Abstract
In this paper, the following facts are stated in the setting of b-metric spaces.
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(1)
The contraction constant in the Banach contraction principle fully extends to [0, 1), but the contraction constants in Reich’s fixed point theorem and many other fixed point theorems do not fully extend to [0, 1), which answers the early stated question on transforming fixed point theorems in metric spaces to fixed point theorems in b-metric spaces.
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(2)
Caristi’s theorem does not fully extend to b-metric spaces, which is a negative answer to a recent Kirk–Shahzad’s question (Remark 12.6) [Fixed Point Theory in Distance Spaces. Springer, 2014].
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To Professor Tran Van An, Vinh University, on the occasion of his 60th birthday
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Dung, N.V., Hang, V.T.L. On relaxations of contraction constants and Caristi’s theorem in b-metric spaces. J. Fixed Point Theory Appl. 18, 267–284 (2016). https://doi.org/10.1007/s11784-015-0273-9
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DOI: https://doi.org/10.1007/s11784-015-0273-9