On ordered-covering mappings and implicit differential inequalities
type: Item
instance of: scholarly article
title: On ordered-covering mappings and implicit differential inequalities
zbMATH Open document ID: 1454.47054
DOI: 10.1134/S0012266116120028
published in: Differential Equations
publication date: 2017-03-28T00:00:00Z
review text: The paper contains three related results. The first and main result covers various extensions of the Knaster-Tarski (Birkhoff-Tarski, and Tarski-Kantorovich) fixed point theorems in partially ordered spaces. This is an existence result for a (minimal) solution of the equation \(T(x,x)=y\) where \(T: X\times X\to Y\) is such that \(T(x,\cdot)\) is antitonic (monotone decreasing) and \(T(\cdot,x)\) has certain subtle covering (surjectivity) properties with respect to the orders on \(X\) and \(Y\). While the general result makes use of Hausdorff's maximality principle, sequential constructions are also provided under additional hypotheses. The second result studies the covering properties for a superposition (Nemytskii) operator defined on \(L_p([a,b],\mathbb R^n)\) with the usual partial order. The two results are appplied to obtain results about the existence and extension of solutions for an initial value problem of an implicit ODE \(f(t,x,x')=0\) in \(\mathbb R^n\) with \(x'\) lying in an order interval \([w_0,v_0']\). The main hypotheses are that \(f(t,\cdot,z)\) is nonincreasing, that \(f(t,x,\cdot)\) is continuous and satisfies (pointwise) some covering type hypotheses on \([w_0(t),v_0'(t)]\), and that \(v_0\) is a ``supersolution'' \(f(t,v_0,v_0')\geq0\).
Mathematics Subject Classification ID: 47H07
Mathematics Subject Classification ID: 06A06
Mathematics Subject Classification ID: 34A09
Mathematics Subject Classification ID: 47H30
zbMATH DE Number: 6697609
zbMATH Keywords: partially ordered set
zbMATH Keywords: monontone map
zbMATH Keywords: Knaster-Tarski fixed point theorem
zbMATH Keywords: implicit differential equation
zbMATH Keywords: minimal solution
zbMATH Keywords: maximal solution
zbMATH Keywords: supersolution
zbMATH Keywords: subsolution
zbMATH Keywords: order interval
zbMATH Keywords: superposition operator
author: E. S. Zhukovskiy
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 6697609