A nonstandard approach to asymptotic fixed point theorems

We use methods from nonstandard analysis to unify and extend results from the theory of asymptotic fixed points on metric spaces. Among the topics we consider are Kirk’s asymptotic fixed point theorem with extensions, Boyd and Wong’s fixed point theorem, and Suzuki’s fixed point theorem for asymptotic contractions of the Meier–Keeler type. We also show how Suzuki’s asymptotic contractions of the final type can be modified to yield uniform convergence on compact and bounded sets.

There is by now a large number of papers on asymptotic results in fixed point theory, many of them inspired by Kirk's seminal paper [29]. The arguments in these papers are often long and complicated, and the purpose of the present paper is to demonstrate that nonstandard analysis provides a powerful method for unifying and simplifying many of the results.
The paper takes two parts. In the first part, we look at what might be called extrinsic fixed point theorems; results where the local contractive behavior of a function T is measured by external functions (φ, or a family φ n ). We give a nonstandard proof of an extension of Kirk's asymptotic fixed point theorem and then turn to some related questions, including Boyd and Wong's [10] classic generalization of Banach's fixed theorem and Suzuki's [39] fixed point theorem for contractions of the Meier-Keeler type.
In the second part, we consider intrinsic conditions on the function T which lead to the existence of a fixed point. We begin with a closer look at Suzuki's [40] asymptotic contractions of the final type (ACFs). We first give a simplified proof of his result that the ACF conditions are equivalent to the existence of an iterative fixed point, and then show how the conditions can be strengthened to guarantee stronger forms of convergence such as uniform convergence on bounded and compact sets.
We shall now describe some simple machinery by which asymptotic fixed points theorems can be proved. The advantages of our machinery are that the conditions are simple and intuitive, and the proofs short and straightforward. The disadvantage is that it requires nonstandard analysis.
Remark. In Kirk's original treatment, the condition that T is continuous was inadvertently left out. Since Kirk's paper there have been numerous extensions and generalizations of the result above. Some of these are summarized in Corollary 1.10 below; mostly they involve dropping some of the continuity conditions, or weakening some of the boundedness conditions. More notably, Suzuki [39] showed that the condition of a bounded orbit is unnecessary and that the continuity conditions on φ and φ n can be weakened. Suzuki's approach is first to show that all asymptotic contractions are actually asymptotic contractions of the Meier-Keeler type (ACMKs), and then prove that all ACMKs have iterative fixed points. We shall instead give a direct proof of (an extension of) Kirk's theorem in this section, and an independent proof of the fixed point theorem for ACMKs in Sect. 1.4. Let us also mention that using estimates from Gerhardy's paper [22], Briseid [12] has given a constructive proof of Kirk's theorem without the boundedness condition and with estimates for the rates of convergence.
In this section we will prove Kirk's theorem without the boundedness condition and without the continuity conditions on φ n and φ: Theorem 1.4. Suppose that X, d, T, φ n , and φ satisfy the usual hypotheses, and that φ n converges to φ uniformly on the range of d. Then T has an iterative fixed point x ∞ .
(We shall see later that even the condition on uniform convergence can be weakened somewhat.) Unless otherwise indicated, X will always denote a complete metric space with metric d. If x ∈ X * , x is finite provided d(x, y) is finite for some standard y ∈ X (equivalently, for all standard y in X). 1 We begin with a condition sufficient for any continuous function on X to have an iterative fixed point.
The converse to the first part of the proposition is easily seen to be true. There are some obvious/trivial generalizations, for example continuity of T can be replaced by continuity of T m for some standard m, or even continuity of T m at x ∞ .
Call any points x and y eventually close if they satisfy condition (1.1) from this proposition.
Proof. Take x = y ∈ X, this shows that {T n x} n is Cauchy, so by completeness converges to some Eventual closeness means that all such orbits converge to the same limit, so the fixed point is unique. Finally, let Y be a bounded subset of X, x ∈ X, y ∈ Y * , and i infinite. Then By the nonstandard characterization of uniform convergence (see Proposition 3.1), this means that the convergence is uniform on Y .
We continue with two useful results. Proposition 1.6. Suppose that T : X → X and that for all infinite i and x, y in X, d(T i x, T i y) ≈ 0. Then: 1. For all infinite i < j and x, y, z, w in X, d( The opposite inequality follows by symmetry. (2) follows from the hypothesis by taking y = T n x. (3) follows from (1) by taking Here s t means that s < t or s ≈ t. Later we will use s t to mean that s < t and s ≈ t.
The next lemma extracts a basic argument from Boyd and Wong's original proof of Theorem 1.12 (below). Lemma 1.7. Suppose that the hypotheses of Proposition 1.6 hold. For all infinite i < j, x, y in X, and hyperreal γ, if 0 γ d(T i x, T j y) then there is an infinite k, i < k < j, with γ ≈ d(T k x, T j y).
Proof. Let k be the greatest hyperinteger in [i, j] with γ < d(T k x, T j y).
Vol. 25 (2023) A nonstandard approach to asymptotic fixed point theorems Page 5 of 22 35 Note that in the proof we also have that γ < d(T k x, T j y); this will be used in the proof of Theorem 1.12.
Proof of Theorem 1.4. By Proposition 1.5, it suffices to show that for any x, y in X and any infinite i and j, d(T i x, T j y) ≈ 0. We start with three claims: To see this, let γ = sup{ d(T i x, T i y) is infinite}, which exists by Claim 1. Suppose (for a contradiction) that γ > 0. Then there is an infinite i with d(T i x, T i y) ≈ γ (Proposition 3.1(10)). By Proposition 3.1(11) there exist infinite j, k ∈ N * with j + k = i. Then: a contradiction. (For the strict inequality in the last line we used the fact that d(T j x, T j y) • = 0, which follows since otherwise γ = 0.) So γ = 0, proving Claim 2.

Claim 3.
For all x = y in X and infinite i and j, there is an infinite n with d(T i x, T j y) ≈ d(T i+n x, T j+n y).
Indeed, by Claim 2 and Proposition 1.6, for any finite n we have for every finite n. The claim now follows by overspill (Proposition 3.1(8)). Now, let x, y be elements of X, and i, j be infinite. Suppose for a contradiction that d(T i x, T j y) ≈ 0. Then x = y, we may assume for definiteness that i < j, and there is a standard γ with 0 γ d(T i x, T j y). By Claim 2 and Lemma 1.7 there is some k, i < k < j, with γ ≈ d(T k x, T j y). By Claim 3 there is an infinite n such that: This is a contradiction, completing the proof.

Uniform convergence of the iterates
In this section we discuss conditions under which the iterates converge uniformly to the fixed point.
The following lemma is a straightforward nonstandard result about any function on any metric space. Lemma 1.8. Let X be a metric space (not necessarily complete), and T : X → X (not necessarily continuous). Suppose x, y ∈ X * are finite. Consider the statements: Suppose in addition that X, d, T, φ n , and φ satisfy the usual hypotheses, and that either (a) φ n converges to φ uniformly on bounded subsets of the range of d; or (b) x, y ∈ X, and φ n converges to φ uniformly on the set  10)). Let k < i, j be infinite but sufficiently small that i − k and j − k are both infinite (Proposition 3.1(11)). It follows: We can now strengthen Theorem 1.4. Proof. For the proof of existence we examine some details in the proof of Theorem 1.4. Uniform convergence of φ n to φ was only used twice (Eqs. 1.2 and 1.3), both times so that we could assume convergence on a possibly nonstandard element d(T i x, T j y). For standard x, y in X both d(x, y) and , so both 1.2 and 1.3 still hold under hypothesis (b). If instead we assume hypothesis (a), 1.2 still holds because d( It remains to prove the assertion about uniform convergence under hypothesis (a). Let x in X, y be a finite (but not necessarily standard) element of X * , and let i, Suppose that X, d, T, φ n , and φ satisfy the usual hypotheses, and that for all x, y ∈ X, φ n converges to φ uniformly on bounded subsets of the range of d. Suppose that any of the following additional hypotheses hold: (i) T has a bounded orbit, The proof is by ignoring conditions (i)-(iii). To prove uniformity of the convergence Arandelović uses in his proof the following result of Arav, Santos, Reich and Zaslavski [4]: Proposition 1.11. Suppose that X, d, T, φ n , and φ satisfy the usual hypotheses, and that φ n converges to φ uniformly on bounded intervals [0, b). If there exists a fixed point x ∞ ∈ X then all sequences {T n x} n converge to x ∞ , uniformly on bounded subsets of X.
This result also follows immediately from Corollary 1.9.

Theorem of Boyd and Wong
Kirk originally formulated his result as an asymptotic version of the following theorem of Boyd and Wong [10]. , y)) for all x, y ∈ X Then T has an iterative fixed point x ∞ . Here upper semicontinuous from the right means that for all s, In nonstandard terms, this is easily seen to be equivalent to: for all standard s, if t > s and t ≈ s then φ(t) φ(s).
In this section we give a simple nonstandard proof of Theorem 1.12. We will abstract from this in the next section to simplify and improve other results from the literature.
Proof of Theorem 1.12. Let s > 0 and consider the sequence . . . This monotonically decreases to some (standard) limit t, which is less than but infinitesimally close to Thus, for all infinite i and standard s > 0, φ (i) (s) ≈ 0. It follows that for all infinite i and x, y ∈ X, We need to show that any x and y in X are eventually close. Let i < j be infinite, and suppose d( a contradiction.

Suzuki and ACMK
In this section we give a simple proof of Suzuki's results [39] that asymptotic contractions of Meier-Keeler (ACMK) type have iterative fixed points. We note that Suzuki's proof of Theorem 1.14 required a careful approximation argument. Our arguments, by contrast, are simple and straightforward adaptations of the proof of Theorem 1.12. , y)) for x = y and n ∈ N.
Nonstandardly, ACM K 1 says that φ i (s) s for infinite i, and ACM K 2 says that φ ν maps the right half of monads of α to the left of or equal to α.
Given α write δ α , ν α for the values given in the second condition. As with all the earlier results, we begin with a proof that d(T i x, T i y) ≈ 0 for infinite i. Lemma 1.13. Suppose that X is complete metric and T : X → X is ACMK. Then for all infinite i and x, y in X, d(T i x, T i y) ≈ 0.
Proof. Fix x = y in X. Let i be infinite and m an arbitrary but standard natural number. Then d( This last inequality is standard and true for all m, so by transfer it holds as well for nonstandard m. For infinite i and m = i Theorem 1.14. Suppose that X is complete metric and T : X → X is continuous and ACMK. Then T has an iterative fixed point x ∞ .
The result now follows from Proposition 1.5.
Remark. In [39], Suzuki showed that all asymptotic contractions are actually ACMKs, and then used the result above to prove his version of Kirk's theorem. We could have copied this approach here to give a second nonstandard proof of Kirk's theorem.

Intrinsic fixed point theorems
In the second part of the paper we shall use nonstandard analysis to study mappings such as Suzuki's asymptotic contractions of the final type (ACFs) introduced in [40]. In this setting there are no "asymptotic moduli of contraction" φ n as above, and instead the conditions are put directly on the iterates T n of the original function T . Suzuki's main result is that a continuous map T : X → X on a complete metric space X has an iterative fixed point if and only if it is an ACF. We first provide a simple nonstandard version of Suzuki's proof, and then show how we can strengthen the conditions of an ACF to get stronger forms of convergence such as uniform convergence on compact and bounded sets. These result are quite close in spirit to the uniformity results Briseid has studied in the extrinsic case using proof mining techniques (see [11,12,16]), and it would be interesting to see if proof mining can also be used to obtain rates of convergence in the present setting.
For another extension of Suzuki's results, see [17].

ACFs and pointwise convergence
We assume throughout that T : X → X is a continuous function on a complete metric space X. It is possible to use our techniques to obtain results even when these conditions are not satisfied, but we want to focus on the central ideas. Let us begin with the main definition: For all x ∈ X and ∈ R + there exists a δ ∈ R + such that if d(T k x, T k y) < δ for some y ∈ X and some k ∈ N, then lim sup n→∞ d(T n x, T n y) < . ACF 2 For all α ∈ R + and all bounded sets B ⊆ X, there exists a δ > 0 such that for all x, y ∈ B with α ≤ d(x, y) < α + δ, there exists a ν ∈ N such that d(T ν x, T ν y) < α. ACF 3 For all x ∈ X and all α ∈ R + , there exist a δ > 0 and a ν ∈ N such that if α < d(T n x, T m x) < α + δ for some n, m ∈ N, then d(T ν+n x, T ν+m x) ≤ α.
As we later want to strengthen these conditions to get stronger forms of convergence, we have changed Suzuki's original definition somewhat to make the relationship between the different forms easier to see. To compare our conditions above to Suzuki's original conditions C1-C4, note that his C4 is identical to our ACF 3 ; the combination of his C2 and C3 implies our ACF 2 ; and his C1 implies our ACF 1 . The reverse implications are less obvious, but the systems as a whole must still be equivalent as they are both equivalent to the existence of an iterative fixed point.
Remark. The strength of the ACF approach is that the conditions only compare the same iterates T n x and T n y of x and y. It's easier to find sufficient and necessary conditions if we allow conditions on T n x and T m y for different iterates, but such conditions are much harder to check; see Section 5 of [40] and the earlier papers [26,33] for more information. Latif's survey chapter [32] puts Suzuki's contribution in the context of various kinds of generalizations of Banach's fixed point theorem.
To prove Suzuki's theorem, we first observe that as the existence of an iterative fixed point clearly implies ACF 1 -ACF 3 , we may concentrate on the converse. We need four easy lemmas to prove the result. and assume for contradiction that α = 0. Choose a bounded set B containing x and y, and let δ ∈ R + be the number guaranteed by ACF 2 . There obviously exists an i ∈ N such that α ≤ d(T i x, T i y) < α + δ, and hence by (ii) a ν ∈ N such that d(T i+ν x, T i+ν y) < α, contradicting the definition of α.
For the second part, note that if the infimum 0 is attained, then d(T i x, T i y) is zero for all infinite i, and if the infimum is not attained, the internal set Remark. Note that the proof of Lemma 2.2 requires that {T n x} and {T n y} are standard sequences, and hence that x, y are standard elements. The proof doesn't work for nonstandard elements x, y ∈ X * , not even if they are finite or nearstandard.
If we add ACF 1 , we can get the lemma above to hold for all infinite i: Lemma 2.3. Assume that T : X → X satisfies ACF 1 and ACF 2 , and that x, y ∈ X. Then d( Proof. It clearly suffices to prove that given an ∈ R + , we have d(T n x, T n y) < for all infinite n. By ACF 1 there is a δ ∈ R + such that if there exists a k with d(T k x, T k y) < δ, then lim sup n→∞ d(T n x, T n y) < . By Lemma 2.2 there exists a finite k such that d(T k x, T k y) < δ, and hence d(T n x, T n y) < for all infinite n.

Proof. Use Lemma 2.3 with y = T ν x.
Note that the proof above doesn't work when ν is infinite as Lemma 2.3 requires y = T ν x to be a standard element. If we add ACF 3 , we can extend the result to infinite ν and hence prove: Proof. It clearly suffices to show that if i is infinite, then d(T i x, T i+n x) ≈ 0 for all n ∈ N * . Choose α ∈ R + and let ∈ R + be any number strictly less than the δ produced by ACF 3 . As we can choose α and arbitrarily small, it obviously suffices to prove that d(T i x, T i+n x) ≤ α + for all n ∈ N * . Assume not, and let n be the smallest number such that d( which is impossible since by Lemma 2.4, T i+ν x is only infinitesimally away from T i x and T i+n+ν x is only infinitesimally away from T i+n x. In formulas, and we have a contradiction.
We can now prove Suzuki's theorem: Theorem 2.6. The following are equivalent: Proof. According to Proposition 1.5, it suffices to show that d(T i x, T j y) ≈ 0 for all x, y ∈ X and all infinite i, j ∈ N * . But where the first term on the left is infinitesimal by Lemma 2.5 and the second term is infinitesimal by Lemma 2.3.
The arguments we have given in this section are basically Suzuki's, but nonstandard techniques have made it possible to simplify some of the steps, especially the proof of Lemma 2.5 where Suzuki had to use a quite intricate induction argument.

Modes of convergence
Pointwise convergence to a fixed point is too weak for many applications, and in the rest of the paper we shall see how we can strengthen the ACF conditions to get stronger forms of (uniform) convergence. This section is devoted to the preliminary tasks of finding the nonstandard characterizations of these modes of convergence and of comparing their relative strength. Throughout the section, X and Y are metric spaces and f and f n , n ∈ N, are functions from X to Y .
For our purposes, a convenient notion of convergence is the following: Definition 2.7. The sequence {f n } converges continuously to f if for all convergent sequences x n → x in X, the images f n (x n ) converge to f (x) in Y .
Remark. The notion of continuous convergence is an old one, and the reader may want to consult the classical paper by Arens and Dugundji [7] for a thorough study of it's relationship to various topologies on the space of continuous functions (in the setting of net convergence on topological spaces).
Note that if f is discontinuous, the constant sequence {f } does not converge continuously to f . In fact, it turns out that if a sequence {f n } converges continuously to a function f , then f is necessarily continuous (see the final Sect. 3 for a proof), and hence we shall always assume that the limit function is continuous when we compare continuous convergence to other forms of convergence.
It is well-known that uniform convergence implies continuous convergence, but that pointwise convergence does not. We shall soon show that continuous convergence is, in fact, strictly between locally uniform convergence and uniform convergence on compacts, but before we prove this, it will be convenient to establish a nonstandard criterion for continuous convergence. Recall that a point y in X * is nearstandard if it is infinitely close to some standard point x ∈ X. This standard point x is called the standard part of y, and is denoted by y • . We write Ns( X * ) for the collection of all nearstandard points.

Proposition 2.8. A sequence {f n } of functions converges continuously to f if and only if for all
Proof. Assume first that for all nearstandard points y ∈ X N , and hence f N (x N ) ≈ f (x). This proves that {f n (x n )} converges to f (x), and hence the convergence is continuous.
Assume now that there is a point x ∈ X and a point y ∈ X * such that y For each nonzero n ∈ N, the sentence clearly holds in the nonstandard universe, and by transfer the sentence holds in the standard universe. This means that for any n ∈ N, there is a point y n ∈ X and a standard integer N n > n such that d(x, y n ) < 1 n and d(f (x), f Nn (y n )) > . We now create a sequence {x n } in X by letting x Nn = y n for an increasing sequence of N n 's and letting x n = x for the remaining terms. Clearly x n → x, but d(f (x), f n (x n )) > for an infinite number of n's, and hence the convergence is not continuous.
The proposition above should be compared to the following well-known, nonstandard description of uniform convergence (see, e.g., Proposition 13.1 in Hurd and Loeb [24] or Theorem 4.6.1 in Robinson's book [38]). Recall that a point in X * is called compact if it is in the nonstandard version K * of some compact set K. The collection of compact points is denoted by Cpt( X * ). It is well-known (see, e.g., Theorem 2.2 in [24] or Proposition 2.1.6 in [2]) that Cpt( X * ) ⊆ Ns( X * ) with equality if and only if X is locally compact. We have:

Corollary 2.10. A sequence {f n } of functions converges to f uniformly on compact sets if and only if for all
Proof. Assume first that {f n } converges uniformly to f on compact sets, and that y ∈ Cpt( X * ). Then there is a compact set K such that y ∈ K * , and since {f n } converges uniformly on the compact K, we get f N (y) ≈ f (y) by Proposition 2.9.
For the contrary, assume that for all y ∈ Cpt( K * ), f N (y) ≈ f (y) for all infinitely large N ∈ N * . If K is a compact set, K * ⊆ Cpt( X * ), and hence for all y ∈ K * , we have f N (y) ≈ f (y) for all infinitely large N ∈ N * . By Proposition 2.9, it follows that {f n } converges uniformly to f on K.
The last statement follows by the nonstandard characterization of continuity (see Proposition 3.1), since f (y) ≈ f ( y • ) at all nearstandard (and hence all compact) points y.
We now use the characterizations above to give an easy proof of: Proposition 2.11. Assume that {f n } is a sequence of functions converging pointwise to a continuous function f . If the convergence is locally uniform, then it is continuous. If the convergence is continuous, then it is uniform on compact sets.
Proof. Assume that the convergence is locally uniform, i.e. for every x ∈ X there is a ball B(x; r) where f n converges uniformly to f . If {x n } converges to x, then x n ∈ B(x; r) for all sufficiently large n ∈ N, and hence for infinite where the last step uses the continuity of f . This shows that {f n } converges continuously to f .
To prove that continuous convergence implies uniform convergence on compacts, assume that {f n } converges continuously to f . By Proposition 2.8, this means that f N (y) ≈ f ( y • ) for all nearstandard y and all infinite N . As the compact points are a subset of the nearstandard points, we get f N (y) ≈ f ( y • ) for all compact points y, and by Corollary 2.10 {f n } converges uniformly to f on compact sets.
Remark. The three notions of convergence in the proposition clearly coincide when the space is locally compact, but in general neither of the implications can be reversed. The arguments in Section 2 of Render's paper [37] proves that continuous convergence and uniform convergence on compact sets only coincide in the locally compact case, and Example 1 below gives an example of continuous convergence that is not locally uniform.
To see that the convergence is not locally uniform, observe that in any ball B(0, r) around the origin, there will for all n ∈ N be an element v n = {0, 0, . . . , 0, r 2 , 0, . . .} ∈ B(0, r) such that ||f n (v n )|| = r 2 , and hence {f n } does not converge uniformly to 0 on B(0, r).

Continuous convergence and stable ACFs
We shall now show how to strengthen the conditions of an ACF to get a characterization of continuous convergence to the fixed point x ∞ . The following simple example illustrates the main challenge. Example 2. Let f : [0, 2π] → [0, 2π] be a strictly increasing, continuous function such that f (0) = 0, f (2π) = 2π and f (θ) > θ for all θ ∈ (0, 2π). Note that if we iterate f , f (n) (θ) increases strictly to 2π for all θ ∈ (0, 2π), and 0 is a repelling and 2π an attracting fixed point. We may move f to the unit circle T by defining T : T → T by T (cos θ, sin θ) = (cos f (θ), sin f (θ)). Obviously, x ∞ = (1, 0) is the unique fixed point for T , and T n (x) → x ∞ for all x ∈ T. It is easy to check directly that T is an ACF (this also follows immediately from Theorem 2.6), but the convergence is clearly not continuous as it not even uniform on the compact space T (we can make the convergence as slow as we want by making the initial θ small).
The basic lesson of Example 2 is that to get stronger forms of convergence, we need to strengthen condition (i) in the definition of an ACF. This condition allows orbits that start close together to wander far apart as long as they wind up close together in the end, and this is clearly not good enough for continuous convergence. An obvious idea is to strengthen our first condition from ACF 1 For all x ∈ X and ∈ R + there exists a δ ∈ R + such that if d(T k x, T k y) < δ for some y ∈ X and some k ∈ N, then lim sup n→∞ d(T n x, T n y) < to ACF 1 For all x ∈ X and ∈ R + there exists a δ ∈ R + such that if d(T k x, T k y) < δ for some y ∈ X and some k ∈ N, then sup n∈N d(T k+n x, T k+n y) < , i.e., simply replace the lim sup by a sup.

Definition 2.12.
A stable ACF is a map T : X → X satisfying conditions ACF 1 , ACF 2 and ACF 3 .
We are ready for the main result of this section. Theorem 2.13. Assume that X is complete and that T : X → X is continuous. The following are equivalent: Proof. (i)⇒(ii): From Suzuki's theorem we know that for every x ∈ X, {T n (x)} converges to an iterative fixed point x ∞ . To prove that the convergence is continuous, we shall use the nonstandard characterization in Proposition 2.8; hence we need to show that if x ∈ X and y ∈ X * is infinitely close to x, then T N y ≈ x ∞ for all infinite N . Given x and an ∈ R + , ACF 1 produces a δ ∈ R + such that if for any k, d(T k x, T k y) < δ, then d(T k+n x, T k+n y) < for all n ∈ N. Since d(T 0 x, T 0 y) = d(x, y) < δ, this implies (using transfer) that d(T n x, T n y) < for all n ∈ N * . Since ∈ R + is arbitrary, we have T n x ≈ T n y for all n ∈ N * , and consequently T N y ≈ T N x ≈ x ∞ for all infinite N . (ii)⇒(i): By Suzuki's theorem we know that T is an ACF, and we only have to prove that ACF 1 is satisfied. This condition is equivalent to saying that for every x ∈ X and every ∈ R + : By the Transfer Principle, it suffices to prove the transferred formula To do so, choose δ to be any positive infinitesimal. It clearly suffices to show that if d(T k x, T k y) < δ for some k ∈ N * , then T k+n x ≈ T k+n y for all n ∈ N * . There are two cases to consider.
k is finite: In this case T k x is a standard point, and by continuous convergence we have T n (T k x) ≈ T n (T k y) for all infinite n. For finite n, the same formula follows from the continuity of T n . Hence T k+n x ≈ T k+n y for all n ∈ N * . k is infinite: Since x is a standard point, we know that x ∞ ≈ T k x ≈ T k y. By continuous convergence we see that for infinite n, x ∞ = T n x ∞ ≈ T n (T k x) and x ∞ = T n x ∞ ≈ T n (T k y). Thus T n (T k x) ≈ T n (T k y) for all infinite n ∈ N * . For finite n, the same formula follows from continuity of T n (using that T k x and T k y are both infinitely close to the standard point x ∞ ), and hence we have also in this case proved that T k+n x ≈ T k+n y for all n ∈ N * . By Proposition 2.11, we now have: Corollary 2.14. Stable ACFs converge uniformly on compact sets.

Uniform convergence on bounded sets
Recall that to show uniform convergence on bounded sets, we need to show that for all finite y ∈ * X, we have T n y ≈ x ∞ for all infinite n. To obtain this, we replace ACF 2 and ACF 3 by a single new condition ACF 2 For all x ∈ X, all bounded B ⊆ X and all α ∈ R + , there exist a δ ∈ R + and a ν ∈ N such that for all y ∈ B and all n ∈ N: α ≤ d(T n x, T n y) < α + δ =⇒ d(T n+ν x, T n+ν y) < α − δ Definition 2.15. We call T a strictly stable ACF if it satisfies ACF 1 and ACF 2 .
Note that although ACF 2 implies ACF 2 , it doesn't in an obvious way imply ACF 3 . Hence we cannot automatically base our arguments on the pointwise convergence of {T n x} for standard x as we could in the previous section.
Vol. 25 (2023) A nonstandard approach to asymptotic fixed point theorems Page 17 of 22 35 It is easy to see that if {T n } converges uniformly on bounded sets to a fixed point x ∞ , then ACF 1 and ACF 2 hold. To prove the converse, we start with a version of Lemma 2.2 that also applies to nonstandard points y. Lemma 2.16. Assume that T is continuous and satisfies ACF 2 . If x ∈ X and y ∈ X * is finite, then and assume for contradiction that α > 0. Choose a bounded set B such that x, y ∈ B * and use ACF 2 to pick δ and ν. Since there must be n ∈ N such that d(T n x, T n y) < α + δ, we have d(T n+ν x, T n+ν y) < α − δ, contradiction. To show the second part, note first that if the infimum is attained, there is a finite n such that T n x ≈ T n y. By continuity of T , we see that T n+N x ≈ T n+N y for all finite N . Since the set We also have a parallel to Lemma 2.3: Lemma 2.17. Assume that T is continuous and satisfies ACF 1 and ACF 2 . If x ∈ X and y ∈ X * is finite, then d(T i x, T i y) ≈ 0 for all infinite i ∈ N * .
Proof. Given ∈ R + , we use ACF 1 to find a δ such that if d(T k x, T k y) < δ, then d(T k+n x, T k+n y) < for all n ∈ N. By the previous lemma, we can find a finite k such that d(T k x, T k y) < δ, and by transfer, d(T k+n x, T k+n y) < for all n ∈ N * . Since is arbitrary, it follows that d(T i x, T i y) ≈ 0 for all infinite i ∈ N * .
Mimicking the proof of Lemma 2.5, replacing ACF 3 by ACF 2 in an obvious way, it is now easy to see that if x ∈ X, then {T n x} is a Cauchy sequence.
Theorem 2.18. Assume X is complete and T : X → X is continuous. The following are equivalent.
(i) T is a strictly stable ACF.
(ii) The sequence {T n } converges uniformly on bounded sets to an iterative fixed point x ∞ .