Let be a complete metric space with metric , which is partially ordered. A mapping is called mixed monotone if is monotone nondecreasing in and monotone nonincreasing in . A pair is called a coupled fixed point of if , . The main result of the paper is the following theorem.
Theorem. Let be a partialy ordered complete metric space, let be mixed monotone and such that
(i) There is a constant such that for each ,
(ii) There exist with
Then has a coupled fixed point .
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