Alternating Points of Set-Valued Mappings in Strong Partial b-Metric Spaces with Applications to Economic and Linguistic Adjustment

Abstract

This paper introduces and studies alternating points of two set-valued mappings acting between two 0-complete strong partial b-metric spaces. An alternating point is a mutually compatible pair of states, each of which belongs to the image of the other under the corresponding mapping. Such a pair simultaneously generates fixed points of the two set-valued compositions. We establish a local existence theorem under nonlinear contractive conditions expressed by one-sided excess functionals on prescribed closed balls. The two successive transitions are controlled by different comparison functions, while the Bianchini–Grandolfi summability condition is imposed only on their composition, which represents one complete alternating cycle. The proof constructs two interdependent sequences, shows that they remain within their respective localization regions, and establishes their zero-Cauchy convergence by means of chain estimates adapted to strong partial b-metrics. A finite discrete model is developed to verify the assumptions of the theorem explicitly and to demonstrate that several alternating points may coexist. The same abstract model is then interpreted in two different settings: as a set-valued price–demand adjustment process and as an alternating process of lexical enrichment and grammatical stabilization. These applications illustrate how the proposed framework can describe nonunique adjustment and refinement processes with multiple mutually compatible terminal configurations.

1. Introduction

Fixed point theory provides a general mathematical framework for studying equilibrium, recursive adjustment, iterative approximation, and stabilization processes. In its classical form, the theory concerns a self-mapping acting on a single space. Many applications, however, involve two different families of states and two transformations acting in opposite directions. A state in the first space generates one or more admissible states in the second space, and the resulting state generates one or more admissible continuations in the first space. Such processes are naturally described by a pair of set-valued mappings

$$F:X\rightrightarrows Y\mathrm{and}G:Y\rightrightarrows X.$$

The appropriate terminal configuration for this alternating process is a pair $(x^{*},y^{*})\in X\times Y$ satisfying $y^{*}\in F\left(x^{*}\right)$ and $x^{*}\in G\left(y^{*}\right)$. We call such a pair an alternating point of F and G. This concept is closely related to, but more informative than, the ordinary fixed point formulation. Indeed, every alternating point simultaneously generates the fixed point relations $x^{*}\in (G\circ F)\left(x^{*}\right)$ and $y^{*}\in (F\circ G)\left(y^{*}\right)$, while retaining the compatibility relation between its two components. Conversely, a fixed point of either set-valued composition produces an alternating point through a suitable intermediate selection. Thus, the alternating formulation makes explicit the two-stage structure that is hidden when only the compositions are considered.

The study is carried out in two 0-complete strong partial b-metric spaces. Strong partial b-metrics combine the possibility of positive self-distances, inherited from partial metric spaces, with a relaxed triangle inequality governed by a coefficient $K\ge 1$. Partial metrics were introduced by Matthews in connection with computational semantics and processes involving partially defined information [1]. Their b-metric extensions permit greater flexibility in measuring transitions between states [2,3]. These features are particularly useful when the self-distance of a state is interpreted as a residual discrepancy, unresolved information, or the amount of adjustment that remains before a prescribed terminal family is reached.

Set-valued mappings are essential when a state admits several possible continuations. They arise naturally in optimization, equilibrium analysis, control, differential inclusions, and models involving heterogeneity or incomplete information [4,5]. Unlike single-valued iterations, a set-valued process need not prescribe a unique trajectory. The mathematical problem is therefore to provide conditions ensuring that at least one admissible sequence remains in the prescribed region, converges in the appropriate generalized sense, and terminates at a pair that is compatible with both mappings.

Contractive fixed point principles in generalized metric spaces have been developed under a variety of linear and nonlinear assumptions. An important nonlinear approach is based on Bianchini–Grandolfi gauge functions, whose iterates satisfy a summability condition suitable for controlling the total length of an iterative chain [6]. Local multivalued contraction principles in b-metric-type spaces have also been obtained through one-sided set excesses and localization on closed balls [7]. These ideas make it possible to replace global assumptions by conditions imposed only on the part of the space visited by the constructed sequence.

A local fixed point theorem for a single multivalued mapping in a 0-complete strong partial b-metric space was established in [8]. That result combines localization, set-valued excess estimates, positive self-distances, and nonlinear Bianchini–Grandolfi contractivity. The present paper develops a different problem. Instead of one multivalued self-mapping, we consider two mappings acting alternately between two possibly different strong partial b-metric spaces: $X\stackrel{F}{\to }Y\stackrel{G}{\to }X$. Consequently, the two individual transitions are controlled by different comparison functions and different generalized distances. The contraction governing one complete alternating step is not either comparison function separately, but their composition.

More precisely, in [8], a single multivalued self-mapping is localized in one prescribed closed ball and is controlled by one Bianchini–Grandolfi gauge function. In the present setting, localization is simultaneous in two prescribed balls, one in each space, and the two directional transitions are controlled by functions $\phi$ and $\psi$, while the Bianchini–Grandolfi summability condition is imposed only on their composition $\theta =\psi \circ \phi$. The conclusion is correspondingly different: instead of a fixed point of one multivalued self-mapping, the theorem produces an alternating pair $(x^{*},y^{*})$, which in turn generates fixed points of both set-valued compositions $G\circ F$ and $F\circ G$.

The main result establishes the existence of an alternating point for two locally contractive set-valued mappings. The assumptions are imposed on prescribed closed balls in the two spaces and are expressed through one-sided excess functionals. If the transition from X to Y is controlled by $\phi$, the transition from Y to X is controlled by $\psi$, and $\theta =\psi \circ \phi$ is a Bianchini–Grandolfi gauge function, then an alternating sequence can be constructed and localized in the prescribed product of closed balls. The two component sequences are shown to be 0-Cauchy, and their limits form an alternating point. An important feature of the result is that $\phi$ and $\psi$ need not themselves be Bianchini–Grandolfi gauge functions; the summability requirement is imposed only on their composition, which represents one complete alternating cycle.

The principal contributions of the paper can be summarized as follows:

• We introduce the alternating-point formulation for a pair of set-valued mappings acting between two strong partial b-metric spaces and establish its precise connection with the fixed points of the compositions $G\circ F$ and $F\circ G$.
• We prove a local existence theorem in which the two mappings are controlled by distinct nonlinear comparison functions, whereas the Bianchini–Grandolfi summability condition is required only for their composition.
• We construct the two alternating sequences explicitly and use chain estimates adapted to strong partial b-metrics to prove both localization and 0-Cauchy convergence.
• We provide a common finite discrete framework that satisfies all assumptions of the theorem and possesses multiple alternating points, thereby showing that the result does not imply uniqueness.
• We give two interpretations of the same discrete framework. The first describes an alternating price–demand adjustment process, while the second models alternating lexical enrichment and grammatical stabilization of a text.

The applications are intended as mathematically transparent interpretations of the abstract construction rather than as complete empirical models. In the economic application, the two mappings represent a demand correspondence and a producer’s price-adjustment correspondence. The resulting terminal configurations should not be identified with competitive equilibria in the full Arrow–Debreu sense or with dynamically stable states of a classical Walrasian tâtonnement process [9,10,11]. They are alternating price–demand configurations generated by the particular set-valued rules of the model.

In the linguistic application, the two mappings represent grammatical stabilization and renewed lexical enrichment. The set-valued character of the mappings reflects the fact that a text may admit several grammatically acceptable revisions and several lexically or stylistically enriched realizations. This interpretation is consistent with process-oriented and complex dynamic approaches to writing and language development, which emphasize recursion, variability, and the possibility of multiple admissible outcomes [12,13,14,15]. The application does not claim that linguistic development is uniquely determined by a metric contraction; rather, it shows how the abstract alternating-point framework can represent a finite revision process with several mutually compatible terminal formulations.

The remainder of the paper is organized as follows. Section 2 recalls the necessary background on strong partial b-metric spaces, localization, 0-completeness, set-valued excesses, Bianchini–Grandolfi gauge functions, and the chain estimates used in the proof. Section 3 introduces alternating points and establishes the main existence theorem. The subsequent discrete construction verifies the assumptions of the theorem explicitly and is then interpreted as an economic adjustment process and as an alternating linguistic-enrichment process.

2. Materials and Methods

This section collects the definitions, notation, and previously established results that form the methodological basis of the present study. We first recall the main properties of strong partial b-metric spaces and explain the roles of generalized distances and positive self-distances. We then introduce the localization sets, point-to-set distances, excess functionals, 0-closed sets, 0-completeness, and Bianchini–Grandolfi gauge functions used in the subsequent analysis. The interaction between these notions is illustrated by a discrete example. Finally, we recall the chain estimate required for the construction and convergence analysis of the alternating sequences. The results collected in this section are known and are stated without proof, with appropriate references to their original sources and to the preliminary framework developed in [8].

2.1. Strong Partial b-Metric Spaces

Strong partial b-metric spaces combine two different extensions of ordinary metric spaces. First, the distance of a point from itself may be positive, as in partial metric spaces. Second, the triangle inequality is relaxed by means of a coefficient $K\ge 1$, as in strong b-metric spaces. These two features make the resulting structure suitable for processes in which a state may contain a nonzero amount of unresolved information and in which transitions between states need not satisfy the ordinary triangle inequality.

We begin by recalling the definition introduced in [3].

Definition 1

([3]).  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.

Condition II shows that the self-distance $p(x,x)$ is the smallest distance measured from x. Unlike the situation in an ordinary metric space, it is not required to vanish. Nevertheless, condition I guarantees that the generalized distance still distinguishes points: two points coincide precisely when their mutual distance agrees with both self-distances.

The coefficient K measures the deviation from the partial metric triangle inequality. When $K=1$, condition IV becomes $p(x,y)\le p(x,z)+p(z,y)-p(z,z)$, which is the ordinary partial metric triangle inequality. If all self-distances vanish, then condition IV reduces to $p(x,y)\le p(x,z)+Kp(z,y)$, and p is a strong b-metric. If both $K=1$ and $p(x,x)=0$ for every $x\in X$, then p is an ordinary metric.

Thus, the following specializations hold:

$$\begin{array}{ccc}\mathrm{Condition}\mathrm{on}K& \mathrm{Condition}\mathrm{on}p(x,x)& \mathrm{Resulting}\mathrm{structure}\\ K=1& p(x,x)\ge 0& \mathrm{partial}\mathrm{metric}\\ K\ge 1& p(x,x)=0& \mathrm{strong}b\text{-}\mathrm{metric}\\ K=1& p(x,x)=0& \mathrm{ordinary}\mathrm{metric}.\end{array}$$

Although condition IV appears asymmetric because the coefficient K multiplies only one of the two distances involving the intermediate point, the symmetry of p allows the inequality to be written in an equivalent form. We recall the corresponding result from [8].

Proposition 1

([8]). Let X be a nonempty set, $K\ge 1$, and $p:X\times X⟶{\mathbb{R}}_{+}$ be symmetric.

Then the inequality $p(x,y)\le p(x,z)+Kp(z,y)-p(z,z)$ holds for all $x,y,z\in X$ if and only if $p(x,y)\le Kp(x,z)+p(z,y)-p(z,z)$ holds for all $x,y,z\in X$.

Proposition 1 shows that the apparent asymmetry of the strong partial b-metric inequality is only formal. Indeed, the coefficient K may be placed before either of the two distances involving the intermediate point. Consequently, both estimates may be used in applications, depending on which one provides the sharper bound.

Combining the two equivalent orientations gives a useful minimum form of the strong partial b-metric inequality.

Proposition 2

([8]). Let X be a nonempty set, $K\ge 1$, and $p:X\times X⟶{\mathbb{R}}_{+}$ be symmetric.

Then $p(x,y)\le p(x,z)+Kp(z,y)-p(z,z)$ holds for all $x,y,z\in X$ if and only if

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

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

If, in addition, $p(x,x)=0$ for every $x\in X$, then the same conclusion holds for strong b-metrics. In this case, $p(x,y)\le p(x,z)+Kp(z,y)$ is equivalent to

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

The minimum formulation will be useful when the generalized triangle inequality is applied repeatedly along finite chains. It makes it possible to choose the more appropriate orientation at every stage and, in some cases, to obtain a sharper estimate than the one resulting from a fixed orientation.

Before introducing the remaining notions needed in the alternating construction, we illustrate Definition 1 and Propositions 1 and 2 by means of finite discrete spaces.

2.2. Construction Principles for Strong Partial b-Metrics

The preceding definition describes the axiomatic properties of a strong partial b-metric but does not, by itself, provide a systematic procedure for constructing such generalized distances. For the discrete models considered below, we use the construction principles established in [8].

The construction consists of two steps. First, a bounded metric is transformed nonlinearly into a strong b-metric. Second, a prescribed nonempty target set is used to introduce nonzero self-distances. This leads to the scheme

$$\mathrm{bounded}\mathrm{metric}⟶\mathrm{strong}b\text{-}\mathrm{metric}⟶\mathrm{strong}\mathrm{partial}b\text{-}\mathrm{metric}.$$

The first step is described by the following theorem.

Theorem 1

([8]). Let $(X,d)$ be a bounded metric space, i.e, $D=sup\left\{d\right(x,y):x,y\in X\}<\infty$. For $q,\lambda \in \mathbb{R}$ with $q>1$ and $\lambda >0$, we define

$$\rho (x,y)=d(x,y)+\lambda d{(x,y)}^q,x,y\in X.$$

(1)

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

In particular, $\rho (x,z)\le \rho (x,y)+K\rho (y,z)$ for all $x,y,z\in X$.

The nonlinear term $\lambda d{(x,y)}^q$ assigns greater relative weight to transitions between more distant points. In general, the resulting function $\rho$ need not satisfy the ordinary triangle inequality and may therefore define a genuinely nonmetric strong b-metric structure.

The second step introduces self-distances determined by a prescribed target set.

Theorem 2

([8]). Let $(X,\rho ,K)$ be a strong b-metric space and $A\subseteq X$ be a nonempty set.

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 and there holds, $p_A(x,x)=\rho (x,A)$ for every $x\in X$.

Theorem 2 gives a geometric meaning to the self-distance. The quantity $p_A(x,x)$ is not assigned arbitrarily; it is precisely the strong b-metric distance from x to the target set A. Thus, the self-distance measures the remaining transition required for x to reach the prescribed family of stable states.

If A is zero-closed with respect to $\rho$, then $p_A(x,x)=0$ if and only if $x\in A$. Consequently, the elements of A are exactly the states having zero self-distance.

Combining the preceding two theorems gives a direct construction from a bounded metric space.

Corollary 1

([8]). Let $(X,d)$ be a metric space with finite diameter $D=diam\left(X\right)$. Let $q>1$, $\lambda >0$, and $A\subseteq X$ 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 $K=1+\lambda q{\left(2D\right)}^{q-1}$.

Furthermore, there holds

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

Since every finite metric space is bounded, these construction principles apply directly to finite discrete spaces. In particular, one may begin with a finite weighted graph, use its shortest-path distance as the metric d, apply the nonlinear transformation in (1), and finally choose a nonempty target set A. The resulting function $p_A$ is then a strong partial b-metric without the need to verify its defining axioms separately.

2.3. A Discrete Construction from a Metric to a Strong Partial b-Metric

We now illustrate the preceding construction principles on a finite discrete space. The example is developed in three successive stages. We begin with an ordinary metric, transform it into a strong b-metric, and finally introduce nonzero self-distances by prescribing a target set.

Example 1 (A three-stage discrete construction).

Let $X=\{x_0,x_1,x_2\}$.

Stage 1: An ordinary metric.

Consider the path graph $x_0⟷x_1⟷x_2$, in which both edges have weight 1. The corresponding shortest-path metric is $d(x_i,x_j)=|i-j|$ for $i,j\in \{0,1,2\}$.

The values of d are presented in the following table:

$$\begin{array}{cccc}d& x_0& x_1& x_2\\ x_0& 0& 1& 2\\ x_1& 1& 0& 1\\ x_2& 2& 1& 0\end{array}.$$

The diameter of $(X,d)$ is $D=diam\left(X\right)=2$.

The function d is an ordinary metric. In particular, $d(x_0,x_2)=d(x_0,x_1)+d(x_1,x_2)=2$.

Stage 2: A strong b-metric.

We apply Theorem 1 with $q=2$ and $\lambda ={\displaystyle \frac{1}{8}}$. Define $\rho (x,y)=d(x,y)+{\displaystyle \frac{1}{8}}d{(x,y)}^2$.

Since $D=2$, the coefficient supplied by Theorem 1 is

$$K=1+\lambda q{\left(2D\right)}^{q-1}=1+{\displaystyle \frac{1}{8}}·2·4=2.$$

Consequently, ρ is a strong b-metric with coefficient $K=2$. Its values are

$$\begin{array}{cccc}\rho & x_0& x_1& x_2\\ x_0& 0& {\displaystyle \frac{9}{8}}& {\displaystyle \frac{5}{2}}\\ x_1& {\displaystyle \frac{9}{8}}& 0& {\displaystyle \frac{9}{8}}\\ x_2& {\displaystyle \frac{5}{2}}& {\displaystyle \frac{9}{8}}& 0\end{array}.$$

This strong b-metric is not an ordinary metric. Indeed, the triangle inequality fails

$$\rho (x_0,x_2)={\displaystyle \frac{5}{2}}>{\displaystyle \frac{9}{8}}+{\displaystyle \frac{9}{8}}=\rho (x_0,x_1)+\rho (x_1,x_2).$$

On the other hand, the strong b-metric inequality with $K=2$ holds. For example,

$$\rho (x_0,x_2)={\displaystyle \frac{5}{2}}\le {\displaystyle \frac{9}{8}}+2·{\displaystyle \frac{9}{8}}=\rho (x_0,x_1)+2\rho (x_1,x_2).$$

Stage 3: A strong partial b-metric.

Let $A=\left\{x_2\right\}$ be the prescribed target set. The distances from the points of X to A are

$$\rho (x_0,A)={\displaystyle \frac{5}{2}},\rho (x_1,A)={\displaystyle \frac{9}{8}},\rho (x_2,A)=0.$$

Following Theorem 2, 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 $K=2$. Its values are

$$\begin{array}{cccc}{p}_A& x_0& x_1& x_2\\ x_0& {\displaystyle \frac{5}{2}}& {\displaystyle \frac{29}{8}}& 5\\ x_1& {\displaystyle \frac{29}{8}}& {\displaystyle \frac{9}{8}}& {\displaystyle \frac{9}{4}}\\ x_2& 5& {\displaystyle \frac{9}{4}}& 0\end{array}.$$

(2)

In particular,

$$p_A(x_0,x_0)={\displaystyle \frac{5}{2}},p_A(x_1,x_1)={\displaystyle \frac{9}{8}},p_A(x_2,x_2)=0.$$

Therefore, the self-distance of each point coincides with its remaining strong b-metric distance from the target set: $p_A(x_i,x_i)=\rho (x_i,A)$ for $i\in \{0,1,2\}$.

The target point $x_2$ has zero self-distance, whereas the intermediate points $x_0$ and $x_1$ have positive self-distances. Moreover,

$$p_A(x_0,x_0)>p_A(x_1,x_1)>p_A(x_2,x_2),$$

which reflects the successive reduction of the distance to the target state along the path

$$x_0⟶x_1⟶x_2.$$

This example is intended only to illustrate the underlying strong partial b-metric geometry and the construction of positive self-distances; it is not yet an application of the alternating-point theorem. A complete discrete model satisfying the assumptions of the main theorem is developed later in Section 4.

Remark 1.

The same construction allows the target set to contain more than one point. For example, if $A=\{x_0,x_2\}$, then $\rho (x_0,A)=\rho (x_2,A)=0$ and $\rho (x_1,A)={\displaystyle \frac{9}{8}}$.

The resulting strong partial b-metric therefore satisfies $p_A(x_0,x_0)=p_A(x_2,x_2)=0$ and $p_A(x_1,x_1)={\displaystyle \frac{9}{8}}$. Thus, the construction may describe processes with several distinct stable target states.

2.4. Interpretation of the Generalized Distance

The construction described above separates three different kinds of information: the original separation between states, the nonlinear weight assigned to transitions of different sizes, and the remaining distance from a prescribed target family. These three components are represented, respectively, by the metric d, the strong b-metric $\rho$, and the strong partial b-metric $p_A$.

The original metric $d(x,y)$ measures the elementary separation between the states x and y. In a discrete model, it may be generated by a weighted graph, in which case $d(x,y)$ represents the length of a shortest path connecting x and y. Thus, $d(x,y)$ measures the minimum total cost of passing from one state to the other through the admissible elementary transitions.

The nonlinear transformation $\rho (x,y)=d(x,y)+\lambda d{(x,y)}^q$ for $q>1$, preserves the ordering of the original distances but assigns a greater relative weight to larger transitions. Indeed, if there hollds $0\le d(x,y)<d(u,v)$, then the inequality $\rho (x,y)<\rho (u,v)$ is true.

The parameters q and $\lambda$ determine the strength of this nonlinear reweighting. The exponent $q>1$ controls how rapidly the nonlinear contribution grows with the underlying distance, whereas $\lambda >0$ controls its relative magnitude. Their choice is therefore application-dependent and may reflect how strongly larger transitions are to be penalized. In the discrete examples below, the choice $q=2$ is made for algebraic simplicity and transparent computations; it is not intended as a canonical or optimal choice.

However, because of the nonlinear term, the difference between the transformed distances may be greater than the corresponding difference between the original distances. Consequently, a transition across several levels may be assigned a greater cost than the sum of the costs of several smaller transitions.

This phenomenon explains why $\rho$ need not satisfy the ordinary triangle inequality. The coefficient K in the strong b-metric inequality controls the admissible deviation from that inequality. It should not be interpreted as an additional distance. Rather, it measures the extent to which the nonlinear geometry permits a direct transition to be more costly than a sequence of intermediate transitions.

The introduction of a nonempty target set $A\subseteq X$ adds a second level of information. The quantity $\rho (x,A)={inf}_{a\in A}\rho (x,a)$ measures the smallest remaining transition required for x to reach one of the admissible target states. The resulting strong partial b-metric

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

therefore combines two components.

The first component, $\rho (x,y)$, measures the separation between the particular states x and y.

The second component, $max\left\{\rho \right(x,A),\rho (y,A\left)\right\}$, records the larger of their remaining distances from the target family.

In particular, $p_A(x,x)=\rho (x,A)$ Thus, the self-distance $p_A(x,x)$ has a direct geometric interpretation. It measures the remaining distance from x to the target set and may be viewed as the residual amount of change needed for x to reach an admissible terminal state. A larger self-distance indicates that a larger transition is still required, whereas a smaller self-distance indicates that the state is closer to the target family. The value $p_A(x,x)$ should not be interpreted as an error in the definition of the distance or as uncertainty concerning the identity of x. Positive self-distances are an intrinsic feature of partial metric-type structures. They allow the geometry to record not only the difference between two states but also the degree to which each individual state remains separated from the prescribed target family.

If A is zero-closed with respect to $\rho$, then $p_A(x,x)=0$ if and only if $x\in A$. In this case, the points of A are precisely the states requiring no further transition. The construction does not require A to consist of a single point. Hence, several distinct states may have zero self-distance and may represent different admissible terminal outcomes.

The mutual distance $p_A(x,y)$ contains more information than the original metric distance $d(x,y)$. Two states may be close with respect to d or $\rho$ but may still have a comparatively large $p_A$-distance if at least one of them is far from the target set. Conversely, two distinct target states may both have zero self-distance, although their mutual distance remains positive.

More precisely, if $x,y\in A$, then $p_A(x,x)=p_A(y,y)=0$, while $p_A(x,y)=\rho (x,y)>0$ whenever $x\ne y$. Thus, zero self-distance characterizes admissibility as a terminal state and does not imply that all terminal states coincide.

In the discrete construction from Example 1, the target set is $A=\left\{x_2\right\}$, and $p_A(x_0,x_0)={\displaystyle \frac{5}{2}}$, $p_A(x_1,x_1)={\displaystyle \frac{9}{8}}$, and $p_A(x_2,x_2)=0$. These values describe the successive reduction of the remaining distance to the target along the path $x_0⟶x_1⟶x_2$. The self-distance therefore provides a numerical description of the progress of the states toward the prescribed target.

The same interpretation will be used in both spaces occurring in the alternating construction. In $(X,p,K_1)$, the value $p(x,x)$ measures the remaining distance of x from the target family in X, whereas in $(Y,\sigma ,K_2)$, the value $\sigma (y,y)$ plays the analogous role for the target family in Y. The alternating mappings $F:X\rightrightarrows Y$ and $G:Y\rightrightarrows X$ describe admissible transitions between the two state spaces, while the generalized distances record both the sizes of these transitions and the remaining distances from the corresponding target families.

2.5. Balls and Localization Sets

Local fixed point and alternating point principles require the contractive assumptions to hold only in a prescribed neighborhood of an initial point. In the present setting, these neighborhoods are described by balls generated by the corresponding strong partial b-metric.

Definition 2

([8,16]). Let $(X,p,K)$ be a strong partial b-metric space, let $x\in X$, and let $r>0$. The open ball with center x and radius r is defined by

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

(3)

whereas the corresponding ball defined by a non-strict inequality is

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

(4)

The term $p(x,x)$ in (3) and (4) compensates for the possibly positive self-distance of the center.

Since $p(x,x)<p(x,x)+r$, the center x belongs to both $B_p(x,r)$ and ${\overline{B}}_p(x,r)$ for every $r>0$.

If $p(x,x)=0$, then (3) and (4) reduce to $B_p(x,r)=\{y\in X:p(x,y)<r\}$ and ${\overline{B}}_p(x,r)=\{y\in X:p(x,y)\le r\}$, respectively, which have the usual form of balls in metric and strong b-metric spaces.

The notation ${\overline{B}}_p(x,r)$ indicates only that the defining inequality is non-strict. It does not assert that this set is closed with respect to a topology generated by p. In particular, the results below do not require the family of p-balls to form a base for a topology, nor do they use any topological closedness property of ${\overline{B}}_p(x,r)$. The ball is used only as a localization set for the iterative construction.

This distinction is important in strong partial b-metric spaces. The generalized triangle inequality and the positive self-distances may produce ball structures that differ substantially from those of ordinary metric spaces. In particular, a point y may be close to x relative to the self-distance of x, even when the numerical value $p(x,y)$ is not small in the ordinary sense.

For the strong partial b-metric constructed from a strong b-metric $\rho$ and a target set A, condition $y\in {\overline{B}}_{p_A}(x,r)$ is equivalent to

$$\rho (x,y)+max\left\{\rho \right(x,A),\rho (y,A\left)\right\}\le \rho (x,A)+r.$$

(5)

Thus, membership in the ball depends simultaneously on the mutual distance $\rho (x,y)$ and on the distances of x and y from the target set.

Equation (5) immediately implies $\rho (x,y)\le r$ and $\rho (y,A)\le \rho (x,A)+r$.

Consequently, a localized $p_A$-ball controls both the size of the transition from x to y and the distance of y from the target family.

We illustrate these observations by returning to Example 1.

Example 2 (Balls in the discrete strong partial b-metric).

Consider the strong partial b-metric $p_A$ on $X=\{x_0,x_1,x_2\}$ constructed in Example 1. Its values are in (2)

Let the center be $x_1$ and let $r=9/8$. Since $p_A(x_1,x_1)+r={\displaystyle \frac{9}{8}}+{\displaystyle \frac{9}{8}}={\displaystyle \frac{9}{4}}$, we obtain $B_{p_A}\left(x_1,{\displaystyle \frac{9}{8}}\right)=\left\{x_1\right\}$ and ${\overline{B}}_{p_A}\left(x_1,{\displaystyle \frac{9}{8}}\right)=\{x_1,x_2\}$.

Indeed, $p_A(x_1,x_2)={\displaystyle \frac{9}{4}}$, so $x_2$ belongs to the ball defined by the non-strict inequality but not to the open ball. On the other hand, $p_A(x_1,x_0)={\displaystyle \frac{29}{8}}>{\displaystyle \frac{9}{4}}$, and therefore $x_0$ belongs to neither of these balls.

If the center is the target point $x_2$, then $p_A(x_2,x_2)=0$. For the radius $r=9/4$, we similarly obtain $B_{p_A}\left(x_2,{\displaystyle \frac{9}{4}}\right)=\left\{x_2\right\}$ and ${\overline{B}}_{p_A}\left(x_2,{\displaystyle \frac{9}{4}}\right)=\{x_1,x_2\}$.

Thus, even though the two centers $x_1$ and $x_2$ have different self-distances, suitable choices of the radii produce the same localization set:

$${\overline{B}}_{p_A}\left(x_1,{\displaystyle \frac{9}{8}}\right)={\overline{B}}_{p_A}\left(x_2,{\displaystyle \frac{9}{4}}\right)=\{x_1,x_2\}.$$

This demonstrates that the radius of a strong partial b-metric ball must be interpreted relative to the self-distance of its center.

In the alternating framework, two different localized balls will generally be used:

$${\overline{B}}_p(x^0,r_X)\subseteq X\mathrm{and}{\overline{B}}_{\sigma }(y^0,r_Y)\subseteq Y.$$

The contractive assumptions on F and G will be imposed only on the appropriate pairs of points belonging to these sets. The localization conditions will then guarantee that all points selected during the alternating construction remain inside the corresponding balls.

2.6. Point-to-Set Distance and Excess

The alternating mappings considered below are set-valued. Therefore, besides the generalized distance between two points, we need a distance from a point to a set and a one-sided measure of the separation between two sets. We use the corresponding notions associated with a strong partial b-metric, following [8].

Definition 3

([8]). Let $(X,p,K)$ be a strong partial b-metric space, let $x\in X$, and let $A\subseteq X$ be nonempty. 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 ${\mathbb{R}}_{+}$.

Condition II implies

$$p(x,x)\le p(x,A)$$

(6)

for every $x\in X$ and every nonempty set $A\subseteq X$.

Indeed, $p(x,x)\le p(x,a)$ for every $a\in A$, and taking the infimum over $a\in A$ gives (6).

If $x\in A$, then

$$p(x,A)\le p(x,x).$$

(7)

Combining (7) with (6), we obtain

$$x\in A⟹p(x,A)=p(x,x).$$

(8)

Thus, unlike the situation in an ordinary metric space, membership of x in A does not necessarily imply $p(x,A)=0$. It implies only that the distance from x to A coincides with the self-distance of x. In particular, $x\in A$ and $p(x,x)>0$ give $p(x,A)=p(x,x)>0$.

The equality $p(x,A)=0$ is therefore a stronger condition.

By (6), it necessarily implies $p(x,x)=0$. Additional assumptions on A will be required to conclude that $x\in A$. This leads to the notion of a zero-closed set, which will be introduced in a subsequent subsection.

To compare two nonempty sets, we use the one-sided excess generated by p.

Definition 4

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

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

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

Lemma 1.

Let $A\subseteq X$ be nonempty. If $p(x,A)<\eta$ for some $\eta >0$, then there exists $a\in A$ such that $p(x,a)<\eta$.

Proof. 

This follows immediately from the definition of the infimum. □

The quantity ${\delta }_p(A,B)$ measures, in a one-sided sense, how far the elements of A are from the set B. In general, ${\delta }_p(A,B)\ne {\delta }_p(B,A)$.

This asymmetry is essential in the study of multivalued mappings, because a contractive condition usually compares the image of one point with the image of another in a prescribed direction.

For singleton sets, Definition 4 gives ${\delta }_p(\left\{x\right\},B)=p(x,B)$ and ${\delta }_p(\left\{x\right\},\left\{y\right\})=p(x,y)$.

Positive self-distances introduce an important distinction from the ordinary metric case. Even the excess of a set over itself need not vanish.

Indeed, $p(a,A)=p(a,a)$ whenever $a\in A$, in the sense of (8), and hence ${\delta }_p(A,A)={sup}_{a\in A}p(a,a)$.

Consequently, ${\delta }_p(A,A)=0$ if all elements of A have zero self-distance, but this equality may fail when A contains points with positive self-distance.

We now illustrate these notions using the discrete strong partial b-metric from Example 1.

Example 3 (Point-to-set distances and excesses in a discrete space).

Let $X=\{x_0,x_1,x_2\}$ and let $p_A$ be the strong partial b-metric whose values are in (2).

To avoid confusing the target set used in the construction of $p_A$ with the sets used in the present calculations, put $U=\{x_0,x_1\}$ and $V=\{x_1,x_2\}$.

The distance from $x_0$ to V is

$$p_A(x_0,V)=min\{p_A(x_0,x_1),p_A(x_0,x_2)\}=min\left\{{\displaystyle \frac{29}{8}},5\right\}={\displaystyle \frac{29}{8}}.$$

Similarly, $p_A(x_1,V)=min\left\{{\displaystyle \frac{9}{8}},{\displaystyle \frac{9}{4}}\right\}={\displaystyle \frac{9}{8}}$.

Therefore,

$${\delta }_{p_A}(U,V)=max\{p_A(x_0,V),p_A(x_1,V)\}=max\left\{{\displaystyle \frac{29}{8}},{\displaystyle \frac{9}{8}}\right\}={\displaystyle \frac{29}{8}}.$$

In the reverse direction,

$$p_A(x_1,U)=min\left\{{\displaystyle \frac{29}{8}},{\displaystyle \frac{9}{8}}\right\}={\displaystyle \frac{9}{8}}\mathit{and}{p}_A(x_2,U)=min\left\{5,{\displaystyle \frac{9}{4}}\right\}={\displaystyle \frac{9}{4}}.$$

Hence, ${\delta }_{p_A}(V,U)=max\left\{{\displaystyle \frac{9}{8}},{\displaystyle \frac{9}{4}}\right\}={\displaystyle \frac{9}{4}}$

Consequently, ${\delta }_{p_A}(U,V)={\displaystyle \frac{29}{8}}\ne {\displaystyle \frac{9}{4}}={\delta }_{p_A}(V,U)$, which demonstrates explicitly that the excess is not symmetric.

The self-excess of V is also positive. Indeed, $p_A(x_1,V)={\displaystyle \frac{9}{8}}$, $p_A(x_2,V)=0$, and therefore

$${\delta }_{p_A}(V,V)=max\left\{{\displaystyle \frac{9}{8}},0\right\}={\displaystyle \frac{9}{8}}.$$

Thus, even though every point of V belongs to V, the excess of V over itself does not vanish because $x_1$ has positive self-distance.

For the mappings used in the alternating construction, we shall write $F\left(x\right)\subseteq Y$ and $G\left(y\right)\subseteq X$. The corresponding excesses will be denoted by ${\delta }_{\sigma }(F\left(x\right),F\left(u\right))$ and ${\delta }_p(G\left(y\right),G\left(v\right))$. These quantities measure the one-sided separation of the values of the multivalued mappings and will appear in the contractive assumptions of the main theorem.

2.7. Zero-Closed Sets and Zero-Completeness

The iterative sequences arising in the alternating construction will have mutual distances converging to zero. Therefore, the appropriate notions of closedness and completeness are formulated in terms of zero distance rather than in terms of an unspecified topology generated by the strong partial b-metric.

We begin with the notion of a zero-closed set.

Definition 5

([3,8]). Let $(X,p,K)$ be a strong partial b-metric space. A nonempty subset $A\subseteq X$ is called zero-closed with respect to p if for every $x\in X$, the equality $p(x,A)=0$ implies that $x\in A$.

The family of all nonempty subsets of X that are zero-closed with respect to p is denoted by $C_0^p\left(X\right)$.

Zero-closedness should not be confused with topological closedness. Definition 5 is formulated entirely in terms of the point-to-set distance. No topology generated by the p-balls is required.

Since $p(x,x)=0$, provided that $p(x,A)=0$, by (6), the zero-closedness condition is relevant only for points having zero self-distance. In particular, a point with positive self-distance cannot have zero distance from any nonempty subset of X.

The following sequential characterization was established in [8].

Proposition 3

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

(i)

A is zero-closed with respect to p;

(ii)

whenever $\left\{a_n\right\}\subseteq A$ and $x\in X$ satisfy ${lim}_{n\to \infty }p(a_n,x)=0$, 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 ${lim}_{n\to \infty }p(x,a_n)=0$.

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

Proposition 3 will be used at the final stage of the alternating construction. After obtaining a limit point and proving that its distance from the appropriate value of a multivalued mapping is zero, zero-closedness allows us to conclude that the limit belongs to that value.

We next recall the notions of zero-Cauchy sequence and zero-completeness.

Definition 6

([3,8]). Let $(X,p,K)$ be a strong partial b-metric space. A sequence $\left\{x_n\right\}\subseteq X$ is called a zero-Cauchy sequence if

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

(9)

Every zero-Cauchy sequence satisfies ${lim}_{n\to \infty }p(x_n,x_n)=0$.

Indeed, condition (II) gives $0\le p(x_n,x_n)\le p(x_n,x_m)$, and the conclusion follows from (9).

Thus, although a strong partial b-metric space may contain points with positive self-distance, a zero-Cauchy sequence must asymptotically approach the part of the space on which the self-distance vanishes.

Definition 7

([3,8]). A sequence $\left\{x_n\right\}\subseteq X$ is said to be a zero-convergent to $x\in X$ if there holds

$$p(x_n,x)⟶0.$$

(10)

In this case, we write $x_n\stackrel{0}{\to }x$.

If $x_n\stackrel{0}{\to }x$, then $p(x,x)=0$. This follows from $0\le p(x,x)\le p(x,x_n)=p(x_n,x)$ and (10).

Definition 8

([3,8]). A strong partial b-metric space $(X,p,K)$ is called zero-complete if every zero-Cauchy sequence $\left\{x_n\right\}\subseteq X$ zero-converges to some $x\in X$ such that

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

(11)

Zero-completeness is precisely the completeness property needed below. The alternating construction will produce two sequences, one in X and one in Y. The contractive estimates and the summability of the corresponding gauge functions will show that both sequences are zero-Cauchy. Zero-completeness will then provide their respective limits.

Ordinary completeness is generally stronger than the property required for this argument. The proof does not need the convergence of every Cauchy sequence in a more general partial metric sense; it uses only the convergence of sequences whose mutual distances tend to zero.

We now return to the discrete model.

Example 4 (Zero-closedness and zero-completeness of the discrete model).

Consider the space $X=\{x_0,x_1,x_2\}$ equipped with the strong partial b-metric (2) $p_A$ from Example 1:

The only zero value in this table is $p_A(x_2,x_2)=0$.

Consequently, for every nonempty subset $B\subseteq X$, the equality $p_A(x,B)=0$ can hold only when there hold simultaneously $x=x_2$ and $x_2\in B$.

Therefore, if $p_A(x,B)=0$, then $x\in B$, and every nonempty subset of X is zero-closed. Hence,

$$C_0^{p_A}\left(X\right)=\mathcal{P}\left(X\right)\setminus \{⌀\}.$$

Let $\left\{z_n\right\}\subseteq X$ be a zero-Cauchy sequence. The smallest positive value of $p_A$ is ${\displaystyle \frac{9}{8}}$. Choose any $0<\epsilon <{\displaystyle \frac{9}{8}}$.

Since $\left\{z_n\right\}$ is zero-Cauchy, there exists $N\in \mathbb{N}$ such that the inequality $p_A(z_m,z_n)<\epsilon$ holds for all $m,n\ge N$.

The equation (2) shows that this inequality is possible only if $z_m=z_n=x_2$.

Thus, $z_n=x_2$ for every $n\ge N$ and consequently it follows that

$$\underset{n\to \infty }{lim}{p}_A(z_n,x_2)=0\mathit{and}{p}_A(x_2,x_2)=0.$$

Therefore, every zero-Cauchy sequence zero-converges to $x_2$, and $(X,p_A,2)$ is zero-complete.

The same notation will be used for the second space:

$$C_0^{\sigma }\left(Y\right)=\{B\subseteq Y:B\ne ⌀\mathrm{and}B\mathrm{is}\mathrm{zero}\text{-}\mathrm{closed}\mathrm{with}\mathrm{respect}\mathrm{to}\sigma \}.$$

Accordingly, the multivalued mappings in the main theorem will have the form

$$F:X⟶C_0^{\sigma }\left(Y\right)\mathrm{and}G:Y⟶C_0^p\left(X\right).$$

The zero-closedness of their values will permit the passage from zero-distance relations to the membership relations required for an alternating point.

2.8. Bianchini–Grandolfi Gauge Functions

The contractive conditions used in the alternating construction are expressed through nonlinear control functions. Instead of requiring a fixed linear contraction coefficient, we use Bianchini–Grandolfi gauge functions. Their defining property guarantees not only that the successive distances tend to zero but also that the series of these distances is convergent. In the original definition, the function is required to be only nondecreasing; here we impose the stronger requirement that it be strictly increasing, as this property is needed in the proof of the main theorem.

Definition 9

([6]). Let $J\subseteq {\mathbb{R}}_{+}$ be an interval containing 0. A strictly increasing function

$$\phi :J⟶J$$

that is continuous at 0 is called a Bianchini–Grandolfi gauge function if

$$S_{\phi }\left(t\right):={\displaystyle \sum _{n=0}^{\infty }}{\phi }^n\left(t\right)<\infty$$

(12)

for every $t\in J$, where ${\phi }^0\left(t\right)=t$ and ${\phi }^{n+1}\left(t\right)=\phi \left({\phi }^n\left(t\right)\right)$ for every $n\ge 0$.

The function $S_{\phi }$ will be called the sum function associated with $\phi$. By separating the first term of the series, we obtain

$$S_{\phi }\left(t\right)=t+{\displaystyle \sum _{n=1}^{\infty }}{\phi }^n\left(t\right).$$

The summability condition imposes several important restrictions on $\phi$. We recall the standard consequences used below.

Proposition 4

([6,8]). Let $\phi :J\to J$ be a Bianchini–Grandolfi gauge function. Then:

(i)

$\phi \left(0\right)=0$;

(ii)

$\phi \left(t\right)<t$ for every $t\in J\setminus \left\{0\right\}$;

(iii)

${lim}_{n\to \infty }{\phi }^n\left(t\right)=0$ for every $t\in J$;

(iv)

${lim}_{m\to \infty }{\sum }_{n=m}^{\infty }{\phi }^n\left(t\right)=0$ for every $t\in J$.

The inequality $\phi \left(t\right)<t$ shows that every application of the gauge function decreases a positive distance. The stronger summability condition in (12) guarantees that the total accumulated distance remains finite. This property is essential for proving that the sequences generated by the multivalued mappings are zero-Cauchy.

If $K\ge 1$, then

$$K{S}_{\phi }\left(t\right)-(K-1)t=t+K{\displaystyle \sum _{n=1}^{\infty }}{\phi }^n\left(t\right).$$

(13)

The right-hand side of (13) naturally appears when the chain estimate for a strong partial b-metric is applied to a sequence of successive transitions. The first transition is counted with coefficient 1, whereas the remaining transitions are multiplied by the strong partial b-metric coefficient K.

The following elementary selection result will be useful when two different gauge functions are used for the two directions of the alternating process.

Proposition 5.

Let $\phi ,\psi :J\to J$ be Bianchini–Grandolfi gauge functions and define $\theta =\psi \circ \phi$. Then $\theta :J\to J$ is also a Bianchini–Grandolfi gauge function.

Moreover, there hold

• $\theta \left(t\right)<\phi \left(t\right)<t$ for every $t\in J\setminus \left\{0\right\}$
• ${\theta }^n\left(t\right)\le {\phi }^n\left(t\right)$ for every $n\ge 0$
• $S_{\theta }\left(t\right)\le S_{\phi }\left(t\right)<\infty$.

In the alternating framework, $\phi$ will control transitions from X to Y, whereas $\psi$ will control transitions from Y back to X. A complete alternating step is therefore controlled by the composite gauge $\theta =\psi \circ \phi$. The iterates of $\theta$ describe the reduction of the distances between successive points belonging to the same space.

The simplest class of Bianchini–Grandolfi gauge functions is given by linear contractions.

Example 5 (Linear gauge functions).

Let $c\in (0,1)$ and define $\phi \left(t\right)=ct$ for $t\in {\mathbb{R}}_{+}$.

Then ${\phi }^n\left(t\right)=c^{n}t$ and $S_{\phi }\left(t\right)={\sum }_{n=0}^{\infty }{c}^{n}t={\displaystyle \frac{t}{1-c}}$.

Therefore, every linear contraction with coefficient $c\in (0,1)$ is a Bianchini–Grandolfi gauge function.

In particular, if $\phi \left(t\right)={\displaystyle \frac{1}{2}}t$, then ${\phi }^n\left(t\right)={\displaystyle \frac{t}{2^n}}$ and $S_{\phi }\left(t\right)=2t$.

For a strong partial b-metric with coefficient $K=2$, the localization expression becomes $2S_{\phi }\left(t\right)-(2-1)t=4t-t=3t$.

Equivalently, $t+2{\sum }_{n=1}^{\infty }{\displaystyle \frac{t}{2^n}}=t+2t=3t$.

The two directions of the alternating construction may be controlled by different linear gauges.

Example 6 (Two directional gauge functions).

Let $\phi \left(t\right)={\displaystyle \frac{1}{2}}t$ and $\psi \left(t\right)={\displaystyle \frac{1}{3}}t$. Then both φ and ψ are Bianchini–Grandolfi gauge functions. Their composition is $\theta \left(t\right)=(\psi \circ \phi )\left(t\right)={\displaystyle \frac{1}{6}}t$.

Consequently, ${\theta }^n\left(t\right)={\displaystyle \frac{t}{6^n}}$ and $S_{\theta }\left(t\right)={\sum }_{n=0}^{\infty }{\displaystyle \frac{t}{6^n}}={\displaystyle \frac{6}{5}}t$.

Thus, although the transition from X to Y reduces a distance by the factor $1/2$ and the transition from Y to X reduces it by the factor $1/3$, a complete alternating cycle reduces the corresponding distance by the factor $1/6$.

More generally, if $\phi \left(t\right)=ct$ and $\psi \left(t\right)=dt$, where $c,d\in (0,1)$, then $\theta \left(t\right)=cdt$ and $S_{\theta }\left(t\right)={\displaystyle \frac{t}{1-cd}}$. The usual linear contractive conditions are therefore included as a special case of the nonlinear gauge framework.

2.9. The Chain Estimate

The proof of the main result requires estimates for the generalized distance between two nonconsecutive terms of an iterative sequence. Because the ordinary triangle inequality is not available, these estimates are obtained by repeatedly applying the strong partial b-metric inequality along a finite chain.

The following result was established in [8].

Lemma 2

(Chain estimate [8]). Let $(X,p,K)$ be a strong partial b-metric space and let

$$x_m,x_{m+1},\dots ,x_n\in X,n>m.$$

Then there holds the inequality

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

(14)

Consequently,

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

(15)

The negative self-distance terms in (14) may provide a sharper estimate. However, the simpler form (15) is sufficient for proving that the iterative sequences constructed below are zero-Cauchy.

Owing to the equivalent orientation of the strong partial b-metric inequality established in Proposition 1, one also obtains the reverse chain estimate

$$p(x_m,x_n)\le K{\displaystyle \sum _{j=m}^{n-2}}p(x_j,x_{j+1})+p(x_{n-1},x_n)-{\displaystyle \sum _{j=m+1}^{n-1}}p(x_j,x_j).$$

(16)

Combining (14) and (16) gives

$$\begin{array}{cc}\hfill p(x_m,x_n)& \le min\left\{p(x_m,x_{m+1})+K{\displaystyle \sum _{j=m+1}^{n-1}}p(x_j,x_{j+1}),K{\displaystyle \sum _{j=m}^{n-2}}p(x_j,x_{j+1})+p(x_{n-1},x_n)\right\}\hfill \\ & -{\displaystyle \sum _{j=m+1}^{n-1}}p(x_j,x_j).\hfill \end{array}$$

Thus, the orientation yielding the smaller bound may be selected in a particular application. In the alternating construction, we shall mainly use (14), because the first transition is naturally separated from the remaining terms of the chain.

The chain estimate interacts directly with the summability property of a Bianchini–Grandolfi gauge function.

Corollary 2.

Let $(X,p,K)$ be a strong partial b-metric space, let $\phi :J\to J$ be a Bianchini–Grandolfi gauge function, and let $\left\{x_n\right\}\subseteq X$ satisfy $p(x_j,x_{j+1})\le {\phi }^j\left(\alpha \right)$ for some $\alpha \in J$ and every $j\ge 0$.

Then, for every $n>m$ there holds the inequality

$$p(x_m,x_n)\le {\phi }^m\left(\alpha \right)+K{\displaystyle \sum _{j=m+1}^{n-1}}{\phi }^j\left(\alpha \right).$$

(17)

In particular,

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

(18)

Consequently, $\left\{x_n\right\}$ is a zero-Cauchy sequence.

Indeed, both terms on the right-hand side of (18) tend to zero as $m\to \infty$.

More precisely, there holds ${lim}_{m\to \infty }{\phi }^m\left(\alpha \right)=0$ and ${lim}_{m\to \infty }{\sum }_{j=m+1}^{\infty }{\phi }^j\left(\alpha \right)=0$ by Proposition 4.

For $m=0$, estimate (17) leads to

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

Using (13), we obtain $p(x_0,x_n)\le K{S}_{\phi }\left(\alpha \right)-(K-1)\alpha$.

Therefore, if there holds the inequality $K{S}_{\phi }\left(\alpha \right)-(K-1)\alpha \le p(x_0,x_0)+r$, then there holds the inclusion $x_n\in {\overline{B}}_p(x_0,r)$ for every $n\ge 1$. This explains the form of the localization conditions used in the main theorem.

We conclude with a calculation in the discrete model.

Example 7 (The chain estimate in the discrete model).

Consider the chain $x_0,x_1,x_2$ in the strong partial b-metric space $(X,p_A,2)$ from Example 1. We have

$$p_A(x_0,x_1)={\displaystyle \frac{29}{8}},p_A(x_1,x_2)={\displaystyle \frac{9}{4}},p_A(x_1,x_1)={\displaystyle \frac{9}{8}}{p}_A(x_0,x_2)=5.$$

Applying (14), we obtain

$$5=p_A(x_0,x_2)\le p_A(x_0,x_1)+2p_A(x_1,x_2)-p_A(x_1,x_1)={\displaystyle \frac{29}{8}}+2·{\displaystyle \frac{9}{4}}-{\displaystyle \frac{9}{8}}=7.$$

If the negative self-distance term is omitted, the weaker estimate is

$$p_A(x_0,x_2)\le p_A(x_0,x_1)+2p_A(x_1,x_2)={\displaystyle \frac{65}{8}}.$$

This calculation shows that retaining the self-distance correction may substantially sharpen the chain estimate.

In the proof of the main result, the chain estimate will be applied separately in $(X,p,K_1)$ and $(Y,\sigma ,K_2)$. The corresponding gauge-controlled bounds will show that the two sequences generated by the alternating mappings are zero-Cauchy and remain inside their prescribed localization sets.

3. Main Results

Definition 10.

Let $F:X\rightrightarrows Y$ and $G:Y\rightrightarrows X$ be set-valued mappings. A pair $(x^{*},y^{*})\in X\times Y$ is called an alternating point of F and G if $y^{*}\in F\left(x^{*}\right)$ and $x^{*}\in G\left(y^{*}\right)$.

In the proof of the main result, we use the chain estimate from Lemma 2, applied separately in the spaces $(X,p,K_1)$ and $(Y,\sigma ,K_2)$.

Remark 2.

Every alternating point $(x^{*},y^{*})$ of F and G generates a pair of fixed points of the set-valued compositions $G\circ F$ and $F\circ G$. Indeed,

$$x^{*}\in G\left(y^{*}\right)\mathit{and}{y}^{*}\in F\left(x^{*}\right)$$

imply

$$x^{*}\in \bigcup _{y\in F\left(x^{*}\right)}G\left(y\right)=(G\circ F)\left(x^{*}\right)\mathit{and}{y}^{*}\in \bigcup _{x\in G\left(y^{*}\right)}F\left(x\right)=(F\circ G)\left(y^{*}\right).$$

Thus, $x^{*}$ is a fixed point of $G\circ F$, while $y^{*}$ is a fixed point of $F\circ G$.

Conversely, if $x^{*}\in (G\circ F)\left(x^{*}\right)$, then there exists $y^{*}\in F\left(x^{*}\right)$ such that $x^{*}\in G\left(y^{*}\right)$. Hence, $(x^{*},y^{*})$ is an alternating point of F and G. An analogous statement holds for $F\circ G$.

Theorem 3.

Let $(X,p,K_1)$ and $(Y,\sigma ,K_2)$ be 0-complete strong partial b-metric spaces, and let us put $K=max\{K_1,K_2\}$. Let $F:X\rightrightarrows C_0^{\sigma }\left(Y\right)$ and $G:Y\rightrightarrows C_0^p\left(X\right)$ be set-valued mappings. Let $J\subseteq [0,\infty )$ be an interval containing 0, and let $\phi ,\psi :J\to J$ be strictly increasing functions, continuous at 0, such that $\phi \left(0\right)=\psi \left(0\right)=0,$ and $\phi \left(t\right)\le t$ for $t\in J$. Suppose that $\theta =\psi \circ \phi$ is a Bianchini–Grandolfi gauge function on J.

Let there exist $\overline{x}\in X$, $\overline{y}\in Y$, $r_X>0$, $r_Y>0$, and $\alpha \in J\setminus \left\{0\right\}$ such that the following assumptions are satisfied:

• $\overline{y}\in F\left(\overline{x}\right)$
• $p(\overline{x},G\left(\overline{y}\right))<\alpha$
• $K{S}_{\theta }\left(\alpha \right)-(K-1)\alpha \le min\left\{p(\overline{x},\overline{x})+r_X,\sigma (\overline{y},\overline{y})+r_Y\right\}$
• ${\delta }_{\sigma }\left(F\left(x_1\right)\cap {\overline{B}}_{\sigma }(\overline{y},r_Y),F\left(x_2\right)\right)\le \phi \left(p(x_1,x_2)\right)$ for all $x_1,x_2\in {\overline{B}}_p(\overline{x},r_X)$ such that $p(x_1,x_2)\in J$ and $F\left(x_1\right)\cap {\overline{B}}_{\sigma }(\overline{y},r_Y)\ne ⌀$
• ${\delta }_p\left(G\left(y_1\right)\cap {\overline{B}}_p(\overline{x},r_X),G\left(y_2\right)\right)\le \psi \left(\sigma (y_1,y_2)\right)$ for all $y_1,y_2\in {\overline{B}}_{\sigma }(\overline{y},r_Y)$ such that $\sigma (y_1,y_2)\in J$ and $G\left(y_1\right)\cap {\overline{B}}_p(\overline{x},r_X)\ne ⌀.$

Then F and G have an alternating point $(x^{*},y^{*})\in {\overline{B}}_p(\overline{x},r_X)\times {\overline{B}}_{\sigma }(\overline{y},r_Y)$, that is, $y^{*}\in F\left(x^{*}\right)$ and $x^{*}\in G\left(y^{*}\right)$.

Consequently, $x^{*}\in (G\circ F)\left(x^{*}\right)$ and $y^{*}\in (F\circ G)\left(y^{*}\right)$.

We emphasize that $\phi$ and $\psi$ are not required to be Bianchini–Grandolfi gauge functions separately. The summability assumption is imposed only on the composite function $\theta =\psi \circ \phi$, which controls one complete alternating step. The condition $\phi \left(t\right)\le t$ is used to localize the sequence in Y; no analogous condition on $\psi$ is needed because the successive transitions in X are controlled by $\theta$.

Proof. 

Set $x^0=\overline{x}$ and $y^0=\overline{y}$. By assumption, $y^0\in F\left(x^0\right)$. Moreover, $y^0\in {\overline{B}}_{\sigma }(\overline{y},r_Y)$. Since $p(x^0,G\left(y^0\right))<\alpha$, Lemma 1 implies that there exists $x^1\in G\left(y^0\right)$ such that $p(x^0,x^1)<\alpha$. Since $S_{\theta }\left(\alpha \right)=\alpha +{\sum }_{n=1}^{\infty }{\theta }^n\left(\alpha \right)$, we have $K{S}_{\theta }\left(\alpha \right)-(K-1)\alpha =\alpha +K{\sum }_{n=1}^{\infty }{\theta }^n\left(\alpha \right)\ge \alpha .$ Consequently, there holds $p(\overline{x},x^1)<\alpha \le K{S}_{\theta }\left(\alpha \right)-(K-1)\alpha \le p(\overline{x},\overline{x})+r_X$ and hence, $x^1\in G\left(y^0\right)\cap {\overline{B}}_p(\overline{x},r_X)$.

Since $0\le p(x^0,x^1)<\alpha$, $\alpha \in J$, and J is an interval containing 0, it follows that

$$p(x^0,x^1)\in J.$$

Moreover, $x^0,x^1\in {\overline{B}}_p(\overline{x},r_X)$ and $y^0\in F\left(x^0\right)\cap {\overline{B}}_{\sigma }(\overline{y},r_Y)$. Thus, all assumptions needed to apply the contractive condition for F are satisfied. The contractive condition for F gives the chain of inequalities

$$\sigma (y^0,F\left(x^1\right))\le {\delta }_{\sigma }\left(F\left(x^0\right)\cap {\overline{B}}_{\sigma }(\overline{y},r_Y),F\left(x^1\right)\right)\le \phi \left(p(x^0,x^1)\right)<\phi \left(\alpha \right).$$

By Lemma 1, there exists $y^1\in F\left(x^1\right)$ such that $\sigma (y^0,y^1)<\phi \left(\alpha \right)\le \alpha$. Indeed,

$$\sigma (\overline{y},y^1)<\phi \left(\alpha \right)\le \alpha \le K{S}_{\theta }\left(\alpha \right)-(K-1)\alpha \le \sigma (\overline{y},\overline{y})+r_Y.$$

Consequently, $y^1\in F\left(x^1\right)\cap {\overline{B}}_{\sigma }(\overline{y},r_Y)$.

We now construct the two sequences inductively. Suppose that, for some $n\ge 0$,

$$x^{n+1}\in G\left(y^n\right)\cap {\overline{B}}_p(\overline{x},r_X),$$

$$y^{n+1}\in F\left(x^{n+1}\right)\cap {\overline{B}}_{\sigma }(\overline{y},r_Y),$$

and

$$p(x^n,x^{n+1})<{\theta }^n\left(\alpha \right),$$

$$\sigma (y^n,y^{n+1})<\phi \left({\theta }^n\left(\alpha \right)\right)\le {\theta }^n\left(\alpha \right).$$

Since ${\theta }^n\left(\alpha \right)\in J$, $\phi \left({\theta }^n\left(\alpha \right)\right)\in J$, and J is an interval containing 0, the preceding inequalities imply $p(x^n,x^{n+1})\in J$ and $\sigma (y^n,y^{n+1})\in J$. Together with the displayed membership relations, this verifies all assumptions needed to apply the contractive condition for G.

From the contractive condition for G, we obtain

$$\begin{array}{ccc}p(x^{n+1},G\left(y^{n+1}\right))\hfill & \le \hfill & {\delta }_p\left(G\left(y^n\right)\cap {\overline{B}}_p(\overline{x},r_X),G\left(y^{n+1}\right)\right)\le \psi \left(\sigma (y^n,y^{n+1})\right)\hfill \\ & <\hfill & \psi \left(\phi \left({\theta }^n\left(\alpha \right)\right)\right)={\theta }^{n+1}\left(\alpha \right).\hfill \end{array}$$

By Lemma 1, there exists $x^{n+2}\in G\left(y^{n+1}\right)$ such that $p(x^{n+1},x^{n+2})<{\theta }^{n+1}\left(\alpha \right)$. By Lemma 2 and $K_1\le K$, we obtain

$$\begin{array}{cc}\hfill p(\overline{x},x^{n+2})& \le p(\overline{x},x^1)+K{\displaystyle \sum _{j=1}^{n+1}}p(x^j,x^{j+1})<\alpha +K{\displaystyle \sum _{j=1}^{n+1}}{\theta }^j\left(\alpha \right)\hfill \\ & \le K{S}_{\theta }\left(\alpha \right)-(K-1)\alpha \le p(\overline{x},\overline{x})+r_X.\hfill \end{array}$$

Thus, $x^{n+2}\in G\left(y^{n+1}\right)\cap {\overline{B}}_p(\overline{x},r_X)$. Since $0\le p(x^{n+1},x^{n+2})<{\theta }^{n+1}\left(\alpha \right)$, ${\theta }^{n+1}\left(\alpha \right)\in J$, and J is an interval containing 0, we have $p(x^{n+1},x^{n+2})\in J$.

Furthermore, $y^{n+1}\in F\left(x^{n+1}\right)\cap {\overline{B}}_{\sigma }(\overline{y},r_Y)$ and thus, all assumptions needed to apply the contractive condition for F are satisfied.

Next, the contractive condition for F gives

$$\begin{array}{cc}\hfill \sigma (y^{n+1},F\left(x^{n+2}\right))& \le {\delta }_{\sigma }\left(F\left(x^{n+1}\right)\cap {\overline{B}}_{\sigma }(\overline{y},r_Y),F\left(x^{n+2}\right)\right)\hfill \\ & \le \phi \left(p(x^{n+1},x^{n+2})\right)<\phi \left({\theta }^{n+1}\left(\alpha \right)\right).\hfill \end{array}$$

By Lemma 1, there exists $y^{n+2}\in F\left(x^{n+2}\right)$ such that

$$\sigma (y^{n+1},y^{n+2})<\phi \left({\theta }^{n+1}\left(\alpha \right)\right)\le {\theta }^{n+1}\left(\alpha \right).$$

Furthermore, Lemma 2 and $K_2\le K$ give

$$\begin{array}{cc}\hfill \sigma (\overline{y},y^{n+2})& \le \sigma (\overline{y},y^1)+K{\displaystyle \sum _{j=1}^{n+1}}\sigma (y^j,y^{j+1})<\phi \left(\alpha \right)+K{\displaystyle \sum _{j=1}^{n+1}}\phi \left({\theta }^j\left(\alpha \right)\right)\hfill \\ & \le \alpha +K{\displaystyle \sum _{j=1}^{n+1}}{\theta }^j\left(\alpha \right)\le K{S}_{\theta }\left(\alpha \right)-(K-1)\alpha \le \sigma (\overline{y},\overline{y})+r_Y.\hfill \end{array}$$

Hence, $y^{n+2}\in F\left(x^{n+2}\right)\cap {\overline{B}}_{\sigma }(\overline{y},r_Y)$, and therefore, the induction is complete.

Let $n>m$. By Lemma 2 and $K_1\le K$, we obtain

$$p(x^m,x^n)\le p(x^m,x^{m+1})+K{\displaystyle \sum _{j=m+1}^{n-1}}p(x^j,x^{j+1})<K{\displaystyle \sum _{j=m}^{\infty }}{\theta }^j\left(\alpha \right).$$

By the properties recorded after the definition of a Bianchini–Grandolfi gauge function, the last tail tends to 0 as $m\to \infty$ and consequently, ${lim}_{m,n\to \infty }p(x^m,x^n)=0$. Thus, $\left\{x^n\right\}$ is 0-Cauchy. Similarly, from the inequality

$$\sigma (y^m,y^n)\le \sigma (y^m,y^{m+1})+K{\displaystyle \sum _{j=m+1}^{n-1}}\sigma (y^j,y^{j+1})<K{\displaystyle \sum _{j=m}^{\infty }}{\theta }^j\left(\alpha \right)$$

it follows ${lim}_{m,n\to \infty }\sigma (y^m,y^n)=0$ and thus, $\left\{y^n\right\}$ is also 0-Cauchy.

By 0-completeness, there exist $x^{*}\in X$ and $y^{*}\in Y$ such that

$$p(x^{*},x^{*})=\underset{n\to \infty }{lim}p(x^n,x^{*})=0$$

and

$$\sigma (y^{*},y^{*})=\underset{n\to \infty }{lim}\sigma (y^n,y^{*})=0.$$

Using condition IV, we obtain

$$p(\overline{x},x^{*})\le p(\overline{x},x^n)+K_{1}p(x^n,x^{*})-p(x^n,x^n).$$

Since $x^n\in {\overline{B}}_p(\overline{x},r_X)$ and $p(x^n,x^n)\ge 0$, the preceding inequality gives

$$p(\overline{x},x^{*})\le p(\overline{x},\overline{x})+r_X+K_{1}p(x^n,x^{*}).$$

Letting $n\to \infty$, we obtain $p(\overline{x},x^{*})\le p(\overline{x},\overline{x})+r_X$. Therefore, $x^{*}\in {\overline{B}}_p(\overline{x},r_X)$.

Analogously, since $y^n\in {\overline{B}}_{\sigma }(\overline{y},r_Y)$, we have

$$\sigma (\overline{y},y^{*})\le \sigma (\overline{y},\overline{y})+r_Y+K_2\sigma (y^n,y^{*}).$$

Letting $n\to \infty$, we obtain $\sigma (\overline{y},y^{*})\le \sigma (\overline{y},\overline{y})+r_Y$, and consequently $y^{*}\in {\overline{B}}_{\sigma }(\overline{y},r_Y)$.

For all sufficiently large n, we have $\sigma (y^n,y^{*})\in J$. Indeed, $\sigma (y^n,y^{*})\to 0$, while J is an interval containing both 0 and the positive number $\alpha$; hence $[0,\alpha ]\subseteq J$, and eventually $\sigma (y^n,y^{*})<\alpha$. Since $x^{n+1}\in G\left(y^n\right)\cap {\overline{B}}_p(\overline{x},r_X)$, the contractive condition for G gives $p(x^{n+1},G\left(y^{*}\right))\le \psi \left(\sigma (y^n,y^{*})\right)$. For every $u\in G\left(y^{*}\right)$, condition (IV) gives

$$p(x^{*},u)\le p(x^{*},x^{n+1})+K_{1}p(x^{n+1},u)-p(x^{n+1},x^{n+1}).$$

Taking the infimum over $u\in G\left(y^{*}\right)$, we obtain

$$\begin{array}{cc}\hfill p(x^{*},G\left(y^{*}\right))& \le p(x^{*},x^{n+1})+K_{1}p(x^{n+1},G\left(y^{*}\right))-p(x^{n+1},x^{n+1})\hfill \\ & \le p(x^{*},x^{n+1})+K_1\psi \left(\sigma (y^n,y^{*})\right).\hfill \end{array}$$

Since $\sigma (y^n,y^{*})⟶0$, $\psi$ is continuous at 0, and $\psi \left(0\right)=0$, we have $\psi \left(\sigma (y^n,y^{*})\right)⟶0$. Letting $n\to \infty$ in the preceding inequality, we obtain $p(x^{*},G\left(y^{*}\right))=0$. On the other hand, the 0-completeness gives $p(x^{*},x^{*})=0$. Thus, $p(x^{*},G\left(y^{*}\right))=p(x^{*},x^{*})$. Since $G\left(y^{*}\right)$ is 0-closed with respect to p, there holds $x^{*}\in G\left(y^{*}\right)$.

Similarly from ${lim}_{n\to \infty }p(x^{n+1},x^{*})=0$ and the same interval argument it follows that for all sufficiently large n, there holds $p(x^{n+1},x^{*})\in J$. Since $y^{n+1}\in F\left(x^{n+1}\right)\cap {\overline{B}}_{\sigma }(\overline{y},r_Y)$, we have $\sigma (y^{n+1},F\left(x^{*}\right))\le \phi \left(p(x^{n+1},x^{*})\right)$. For every $v\in F\left(x^{*}\right)$, condition (IV) gives

$$\sigma (y^{*},v)\le \sigma (y^{*},y^{n+1})+K_2\sigma (y^{n+1},v)-\sigma (y^{n+1},y^{n+1}).$$

Taking the infimum over $v\in F\left(x^{*}\right)$, we obtain

$$\begin{array}{cc}\hfill \sigma (y^{*},F\left(x^{*}\right))& \le \sigma (y^{*},y^{n+1})+K_2\sigma (y^{n+1},F\left(x^{*}\right))-\sigma (y^{n+1},y^{n+1})\hfill \\ & \le \sigma (y^{*},y^{n+1})+K_2\phi \left(p(x^{n+1},x^{*})\right).\hfill \end{array}$$

Since $p(x^{n+1},x^{*})\to 0$, $\phi$ is continuous at 0, and $\phi \left(0\right)=0$, we get $\phi \left(p(x^{n+1},x^{*})\right)\to 0$. Passing to the limit in the preceding inequality gives $\sigma (y^{*},F\left(x^{*}\right))=0$. Since $\sigma (y^{*},y^{*})=0$, it follows that $\sigma (y^{*},F\left(x^{*}\right))=\sigma (y^{*},y^{*})$. Since $F\left(x^{*}\right)$ is 0-closed with respect to $\sigma$, $y^{*}\in F\left(x^{*}\right)$.

Therefore, $y^{*}\in F\left(x^{*}\right)$ and $x^{*}\in G\left(y^{*}\right)$, so $(x^{*},y^{*})$ is an alternating point of F and G. The conclusion $x^{*}\in (G\circ F)\left(x^{*}\right)$ and $y^{*}\in (F\circ G)\left(y^{*}\right)$, concerning the compositions follows from Remark 2. This completes the proof. □

The conclusion of Theorem 3 is an existence statement obtained by constructing one admissible alternating selection whose two component sequences converge to a compatible pair. It does not, in general, assert invariance of whole families of states or uniqueness of the resulting alternating point.

Corollary 3.

Under the assumptions of Theorem 3, suppose that $F:X\to Y$ and $G:Y\to X$ are single-valued mappings, regarded as set-valued mappings with singleton values $\left\{F\right(x\left)\right\}$ and $\left\{G\right(y\left)\right\}$, respectively, and $K_{1}p(\overline{x},\overline{x})+(K_1+1)r_X\in J.$

Then F and G have a unique alternating point $(x^{*},y^{*})\in {\overline{B}}_p(\overline{x},r_X)\times {\overline{B}}_{\sigma }(\overline{y},r_Y)$.

In particular, $x^{*}=(G\circ F)\left(x^{*}\right)$ and $y^{*}=(F\circ G)\left(y^{*}\right)$.

If, in addition, $F\left({\overline{B}}_p(\overline{x},r_X)\right)\subseteq {\overline{B}}_{\sigma }(\overline{y},r_Y)$ and $G\left({\overline{B}}_{\sigma }(\overline{y},r_Y)\right)\subseteq {\overline{B}}_p(\overline{x},r_X)$, then $x^{*}$ is the unique fixed point of $G\circ F$ in ${\overline{B}}_p(\overline{x},r_X)$, and $y^{*}$ is the unique fixed point of $F\circ G$ in ${\overline{B}}_{\sigma }(\overline{y},r_Y)$.

Proof. 

Existence follows directly from Theorem 3.

Suppose that $(x,y)$ and $(x^{*},y^{*})$ are two alternating points in ${\overline{B}}_p(\overline{x},r_X)\times {\overline{B}}_{\sigma }(\overline{y},r_Y)$.

Then $(y,x)=\left(F\right(x),G(y\left)\right)$ and $(y^{*},x^{*})=(F\left(x^{*}\right),G\left(y^{*}\right))$.

By (IV),

$$\begin{array}{cc}\hfill p(x,x^{*})& \le p(x,\overline{x})+K_{1}p(\overline{x},x^{*})-p(\overline{x},\overline{x})\hfill \\ & \le p(\overline{x},\overline{x})+r_X+K_1\left(p(\overline{x},\overline{x})+r_X\right)-p(\overline{x},\overline{x})\hfill \\ & =K_{1}p(\overline{x},\overline{x})+(K_1+1)r_X.\hfill \end{array}$$

Therefore, $p(x,x^{*})\in J$.

Suppose that $x\ne x^{*}$. Then $p(x,x^{*})>0$. By the contractive condition for F, we can write $\sigma (y,y^{*})=\sigma (F\left(x\right),F\left(x^{*}\right))\le \phi \left(p(x,x^{*})\right)$. Since J is an interval containing 0, it follows that $\sigma (y,y^{*})\in J$. The contractive condition for G now yields

$$p(x,x^{*})=p(G\left(y\right),G\left(y^{*}\right))\le \psi \left(\sigma (y,y^{*})\right)\le \psi \left(\phi \left(p(x,x^{*})\right)\right)=\theta \left(p(x,x^{*})\right).$$

By the properties recorded after the definition of a Bianchini–Grandolfi gauge function, $\theta \left(t\right)<t$ for every $t>0$. Hence, $p(x,x^{*})\le \theta \left(p(x,x^{*})\right)<p(x,x^{*})$, which is impossible. Therefore, $x=x^{*}$.

Since F is single-valued, $y=F\left(x\right)=F\left(x^{*}\right)=y^{*}$. Thus, the alternating point is unique.

Finally, assume the two invariance conditions. If $x\in {\overline{B}}_p(\overline{x},r_X)$ is a fixed point of $G\circ F$, then $x=G\left(F\right(x\left)\right)$. By the invariance of F, there holds $F\left(x\right)\in {\overline{B}}_{\sigma }(\overline{y},r_Y)$.

Therefore, $(x,F\left(x\right))\in {\overline{B}}_p(\overline{x},r_X)\times {\overline{B}}_{\sigma }(\overline{y},r_Y)$ is an alternating point. Its uniqueness gives $x=x^{*}$.

The proof for $F\circ G$ is analogous. Hence, $x^{*}$ and $y^{*}$ are the unique fixed points of the respective compositions. □

4. Applications

4.1. A Common Discrete Framework

Both applications developed below are based on the same abstract discrete structure. The first interpretation describes alternating lexical enrichment and grammatical stabilization, whereas the second describes an alternating adjustment between price decisions and demand responses. The mathematical spaces, generalized distances, and set-valued mappings are identical in both models; only the interpretation of the individual states is different.

Let $X=X_0\cup X_1\cup X_2$ and $Y=Y_0\cup Y_1\cup Y_2$, where the unions are disjoint and $|X_0|=2$, $|X_1|=4$, $|X_2|=6$, while $|Y_0|=3$, $|Y_1|=5$, $|Y_2|=7$.

Thus, $\left|X\right|=12$ and $\left|Y\right|=15$.

We use the notations

$$X_i=\{x_i^{\left(1\right)},\dots ,x_i^{\left(m_i\right)}\},(m_0,m_1,m_2)=(2,4,6),i=0,1,2$$

and

$$Y_i=\{y_i^{\left(1\right)},\dots ,y_i^{\left(n_i\right)}\},(n_0,n_1,n_2)=(3,5,7),i=0,1,2.$$

The sets $X_0$ and $Y_0$ represent initial states, $X_1$ and $Y_1$ represent intermediate states, and $X_2$ and $Y_2$ represent stable states.

4.1.1. The Underlying Metrics

For distinct $x\in X_i$ and $u\in X_j$, define $d_X(x,u)=4^{-min\{i,j\}}$, and put $d_X(x,x)=0$. Thus, the possible positive distances are $\left\{1,{\displaystyle \frac{1}{4}},{\displaystyle \frac{1}{16}}\right\}$.

Similarly, for distinct $y\in Y_i$ and $v\in Y_j$, define $d_Y(y,v)={\displaystyle \frac{4^{-min\{i,j\}}}{2}}$, and let us put $d_Y(y,y)=0$. The possible positive distances in Y are therefore $\left\{{\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{8}},{\displaystyle \frac{1}{32}}\right\}$.

Both $d_X$ and $d_Y$ are ultrametrics and hence ordinary metrics.

Moreover, $diam(X,d_X)=1$ and $diam(Y,d_Y)={\displaystyle \frac{1}{2}}$.

4.1.2. Construction of the Strong b-Metrics

Following Theorem 1, define

$${\rho }_X(x,u)=d_X(x,u)+{\displaystyle \frac{1}{4}}{d}_X{(x,u)}^2\mathrm{and}{\rho }_Y(y,v)=d_Y(y,v)+{\displaystyle \frac{1}{2}}{d}_Y{(y,v)}^2.$$

For ${\rho }_X$, we have $q=2$, ${\lambda }_X={\displaystyle \frac{1}{4}}$, $D_X=1$, and hence $K_X=1+4{\lambda }_X{D}_X=2$.

For ${\rho }_Y$, we have $q=2$, ${\lambda }_Y={\displaystyle \frac{1}{2}}$, $D_Y={\displaystyle \frac{1}{2}}$, and therefore $K_Y=1+4{\lambda }_Y{D}_Y=2$.

Consequently, both ${\rho }_X$ and ${\rho }_Y$ are strong b-metrics with coefficient 2.

For convenience, let us put $r_i=4^{-i}+{\displaystyle \frac{1}{4}}{16}^{-i}$ for $i=0,1,2$.

Then $r_0={\displaystyle \frac{5}{4}}$, $r_1={\displaystyle \frac{17}{64}}$, $r_2={\displaystyle \frac{65}{1024}}$.

If $x\in X_i$, $u\in X_j$, and $x\ne u$, then ${\rho }_X(x,u)=r_{min\{i,j\}}$.

Similarly, if $y\in Y_i$, $v\in Y_j$, and $y\ne v$, then ${\rho }_Y(y,v)={\displaystyle \frac{r_{min\{i,j\}}}{2}}$.

4.1.3. Construction of the Strong Partial b-Metrics

Let us choose $A_X=X_2$ and $A_Y=Y_2$ as the target families. Define $w_X\left(x\right)={\rho }_X(x,X_2)$ and $w_Y\left(y\right)={\rho }_Y(y,Y_2)$.

For $x\in X_i$ and $y\in Y_i$, we have, respectively

$$w_X\left(x\right)=\left\{\begin{array}{cc}{r}_0,\hfill & i=0,\hfill \\ r_1,\hfill & i=1,\hfill \\ 0,\hfill & i=2,\hfill \end{array}\right.w_Y\left(y\right)=\left\{\begin{array}{cc}{\displaystyle \frac{1}{2}}{r}_0,\hfill & i=0,\hfill \\ {\displaystyle \frac{1}{2}}{r}_1,\hfill & i=1,\hfill \\ 0,\hfill & i=2.\hfill \end{array}\right.$$

Now define

$$p(x,u)={\rho }_X(x,u)+max\{w_X\left(x\right),w_X\left(u\right)\}$$

and

$$\sigma (y,v)={\rho }_Y(y,v)+max\{w_Y\left(y\right),w_Y\left(v\right)\}.$$

By Theorem 2, $(X,p,2)$ and $(Y,\sigma ,2)$ are strong partial b-metric spaces.

Their self-distances are $p(x,x)=w_X\left(x\right)$ and $\sigma (y,y)=w_Y\left(y\right)$.

Therefore, $p(x,x)=0$ if and only if $x\in X_2$, and $\sigma (y,y)=0$ if and only if $y\in Y_2$.

4.1.4. The Set-Valued Mappings

Define $F:X\rightrightarrows Y$ by $F\left(x\right)=Y_i$ for every $x\in X_i$, $i=0,1,2$.

Define $\gamma :\{0,1,2\}\to \{1,2\}$ by $\gamma \left(i\right)=min\{i+1,2\}$ for $i=0,1,2$, and let $G:Y\rightrightarrows X$ be given by $G\left(y\right)=X_{\gamma \left(i\right)}$ for every $y\in Y_i$.

Equivalently,

$$F\left(x_0^{\left(j\right)}\right)=Y_0,j=1,2,$$

$$G\left(y_0^{\left(k\right)}\right)=X_1,k=1,2,3,$$

$$F\left(x_1^{\left(j\right)}\right)=Y_1,j=1,\dots ,4,$$

$$G\left(y_1^{\left(k\right)}\right)=X_2,k=1,\dots ,5,$$

and, at the stable level,

$$F\left(x_2^{\left(j\right)}\right)=Y_2,j=1,\dots ,6,$$

$$G\left(y_2^{\left(k\right)}\right)=X_2,k=1,\dots ,7.$$

Thus, the common transition structure is

$$X_0\stackrel{F}{\to }{Y}_0\stackrel{G}{\to }{X}_1\stackrel{F}{\to }{Y}_1\stackrel{G}{\to }{X}_2\stackrel{F}{\to }{Y}_2\stackrel{G}{\to }{X}_2.$$

At the terminal level, the two families are invariant under the alternating rules in the sense that $F\left(x\right)=Y_2$ for every $x\in X_2$ and $G\left(y\right)=X_2$ for every $y\in Y_2$. This invariance is a special property of the present finite model and is distinct from the general existence conclusion of Theorem 3.

4.1.5. Zero-Completeness and Zero-Closedness

The spaces X and Y are finite, and the distance between any two distinct points is positive. Hence, every zero-Cauchy sequence in X is eventually constant at a point of $X_2$, and every zero-Cauchy sequence in Y is eventually constant at a point of $Y_2$. Consequently, $(X,p,2)$ and $(Y,\sigma ,2)$ are zero-complete.

Moreover, every nonempty subset of X is zero-closed with respect to p, and every nonempty subset of Y is zero-closed with respect to $\sigma$.

Therefore, $F\left(x\right)\in C_0^{\sigma }\left(Y\right)$ and $G\left(y\right)\in C_0^p\left(X\right)$ for all $x\in X$ and $y\in Y$.

4.1.6. The Control Functions

Let $J=[0,\infty )$ and define $\phi \left(t\right)={\displaystyle \frac{1}{2}}t$ and $\psi \left(t\right)={\displaystyle \frac{1}{2}}t$.

Then $\theta \left(t\right)=(\psi \circ \phi )\left(t\right)={\displaystyle \frac{1}{4}}t$. Consequently, $S_{\theta }\left(t\right)={\sum }_{n=0}^{\infty }{\theta }^n\left(t\right)={\displaystyle \frac{4}{3}}t$, and $\theta$ is a Bianchini–Grandolfi gauge function.

4.1.7. Verification of the Contractive Condition for F

Let $x\in X_i$ and $u\in X_j$.

If $i=j$, then $F\left(x\right)=F\left(u\right)=Y_i$.

Since all points of $Y_i$ have the same self-distance, we obtain

$${\delta }_{\sigma }(F\left(x\right),F\left(u\right))={\delta }_{\sigma }(Y_i,Y_i)=\underset{y\in Y_i}{sup}\sigma (y,y)=\left\{\begin{array}{cc}{\displaystyle \frac{1}{2}}{r}_i,\hfill & i=0,1,\hfill \\ 0,\hfill & i=2.\hfill \end{array}\right.$$

On the other hand,

$$p(x,u)\ge p(x,x)=w_X\left(x\right)=\left\{\begin{array}{cc}{r}_i,\hfill & i=0,1,\hfill \\ 0,\hfill & i=2.\hfill \end{array}\right.$$

Therefore, for $i=0,1$,

$${\delta }_{\sigma }(F\left(x\right),F\left(u\right))={\displaystyle \frac{1}{2}}{r}_i\le {\displaystyle \frac{1}{2}}p(x,u)=\phi \left(p(x,u)\right),$$

whereas for $i=2$,

$${\delta }_{\sigma }(F\left(x\right),F\left(u\right))=0\le {\displaystyle \frac{1}{2}}p(x,u)=\phi \left(p(x,u)\right).$$

Suppose that $i\ne j$. Since the distance between any point of $Y_i$ and any point of $Y_j$ is constant, we have

$${\delta }_{\sigma }(Y_i,Y_j)={\displaystyle \frac{1}{2}}{r}_{min\{i,j\}}+{\displaystyle \frac{1}{2}}max\{w_i,w_j\}={\displaystyle \frac{1}{2}}\left(r_{min\{i,j\}}+max\{w_i,w_j\}\right),$$

where $w_0=r_0$, $w_1=r_1$, $w_2=0$.

On the other hand, $p(x,u)=r_{min\{i,j\}}+max\{w_i,w_j\}$. Therefore,

$${\delta }_{\sigma }(F\left(x\right),F\left(u\right))={\displaystyle \frac{1}{2}}p(x,u)=\phi \left(p(x,u)\right).$$

Thus, ${\delta }_{\sigma }(F\left(x\right),F\left(u\right))\le \phi \left(p(x,u)\right)$ for all $x,u\in X$.

4.1.8. Verification of the Contractive Condition for G

Let $y\in Y_i$ and $v\in Y_j$.

If $\gamma \left(i\right)=\gamma \left(j\right)$, then the image sets coincide. If their common value is $X_2$, then

$${\delta }_p(X_2,X_2)=0.$$

If $i=j=0$, then the common image is $X_1$, and

$${\delta }_p(X_1,X_1)=r_1.$$

Since $y,v\in Y_0$, condition II gives $\sigma (y,v)\ge \sigma (y,y)={\displaystyle \frac{1}{2}}{r}_0$. Consequently, ${\displaystyle \frac{1}{2}}\sigma (y,v)\ge {\displaystyle \frac{1}{4}}{r}_0$.

Since

$$r_1={\displaystyle \frac{17}{64}}\le {\displaystyle \frac{1}{4}}{r}_0={\displaystyle \frac{5}{16}},$$

we obtain

$${\delta }_p(G\left(y\right),G\left(v\right))\le {\displaystyle \frac{1}{2}}\sigma (y,v)=\psi \left(\sigma (y,v)\right).$$

It remains to consider $\gamma \left(i\right)\ne \gamma \left(j\right)$. This is possible only when one of $i,j$ is 0 and the other belongs to $\{1,2\}$.

Because the excess is one-sided, the two possible orientations must be distinguished:

$${\delta }_p(G\left(y\right),G\left(v\right))=\left\{\begin{array}{cc}{\delta }_p(X_1,X_2),\hfill & i=0,j\in \{1,2\},\hfill \\ {\delta }_p(X_2,X_1),\hfill & i\in \{1,2\},j=0.\hfill \end{array}\right.$$

For every $a\in X_1$ and $b\in X_2$, we have

$$p(a,b)=2r_1.$$

Therefore,

$${\delta }_p(X_1,X_2)={\delta }_p(X_2,X_1)=2r_1.$$

Hence,

$${\delta }_p(G\left(y\right),G\left(v\right))=2r_1$$

in either orientation.

Moreover, $\sigma (y,v)=r_0$. Since $2r_1={\displaystyle \frac{17}{32}}\le {\displaystyle \frac{5}{8}}={\displaystyle \frac{1}{2}}{r}_0$, we obtain

$${\delta }_p(G\left(y\right),G\left(v\right))\le {\displaystyle \frac{1}{2}}\sigma (y,v)=\psi \left(\sigma (y,v)\right).$$

Therefore, ${\delta }_p(G\left(y\right),G\left(v\right))\le \psi \left(\sigma (y,v)\right)$ for all $y,v\in Y$.

4.1.9. Verification of the Localization Condition

Choose $\overline{x}=x_0^{\left(1\right)}$, $\overline{y}=y_0^{\left(1\right)}$, $\alpha =3$, $r_X=r_Y={\displaystyle \frac{35}{8}}$.

Since $F\left(\overline{x}\right)=Y_0$, we have $\overline{y}\in F\left(\overline{x}\right)$.

Furthermore, $G\left(\overline{y}\right)=X_1$.

For every $u\in X_1$, there holds $p(\overline{x},u)=2r_0={\displaystyle \frac{5}{2}}$. Hence,

$$p(\overline{x},G\left(\overline{y}\right))={\displaystyle \frac{5}{2}}<3=\alpha .$$

Since $K=2$ and $S_{\theta }\left(\alpha \right)={\displaystyle \frac{4}{3}}\alpha =4$, we have

$$K{S}_{\theta }\left(\alpha \right)-(K-1)\alpha =2·4-3=5.$$

On the other hand, $p(\overline{x},\overline{x})=r_0={\displaystyle \frac{5}{4}}$ and $\sigma (\overline{y},\overline{y})={\displaystyle \frac{1}{2}}{r}_0={\displaystyle \frac{5}{8}}$. Consequently,

$$min\left\{p(\overline{x},\overline{x})+r_X,\sigma (\overline{y},\overline{y})+r_Y\right\}={\displaystyle \frac{5}{8}}+{\displaystyle \frac{35}{8}}=5.$$

Thus,

$$K{S}_{\theta }\left(\alpha \right)-(K-1)\alpha \le min\left\{p(\overline{x},\overline{x})+r_X,\sigma (\overline{y},\overline{y})+r_Y\right\}.$$

The selected radius contains all states: ${\overline{B}}_p(\overline{x},r_X)=X$ and ${\overline{B}}_{\sigma }(\overline{y},r_Y)=Y$.

Therefore, the intersections with the localization balls do not alter the image sets:

$$F\left(x\right)\cap {\overline{B}}_{\sigma }(\overline{y},r_Y)=F\left(x\right)\mathrm{for}\mathrm{every}x\in X,$$

and

$$G\left(y\right)\cap {\overline{B}}_p(\overline{x},r_X)=G\left(y\right)\mathrm{for}\mathrm{every}y\in Y.$$

Consequently, the global contractive estimates established above imply precisely the two localized contractive conditions required in Theorem 3. All assumptions of that theorem are therefore satisfied.

4.1.10. The Alternating Points

For every $j\in \{1,\dots ,6\}$ and $k\in \{1,\dots ,7\}$, we have $y_2^{\left(k\right)}\in F\left(x_2^{\left(j\right)}\right)$ and $x_2^{\left(j\right)}\in G\left(y_2^{\left(k\right)}\right)$.

Hence, all pairs $(x_2^{\left(j\right)},y_2^{\left(k\right)})$ for $j=1,\dots ,6$ and $k=1,\dots ,7$, are alternating points.

The common discrete model therefore possesses at least $6·7=42$ alternating points.

Conversely, suppose that $(x,y)$ is an alternating point. If $x\in X_0$, then $y\in F\left(x\right)=Y_0$, and hence $G\left(y\right)=X_1$, which does not contain x. Thus, no alternating point has its first component in $X_0$. Similarly, if $x\in X_1$, then $y\in F\left(x\right)=Y_1$, whereas $G\left(y\right)=X_2$, which again does not contain x. Consequently, every alternating point belongs to $X_2\times Y_2$.

Therefore, the set of all alternating points is exactly $X_2\times Y_2$, and the common discrete model has precisely $|X_2\left|\right|Y_2|=6·7=42$ alternating points.

4.2. Application to Economic Adjustment

We interpret the common discrete framework as a market adjustment process between prices selected by a producer and quantities demanded by consumers. The elements of X represent admissible price decisions, whereas the elements of Y represent possible demand responses. This interpretation is motivated by the classical theory of competitive equilibrium, in which equilibrium is characterized by the compatibility of consumers’ demand decisions, producers’ production decisions, and a system of market-clearing prices [9,10].

The Arrow–Debreu existence theory establishes, under appropriate continuity, convexity, and compactness assumptions, the existence of a competitive equilibrium for an integrated model of production, exchange, and consumption [10]. The present application does not provide an alternative proof of that general existence result. Instead, it gives a finite and set-valued model of an alternating local adjustment between admissible price states and demand-response states.

The idea that observed excess demand may induce subsequent price adjustments is classical in the theory of competitive-market dynamics [11,17]. The present construction is related to this tradition but is not a Walrasian tâtonnement process: it alternates between two set-valued mappings on finite state spaces and is governed by the contractive assumptions of Theorem 3.

The alternating process has the form

$$X_0\stackrel{F}{\to }{Y}_0\stackrel{G}{\to }{X}_1\stackrel{F}{\to }{Y}_1\stackrel{G}{\to }{X}_2\stackrel{F}{\to }{Y}_2\stackrel{G}{\to }{X}_2.$$

Here, $X_0$ and $Y_0$ contain the initial price and demand states, $X_1$ and $Y_1$ contain intermediate adjustment states, and $X_2$ and $Y_2$ describe terminal price and demand regions for the present alternating process. Their stability is understood with respect to the mappings F and G and the generalized distance structure introduced above; it should not be identified with the stability of a Walrasian price-adjustment process. Classical competitive-equilibrium theory shows that existence and dynamic stability are logically distinct questions [10,11,18].

4.2.1. Numerical Realization of the States

The numerical values introduced below are illustrative and are not estimated from empirical market data.

Let $P\subseteq {\mathbb{R}}_{+}$ and $Q\subseteq {\mathbb{R}}_{+}$ denote the admissible price and quantity sets. The abstract states are related to their numerical values by the realization mappings

$$\pi :X⟶P\mathrm{and}\kappa :Y⟶Q.$$

The price realization is defined by

$$\pi \left(x_0^{\left(j\right)}\right)=12.2-0.2j,j=1,2,$$

$$\pi \left(x_1^{\left(j\right)}\right)=11.1-0.2j,j=1,\dots ,4,$$

and

$$\pi \left(x_2^{\left(j\right)}\right)=10.3-0.1j,j=1,\dots ,6.$$

Explicitly,

$$\begin{array}{cccc}\mathrm{Level}& 0& 1& 2\\ \mathrm{State}& x_0^{\left(1\right)} x_0^{\left(2\right)}& x_1^{\left(1\right)} x_1^{\left(2\right)} x_1^{\left(3\right)} x_1^{\left(4\right)}& x_2^{\left(1\right)} x_2^{\left(2\right)} x_2^{\left(3\right)} x_2^{\left(4\right)} x_2^{\left(5\right)} x_2^{\left(6\right)}\\ \pi \left(x\right)& 12.00 11.80& 10.90 10.70 10.50 10.30& 10.20 10.10 10.00 9.90 9.80 9.70\end{array}$$

Thus, $\pi \left(X_0\right)=\{11.80,12.00\}$, $\pi \left(X_1\right)=\{10.30,10.50,10.70,10.90\}$, and $\pi \left(X_2\right)=\{9.70,9.80,9.90,10.00,10.10,10.20\}$.

The quantity realization is defined by

$$\kappa \left(y_0^{\left(k\right)}\right)=68+2k,k=1,2,3,$$

$$\kappa \left(y_1^{\left(k\right)}\right)=82+2k,k=1,\dots ,5,$$

and

$$\kappa \left(y_2^{\left(k\right)}\right)=94+2k,k=1,\dots ,7.$$

Explicitly,

$$\begin{array}{cccccccccccccccc}\mathrm{Level}& & 0& & & & 1& & & & & & 2& & & \\ \mathrm{State}& y_0^{\left(1\right)}& y_0^{\left(2\right)}& y_0^{\left(3\right)}& y_1^{\left(1\right)}& y_1^{\left(2\right)}& y_1^{\left(3\right)}& y_1^{\left(4\right)}& y_1^{\left(5\right)}& y_2^{\left(1\right)}& y_2^{\left(2\right)}& y_2^{\left(3\right)}& y_2^{\left(4\right)}& y_2^{\left(5\right)}& y_2^{\left(6\right)}& y_2^{\left(7\right)}\\ \kappa \left(y\right)& 70& 72& 74& 84& 86& 88& 90& 92& 96& 98& 100& 102& 104& 106& 108\end{array}$$

Consequently, $\kappa \left(Y_0\right)=\{70,72,74\}$, $\kappa \left(Y_1\right)=\{84,86,88,90,92\}$, and $\kappa \left(Y_2\right)=\{96,98,100,102,104,106,108\}$.

4.2.2. Demand and Price-Adjustment Correspondences

Let

$$P_d=[11.80,12.00]\cup [10.30,10.90]\cup [9.70,10.20]\mathrm{and}{Q}_d=[70,74]\cup [84,92]\cup [96,108].$$

Demand and excess-demand mappings play a central role in the mathematical formulation of competitive equilibrium. In particular, the general shape and aggregation properties of market excess demand have been studied by Sonnenschein and Debreu [19,20]. In the present simplified application, D is a demand correspondence rather than a complete excess-demand correspondence, because the supply side is represented indirectly through the producer’s price-adjustment rule R.

The demand correspondence $D:P_d\rightrightarrows Q_d$ is defined by

$$D\left(p\right)=\left\{\begin{array}{cc}[70,74],\hfill & p\in [11.80,12.00],\hfill \\ [84,92],\hfill & p\in [10.30,10.90],\hfill \\ [96,108],\hfill & p\in [9.70,10.20].\hfill \end{array}\right.$$

The producer’s price-adjustment correspondence $R:Q_d\rightrightarrows P_d$ is defined by

$$R\left(q\right)=\left\{\begin{array}{cc}[10.30,10.90],\hfill & q\in [70,74],\hfill \\ [9.70,10.20],\hfill & q\in [84,92],\hfill \\ [9.70,10.20],\hfill & q\in [96,108].\hfill \end{array}\right.$$

Price-adjustment mechanisms based on observed market imbalances have a long history in the study of competitive equilibrium dynamics [11,17,21]. The correspondence R used here is a finite, set-valued adjustment rule constructed for the present example and is not claimed to coincide with any particular classical tâtonnement mechanism.

Thus, an initial price in $[11.80,12.00]$ may lead to a demanded quantity in $[70,74]$. After observing such a quantity, the producer selects an adjusted price in $[10.30,10.90]$. This price interval may generate a demanded quantity in $[84,92]$, leading to a further adjustment in $[9.70,10.20]$. Prices in the last interval generate quantities in $[96,108]$, and the response rule then keeps the price inside the same terminal interval of the alternating process.

The correspondence R is a price-adjustment rule based on observed demand. It is not identified with a classical supply correspondence, which associates admissible production plans or quantities supplied with a given price system. The distinction between consumers’ demand correspondences, producers’ supply correspondences, and market-clearing prices is fundamental in the classical theory of competitive equilibrium [10].

The discrete mappings F and G are induced by D and R through $\pi$ and $\kappa$:

$$F\left(x\right)=\{y\in Y:\kappa (y)\in D(\pi \left(x\right)\left)\right\}\mathrm{and}G\left(y\right)=\{x\in X:\pi (x)\in R(\kappa \left(y\right)\left)\right\}.$$

It follows directly that

$$F\left(x\right)=\left\{\begin{array}{cc}{Y}_0,\hfill & x\in X_0,\hfill \\ Y_1,\hfill & x\in X_1,\hfill \\ Y_2,\hfill & x\in X_2,\hfill \end{array}\right.G\left(y\right)=\left\{\begin{array}{cc}{X}_1,\hfill & y\in Y_0,\hfill \\ X_2,\hfill & y\in Y_1,\hfill \\ X_2,\hfill & y\in Y_2.\hfill \end{array}\right.$$

Hence, these economically interpreted correspondences coincide exactly with the mappings used in the common discrete framework.

4.2.3. Economic Interpretation of the Levels

The two prices in $X_0$ represent initial pricing policies for which demand is substantially below the producer’s target sales region. The three quantities in $Y_0$ represent possible consumer responses to either initial policy.

After observing a demand state in $Y_0$, the producer may choose among the four prices in $X_1$. These states describe different intensities of the first price reduction. The corresponding quantities in $Y_1$ are higher than the initial demand levels but remain below the locally acceptable sales region.

The six prices in $X_2$ form a terminal price region rather than a single uniquely determined price. Their multiplicity represents the possibility that several price states may be compatible with the same set-valued adjustment rule. Similarly, the seven quantities in $Y_2$ form the corresponding terminal demand region. Such multiplicity is consistent with the use of correspondences in equilibrium models with heterogeneous agents or uncertainty [22,23].

In agreement with the common strong partial b-metric construction, $p(x,x)=0$ for every $x\in X_2$, and $\sigma (y,y)=0$ for every $y\in Y_2$.

Thus, no further essential adjustment is required once both variables belong to their respective stable regions.

4.2.4. The Complete Market Adjustment Process

Unlike the continuous-time or discrete-time price dynamics studied in classical tâtonnement theory [11,17,21], the following scheme records transitions between finitely many admissible price and demand regions.

The numerical adjustment scheme is

$$[11.80,12.00]\to [70,74]\to [10.30,10.90]\to [84,92]\to [9.70,10.20]\to [96,108]\to [9.70,10.20].$$

For example, one admissible path is $x_0^{\left(1\right)}\to y_0^{\left(2\right)}\to x_1^{\left(3\right)}\to y_1^{\left(4\right)}\to x_2^{\left(3\right)}\to y_2^{\left(4\right)}$, or, numerically, $12.00\to 72\to 10.50\to 90\to 10.00\to 102$.

Other selections from the same multivalued correspondences give other admissible adjustment paths.

4.2.5. Stable Market Configurations

The verification in Section 4.1 applies without change. Hence, for every $j=1,\dots ,6$ and $k=1,\dots ,7$, we have $y_2^{\left(k\right)}\in F\left(x_2^{\left(j\right)}\right)$ and $x_2^{\left(j\right)}\in G\left(y_2^{\left(k\right)}\right)$.

Consequently, all pairs $(x_2^{\left(j\right)},y_2^{\left(k\right)})$ for $j=1,\dots ,6$ and $k=1,\dots ,7$, are alternating points. The model therefore admits $6·7=42$ alternating price–demand configurations.

Moreover, $x_2^{\left(j\right)}\in (G\circ F)\left(x_2^{\left(j\right)}\right)$ for $j=1,\dots ,6$, and $y_2^{\left(k\right)}\in (F\circ G)\left(y_2^{\left(k\right)}\right)$ for $k=1,\dots ,7$.

Thus, the six stable price states are fixed points of $G\circ F$, while the seven stable demand states are fixed points of $F\circ G$. Their multiplicity reflects the coexistence of several terminal outcomes that are compatible with the same set-valued alternating adjustment rule.

4.3. Application to Alternating Linguistic Enrichment

We now interpret the common discrete framework as a model of linguistic enrichment and text revision. The purpose of the model is not to identify a single universally correct formulation, but to describe how a text may pass through several admissible stages of lexical, grammatical, syntactic, and stylistic development. This interpretation is consistent with process-oriented accounts of writing and revision, in which text production is treated as a recursive rather than a strictly linear activity [12,15].

The starting point is a fixed semantic core. The principal participants, events, temporal relations, and causal connections remain unchanged throughout the process. What changes is the linguistic realization of this content: the selection of words and collocations, the grammatical organization of the sentences, the degree of explicitness, the distribution of information, and the stylistic emphasis. This distinction between an underlying representation of meaning and its different textual realizations is consistent with established models of text comprehension and with contemporary accounts of paraphrase [24,25].

The model distinguishes between two complementary revision activities. The first is lexical enrichment. It introduces more precise words, synonyms, collocations, descriptive expressions, and alternative ways of presenting the same content. The second is grammatical stabilization. It integrates the available lexical material into grammatically acceptable, syntactically coherent, and stylistically appropriate sentences. The distinction is also related to research on lexical richness and lexical choice, which shows that vocabulary range and the context-sensitive selection of near-synonyms constitute dimensions of textual development that cannot be reduced to grammatical correctness alone [26,27].

These activities are related but should not be identified. A text may be grammatically correct while remaining lexically elementary, repetitive, or insufficiently precise. Conversely, the introduction of new vocabulary may create difficulties involving agreement, tense, word order, complementation, sentence structure, or register. Lexical enrichment may therefore require subsequent grammatical revision, while grammatical stabilization may reveal possibilities for further lexical improvement. The interaction between revision, error correction, and language development has likewise been emphasized in research on written corrective feedback, although successful revision of a particular text should not automatically be identified with permanent language acquisition [28].

For this reason, linguistic development is represented as an alternating rather than a single-stage process. A lexically enriched version is followed by one of several admissible grammatical revisions. The resulting grammatically stabilized version may then undergo one of several further lexical enrichments. Within the present mathematical model, the process continues until a pair of mutually compatible formulations is reached. The representation of language development as an interconnected and continuously reorganizing process is consistent with complex dynamic approaches to second-language development [13,14,29,30].

4.3.1. Lexically Enriched and Grammatically Stabilized States

The elements of X represent textual states considered immediately after a stage of lexical enrichment. Such a state may contain newly introduced vocabulary and more informative expressions, but these resources need not yet be fully integrated into the most appropriate grammatical and syntactic structures.

The elements of Y represent textual states obtained after grammatical and syntactic stabilization. In these states, the lexical material has been reorganized into acceptable sentence structures. However, grammatical acceptability does not exclude the possibility of further lexical or stylistic enrichment.

Accordingly, the distinction between X and Y is functional rather than absolute. It does not establish an absolute division between “incorrect” and “correct” formulations. Instead, it records which type of revision has most recently been applied and which type of improvement may follow.

The state families are organized into three successive levels:

$$X=X_0\cup X_1\cup X_2\mathrm{and}Y=Y_0\cup Y_1\cup Y_2.$$

The initial families contain $|X_0|=2$ and $|Y_0|=3$ possible realizations. The intermediate families contain $|X_1|=4$ and $|Y_1|=5$ realizations, whereas the stable families contain $|X_2|=6$ and $|Y_2|=7$.

Within this illustrative construction, the increasing numbers of states,

$$2<3<4<5<6<7,$$

are stipulated to represent an expansion of the available linguistic repertoire. As the process develops, the same semantic content is represented by a larger number of lexically, grammatically, and stylistically admissible formulations. This interpretation is consistent with research treating lexical diversity and variability as relevant dimensions of language development [13,14,26].

4.3.2. The Alternating Revision Process

The mapping $F:X\rightrightarrows Y$ represents grammatical stabilization. For a lexically enriched state $x\in X$, the set $F\left(x\right)$ contains the grammatically and syntactically admissible revisions of that state.

The mapping $G:Y\rightrightarrows X$ represents renewed lexical enrichment. For a grammatically stabilized state $y\in Y$, the set $G\left(y\right)$ contains the admissible lexically enriched versions that may be developed from it.

Both mappings are set-valued because linguistic revision is generally nonunique. A sentence may admit several grammatically acceptable reformulations, and approximately the same content may be expressed through different choices of vocabulary, collocations, sentence patterns, and stylistic emphasis. Research on paraphrase and near-synonymy likewise shows that alternative formulations may preserve a common semantic core while differing in lexical nuance, information structure, or communicative effect [25,27]. The model therefore treats variation as an intrinsic feature of the revision process rather than as an error or an undesirable ambiguity.

In the common discrete framework, the revision mappings satisfy

$$F\left(x\right)=\left\{\begin{array}{cc}{Y}_0,\hfill & x\in X_0,\hfill \\ Y_1,\hfill & x\in X_1,\hfill \\ Y_2,\hfill & x\in X_2,\hfill \end{array}\right.\mathrm{and}G\left(y\right)=\left\{\begin{array}{cc}{X}_1,\hfill & y\in Y_0,\hfill \\ X_2,\hfill & y\in Y_1,\hfill \\ X_2,\hfill & y\in Y_2.\hfill \end{array}\right.$$

The complete revision process is therefore

$$X_0\stackrel{F}{\to }{Y}_0\stackrel{G}{\to }{X}_1\stackrel{F}{\to }{Y}_1\stackrel{G}{\to }{X}_2\stackrel{F}{\to }{Y}_2\stackrel{G}{\to }{X}_2.$$

The generalized distance between two states measures the extent of the revision needed to pass from one formulation to the other. The self-distance of a state represents the amount of essential linguistic work that remains before a stable formulation is reached.

A positive self-distance does not mean that the corresponding text is meaningless or entirely unacceptable. It indicates only that, according to the criteria adopted in the model, further lexical, grammatical, or stylistic revision is still required.

The states in $X_2$ and $Y_2$ have zero self-distance. They are therefore interpreted as stable formulations: no further essential revision is required, although additional optional stylistic changes may still be possible.

All states preserve the same narrative core:

• Tom approaches an old house at dusk;
• the house is dark;
• Tom enters the house;
• he finds a small wooden box on a table;
• he takes the box;
• he returns home.

Every text contains fewer than 150 words.

4.3.3. The Initial Lexical Family $X_0$

The family $X_0=\left\{x_0^{\left(1\right)},x_0^{\left(2\right)}\right\}$ contains two elementary narratives.

State $x_0^{\left(1\right)}$: Tom go to the old house. The house was dark. He open the door and go inside. He see a small box on a table. He take the box and go home.

State $x_0^{\left(2\right)}$: At evening, Tom walk to an old house. It was dark. He enter the house and find a small box on the table. Tom pick the box and return to home.

Both narratives use elementary vocabulary. The second state introduces the alternatives walk, enter, find, and return, but the learner has not yet mastered tense formation, articles, prepositions, or the collocation pick up.

4.3.4. The First Grammatically Stabilized Family $Y_0$

The mapping F produces $Y_0=\left\{y_0^{\left(1\right)},y_0^{\left(2\right)},y_0^{\left(3\right)}\right\}$.

State $y_0^{\left(1\right)}$: Tom went to the old house. The house was dark. He opened the door and went inside. He saw a small box on a table. He took the box and went home

State $y_0^{\left(2\right)}$: In the evening, Tom walked to an old house. It was dark inside. He entered the house and found a small box on a table. Tom picked up the box and returned home.

State $y_0^{\left(3\right)}$: When evening came, Tom went to an old house. Finding it dark, he opened the door and entered. A small box was lying on a table. He took it and walked back home.

The three states are grammatically acceptable but remain lexically limited. They already provide the learner with several grammatical ways of expressing temporal order: In the evening..., When evening came..., and Finding it dark...

4.3.5. The First Lexically Enriched Family $X_1$

The mapping G introduces new synonyms and descriptive expressions: $X_1=\left\{x_1^{\left(1\right)},x_1^{\left(2\right)},x_1^{\left(3\right)},x_1^{\left(4\right)}\right\}$.

State $x_1^{\left(1\right)}$: At dusk, Tom approach the abandoned house. The building stood dark and silent. He push open the door, enter the room, and notice a small wooden box resting on a table. He pick it up and return home.

State $x_1^{\left(2\right)}$: As night approached, Tom head toward the deserted house. Its windows was dark. He step through the door and spot a small wooden box on a table. Taking the box, he walk back home.

State $x_1^{\left(3\right)}$: Near nightfall, Tom make his way to the neglected house. No light shine from its windows. After open the door, he cross the threshold and discover a small wooden box. He collect it and return home.

State $x_1^{\left(4\right)}$: As evening fall, Tom arrive at the long-abandoned house. The building appear empty and silent. Having opened the door, he enter a dim room and find a small wooden box lying on a table. He carry it home.

The family $X_1$ contains a substantially richer vocabulary: approach, head toward, make one’s way to, arrive at, abandoned, deserted, neglected, long-abandoned, notice, spot, discover, find, pick up, take, collect, and carry.

The alternatives are not assumed to be perfectly interchangeable: near-synonyms commonly differ in semantic nuance, collocational behavior, register, and contextual appropriateness [27].

However, the learner has not yet mastered the grammatical structures required for all these expressions. The states contain errors in tense, agreement, participial constructions, and verb complementation.

4.3.6. The Second Grammatically Stabilized Family $Y_1$

The mapping F preserves the enriched vocabulary and produces five grammatically admissible narratives: $Y_1=\left\{y_1^{\left(1\right)},y_1^{\left(2\right)},y_1^{\left(3\right)},y_1^{\left(4\right)},y_1^{\left(5\right)}\right\}$.

State $y_1^{\left(1\right)}$: At dusk, Tom approached the abandoned house. The building stood dark and silent. He pushed open the door, entered the room, and noticed a small wooden box resting on a table. He picked it up and returned home.

State $y_1^{\left(2\right)}$: As night approached, Tom headed toward the deserted house. Its windows were dark. He stepped through the door and spotted a small wooden box on a table. Taking the box with him, he walked back home.

State $y_1^{\left(3\right)}$: Near nightfall, Tom made his way to the neglected house. No light shone from its windows. After opening the door, he crossed the threshold and discovered a small wooden box on a table. He collected it and returned home.

State $y_1^{\left(4\right)}$: As evening fell, Tom arrived at the long-abandoned house. The building appeared empty and silent. Having opened the door, he entered a dim room and found a small wooden box lying on a table. He carried it home.

State $y_1^{\left(5\right)}$: When Tom reached the deserted house at dusk, he found its windows dark and its rooms silent. He eased open the door, stepped inside, and saw a small wooden box resting on a table. He lifted it carefully and made his way home.

The fifth state is not merely a correction of one particular $X_1$-state. It combines lexical and grammatical resources appearing in several preceding states. This illustrates that a set-valued revision process may create a new admissible formulation rather than only correct an existing text word for word.

4.3.7. The Stable Lexically Enriched Family $X_2$

The next application of G produces $X_2=\left\{x_2^{\left(1\right)},x_2^{\left(2\right)},x_2^{\left(3\right)},x_2^{\left(4\right)},x_2^{\left(5\right)},x_2^{\left(6\right)}\right\}$.

State $x_2^{\left(1\right)}$: neutral enriched version At twilight, Tom approached the abandoned house. Its windows were dark, and the rooms beyond them seemed silent. He pushed open the door and entered. A small wooden box rested on a dusty table. Tom lifted it, left the house, and walked home through the fading light.

State $x_2^{\left(2\right)}$: concise version. Near nightfall, Tom reached the deserted house. Finding it dark and silent, he opened the door and stepped inside. A small wooden box lay on a dusty table. He picked it up, left the building, and returned home.

State $x_2^{\left(3\right)}$: descriptive version. Evening shadows surrounded the long-abandoned house as Tom approached. No light shone from its windows. He eased open the door and entered the dim room, where a small wooden box rested on a dust-covered table. Tom took it and followed the dark road home.

State $x_2^{\left(4\right)}$: action-oriented version. Tom arrived at the old house just before darkness fell. He opened the door, crossed the silent room, and discovered a small wooden box on a dusty table. Without delaying, he picked it up, stepped outside, and made his way home.

State $x_2^{\left(5\right)}$: syntactically developed version. By the time Tom reached the neglected house, the last daylight was fading. Although the building was dark and silent, he opened the door and entered. On a table in the shadowy room, he found a small wooden box, which he carried home.

State $x_2^{\left(6\right)}$: atmospheric version. As twilight faded from the empty road, Tom made his way toward the abandoned house. Its unlit windows faced the gathering darkness. He stepped through the door and found a small wooden box on a dusty table. Holding it securely, he left the house and returned home.

All six states preserve the same invariant events. Nevertheless, they provide different linguistic information. A learner reading the family $X_2$ encounters several synonymous expressions for movement, perception, darkness, and returning home, together with different degrees of descriptive and syntactic complexity.

They may therefore be regarded as related paraphrastic realizations rather than as lexically interchangeable texts [25,27].

4.3.8. The Stable Grammatically and Stylistically Stabilized Family $Y_2$

The final application of F produces seven stable versions:

$$Y_2=\left\{y_2^{\left(1\right)},y_2^{\left(2\right)},y_2^{\left(3\right)},y_2^{\left(4\right)},y_2^{\left(5\right)},y_2^{\left(6\right)},y_2^{\left(7\right)}\right\}.$$

State $y_2^{\left(1\right)}$: balanced version. As twilight settled over the road, Tom approached the abandoned house. Its windows were dark, and silence filled the rooms beyond them. He pushed open the door and entered. A small wooden box rested on a dusty table. Tom lifted it carefully, left the house, and walked home through the gathering darkness.

State $y_2^{\left(2\right)}$: compact version. Tom reached the deserted house near nightfall. It was dark and silent. After opening the door, he stepped inside and found a small wooden box on a dusty table. He picked it up, left the building, and returned home.

State $y_2^{\left(3\right)}$: descriptive version. By the time Tom approached the long-abandoned house, the last daylight was fading. The windows showed no light, and the interior lay silent. He eased open the door and entered the dim room. There, on a dust-covered table, rested a small wooden box. Tom took it and followed the dark road home.

State $y_2^{\left(4\right)}$: dynamic version. Just before nightfall, Tom arrived at the old house. He opened the door and crossed the silent room. Spotting a small wooden box on a dusty table, he picked it up without delay, stepped outside, and made his way home.

State $y_2^{\left(5\right)}$: complex-sentence version. When Tom reached the neglected house, evening shadows already covered the road. Although no light shone from the windows, he opened the door and entered. In the shadowy room, he found a small wooden box, which he lifted from the table before returning home.

State $y_2^{\left(6\right)}$: participial version. Approaching the abandoned house at twilight, Tom noticed that its windows were unlit. Having eased open the door, he entered the silent room and discovered a small wooden box resting on a dusty table. Taking it with him, he left the house and walked home.

State $y_2^{\left(7\right)}$: explanatory literary version. The last light was fading when Tom reached the old, deserted house. Everything inside was dark and silent. He carefully opened the door, stepped into the room, and noticed a small wooden box on a table. After picking it up, he left the house and followed the road home.

The seven narratives preserve the same stipulated event structure and communicate approximately the same narrative content. They nevertheless differ in lexical choice, grammatical organization, information distribution, rhythm, and stylistic emphasis. They may therefore be treated as distinct linguistic states even though they support closely related interpretations of the narrated events [24,25].

4.3.9. Stable Linguistic Realizations

The final level does not contain a single prescribed text. Instead, it contains six lexically enriched states and seven grammatically stabilized states. Each state preserves the same semantic core, but the states differ in lexical selection, syntactic organization, degree of explicitness, narrative rhythm, and stylistic emphasis.

For every $j\in \{1,\dots ,6\}$ and $k\in \{1,\dots ,7\}$, we have $y_2^{\left(k\right)}\in F\left(x_2^{\left(j\right)}\right)$ and $x_2^{\left(j\right)}\in G\left(y_2^{\left(k\right)}\right)$.

Thus, every pair $(x_2^{\left(j\right)},y_2^{\left(k\right)})$ is a mutually compatible stable pair. The model consequently possesses $6·7=42$ stable alternating pairs.

These 42 pairs should not be interpreted as identical or freely interchangeable texts. They represent different combinations of lexical and grammatical choices that satisfy the semantic and communicative criteria stipulated in the model. Their multiplicity does not indicate uncertainty about the prescribed narrative core; within the present construction, it represents an expanded repertoire of admissible linguistic realizations. The model therefore describes stabilization without imposing linguistic uniformity, consistently with approaches that regard variability and the coexistence of alternative forms as intrinsic features of language development [13,14,30].

5. Discussion

The main result of this study extends the local fixed point methodology for multivalued mappings in strong partial b-metric spaces from a single self-mapping to an alternating process involving two mappings between possibly different spaces. This distinction is mathematically important. A self-mapping describes repeated transformations within one class of states, whereas the present framework allows two qualitatively different types of states and two directional transitions, $X\stackrel{F}{\to }Y$ and $Y\stackrel{G}{\to }X$. The resulting alternating point retains the intermediate compatibility relation between the two components, which is no longer visible if the analysis is reduced immediately to the compositions $G\circ F$ and $F\circ G$.

The contractive structure also reflects the two-stage character of the process. The mappings F and G are controlled by different comparison functions, $\phi$ and $\psi$, while the Bianchini–Grandolfi summability condition is imposed only on their composition $\theta =\psi \circ \phi$. Thus, contraction is required for one complete alternating cycle rather than independently for each of its two transitions. This makes the assumptions more flexible than conditions requiring both directional control functions to satisfy the full summability property separately. At the same time, the use of one-sided excesses is adapted to the directional nature of the set-valued mappings and avoids imposing an unnecessary symmetric Hausdorff-type condition.

Another relevant feature is the local character of the theorem. The contractive assumptions are required only within prescribed closed balls, and the proof establishes that the constructed sequences remain inside these localization regions. Consequently, the behavior of the mappings outside the portion of the spaces visited by the alternating process is irrelevant to the existence argument. This is useful in applications in which the transition rules are contractive only near the admissible or stable states and may exhibit substantially different behavior elsewhere.

The strong partial b-metric setting provides two additional degrees of flexibility. The coefficient in the relaxed triangle inequality allows transition distances that need not satisfy the ordinary metric triangle inequality, while a positive self-distance can represent a residual discrepancy associated with an individual state. Within the applications considered here, this residual quantity is interpreted as the amount of adjustment or revision that remains before a terminal family is reached. The zero self-distances assigned to the final levels therefore encode stability relative to the criteria of the model. They should not be understood as asserting that the corresponding economic or linguistic states are optimal in every possible sense.

The common discrete construction demonstrates that the hypotheses of the theorem can be verified explicitly and that existence does not imply uniqueness in the set-valued case. The model has exactly 42 alternating points, all belonging to the product of the two terminal families. This multiplicity is not a defect of the construction. Rather, it illustrates the distinction between stabilization and selection: the contractive alternating process reaches a stable region, but the set-valued mappings may admit several mutually compatible terminal configurations. By contrast, the single-valued corollary shows that uniqueness can be recovered when each state has a uniquely determined successor and the additional localization assumptions are satisfied.

The economic and linguistic applications emphasize different aspects of this nonuniqueness. In the economic interpretation, the alternating points represent price–demand configurations compatible with the specified demand and price-adjustment correspondences. They are not claimed to be competitive equilibria in the full Arrow–Debreu sense, nor does the discrete process reproduce a classical tâtonnement mechanism. Its purpose is to illustrate how a set-valued adjustment rule can lead to a family of locally stable configurations. In the linguistic interpretation, the multiplicity of terminal states reflects the fact that a fixed semantic core may admit several lexically, grammatically, and stylistically acceptable realizations. Stabilization therefore does not require convergence to a unique prescribed text.

These interpretations also clarify the limitations of the present study. The numerical levels, transition correspondences, generalized distances, and self-distances in the discrete example are stipulated so that the mathematical assumptions can be checked transparently. They are not estimated from empirical market observations or from a corpus of learner texts. Accordingly, the applications should be regarded as proof-of-concept models rather than as predictive economic or linguistic systems. In particular, the generalized linguistic distance does not yet incorporate independently validated measures of semantic similarity, grammatical accuracy, lexical diversity, or stylistic appropriateness.

A possible direction for future work is to extend this interpretation to machine-generated text. Iterative feedback and refinement of language-model outputs have been studied in Self-Refine [31], which operates at inference time without additional model training. In an alternating revision scheme, F could represent grammatical stabilization and G lexical enrichment, each returning a set of admissible revisions preserving a stipulated semantic core. Applying Theorem 3 would require independent verification of all its hypotheses, including 0-completeness, zero-closedness, localization, and the contractive conditions. This remains a direction for future investigation, not a claim that existing language models satisfy these hypotheses or that the present theorem guarantees improvements in text quality.

6. Conclusions

This paper introduced the concept of an alternating point for two set-valued mappings acting between two 0-complete strong partial b-metric spaces. Such a point consists of a mutually compatible pair of states and simultaneously induces fixed points of the two set-valued compositions. In this way, the proposed formulation preserves the two-stage structure of the underlying process instead of describing it only through a composed self-mapping.

A local existence theorem was established under nonlinear contractive conditions formulated by means of one-sided excess functionals. The two directional transitions are controlled by distinct functions $\phi$ and $\psi$, whereas the Bianchini–Grandolfi summability condition is required only for the composite function $\theta =\psi \circ \phi$, which represents one complete alternating cycle. The proof constructs two interdependent sequences, verifies that they remain in the prescribed localization regions, and establishes their 0-Cauchy convergence using chain estimates adapted to strong partial b-metrics. For single-valued mappings, additional assumptions were shown to ensure uniqueness of the alternating point and of the corresponding fixed points of the compositions.

The abstract result was illustrated by a finite discrete construction in which all hypotheses can be checked explicitly. The example admits exactly 42 alternating points and therefore demonstrates that existence and localization do not imply uniqueness for set-valued mappings. Instead, the process may stabilize in a family of mutually compatible terminal configurations.

Two interpretations of the same discrete structure were presented. In the economic application, the alternating mappings describe transitions between price decisions and demand responses. In the linguistic application, they represent successive stages of grammatical stabilization and lexical enrichment. Although these models are illustrative rather than empirical, they show how positive self-distances, set-valued transitions, and multiple alternating points can be used to represent residual adjustment, nonunique development paths, and stabilization without uniformity.

The results provide a basis for further study of multistage and cyclic processes in generalized metric spaces. Possible extensions include alternating systems with more than two mappings, perturbation and stability analysis, approximate alternating points, and selection criteria for distinguishing among multiple terminal configurations. Empirically calibrated generalized distances and transition correspondences may also lead to more detailed applications in economic adjustment, language development, and other processes governed by successive nonunique transformations.

A possible data-driven extension is to select $q>1$ and $\lambda >0$ by optimizing a task-specific objective on training data and evaluating the resulting choice on held-out data, with a fixed normalization of the underlying metric d on a bounded state space. Such calibration would need to account for the admissible coefficient $K=1+\lambda q{\left(2D\right)}^{q-1}$, where D is the diameter of that space. For the selected parameters, all hypotheses of Theorem 3, including the contractive and localization conditions, would still have to be verified on the intended state spaces; empirical performance on a finite sample would not establish them. Developing such a calibration procedure and assessing its possible predictive benefits remain topics for future work.