Weakly cofiniteness of local cohomology modules

Let R be a commutative Noetherian ring, Phi a system of ideals of R and I is an element of Phi. Let M be an R-module (not necessary I-torsion) such that dim M <= 1, then the R-module Ext(R)(i) (R/I, M) is weakly Laskerian, for all i >= 0, if and only if the R-module Ext(R)(i) (R/I, M) is weakly Laskerian for i = 0, 1. Let t is an element of N-0 be an integer and M an R-module such that Ext(R)(i) (R/I, M) is weakly Laskerian for all i <= t + 1. We prove that if the R-module H-Phi(i) (M) is FD <= 1 for all i < t, then H-Phi(i) (M) is Phi-weakly cofinite for all i < t, and for any FD <= 0 (or minimax) submodule N of H-Phi(t) (M), the R-modules Hom(R)(R/I, H-Phi(t) (M)/N) and Ext(R)(1) (R/I, H-Phi(t) (M)/N) are weakly Laskerian. Let N be a finitely generated R-module. We also prove that Ext(R)(j) (N, H-Phi(i) (M)) and Tor(j)(R) (N, H-Phi(i) (M)) are Phi-weakly cofinite for all i and j whenever M is weakly Laskerian and H-Phi(i) (M) is FD <= 1 for all i. Similar results are true for ordinary local cohomology modules and local cohomology modules defined by a pair of ideals.