Alternating Points of Set-Valued Mappings in Strong Partial b-Metric Spaces with Applications to Economic and Linguistic Adjustment
4 affiliations
- 1Department of Mathematical Analysis, Faculty of Mathematics and Informatics, University of Plovdiv Paisii Hilendarski, 4000 Plovdiv, Bulgaria
- 2Department of Computer Systems, Faculty of Mathematics and Informatics, University of Plovdiv Paisii Hilendarski, 4000 Plovdiv, Bulgaria
- 3Department of Mathematics and Physics, Technical University of Varna, 1 Studentska Str., 9000 Varna, Bulgaria
- 4Faculty of Philology, University of Plovdiv Paisii Hilendarski, 4000 Plovdiv, Bulgaria
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.
Related
- Existence and Approximation of Fixed Points for a Novel Class of Mappings with Applications
- Monotonicity and bounded variation for harmonic mappings with applications to special functions
- Analytic Combinatorics of d-Set Mappings and Their Applications
- A Composite Representation Theorem for Solution Mappings in Regularized Composite Optimization with Applications to Hausdorff Lipschitz Stability
- Applications of the fixed point theorem for mappings contracting perimeters of triangles: ODEs, PDEs, integer and difference equations
BibTeX
@article{ilchev2026alternating,
title = {{Alternating Points of Set-Valued Mappings in Strong Partial b-Metric Spaces with Applications to Economic and Linguistic Adjustment}},
author = {Atanas Ilchev and Vanya Ivanova and Rumen Marinov and Diana Nedelcheva and Angel Todorov and Boyan Zlatanov},
journal = {Axioms},
year = {2026},
month = oct,
doi = {10.3390/axioms15100740},
url = {https://www.mdpi.com/2075-1680/15/10/740},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - Alternating Points of Set-Valued Mappings in Strong Partial b-Metric Spaces with Applications to Economic and Linguistic Adjustment
AU - Atanas Ilchev
AU - Vanya Ivanova
AU - Rumen Marinov
AU - Diana Nedelcheva
AU - Angel Todorov
AU - Boyan Zlatanov
PY - 2026
DA - 2026/10/02
JO - Axioms
DO - 10.3390/axioms15100740
UR - https://www.mdpi.com/2075-1680/15/10/740
AB - 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.
SN - 2075-1680
ER -