Best Proximity Points of Generalized Semicyclic Impulsive Self-Mappings: Applications to Impulsive Differential and Difference Equations

Abstract

This paper is devoted to the study of convergence properties of distances between points and the existence and uniqueness of best proximity and fixed points of the so-called semicyclic impulsive self-mappings on the union of a number of nonempty subsets in metric spaces. The convergences of distances between consecutive iterated points are studied in metric spaces, while those associated with convergence to best proximity points are set in uniformly convex Banach spaces which are simultaneously complete metric spaces. The concept of semicyclic self-mappings generalizes the well-known one of cyclic ones in the sense that the iterated sequences built through such mappings are allowed to have images located in the same subset as their pre-image. The self-mappings under study might be in the most general case impulsive in the sense that they are composite mappings consisting of two self-mappings, and one of them is eventually discontinuous. Thus, the developed formalism can be applied to the study of stability of a class of impulsive differential equations and that of their discrete counterparts. Some application examples to impulsive differential equations are also given.

Some Results on Best Proximity Points of Cyclic Contractions in Probabilistic Metric Spaces

“…It has also to be pointed out that the parallel background literature related to best proximity points and fixed points in cyclic mappings in metric and Banach spaces is exhaustive. See, for instance, [15][16][17][18][19][20][21][22][23][24][25][26][27][28] and references therein. Fixed point theory has also been widely applied to stability and equilibrium problems since, even based on intuition, the convergence of trajectory-solutions of differential or difference equations or dynamic systems to a point can be typically associated to the convergence of sequences to fixed points; see, for instance, [27,29,30] and references therein, and to ergodic processes [31].…”

Best Proximity Point Theorems for Multi–valued Mappings in Complete Metric Spaces

“…In 2006, Eldred and Veeramani [3] proved the following existence theorem. Recently, best proximity point theorems for various types of contractions have been obtained in [1,8,9,10,14,16]. Definition 1.1.…”

On Weak Contractive Cyclic Maps in Generalized Metric Spaces and Some Related Results on Best Proximity Points and Fixed Points

“…The literature includes related studies on cyclic contractions and cyclic weak contractions and proximal contractions [1][2][3][4][5][6][7][8][9][10][11][12][13][14][18][19][20][21] and proximal weak contractions [15][16][17]. See also [22][23][24][25] for related results. On the other hand, fixed point theory has a wide amount of applications, for instance, in the study of stability of dynamic systems and differential and difference equations.…”

“…On the other hand, fixed point theory has a wide amount of applications, for instance, in the study of stability of dynamic systems and differential and difference equations. See, for instance, [21,22,26]. In this context, the relevance of cyclic contractions and cyclic nonexpansive mappings is also of interest when strips of the solutions of dynamic systems or difference equations have to lie in different time intervals or due to control actions or external events in distinct defined sets.…”