Caristi-Like Condition and the Existence of Minima of Mappings in Partially Ordered Spaces
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In this paper, we study mappings acting in partially ordered spaces. For these mappings, we introduce a condition, analogous to the Caristi-like condition, used for functions defined on metric spaces. A proposition on the achievement of a minimal point by a mapping of partially ordered spaces is proved. It is shown that a known result on the existence of the minimum of a lower semicontinuous function defined on a complete metric space follows from the obtained proposition. New results on coincidence points of mappings of partially ordered spaces are obtained.
KeywordsPartially ordered space Caristi-like condition Coincidence point Orderly covering mapping
Mathematics Subject Classification06A06 65K10
The publication was supported by a grant from the Russian Science Foundation (Project No. 17-11-01168). Authors are grateful to anonymous referees for useful comments and remarks.
- 3.Arutyunov, A.V., Gel’man, B.D., Zhukovskiy, E.S., Zhukovskiy, S.E.: Caristi-like condition. Existence of solutions to equations and minima of functions in metric spaces. Fixed Point Theory (2019, to appear)Google Scholar
- 8.Turinici, M.: Contraction maps in ordered metrical structures. In: Pardalos, P., Rassias, T. (eds.) Mathematics Without Boundaries, pp. 533–575. Springer, New York (2014)Google Scholar
- 14.Kolmogorov, A.N., Fomin, S.V.: Elements of the Theory of Functions and Functional Analysis. Dover Publications Inc., New York (1999)Google Scholar