Remarks on Nonlinear Contraction and Comparison Principle in Abstract Cones
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The contraction mapping principle and the Schauder principle can both be viewed as a comparison of maps. For the former one has a condition of the type [see pdf for notation] and for the latter one has a condition of the type [see pdf for notation] where p is the metric and y is the Kuratowski measure of noncompactness. If p is a linear map [see pdf for notation] from the nonnegative reals [see pdf for notation] into itself then the map T satisfying (1.1) is said to be k-contractive and the map satisfying (1.2) is said to be k-set contractive. It is also usually assumed that k < 1 in which case the adjective "strict" is used to describe the contractive property. Instead of taking Y to be a linear map on the cone [see pdf for notation], can be chosen as a nonlinear map from a cone of a Banach space into itself , [4). This innovation provides for greater flexibility in the choice of and it also has the advantage of stronger convergence properties and more accurate estimates. The comparison map is: positive (in the sense that it takes values in a cone), monotone (nondecreasing) and has a unique fixed point which is the zero element of the cone. For a regular cone (such as cones in [see pdf for notation] needs only satisfy the weak continuity condition: upper semi-continuous from above (or from the right). However, in the case of a normal cone which is not regular (such as [see pdf for notation]) it is assumed in ,  that y is completely continuous. The complete continuity condition which is also used by Krasnoselskii [7,p.127] may be replaced by a weaker compactness-type condition in terms of measure of noncompactness along with upper semi-continuity from above. We also manage to avoid strict contractive conditions. The paper is organized as follows. In Section 2 we state definitions regarding the theory of cones and some propositions which are used as lemmas or to amplify results proved later on. In Section 3 we present some results dealing with maximal fixed points of monotone maps. As a consequence we obtain a generalized Bellman-Gronwall-Reid inequality. In Section 4 we present a generalization of the contraction mapping principle. For applications see  -  and . Also see  for modifications using minimal solutions in place of maximal solutions.