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In many nonlinear differential and integral equations which formulate models of chemical reactors, chemical engineering, population biology, infectious diseases, economics, neutron transport, and other equations, we need to discuss the existence of positive solutions of nonlinear operators leaving invariant a cone in ordered Banach space. However, in the study of these operators it is often convenient to make use of majorants and minorants in order to establish the existence of a nontrivial fixed point. In this direction, it is the purpose of this book to expand the results introduced by Krasnosel'skii in "positive solutions of operators equations" concerning the utilization of differentiable operators. The method used in the present book is the topological methods, more precisely, the fixed point index. In order to demonstrate the importance of the abstract results applications to the existence of positive solutions to nonlinear boundary value problems are presented here. But, of course, the abstract techniques and results of this book apply also to a variety of other problems.