Fixed point theorem for a kind of \'{C}iri\'{c} type contractions in complete metric spaces

Year 2018, , 33 - 41, 25.03.2018

Tomonari Suzuki

https://doi.org/10.31197/atnaa.375870

Cited By: 4

Abstract

We prove a fixed point theorem  for a kind of C^'iric^' type contractions  in complete metric spaces. In order to demonstrate the assumption of the fixed point theorem,  we give an example. We also clarify the mathematical structure of  some fixed point theorem  proved by Mınak-Helvacı-Altun and  Wardowski-Dung independently.

Keywords

Fixed point, $F$-contraction, contractive condition

References

• F. E. Browder, {\it On the convergence of successive approximations for nonlinear functional equations}, Nederl.\ Akad.\ Wetensch.\ Proc.\ Ser.\ A 71=Indag.\ Math.\ 30 (1968), 27--35. % MR0230180 % 10.1016/S1385-7258(68)50004-0
• Lj. B. \'Ciri\'c, {\it A new fixed-point theorem for contractive mappings}, Publ.\ Inst.\ Math.\ (Beograd), 30 (1981), 25--27. % MR0672538
• M. Heged\"{u}s and T. Szil\'{a}gyi, {\it Equivalent conditions and a new fixed point theorem in the theory of contractive type mappings}, Math.\ Japon., 25 (1980), 147--157. % MR0571276
• J. Jachymski, {\it Equivalent conditions and the Meir-Keeler type theorems}, J.\ Math.\ Anal.\ Appl., 194 (1995), 293--303. % MR1353081 % 10.1006/jmaa.1995.1299
• J. Matkowski, {\it Fixed point theorems for contractive mappings in metric spaces}, \v Casopis P\v est.\ Mat., 105 (1980), 341--344. % MR0597909
• G. M\i{}nak, A. Helvac\i{} and \.{I}. Altun, {\it \'{C}iri\'{c} type generalized $F$-contractions on complete metric spaces and fixed point results}, Filomat, 28 (2014), 1143--1151. % MR3360088
• T. Suzuki, {\it Discussion of several contractions by Jachymski's approach}, Fixed Point Theory Appl., 2016, 2016:91. % MR3548683 % 10.1186/s13663-016-0581-9
• T. Suzuki, {\it Characterizations of contractive conditions by using convergent sequences}, Fixed Point Theory Appl., 2017, 2017:30. % 10.1186/s13663-017-0623-y
• T. Suzuki, {\it A generalization of Heged\"{u}s-Szil\'{a}gyi's fixed point theorem in complete metric spaces}, Fixed Point Theory Appl., 2018, 2018:1. % 10.1186/s13663-017-0625-9
• D. Wardowski, {\it Fixed points of a new type of contractive mappings in complete metric spaces}, Fixed Point Theory Appl., 2012, 2012:94. % MR2949666
• D. Wardowski and N. V. Dung, {\it Fixed points of $F$-weak contractions on complete metric spaces}, Demonstr.\ Math., 47 (2014), 146--155. % MR3200192