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In many systems of differential or integral equations that arise from models of population dynamics, chimical engeenering, infectious diseases, economics, neutron transport and other systems, it is used to find coexistence states for systems of integral or differential equations in ordered Banach spaces. The so called "coexistence states" are of special importance: these are solutions (x,y) with both components nonnegative and nontrivial. Semitrivial solutions i.e., solutions (x,y) with exactly one component nonnegative and nontrivial, are also of interest. Note that a direct application of the corresponding Amann's results implies the existence of a solution (x,y) but some component of the fixed point (x,y) may be trivial. To solve this problem it is the purpose of the book to expand and generalize the results by Amann and Krasnosel'skii concerning the existence of fixed points of cone expansion and compression to assure the existence of coexistence states. We also give important applications to demonstrate the importance of the abstract theorems given here.