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

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  • M. De la Sen
  • E. Karapinar

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.

Suggested Citation

  • M. De la Sen & E. Karapinar, 2013. "Best Proximity Points of Generalized Semicyclic Impulsive Self‐Mappings: Applications to Impulsive Differential and Difference Equations," Abstract and Applied Analysis, John Wiley & Sons, vol. 2013(1).
    • Handle: RePEc:wly:jnlaaa:v:2013:y:2013:i:1:n:505487
    DOI: 10.1155/2013/505487
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    References listed on IDEAS

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    Cited by:

    1. Marta Demma & Mohamed Jleli & Bessem Samet & Calogero Vetro, 2014. "A Note on Best Approximation in 0‐Complete Partial Metric Spaces," Abstract and Applied Analysis, John Wiley & Sons, vol. 2014(1).
    2. M. De la Sen, 2016. "On Weak Contractive Cyclic Maps in Generalized Metric Spaces and Some Related Results on Best Proximity Points and Fixed Points," Discrete Dynamics in Nature and Society, John Wiley & Sons, vol. 2016(1).
    3. Manuel De la Sen & Asier Ibeas, 2014. "Convergence Properties and Fixed Points of Two General Iterative Schemes with Composed Maps in Banach Spaces with Applications to Guaranteed Global Stability," Abstract and Applied Analysis, John Wiley & Sons, vol. 2014(1).

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