On ordered-covering mappings and implicit differential inequalities
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'' \(f(t,v_0,v_0')\geq0\).
scientific article; zbMATH DE number 6697609
scientific article