Let S be an inverse semigroup with an upward directed set of idempotents E. In this paper we define the module topological center of second dual of a Banach algebra which is a Banach module over another Banach algebra with compatible actions, and find it for ℓ 1 (S) * * (as an ℓ 1 (E)-module). We also prove that ℓ 1 (S) * * is ℓ 1 (E)-module amenable if and only if an appropriate group homomorphic image of S is finite.2000 Mathematics Subject Classification. 46H25.

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In this paper, we introduce new concept of orthogonal cone metric spaces. We stablish new versions of fixed point theorems in incomplete orthogonal cone metric spaces. As an application, we show the existence and uniqueness of solution of the periodic boundry value problem.