TY - JOUR

T1 - On Some Properties of a Class of Eventually Locally Mixed Cyclic/Acyclic Multivalued Self-Mappings with Application Examples

AU - Sen, Manuel De la

AU - Ibeas, Asier

N1 - Publisher Copyright:
© 2022 by the authors.

PY - 2022/7/11

Y1 - 2022/7/11

N2 - In this paper, a multivalued self-mapping is defined on the union of a finite number of subsets (Formula presented.) of a metric space which is, in general, of a mixed cyclic and acyclic nature in the sense that it can perform some iterations within each of the subsets before executing a switching action to its right adjacent one when generating orbits. The self-mapping can have combinations of locally contractive, non-contractive/non-expansive and locally expansive properties for some of the switching between different pairs of adjacent subsets. The properties of the asymptotic boundedness of the distances associated with the elements of the orbits are achieved under certain conditions of the global dominance of the contractivity of groups of consecutive iterations of the self-mapping, with each of those groups being of non-necessarily fixed size. If the metric space is a uniformly convex Banach one and the subsets are closed and convex, then some particular results on the convergence of the sequences of iterates to the best proximity points of the adjacent subsets are obtained in the absence of eventual local expansivity for switches between all the pairs of adjacent subsets. An application of the stabilization of a discrete dynamic system subject to impulsive effects in its dynamics due to finite discontinuity jumps in its state is also discussed.

AB - In this paper, a multivalued self-mapping is defined on the union of a finite number of subsets (Formula presented.) of a metric space which is, in general, of a mixed cyclic and acyclic nature in the sense that it can perform some iterations within each of the subsets before executing a switching action to its right adjacent one when generating orbits. The self-mapping can have combinations of locally contractive, non-contractive/non-expansive and locally expansive properties for some of the switching between different pairs of adjacent subsets. The properties of the asymptotic boundedness of the distances associated with the elements of the orbits are achieved under certain conditions of the global dominance of the contractivity of groups of consecutive iterations of the self-mapping, with each of those groups being of non-necessarily fixed size. If the metric space is a uniformly convex Banach one and the subsets are closed and convex, then some particular results on the convergence of the sequences of iterates to the best proximity points of the adjacent subsets are obtained in the absence of eventual local expansivity for switches between all the pairs of adjacent subsets. An application of the stabilization of a discrete dynamic system subject to impulsive effects in its dynamics due to finite discontinuity jumps in its state is also discussed.

KW - impulsive dynamic systems

KW - uniformly convex Banach space

KW - stabilization

KW - cyclic self-mappings

KW - mixed cyclic/acyclic self-mappings

KW - cyclic contractions

KW - DELAY SYSTEMS

KW - EXISTENCE

KW - STABILITY

KW - CYCLIC QUASI-CONTRACTIONS

KW - PROXIMITY POINT THEOREMS

UR - https://doi.org/10.3390/math10142415

UR - http://www.scopus.com/inward/record.url?scp=85136918243&partnerID=8YFLogxK

UR - https://www.mendeley.com/catalogue/569e82bb-fd23-3005-ab3f-b46b220e349f/

U2 - 10.3390/math10142415

DO - 10.3390/math10142415

M3 - Article

VL - 10

IS - 14

M1 - 2415

ER -