Abstract:
Some of the properties of vector-valued fuzzy multifunctions are studied. The notion of sum fuzzy multifunction, convex hull fuzzy multifunction, close convex hull fuzzy multifunction, and upper demicontinuous are given, and some of the properties of these fuzzy multifunctions are investigated.

Abstract:
In this paper we prove random fixed point theorems in reflexive Banach spaces for nonexpansive random operators satisfying inward or Leray-Schauder condition and establish a random approximation theorem.

Abstract:
We define the upper and lower inverse of a fuzzy multifunction and prove basic identities. Then by using these ideas we introduce the concept of hemicontinuity and obtain many interesting properties of lower and upper hemicontinuous fuzzy multifunctions. Using the notion of hemicontinuity, we also characterize closed and open fuzzy mappings.

Abstract:
We construct a random iteration scheme and study necessary conditions for its convergence to a common random fixed point of two pairs of compatible random operators satisfying Meir-Keeler type conditions in Polish spaces. Some random fixed point theorems for weakly compatible random operators under generalized contractive conditions in the framework of symmetric spaces are also proved.

Abstract:
We explore how judgment aggregation and belief merging in the framework of fuzzy logic can help resolve the “Doctrinal Paradox.” We also illustrate the use of fuzzy aggregation functions in social choice theory. 1. Introduction Social choice theory defines “preference aggregation” as forming collective preferences over a given set of alternatives. Likewise, “judgment aggregation” pertains to forming collective judgments on a given set of logically interrelated propositions. This paper extends beyond classical propositional logic into the realm of general multivalued logic, so that we can handle realistic collective decision problems (see Dietrich and List [1, 2], List [3], Beg and Butt [4], and Manzini and Mariotti [5]). List and Pettit [6, 7] were the first to give an axiomatic treatment to the problem associated with judgment aggregation. In their classic example, a set of propositions is expressed in propositional calculus as . The set consists of all assignments of 0 or 1 to the propositions in that are logically consistent. A procedure for judges to decide on the truthfulness of each proposition in amounts to an aggregator that maps . The “Doctrinal Paradox” illustrates that proposition-wise majority rule leads to inconsistent collective decisions. This paradox has made the literature on “judgment aggregation” grow appreciably. Most of the discussions on this paradox have been in the domain of social choice theory, and a number of “(im)possibility theorems,” similar to those of Arrow [8] and Sen [9] have been proved. In fact, these theorems show that there cannot exist any judgment aggregation procedure that simultaneously satisfies certain minimal consistency requirements (see Dietrich [10]). List and Pettit [6] have shown that the majority rule is but one member of a class of aggregation procedures that fails to ensure consistency in the set of collective judgments. Van Hees [11] has further generalized the paradox by showing that there is even a larger class of aggregation procedures for which this is true. The aim of this paper is to resolve the paradox and also to illustrate optimal judgment aggregation. We abandon the assumption that individual and collective beliefs necessarily have a binary nature (true or false) and so our analysis is in a fuzzy logic framework. Pigozzi [12] discusses a possibility result in binary logic in which the paradox is avoided at the price of “indecision.” Distance-based aggregation procedures like that of Pigozzi [12] often result in dictatorship. Accordingly, aggregation procedures in fuzzy logic can help us

Abstract:
Probabilistic version of the invariance of domain for contractive field and Schauder invertibility theorem are proved. As an application, the stability of probabilistic open embedding is established.

Abstract:
We generate a sequence of measurable mappings iteratively and study necessary conditions for its strong convergence to a random fixed point of strongly pseudocontractive random operator. We establish the weak convergence of an implicit random iterative procedure to common random fixed point of a finite family of nonexpansive random operators in Hilbert spaces. We prove the equivalence between the convergence of random Ishikawa and random Mann iterative schemes for contraction random operator and strongly pseudocontractive random operator. We also examine the stability of random fixed point iterative procedures for the random operators satisfying certain contractive conditions in the context of metric spaces.

Abstract:
We prove the existence of a common random fixed point of two asymptotically nonexpansive random operators through strong and weak convergences of an iterative process. The necessary and sufficient condition for the convergence of sequence of measurable functions to a random fixed point of asymptotically quasi-nonexpansive random operators in uniformly convex Banach spaces is also established.

Abstract:
Let be a metric space and S, T be mappings from X to a set of all fuzzy subsets of X. We obtained sufficient conditions for the existence of a common α-fuzzy fixed point of S and T.