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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 $$F: X\times X\to X$$ is called mixed monotone if $$F(x,y)$$ is monotone nondecreasing in $$x$$ and monotone nonincreasing in $$y$$. A pair $$(\overline x,\overline y)\in X\times X$$ is called a coupled fixed point of $$F$$ if $$F(\overline x,\overline y)=\overline x$$, $$F(\overline y,\overline x)=\overline y$$. The main result of the paper is the following theorem.
Theorem. Let $$X$$ be a partialy ordered complete metric space, let $$F: X\times X\to X$$ be mixed monotone and such that
(i) There is a constant $$k\in [0,1)$$ such that for each $$x\geq u$$, $$y\leq v$$ $d(F(x,y), F(u,v))+ d(F(y, x), F(v,u))\leq k[d(x, u)+ d(y,v)].$ (ii) There exist $$x_0,y_0\in X$$ with $x_0\leq F(x_0, y_0)\quad\text{and}\quad y_0\leq F(y_0, x_0)$ or $x_0\geq F(x_0, y_0)\quad\text{and}\quad y_0\leq F(y_0, x_0).$ Then $$F$$ has a coupled fixed point $$(\overline x,\overline y)$$.
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 $u'= h(t,u),\quad t\in (0,T),\quad u(0)= u(T)$ with $$h(t,u)= f(t,u)+ g(t,u)$$.

##### MSC:
 54H25 Fixed-point and coincidence theorems (topological aspects) 54E50 Complete metric spaces 54F05 Linearly ordered topological spaces, generalized ordered spaces, and partially ordered spaces 34B15 Nonlinear boundary value problems for ordinary differential equations
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