- Open Access
Erratum to: Common Fixed Point Theorems for Hybrid Pairs of Occasionally Weakly Compatible Mappings Satisfying Generalized Contractive Condition of Integral Type Revisited
© M. Abbas and B. E. Rhoades. 2008
- Received: 3 September 2008
- Accepted: 30 September 2008
- Published: 11 November 2008
The original article was published in Fixed Point Theory and Applications 2007 2007:054101
We are indebted to Valeriu Popa for pointing out our error in . In looking again at the paper, we came up with the following example.
Let with the usual metric, and define by . Since , every point is a coincidence point, and . Also, for all and , and for , so and satisfy the hypotheses of all theorems and corollaries in , but and have no common fixed point.
Thus, it is not surprising that there are a number of papers involving hybrid pairs in which the conclusion of the theorems is not a common fixed point, but a common coincidence point (see, e.g., [2–10]). To obtain a common fixed point, a number of theorems assume the strong condition that the common coincidence point is also a fixed point of one of the maps.
- Abbas M, Rhoades BE: Common fixed point theorems for hybrid pairs of occasionally weakly compatible mappings satisfying generalized contractive condition of integral type. Fixed Point Theory and Applications 2007, 2007:-9.Google Scholar
- Constantin A: Coincidence point theorems for multivalued contraction mappings. Mathematica Japonica 1991, 36(5):925–933.MathSciNetMATHGoogle Scholar
- Imdad M, Ahmad A, Kumar S: On nonlinear nonself hybrid contractions. Radovi Matematički 2001, 10(2):233–244.MathSciNetMATHGoogle Scholar
- Kamran T: Coincidence and fixed points for hybrid strict contractions. Journal of Mathematical Analysis and Applications 2004, 299(1):235–241. 10.1016/j.jmaa.2004.06.047MathSciNetView ArticleMATHGoogle Scholar
- Kubiaczyk I, Deshpande B: Coincidence point for noncompatible multivalued maps satisfying an implicit relation. Demonstratio Mathematica 2006, 39(4):855–862.MathSciNetMATHGoogle Scholar
- Naidu SVR: Fixed points and coincidence points for multimaps with not necessarily bounded images. Fixed Point Theory and Applications 2004, 2004(3):221–242. 10.1155/S1687182004308090MathSciNetView ArticleMATHGoogle Scholar
- Pathak HK, Mishra SN: Coincidence points for hybrid mappings. Rostocker Mathematisches Kolloquium 2004, (58):67–85.Google Scholar
- Singh SL, Giniswamy : Concidences and fixed point theorems for single valued and multivalued maps. Fixed Point Theory and Applications 2004, 2004(5):127–139.MathSciNetGoogle Scholar
- Singh SL, Ha KS, Cho YJ: Coincidence and fixed points of nonlinear hybrid contractions. International Journal of Mathematics and Mathematical Sciences 1989, 12(2):247–256. 10.1155/S0161171289000281MathSciNetView ArticleMATHGoogle Scholar
- Singh SL, Mishra SN: Coincidences and fixed points of nonself hybrid contractions. Journal of Mathematical Analysis and Applications 2001, 256(2):486–497. 10.1006/jmaa.2000.7301MathSciNetView ArticleMATHGoogle Scholar
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