On generalizations of the Meir-Keeler type contraction maps

In 1969, Meir and Keeler [29] obtained a remarkable generalization of the Banach contraction principle. Since then, there have appeared a number of generalizations of their result. In 1981, the second author and Bae [33] extended the Meir-Keeler theorem to two commuting maps by adopting Jungck’s method. This influenced many authors, and, consequently, a number of new results in this line followed. Recent works of Sessa and others [46,47] contain common fixed point theorems of four maps satisfying certain contractive type conditions. In the present paper, we give a new result which encompasses most of such generalizations of the Meir-Keeler theorem. Further our result also includes many other generalizations of the Banach contraction principle. Some authors have obtained their results on 2-metric spaces. However, 2-metric versions are easily obtained from metric ones by an obvious

[1]  G. E. Hardy,et al.  A Generalization of a Fixed Point Theorem of Reich , 1973, Canadian Mathematical Bulletin.

[2]  Tudor Zamfirescu,et al.  Fix point theorems in metric spaces , 1972 .

[3]  Sehie Park,et al.  Extensions of a fixed point theorem of Meir and Keeler , 1981 .

[4]  On Common Fixed Points of Commuting Mappings , 1977 .

[5]  B. K. Ray.  On Ciric's fixed point theorem , 1977 .

[6]  Simeon Reich,et al.  Some Remarks Concerning Contraction Mappings , 1971, Canadian Mathematical Bulletin.

[7]  G. Jungck.  Compatible mappings and common fixed points , 1986 .

[8]  L. Ciric,et al.  Generalized contractions and fixed-point theorems. , 1971 .

[9]  D. R. Smart.  Fixed Point Theorems , 1974 .

[10]  R. Kannan,et al.  Some Results on Fixed Points—II , 1969 .

[11]  B. Rhoades,et al.  A comparison of various definitions of contractive mappings , 1977 .

[12]  T. Kubiak.  Common Fixed Points of Pairwise Commuting Mappings , 1984 .

[13]  S. K. Chatterjea.  Fixed Point Theorems For A Sequence Of Mappings With Contractive Iterates , 1972 .

[14]  V. Sehgal.  Some Fixed and Common Fixed Point Theorems in Metric Spaces , 1974, Canadian Mathematical Bulletin.

[15]  J. S. W. Wong,et al.  On nonlinear contractions , 1969 .

[16]  Emmett B. Keeler,et al.  A theorem on contraction mappings , 1969 .

[17]  E. Begle.  A FIXED POINT THEOREM , 1950 .

[18]  B. Fisher.  Mappings with a Common Fixed Point , 1979 .

[19]  B. Fisher,et al.  Some fixed point theorems for commuting mappings , 1982 .

[20]  Kiyoshi Iseki,et al.  On common fixed points of mappings , 1974, Bulletin of the Australian Mathematical Society.

[21]  F. Browder.  On the convergence of successive applications for nonlinear functional equations , 1968 .

[22]  B. Fisher,et al.  Two common fixed point theorems for weakly commuting mappings , 1989 .

[23]  Salvatore Sessa,et al.  ON A WEAK COMMUTATIVITY CONDITION OF MAPPINGS IN FIXED POINT CONSIDERATIONS , 1982 .