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Existence theorems of coupled fixed points. (English) Zbl 0719.47041
Let D be a subset of a real partially ordered Banach space E. Let A: $D\times D\to E$ be a mixed monotone operator (i.e. $A(\cdot,y)$ is nondecreasing and $A(x,\cdot)$ is nonincreasing). A point $(x\sp*,y\sp*)\in D\times D$ is called a coupled fixed point of A if $x\sp*=A(x\sp*,y\sp*)$ and $y\sp*=A(y\sp*,x\sp*)$. Let $\tilde A:$ $D\times D\to E\times E$ be given by $\tilde A(u,v)=(A(u,v),A(v,u))$. Note that $\tilde A$ is increasing and the fixed point of $\tilde A$ is the coupled fixed point of A. By using this observation and known results on fixed points, the author gives several existence theorems for coupled fixed points.

47H10Fixed-point theorems for nonlinear operators on topological linear spaces
47H07Monotone and positive operators on ordered topological linear spaces
Full Text: DOI
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