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Generalized coupled fixed point theorems for mixed monotone mappings in partially ordered metric spaces. (English) Zbl 1235.54024

Let $X$ be a complete metric space with metric $d$, which is partially ordered. A mapping $F:X×X\to X$ is called mixed monotone if $F\left(x,y\right)$ is monotone nondecreasing in $x$ and monotone nonincreasing in $y$. A pair $\left(\overline{x},\overline{y}\right)\in X×X$ is called a coupled fixed point of $F$ if $F\left(\overline{x},\overline{y}\right)=\overline{x}$, $F\left(\overline{y},\overline{x}\right)=\overline{y}$. The main result of the paper is the following theorem.

Theorem. Let $X$ be a partialy ordered complete metric space, let $F:X×X\to X$ be mixed monotone and such that

(i) There is a constant $k\in \left[0,1\right)$ such that for each $x\ge u$, $y\le v$

$d\left(F\left(x,y\right),F\left(u,v\right)\right)+d\left(F\left(y,x\right),F\left(v,u\right)\right)\le k\left[d\left(x,u\right)+d\left(y,v\right)\right]·$

(ii) There exist ${x}_{0},{y}_{0}\in X$ with

${x}_{0}\le F\left({x}_{0},{y}_{0}\right)\phantom{\rule{1.em}{0ex}}\text{and}\phantom{\rule{1.em}{0ex}}{y}_{0}\le F\left({y}_{0},{x}_{0}\right)$

or

${x}_{0}\ge F\left({x}_{0},{y}_{0}\right)\phantom{\rule{1.em}{0ex}}\text{and}\phantom{\rule{1.em}{0ex}}{y}_{0}\le F\left({y}_{0},{x}_{0}\right)·$

Then $F$ has a coupled fixed point $\left(\overline{x},\overline{y}\right)$.

The author also gives conditions under which there exists a unique coupled fixed point. Finally, he applies this theorems to the periodic boundary value problem

${u}^{\text{'}}=h\left(t,u\right),\phantom{\rule{1.em}{0ex}}t\in \left(0,T\right),\phantom{\rule{1.em}{0ex}}u\left(0\right)=u\left(T\right)$

with $h\left(t,u\right)=f\left(t,u\right)+g\left(t,u\right)$.

##### MSC:
 54H25 Fixed-point and coincidence theorems in topological spaces 54E50 Complete metric spaces 54F05 Linearly, generalized, and partial ordered topological spaces 34B15 Nonlinear boundary value problems for ODE