Meyer-Nieberg in (Banach Lattices, Springer-Verlag, Berlin, Heidelberg, New York, 1991) present some characterizations of the class of order weakly compact operators. The main focus of this paper is to invest these results in order to introduce the order weakly demicompact operators. Suppose that E is a Banach lattice. An operator T from E into E is said to be order weakly demicompact if, for every order bounded sequence {xn}\documentclass[12pt]{minimal}
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\begin{document}$$\{x_{n}\}$$\end{document} in E+\documentclass[12pt]{minimal}
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\begin{document}$$x_{n}\rightarrow 0$$\end{document} in σ(E,E′)\documentclass[12pt]{minimal}
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\begin{document}$$\sigma (E,E')$$\end{document} and ‖xn-Txn‖→0\documentclass[12pt]{minimal}
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\begin{document}$$\Vert x_{n}-Tx_{n}\Vert \rightarrow 0$$\end{document} as n→∞\documentclass[12pt]{minimal}
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\begin{document}$$n\rightarrow \infty $$\end{document}, we have ‖xn‖→0\documentclass[12pt]{minimal}
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\begin{document}$$n\rightarrow \infty $$\end{document}. In addition, we establish some properties of this class of operators. We also introduce the order positive Schur property of Banach lattice in order to give a condition under which each operator is order weakly demicompact.
