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.

In this short note we present a new proof of Ekeland's variational principle and Caristi's fixed point theorem using a recently proved constrained variational principle in completely regular topological spaces.

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.