Abstract
AbstractIn this paper, we obtain a fixed point theorem for mappings satisfying cyclic φ-contractive conditions in complete metric spaces, which gives a positive answer to the question raised by Radenović (Fixed Point Theory Appl. 2015:189, 2015). We also find that this result and the fixed point result satisfying cyclic weak ϕ-contractions given by Karapınar (Appl. Math. Lett. 24:822-825, 2011) are independent of each other. Furthermore, when the number of cyclic sets is odd, we obtain fixed point theorems satisfying cyclic weak ϕ-contractions and cyclic φ-contractions in the setting of generalized metric spaces.
Highlights
Introduction and preliminariesThe main purpose of this paper is to answer an open question raised by Radenović in [ ]
In order to go further, we attempt to extend our result and the result established by Karapınar [, ] to the setting of generalized metric spaces
We show these results are valid in generalized metric spaces when the number of cyclic sets is odd
Summary
Φ is a comparison function, but it is not a strong comparison function. Many authors considered fixed point results about cyclic φ-contractions in setting of different type of spaces; see, for example, [ – ]. In [ ], Radenović obtained a fixed point theorem for non-cyclic φ-contraction, where φ is comparison function, and raised the following question. Let {Ai}pi be nonempty closed subsets of a complete metric space, and suppose f :. (ii) there exists a comparison function φ : [ , ∞) → [ , ∞) such that d(fx, fy) ≤ φ d(x, y) , for any x ∈ Ai, y ∈ Ai+ , ≤ i ≤ p
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