Fixed Point Theorems Through Abstract Cones
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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  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 , , . In this paper, we continue our work further by means of Kuratowski's measure of noncompactness of a set [see pdf for notation] . 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  and which uses the condition (1.4) [see pdf for notation]. In the spirit of  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 . 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.