General Higher-Order Lipschitz Mappings

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.


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: for all x, y ∈ X, where α ≥ 0 is a real number, usually referred 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.
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 Japón 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, ∞), 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 mappings 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.
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 where r is a natural number and c k , for all 0 ≤ k ≤ r − 1, are nonnegative real numbers, whereas Lipschitz mappings are (uniformly) continuous mappings; this needs not be the case for higher-order Lipschitz mappings; they may not even be continuous as given by the following example taken from [4].
with the metric induced by the usual absolute value on R.
Clearly, T is discontinuous at x � 0 but a second-order Lipschitz (indeed, a second-order nonexpansive) mapping.
In this paper, we introduce the following new class of mappings which generalizes both inequalities (3) and (5).
It is clear that inequality (7) reduces to inequality (3) when l � 0. Similarly, inequality (7) reduces to inequality (5) when p � 1 and l � r − 1. Generally, a (r, p)-general higher-order Lipschitz mapping may not be continuous as already seen in examples 1 and 2. Now, to every (r, p)-general higher-order Lipschitz mapping, we associate a polynomial defined as just as Ezearn did in his paper [4]. Now, we classify (r, p)-general higher-order Lipschitz mappings into three main categories, thus Now, we provide some fixed point results as it is presently understood and known and their connections to the research focus of this paper.

Theorem 1. (Banach contraction mapping theorem). Let
(X, d) be a complete metric space and let T: X ⟶ X be a contraction mapping. en, T has a unique fixed point and lim n⟶∞ T n x converges to this fixed point for any x ∈ X.
Gordon [5] provided a version of eorem 1 in a general Banach space using the concept of the normalized duality mappings and the generalized projection functional. He defined the following set of mappings.
Definition 2. (monotone contraction mapping). Let X be a smooth Banach space and let C be a closed subset of X.
en, the mapping T: C ⟶ C is said to be a monotone contraction mapping if there exists 0 ≤ c < 1 such that for all x, y ∈ C, the following two conditions are satisfied: where R denotes the real part of a complex number and J is the normalized duality mapping for all m, n ≥ 0 with m ≠ n.
His result is summarized in the following theorem.

Theorem 2. (monotone contraction mapping theorem). Let
C be a closed subset of a uniformly convex smooth Banach space X and let T: C ⟶ C be a monotone contraction mapping. en, T has a unique fixed point, that is, F(T) � p and that the Picard iteration associated to T, that is, the sequence defined by x n � T(x n− 1 ) � T n (x 0 ) for all n ≥ 1 converges to p for any initial guess x 0 ∈ X.
Ezearn [4] extended the conclusion of the Banach contraction mapping theorem ( eorem 1) to higher-order contraction mappings as summarized in the following theorem.

Theorem 3. (higher-order contraction mapping theorem).
Let (X, d) be a complete metric space and let T: X ⟶ X be an rth-order contraction mapping. en, T has a unique fixed point and lim n⟶∞ T n x converges to this fixed point for arbitrary x ∈ X.
Ezearn later provided a remetrisation argument that relates his higher-order Lipschitz mappings to (first-order) 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. Before stating this theorem, let us introduce the following: a new metric on the space X as already defined by Ezearn in his paper, Journal of Mathematics His theorem (already stated as eorem 3.5 in his paper) is stated as follows.

Theorem 4. Define the mapping,
T: X ⟶ X, en, we have D T y n , T x n ≤ λD y n , x n , where y n n ≥ 1 , x n n ≥ 1 are Cauchy sequences in (X, D) and In light of eorem 4, this paper carefully uses ideas of the auxiliary results leading to the proof of eorem 4 to establish entirely similar proof of eorem 4 as far as (r, p)-general higher-order contraction mappings are concerned.
An approximate fixed point sequence of a nonexpansive self-map T on a closed convex subset C of Banach space X is any sequence x n n ≥ 1 such that lim n⟶∞ x n − Tx n � � � � � � � � � 0.
When C is bounded, then such a sequence always exists. 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. [10]). If a continuous real function defined on an interval is sometimes positive and sometimes negative, then it must be 0 at some point in the interval. Theorem 6. (Descartes' rule of signs, see for instance, [11]). Let f(z): � r k�0 a k z k be an rth degree polynomial over the real numbers a k . en, the number of positive real roots of f is bounded above by the number of sign changes of the coefficients a k as one proceeds from k � 0 to k � r (ignoring zero coefficients).

Main Result
We give the proof of the main result of this paper, which is accomplished in eorems 7 and 8. e following lemma, corollary, and proposition shall aid us in arriving at the conclusion of the main result.

Fixed Point eorem for General Higher-Order Contraction Mappings.
Let T be a (r, p)-general higher-order Lipschitz mapping on a complete metric space (X, d) as given in inequality (13) and let λ be the unique root of the polynomial h(z) � r k�l+1 α k z k − l k�0 α k z k as guaranteed by Proposition 1. Now, let us define the following on the space X: and p ≥ 1 for all x, y ∈ X, 0 ≤ k ≤ r − 1 and

Corollary 1. Let b k be defined in equation (13). en, b k is nonnegative.
Proof. To see this, if k ≤ l, then λ k+1 b k � k j�0 α j λ j ≥ 0. On the other hand, if k ≥ l + 1, then and that completes the proof. □ Lemma 1. Let D p (x, y) be defined in equation (13). en, D p is a metric on the space X.
Proof. By Corollary 1, b k is nonnegative. Now, for p � 1 is straightforward. For p > 1, it is obvious that D p is nonnegative since d and b k are nonnegative and it is subadditive since d is subadditive. We have also that D p (x, y) � 0 if and only if x � y since b k ≠ 0 because of the fact that α 0 ≠ 0. Also, D p (x, y) � D p (y, x). Finally, without loss of generality, suppose x ≠ z. en, we have the following evaluation: Using the Hölder's inequality on each of the terms on the right hand side, we have and that completes the proof. □ Proposition 2. Let b k be defined in equation (13). en, the following recurrence relations hold: Journal of Mathematics and multiplying through again by λ completes the proof. Finally, for k � r − 1, we have (20) Let (X, d) be a (not necessarily complete) metric space and let T: X ⟶ X be a (r, p)-general higher-order Lipschitz mapping. Let D p be the new metric defined in equation (13). en, remedies the case when T is discontinuous on (X, d). Here, our proof follows exactly as in Ezearn's eorem 3.5, but we give the proof for the sake of completeness. To begin, let (X, D p ) be the canonical completion of the metric space (X, D p ); that is, where y n n ≥ 1 , x n n ≥ 1 are Cauchy sequences in (X, D p ) and [x n ], [y n ] denote the equivalence class of x n n ≥ 1 and y n n ≥ 1 in (X, D p ), where y n n ≥ 1 is equivalent to x n n ≥ 1 if lim n⟶∞ D p (y n , x n ) � 0.

Theorem 7. Define the mapping,
T: X ⟶ X, Proof. Since x n n ≥ 1 is Cauchy in (X, D p ), then, by Lemma 2, D p Tx n , Tx m ≤ λ (1/p) D p x n , x m , and so Tx n n ≥ 1 is Cauchy in (X, D p ); thus, T is well defined. Now, given Cauchy sequences x n n ≥ 1 , y n n ≥ 1 in (X, D p ), then we have which completes the proof.

Approximate Fixed Point Property for Positively Homogeneous General Higher-Order Nonexpansive Mappings.
In this subsection, we prove the approximate fixed point property for (r, p)-general higher-order nonexpansive mapping in a Banach space, which is accomplished in the following theorem.
Theorem 8. Let C be a closed bounded convex subset containing the origin of a Banach space X and let T: C ⟶ C be a positively homogeneous general higherorder nonexpansive mapping. en, T has an approximate fixed point sequence.