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The geometry of pseudo-slant submanifolds of nearly quasi Sasakian manifold is studied. It is proved that totally umbilical proper-slant submanifold of nearly quasi Sasakian manifold admits totally geodesic if the mean curvature vector ∈ µ. The integrability conditions of the distributions of pseudo-slant submanifolds of nearly quasi Sasakian manifold are also obtained.

Let be a Banach space, let K be a non-empty closed subset of and let Ì Ã be a non-self mapping.The main result of this paper is that if Ì satisfies the contractive-type condition (1.1) below and maps Ã( à the boundary of Ã) into à then Ì has a unique fixed point in à 0 Intoduction If´ µ is a complete metric space, à is a nonempty closed subset of and Ì Ã Ã a self-mappingthen Banach Contraction Principle states that Ì has a unique fixed point, say Þ in à and the Picard iterations Ì Ò Ü converge to Þ for all Ü ¾ à Rhoades [13] pointed out that one of the most general contractive-type definitions of mappings for which such theorems have been proved is introduced in [6]: for all Ü Ý ¾ à ´Ì Ü Ì Ý µ Å´Ü Ýµ (0.2) where ¼ ½ and Å´Ü Ýµ Ñ Ü ´Ü ݵ ´Ü Ìܵ ´Ý Ìݵ ´Ü Ìݵ ´Ý Ìܵ However, in many applications, the functions involved are not self-mappings of K. So it is of an interest to prove fixed point theorems for as wide a class of non-self mappings as possible. In 1972 Assad and Kirk [3] extended Banach fixed point theorem to non-self contraction multi-valued mappings Ì Ã which maps Ã( Ãthe boundary of Ã) into à where´ µ is a convex metric space in the sense of Menger (c.f. [3], [4]). Rhoades [12] proved the corresponding fixed point theorem in Banach spaces for nonself mappings which, instead of the contraction property (0.1), satisfy the condition ´Ì Ü Ì Ý µ Ñ Ü ´Ü ݵ ¾ ´Ü Ìܵ ´Ý Ìݵ ´Ü Ìݵ · ´Ý Ìܵ ½ · ¾ (0.3) £

In this paper, making use of some well-known summation formulae and generating relations due to Qureshi, Khan and Pathan, an attempt has been made to establish some transformation formulae of ordinary hyper geometric series which are seemed to be new and in different form. We have also given some special cases.

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