We have obtained a new class of ordered pairs of multivalued maps that have pairs of coupled fixed points. We illustrate the main result with two examples that cover a wide range of models. We apply the main result in models in duopoly markets to get a market equilibrium and in aquatic ecosystems, also to get an equilibrium.

We generalize the notion of coupled fixed (or best proximity) points for cyclic ordered pairs of maps to p-cyclic ordered pairs of maps. We find sufficient conditions for the existence and uniqueness of the coupled fixed (or best proximity) points. We illustrate the results with an example that covers a wide class of maps.

This paper presents a new coupled fixed-point theorem for a pair of set-valued mappings acting on the Cartesian product of $(m_1, m_2)$- and $(n_1, n_2)$-quasi-metric spaces. Within the general, non-symmetric quasi-metric setting, we establish the existence of an approximate coupled fixed point. Moreover, under the additional assumption of $q_0$-symmetry, we guarantee the existence of a coupled fixed point. Together, these results extend and unify several known theorems in fixed-point theory for quasi-metric and asymmetric spaces. We illustrate the obtained results regarding fixed points when the underlying space is equipped with a graph structure and, thus, sufficient conditions are found to guarantee the existence of a subgraph with a loop with a length greater than or equal to 2.

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

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.