A Local Fixed Point Theorem for Multivalued Mappings in Strong Partial b-Metric Spaces with Potential Applications to Language Dynamics

Abstract

This paper establishes a local fixed point theorem for multivalued mappings in zero-complete strong partial b-metric spaces and develops a framework for modelling multistage processes with several admissible terminal states. The contractive condition is formulated using a Bianchini–Grandolfi gauge function and the associated excess functional. Unlike global principles, the theorem requires contractive assumptions only within a prescribed closed ball. A localization condition keeps the successive approximations inside this ball, while an adapted chain estimate proves that the iterative sequence is zero-Cauchy. The proof does not require the family of partial b-metric balls to form a topological basis; the ball serves only as a localization set, and the limiting argument relies on zero-completeness and the zero-closedness of the mapping values. The main theorem guarantees a fixed point for a multivalued mapping and, under an additional condition, uniqueness in the single-valued case. Its consequences include local and global principles for linear set-valued contractions, a partial metric version, and a local Banach-type theorem. A further contribution is a method for constructing strong partial b-metrics from bounded metric spaces and prescribed nonempty target families. A nonlinear transformation of the original metric is combined with the transformed distances from the target family. Thus, the self-distance of each point is determined by its position relative to that family and vanishes precisely on it. This provides a natural model for processes with several distinct but equally stable terminal states. The theory is illustrated through a finite model of linguistic enrichment. Twenty formulations of the same mathematical statement are arranged into successive levels of grammatical, terminological, logical, and stylistic refinement. A weighted revision graph generates the generalized distance, while the self-distance represents the remaining effort required to reach a stable formulation. A multivalued revision mapping allows several admissible improvements at every stage. The model admits two distinct stable formulations: a concise formal version and a more explanatory, pedagogically oriented version. Both require no further essential revision but remain distinct. This demonstrates that stabilization need not imply uniqueness and that different enrichment trajectories may lead to different acceptable terminal texts. More generally, the example shows how generalized fixed point methods can describe local, nonunique, and multistage stabilization processes in language dynamics and related nonlinear systems.

1. Introduction

Fixed point theory provides one of the fundamental mathematical tools for analyzing the existence and stability of equilibrium states, iterative algorithms, optimization procedures, and learning processes. In many applications, however, the underlying dynamics are meaningful only in a prescribed neighborhood of an initial state, while global contractive assumptions are either unavailable or unnecessarily restrictive. This has led to the development of local fixed point principles, where the iteration is required to remain inside a suitable ball and converge to a fixed point without relying on global properties of the entire space.

The study of generalized metric structures has substantially enlarged the scope of classical fixed point theory. Besides ordinary metric spaces, numerous extensions have been introduced in order to model situations in which the standard notion of distance is no longer appropriate. Among the best known are partial metric spaces, introduced by Matthews, b-metric spaces, strong b-metric spaces, partial b-metric spaces, and strong partial metric spaces. The class of strong partial b-metric spaces forms an intermediate class between strong b-metric spaces and partial b-metric spaces. It combines the possibility of positive self-distances with the asymmetric coefficient K appearing in the strong generalized triangle inequality. Partial metric spaces are recovered when $K=1$, while ordinary metric spaces are recovered when, in addition, all self-distances vanish.

The position of strong partial b-metric spaces between strong b-metric spaces and partial b-metric spaces was established by Moshokoa and Ncongwane; see [1].

Unlike classical metric spaces, positive self-distances may carry meaningful information rather than merely representing a technical generalization. In many situations, they naturally quantify the amount of incompleteness, uncertainty, or residual discrepancy associated with a given state. Consequently, a point with a positive self-distance should not necessarily be interpreted as being “far from itself,” but rather as representing an object that has not yet reached its final stable form. This interpretation makes strong partial b-metric spaces particularly suitable for describing progressive refinement processes.

A subtle issue in partial b-metric spaces concerns the topological behavior of their open balls. Ge and Lin showed that, in general, the family of partial b-metric open balls need not form a basis for a topology; see [2]. Since strong partial b-metric spaces constitute a more restrictive class, such a topological failure should not be transferred to the present setting without additional justification. Accordingly, our arguments do not require the family of p-balls to form a topological basis: the closed ball is used only as a localization set for the iterative construction. Moreover, because the iterative sequence constructed below is 0-Cauchy, 0-completeness is the natural completeness assumption for the fixed point argument developed here.

The purpose of the present paper is to establish a local fixed point principle for multivalued mappings acting on 0-complete strong partial b-metric spaces. The contraction condition is formulated by means of Bianchini–Grandolfi gauge functions and is expressed through a set-valued excess generated by the strong partial b-metric. This approach simultaneously combines localization, multivalued mappings, nonlinear contractions, and generalized metric structures within a single framework.

The main original mathematical contribution of the present paper is a local fixed point theorem for multivalued mappings in 0-complete strong partial b-metric spaces. Its contractive condition is imposed only on a prescribed localization ball and is formulated by means of a Bianchini–Grandolfi gauge function rather than a constant linear contraction factor. The set-valued condition is expressed through a one-sided excess functional, which is weaker than the corresponding symmetric Hausdorff-type contraction and is sufficient for constructing an admissible iterative selection. The argument relies on 0-completeness and 0-closedness and does not require the family of partial b-metric balls to form a topological basis. The theorem also yields local and global consequences, including linear set-valued contractions, the partial metric case, and a local Banach-type principle.

A closely related local multivalued fixed point framework was obtained by Slimani, Graef, and Ouahab [3] for complete strong b-metric spaces. Their local results employ a one-sided set excess on a closed ball together with linear contractive conditions. The present setting differs in several essential respects: the underlying space is a 0-complete strong partial b-metric space, positive self-distances are allowed, the values of the multifunction are treated through 0-closedness, and the linear contraction modulus is replaced by a Bianchini–Grandolfi gauge. The localization argument must therefore be adapted to the strong partial b-metric inequality and to the corresponding notion of 0-convergence.

The obtained theorem yields, as immediate consequences, local versions of several familiar contractive principles, including the linear contraction case. The proof is based on an explicit construction of an iterative sequence inside a prescribed ball together with chain estimates adapted to the asymmetric role of the coefficient K in the strong partial b-metric inequality. The argument explicitly accommodates positive self-distances and uses the corresponding notion of 0-completeness in order to obtain the required limiting behavior.

Although the present paper is purely theoretical, the proposed framework admits natural interpretations in learning and refinement processes. One motivating example comes from mathematical models of linguistic enrichment. In such models, a text is regarded as a state of an iterative editing process rather than as a static object. The positive self-distance of a formulation may then be interpreted as a quantitative measure of its remaining discrepancy from a prescribed stable target family, while a multivalued mapping associates with each text the family of all acceptable subsequent revisions. The multivalued character of the model allows several admissible revisions of the same formulation and may also permit more than one stable formulation, without requiring every possible revision path to converge.

To illustrate this interpretation, the paper develops an abstract finite enrichment model, while its complete linguistic realization is provided in Appendix B. Within this model, the self-distance is generated by the transformed shortest distance from the current state to the prescribed stable target family, thereby providing an intrinsic geometric interpretation of positive self-distances in strong partial b-metric spaces.

A recently proposed linguistic enrichment framework interprets language development not merely as the accumulation of new lexical and grammatical resources, but as a broader process involving correction, selection, replacement, adaptation, and stylistic refinement [4]. This interpretation is consistent with usage-based and complex dynamic approaches, according to which linguistic systems are continuously reorganized through interaction, experience, and changing communicative conditions [5,6,7,8].

Within this framework, linguistic smoothing refers to the correction of grammatical, lexical, terminological, and stylistic deficiencies, whereas enrichment includes the acquisition and functional differentiation of expressions appropriate to particular discourse contexts. Repeated smoothing and enrichment operations may lead to several stable linguistic outcomes rather than to a unique final text [4]. The present paper develops a complementary metric interpretation of this process. Linguistic states are organized into successive refinement levels, their mutual separation is measured by a revision metric, and their self-distances quantify the remaining effort required to reach the family of stable formulations.

2. Materials and Methods

Throughout the paper, ${\mathbb{R}}_{+}=[0,\infty )$, and all sets under consideration are assumed to be nonempty unless stated otherwise.

This section introduces the generalized metric framework and the auxiliary results required for the proof of the main theorem. Since the local fixed point principle established in this paper is formulated in the setting of strong partial b-metric spaces, we begin with the definition of this class and explain its relation to several classical metric-type structures. We then discuss a number of elementary properties that will play a fundamental role in the construction of the local iterative process and in the subsequent chain estimates.

2.1. Strong Partial B-Metric Spaces

Definition 1

([1]). Let X be a nonempty set and let $K\ge 1$. A mapping $p:X\times X⟶{\mathbb{R}}_{+}$ is called a strong partial b-metric if, for every $x,y,z\in X$, the following conditions are satisfied:

I 

$x=y$, provided that $p(x,x)=p(x,y)=p(y,y)$;

II 

$p(x,x)\le p(x,y)$;

III 

$p(x,y)=p(y,x)$;

IV 

$p(x,y)\le p(x,z)+Kp(z,y)-p(z,z)$.

The triple $(X,p,K)$ is called a strong partial b-metric space.

The parameter K measures the deviation from the classical triangle inequality, whereas positive self-distances distinguish strong partial b-metrics from ordinary metric-type structures. As will become clear in the subsequent sections, both features are explicitly accommodated by the local fixed point framework developed in this paper.

The definition of a strong partial b-metric is closely related to several well-known generalized metric structures. Some of them are obtained by imposing additional restrictions on the above definition, whereas partial b-metrics arise after replacing the strong triangle inequality by the corresponding weaker b-metric-type inequality. The parameter K measures the deviation from the classical triangle inequality, whereas positive self-distances distinguish strong partial b-metrics from ordinary metric-type structures. The relation with the principal classical structures can be summarized as follows.

1.

If $p(x,x)=0$ for any $x\in X$, then condition (IV) becomes $p(x,y)\le p(x,z)+Kp(z,y)$, and $(X,p,K)$ is a strong b-metric space.

2.

If, in addition, $K=1$, then $p(x,y)\le p(x,z)+p(z,y)$, so p is an ordinary metric.

3.

If $K=1$, then condition (IV) becomes the usual partial metric triangle inequality, and one recovers the class of partial metric spaces.

4.

Replacing (IV) by $p(x,y)\le K\left(p(x,z)+p(z,y)\right)-p(z,z)$ yields the partial b-metric introduced by Shukla [9].

5.

Finally, choosing $K=1$ in the previous inequality gives the classical partial metric introduced by Matthews [10].

Hence, strong partial b-metric spaces should be viewed as an intermediate class between strong b-metric spaces and partial b-metric spaces. In particular, imposing $p(x,x)=0$ yields a strong b-metric, while setting $K=1$ yields a partial metric. Imposing both conditions recovers the ordinary metric case. On the other hand, every strong partial b-metric satisfies the corresponding partial b-metric inequality, but a general partial b-metric need not satisfy the stronger inequality required of a strong partial b-metric. From this point onward, all results are formulated in the setting of strong partial b-metric spaces. Consequently, they immediately specialize to strong b-metric spaces, partial metric spaces, and ordinary metric spaces under the corresponding additional assumptions. They should not, however, be regarded as automatic results for arbitrary partial b-metric spaces.

Although condition (IV) appears asymmetric because the coefficient K is attached only to one of the two terms involving the intermediate point, this asymmetry is only apparent. Owing to the symmetry of the distance function, the coefficient may equivalently be placed before either summand. This simple observation will be useful later in the derivation of the chain estimates.

Proposition 1.

Let X be a nonempty set, let $K\ge 1$, and let $p:X\times X⟶{\mathbb{R}}_{+}$ be symmetric. Then the following two inequalities are equivalent:

$$p(x,y)\le p(x,z)+Kp(z,y)-p(z,z),$$

(1)

and

$$p(x,y)\le Kp(x,z)+p(z,y)-p(z,z),$$

(2)

for all $x,y,z\in X$.

Proof. 

Assume first that (1) holds. Let $x,y,z\in X$. Applying (1) to the ordered triple $(y,x,z)$, we obtain $p(y,x)\le p(y,z)+Kp(z,x)-p(z,z)$. By the symmetry of p, we have $p(y,x)=p(x,y)$, $p(y,z)=p(z,y)$, and $p(z,x)=p(x,z)$.

Therefore, $p(x,y)\le p(z,y)+Kp(x,z)-p(z,z)$, which is precisely $p(x,y)\le Kp(x,z)+p(z,y)-p(z,z)$.

Hence (2) holds.

Conversely, suppose that (2) holds. Applying it to the ordered triple $(y,x,z)$, we obtain

$$p(y,x)\le Kp(y,z)+p(z,x)-p(z,z).$$

Using again the symmetry of p, it follows that $p(x,y)\le Kp(z,y)+p(x,z)-p(z,z)$ or equivalently, $p(x,y)\le p(x,z)+Kp(z,y)-p(z,z)$.

Thus (1) holds, and the two inequalities are equivalent. □

Proposition 1 shows that the apparent asymmetry in the strong partial b-metric inequality is only formal. Although the coefficient K is placed before one of the two terms, the symmetry of p allows the inequality to be used in either orientation.

This observation may lead to two different estimates when the inequality is applied repeatedly along a finite chain $x_0,x_1,\dots ,x_n$. Depending on the chosen orientation, the powers of K may accumulate from the beginning or from the end of the chain. Consequently, both estimates are valid, and in applications one may use the sharper of the two.

Proposition 2.

Let X be a nonempty set, let $K\ge 1$, and let $p:X\times X⟶{\mathbb{R}}_{+}$ satisfy the symmetry condition $p(x,y)=p(y,x)$ for all $x,y\in X$. Then the strong partial b-metric inequality

$$p(x,y)\le p(x,z)+Kp(z,y)-p(z,z)$$

(3)

holds for all $x,y,z\in X$ if and only if

$$p(x,y)\le \min \left\{p(x,z)+Kp(z,y),Kp(x,z)+p(z,y)\right\}-p(z,z)$$

(4)

holds for all $x,y,z\in X$.

Moreover, if $p(x,x)=0$ for all $x\in X$, then the same conclusion holds for strong b-metrics, i.e, $p(x,y)\le p(x,z)+Kp(z,y)$ is equivalent to

$$p(x,y)\le \min \left\{p(x,z)+Kp(z,y),Kp(x,z)+p(z,y)\right\}.$$

Remark 1.

Consequently, condition (IV) in the definition of a strong partial b-metric may equivalently be replaced by (4).

Proof. 

Assume first that (3) holds. Applying this inequality to the ordered triple $(y,x,z)$, we obtain $p(y,x)\le p(y,z)+Kp(z,x)-p(z,z)$.

Using the symmetry of p, it follows that $p(x,y)\le Kp(x,z)+p(z,y)-p(z,z)$.

Therefore, $p(x,y)$ is bounded above by both

$$p(x,z)+Kp(z,y)-p(z,z)$$

and

$$Kp(x,z)+p(z,y)-p(z,z).$$

Hence $p(x,y)\le \min \left\{p(x,z)+Kp(z,y),Kp(x,z)+p(z,y)\right\}-p(z,z)$.

Conversely, suppose that (4) holds. Since

$$\min \left\{p(x,z)+Kp(z,y),Kp(x,z)+p(z,y)\right\}\le p(x,z)+Kp(z,y),$$

we obtain $p(x,y)\le p(x,z)+Kp(z,y)-p(z,z)$. Thus (3) holds.

Finally, if $p(x,x)=0$ for every $x\in X$, then in particular $p(z,z)=0$. Therefore, the strong partial b-metric inequalities reduce respectively to $p(x,y)\le p(x,z)+Kp(z,y)$ and

$$p(x,y)\le \min \left\{p(x,z)+Kp(z,y),Kp(x,z)+p(z,y)\right\}.$$

This proves the corresponding equivalence for strong b-metrics. □

Remark 2.

The minimum formulation is therefore valid both in strong partial b-metric spaces and in strong b-metric spaces. In the latter case, the self-distance correction term disappears, while the symmetry argument remains unchanged.

The previous definition is closely related to several well-established generalized metric structures. The following examples, taken from the literature, illustrate these spaces and emphasize the progressive transition from ordinary metrics to strong partial b-metrics. These examples will be revisited in the subsequent subsections in order to compare the corresponding balls, point-to-set distances, and excesses.

Example 1

(Classical examples). The following examples illustrate the generalized metric structures discussed above.

1. 

Metric space: Let $X=\mathbb{R}$ and define $d(x,y)=|x-y|$. Then $(\mathbb{R},d)$ is a metric space.

2. 

b-metric space: Let $X=\mathbb{R}$ and let $d(x,y)={|x-y|}^2$. Then $d(x,z)\le 2\left(d\right(x,y)+d(y,z\left)\right)$, so $(\mathbb{R},d,2)$ is a b-metric space in the sense of Czerwik [11]. It is not a metric space since the ordinary triangle inequality fails.

3. 

Strong b-metric space: Let $X=\mathbb{R}$, define $h\left(t\right)=t+{\displaystyle \frac{t^2}{1+t}}$ for $t\ge 0$, and put $d(x,y)=h\left(\right|x-y\left|\right)$. Since h is increasing and $h^{\prime }\left(t\right)=2-{\displaystyle \frac{1}{{(1+t)}^2}}\le 2$, the mean value theorem gives $h(a+b)-h\left(a\right)\le 2b\le 2h\left(b\right)$ for all $a,b\ge 0$. Consequently, $d(x,z)\le d(x,y)+2d(y,z)$, so $(\mathbb{R},d,2)$ is a strong b-metric space. It is not a metric space, since

$$d(0,2)=h\left(2\right)={\displaystyle \frac{10}{3}}>3=2h\left(1\right)=d(0,1)+d(1,2).$$

4. 

Partial metric space: Let $X={\mathbb{R}}_{+}$ and define $p(x,y)=\max \{x,y\}$. Then p is the classical partial metric introduced by Matthews [10]. Here $p(x,x)=x$. so every point possesses a positive self-distance.

5. 

Partial b-metric space: Let $X={\mathbb{R}}_{+}$ and define $p(x,y)={\left(\max \{x,y\}\right)}^2$. This is one of the standard examples of a partial b-metric given by Shukla [9]. The self-distances are positive, and the generalized triangle inequality holds with a coefficient $K>1$.

6. 

Strong partial b-metric space: Let $X={\mathbb{R}}_{+}$, and define

$$p(x,y)=\left\{\begin{array}{cc}{x}^2,\hfill & x=y,\hfill \\ \max \{x^2,y^2\},\hfill & x\ne y.\hfill \end{array}\right.$$

This is a standard example of a strong partial b-metric introduced by Moshokoa and Ncongwane [1]. It illustrates simultaneously the presence of positive self-distances and the strong form of the generalized triangle inequality.

2.2. Interpretation of the Generalized Distance

Although the definition of a strong partial b-metric is purely axiomatic, each of its components admits a natural interpretation. Taken together, these axioms describe two independent phenomena. The first concerns the intrinsic state of the objects themselves, whereas the second concerns the process relating different objects.

Condition (II), $p(x,x)\le p(x,y)$, states that every object is at least as close to itself as to any other object. In an ordinary metric space this simply reduces to the equality $p(x,x)=0$. Allowing positive self-distances enriches this picture by assigning to every object an additional numerical characteristic. Thus, the quantity $p(x,x)$ should not be interpreted as the distance from an object to itself in the usual geometric sense, but rather as a measure of its intrinsic incompleteness or the amount of information still missing before reaching a stable state.

The symmetry condition (III) $p(x,y)=p(y,x)$ expresses that the mutual discrepancy between two objects is independent of the direction in which it is measured. Owing to this symmetry, the generalized triangle inequality may be written in either of its equivalent forms established in Proposition 1.

The generalized triangle inequality

$$p(x,y)\le p(x,z)+Kp(z,y)-p(z,z)$$

(5)

contains two additional mechanisms that distinguish strong partial b-metrics from ordinary metrics.

The correction term $“-p(z,z)”$ in (5) prevents the intrinsic incompleteness of the intermediate object from being counted twice. Indeed, both quantities $p(x,z)$ and $p(z,y)$ already contain the contribution of $p(z,z)$. Subtracting this value therefore compensates for this duplication and preserves the intended interpretation of the distance.

The coefficient $K\ge 1$ describes a possible amplification of the transition through the intermediate object. When $K=1$, the generalized triangle inequality reduces to its partial metric counterpart. Larger values of K reflect situations in which indirect transitions may accumulate additional distortion or uncertainty.

These two mechanisms are independent. Positive self-distances describe properties of the objects themselves, whereas the coefficient K describes properties of the transition between objects. Consequently, different generalized metric structures arise naturally by suppressing one or both of these effects.

Indeed, imposing $p(x,x)=0$ eliminates intrinsic incompleteness and yields the class of strong b-metric spaces. Setting $K=1$ eliminates the amplification effect and gives partial metric spaces. Removing both features simultaneously recovers the classical metric. Replacing the strong triangle inequality by

$$p(x,y)\le K\left(p(x,z)+p(z,y)\right)-p(z,z)$$

produces partial b-metric spaces, while choosing $K=1$ in this relation yields the classical partial metric of Matthews.

The interpretation presented above is intentionally independent of any particular application. Nevertheless, it provides a useful conceptual framework for understanding the examples developed later in the paper. In the linguistic model introduced in Section 3.1, positive self-distances are interpreted as transformed distances from the current linguistic state to the prescribed stable target family, whereas the multivalued mapping represents the family of all acceptable subsequent revisions. Thus, the abstract axioms of a strong partial b-metric acquire a natural interpretation within an iterative enrichment process.

2.3. Balls and Induced Topology

Since the main result of this paper is a local fixed point theorem, the notion of a ball generated by a strong partial b-metric plays a central role. In the broader setting of partial b-metric spaces, families of open balls may exhibit topological behavior different from that of ordinary metric balls. We therefore recall the relevant definitions while emphasizing that, in the present paper, closed balls are used only as localization sets.

Definition 2.

Let $(X,p,K)$ be a strong partial b-metric space, let $x\in X$, and let $r>0$.

The open ball centered at x with radius r is defined by

$$B_p(x,r)=\{y\in X:p(x,y)<p(x,x)+r\}.$$

The closed ball centered at x with radius r is defined by

$${\overline{B}}_p(x,r)=\{y\in X:p(x,y)\le p(x,x)+r\}.$$

The presence of the self-distance $p(x,x)$ distinguishes these balls from the ordinary metric balls. When $p(x,x)=0$ for every $x\in X$, the above definitions reduce to the usual balls in a strong b-metric space and, in particular, to the classical metric balls when $K=1$.

Remark 3.

The family $\mathcal{B}=\{B_p(x,r):x\in X,r>0\}$ does not, in general, form a basis for the topology generated by a partial b-metric. Ge and Lin showed that the corresponding claim in the literature is false by constructing an explicit counterexample. Nevertheless, the family of open balls always forms a subbasis for a natural topology generated by the partial b-metric.

Every strong partial b-metric satisfies the corresponding partial b-metric inequality, since $K\ge 1$. However, a counterexample established for the larger class of partial b-metric spaces does not, by itself, imply the same topological failure for the narrower class of strong partial b-metric spaces. No such transfer is required in the present paper. Throughout the proof, the closed ball is used only to localize the iterative process, and no assumption that the family of p-balls forms a topological basis is invoked.

The geometry induced by the six metric-type structures introduced above can be compared by computing their open and closed balls explicitly. To avoid interrupting the development of the main results with several similar calculations, the resulting formulas are included in the summary presented in

Table 1. Comparison of open balls, point-to-set distances, and one-sided excesses in the six metric-type structures.

Metric-Type Structure	Open Ball	$\mathit{p}(1,\mathit{A})$	${\mathit{\delta }}_{\mathit{p}}({\mathit{A}}_0,{\mathit{B}}_0)$
Metric	$(x-r,x+r)$	1	1
b-metric	$(x-\sqrt{r},x+\sqrt{r})$	1	1
Strong b-metric	$(x-\eta (r),x+\eta (r\left)\right)$	${\displaystyle \frac{3}{2}}$	${\displaystyle \frac{3}{2}}$
Partial metric	$[0,x+r)$	2	2
Partial b-metric	$[0,\sqrt{x^2+r})$	4	4
Strong partial b-metric	$[0,\sqrt{x^2+r})$	4	4

, while the complete computations and a schematic representation of the corresponding balls are given in Appendix A.

The comparison shows that, in the metric, b-metric, and strong b-metric cases, the balls occurring in the selected examples are intervals symmetric about their centres. In the partial metric-type cases, the positive self-distance enters the defining inequality and the centre need not be the geometric midpoint of the ball. This illustrates one of the principal geometric differences between ordinary metric spaces and partial metric-type spaces.

2.4. Point-to-Set Distance and Excess

The contraction condition in the main theorem is formulated for multivalued mappings. For this reason, besides the distance between two points, we also need the distance from a point to a set and the corresponding excess between two sets. These notions play the role of the point-to-set distance and the one-sided Hausdorff excess adapted to strong partial b-metric spaces.

Definition 3.

Let $(X,p,K)$ be a strong partial b-metric space, let $x\in X$, and let $⌀\ne A\subseteq X$. The distance from x to A is defined by $p(x,A)={\inf }_{a\in A}p(x,a)$.

Since $A\ne ⌀$ and $p(x,a)\ge 0$ for every $a\in A$, the quantity $p(x,A)$ is well defined and belongs to ${\mathbb{R}}_{+}$.

For comparison with the different metric-type structures introduced above, consider the point $x=1$ and the set $A=[2,\infty )$. The corresponding values of $p(1,A)$ are summarized in Table 1. The detailed computations are provided in Appendix A.2.

The following definition is motivated by the classical one-sided Pompeiu–Hausdorff excess used in set-valued analysis; see, for example, Aubin and Frankowska [12]. Here, the ordinary metric is replaced by the strong partial b-metric p. A closely related one-sided excess in the setting of strong b-metric spaces is employed in [3].

Definition 4.

Let $(X,p,K)$ be a strong partial b-metric space and let $⌀\ne A,B\subseteq X$. The excess of A over B is defined by ${\delta }_p(A,B)={\sup }_{a\in A}p(a,B)$, where $p(a,B)={\inf }_{b\in B}p(a,b)$.

Equivalently, ${\delta }_p(A,B)={\sup }_{a\in A}{\inf }_{b\in B}p(a,b)$.

The value $+\infty$ is allowed whenever the supremum is unbounded.

A symmetric Hausdorff-type functional may be defined by

$$H_p(A,B)=\max \{{\delta }_p(A,B),{\delta }_p(B,A)\}.$$

An analogue of Theorem 1 follows immediately if its one-sided contractive assumption is replaced by the stronger condition $H_p(F\left(x\right),F\left(y\right))\le \phi \left(p(x,y)\right)$, because this estimate implies the one-sided bound required in the construction of the iterative selection.

The formulation in terms of ${\delta }_p$ is genuinely more general. The proof only requires that each selected element of the first image can be approximated by an element of the second image. It does not require the reverse approximation simultaneously. This directional property is particularly natural for multistage processes, in which admissible transitions are oriented from a current state toward its subsequent refinements.

The excess ${\delta }_p(A,B)$ measures how far the set A extends beyond the set B. In particular, since $p(a,B)\ge 0$ for every $a\in A$, one has ${\delta }_p(A,B)=0$ if and only if $p(a,B)=0$ for every $a\in A$.

Moreover, ${\delta }_p(A,B)$ is not symmetric, so, in general, ${\delta }_p(A,B)\ne {\delta }_p(B,A)$. This asymmetry is natural for multivalued contractions, where one compares the image of one point with the image of another.

For nonempty sets $A,B,C\subseteq X$, the excess satisfies the generalized one-sided triangle estimate ${\delta }_p(A,C)\le {\delta }_p(A,B)+K{\delta }_p(B,C)$, whenever the quantities involved are finite.

Indeed, for every $a\in A$ and $b\in B$, taking the infimum over $c\in C$ gives us the inequality $p(a,C)\le p(a,b)+Kp(b,C)$. Since $p(b,C)\le {\delta }_p(B,C)$, taking the infimum over $b\in B$ yields $p(a,C)\le p(a,B)+K{\delta }_p(B,C)$.

Finally, taking the supremum over $a\in A$ gives ${\delta }_p(A,C)\le {\delta }_p(A,B)+K{\delta }_p(B,C)$.

The relation with 0-closedness differs from the corresponding relation in an ordinary metric space. If B is 0-closed and ${\delta }_p(A,B)=0$, then $p(a,B)=0$ for every $a\in A$, and hence $A\subseteq B$. Moreover, the small self-distance axiom gives $p(a,a)\le p(a,B)=0$, so $p(a,a)=0$ for every $a\in A$. The converse is generally false: $A\subseteq B$ does not necessarily imply ${\delta }_p(A,B)=0$, because, for $a\in B$, one has $p(a,B)=p(a,a)$, which may be positive.

Throughout the paper, all excesses appearing in the contractive conditions are finite. This follows automatically from the hypotheses of the main theorem.

Example 2.

We summarize the preceding metric-type examples in a single table. For the point-to-set distance, take $x=1$ and $A=[2,\infty )$. For the one-sided excess, take $A_0=[1,2]$ and $B_0=[2,3]$. In the strong b-metric example,

$$h\left(t\right)=t+{\displaystyle \frac{t^2}{1+t}}\mathit{and}\eta \left(r\right)=h^{-1}\left(r\right)={\displaystyle \frac{r-1+\sqrt{r^2+6r+1}}{4}}.$$

The resulting open balls, point-to-set distances, and excesses are listed in Table 1.

In the first three rows, the selected open balls are symmetric about their centers. In contrast, the partial metric-type examples produce intervals whose left endpoint is fixed at zero, and the center x is generally not their geometric midpoint. The values in the last two columns also show explicitly how positive self-distances modify point-to-set distances and one-sided excesses.

The corresponding closed balls are obtained by replacing each open right endpoint by a closed endpoint and, in the first three rows, by closing both endpoints. Complete derivations of all entries in Table 1, together with the corresponding schematic comparison, are provided in Appendix A.

2.5. 0-Completeness

The convergence concept naturally associated with strong partial b-metric spaces differs from the classical notion of Cauchy completeness. The contractive iteration constructed in the proof of the main theorem produces a sequence whose mutual distances converge to zero. Therefore, the appropriate assumption for the present result is 0-completeness rather than full Cauchy completeness.

We first recall the notions of a 0-closed set, a 0-Cauchy sequence, and a 0-complete space.

Definition 5

([1]). A nonempty subset A of a strong partial b-metric space $(X,p,K)$ is called 0-closed with respect to p if $p(x,A)=0$ implies $x\in A$ for every $x\in X$.

We denote by ${\mathcal{C}}_0^p\left(X\right)$ the family of all nonempty subsets of X that are 0-closed with respect to p.

The term 0-closed is used here in the zero-distance sense. It does not refer to closedness with respect to a topology generated by p-balls.

Definition 6

([1]). A sequence $\left\{x_n\right\}$ in a strong partial b-metric space $(X,p,K)$ is called 0-Cauchy if ${\lim }_{m,n\to \infty }p(x_m,x_n)=0$.

Definition 7

([1]). A strong partial b-metric space $(X,p,K)$ is called 0-complete if every 0-Cauchy sequence $\left\{x_n\right\}$ admits a point $x\in X$ such that $p(x,x)={\lim }_{n\to \infty }p(x_n,x)={\lim }_{m,n\to \infty }p(x_m,x_n)=0$.

In this case, we say that $\left\{x_n\right\}$ 0-converges to x and write $x_n\stackrel{0}{\to }x$.

For comparison, recall that a sequence $\left\{x_n\right\}$ in $(X,p,K)$ is called Cauchy if the finite double limit ${\lim }_{m,n\to \infty }p(x_m,x_n)$ exists. The space $(X,p,K)$ is called Cauchy complete if every Cauchy sequence $\left\{x_n\right\}$ admits a point $x\in X$ satisfying $p(x,x)={\lim }_{n\to \infty }p(x_n,x)={\lim }_{m,n\to \infty }p(x_m,x_n)$.

Proposition 3.

Every Cauchy complete strong partial b-metric space is 0-complete.

Proof. 

Let $(X,p,K)$ be a Cauchy complete strong partial b-metric space, and let $\left\{x_n\right\}$ be an arbitrary 0-Cauchy sequence in X. By Definition 6, ${\lim }_{m,n\to \infty }p(x_m,x_n)=0$. In particular, this double limit exists and is finite, so $\left\{x_n\right\}$ is a Cauchy sequence.

Since $(X,p,K)$ is Cauchy complete, there exists $x\in X$ such that $p(x,x)={\lim }_{n\to \infty }p(x_n,x)={\lim }_{m,n\to \infty }p(x_m,x_n)$. The last limit is zero. Consequently, $p(x,x)=0$ and ${\lim }_{n\to \infty }p(x_n,x)=0$. Hence, $p(x,x)={\lim }_{n\to \infty }p(x_n,x)={\lim }_{m,n\to \infty }p(x_m,x_n)=0$, which means that $x_n\stackrel{0}{\to }x$.

Since the 0-Cauchy sequence $\left\{x_n\right\}$ was arbitrary, $(X,p,K)$ is 0-complete. □

The use of 0-completeness is intrinsic to the fixed point argument developed below. Indeed, the iterative sequence constructed in the proof of Theorem 1 satisfies ${\lim }_{m,n\to \infty }p(x_m,x_n)=0$ and is therefore 0-Cauchy. The proof requires only the existence of a point $x^{*}\in X$ such that $p(x^{*},x^{*})={\lim }_{n\to \infty }p(x_n,x^{*})={\lim }_{m,n\to \infty }p(x_m,x_n)=0$. Consequently, the assumption of Cauchy completeness would be more than is required for the present result.

Proposition 3 follows directly from the definitions and does not depend on the value of the coefficient $K\ge 1$. Although the strong partial b-metric triangle inequality differs from the corresponding partial metric inequality when $K>1$, this difference plays no role in the proof of the implication.

2.6. Bianchini–Grandolfi Gauge Functions

The following notion goes back to the work of Bianchini and Grandolfi and has become standard in nonlinear fixed point theory.

In the original formulation, the gauge function is assumed to be nondecreasing. In the present paper, we adopt the stronger requirement that it be strictly increasing, as this property is used in the strict iterative estimates in the proof of the main theorem.

Definition 8

([13]). Let $J\subseteq {\mathbb{R}}_{+}$ be an interval containing 0. A strictly increasing mapping $\phi :J⟶J$ will be called a Bianchini–Grandolfi gauge function if it is continuous at 0 and

$$S_{\phi }\left(t\right)={\displaystyle \sum _{k=0}^{\infty }}{\phi }^k\left(t\right)<\infty ,t\in J,$$

where ${\phi }^0\left(t\right)=t$ and ${\phi }^{k+1}\left(t\right)=\phi \left({\phi }^k\left(t\right)\right)$.

The summability condition in Definition 8 implies $\phi \left(0\right)=0$ and $\phi \left(t\right)<t$ for every $t\in J\setminus \left\{0\right\}$.

Indeed, otherwise the terms of the corresponding convergent series would not tend to zero.

3. Auxiliary Estimates

Lemma 1.

Let $(X,p,K)$ be a strong partial b-metric space and let $⌀\ne A\subseteq X$. The following conditions are equivalent:

(i) 

A is 0-closed with respect to p;

(ii) 

whenever $\left\{a_n\right\}\subseteq A$ and $x\in X$ satisfy $p(a_n,x)⟶0$, then $x\in A$.

Moreover, for every $x\in X$, there holds $p(x,A)=0$ if and only if there exists a sequence $\left\{a_n\right\}\subseteq A$ such that $p(x,a_n)⟶0$.

Whenever these equivalent conditions hold for a given x, one necessarily has $p(x,x)=0$.

Proof. 

Assume first that A is 0-closed and let $\left\{a_n\right\}\subseteq A$ satisfy $p(a_n,x)\to 0$ for some $x\in X$. By symmetry and the definition of the distance from a point to a set,

$$0\le p(x,A)\le p(x,a_n)=p(a_n,x).$$

Passing to the limit gives $p(x,A)=0$. The 0-closedness of A therefore implies $x\in A$.

Conversely, suppose that (ii) holds and let $x\in X$ satisfy $p(x,A)=0$. By the definition of the infimum, for every $n\in \mathbb{N}$ there exists $a_n\in A$ such that $p(x,a_n)<{\displaystyle \frac{1}{n}}$.

By symmetry, $p(a_n,x)=p(x,a_n)\to 0$. Condition (ii) yields $x\in A$, and hence A is 0-closed.

The same argument proves the asserted sequential characterization of $p(x,A)=0$. Finally, by (Definition 1(II)), $0\le p(x,x)\le p(x,a_n)$. From ${\lim }_{n\to \infty }p(x,a_n)=0$ we conclude that $p(x,x)=0$. □

Lemma 2.

Let $x_0,x_1,\dots ,x_n\in X$, where $n\ge 2$. Then

$$p(x_0,x_n)\le p(x_0,x_1)+K{\displaystyle \sum _{j=1}^{n-1}}p(x_j,x_{j+1})-{\displaystyle \sum _{j=1}^{n-1}}p(x_j,x_j)\le p(x_0,x_1)+K{\displaystyle \sum _{j=1}^{n-1}}p(x_j,x_{j+1}).$$

Proof. 

We will prove the inequality by induction by repeatedly applying Definition 1(IV) to the first term of the preceding estimate.

Step 1: Applying Definition 1(IV) with $x=x_0$, $y=x_n$, and $z=x_{n-1}$, we obtain

$$\begin{array}{cc}\hfill p(x_0,x_n)& \le p(x_0,x_{n-1})+Kp(x_{n-1},x_n)-p(x_{n-1},x_{n-1}).\hfill \end{array}$$

Step 2: Now apply Definition 1(IV) only to the term $p(x_0,x_{n-1})$, using $z=x_{n-2}$:

$$\begin{array}{cc}\hfill p(x_0,x_{n-1})& \le p(x_0,x_{n-2})+Kp(x_{n-2},x_{n-1})-p(x_{n-2},x_{n-2}).\hfill \end{array}$$

Substituting this estimate into the previous inequality gives

$$p(x_0,x_n)\le p(x_0,x_{n-2})+Kp(x_{n-2},x_{n-1})+Kp(x_{n-1},x_n)-p(x_{n-2},x_{n-2})-p(x_{n-1},x_{n-1}).$$

Step k: Assume that after $k-1$ substitutions we have reached

$$p(x_0,x_n)\le p(x_0,x_{n-k+1})+K{\displaystyle \sum _{j=n-k+1}^{n-1}}p(x_j,x_{j+1})-{\displaystyle \sum _{j=n-k+1}^{n-1}}p(x_j,x_j).$$

Applying Definition 1(IV) only to the remaining term $p(x_0,x_{n-k+1})$, with intermediate point $x_{n-k}$, yields

$$p(x_0,x_{n-k+1})\le p(x_0,x_{n-k})+Kp(x_{n-k},x_{n-k+1})-p(x_{n-k},x_{n-k}).$$

Substituting this estimate gives the next step of the iteration.

After $n-1$ iterations we arrive at

$$p(x_0,x_n)\le p(x_0,x_1)+K{\displaystyle \sum _{j=1}^{n-1}}p(x_j,x_{j+1})-{\displaystyle \sum _{j=1}^{n-1}}p(x_j,x_j),$$

which proves the first inequality.

Finally, since $p(x_j,x_j)\ge 0$ for every j, $-{\sum }_{j=1}^{n-1}p(x_j,x_j)\le 0$, and therefore

$$p(x_0,x_n)\le p(x_0,x_1)+K{\displaystyle \sum _{j=1}^{n-1}}p(x_j,x_{j+1}).$$

□

The subtracted self-distance terms in the chain estimate can only improve the resulting upper bound. After omitting them, one obtains the uniform estimate $p(x_m,x_n)\le K{\sum }_{j=m}^{\infty }{\phi }^j\left(\alpha \right)$ for $n>m$, which is sufficient to prove that the iterative sequence is 0-Cauchy. If quantitative lower information about the self-distances $p(x_j,x_j)$ is available, retaining the negative terms may give a sharper convergence estimate. Without such additional information, however, no uniform improvement over the displayed tail bound can be asserted. In the strong b-metric case, all self-distances vanish, and the estimate reduces to the familiar one without correction terms.

Remark 4.

Definitions 5 and 7, as well as Lemmas 1 and 2, are purely metric. They do not require the family of open p-balls to form a base for a topology. In particular, ${\overline{B}}_p(x_0,r)$ is used only as a localization set, and no topological closure argument is used below.

3.1. A Preliminary Linguistic Interpretations

Before presenting the complete finite model, we briefly illustrate the difference between a strong b-metric and a strong partial b-metric in linguistic terms.

Let us denote $X=\{u_0,u_1,u_2,u_3\}$ be a finite family of texts arranged according to increasing linguistic refinement, and define $q(u_i,u_j)={|i-j|}^2$.

The value $q(u_i,u_j)$ measures the structural separation between two refinement levels. Since X is finite, one verifies that $q(u_i,u_j)\le q(u_i,u_k)+5q(u_k,u_j)$, so $(X,q,5)$ is a strong b-metric space.

In this model, only the discrepancy between different texts is measured. To account also for the remaining incompleteness of each individual text, let $w\left(u_0\right)={\displaystyle \frac{3}{2}}$, $w\left(u_1\right)=1$, $w\left(u_2\right)={\displaystyle \frac{1}{2}}$, $w\left(u_3\right)=0$, and put $p(u_i,u_j)=q(u_i,u_j)+w\left(u_i\right)+w\left(u_j\right)$.

Then $p(u_i,u_i)=2w\left(u_i\right)$, so the self-distance represents the amount of linguistic refinement that remains necessary. A direct verification shows that $(X,p,5)$ is a strong partial b-metric space. Indeed, the symmetry of p is immediate. Moreover, the selected values of w satisfy $w\left(u_i\right)-w\left(u_j\right)\le q(u_i,u_j)$, and hence

$$p(u_i,u_i)=2w\left(u_i\right)\le q(u_i,u_j)+w\left(u_i\right)+w\left(u_j\right)=p(u_i,u_j).$$

The separation axiom follows from the fact that $q(u_i,u_j)=0$ only when $i=j$, while the strong partial b-metric inequality follows from the corresponding strong b-metric inequality for q and the nonnegativity of w.

Thus, the self-distance $p(u_i,u_i)=2w\left(u_i\right)$ represents the amount of linguistic refinement that remains necessary. These self-distances decrease along the refinement levels and vanish at the stable state $u_3$. The values decrease along the refinement levels and vanish at the stable state $u_3$.

This elementary interpretation motivates the more systematic target-family construction developed in Section 4.2. The complete twenty-state linguistic realization is provided in Appendix B.

Example 3

(A linguistic interpretation of the excess functional). Let X denote a collection of English texts equipped with a strong partial b-metric p. Assume that the value $p(x,y)$ measures the overall linguistic dissimilarity between two texts, taking into account their grammatical correctness, lexical richness, structural complexity, and the remaining linguistic incompleteness represented by the self-distance $p(x,x)$.

Let $A=\{a_1,a_2,a_3\}$ be a collection of texts produced by a learner,

$$\begin{array}{cc}\hfill a_1& =“\mathit{The}\mathit{student}\mathit{solve}\mathit{the}\mathit{problem}.”,\hfill \\ \hfill a_2& =“\mathit{The}\mathit{student}\mathit{solved}\mathit{problem}\mathit{by}\mathit{induction}.”,\hfill \\ \hfill a_3& =“\mathit{Student}\mathit{use}\mathit{theorem}\mathit{because}\mathit{function}\mathit{continuous}.”,\hfill \end{array}$$

and let $B=\{b_1,b_2,b_3\}$ be a family of linguistically acceptable target texts,

$$\begin{array}{cc}\hfill b_1& =“\mathit{The}\mathit{student}\mathit{solves}\mathit{the}\mathit{problem}.”,\hfill \\ \hfill b_2& =“\mathit{The}\mathit{student}\mathit{solved}\mathit{the}\mathit{problem}\mathit{by}\mathit{induction}.”,\hfill \\ \hfill b_3& =“\mathit{The}\mathit{student}\mathit{uses}\mathit{the}\mathit{theorem}\mathit{because}\mathit{the}\mathit{function}\mathit{is}\mathit{continuous}.”.\hfill \end{array}$$

Suppose that the linguistic distances between the learner texts and the target texts are given by

$$\begin{array}{cccc}p& b_1& b_2& b_3\\ a_1& 1& 5& 8\\ a_2& 4& 2& 7\\ a_3& 8& 6& 3\end{array}$$

For the three learner texts, the corresponding point-to-set distances are

$$p(a_1,B)=1,p(a_2,B)=2,p(a_3,B)=3.$$

Therefore,

$${\delta }_p(A,B)=\underset{a\in A}{\sup }p(a,B)=\max \{1,2,3\}=3.$$

Thus, the excess records the largest distance of a learner text from the family of acceptable target texts. In this example, the largest remaining discrepancy is attained by $a_3$.

4. Main Result

4.1. A Local Set-Valued Fixed Point Theorem

Theorem 1.

Let $(X,p,K)$ be a 0-complete strong partial b-metric space. Let $x_0\in X$, $r>0$, and let $F:{\overline{B}}_p(x_0,r)⟶{\mathcal{C}}_0^p\left(X\right)$ be a set-valued mapping. Let $J\subseteq {\mathbb{R}}_{+}$ be an interval containing 0, and let $\phi :J⟶J$ be a Bianchini–Grandolfi gauge function.

Assume that there exists $\alpha \in J\setminus \left\{0\right\}$ such that

$$p(x_0,F\left(x_0\right))<\alpha$$

(6)

and

$$K{S}_{\phi }\left(\alpha \right)-(K-1)\alpha \le p(x_0,x_0)+r.$$

(7)

Suppose further that

$${\delta }_p\left(F\left(x\right)\cap {\overline{B}}_p(x_0,r),F\left(y\right)\right)\le \phi \left(p(x,y)\right)$$

(8)

for all distinct $x,y\in {\overline{B}}_p(x_0,r)$ such that $p(x,y)\in J$ and $F\left(x\right)\cap {\overline{B}}_p(x_0,r)\ne ⌀$.

Then F has a fixed point $x^{*}\in {\overline{B}}_p(x_0,r)$, i.e., $x^{*}\in F\left(x^{*}\right)$.

If F is single-valued and

$$Kp(x_0,x_0)+(K+1)r\in J,$$

(9)

then $x^{*}$ is the unique fixed point of F in ${\overline{B}}_p(x_0,r)$.

Proof. 

Since $S_{\phi }\left(\alpha \right)=\alpha +{\sum }_{j=1}^{\infty }{\phi }^j\left(\alpha \right)$, condition (7) is equivalent to

$$\alpha +K{\displaystyle \sum _{j=1}^{\infty }}{\phi }^j\left(\alpha \right)\le p(x_0,x_0)+r.$$

(10)

By (6) and the definition of the infimum, there exists $x_1\in F\left(x_0\right)$ such that

$$p(x_0,x_1)<\alpha .$$

(11)

Condition (10) gives $p(x_0,x_1)<p(x_0,x_0)+r$, so $x_1\in {\overline{B}}_p(x_0,r)$. If $x_1=x_0$, then $x_0\in F\left(x_0\right)$, and the existence assertion follows. Assume therefore that $x_1\ne x_0$.

Since $x_1\in F\left(x_0\right)\cap {\overline{B}}_p(x_0,r)$, condition (8) yields

$$p(x_1,F\left(x_1\right))\le {\delta }_p\left(F\left(x_0\right)\cap {\overline{B}}_p(x_0,r),F\left(x_1\right)\right)\le \phi \left(p(x_0,x_1)\right)<\phi \left(\alpha \right).$$

Hence, there exists $x_2\in F\left(x_1\right)$ such that $p(x_1,x_2)<\phi \left(\alpha \right)$.

We construct the sequence and verify its localization simultaneously. Suppose that $x_0,\dots ,x_n$ have been chosen so that

$$x_{j+1}\in F\left(x_j\right),p(x_j,x_{j+1})<{\phi }^j\left(\alpha \right),0\le j\le n-1,$$

and

$$x_j\in {\overline{B}}_p(x_0,r),0\le j\le n.$$

Case (I) If $x_n=x_{n-1}$, then $x_n\in F\left(x_{n-1}\right)=F\left(x_n\right)$, and a fixed point has been obtained.

Case (II) Assume that $x_n\ne x_{n-1}$. Since $0\le p(x_{n-1},x_n)<{\phi }^{n-1}\left(\alpha \right)\le \alpha$ and J is an interval containing 0 and $\alpha$, we have $p(x_{n-1},x_n)\in J$.

Moreover, $x_n\in F\left(x_{n-1}\right)\cap {\overline{B}}_p(x_0,r)$ and thus,

$$p(x_n,F\left(x_n\right))\le {\delta }_p\left(F\left(x_{n-1}\right)\cap {\overline{B}}_p(x_0,r),F\left(x_n\right)\right)\le \phi \left(p(x_{n-1},x_n)\right)<{\phi }^n\left(\alpha \right).$$

We may therefore choose $x_{n+1}\in F\left(x_n\right)$ such that

$$p(x_n,x_{n+1})<{\phi }^n\left(\alpha \right).$$

(12)

By Lemma 2,

$$\begin{array}{cc}\hfill p(x_0,x_{n+1})& \le p(x_0,x_1)+K{\displaystyle \sum _{j=1}^n}p(x_j,x_{j+1})-{\displaystyle \sum _{j=1}^n}p(x_j,x_j)\hfill \\ & \le p(x_0,x_1)+K{\displaystyle \sum _{j=1}^n}p(x_j,x_{j+1})<\alpha +K{\displaystyle \sum _{j=1}^n}{\phi }^j\left(\alpha \right)\hfill \\ & \le p(x_0,x_0)+r.\hfill \end{array}$$

Consequently, $x_{n+1}\in {\overline{B}}_p(x_0,r)$.

Thus, either a fixed point is obtained after finitely many steps, or there exists a sequence $\left\{x_n\right\}$ such that

$$x_{n+1}\in F\left(x_n\right),p(x_n,x_{n+1})<{\phi }^n\left(\alpha \right),n\ge 0,$$

(13)

and $x_n\in {\overline{B}}_p(x_0,r)$ for every $n\ge 0$.

Let $n>m$. By Lemma 2,

$$\begin{array}{cc}\hfill p(x_m,x_n)& \le p(x_m,x_{m+1})+K{\displaystyle \sum _{j=m+1}^{n-1}}p(x_j,x_{j+1})-{\displaystyle \sum _{j=m+1}^{n-1}}p(x_j,x_j)\hfill \\ & \le p(x_m,x_{m+1})+K{\displaystyle \sum _{j=m+1}^{n-1}}p(x_j,x_{j+1})<{\phi }^m\left(\alpha \right)+K{\displaystyle \sum _{j=m+1}^{n-1}}{\phi }^j\left(\alpha \right)\hfill \\ & \le K{\displaystyle \sum _{j=m}^{\infty }}{\phi }^j\left(\alpha \right).\hfill \end{array}$$

Because of $S_{\phi }\left(\alpha \right)<\infty$, we have ${\lim }_{m\to \infty }{\sum }_{j=m}^{\infty }{\phi }^j\left(\alpha \right)=0$ and thus

$$\underset{m,n\to \infty }{\lim }p(x_m,x_n)=0.$$

Hence, $\left\{x_n\right\}$ is 0-Cauchy.

By the 0-completeness of $(X,p,K)$, there exists $x^{*}\in X$ such that

$$p(x^{*},x^{*})=0\mathrm{and}p(x_n,x^{*})⟶0.$$

(14)

By (IV), we can write $p(x_0,x^{*})\le p(x_0,x_n)+Kp(x_n,x^{*})-p(x_n,x_n)$. Since $p(x_n,x_n)\ge 0$, passing to the limit gives $p(x_0,x^{*})\le p(x_0,x_0)+r$ and therefore, $x^{*}\in {\overline{B}}_p(x_0,r)$.

It remains to prove that $x^{*}\in F\left(x^{*}\right)$.

Case (I) If $x_{n-1}=x^{*}$ for all sufficiently large n, then also $x_n=x^{*}$ for all sufficiently large n. Since $x_n\in F\left(x_{n-1}\right)$, we obtain $x^{*}\in F\left(x^{*}\right)$.

Case (II) Otherwise, there exists a subsequence, still denoted by $\left\{x_{n-1}\right\}$, such that $x_{n-1}\ne x^{*}$.

Since $p(x_{n-1},x^{*})⟶0$ and J is an interval containing 0 and $\alpha >0$, we have $p(x_{n-1},x^{*})\in J$ for all sufficiently large n.

Moreover, $x_n\in F\left(x_{n-1}\right)\cap {\overline{B}}_p(x_0,r)$.

Condition (8) therefore gives

$$p(x_n,F\left(x^{*}\right))\le {\delta }_p\left(F\left(x_{n-1}\right)\cap {\overline{B}}_p(x_0,r),F\left(x^{*}\right)\right)\le \phi \left(p(x_{n-1},x^{*})\right).$$

For every $u\in F\left(x^{*}\right)$, condition (IV) implies

$$p(x^{*},u)\le p(x^{*},x_n)+Kp(x_n,u)-p(x_n,x_n).$$

Taking the infimum over $u\in F\left(x^{*}\right)$, and then omitting the nonpositive term “$-p(x_n,x_n)$”, we obtain

$$p(x^{*},F\left(x^{*}\right))\le p(x^{*},x_n)+Kp(x_n,F\left(x^{*}\right)).$$

Consequently,

$$p(x^{*},F\left(x^{*}\right))\le p(x^{*},x_n)+K\phi \left(p(x_{n-1},x^{*})\right).$$

Letting $n\to \infty$ in the above inequality, using (14), the continuity of $\phi$ at 0, and $\phi \left(0\right)=0$, we obtain $p(x^{*},F\left(x^{*}\right))=0$.

Since $F\left(x^{*}\right)$ is 0-closed with respect to p, Definition 5 yields $x^{*}\in F\left(x^{*}\right)$. This proves the existence assertion.

Finally, suppose that F is single-valued and let $u,v\in {\overline{B}}_p(x_0,r)$ be fixed points of F. By (IV),

$$\begin{array}{cc}\hfill p(u,v)& \le p(u,x_0)+Kp(x_0,v)-p(x_0,x_0)\hfill \\ & \le p(x_0,x_0)+r+K\left(p(x_0,x_0)+r\right)-p(x_0,x_0)=Kp(x_0,x_0)+(K+1)r.\hfill \end{array}$$

Since J is an interval containing 0, condition (9) implies $p(u,v)\in J$.

If $u\ne v$, condition (8) yields $p(u,v)={\delta }_p(\left\{u\right\},\left\{v\right\})\le \phi \left(p(u,v)\right)<p(u,v)$, which is impossible.

Therefore $u=v$, and the fixed point is unique. □

Remark 5.

Condition (9) is a sufficient domain condition ensuring that $p(u,v)\in J$ for any two possible fixed points $u,v\in {\overline{B}}_p(x_0,r)$. It is automatic when $J={\mathbb{R}}_{+}$ and is not claimed to be necessary.

The role of the localization condition can be clarified directly from the chain estimate used to keep the iterative sequence inside the localization ball. Indeed,

$$K{S}_{\phi }\left(\alpha \right)-(K-1)\alpha =\alpha +K{\displaystyle \sum _{j=1}^{\infty }}{\phi }^j\left(\alpha \right),$$

and the right-hand side bounds the distance of every iterate from the initial point. The subtraction of the term $(K-1)\alpha$ reflects the fact that the first step has coefficient 1, whereas the subsequent steps enter the chain estimate with coefficient K.

When $K=1$, the condition reduces to $S_{\phi }\left(\alpha \right)\le p(x_0,x_0)+r$, which is the corresponding localization condition in a partial metric space. For $K>1$, the additional factor reflects the possible amplification of the later steps under the strong partial b-metric inequality.

The condition is used here as a sufficient invariance condition. No claim of sharpness is made. In particular, the proof discards the non-positive self-distance terms occurring in the chain estimate, and for a particular space or iterative trajectory these terms may yield a smaller bound. Thus, the stated condition is uniform and convenient, but it need not be necessary in every concrete example.

Example 4

(A local model of linguistic enrichment). Let X denote the collection of all linguistic productions that may occur during a learning process. Each element of X is regarded as a complete text rather than an isolated sentence. Assume that $(X,p,K)$ is a 0-complete strong partial b-metric space, where the distance $p(x,y)$ measures the overall linguistic discrepancy between two texts, while the self-distance $p(x,x)$ represents the remaining linguistic incompleteness of the text x.

Suppose that a learner starts from an initial linguistic state $x_0\in X$. For a prescribed radius $r>0$, the closed ball ${\overline{B}}_p(x_0,r)$ contains all texts that can realistically be reached during the current stage of instruction. Thus, the learning process is assumed to remain within a local neighbourhood of the initial linguistic competence.

For every text $x\in {\overline{B}}_p(x_0,r)$, let $F\left(x\right)\subseteq X$ denote the family of all linguistically acceptable immediate revisions of x. Since a text may admit several equally appropriate improvements, the mapping F is naturally set-valued.

The initial condition (6) expresses the existence of at least one feasible linguistic improvement of the initial production.

The condition (7) guarantees that the cumulative effect of successive linguistic refinements never leaves the prescribed local learning region.

The contractive condition (8) has a natural educational interpretation. It requires that whenever two texts are linguistically similar, their corresponding families of acceptable revisions are also close. In other words, similar linguistic productions should generate similar pedagogical recommendations. This condition ensures the coherence and stability of the learning strategy.

The conclusion of Theorem 1 asserts the existence of a text $x^{*}\in {\overline{B}}_p(x_0,r)$ such that $x^{*}\in F\left(x^{*}\right)$. From the linguistic point of view, this means that the learning process reaches a locally stable linguistic state. At this stage, the current text already belongs to the family of its own admissible revisions, so no further essential linguistic modification is required within the given learning environment.

If the mapping F is single-valued, then the theorem guarantees that this stable linguistic state is unique inside the prescribed local region. For the prescribed initial state satisfying the hypotheses of the theorem, the single-valued iterative trajectory constructed in the proof is uniquely determined and 0-converges to this local fixed point. The theorem does not assert convergence from every arbitrary initial state of the localization region.

4.2. Construction of a Strong Partial B-Metric for Linguistic Enrichment

We now develop a finite linguistic model illustrating the abstract framework of the preceding theorem. The construction is carried out in three steps.

First, the linguistic states are organized as the vertices of a weighted graph, and the corresponding shortest-path distance provides an ordinary metric. Second, this metric is transformed into a strong b-metric by means of a nonlinear increasing function. Third, a prescribed family of stable linguistic states is used to introduce positive self-distances, thereby producing a strong partial b-metric.

The resulting construction has a direct linguistic interpretation. The graph metric measures the minimum number of normalized revision steps between two formulations. The strong b-metric assigns additional weight to larger linguistic transitions, while the self-distance of a state measures its remaining distance from the stable target family.

The construction may be summarized as

$$\mathrm{weighted}\mathrm{revision}\mathrm{graph}⟶\mathrm{metric}⟶\mathrm{strong} b\text{-}\mathrm{metric}⟶\mathrm{strong}\mathrm{partial} b\text{-metric}.$$

We begin with two general construction results and then apply them to a finite family of progressively enriched linguistic states.

4.2.1. From a Metric to a Strong B-Metric

Every metric is trivially a strong b-metric with any coefficient $K\ge 1$. However, this observation does not produce a genuinely new distance. The following result gives a nonlinear construction that may fail to satisfy the ordinary triangle inequality while retaining the strong b-metric inequality.

Theorem 2.

Let $(X,d)$ be a bounded metric space and let $D=\sup \left\{d\right(x,y):x,y\in X\}<\infty$. Let $q>1$ and $\lambda >0$. Define $d_{q,\lambda }(x,y)=d(x,y)+\lambda d{(x,y)}^q$ for $x,y\in X$.

Then $d_{q,\lambda }$ is a strong b-metric on X with coefficient $K=1+\lambda q{\left(2D\right)}^{q-1}$, i.e.,

$$d_{q,\lambda }(x,z)\le d_{q,\lambda }(x,y)+K{d}_{q,\lambda }(y,z)$$

for all $x,y,z\in X$.

In general, $d_{q,\lambda }$ need not be a metric.

Proof. 

Put $h\left(t\right)=t+\lambda t^q$ for $t\ge 0$. Then $\rho (x,y)=h\left(d\right(x,y\left)\right)$.

The function h is strictly increasing, $h\left(0\right)=0$, and therefore $\rho$ is symmetric and $\rho (x,y)=0$ if and only if $x=y$.

Let $a=d(x,y)$ and $b=d(y,z)$. By the metric triangle inequality, we have $d(x,z)\le a+b$. Since $a,b\le D$, we have $a+b\le 2D$. By the mean value theorem, the equality

$$h(a+b)-h\left(a\right)=h^{\prime }\left(\xi \right)b$$

holds for some $\xi \in [a,a+b]\subseteq [0,2D]$. Since $h^{\prime }\left(t\right)=1+\lambda q{t}^{q-1}$ it follows that

$$h^{\prime }\left(\xi \right)\le 1+\lambda q{\left(2D\right)}^{q-1}=K.$$

Therefore, $h(a+b)\le h\left(a\right)+Kb$. From $b\le h\left(b\right)$ we get $h(a+b)\le h\left(a\right)+Kh\left(b\right)$.

Consequently,

$$\rho (x,z)=h\left(d\right(x,z\left)\right)\le h(a+b)\le h\left(a\right)+Kh\left(b\right)=\rho (x,y)+K\rho (y,z).$$

Thus, $\rho$ is a strong b-metric with coefficient K. □

The boundedness assumption is automatic in the finite linguistic model considered below. The transformation $t⟼t+\lambda t^q$ preserves the order of the original distances but gives additional weight to larger separations. Thus, a transition across several linguistic levels is penalized more strongly than a transition between neighboring levels.

Corollary 1.

Let $(X,d)$ be a metric space of finite diameter D, and let $\lambda >0$. Then

$$\rho (x,y)=d(x,y)+\lambda d{(x,y)}^2$$

is a strong b-metric with coefficient $K=1+4\lambda D$.

4.2.2. From a Strong B-Metric to a Strong Partial B-Metric

The preceding construction measures the distance between distinct states but assigns zero self-distance to every state. To model the remaining incompleteness of an intermediate linguistic formulation, we now prescribe a nonempty target family A of stable states.

The distance $\rho (x,A)={\inf }_{a\in A}\rho (x,a)$ from x to A, measures the smallest remaining linguistic transition required to reach an admissible stable formulation. We use this value as the self-distance of x.

Theorem 3.

Let $(X,\rho ,K)$ be a strong b-metric space and $A\subseteq X$ be nonempty. Put $\rho (x,A)={\inf }_{a\in A}\rho (x,a)$ and define

$$p_A(x,y)=\rho (x,y)+\max \{\rho (x,A),\rho (y,A)\}.$$

Then $p_A$ is a strong partial b-metric on X with coefficient K.

Moreover, $p_A(x,x)=\rho (x,A)$ for every $x\in X$.

Proof. 

For brevity, put $w\left(x\right)=\rho (x,A)$. Since $\rho (x,x)=0$, we have

$$p_A(x,x)=\rho (x,x)+\max \{w\left(x\right),w\left(x\right)\}=w\left(x\right)=\rho (x,A).$$

We verify the axioms of a strong partial b-metric.

First,

$$p_A(x,y)=\rho (x,y)+\max \{w\left(x\right),w\left(y\right)\}\ge w\left(x\right)=p_A(x,x).$$

Thus, the small self-distance condition is satisfied.

The symmetry of $p_A$ follows immediately from the symmetry of $\rho$:

$$p_A(x,y)=p_A(y,x).$$

Suppose that $p_A(x,x)=p_A(x,y)=p_A(y,y)$. Then $w\left(x\right)=w\left(y\right)$. Furthermore,

$$w\left(x\right)=\rho (x,y)+\max \left\{w\right(x),w(y\left)\right\}=\rho (x,y)+w\left(x\right).$$

Hence, $\rho (x,y)=0$. Since $\rho$ separates points, it follows that $x=y$. The converse implication is immediate.

It remains to verify the strong partial b-metric inequality. Since $\rho$ is a strong b-metric,

$$\rho (x,z)\le \rho (x,y)+K\rho (y,z).$$

For arbitrary nonnegative numbers $a,b,c$, we have

$$\max \{a,c\}\le \max \{a,b\}+\max \{b,c\}-b.$$

(15)

Indeed, the right-hand side in (15) equals

$$b+{(a-b)}_{+}+{(c-b)}_{+}\ge \max \{a,c\},$$

(16)

where $t_{+}=\max \{t,0\}$.

Applying (16) to $a=w\left(x\right)$, $b=w\left(y\right)$, and $c=w\left(z\right)$, we obtain

$$\max \left\{w\right(x),w(z\left)\right\}\le \max \left\{w\right(x),w(y\left)\right\}+\max \left\{w\right(y),w(z\left)\right\}-w\left(y\right).$$

Since $K\ge 1$, there holds

$$\max \left\{w\right(y),w(z\left)\right\}\le K\max \left\{w\right(y),w(z\left)\right\}.$$

Consequently,

$$\begin{array}{cc}\hfill p_A(x,z)& =\rho (x,z)+\max \left\{w\right(x),w(z\left)\right\}\hfill \\ & \le \rho (x,y)+K\rho (y,z)+\max \left\{w\right(x),w(y\left)\right\}+K\max \left\{w\right(y),w(z\left)\right\}-w\left(y\right)\hfill \\ & =p_A(x,y)+K{p}_A(y,z)-p_A(y,y).\hfill \end{array}$$

Therefore, $p_A$ is a strong partial b-metric with coefficient K. □

The construction separates two kinds of information. The term $\rho (x,y)$ measures the difference between the particular formulations x and y, whereas $\max \left\{\rho \right(x,A),\rho (y,A\left)\right\}$ records their remaining distance from the stable target family.

In particular, $p_A(x,x)=\rho (x,A)$. Thus, positive self-distance is no longer introduced by an arbitrary assignment of numerical weights. It is generated by the geometry of the underlying revision process.

Corollary 2.

Under the assumptions of Theorem 3, suppose additionally that A is closed in the topology induced by ρ. Then $p_A(x,x)=0$ if and only if $x\in A$.

Proof. 

By the definition of $p_A$, we have $p_A(x,x)=\rho (x,A)$. If $x\in A$, then

$$0\le \rho (x,A)\le \rho (x,x)=0.$$

Conversely, if $p_A(x,x)=0$, then $\rho (x,A)=0$, and the assumed closedness of A gives $x\in A$. □

Corollary 3.

Let $(X,d)$ be a metric space with finite diameter $D=diam\left(X\right)$. Let $q>1$, $\lambda >0$, and let $A\subseteq X$ be nonempty. Define $\rho (x,y)=d(x,y)+\lambda d{(x,y)}^q$ and

$$p_A(x,y)=\rho (x,y)+\max \{\rho (x,A),\rho (y,A)\}.$$

Then $p_A$ is a strong partial b-metric with coefficient $K=1+\lambda q{\left(2D\right)}^{q-1}$ and there holds

$$p_A(x,x)=\rho (x,A)=\underset{a\in A}{\inf }\left(d(x,a)+\lambda d{(x,a)}^q\right).$$

The preceding results apply to any bounded metric space and any nonempty target family. We now specialize them to the twenty linguistic states used in the enrichment model. Their organization into successive levels determines a weighted revision graph and hence an ordinary shortest-path metric.

4.2.3. A Graph Metric on the Linguistic States

Let $\mathcal{A}=\{a_1,\dots ,a_{20}\}$ and consider the partition $\mathcal{A}=L_0\cup L_1\cup L_2\cup L_3\cup L_4\cup L_5$, where

$$\begin{array}{cc}\hfill L_0& =\{a_1,a_2,a_3,a_4\},\hfill \\ \hfill L_1& =\{a_5,a_6,a_7,a_8\},\hfill \\ \hfill L_2& =\{a_9,a_{10},a_{11},a_{12}\},\hfill \\ \hfill L_3& =\{a_{13},a_{14},a_{15},a_{16}\},\hfill \\ \hfill L_4& =\{a_{17},a_{18}\},\hfill \\ \hfill L_5& =\{a_{19},a_{20}\}.\hfill \end{array}$$

Construct an undirected weighted graph whose vertex set is $\mathcal{A}$. Every vertex in $L_i$ is joined to every vertex in $L_{i+1}$, for $i=0,\dots ,4$, and every edge has length ${\displaystyle \frac{1}{5}}$.

Let $d(a,b)$ be the length of a shortest path joining a and b. Then d is a metric on $\mathcal{A}$.

More explicitly, if $a\in L_i$ and $b\in L_j$, then

$$d(a,b)=\left\{\begin{array}{cc}0,\hfill & a=b,\hfill \\ {\displaystyle \frac{|i-j|}{5}},\hfill & i\ne j,\hfill \\ {\displaystyle \frac{2}{5}},\hfill & i=j,a\ne b.\hfill \end{array}\right.$$

Since the largest separation occurs between $L_0$ and $L_5$, we have $diam(\mathcal{A},d)=1$. The metric d therefore records the normalized number of refinement levels separating two linguistic states.

4.2.4. The Stable Target Family and the Induced Self-Distances

We now apply the constructions in two preceding theorems and Section 4.2.3. Choose $q=2$, $\lambda =1$, and define $\rho (a,b)=d(a,b)+d{(a,b)}^2$.

Since $diam(\mathcal{A},d)=1$, Corollary 1 shows that $\rho$ is a strong b-metric with coefficient $K=1+4diam(\mathcal{A},d)=5$.

Let $A=L_5=\{a_{19},a_{20}\}$ be the prescribed stable target family. Define $p(a,b)=\rho (a,b)+\max \left\{\rho \right(a,A),\rho (b,A\left)\right\}$. By Theorem 3, p is a strong partial b-metric with coefficient $K=5$.

Indeed, states belonging to different levels are connected by a shortest path passing through the intermediate levels. Two distinct states at the same level are connected by a path through an adjacent level and therefore have distance $2/5$.

Consequently,

$$p(a,a)=\left\{\begin{array}{cc}2,\hfill & a\in L_0,\hfill \\ {\displaystyle \frac{36}{25}},\hfill & a\in L_1,\hfill \\ {\displaystyle \frac{24}{25}},\hfill & a\in L_2,\hfill \\ {\displaystyle \frac{14}{25}},\hfill & a\in L_3,\hfill \\ {\displaystyle \frac{6}{25}},\hfill & a\in L_4,\hfill \\ 0,\hfill & a\in L_5.\hfill \end{array}\right.$$

The successive values 2, ${\displaystyle \frac{36}{25}}$, ${\displaystyle \frac{24}{25}}$, ${\displaystyle \frac{14}{25}}$, ${\displaystyle \frac{6}{25}}$, and 0 measure the decreasing distance from the corresponding refinement levels to the stable target family. They are not assigned independently to the states; they are obtained from the graph metric and the chosen nonlinear transformation.

Thus, the linguistic interpretation of the self-distance is determined by the structure of the revision graph:

$$p(a,a)=\mathrm{the}\mathrm{transformed}\mathrm{shortest}\mathrm{distance}\mathrm{from}a\mathrm{to}\mathrm{a}\mathrm{stable}\mathrm{text}.$$

The graph metric measures the minimum number of normalized revision steps required to move between two linguistic states. The nonlinear transformation assigns a larger weight to transitions between more distant refinement levels, while the resulting self-distance measures the remaining linguistic effort required to reach at least one stable formulation.

Thus, the displayed self-distances decrease strictly along the refinement levels.

The two states $a_{19}$ and $a_{20}$ have zero self-distance because they are stable. Nevertheless,

$$p(a_{19},a_{20})=\rho (a_{19},a_{20})={\displaystyle \frac{2}{5}}+{\left({\displaystyle \frac{2}{5}}\right)}^2={\displaystyle \frac{14}{25}}>0.$$

Therefore, the two stable formulations remain distinct despite having zero self-distance.

4.2.5. The Finite Refinement Mapping and Verification of the Contractive Conditions

We now complete the finite mathematical model by defining the set-valued refinement mapping and verifying the hypotheses of Theorem 1. We retain the state space $X=\mathcal{A}$, partitioned into the levels $L_0,\dots ,L_5$, the stable target family $A=L_5$, and the strong partial b-metric p constructed in Section 4.2.3 and Section 4.2.4. The concrete linguistic formulations assigned to the abstract states are presented separately in Appendix B.

We first verify that $(X,p,5)$ is 0-complete. Let $\left\{x_n\right\}\subseteq X$ be a 0-Cauchy sequence. Since X is finite and $p(x,y)>0$ whenever $x\ne y$, the sequence must eventually be constant. If its eventual value is x, then the 0-Cauchy property gives $p(x,x)=0$. Therefore $x\in L_5$. Hence, every 0-Cauchy sequence 0-converges to a point of $L_5$, and $(X,p,5)$ is 0-complete.

Define the set-valued revision mapping $F:X⟶{\mathcal{C}}_0^p\left(X\right)$ by $F\left(a\right)=L_{i+1}$, provided that $a\in L_i$ for $0\le i\le 4$, and $F\left(a\right)=L_5$ for $a\in L_5$.

Thus, every nonterminal state is mapped to the entire next refinement level, while every stable state is mapped to the stable target family. The mapping is genuinely set-valued.

Every nonempty subset of X is 0-closed with respect to p. Indeed, suppose that $B\subseteq X$ is nonempty and $p(x,B)=0$.

Since B is finite, there exists $b\in B$ such that $p(x,b)=0$.

The definition of p implies $\rho (x,b)=0$. Since $\rho$ separates points, $x=b$, and hence $x\in B$. Therefore, $F\left(a\right)\in {\mathcal{C}}_0^p\left(X\right)$ for every $a\in X$.

Let $J=[0,\infty )$ and define $\phi \left(t\right)={\displaystyle \frac{6}{7}}t$.

Then $\phi :J\to J$ is strictly increasing, satisfies $\phi \left(t\right)<t$ for every $t>0$, and ${\phi }^n\left(t\right)={\left({\displaystyle \frac{6}{7}}\right)}^{n}t$.

Moreover, $S_{\phi }\left(t\right)={\sum }_{n=0}^{\infty }{\phi }^n\left(t\right)={\displaystyle \frac{t}{1-\frac{6}{7}}}=7t$ and thus $\phi$ is a Bianchini–Grandolfi gauge function.

We now verify the contractive condition. Put $s\left(i\right)=\min \{i+1,5\}$ for $i=0,\dots ,5$. Then $F\left(x\right)=L_{s\left(i\right)}$ whenever $x\in L_i$.

Just for convenience, let us put $r_i=\rho (x,A)$ for $x\in L_i$. Therefore, $r_i$ is precisely the self-distance already listed for the level $L_i$.

If $x\in L_i$, $y\in L_j$, and $i\ne j$, then $p(x,y)=h\left({\displaystyle \frac{|i-j|}{5}}\right)+\max \{r_i,r_j\}$, where $h\left(t\right)=t+t^2$.

If $x,y\in L_i$ are distinct, then

$$p(x,y)=h\left({\displaystyle \frac{2}{5}}\right)+r_i={\displaystyle \frac{14}{25}}+r_i.$$

Similarly,

$${\delta }_p(F\left(x\right),F\left(y\right))=\left\{\begin{array}{cc}{r}_{s\left(i\right)},\hfill & s\left(i\right)=s\left(j\right),\hfill \\ h\left({\displaystyle \frac{\left|s\right(i)-s(j\left)\right|}{5}}\right)+\max \{r_{s\left(i\right)},r_{s\left(j\right)}\},\hfill & s\left(i\right)\ne s\left(j\right).\hfill \end{array}\right.$$

For distinct $x\in L_i$ and $y\in L_j$, define

$$R_{ij}={\displaystyle \frac{{\delta }_p(F\left(x\right),F\left(y\right))}{p(x,y)}}.$$

Since both $p(x,y)$ and the images $F\left(x\right)$ and $F\left(y\right)$ depend only on the corresponding refinement levels, the quotient $R_{ij}$ also depends only on i and j.

A direct computation gives ${\max }_{0\le i,j\le 5}{R}_{ij}=R_{04}={\displaystyle \frac{36}{43}}<{\displaystyle \frac{6}{7}}$. The maximum is attained for $x\in L_0$ and $y\in L_4$.

Consequently, ${\delta }_p(F\left(x\right),F\left(y\right))\le {\displaystyle \frac{6}{7}}p(x,y)=\phi \left(p(x,y)\right)$ for all distinct $x,y\in X$. The complete ratio table and the explicit calculation of its maximal entry are provided in Appendix A.4.

The linear gauge $\phi \left(t\right)={\displaystyle \frac{6}{7}}t$ has been selected only to make the numerical verification transparent; its linearity is not imposed by the finite graph model. For example, the same contractive estimate also holds for the nonlinear gauge

$$\tilde{\phi }\left(t\right)={\displaystyle \frac{21}{25}}t+{\displaystyle \frac{1}{100}}{\displaystyle \frac{t^2}{1+t}},t\ge 0.$$

Indeed, ${\displaystyle \frac{36}{43}}<{\displaystyle \frac{21}{25}}$ and hence ${\delta }_p(F\left(x\right),F\left(y\right))\le {\displaystyle \frac{36}{43}}p(x,y)<\tilde{\phi }\left(p(x,y)\right)$ for all distinct $x,y\in X$.

Moreover, the inequalities $0\le \tilde{\phi }\left(t\right)\le {\displaystyle \frac{17}{20}}t<t$ hold for any $t>0$.

Therefore, ${\tilde{\phi }}^n\left(t\right)\le {\left({\displaystyle \frac{17}{20}}\right)}^{n}t$, and consequently,

$${\displaystyle \sum _{n=0}^{\infty }}{\tilde{\phi }}^n\left(t\right)\le {\displaystyle \sum _{n=0}^{\infty }}{\left({\displaystyle \frac{17}{20}}\right)}^{n}t<\infty .$$

Thus, $\tilde{\phi }$ is a nonlinear Bianchini–Grandolfi gauge function that can be used for the same finite model.

To demonstrate the genuinely local character of Theorem 1, we now apply it on a proper subset of X. Choose $x_0=a_{17}$, $\alpha ={\displaystyle \frac{1}{2}}$, and $r={\displaystyle \frac{11}{20}}$. Since $a_{17}\in L_4$ and $F\left(a_{17}\right)=L_5$, we have

$$p(a_{17},F\left(a_{17}\right))=p(a_{17},L_5)={\displaystyle \frac{12}{25}}<{\displaystyle \frac{1}{2}}=\alpha .$$

There holds $p(a_{17},a_{17})+r={\displaystyle \frac{6}{25}}+{\displaystyle \frac{11}{20}}={\displaystyle \frac{79}{100}}$.

For $a_{19},a_{20}\in L_5$, the inequality $p(a_{17},a_{19})=p(a_{17},a_{20})={\displaystyle \frac{12}{25}}<{\displaystyle \frac{79}{100}}$ holds.

For every $x\in L_0\cup L_1\cup L_2\cup L_3$, we have

$$p(a_{17},a_{18})={\displaystyle \frac{4}{5}},p(a_{17},x)\ge {\displaystyle \frac{4}{5}}$$

Consequently, ${\overline{B}}_p\left(a_{17},{\displaystyle \frac{11}{20}}\right)=\{a_{17},a_{19},a_{20}\}⊊X$.

For this local application, define the Bianchini–Grandolfi gauge $\psi \left(t\right)={\displaystyle \frac{1}{10}}t$ for $t\in [0,\infty )$.

Then $S_{\psi }\left(t\right)={\displaystyle \frac{t}{1-\frac{1}{10}}}={\displaystyle \frac{10}{9}}t$.

For every $x\in {\overline{B}}_p(a_{17},11/20)$, one has $F\left(x\right)=L_5$. Hence, for all distinct $x,y\in {\overline{B}}_p(a_{17},11/20)$, we have

$${\delta }_p\left(F\left(x\right)\cap {\overline{B}}_p(a_{17},11/20),F\left(y\right)\right)={\delta }_p(L_5,L_5)=0\le \psi \left(p(x,y)\right).$$

Finally, since $K=5$, we get $K{S}_{\psi }\left(\alpha \right)-(K-1)\alpha =5·{\displaystyle \frac{10}{9}}·{\displaystyle \frac{1}{2}}-4·{\displaystyle \frac{1}{2}}={\displaystyle \frac{7}{9}}$ and furthermore, ${\displaystyle \frac{7}{9}}<{\displaystyle \frac{79}{100}}=p(a_{17},a_{17})+{\displaystyle \frac{11}{20}}$.

Thus, all assumptions of Theorem 1 are satisfied on the proper localization set

$${\overline{B}}_p\left(a_{17},{\displaystyle \frac{11}{20}}\right)=\{a_{17},a_{19},a_{20}\}.$$

Consequently, F has a fixed point in this proper local ball.

In fact, $F\left(a_{19}\right)=F\left(a_{20}\right)=L_5=\{a_{19},a_{20}\}$ and hence

$$a_{19}\in F\left(a_{19}\right),a_{20}\in F\left(a_{20}\right).$$

No state in $L_0\cup \dots \cup L_4$ belongs to its own image. Therefore, $Fix\left(F\right)=\{a_{19},a_{20}\}$.

The mapping describes the multistage enrichment process

$$L_0⟶L_1⟶L_2⟶L_3⟶L_4⟶L_5.$$

At every nonterminal stage, the image $F\left(a\right)$ contains all admissible formulations at the next refinement level. Different choices may therefore generate different enrichment trajectories, but every trajectory eventually reaches the stable family $L_5$.

The self-distance $p(a,a)=\rho (a,L_5)$ measures the transformed shortest distance from the current linguistic state to the stable target family. It decreases strictly from one refinement level to the next and vanishes exactly on the stable states $a_{19}$ and $a_{20}$.

From the linguistic point of view, the construction above models linguistic enrichment as a multistage process in which a formulation may admit several equally acceptable revisions at each stage. Consequently, the mapping F is multivalued: it does not prescribe a unique successor, but allows different revision trajectories leading toward the stable target family.

The self-distance has a global rather than a local interpretation. It measures the remaining linguistic effort required to reach a stable formulation, not merely the effort associated with the next revision step. Its strict decrease along the refinement levels therefore represents the progressive reduction in the grammatical, terminological, logical, and stylistic deficiencies of the text.

The presence of two terminal states also distinguishes stabilization from uniqueness. The states $a_{19}$ and $a_{20}$ may represent, for example, a concise formal formulation and a more explanatory, pedagogically oriented formulation of the same mathematical statement. Both are stable because no further essential revision is required within the adopted model, but they remain distinct because they serve different communicative purposes. Thus, convergence to the stable target family does not require all admissible enrichment trajectories to produce the same final text.

The complete assignment of linguistic formulations to the abstract states $a_1,\dots ,a_{20}$, together with their classification into the levels $L_0,\dots ,L_5$, is presented in Appendix B.

5. Special Cases of the Main Theorem

The following corollaries show that several well-known classes of contractive mappings arise as direct special cases of Theorem 1.

Corollary 4

(Local linear set-valued contraction). Let $(X,p,K)$ be a 0-complete strong partial b-metric space, let $x_0\in X$ and $r>0$, and let $F:{\overline{B}}_p(x_0,r)⟶{\mathcal{C}}_0^p\left(X\right)$. Suppose that there exist $q\in (0,1)$ and $\alpha >0$ such that

$$p(x_0,F\left(x_0\right))<\alpha$$

(17)

and

$$\alpha +{\displaystyle \frac{Kq\alpha }{1-q}}\le p(x_0,x_0)+r.$$

(18)

Assume further that

$${\delta }_p\left(F\left(x\right)\cap {\overline{B}}_p(x_0,r),F\left(y\right)\right)\le qp(x,y)$$

(19)

for all distinct $x,y\in {\overline{B}}_p(x_0,r)$ such that $F\left(x\right)\cap {\overline{B}}_p(x_0,r)\ne ⌀$.

Then F has a fixed point $x^{*}\in {\overline{B}}_p(x_0,r)$.

If F is single-valued, then $x^{*}$ is the unique fixed point of F in ${\overline{B}}_p(x_0,r)$.

Proof. 

Define $\phi \left(t\right)=qt$ for $t\in {\mathbb{R}}_{+}$. Since $q\in (0,1)$, the mapping $\phi$ is strictly increasing, continuous at 0, and ${\phi }^n\left(t\right)=q^{n}t$.

Consequently, $S_{\phi }\left(t\right)={\sum }_{n=0}^{\infty }{q}^{n}t={\displaystyle \frac{t}{1-q}}$. Therefore,

$$K{S}_{\phi }\left(\alpha \right)-(K-1)\alpha ={\displaystyle \frac{K\alpha }{1-q}}-(K-1)\alpha =\alpha +{\displaystyle \frac{Kq\alpha }{1-q}}.$$

Thus, condition (18) is precisely the localization condition in Theorem 1. Moreover, (19) becomes

$${\delta }_p\left(F\left(x\right)\cap {\overline{B}}_p(x_0,r),F\left(y\right)\right)\le \phi \left(p(x,y)\right).$$

Hence, all the assumptions of Theorem 1 are satisfied.

Since here $J={\mathbb{R}}_{+}$, the additional condition for uniqueness is automatic. Therefore, if F is single-valued, its fixed point is unique in ${\overline{B}}_p(x_0,r)$. □

Example 5

(Linear linguistic improvement). Corollary 4 describes a learning process in which each instructional step produces a uniform proportional improvement of the learner’s linguistic competence. Let X denote the collection of all possible linguistic productions, equipped with a strong partial b-metric p. For every text x, the set $F\left(x\right)$ consists of all linguistically acceptable revisions that may be obtained after one stage of instruction.

The contractive condition (19) expresses a one-sided contractive relation between the corresponding families of admissible revisions. More precisely, every element of $F\left(x\right)\cap {\overline{B}}_p(x_0,r)$ has point-to-set distance to $F\left(y\right)$ bounded above by $qp(x,y)$.

Consequently, the condition does not assert that every possible pair of revisions is at distance at most $qp(x,y)$. Rather, each admissible revision associated with the first state can be approximated by a revision associated with the second state in accordance with the prescribed one-sided point-to-set estimate.

Along the iterative selection constructed in the proof, the successive step sizes are controlled geometrically, while the chain estimate ensures that the selected trajectory remains inside the prescribed local learning region.

Corollary 4 guarantees the existence of a stable linguistic state $x^{*}$, while its proof constructs an admissible iterative selection that 0-converges to such a fixed point. In this state, $x^{*}\in F\left(x^{*}\right)$, meaning that the current linguistic production already satisfies the requirements of the instructional model and no further essential local revision is required.

If the revision operator is single-valued, then this stable linguistic state is unique. For the prescribed initial state satisfying the hypotheses of the corollary, the resulting single-valued iterative trajectory converges to this unique local fixed point. No assertion is made here about convergence from every arbitrary initial state of the local region.

Corollary 5

(Partial metric case). Let $(X,p)$ be a 0-complete partial metric space, let $x_0\in X$ and $r>0$, and let $F:{\overline{B}}_p(x_0,r)⟶{\mathcal{C}}_0^p\left(X\right)$. Let $J\subseteq {\mathbb{R}}_{+}$ be an interval containing 0, and let $\phi :J⟶J$ be a Bianchini–Grandolfi gauge function.

Assume that there exists $\alpha \in J\setminus \left\{0\right\}$ such that

$$p(x_0,F\left(x_0\right))<\alpha$$

(20)

and

$$S_{\phi }\left(\alpha \right)\le p(x_0,x_0)+r.$$

(21)

Suppose further that

$${\delta }_p\left(F\left(x\right)\cap {\overline{B}}_p(x_0,r),F\left(y\right)\right)\le \phi \left(p(x,y)\right)$$

(22)

for all distinct $x,y\in {\overline{B}}_p(x_0,r)$ such that $p(x,y)\in J$ and $F\left(x\right)\cap {\overline{B}}_p(x_0,r)\ne ⌀$.

Then F has a fixed point $x^{*}\in {\overline{B}}_p(x_0,r)$.

If F is single-valued and

$$p(x_0,x_0)+2r\in J,$$

(23)

then $x^{*}$ is the unique fixed point of F in ${\overline{B}}_p(x_0,r)$.

Proof. 

A partial metric is a strong partial b-metric with coefficient $K=1$. Therefore,

$$K{S}_{\phi }\left(\alpha \right)-(K-1)\alpha =S_{\phi }\left(\alpha \right),$$

while the uniqueness condition in Theorem 1 becomes

$$Kp(x_0,x_0)+(K+1)r=p(x_0,x_0)+2r.$$

The conclusion now follows directly from Theorem 1. □

Corollary 6

(Local Banach-type principle). Let $(X,p,K)$ be a 0-complete strong partial b-metric space, let $x_0\in X$ and $r>0$, and let $T:{\overline{B}}_p(x_0,r)⟶{\overline{B}}_p(x_0,r)$ be a single-valued mapping. Suppose that there exist $q\in (0,1)$ and $\alpha >0$ such that

$$p(x_0,T{x}_0)<\alpha$$

(24)

and

$$\alpha +{\displaystyle \frac{Kq\alpha }{1-q}}\le p(x_0,x_0)+r.$$

(25)

If

$$p(Tx,Ty)\le qp(x,y)$$

(26)

for all distinct $x,y\in {\overline{B}}_p(x_0,r)$, then T has a unique fixed point in ${\overline{B}}_p(x_0,r)$.

Proof. 

Define the set-valued mapping $F\left(x\right)=\left\{Tx\right\}$ for $x\in {\overline{B}}_p(x_0,r)$.

Every singleton is 0-closed with respect to p. Indeed, if $p(z,\{Tx\left\}\right)=p(z,Tx)=0$, then, by (II), there holds $p(z,z)=p(Tx,Tx)=0$. By (I) we conclude that $z=Tx$.

Since T maps ${\overline{B}}_p(x_0,r)$ into itself, $F\left(x\right)\cap {\overline{B}}_p(x_0,r)=\left\{Tx\right\}$. Consequently,

$${\delta }_p\left(F\left(x\right)\cap {\overline{B}}_p(x_0,r),F\left(y\right)\right)={\delta }_p(\left\{Tx\right\},\left\{Ty\right\})=p(Tx,Ty)\le qp(x,y).$$

The result now follows from Corollary 4. □

Corollary 7

(Global set-valued fixed point principle). Let $(X,p,K)$ be a 0-complete strong partial b-metric space, $F:X⟶{\mathcal{C}}_0^p\left(X\right)$, and $\phi :{\mathbb{R}}_{+}⟶{\mathbb{R}}_{+}$ be a Bianchini–Grandolfi gauge function. Suppose that

$${\delta }_p(F\left(x\right),F\left(y\right))\le \phi \left(p(x,y)\right)$$

(27)

for all distinct $x,y\in X$.

Then F has a fixed point in X. If F is single-valued, then the fixed point is unique.

Proof. 

Fix an arbitrary point $x_0\in X$. Since $F\left(x_0\right)$ is nonempty and p is finite-valued, we have $p(x_0,F\left(x_0\right))<\infty$.

Choose $\alpha >0$ such that $p(x_0,F\left(x_0\right))<\alpha$. Since $S_{\phi }\left(\alpha \right)<\infty$, we may choose $r>0$ sufficiently large so that $K{S}_{\phi }\left(\alpha \right)-(K-1)\alpha \le p(x_0,x_0)+r$.

Consider the restriction of F to ${\overline{B}}_p(x_0,r)$. For all distinct $x,y\in {\overline{B}}_p(x_0,r)$ such that $F\left(x\right)\cap {\overline{B}}_p(x_0,r)\ne ⌀$, we have $F\left(x\right)\cap {\overline{B}}_p(x_0,r)\subseteq F\left(x\right)$. Therefore,

$${\delta }_p\left(F\left(x\right)\cap {\overline{B}}_p(x_0,r),F\left(y\right)\right)\le {\delta }_p(F\left(x\right),F\left(y\right))\le \phi \left(p(x,y)\right).$$

Thus, all the assumptions of Theorem 1 are satisfied with $J={\mathbb{R}}_{+}$. Consequently, F has a fixed point $x^{*}\in {\overline{B}}_p(x_0,r)\subseteq X$.

Suppose now that F is single-valued and let $u,v\in X$ be fixed points of F. If $u\ne v$, then

$$p(u,v)={\delta }_p(\left\{u\right\},\left\{v\right\})={\delta }_p(F\left(u\right),F\left(v\right))\le \phi \left(p(u,v)\right)<p(u,v),$$

which is impossible. Therefore, $u=v$. □

Corollary 8

(Global linear set-valued contraction). Let $(X,p,K)$ be a 0-complete strong partial b-metric space, and let $F:X⟶{\mathcal{C}}_0^p\left(X\right)$. Suppose that there exists $q\in (0,1)$ such that ${\delta }_p(F\left(x\right),F\left(y\right))\le qp(x,y)$ for all distinct $x,y\in X$.

Then F has a fixed point in X. If F is single-valued, then the fixed point is unique.

Proof. 

Set $\phi \left(t\right)=qt$ for $t\in {\mathbb{R}}_{+}$. Then $S_{\phi }\left(t\right)={\sum }_{n=0}^{\infty }{q}^{n}t={\displaystyle \frac{t}{1-q}}<\infty$.

Thus, $\phi$ is a Bianchini–Grandolfi gauge function, and the conclusion follows from Corollary 7. □

6. Discussion

The main result extends local fixed point theory for multivalued mappings to 0-complete strong partial b-metric spaces. The combination of a nonzero self-distance, a relaxed triangle inequality, and set-valued images requires a proof strategy that differs from the standard arguments used in metric spaces. In particular, the iterative sequence must be constructed in such a way that all its elements remain inside the prescribed localization set. The chain estimate developed in this paper serves two purposes simultaneously: it controls the position of the iterates relative to the initial point and proves that the sequence is (0)-Cauchy.

An important feature of the theorem is that the closed ball is used only as a localization set. No assumption is imposed that the family of (p)-balls forms a basis for a topology, and no topological closure argument is required in the final step of the proof. Instead, the existence of a fixed point follows from 0-completeness and the (0)-closedness of the values of the mapping. This distinction is relevant because the geometric and topological behavior of balls in partial b-metric-type spaces may differ substantially from that of ordinary metric balls. The result therefore separates the localization mechanism from additional topological properties that are not needed for the fixed point argument.

The use of a Bianchini–Grandolfi gauge function provides greater flexibility than a contraction condition determined by a single constant. The summability of the iterates of the gauge function gives the quantitative control required for both localization and convergence. Linear contractions appear as a particularly transparent special case, but the general formulation also covers nonlinear rates of contraction. The corollaries derived from the main theorem clarify its relation to partial metric, Banach-type, and global set-valued fixed point principles.

The construction of a strong partial b-metric from a bounded metric space and a prescribed target family constitutes a second contribution of the paper. It provides a systematic way to generate nonzero self-distances with a direct interpretation. The self-distance of a point is determined by its remaining transformed distance to the target family and vanishes precisely on that family. Consequently, the construction is suitable for processes in which stabilization is associated with reaching any member of a collection of admissible terminal states rather than one uniquely prescribed endpoint.

This target-family approach also clarifies the distinction between stability and uniqueness. Several elements may have zero self-distance and may simultaneously be fixed points of a multivalued mapping while remaining mutually distinct. Thus, zero self-distance should not be interpreted as an identification of all stable elements. It indicates only that no further essential transition is required relative to the chosen model. This feature may be useful in applications involving alternative solutions, equivalent representations, or several acceptable outcomes.

The linguistic example illustrates this interpretation in a finite and explicit setting. Alternative formulations of the same mathematical statement are arranged into successive refinement levels, and the revision graph represents admissible transitions between them. The multivalued mapping reflects the fact that linguistic revision is generally nonunique: at a given stage, several grammatically or stylistically acceptable improvements may be available. The two stable states demonstrate that different final texts may express the same mathematical content while serving different communicative purposes. For instance, one formulation may prioritize formal concision, whereas another may emphasize explanation and pedagogical accessibility.

The linguistic construction should be regarded as a mathematical model of the structure of a revision process rather than as an empirical model of language acquisition or text editing. The refinement levels, edge weights, and admissible transitions are prescribed in advance, and the model does not claim that linguistic effort can be measured universally by fixed numerical values. Its purpose is to demonstrate how the abstract notions of self-distance, multivalued transition, and nonunique stabilization can receive a coherent interpretation in language dynamics. An empirical application would require criteria for classifying texts, estimating transition costs, and validating the resulting refinement structure using linguistic data or expert assessment.

Several extensions are possible. On the theoretical side, the local principle may be investigated for other classes of gauge functions, alternative set-distance functionals, coupled or tripled fixed points, common fixed points, and best proximity points. It would also be useful to determine whether some of the localization assumptions can be weakened while preserving the invariance of the iterative sequence. On the applied side, the finite linguistic model could be extended by allowing transitions between nonconsecutive levels, assigning nonuniform revision costs, or permitting the target family to change according to genre, audience, or communicative purpose. Such developments may provide a basis for more detailed mathematical models of linguistic refinement, automated text revision, and other multistage processes with multiple acceptable stable outcomes.

7. Conclusions

A local set-valued fixed point principle was established in a 0-complete strong partial b-metric space. The proof avoids topological closure arguments and relies instead on 0-closed values of the mapping. The chain estimate in Lemma 2 is essential: it simultaneously guarantees that the iterative sequence remains in the prescribed localization set and that the sequence is 0-Cauchy. Further developments may include coupled, tripled, or n-tupled fixed point results and best proximity point versions of the local principle.