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On ordered-covering mappings and implicit differential inequalities. (English) Zbl 1454.47054

Differ. Equ. 52, No. 12, 1539-1556 (2016); translation from Differ. Uravn. 52, No. 12, 1610-1627 (2016).
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” \(f(t,v_0,v_0')\geq0\).

MSC:

47H07 Monotone and positive operators on ordered Banach spaces or other ordered topological vector spaces
06A06 Partial orders, general
34A09 Implicit ordinary differential equations, differential-algebraic equations
47H30 Particular nonlinear operators (superposition, Hammerstein, Nemytskiĭ, Uryson, etc.)
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