Some results on fixed points (Q5896344)

We first show several fixed point and/or common fixed point theorems for point-valued and set-valued contractive type mappings in compact metric spaces, which satisfy very general contractivity condition with strict inequality and may be discontinuous. These theorems improve and generalize some main results of \textit{M. P. Chen} and \textit{M. H. Shih} [Math. Jap. 21, 281-282 (1976; Zbl 0356.54049)], \textit{B. E. Rhoades} [Trans. Am. Math. Soc. 226, 257-290 (1977; Zbl 0365.54023)], \textit{S. Kasahara} and \textit{B. E. Rhoades} [Math. Jap. 23, 227-229 (1978; Zbl 0402.54048)] and the author [Sichuan Shiyuan Xuebao, A special issue of Math. 26-34 (1981)]. Further we provide some new results for quite general orbitally contractive and quasi-contractive mappings. They unify and extend results of the author [Chin. Ann. Math. 2, 511-517 (1981; Zbl 0486.54039)], \textit{F. E. Browder} [Nonlinear Anal., Theory Methods Appl. 3, 657-661 (1979; Zbl 0427.47037)], \textit{W. Walter} [ibid. 5, 21-25 (1981; Zbl 0461.47032)], \textit{S. Park} and \textit{B. E. Rhoades} [Acta Sci. Math. 42, 299-304 (1980; Zbl 0449.54045)] and others.