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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×XX is called mixed monotone if F(x,y) is monotone nondecreasing in x and monotone nonincreasing in y. A pair (x ¯,y ¯)X×X is called a coupled fixed point of F if F(x ¯,y ¯)=x ¯, F(y ¯,x ¯)=y ¯. The main result of the paper is the following theorem.

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

(i) There is a constant k[0,1) such that for each xu, yv

d(F(x,y),F(u,v))+d(F(y,x),F(v,u))k[d(x,u)+d(y,v)]·

(ii) There exist x 0 ,y 0 X with

x 0 F(x 0 ,y 0 )andy 0 F(y 0 ,x 0 )

or

x 0 F(x 0 ,y 0 )andy 0 F(y 0 ,x 0 )·

Then F has a coupled fixed point (x ¯,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),t(0,T),u(0)=u(T)

with h(t,u)=f(t,u)+g(t,u).


MSC:
54H25Fixed-point and coincidence theorems in topological spaces
54E50Complete metric spaces
54F05Linearly, generalized, and partial ordered topological spaces
34B15Nonlinear boundary value problems for ODE
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