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Coupled fixed points of nonlinear operators with applications. (English) Zbl 0635.47045

Let \(D\) be a subset of a real Banach space \(E\), which is partially ordered by a cone \(P\) of \(E\). The operator \(A: D\times D\to E\) is mixed monotone if \(A(x,y)\) is nondecreasing in \(x\) and nonincreasing in \(x\). The point \((x',y')\) in \(D\times D\) is a coupled fixed point of \(A\) if \(A(x',y')=x'\) and \(A(y',x')=y'\). This extends the notion of fixed point \(z\), for which \(A(z,z)=z\). The authors prove existence theorems for coupled fixed points both for continuous and discontinuous \(A\). Applications include the existence of solutions of initial value problems for some systems of ordinary differential equations with discontinuous right hand side.

MSC:

47H10 Fixed-point theorems
47H07 Monotone and positive operators on ordered Banach spaces or other ordered topological vector spaces
34G10 Linear differential equations in abstract spaces
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References:

[1] Ladde, G. S.; Lakshmikantham, V.; Vatsala, A. S., Monotone Iterative Techniques for Nonlinear Differential Equations (1985), Pitman · Zbl 0658.35003
[2] Guo, Dajun, Positive solutions of nonlinear operator equations and its applications to nonlinear integral equations, Adv. Math., 13, 294-310 (1984), (In Chinese.) · Zbl 0571.47044
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