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Unifying a multitude of common fixed point theorems employing an implicit relation. (English) Zbl 1168.54325
Summary: A general common fixed point theorem for two pairs of weakly compatible mappings using an implicit function is proved without any continuity requirement which generalizes the result due to V. Popa [Rad. Mat. 10, No. 2, 245–252 (2001; Zbl 1064.54506), Theorem 3]. In process, several previously known results due to Fisher, Kannan, Jeong and Rhoades, Imdad and Ali, Imdad and Khan, Khan, Shahzad and others are derived as special cases. Some related results and illustrative examples are also discussed. As an application of our main result, we prove an existence theorem for the solution of simultaneous Hammerstein type integral equations.

MSC:
54H25 Fixed-point and coincidence theorems (topological aspects)
47H10 Fixed-point theorems
47H30 Particular nonlinear operators (superposition, Hammerstein, Nemytskiń≠, Uryson, etc.)
65J15 Numerical solutions to equations with nonlinear operators (do not use 65Hxx)
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