Research Article General Higher-Order Lipschitz Mappings Joseph Frank Gordon Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 211106, China Correspondence should be addressed to Joseph Frank Gordon; jgordon@aims.edu.gh Received 20 January 2021; Revised 12 March 2021; Accepted 19 March 2021; Published 1 April 2021 Academic Editor: Ching-Feng Wen Copyright © 2021 Joseph Frank Gordon. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 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. 1. Introduction 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), (1) for all x, y ∈ X, where α ≥ 0 is a real number, usually re- ferred to as the Lipschitz constant of T. Now, for technical and historical reasons, we classify Lipschitz mappings into three categories, thus contraction mappings for the case where 0 ≤ α < 1, nonexpansive mappings for the case where α � 1, and expansive mappings for the case where α > 1. In 2007, Goebel and Japon Pineda [1] introduced the so-called mean nonexpansive mappings which are class of mappings more general than the class of nonexpansive mappings. us, given a Banach space (X, ‖ · ‖) and a nonempty subset C of X, a mapping T: C ⟶ C is called mean nonexpansive (or α-nonexpansive) if, for some α � (α 1 , α 2 , ... , α n ) with  n k�1 α k � 1, α k ≥ 0 for all k, and α 1 , α n > 0, we have  n k�1 α k T k x − T k y � � � � � � � � � � ≤ ‖x − y‖, forall x, y ∈ C. (2) It is obvious that all nonexpansive mappings are mean nonexpansive mappings but the converse may not be true (see for instance [2], examples 2.3 and 2.4). Goebel and Jap´ on Pineda further suggested the class of (α,p)-nonexpansive maps. A self-map T: C ⟶ C is called (α,p)-nonexpansive if, for some α �(α 1 , α 2 , ... , α n ) with  n k�1 α k � 1, α k ≥ 0 for all k, and α 1 , α n > 0 and for some p ∈ [1, ∞),  n k�1 α k T k x − T k y � � � � � � � � � � p ≤ ‖x − y‖ p , (3) for all x, y ∈ C. It is easy to check that (α,p)-nonexpansive map for p > 1 is also α-nonexpansive, but the converse does not hold (see [3] for details), whereas nonexpansive map- pings are uniformly continuous and that continuous, mean nonexpansive mappings may not generally be continuous as seen in the following example taken from [2], example 2.2. Example 1. Let f: [0, 1] ⟶ [0, 1] be given by f(x) ≔ 1, x � 0, 0, x ≠ 0.  (4) Clearly, f is discontinuous but a mean nonexpansive mapping. 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 Hindawi Journal of Mathematics Volume 2021, Article ID 5570373, 7 pages https://doi.org/10.1155/2021/5570373