On a Pseudo-Orthogonality Condition Related to Cyclic Self-Mappings in Metric Spaces and Some of Their Relevant Properties

This paper relies on orthogonal metric spaces related to cyclic self-mappings and some of their relevant properties. The involved binary relation is not symmetric, and then the term pseudo-orthogonality will be used for the relation used in the article to address the established results on cyclic self-mappings. Firstly, some orthogonal binary relations are given through examples to fix some ideas to be followed in the main body of the article. It is seen that the orthogonal elements of the orthogonal sets are not necessarily singletons. Secondly, “ad hoc” specific orthogonality binary relations are also described through examples related to the investigation of stability and controllability problems in dynamic systems. The main objective of this paper is to investigate the properties of cyclic single-valued self-mappings on the union of any finite number p≥2 of nonempty closed subsets of a metric space in a cyclic disposal under an “ad hoc” defined pseudo-orthogonality condition. Such a condition is defined on certain subsequences, referred to as pseudo-orthogonal sequences, rather than on the whole generated sequences under the self-mapping. It basically consists of a cyclic, in general iteration-dependent, contractive condition just for such subsequences which, on the other hand, are not forced as a constraint to be fulfilled by the whole sequences. Furthermore, the whole sequences in which those sequences are contained are allowed to be locally non-contractive or even locally expansive. The boundedness and the convergence properties of distances between pseudo-orthogonal subsequences and sequences are investigated under the condition that one of the subsets has a unique best proximity point to its adjacent subset in the cyclic disposal to which the pseudo-orthogonal subsequences converge. The pseudo-orthogonal metric subspace of the given metric space is proved to be complete although the whole metric space is not assumed to be complete. The pseudo-orthogonal element is seen to be a set of best proximity points, one per subset of the cyclic disposal, although it is not required for all the best proximity sets to be singletons. It is proved that the pseudo-orthogonal subsequences converge to a limit cycle, consisting of a best proximity point per subset of the cyclic disposal, which is also the pseudo-orthogonal element. The whole sequences are also proved to be bounded and the distances between their elements in adjacent subsets are also proved to converge to the distance between adjacent subsets. In the event that the metric space is a uniformly convex Banach space, it suffices that one of the subsets of the cyclic disposal be boundedly compact with its best proximity set being a singleton. In this case, the pseudo-orthogonal sequences converge to their best proximity set to their adjacent subset provided that such a best proximity set is a singleton.