality: d(Tx, Ty) ≤ α d(x, y), (1)for all x, y ∈ X, where α ≥ 0 is a real number, usually referred to as the Lipschitz constant of T
In 2015, Ezearn [4] introduced a new class of mappings called higher-order Lipschitz mappings which are seen as a generalization of inequality (1). us, a mapping Journal of Mathematics
Ezearn later provided a remetrisation argument that relates his higher-order Lipschitz mappings to Lipschitz mappings. His remetrisation of the original metric space does not necessarily result in a complete metric space, and as a result, he provided a completion of the remetrised space and an extension of the higher-order Lipschitz mapping into a complete remetrised space
Summary
Given a complete metric space (X, d), the most well-studied types of self-maps are referred to as Lipschitz mappings (or Lipschitz maps, for short), which are given by the metric inequality: d(Tx, Ty) ≤ α d(x, y),. In 2015, Ezearn [4] introduced a new class of mappings called higher-order Lipschitz mappings which are seen as a generalization of inequality (1). His remetrisation of the original metric space does not necessarily result in a complete metric space, and as a result, he provided a completion of the remetrised space and an extension of the higher-order Lipschitz mapping into a complete remetrised space Before stating this theorem, let us introduce the following: a new metric on the space X as already defined by Ezearn in his paper, r− 1. One of the results in this paper is to show that given any Banach space X and a closed bounded convex subset C containing the origin, an approximate fixed point sequence always exists for positively homogeneous (r, p)-general higher-order nonexpansive mappings
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