Among three fixed point theorems established in the paper there is the following: Let be a complete metric space and let , , be such that for all . If (= all nonempty closed sets of ) is such that is lower-semicontinuous and for any there is with and , then for some .
This theorem generalizes results of D. Klim and D. Wiatrowski [J. Math. Anal. Appl. 334, No. 1, 132–139 (2007; Zbl 1133.54025)], Y. Feng and S. Liu [ibid., 317, No. 1, 103–112 (2006; Zbl 1094.47049)], N. Mizoguchi and W. Takahashi [ibid., 141, No. 1, 177–188 (1989; Zbl 0688.54028)].