If possible, give an example of a continuous function defined on a convex subset of a Banach space $X$ satisfies Kannan contraction but does not satisfy Banach contraction.
Contractions are always continuous and Kannan mappings are not necessarily continuous. Even more general condition in this paper: Górnicki, J. Fixed point theorems for Kannan type mappings. J. Fixed Point Theory Appl. 19, 2145–2152 (2017). https://doi.org/10.1007/s11784-017-0402-8 or here:
Fixed Point Theory and Applications Volume 2008, Article ID 649749, "Some Similarity between Contractions and Kannan Mappings" Misako Kikkawa, Tomonari Suzuki
Mieczyslaw Cichon Thank you for your consideration. I have gone through your suggested articles but unfortunately, I am sorry to say that I couldn't find any fruitful result supporting my question. If you can precisely mention a result(or, results) from those articles that will be very much helpful.
Please read the papers: section 12 in the first paper you have even more than Kannan contraction (new proposed FPT are presented) and some references ([7,8]) are directly referred to your question. If you expect ONLY a simple example, please use scholar.google.com . Please follow the papers by Gornicki or Wlodarczyk. Usually, people expect weaker conditions. If you wish: please check http://www.math.bas.bg/~serdica/2010/2010-001-010.pdf
Dear Friends, I am having one doubt about Kannan Mapping... Let T:X---> X be a Kannan Contraction Mapping and have fixed point x_0. It is possible to find a subset A of X containing x_0 so that the mapping T on A is the usual contraction?
I am having this doubt since any Kanan contraction mappings I have seen so far validate my question@.
https://www.researchgate.net/deref/http%3A%2F%2Fwww.math.bas.bg%2F~serdica%2F2010%2F2010-001-010.pdf
Article Remarks on Sarkhel’s comparison of Banach and Kannan contractions
Article Interpolative Kannan Contractions in T 0 -Quasi-Metric Spaces
Article On Some Interpolative Contractions of Suzuki Type Mappings
Article Fixed point results of T-Reich contraction, T-Kannan contrac...
Article A Common Fixed Point Theorem for Generalised F -Kannan Mappi...
Article Common Fixed-Point Theorems in the Partial -Metric Spaces an...
Article Fixed Points for Banach and Kannan Contractions in Modular S...
Article Kannan-Type Contractions on New Extended -Metric Spaces
Article On The Fixed Points Of Large-Kannan Contraction Mappings And...
Article On The Fixed Points Of Large-Kannan Contraction Mappings And...
Article Discrete Optimization for Ordered Weak Proximal Kannan Contractions
Article Some Remarks on Fixed Point Theorems for Interpolative Kanna...
Article Fixed points of Kannan contractive mappings in relational me...
Article A new extension of Kannan contractions and related fixed point results
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