Fixed Point Theorems Through Abstract Cones

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Fixed Point Theorems Through Abstract Cones

Show simple item record Eisenfeld, Jerome en Lakshmikantham, V. en 2010-05-26T15:43:34Z en 2010-05-26T15:43:34Z en 1974-03 en
dc.identifier.uri en
dc.description.abstract A well-known theorem of Banach states that if T is a mapping on a complete metric space [see pdf for notation] such that for some number [see pdf for notation], the inequality (1.1) [see pdf for notation] holds, then T has a unique fixed point (i.e., a point u such that Tu = u). Extensions of this theorem [1,2] continue to require that T is a contraction i.e., (1.2) [see pdf for notation] This condition is essential if p is a metric but if p takes values in a partially ordered set k, then the condition (1.2) is avoidable. In [3] we assumed the following condition (1.3) [see pdf for notation] as a replacement for (1.1). Here [see pdf for notation] is a mapping of a cone k (in a Banach space) into itself and the k-metric [see pdf for notation] takes values in k. In the event that [see pdf for notation] and [see pdf for notation], then (1.1) emerges as a special case. Several applications to initial value problems have been developed as a result of this extension [3], [4], [5]. In this paper, we continue our work further by means of Kuratowski's measure of noncompactness of a set [see pdf for notation] [6]. We define a measure of noncompactness which is [see pdf for notation]-valued. By means of [see pdf for notation]-valued set functions one can define a [see pdf for notation]-semimetric space, where points are subsets, and thereby obtain interesting results. We also extend a generalization of Schauder's fixed point theorem which is due to Darbo [7] and which uses the condition (1.4) [see pdf for notation]. In the spirit of [3] we replace this condition by (1.5) [see pdf for notation] where y now takes values in a cone. Such a result applied to ordinary differential equations in a Banach space parallels recent work [8,9,11, 12,13]. Several results due to Darbo which are used in this paper may also be found in [16]. Finally, we remark that working in cones provides more accurate estimates by means of the induced partial ordering. For example, in place of convergence and compactness, we obtain stronger conditions like [see pdf for notation]-convergence and [see pdf for notation]-compactness. We adopt a convention of referring to known results as propositions. en
dc.language.iso en_US en
dc.publisher University of Texas at Arlington en
dc.relation.ispartofseries Technical Report;6 en
dc.subject Fixed point en
dc.subject Measure of noncompactness en
dc.subject Banach spaces en
dc.subject Abstract cones en
dc.subject.lcsh Mathematics Research en
dc.title Fixed Point Theorems Through Abstract Cones en
dc.type Technical Report en
dc.publisher.department Department of Mathematics en

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