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In this paper a new definition for Menger probabilistic inner product spaces is presented. In the new setting a Menger probabilistic inner product space can naturally become a Menger probabilistic normed space, and a classical inner product space can be considered as a special case of Menger probabilistic inner product spaces. An example is given to illustrate that, the new definition is a nontrivial generalization for classical inner product spaces, and so it has rich contents in probability. By virtue of this definition, the topological structure is discussed and some elementary properties are described in terms of the families of semi-inner products. Also, some convergence theorems and probabilistic Pythagorean theorem are given. As applications, a new fixed point theorem for nonlinear contractions in Menger probabilistic inner product spaces is established. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, a class of boundary problems for second order semilinear differential inclusions is studied. By applying techniques from multivalued analysis and fixed point theorems due to Covitz–Nadler, Bohnenblust–Karlin and Schaefer, some existence results of solutions are presented for the nonlinear term which is a Lipschitz continuous, upper or lower semicontinuous multivalued map, respectively. As consequences some existence conditions of solutions are given for a class of second order semilinear differential systems. [Copyright &y& Elsevier]

Abstract: In this paper the existence and unicity of fixed points for mappings in fuzzy metric spaces (in the sense of Kaleva and Seikkala) is discussed. Nonlinear contractions of the Boyd–Wong''s type, Alber–Guerre Delabriere''s type and Kannan–Reich''s type are considered, and several new fixed point theorems for these contractions in complete fuzzy metric spaces are presented, respectively. Also, some error estimates are given for iterations to approximate fixed point. Previous work with respect to fixed point in fuzzy metric spaces is based on the t-conorm max. The presented work does away with this restriction, by proposing weaker conditions defining a generic class of suitable binary operations. As applications the corresponding fixed point theorems for Menger probabilistic metric spaces are obtained. [Copyright &y& Elsevier]

In this paper, the Leray–Schauder topological degree theory is developed in a fuzzy normed space. Since the linear topology on this fuzzy normed space is not necessarily locally convex, and since each Menger probabilistic normed space can be considered as a special fuzzy normed space, the degree theory in this paper is different from the degree theory in locally convex linear topological space presented by Nagumo (Amer. J. Math. 73 (1951) 497–511), and it also is an extension of the degree theory in Menger probabilistic normed space studied by Zhang and Chen (Appl. Math. Mech. 10(6) (1989) 477–486). Applying this degree theory, some fixed point theorems for operators are given in fuzzy normed spaces, and some former corresponding results are extended and improved. [Copyright &y& Elsevier]