Invariant subspaces of positive operators

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My primary area of research is the theory of Ordered Banach spaces and positive operators. Positivity is an important field of Functional analysis that has many applications in different disciplines of sciences, including mathematical economics, which is my secondary area of research. One of the most interesting open problems of this theory, is the Invariant Subspace Problem(ISP): Does any positive operator T on a Banach lattice have a non trivial closed invariant subspace? In the proposed research we want to adopt a different and new approach to this problem, which is the use of fixed point theorems of set-valued maps. This approach seems that is able to work without the assumption that T is quasinilpotent, hence it is expected tol bring new insight to this problem. So far, in my research I have been studying problems in Ordered Banach spaces that lack lattice structure. In this vain, I want to study the ISP for positive operators in the latter class of Ordered Banach spaces. Moreover independently with the ISP, I want to further extend the study of reflexive cones by considering solid cones and relate these type of cones with the Radon Nikodym Property.