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The concept of r-semigroup is a generalization of semigroups. In this paper, we briefly introduce the action of left (right) operator semigroups on a r-semigroup and deduce in particular that there exists an inclusion preserving bijection between the set of all right ideals of S and the set all right ideals of L x S. [ABSTRACT FROM AUTHOR]

The purpose of this article is to extend Banach's contraction principle through a new rational expression in the contractive condition to establish the existence and uniqueness of fixed point of a closed subset of Hilbert space to a self mapping. The result is extended to a pair of self mappings and positive integers powers of a pair mapping and further extended to a sequence of mappings in the space. The presented results extend and generalized various known comparable results from the current literature. [ABSTRACT FROM AUTHOR]

This paper presents some fundamental results tacit in [20] precisely on regular Γ-semigroups and operator semigroups of a Γ-semigroup, and an overview of some classes of Γ-semigroups yet to be surveyed. [ABSTRACT FROM AUTHOR]