3.1 A refined extensionality principle and its closed variant

We begin our investigation regarding well-behaved fragments of the full extensionality principle with a uniform continuity principle for set-valued operators based on the so-called Hausdorff-like predicate as introduced in Kohlenbach and Powell (Reference Kohlenbach and Powell2020). Concretely, in Kohlenbach and Powell (Reference Kohlenbach and Powell2020), they introduced a form of uniform continuity for a set-valued operator $T:X\to 2^X$ on a normed space $X$ by assuming the existence of a modulus $\omega :\mathbb{N}\to \mathbb{N}$ such that

\begin{equation*} \forall k\in \mathbb{N}\forall x,y\in \textrm{dom} T( \lVert x-y \rVert \lt 2^{-\omega (k)}\to H^*[Tx,Ty,2^{-k}]) \end{equation*}

where $H^*$ is the aforementioned Hausdorff-like predicate defined by

\begin{equation*} H^*[P,Q,\varepsilon ]=\forall p\in Q\exists q\in Q\left (\lVert p-q \rVert \leq \varepsilon \right ). \end{equation*}

This notion was introduced in Kohlenbach and Powell (Reference Kohlenbach and Powell2020) by logical motivations to avoid the use made of the full Hausdorff metric $H$ , defined by

\begin{equation*} H(P,Q):=\max \left \{\sup _{p\in P}\inf _{q\in Q}\lVert p-q \rVert ,\sup _{q\in Q}\inf _{p\in P}\lVert p-q \rVert \right \} \end{equation*}

for nonempty, closed, and bounded sets $P,Q\subseteq X$ , in the proofs analyzed therein, which features there in the form of a uniform continuity assumption (and hence an associated extensionality statement). Further, the uniform continuity statement also features crucially in the only other previously mentioned proof mining case study from Pischke (Reference Pischke2023b) that had to resolve an extensionality statement for a set-valued operator. We here now want to argue that this uniform continuity statement already represents, or at least indicates, the correct refined extensionality principle for set-valued operators, which in particular then also indicates that the above uniform continuity statement represents the faithful uniform quantitative strengthening of the extensionality of a set-valued operator as suggested by the perspective of proof mining.

For this, we first turn to the associated extensionality principle suggested by the above uniform continuity principle relative to $H^*$ which, following Pischke (Reference Pischke2024c) where this principle was already discussed from a logical perspective (albeit embedded in the context of systems treating monotone and accretive set-valued operators), takes the following form:

\begin{align*} &\forall x^X,y^X\left (x,y\in \textrm{dom}T\land x=_Xy\rightarrow \forall k^0\left ( H^*[Tx,Ty,2^{-k}]\right )\right ) && (E)^*_T\\ &\quad\equiv \;\forall x^X,y^X\left (x,y\in \textrm{dom}T\land x=_Xy\rightarrow \forall k^0\forall u\in Tx\exists v\in Ty\left ( \lVert u-v \rVert \leq _{\mathbb{R}}2^{-k}\right )\right ). &&\end{align*}

Indeed, it can be immediately recognized that the uniform continuity principle suggested by the perspective of the monotone functional interpretation of $(E)^*_T$ amounts to the above uniform continuity statement, actually in a slightly less uniform variant where $\omega$ does additionally depend on a norm upper bound $b$ on the points from $X$ involved. Further, as discussed in Pischke (Reference Pischke2024c), this uniform continuity principle can be formalized in a proof-theoretically tame way over a system treating such operators intensionally as outline above in the following way: A “naive” first formalization of the principle, resolving in particular the hidden quantifiers in $x,y\in \textrm{dom}T$ , yields

\begin{align*} &\forall k^0,b^0,x^X,y^X,z^X,u^X\exists v^X \big(\lVert x \rVert _X,\lVert y \rVert _X,\lVert z \rVert _X,\lVert u \rVert _X\lt _{\mathbb{R}}b\land z\in Ty\land u\in Tx\\ &\quad \land \lVert x-_Xy \rVert _X\lt _{\mathbb{R}}2^{-\omega (k,b)}\rightarrow \big(v\in Ty\land \lVert u-_Xv \rVert _X\leq _{\mathbb{R}}2^{-k}\big)\big) \end{align*}

where $\omega$ is a suitable constant of type $0(0)(0)$ . As any such $v$ naturally satisfies $\lVert v \rVert \leq \lVert u \rVert +\lVert u-v \rVert \leq \lVert u \rVert +1$ , the above statement can be further specified as

\begin{align*} &\forall k^0,b^0,x^X,y^X,z^X,u^X\exists v^X\preceq _X (\lVert u \rVert _X+1)1_X\big(\lVert x \rVert _X,\lVert y \rVert _X,\lVert z \rVert _X,\lVert u \rVert _X\lt _{\mathbb{R}}b\land z\in Ty &&(UC)^*_T\\ &\quad\land u\in Tx\land \lVert x-_Xy \rVert _X\lt _{\mathbb{R}}2^{-\omega (k,b)}\rightarrow \big(v\in Ty\land \lVert u-_Xv \rVert _X\leq _{\mathbb{R}}2^{-k}\big)\big) &&\end{align*}

where $x^X\preceq _Xy^X$ means $\lVert x \rVert _X\leq _{\mathbb{R}}\lVert y \rVert _X$ . As the existential quantifier over $v$ is now bounded in terms of the preceding universal quantifiers and the inner matrix is universal, the principle $(UC)^*_T$ can be recognized as a statement of type $\Delta$ as defined in Günzel and Kohlenbach (Reference Günzel and Kohlenbach2016) for languages involving abstract types (originally stemming from the earliest works on proof mining such as Kohlenbach (Reference Kohlenbach1992), see also Kohlenbach (Reference Kohlenbach2008)), a class of formulas with a particularly trivial monotone functional interpretation, which hence are admissible in the context of systems tailored for the extraction of bounds using the monotone functional interpretation.

Now, the above extensionality principle $(E)^*_T$ seems to suggest a further extensionality principle as follows: If $H^*$ would be “continuous” in its last argument, we could move from $\forall k^0\left ( H^*[Tx,Ty,2^{-k}]\right )$ to $H^*[Tx,Ty,0]$ , whereby the above statement then would in particular imply the following even more concise extensionality principle

\begin{align*} &\forall x^X,y^X\left (x,y\in \textrm{dom}T\land x=_Xy\rightarrow H^*[Tx,Ty,0]\right ) &&(E)_T\\ &\quad\equiv \;\forall x^X,y^X\left (x,y\in \textrm{dom}T\land x=_Xy\rightarrow \forall u\in Tx\exists v\in Ty\left ( u=_Xv\right )\right ). &&\end{align*}

Here, compared to $(E)^*_T$ , the closedness of the image sets of $T$ is already “infused”, in a way, as it does not only allow us to conclude the existence of a sequence in $Ty$ approximating $u$ but actually allows us to conclude the existence of an extensionally equal witness $v$ . Further, the above principle can be thought of as an “extensionalized version” of the principle $(E)^\chi _T$ in the sense that it posits the extensional equality of the set $Tx$ not as formalized by $u\in Tx\equiv \chi (x,u)=_00$ but by the “extensionalized variant” $u\in _E Tx\equiv \exists u'\in Tx\left (u=_Xu'\right )$ . Now, while there is certainly a subtle difference between $(E)_T$ and $(E)_T^*$ , the following result makes their close relationship based on the topology of the set $Tx$ formally precise:

Proposition 1. Over $\mathscr{A}^\omega [X,\lVert \cdot \rVert ,T]$ , the principle $(E)_T$ implies $(E)_T^*$ .

Further, define the closure principle

\begin{align*} &\forall x^X,z^X,y_{(\cdot )}^{X(0)}\left (x\in \textrm{dom}T\land \forall n^0(y_n\in Tx)\land \left (y_n\to _X z\right ) \to \exists w^X\left ( w=_Xz\land w\in Tx\right )\right ), && (\textrm{pClsd})_T \end{align*}

where $y_n\to _X z$ is some formal representation of convergence in $X$ , say

\begin{equation*} \forall k^0\exists N^0\forall n\geq _0 N\left (\lVert y_n-_X z \rVert _X\leq _{\mathbb{R}}2^{-k}\right ), \end{equation*}

expressing that $Tx$ is closed for any $x\in \textrm{dom}T$ . Then over $\mathscr{A}^\omega [X,\lVert \cdot \rVert ,T]+(\textrm{pClsd})_T$ , the principle $(E)^*_T$ implies $(E)_T$ .

So, in essence, both $(E)_T$ and $(E)_T^*$ represent the same extensionality principle which posits the equality of $Tx$ and $Ty$ , seen as extensional sets, for $x=y$ in the domain of $T$ , with the difference that $(E)_T^*$ only requires a weaker approximating sequence to witness this equality which suffices in the context of closed operators.

Naturally, also $(E)_T$ entails its own uniform continuity principle via the perspective of the monotone functional interpretation which takes the form

\begin{align*} &\forall x^X,y^X,z^X,u^X\exists v^X\preceq _X (\lVert u \rVert _X+1)1_X\forall k^0,b^0\Big (\lVert x \rVert _X,\lVert y \rVert _X,\lVert z \rVert _X,\lVert u \rVert _X\lt _{\mathbb{R}}b\land z\in Ty &&(UC)_T\\ &\quad\land u\in Tx\land \lVert x-_Xy \rVert _X\lt _{\mathbb{R}}2^{-\omega (k,b)}\rightarrow \left (v\in Ty\land \lVert u-_Xv \rVert _X\leq _{\mathbb{R}}2^{-k}\right )\Big ), && \end{align*}

where we already have highlighted the natural boundedness of the quantifier over $v$ which illustrates that $(UC)_T$ , similar to $(UC)^*_T$ before, is a statement of type $\Delta$ and so is similarly admissible in the context of systems tailored for the extraction of bounds using the monotone functional interpretation.

In particular, compared to $(E)_T^\chi$ and $(E)_T^{\chi _d}$ , the fragments $(E)_T$ and $(E)_T^*$ are now very applicable as their uniform quantitative versions as guided by the monotone functional interpretation, that is the above uniform continuity principles $(UC)_T$ and $(UC)^*_T$ , are highly nontrivially populated. This also allows us to see formally that $(E)_T$ and hence $(E)_T^*$ are properly weaker than $(E)_T^{\chi _d}$ . For that, we first consider the following result which shows that any suitable operator $T:X\to 2^X$ that is uniformly continuous in the sense of $(UC)^*_T$ is closed in $X\times X$ :

Proof. As $T$ is uniformly continuous in the sense of $(UC)^*_T$ , there exists a $\omega$ with

\begin{align*} &\forall k,b\in \mathbb{N}\forall x,y,z,u\in \overline {B}_b(0)\big (z\in Ty\land u\in Tx \land \lVert x-y \rVert \lt 2^{-\omega (k,b)}\\ &\quad \rightarrow \exists v\in X\big(v\in Ty\land \lVert u-v \rVert \leq 2^{-k}\big)\big ). \end{align*}

Let $(x_n,y_n)\subseteq T$ be a sequence in $T$ such that $(x_n,y_n)\to (x,y)$ for $n\to \infty$ . As $(x_n,y_n)$ converges, the sequence is bounded and thus, using the existence of $\omega$ , we get that for any $n\in \mathbb{N}$ , there exists a $v_n\in Tx$ such that the sequence $v_n$ converges to $y$ . As $Tx$ is closed, we have $y\in Tx$ . Thus, $T$ is closed.

Hence, the results from Section 2 apply in this context and yield that $\mathscr{A}^\omega [X,\lVert \cdot \rVert ,T]+(UC)_T\not \vdash (E)^{\chi _d}_T$ as there are operators which are uniformly continuous in the sense of $(UC)_T$ that are not of the form

\begin{equation*} T:x\mapsto \begin{cases}X&\text{if }x\in \textrm{dom}T,\\\emptyset &\text{otherwise},\end{cases} \end{equation*}

the most trivial example being the operator defined by $T(x):=\{x\}$ on a space that is nontrivial. So, $(E)_T$ is properly weaker than $(E)^{\chi _d}_T$ .

Now, as mentioned before, the predicate $H^*$ and the associated uniform continuity principle for set-valued operators was introduced in Kohlenbach and Powell (Reference Kohlenbach and Powell2020) to avoid formal considerations on the Hausdorff metric. This was in particular possible as the precise value of the Hausdorff metric was not a required quantity in the proof but was only used, by means of a uniform continuity assumption, to derive certain approximation properties of the involved sets. While this was possible in Kohlenbach and Powell (Reference Kohlenbach and Powell2020), there certainly are other proofs from the literature where the value of the Hausdorff metric seems to feature much more essential in the proof and where hence a quantitative treatment thereof would be desirable to allow a more direct access to those proofs as they are found in the literature. In the next section, we provide such an access here by leveraging the strengths of the intensional approach and showing that in such a context, one can indeed treat the Hausdorff metric and its associated uniform continuity principle for a set-valued operator in a proof-theoretically tame way amenable to proof mining metatheorems.

In that section, we in particular further show that the associated extensionality principle is equivalent to $(E)^*_T$ whenever $H(Tx,Ty)$ is well-defined, showing that $(E)_T^*$ and hence $(E)_T$ are very robust as extensionality principles in the sense that small perturbations yields equivalent principles. Based on this robustness and the applicability of the principles $(E)_T$ and $(E)^*_T$ and their associated uniform continuity principles $(UC)_T$ and $(UC)^*_T$ as evidenced from the previous proof mining literature together with the logical motivations of this section, we thereby want to argue in this paper that $(E)_T$ and with that this cluster of related extensionality principles are the faithful and correct representation of the notion of extensionality of a set-valued operator in the context of this intensional approach.

3.2 An extensionality principle based on the Hausdorff metric

We now show how the Hausdorff metric and its associated extensionality and uniform continuity principles can be formally approached in the context of systems providing and intensional treatment of set-valued operators like $\mathscr{A}^\omega [X,\lVert \cdot \rVert ,T]$ (or related systems). For that, we begin with showing that for certain sets $P,Q$ , the Hausdorff distance $H(P,Q)$ can indeed be treated in a proof-theoretically tame way in the context of the systems considered in the context of proof mining over normed linear spaces.Footnote ⁷ For this, we work over the basic system $\mathscr{A}^\omega [X,\lVert \cdot \rVert ]$ for now. Now, to approach the Hausdorff metric, let concretely $P$ now be a set in a normed space $X$ which is bounded, that is $\lVert p \rVert \leq c$ for all $p\in P$ with $c\in \mathbb{N}$ . Then, we can treat the real-valued distance function

\begin{equation*} d(x,P)=\inf _{p\in P} \lVert x-p \rVert \end{equation*}

by adding an additional constant $d(\cdot ,P)$ of type $1(X)$ with the following two axiom schemes:

\begin{align*} &\forall x^X,p^X\left ( P(p)\to d(x,P)\leq _{\mathbb{R}} \lVert x-_Xp \rVert _X\right ) &&(d_P)_1 \end{align*}

as well as (writing $c$ for the real number arising from $c$ seen as a numeral)

\begin{align*} &\forall x^X,k^0\exists p\leq _X c1_X\left ( P(p)\land \lVert x-_Xp \rVert _X\leq _{\mathbb{R}} d(x,P)+2^{-k}\right ) && (d_P)_2 \end{align*}

where $P(p)$ is a predicate describing $p\in P$ . These two axioms schemes completely characterize the facts that, for one, $d(x,P)$ is supposed to be a lower bound on the norm distance $\lVert x-p \rVert$ from $x$ to any element $p\in P$ as governed by $P(p)$ , and for another, that $d(x,P)$ is arbitrarily well approximated by any such norm distance. In other words, the two schemes exactly specify that $d(x,P)$ is the greatest lower bound of all norm distances $\lVert x-p \rVert$ for all $p\in P$ .Footnote ⁸

These schemes become admissible if they are instantiated with a $P$ such that the two axioms have a monotone functional interpretation. This can in particular be guaranteed if the formula $P$ , besides potential parameters, is quantifier-free (as is e.g. naturally the case in the context of an intensional description of a set akin to the way we previously treated set-valued operators). Concretely, in this quantifier-free case (which will actually be the only concrete case occurring in the applications given in this paper), the axiom $(d_P)_2$ is of type $\Delta$ , since the existential quantifier in $(d_P)_2$ is bounded (which crucially uses the boundedness of the set specified by $P$ ), and hence admissible in systems with bound extraction theorems in the style of proof mining.

Similarly, we can add a constant $d(\cdot ,Q)$ of the same type for a second bounded set $Q$ (w.l.o.g. also bounded by $c$ ) together with the following axioms determined as above over a (in all practical circumstances of this paper quantifier-free) predicate $Q(q)$ describing $q\in Q$ :

\begin{align*} &\forall x^X,q^X\left ( Q(q)\to d(x,Q)\leq _{\mathbb{R}} \lVert x-_Xq \rVert _X\right ),&&(d_Q)_1\\ &\forall x^X,k^0\exists q\leq _X c1_X\big( Q(q)\land \lVert x-_Xq \rVert _X\leq _{\mathbb{R}} d(x,Q)+2^{-k}\big). &&(d_Q)_2 \end{align*}

In the context of both $d(x,P)$ and $d(x,Q)$ , we can then introduce the quantities

\begin{equation*} d(P,Q)=\sup _{p\in P}d(p,Q)\text{ and }d(Q,P)=\sup _{q\in Q}d(q,P) \end{equation*}

into the system by following a dual idea as the above approach towards the treatment of the infima $d(x,P)$ and $d(x,Q)$ and hence adding corresponding constants (for simplicity also denoted by) $d(P,Q)$ and $d(Q,P)$ of type $1$ into the language together with another set of similar axiom schemes, concretely taking

\begin{align*}& \forall p^X(P(p)\to d(P,Q)\geq _{\mathbb{R}} d(p,Q)), && (d_{P,Q})_1\\ &\forall k^0\exists p\leq _X c1_X\big( P(p)\land d(p,Q)\geq _{\mathbb{R}} d(P,Q)-2^{-k}\big), && (d_{P,Q})_2 \end{align*}

for the quantity $d(P,Q)$ as well as

\begin{align*} &\forall q^X(Q(q)\to d(Q,P)\geq _{\mathbb{R}} d(q,P)), && (d_{Q,P})_1\\ &\forall k^0\exists q\leq _X c1_X\big( Q(q)\land d(q,P)\geq _{\mathbb{R}} d(Q,P)-2^{-k}\big), && (d_{Q,P})_2 \end{align*}

for the quantity $d(Q,P)$ .

Again, also these axiom schemes are of the form $\Delta$ if the predicates $P$ and $Q$ are both quantifier-free (again making use of the fact that the existential quantifiers can be bounded as the specified sets are assumed to be bounded), and so these schemes are admissible in systems with bound extraction theorems in the style of proof mining.

Lastly, we move to the concrete Hausdorff metric which can now just be introduced by a closed term involving $d(P,Q)$ and $d(Q,P)$ :

\begin{equation*} H(P,Q):=\max _{\mathbb{R}}\{d(P,Q),d(Q,P)\}. \end{equation*}

Of course, this distance can also be introduced uniformly for a family of sets described by formulas $P(p,\underline {x}),Q(q,\underline {x})$ with parameters $\underline {x}$ of type $\underline {\sigma }$ if the sets described by $P(p,\underline {x}),Q(q,\underline {x})$ are bounded by a function $c(\underline {x})$ pointwise in the parameters.

Note that the nonemptiness of the sets $P,Q$ is not needed to define these formulas but the nonemptyness is required on a semantic level in order for these formulas to actually have a model as the objects, mapping to type $1$ , have to be interpreted by a real number (or by a function mapping into real numbers, respectively).

As mentioned before, this abstract treatment is fruitful at least in the context of sets describable by quantifier-free formulas, where these constants and axioms then allow for extending a previous metatheorem of an underlying system via suitable interpretations of the constants in the modelFootnote ⁹ since the axioms are admissible as discussed before. Crucial for this however is the majorizability of the constants. This however can be easily achieved: For $d(\cdot ,P)$ , via the axiom $(d_P)_1$ , we have

\begin{equation*} d(x,P)\leq \lVert x-p \rVert \leq \lVert x \rVert +\lVert p \rVert \leq \lVert x \rVert +c \end{equation*}

where $p$ is some point witnessing that $P$ is nonempty (and thus the nonemptyness is also important for majorization). Further, we have

\begin{equation*} d(Q,P)\leq d(q,P)+1\leq \lVert q \rVert +c+1\leq 2c+1 \end{equation*}

for a suitable $q$ chosen with axiom $(d_{Q,P})_2$ . From this, majorants for $d(\cdot ,P)$ and $d(Q,P)$ are immediate.

By a similar reasoning, $d(\cdot ,Q)$ as well as $d(P,Q)$ are majorizable and this extends to any variant using additional parameters if the sets are nonempty and bounded pointwise for all parameters. Naturally, also the resulting bounding function $c(\underline {x})$ then has to be majorizable as a function of type $0(\underline {\sigma }^t)$ .

We are now in particular interested in using this way of formulating the Hausdorff distance to talk about uniform continuity formulations and extensionality principles for set-valued operators $T$ treated as in the previous basic system $\mathscr{A}^\omega [X,\lVert \cdot \rVert ,T]$ . Then, the sets $P$ and $Q$ can be taken to be of the form $Tx$ with a parameter $x$ of type $X$ for a given set-valued operator $T$ which is represented in the system by an intensional description over its graph via $\chi _T$ as discussed before. Formally, this is naturally represented by taking $P(p,x):=\chi _T(x,p)=_00$ . As this resulting formulation of the set $Tx$ is quantifier-free, the above axioms in particular become admissible for bound extraction results if, as discussed before, the operator $T$ is actually such that all $Tx$ are bounded with a bounding function $c$ of type $0(X)$ that is majorizable. In the language of Pischke (Reference Pischke2024c), the existence of such a $c$ is equivalent to the operator $T$ being uniformly majorizable, that is to being bounded on bounded sets. Thus, to treat such operators in the Hausdorff metric, we consider an additional constant $T^*$ of type $1$ together with the axiom

\begin{align*} &\forall x^X,y^X,b^0\left ( y\in Tx\land \lVert x \rVert _X\lt _{\mathbb{R}} b\to \lVert y \rVert _X\leq _{\mathbb{R}}T^*b\right ) && (T^*) \end{align*}

which serves as a majorant (and hence witness) to $c$ . Then, we can as above introduce constants $d(\cdot ,Tx)$ and $d(Tx,Ty)$ for $x,y\in \textrm{dom}T$ into the language using $\chi _T$ and $T^*$ to form $H$ such that the expression $H(Tx,Ty)$ is represented by a term for any $x$ and $y$ .

With this, an extensionality statement corresponding to the Hausdorff metric now indeed can be written as a formal sentence in this extended language via

\begin{align*} &\forall x^X,y^X\left ( x,y\in \textrm{dom}T\land x=_X y\rightarrow H(Tx,Ty)=_{\mathbb{R}} 0\right ). && (E)_T^H \end{align*}

In that context, this extensionality principle $(E)_T^H$ is provably equivalent to the previous principle $(E)_T^*$ as the following result shows. For that, let $\mathscr{A}^\omega [X,\lVert \cdot \rVert ,T,H]$ refer to the system which results from $\mathscr{A}^\omega [X,\lVert \cdot \rVert ,T]$ by adding the respective constants and axioms for the Hausdorff metric required to introduce $H(Tx,Ty)$ as detailed above.

Proof. To show that $(E)_T^*$ implies $(E)_T^H$ , let $x=y$ for $x,y\in \textrm{dom}T$ . Fixing $k$ of type $0$ , let $u\in Tx$ be such that $d(u,Ty)+2^{-(k+1)}\geq d(Tx,Ty)$ , using the axioms for $d(Tx,Ty)$ . Then use $(E)_T^*$ to pick $v\in Ty$ with $\lVert u-v \rVert \leq 2^{-(k+1)}$ . We now have

\begin{equation*} d(Tx,Ty)\leq d(u,Ty)+2^{-(k+1)}\leq \lVert u-v \rVert +2^{-(k+1)}\leq 2^{-k} \end{equation*}

using the axioms for $d(u,Ty)$ . As $k$ was arbitrary, we have $d(Tx,Ty)=0$ , and similarly we can show $d(Ty,Tx)=0$ . This yields $H(Tx,Ty)=0$ , and we have shown $(E)_T^H$ .

To show that $(E)_T^H$ implies $(E)_T^*$ , again let $x=y$ for $x,y\in \textrm{dom}T$ and fix $k$ of type $0$ as well as $u\in Tx$ . As $(E)_T^H$ implies $H(Tx,Ty)=0$ , we have $d(Tx,Ty)=0$ . Using the axioms for $d(Tx,Ty)$ , we have $d(u,Ty)=0$ and so using the axioms for $d(u,Ty)$ , we have that there exists a $v\in Ty$ with

\begin{equation*} \lVert u-v \rVert \leq d(u,Ty)+2^{-k}=2^{-k}. \end{equation*}

Thus, we have shown $(E)_T^*$ .

Further, the monotone functional interpretation then associates to this a corresponding uniform continuity principle for set-valued operators. Further, this principle is actually the usual notion of uniform continuity for set-valued operators w.r.t. the Hausdorff metric (as commonly used in the analytic literature, see e.g. Moore and Nnoli (Reference Moore and Nnoli2001)). Concretely, the monotone functional interpretation posits the existence of a modulus $\omega$ of type $0(0)(0)$ satisfyingFootnote ¹⁰

\begin{align*} &\forall x^X,y^X,u^X,v^X,k^0,b^0( (x,u),(y,v)\in T\land \lVert x \rVert _X,\lVert y \rVert _X,\lVert u \rVert _X,\lVert v \rVert _X\lt _{\mathbb{R}} b && (UC)^H_T\\ &\quad \land \lVert x-_Xy \rVert _X \lt _{\mathbb{R}} 2^{-\omega (k,b)}\to H(Tx,Ty)\leq _{\mathbb{R}} 2^{-k}). && \end{align*}

As this statement $(UC)^H_T$ is now universal based on our treatment of the Hausdorff metric, it can thus be freely added to a system with bound extraction theorems in the style of proof mining together with an accompanying constant $\omega$ and the preceding treatment of $H$ so that, for this extension, one retains the bound extraction results. Even further, a quantitative analysis of Proposition4 immediately yields that if $\omega (k,b)$ is a modulus of uniform continuity for $T$ in the sense of $(UC)^H_T$ , we have that $\omega (k+1,b)$ is a corresponding modulus for the uniform continuity of $T$ in the sense of $(UC)^*_T$ , and similarly, we conversely have that if $\omega (k,b)$ is a modulus of uniform continuity for $T$ in the sense of $(UC)^*_T$ , then $\omega (k+1,b)$ is a corresponding modulus for the uniform continuity of $T$ in the sense of $(UC)^H_T$ (in this context where $H(Tx,Ty)$ is well-defined).

In the next section, we will illustrate the applicability of this approach towards the Hausdorff metric by analyzing iterative methods related to set-valued mappings which are uniformly continuous w.r.t. the Hausdorff metric.

4.1 The central assumptions and their quantitative content

In this section, we now first discuss the central assumptions present in Theorem2 and in particular discuss (using the underlying logical methodology) what kind of quantitative assumptions they entail to potentially feature in the analysis of the main theorem.

The first important assumption present in Theorem2 is the compactness of the set $K$ . This compactness assumption on $K$ is witnessed in the following by a quantitative modulus of compactness introduced in Gerhardy (Reference Gerhardy2008) under the name of a modulus of total boundedness Footnote ¹¹ which takes the form of a function $\gamma :\mathbb{N}\to \mathbb{N}$ such that for any $k\in \mathbb{N}$ and for any $(x_n)\subseteq K$ :

\begin{equation*} \exists 0\leq i\lt j\leq \gamma (k)\left ( \lVert x_i-x_j \rVert \leq \frac {1}{k+1}\right ). \end{equation*}

Such a modulus exists if, and only if, $K$ is compact and we refer to Kohlenbach et al. (Reference Kohlenbach, Leuştean and Nicolae2018) for various discussions on the construction of such moduli for certain concrete classes of compact sets and spaces.

As a second assumption, we find the nonemptyness of the fixed point set $F(T)$ which will be represented by a concrete witness $p_0$ (i.e. $p_0\in K$ and $p_0\in Tp_0$ ) in the following. As follows by the perspective of majorization, the bounds extracted later will of course only depend on an upper bound on the norm of $p_0$ , which by the compactness and therefore the boundedness of $K$ is in particular represented by any upper bound on the diameter of $K$ .

One of the most crucial assumptions, in some sense, is the single-valuedness of $T$ on actual fixed points, that is the assumption that $Tp=\{p\}$ if $p\in F(T)$ . This implication is equivalent to

\begin{align*} &\forall p\in K\left (d(p,Tp)=0\to H(\{p\},Tp)=0\right ) && ({*}) \end{align*}

which in turn unravels into

\begin{equation*} \forall p\in K\forall k\in \mathbb{N}\exists j\in \mathbb{N}\left ( d(p,Tp)\leq \frac {1}{j+1}\to H(\{p\},Tp)\leq \frac {1}{k+1}\right ) \end{equation*}

and in that way, the logical methodology inducesFootnote ¹² a modulus $\theta :\mathbb{N}\to \mathbb{N}$ bounding (and thus witnessing) such a $j$ in terms of $k$ , that is such thatFootnote ¹³

\begin{equation*} \forall p\in K\forall k\in \mathbb{N}\left ( d(p,Tp)\leq \frac {1}{\theta (k)+1}\to H(\{p\},Tp)\leq \frac {1}{k+1}\right ). \end{equation*}

Note that by a simple compactness argument, possessing such a modulus is equivalent to the property $({*})$ in compact spaces:

Lemma 6. Let $K$ be compact and let $T:K\to CB(K)$ be a nonexpansive operator. Then $T$ satisfies $({*})$ if, and only if,

\begin{align*} & \forall k\in \mathbb{N}\exists j\in \mathbb{N}\forall p\in K\left ( d(p,Tp)\leq \frac {1}{j+1}\to H(\{p\},Tp)\leq \frac {1}{k+1}\right ). && ({**}) \end{align*}

Proof. Clearly, $({**})$ implies $({*})$ . Conversely, suppose that $({**})$ fails, that is suppose there exists a $k\in \mathbb{N}$ such that for any $j\in \mathbb{N}$ :

\begin{equation*} \exists p_j\in K\left ( d(p_j,Tp_j)\leq \frac {1}{j+1}\land H(\{p_j\},Tp_j)\gt \frac {1}{k+1}\right ). \end{equation*}

Then $d(p_j,Tp_j)\leq \frac {1}{j+1}$ implies that for any $j\geq 1$ , there exists a $q_j\in Tp_j$ such that $\lVert p_j-q_j \rVert \leq 1/j$ . Further, $H(\{p_{j}\},Tp_{j})\gt \frac {1}{k+1}$ now implies that there exists a $q_j'\in Tp_j$ such that $\lVert p_j-q_j' \rVert \gt \frac {1}{k+1}$ .

We now pick subsequences $p_{j_i}$ , $q_{j_i}$ and $q_{j_i}'$ such that $p_{j_i}\to p$ , $q_{j_i}\to q$ and $q_{j_i}'\to q'$ with $p,q,$ $q'\in K$ . Then, $\lVert p-q \rVert =0$ and $H(Tp_{j_i},Tp)\to 0$ for $i\to \infty$ as $T$ is nonexpansive. Thus, in particular $d(q_{j_i},Tp),d(q'_{j_i},Tp)\to 0$ which yields

\begin{equation*} d(q,Tp)\leq \lVert q-q_{j_i} \rVert +d(q_{j_i},Tp)\to 0 \end{equation*}

and thus $d(p,Tp)=d(q,Tp)=0$ . Similarly $d(q',Tp)=0$ and thus $q'\in Tp$ . However, we have $\lVert p-q' \rVert \geq \frac {1}{k+1}$ and so $H(\{p\},Tp)\geq \lVert p-q' \rVert \geq \frac {1}{k+1}$ . This is a contradiction to $({*})$ .

In that way, the existence of such a modulus is implied already by the assumptions in Theorem 2.

At last, we consider the assumptions on the auxiliary sequences $\gamma _n$ and $\alpha _n$ . For $\gamma _n$ , where it is assumed that

\begin{equation*} \lim _{n\to \infty } \gamma _n= 0, \end{equation*}

we will later rely on a rate of convergence $\tau$ witnessing this property, i.e. on a $\tau$ satisfying

\begin{equation*} \forall k\in \mathbb{N}\forall n\geq \tau (k)\left ( \gamma _n\leq \frac {1}{k+1}\right ). \end{equation*}

For $\alpha _n$ , the assumption that

\begin{equation*} 0\lt \liminf _{n\to \infty }\alpha _n\leq \limsup _{n\to \infty }\alpha _n\lt 1 \end{equation*}

is witnessed by a value $a\in \mathbb{N}^*$ with the property

\begin{equation*} \forall n\geq a\left ( \frac {1}{a}\leq \alpha _n\leq 1-\frac {1}{a}\right ) \end{equation*}

in similarity to Dinis and Pinto (Reference Dinis and Pinto2020).

4.2 Suzuki’s lemma and its analysis

The main analytical ingredient of the convergence proof from Song and Wang (Reference Song and Wang2009) is a well-known lemma from Suzuki (Reference Suzuki2005):

Lemma 8 (Suzuki Reference Suzuki2005). Let $(x_n),(y_n)$ be bounded sequences in a Banach space $X$ and let $(\alpha _n)\subseteq [0,1]$ be such that $0\lt \liminf _{n\to \infty }\alpha _n\leq \limsup _{n\in \mathbb{N}}\alpha _n\lt 1$ . Suppose that $x_{n+1}=\alpha _nx_n+(1-\alpha _n)y_n$ as well as

\begin{equation*} \limsup _{n\to \infty }(\lVert y_{n+1}-y_n \rVert -\lVert x_{n+1}-x_n \rVert )\leq 0. \end{equation*}

Then $\lim _{n\to \infty }\lVert x_n-y_n \rVert =0$ .

This lemma was analyzed quantitatively in Dinis and Pinto (Reference Dinis and Pinto2020), and we will rely in the following on this analysis:

Lemma 9 (Dinis and Pinto 2020). Let $(x_n),(y_n)$ be sequences in a Banach space $X$ with $\lVert x_n \rVert ,\lVert y_n \rVert \leq b$ for some $b\in \mathbb{N}^*$ and let $(\alpha _n)\subseteq [0,1]$ be such that there exists a $a\in \mathbb{N}^*$ with the property

\begin{equation*} \forall n\geq a\left ( \frac {1}{a}\leq \alpha _n\leq 1-\frac {1}{a}\right ). \end{equation*}

Suppose that $x_{n+1}=\alpha _nx_n+(1-\alpha _n)y_n$ as well as that there exists a monotone function $\tau :\mathbb{N}\to \mathbb{N}$ such that

\begin{equation*} \forall k\in \mathbb{N}\forall n\geq \tau (k)\left(\left(\lVert y_{n+1}-y_n \rVert -\lVert x_{n+1}-x_n \rVert \right)\leq \frac {1}{k+1}\right). \end{equation*}

Then for any $k\in \mathbb{N}$ and any $g:\mathbb{N}\to \mathbb{N}$ :

\begin{equation*} \exists n\leq \varphi _{a,\tau ,b}(k,g)\forall m\in [n;\, n+g(n)]\left ( \lVert x_m-y_m \rVert \leq \frac {1}{k+1}\right ), \end{equation*}

where $\varphi _{a,\tau ,b}(k,g)=\max \{a,\tau (t(2t+1)a^t(k+1)-1)\}+(bt(2t+1)a^t(k+1)-1)t+r_0$ for

\begin{equation*} r_i:=\begin{cases}0&\text{if }i=b(k+1),\\t+r_{i+1}+\widehat {g}(\mathrm{max} \{a,\tau (t(2t+1)a^t(k+1)-1)\}+it+r_{i+1})&\text{if }i\lt b(k+1).\end{cases} \end{equation*}

where $\widehat {g}(m)=t+g(m)$ and $t=2ba(k+1)$ .

4.3 Fejér monotonicity and metastability

We now present the extractions of the quantitative versions of Fejér monotonicity and asymptotic regularity.

For this, we first need to define an appropriate notion of an approximate solution (i.e. of an approximate fixed point) as the results given in Kohlenbach et al. (Reference Kohlenbach, Leuştean and Nicolae2018) rely on uniform reformulations of the respective properties in terms of such approximate solutions. For our concrete situation here, note that $p$ is a fixed point of $T$ if, and only if, $d(p,Tp)=0$ (as $Tp$ is closed since $Tp\in CB(K)$ ). In that vein, we call $p$ a $\frac {1}{k+1}$ -approximate fixed point of $T$ if

\begin{equation*} d(p,Tp)\leq \frac {1}{k+1} \end{equation*}

and define correspondingly

\begin{equation*} AF_k=\left \{p\in K\mid d(p,Tp)\leq \frac {1}{k+1}\right \} \end{equation*}

as the set of approximate solutions which extend the set of full solutions

\begin{equation*} F=\{p\in K\mid d(p,Tp)=0\}=F(T). \end{equation*}

Now, for the Fejér monotonicity of $(x_n)$ , we concretely strive to establish the existence of the following modulus relative to the chosen $AF_k$ :

Definition 10 (Kohlenbach et al. 2018). A function $\chi :\mathbb{N}^3\to \mathbb{N}$ is a modulus of uniform Fejér monotonicity for $(x_n)$ w.r.t. $(AF_k)$ if for any $n,m,r\in \mathbb{N}$ , any $p\in AF_{\chi (k,m,r)}$ and any $l\leq m$ :

\begin{equation*} \lVert x_{n+l}-p \rVert \lt \lVert x_n-p \rVert +\frac {1}{r+1}. \end{equation*}

For this, we can now extract the following from the proof of Fejér monotonicity given in Song and Wang (Reference Song and Wang2009) for the sequence $(x_n)$ defined as in ( $\dagger$ ).

Lemma 11. Let $\theta$ be such that

\begin{equation*} \forall p\in K\forall k\in \mathbb{N}\left ( d(p,Tp)\leq \frac {1}{\theta (k)+1}\to H(\{p\},Tp)\leq \frac {1}{k+1}\right ). \end{equation*}

Then sequence $(x_n)$ defined as in $({\dagger})$ is uniformly Fejér monotone w.r.t. $(AF_k)$ with a modulus

\begin{equation*} \chi (n,m,r)=\theta (m(r+1)+1). \end{equation*}

Proof. Let $p$ be given with $d(p,Tp)\leq \frac {1}{\chi (n,m,r)+1}$ . Then

\begin{align*} \lVert x_{n+1}-p \rVert &\leq (1-\alpha _n)\lVert x_n-p \rVert +\alpha _n\lVert y_n-p \rVert \\ &\leq (1-\alpha _n)\lVert x_n-p \rVert +\alpha _nd(y_n,Tp)+\alpha _n(\lVert y_n-p \rVert -d(y_n,Tp))\\ &\leq (1-\alpha _n)\lVert x_n-p \rVert +\alpha _nH(Tx_n,Tp)+\alpha _n(\lVert y_n-p \rVert -d(y_n,Tp))\\ &\leq \lVert x_n-p \rVert +(\lVert y_n-p \rVert -d(y_n,Tp)) \end{align*}

and by induction we get

\begin{equation*} \lVert x_{n+l}-p \rVert \leq \lVert x_n-p \rVert +\sum _{i=0}^{l-1}(\lVert y_{n+i}-p \rVert -d(y_{n+i},Tp)) \end{equation*}

for any $l\geq 1$ . It is rather immediate to see that in general, for nonempty sets $Y,Z\subseteq X$ and a point $x$ , we have $d(x,Y)\leq d(x,Z)+H(Y,Z)$ and instantiating this yields

\begin{equation*} \lVert y_{n+i}-p \rVert =d(y_{n+i},\{p\})\leq d(y_{n+i},Tp)+H(\{p\},Tp) \end{equation*}

and thus $\lVert y_{n+i}-p \rVert -d(y_{n+i},Tp)\leq H(\{p\},Tp)$ . As now $p\in AF_{\chi (n,m,r)}$ , we get

\begin{equation*} H(\{p\},Tp)\lt \frac {1}{m(r+1)}. \end{equation*}

In particular, in that case we have

\begin{align*} \lVert x_{n+l}-p \rVert &\leq \lVert x_n-p \rVert +mH(\{p\},Tp)\\ &\lt \lVert x_n-p \rVert +\frac {1}{r+1} \end{align*}

for $l\leq m$ .

For the asymptotic behavior, we are interested in the following type of quantitative information:

Definition 13 (Kohlenbach et al. 2018). A function $\Phi$ is an approximate $F$ -point bound for $(x_n)$ w.r.t. $(AF_k)$ if for any $k\in \mathbb{N}$ :

\begin{equation*} \exists n\leq \Phi (k)\left ( x_n\in AF_{k}\right ). \end{equation*}

The construction of such a $\Phi$ for the sequence studied here relies on analyzing the proof of the statement $d(x_n,Tx_n)\to 0$ from Song and Wang (Reference Song and Wang2009), which relies on Suzuki’s lemma. Concretely, we get the following:

Lemma 14. Let $b$ be a bound on the diameter of $K$ and let $(\alpha _n)\subseteq [0,1]$ be such that there exists an $a\in \mathbb{N}^*$ with the property

\begin{equation*} \forall n\geq a\left ( \frac {1}{a}\leq \alpha _n\leq 1-\frac {1}{a}\right ). \end{equation*}

Let $\tau$ be a monotone rate of convergence for $\gamma _n\to 0$ . Let $\varphi _{a,\tau ,b}$ be defined as in Lemma 9. Then $(x_n)$ defined as in $({\dagger})$ has approximate $F$ -points w.r.t. $(AF_k)$ with an approximate $F$ -point bound

\begin{equation*} \Phi (k)=\varphi _{a,\tau ,b}(k,0). \end{equation*}

Proof. As in Song and Wang (Reference Song and Wang2009), we can derive

\begin{equation*} \lVert y_{n+1}-y_n \rVert \leq H(Tx_{n+1},Tx_n)+\gamma _n\leq \lVert x_{n+1}-x_n \rVert +\gamma _n \end{equation*}

which yields that

\begin{equation*} \lVert y_{n+1}-y_n \rVert -\lVert x_{n+1}-x_n \rVert \leq \gamma _n \end{equation*}

and thus $\tau$ satisfies the assumption of Lemma 9. Applying Lemma 9, we get that for any $k\in \mathbb{N}$ and any $g:\mathbb{N}\to \mathbb{N}$ :

\begin{equation*} \exists n\leq \varphi _{a,\tau ,b}(k,g)\forall m\in [n;\,n+g(n)]\left ( \lVert x_m-y_m \rVert \leq \frac {1}{k+1}\right ). \end{equation*}

In particular, we get for any $k\in \mathbb{N}$ that

\begin{equation*} \exists n\leq \varphi _{a,\tau ,b}(k,0)\left ( \lVert x_n-y_n \rVert \leq \frac {1}{k+1}\right ) \end{equation*}

which yields that for this $n$ , we have

\begin{equation*} d(x_n,Tx_n)\leq \lVert x_n-y_n \rVert \leq \frac {1}{k+1}, \end{equation*}

i.e. $x_n\in AF_k$ .

Remark 15. While the full function $\varphi _{a,\tau ,b}$ is rather complex, in the above special case of considering the constant- $0$ function, it simplifies considerably to

\begin{equation*} \varphi _{a,\tau ,b}(k,0)=\max \{a,\tau (t(2t+1)a^t(k+1)-1)\}+(bt(2t+1)a^t(k+1)-1)t+2b(k+1)t \end{equation*}

for $t=2ba(k+1)$ .

Lastly, we show that $F(T)$ is not only closed but that it is even sufficiently uniformly closed respective to the approximations $AF_k$ in a concrete way introduced in Kohlenbach et al. (Reference Kohlenbach, Leuştean and Nicolae2018):

Lemma 17. The set $F=F(T)$ is uniformly closed w.r.t. $(AF_k)$ with moduli

\begin{equation*} \begin{cases} \delta (k)=2k+1,\\ \omega (k)=4k+3. \end{cases} \end{equation*}

Proof. Note that we have

\begin{align*} d(p,Tp)&\leq d(p,Tq)+H(Tp,Tq)\\ &\leq \lVert p-q \rVert +d(q,Tq)+\lVert q-p \rVert \end{align*}

and thus if $q\in AF_{2k+1}$ and $\lVert p-q \rVert \leq \frac {1}{4(k+1)}$ , then $d(p,Tp)\leq \frac {1}{k+1}$ , i.e. $p\in AF_{k}$ .

Combined, we can now apply the general result from Kohlenbach et al. (Reference Kohlenbach, Leuştean and Nicolae2018) to get the following quantitative version of Theorem2.

Theorem 3. Let $\gamma$ be a modulus of total boundedness for $K$ . Let $b$ be a bound on the diameter of $K$ and let $(\alpha _n)\subseteq [0,1]$ be such that there exists an $a\in \mathbb{N}^*$ with the property

\begin{equation*} \forall n\geq a\left ( \frac {1}{a}\leq \alpha _n\leq 1-\frac {1}{a}\right ). \end{equation*}

Let $(\gamma _n)\subseteq (0,\infty )$ be such that $\gamma _n\to 0$ and let $\tau$ be a monotone rate of convergence for $\gamma _n\to 0$ . Let $\theta$ be such that

\begin{equation*} \forall p\in K\forall k\in \mathbb{N}\left ( d(p,Tp)\leq \frac {1}{\theta (k)+1}\to H(\{p\},Tp)\leq \frac {1}{k+1}\right ). \end{equation*}

Let $\varphi _{a,\tau ,b}(k,0)$ be defined as in Remark 15, i.e.

\begin{equation*} \varphi _{a,\tau ,b}(k,0)=\max \{a,\tau (t(2t+1)a^t(k+1)-1)\}+(bt(2t+1)a^t(k+1)-1)t+2b(k+1)t \end{equation*}

for $t=2ba(k+1)$ . Then, $(x_n)$ defined as in $({\dagger})$ is Cauchy and moreover, for all $k\in \mathbb{N}$ and all $g:\mathbb{N}\to \mathbb{N}$ ,

\begin{equation*} \exists N\leq \Psi (k,g)\forall i,j\in [N;\, N+g(N)]\left ( \lVert x_i-x_j \rVert \leq \frac {1}{k+1}\land x_i\in AF_k\right ) \end{equation*}

where $\Psi (k,g)=\Psi _0(P,k,g)$ for $P=\gamma (4k+3)$ and with

\begin{equation*} \begin{cases} \Psi _0(0,k,g)=0,\\ \Psi _0(n+1,k,g)=\varphi _{a,\tau ,b}(\chi _{k,g}^M(\Psi _0(n,k,g),8k+7),0), \end{cases} \end{equation*}

and where

\begin{align*} &\chi (n,m,r)=\theta (m(r+1)+1),\\ &\chi _{k}(n,m,r)=\max \{2k+1,\chi (n,m,r)\},\\ &\chi _{k,g}^M(n,r)=\max \{\chi _k(i,g(i),r)\mid i\leq n\}. \end{align*}

4.4 Moduli of regularity and rates of convergence

In this section, using the results from Kohlenbach et al. (Reference Kohlenbach, López-Acedo and Nicolae2019), we give constructions for rates of convergence based on the assumption of a (very general) kind of regularity notion as discussed in the introduction.

The central notion here is consequently the following instantiation of the abstract notion of a modulus of regularity from Kohlenbach et al. (Reference Kohlenbach, López-Acedo and Nicolae2019):

Definition 19. Let $z\in F(T)$ and $r\gt 0$ . A function $\phi :(0,\infty )\to (0,\infty )$ is called a modulus of regularity for $T$ w.r.t $\overline {B}_r(z)$ if for all $\varepsilon \gt 0$ and all $x\in \overline {B}_r(z)$ :

\begin{equation*} d(p,Tp)\lt \phi (\varepsilon )\to \textrm{dist}(x,F(T))\lt \varepsilon . \end{equation*}

If there is a $z\in F(T)$ such that $\phi$ is a modulus of regularity w.r.t. $\overline {B}_r(z)$ for all $r\gt 0$ , then $\phi$ is just called a modulus of regularity for $T$ .

Note that the function $d(p,Tp)$ is continuous in $p$ if $T$ is nonexpansive as

\begin{align*} d(p,Tp)&\leq d(p,Tq)+H(Tp,Tq)\\ &\leq \lVert p-q \rVert +d(q,Tq)+\lVert q-p \rVert \end{align*}

and thus

\begin{equation*} \vert d(p,Tp)-d(q,Tq)\vert \leq 2\lVert p-q \rVert . \end{equation*}

It follows from Proposition 3.3 of Kohlenbach et al. (Reference Kohlenbach, López-Acedo and Nicolae2019) that any such nonexpansive map $T$ has a modulus of regularity (albeit in general being uncomputable) if $K$ is compact.

Under the assumption of such a modulus, we now get the following result on rates of convergence:

Theorem 4. Let $z\in F(T)\neq \emptyset$ and let $b$ be a bound on the diameter of $K$ . Assume that $K$ is closed. Let $(x_n)$ be defined as in $({\dagger})$ . Assume that $T$ satisfies $({*})$ . Let $(\alpha _n)$ with $a$ and $(\gamma _n)$ with $\tau$ as well as $\varphi _{a,\tau ,b}(k,0)$ be as in Theorem 3 (and Remark 15). Let $\phi$ be a modulus of regularity for $T$ w.r.t. $\overline {B}_b(z)$ . Then, $(x_n)$ is Cauchy with

\begin{equation*} \forall \varepsilon \gt 0\forall i,j\geq \varphi _{a,\tau ,b}\left (\left\lceil {\frac {1}{\phi (\varepsilon /2)}}\right\rceil ,0\right )\left ( \lVert x_i-x_j\rVert \lt \varepsilon \right ). \end{equation*}

and further $(x_n)$ converges to a fixed point of $T$ with a rate of convergence

\begin{equation*} \varphi _{a,\tau ,b}\left (\left\lceil {\frac {1}{\phi (\varepsilon /2)}}\right\rceil ,0\right ). \end{equation*}

Proof. The result is a straightforward instantiation of the general abstract Theorem 4.1 from Kohlenbach et al. (Reference Kohlenbach, López-Acedo and Nicolae2019), using the previous Lemma 14 by which we have that

\begin{equation*} \forall \varepsilon \gt 0\exists n\leq \varphi _{a,\tau ,b}\left (\left\lceil {\frac {1}{\varepsilon }}\right\rceil ,0\right )\left ( d(x_n,Tx_n)\lt \varepsilon \right ). \end{equation*}

Note for this that the sequence $(x_n)$ is Fejér monotone w.r.t. $F(T)$ by Remark 12 since $T$ satisfies $({*})$ . That $(x_n)$ converges to a fixed point of $T$ with the given rate follows from Theorem 4.1, (i) in Kohlenbach et al. (Reference Kohlenbach, López-Acedo and Nicolae2019) for which we need that $K$ is complete (which follows as $X$ is a Banach space and as $K$ is closed) and that $F(T)$ is closed which follows from the fact that $d(p,Tp)$ is uniformly continuous in $p$ and $F(T)=(d(\cdot ,T\cdot ))^{-1}(0)$ .

Finally, we look at a notion for multivalued mappings where simple instances of such moduli of regularity can be derived. Following Senter and Dotson (Reference Senter and Dotson1974), a multivalued mapping $T:K\to \textrm{CB}(K)$ is said to satisfy Condition I if there is a nondecreasing function $f:[0,\infty )\to [0,\infty )$ with $f(0)=0$ , $f(r)\gt 0$ for $r\in (0,\infty )$ and

\begin{equation*} d(x,Tx)\geq f(d(x,F(T))) \end{equation*}

for all $x\in K$ . If the property that $f(r)\gt 0$ for $r\in (0,\infty )$ is witnessed in a uniform and quantitative way by a function $\phi :(0,\infty )\to (0,\infty )$ with

\begin{equation*} f(r)\lt \phi (\varepsilon )\to r\lt \varepsilon \end{equation*}

for any $r,\varepsilon \gt 0$ , then such a $\phi$ is clearly already a modulus of regularity for $T$ . This in particular is true for mappings that satisfy Condition II of Senter and Dotson (Reference Senter and Dotson1974), i.e. where there exists a real $\alpha \gt 0$ such that

\begin{equation*} d(x,Tx)\geq \alpha d(x,\textrm{F}(T)) \end{equation*}

where then $\phi$ can be given by $\phi (\varepsilon )=\alpha \varepsilon$ . Examples of mappings which satisfy Condition II are for instance discussed in Senter and Dotson (Reference Senter and Dotson1974) and for these, the above rates of convergence therefore instantiate immediately.