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Remarks on Nonlinear Contraction and Comparison Principle in Abstract Cones

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Remarks on Nonlinear Contraction and Comparison Principle in Abstract Cones

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dc.contributor.author Lakshmikantham, V. en
dc.contributor.author Eisenfeld, Jerome en
dc.date.accessioned 2010-05-26T18:27:50Z en
dc.date.available 2010-05-26T18:27:50Z en
dc.date.issued 1975-06 en
dc.identifier.uri http://hdl.handle.net/10106/2180 en
dc.description.abstract 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 [1], [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 [1], [4] 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 [1] - [4] and [6]. Also see [6] for modifications using minimal solutions in place of maximal solutions. en
dc.language.iso en_US en
dc.publisher University of Texas at Arlington en
dc.relation.ispartofseries Technical Report;25 en
dc.subject Abstract cones en
dc.subject Bellman-Gronwall-Reid inequality en
dc.subject Contraction mapping principle en
dc.subject Nonlinear map en
dc.subject Schauder principle en
dc.subject.lcsh Mathematics Research en
dc.title Remarks on Nonlinear Contraction and Comparison Principle in Abstract Cones en
dc.type Technical Report en
dc.publisher.department Department of Mathematics en

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