Fixed point theorems for the ΨΨ-set contraction mapping on almost convex sets

In this paper, we first discuss the properties of the almost convex subsets of a Hausdorff topological vector space. Then, in the setting of the almost convex sets, we establish some fixed point theorems for the two KKM∗KKM∗ and BDBD type set-valued mappings with either ΨΨ-set contraction or 1-set contraction. Next, we also establish new cycle point theorems for three set-valued mappings. Our results generalize the results of Chang et al. [T.H. Chang, C.L. Yen, KKM property and fixed point theorems, J. Math. Anal. Appl. 203 (1996), 224–235.] and Park [S. Park, Fixed point theorems for better admissible multimaps on almost convex sets, J. Math. Anal. Appl. 329 (2007) 690–702].