On Annihilated Points and Approximate Fixed Points of General Higher-Order Nonexpansive Mappings

. In this paper, we extend the results obtained by Ezearn on annihilated points for his higher-order nonexpansive mappings to the context of general higher-order nonexpansive mappings. Precisely in his thesis, Ezearn introduced the concept of annihilated points, which extends the notion of ﬁ xed points, and it is only meaningful in the context of higher-order nonexpansive mappings and gave some mild conditions when the annihilated points could exist in strictly convex Banach spaces. In the last direction, we also extend Ezearn ’ s result on the approximate ﬁ xed point sequence for higher-order nonexpansive mappings to general higher-order nonexpansive mappings.


Introduction
Given a complete metric space ðX, dÞ, the most well-studied examples of such mappings are those that can be immediately put in the form For all x, y ∈ X where c > 0 is a fixed real number. Such mappings are referred to as Lipschitz continuous mappings. Lipschitz continuous mappings are generally classified into three categories: T is a (i) contraction mapping if 0 < c < 1 (ii) nonexpansive mapping if c = 1 (iii) expansive mapping if c > 1 In [1], the concept of mean nonexpansive mappings was introduced which is often seen as a generalization of nonex-pansive mappings. Thus, let C be a nonempty subset of a Banach space X, and let T be a self-mapping on C. Then T is called a mean nonexpansive (or α-nonexpansive) if For all x, y ∈ C and for some α = ðα 1 , α 2 , ⋯, α n Þ, we have ∑ n k=1 α k = 1, α k ≥ 0 for all k, and α 1 , α n > 0. Clearly, it is seen that all nonexpansive mappings are mean nonexpansive mappings, but the reverse is not always true, as demonstrated in ( [2], Examples 2.3 and 2.4). A more general class of ðα, pÞ-nonexpansive maps was further introduced in [1]. That is, a self-map T on a subset C of a Banach space X is called ðα, pÞ-nonexpansive if For all x, y ∈ C and for some α = ðα 1 , α 2 , ⋯, α n Þ, we have ∑ n k=1 α k = 1, α k ≥ 0 for all k, α 1 , α n > 0 and for some p ∈ ½1,∞Þ. It is obvious that ðα, pÞ-nonexpansive map for p > 1 is also α-nonexpansive, but the reverse is not always true, as shown in [3]. Now, given a metric space ðX, dÞ, a more general class of mappings which extend inequality (1) can be put in the following form: where r ∈ ℕ and c k ≥ 0, for all 0 ≤ k ≤ r − 1. Such mappings are called higher-order Lipschitz mappings (or rth-order Lipschitz mappings, for short) which was introduced by Ezearn [4] in 2015. Now, to every higher-order Lipschitz mapping, Ezearn associated a polynomial which is defined as and for r th-order nonexpansive mapping, we have pð1Þ = 0. Ezearn [5] in his thesis introduced the concept of annihilated points of a higher-order nonexpansive mapping as defined below: Definition 1 (Annihilated point of T). Let T : S ⟶ X be a higher-order nonexpansive mapping on a subset S of a Banach space X, and let p be the associated polynomial of T. Then x is an annihilated point (respectively, a totally annihilated point) of T if pðTÞ annihilates x (respectively, the Picard iterates of x) that is, pðTÞx = 0 (respectively, pðTÞT n x = 0 for all n ≥ 0).
Ezearn is denoted by AðTÞ (respectively, A ∞ ðTÞ) the set of annihilated (respectively, a totally annihilated) points of T. Ezearn, in an attempt to prove a fixed point result for higher-order nonexpansive mappings, proved the following theorems on sufficient conditions for an annihilated point when the Banach space is strictly convex: a strictly convex Banach space is a Banach space such that whenever x ≠ 0 and y ≠ 0, then kx + yk = kxk + kyk if and only if x = ky for some constant k > 0. Theorem 2. Let C be a convex subset of a strictly Banach space X, and let T : C ⟶ C an rth-order nonexpansive mapping of the form Suppose u, v ∈ AðTÞ and T r x − pðTÞx ∈ convfT r , T r vg. Then, x ∈ AðTÞ. Theorem 3. Let T be an rth-order nonexpansive mapping on a convex subset C of a strictly convex Banach space X. Suppose u, v ∈ FixðTÞ and fT k xg With a mild condition on the set of totally annihilated points, A ∞ ðTÞ, Ezearn proved the following fixed point result in a general Banach space. Theorem 4. Let T be an affine higher-order nonexpansive mapping on a convex subset C of a Banach space X. Then, FixðTÞ = ∅ only if A ∞ ðTÞ = ∅. In particular, the identity A ðTÞ = A ∞ ðTÞ holds, and if x ∈ A ∞ ðTÞ, then Finally, Ezearn proved the following approximate fixed point sequence result for his higher-order nonexpansive mapping in a general Banach space.

Theorem 5.
Let C be a closed bounded star-convex subset of a Banach space, and let T be an affine rth-order nonexpansive self-mapping on C. Then, T has an approximate fixed point sequence in C. That is, there exists fx n g n≥1 ⊂ C such that lim n⟶∞ ðx n − Tx n Þ = 0.
Now to every ðr, pÞ-general higher-order Lipschitz mapping, the author associated the following polynomial: The author classified ðr, pÞ-general higher-order Lipschitz mappings as follows: (i) T is ðr, pÞ-general higher-order contraction mapping if hð1Þ > 0 (ii) T is ðr, pÞ-general higher-order non-expansive mapping if hð1Þ = 0 2 Abstract and Applied Analysis (iii) T is ðr, pÞ-general higher-order expansive mapping if hð1Þ < 0 In this paper, we generalize Theorem 2 and Theorem 3 to ðr, pÞ-general higher-order nonexpansive mapping when p = 1 except that in the second case (Theorem 3), it will not be totally annihilated points but just annihilated points because in Ezearn's case, all the constants are on the right, and therefore, by induction, he could obtain that result for a totally annihilated point. In the other direction, we generalize Theorem 4 to ðr, pÞ-general higher-order nonexpansive mappings, but in the context of an affine subset of a given Banach space. In the last direction, we generalize Theorem 5 to ðr, pÞ-general higher-order nonexpansive mappings. That is, in this paper, we prove the following results: Theorem 7. Let C be a convex subset of a strictly convex Banach space X, and define T : C ⟶ C to be a (r,1)-general-higher order nonexpansive mapping of the form Suppose u, v ∈ AðTÞ and ∑ r k=l+1 Theorem 8. Let T be an (r,1)-general higher-order nonexpansive mapping on a convex subset C of a strictly convex Banach space (X, k·k). Suppose u, v ∈ FixðTÞ and fT k xg Then, x ∈ AðTÞ.
Theorem 9. Let T be an affine general higher-order nonexpansive mapping on an affine subset C of a Banach space X. The FixðTÞ = ∅ only if A ∞ ðTÞ = ∅. In particular, the identity AðTÞ = A ∞ ðTÞ holds and if x ∈ A ∞ ðTÞ, then Theorem 10. Let C be a closed, bounded star-convex subset of a Banach space, and let T be an affine ðr, pÞ-general higher-order nonexpansive self-mapping on C. Then T has an approximate fixed point sequence in C. That is, there exists fx n g n≥1 ⊂ C such that lim n⟶∞ ðx n − Tx n Þ = 0.
From Definition 1, for any (r,p)-general higher-order nonexpansive mapping, the fixed point set is always a subset of the annihilated point set and they coincide when l = 0 and r = 1. To see this, for l = 0 and r = 1, we have the following: Given that TðzÞ = z, then we have In the same vein, since for any (r,p)-general higher-order nonexpansive mapping, we have then, we have the following: Now, since Tz = z ⇒ T k z = z for k ≥ 1, then the above equation reduces to

Preliminaries
Proposition 11. Define T to be an ðr, pÞ-general higher-order Lipschitz mapping, and let hðzÞ be the associated polynomial for T as stated in Definition 6.
(i) If hð1Þ > 0, then we can always find a certain λ ∈ ð0, 1Þ, which is unique and positive if α k ≠ 0, such that hðλÞ = 0 (ii) If hð1Þ = 0, then there exists 1 as the only positive root of h (iii) If hð1Þ < 0, then we can find a unique positive λ > 1 such that hðλÞ = 0 Now, let us define T to be an ðr, pÞ-general higher-order Lipschitz mapping on a complete metric space ðX, dÞ as given in inequality (8) and let λ be the unique root of the polynomial hðzÞ as guaranteed by Proposition 11. Define the following on the space X: where 3 Abstract and Applied Analysis and p ≥ 1 for all x, y ∈ X, 0 ≤ k ≤ r − 1 and Corollary 12. b k stated in equation (17) is non-negative.
Lemma 13. D p ðx, yÞ stated in equation (17) is a metric on the space X.
Proposition 14. Define b k in equation (17). Then the following results hold: Lemma 15. Given a metric space ðX, dÞ (not necessarily complete) and define T : X ⟶ X to be an ðr, pÞ-general higherorder Lipschitz mapping. Then Moreover, a sequence fx n g n≥1 ⊂ ðX, D p Þ is Cauchy in ðX, D p Þ if and only if the sequence fT k x n g n≥1 ⊂ ðX, dÞ is Cauchy in ðX, dÞ for all 0 ≤ k ≤ r − 1.

Theorem 16. Define the mapping,
T : Then, we have D p T y n ½ , In particular, if ðX, dÞ is complete, then T has a fixed point in ðX, dÞ if and only if T has a fixed point in ð X, D p Þ.

Main Result
We prove the main result of this paper, which is already stated in Theorem 7, Theorem 8, Theorem 9, and Theorem 10. The proofs follow similarly as in Ezearn [5] except for few modifications as necessary.
Proof of Theorem 17. Let for some c ∈ ½0, 1. Then, given that ∑ r k=l+1 α k T k − hðTÞ = ∑ l k=0 α k T k , then the following identity holds: Hence, we have Similarly, one can also have Abstract and Applied Analysis Hence, we have From equation (26), it follows that when c = 0, then and that implies that Note also that when c = 0, then ∑ r k=l+1 α k T k x − hðTÞ Combining equation (30) and equation (31), we have that giving hðTÞ = 0 or equivalently x ∈ AðTÞ. Similarly, from equation (28), when c = 1, then and that implies that Note also that when c = 1, then ∑ r k=l+1 α k T k x − hðTÞ x = 1 · ∑ r k=l+1 α k T k u + ð1 − 1Þ∑ r k=l+1 α k T k v and it follows that Combining equation (34) and equation (35), we have that giving hðTÞ = 0 or equivalently x ∈ AðTÞ.
Hence, we assume that c ∈ ð0, 1Þ. We observe that To see this, we note that if ∑ r k=l+1 α k T k u = ∑ r k=l+1 α k T k x, then we have the following leading to the contradiction that c ≥ 1.
leading to the contradiction that c ≤ 1. Now, given that

Abstract and Applied Analysis
It follows from the above that and since Then from the strict convexity of X, there exists λ > 0 such that the following holds: Set λ ≔ β/ð1 − βÞ ðthus β ∈ ð0, 1ÞÞ, and equation (43) becomes equivalent to the following: Consequently, we have and 〠 r k=l+1 Now, since then, we have that β ≤ c. Similarly, since and that gives us β ≥ c and therefore we must have β = c. Hence, we have shown that and so we have hðTÞ = 0 or equivalently x ∈ AðTÞ and that completes the proof.
Proof of Theorem 18. Let c k ∈ ½0, 1 for all 0 ≤ k ≤ r − 1 and T k x ≔ c k u + ð1 − c k Þv, then we have c k ðu − T k xÞ = ð1 − c k ÞðT k x − vÞ, and that also follows that Now, if c 0 = 1, then x = u, and this means that x ∈ FixðTÞ and since by definition FixðTÞ ⊂ AðTÞ, then it follows that x ∈ AðTÞ. Similarly, for c 0 = 0, then x = v and also follows that x ∈ AðTÞ. Hence, we may assume that c 0 ∈ ð0, 1Þ. First and foremost, we may observe that u ≠ Tx ≠ v, and to see this, we note that if u = Tx, then u = T k+1 x for all 0 ≤ k ≤ r − 1 and since by assumption c 0 ≠ 1, then we have leading to the contradiction that D 1 ðx, vÞ < D 1 ðx, vÞ. In the same vein, if v = Tx, then v = T k+1 x for all 0 ≤ k ≤ r − 1. Since by assumption, c 0 ≠ 0, then we have the following: leading to the contradiction that D 1 ðu, xÞ < D 1 ðu, xÞ. Given that T is an (r,1)-general higher-order nonexpansive mapping, we have which implies that D 1 ðu, TxÞ + D 1 ðTx, vÞ = D 1 ðu, vÞ or equivalently Now, given that ku − T k+1 xk + kT k+1 x − vk − ku − vk ≥ 0 and b k > 0ðsince b r−1 = α r ≠ 0Þ, then for all 0 ≤ k ≤ r − 1, we have