The aim of this paper is to present a method for solving of system of first order initial value problems of ordinary differential equation by a semi-analytic technique with constructing polynomial solutions for decreasing dangers of lead. The original problem is concerned using two-point osculatory interpolation with the fit equals  numbers of derivatives at the end points of an interval [0 , 1].

Let R be a Γ-ring and G be an R_Γ-module. A proper R_Γ-submodule S of G is said to be semiprime R_Γ-submodule if for any ideal I of a Γ-ring R and for any R_Γ-submodule A of G such that or which implies that . The purpose of this paper is to introduce interesting results of semiprime R_Γ-submodule of R_Γ-module which represents a generalization of semiprime submodules.

      In this paper, we introduce the concepts of Large-lifting and Large-supplemented modules as a generalization of lifting and supplemented modules.  We also give some results and properties of this new kind of modules.

The concepts of generalized higher derivations, Jordan generalized higher derivations, and Jordan generalized triple higher derivations on Γ-ring M into ΓM-modules X are presented. We prove that every Jordan generalized higher derivation of Γ-ring M into 2-torsion free ΓM-module X, such that aαbβc=aβbαc, for all a, b, c M and α,βΓ, is Jordan generalized triple higher derivation of M into X.

      In this paper, we introduce the concepts of Large-lifting and Large-supplemented modules as a generalization of lifting and supplemented modules.  We also give some results and properties of this new kind of modules.

    The main objective of this research is to study and to introduce a concept of strong fully stable Banach -algebra modules related to an ideal.. Some properties and characterizations of full stability are studied.