Fixed point theorems of local contraction mappings on Menger spaces (Q1192875)

The concept of \(\varepsilon\)-chainable PM-space is introduced, which is a strengthening form of the well-known definition of \((\varepsilon,\lambda)\)-chainable PM-space. By using this concept and under some additional conditions, the author proves some fixed point theorems for single-valued and set-valued local contraction mappings on this kind of spaces. The results presented in this paper extend the corresponding results for local contraction mappings on metric spaces given in [\textit{M. Edelstein}, Proc. Am. Math. Soc. 12, 7-10 (1961; Zbl 0096.171) and \textit{P. K. F. Kuhfittig}, Pac. J. Math. 65, No. 2, 399-403 (1976; Zbl 0335.54035)].