Fixed points of mixed monotone operators with applications. (English) Zbl 0688.47019
Let E be a Banach space partially ordered by a cone P (i.e. \(x\leq y\), iff y-x\(\in P)\). Let D be a subset of E and A: \(D\times D\to E\) a mixed- monotone operator (i.e. A(x,y) is nondecreasing in x and nonincreasing in y).
Under suitable conditions, the authors prove existence and uniqueness theorems of fixed points for A and give applications of these results to the initial value problem of ordinary differential equations.
Under suitable conditions, the authors prove existence and uniqueness theorems of fixed points for A and give applications of these results to the initial value problem of ordinary differential equations.
Reviewer: D.Roux
MSC:
| 47H10 | Fixed-point theorems |
| 34A12 | Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations |
| 34A45 | Theoretical approximation of solutions to ordinary differential equations |
Keywords:
Banach space partially ordered by a cone; mixed-monotone operator; existence and uniqueness theorems of fixed points; initial value problem of ordinary differential equationsReferences:
| [1] | DOI: 10.1016/0362-546X(87)90077-0 · Zbl 0635.47045 · doi:10.1016/0362-546X(87)90077-0 |
| [2] | Guo Dajun, Nonlinear problems in abstract cones (1966) |
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