Probabilistic B-contraction principles for multi-valued mappings
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The inequality F fx,fy (qs) ≥ F x,y (s) (s ≥ 0), where q ∈ (0, 1), is generalized for multi-valued mappings in many directions. In this chapter we consider three generalizations of the above inequality for multi-valued mappings, and for such a kind of mappings some fixed point theorems are proved. In section 4.1 a fixed point theorem is proved for multi-valued mappings which satisfy a multi-valued version of the strict probabilistic (b n ,)-contraction condition introduced in section 3.3. We introduce in section 4.2 the notion of a multi-valued probabilistic Ψ-contraction, and by using the notion of the function of non-compactness a fixed point theorem is proved. Using Hausdorff distance S.B. Nadler obtained in  a generalization of the Banach contraction principle in metric spaces, and in section 4.3 a probabilistic version of Nadler’s fixed point theorem is proved. As a corollary a multi-valued version of Tardiff’s fixed point theorem is obtained. In section 4.4 a probabilistic version of Itoh’s fixed point theorem from  is given, and section 4.5 contains a fixed point result for probabilistic non-expansive multi-valued mappings of Nadler’s type, defined on probabilistic metric spaces with convex structures.
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