Let D be a subset of a real Banach space E, which is partially ordered by a cone P of E. The operator A:D
is mixed monotone if A(x,y) is nondecreasing in x and nonincreasing in y. The point (x’,y’) in
is a coupled fixed point of A if
. This extends the notion of fixed point z, for which
. The authors prove existence theorems for coupled fixed points both for continuous and discontinuous A. Applications include the existence of solutions of initial value problems for some systems of ordinary differential equations with discontinuous right hand side.