Set-valued Leader type contractions, periodic point and endpoint theorems, quasi-triangular spaces, Bellman and Volterra equations

• Main Author: Kazimierz Włodarczyk
• Format: Article
• Language: English
• Published: SpringerOpen 2020-03-01
• Series: Fixed Point Theory and Applications
• Subjects: Quasi-triangular space; Set-valued Leader contraction; Dynamic process; Periodic point; Fixed point; Endpoint
• Online Access: http://link.springer.com/article/10.1186/s13663-020-00673-1

Abstract Set-valued contractions of Leader type in quasi-triangular spaces are constructed, conditions guaranteeing the existence of nonempty sets of periodic points, fixed points and endpoints of such contractions are established, convergence of dynamic processes of these contractions are studied, uniqueness properties are derived, and single-valued cases are considered. Investigated dynamic systems are not necessarily continuous and spaces are not necessarily sequentially complete or Hausdorff. Obtained results suggest, in particular, strategies to new studies of functional Bellman equations and variable discounted Bellman equations in metric spaces and integral Volterra equations in locally convex spaces. Results in this direction are also presented in this paper. More precisely, without continuity of Bellman and Volterra appropriate operators, the sets of solutions of these equations (which are periodic points of these operators) are studied and new and general convergence, existence and uniqueness theorems concerning such equations are proved.