##
**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.

Reviewer: I. N. Baker (London)

### 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 |

### Keywords:

cone and partial ordering; mixed monotone operators; monotone iterative technique; coupled fixed point; existence of solutions of initial value problems for some systems of ordinary differential equations with discontinuous right hand side
PDFBibTeX
XMLCite

\textit{D. Guo} and \textit{V. Lakshmikantham}, Nonlinear Anal., Theory Methods Appl. 11, 623--632 (1987; Zbl 0635.47045)

Full Text:
DOI

### 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 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.