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Some common fixed point theorems in Menger PM spaces. (English) Zbl 1204.54032
Summary: Employing the common property (E.A), we prove some common fixed point theorems for weakly compatible mappings via an implicit relation in Menger PM spaces. Some results on similar lines satisfying quasicontraction conditions as well as $\psi$-type contraction conditions are also proved in Menger PM spaces. Our results substantially improve the corresponding theorems contained in A. Branciari [Int. J. Math. Math. Sci. 29, No. 9, 531–536 (2002; Zbl 0993.54040)], B. E. Rhoades [ibid. 2003, No. 63, 4007–4013 (2003; Zbl 1052.47052)], and P. Vijayaraju, B. E. Rhoades and R. Mohanraj [ibid. 2005, No. 15, 2359–2364 (2005; Zbl 1113.54027)] and also some others in Menger as well as metric spaces. Some related results are also derived besides furnishing illustrative examples.
##### MSC:
 54H25 Fixed-point and coincidence theorems in topological spaces 47H10 Fixed point theorems for nonlinear operators on topological linear spaces 54E70 Probabilistic metric spaces