Summary: We prove some coupled fixed point theorems for mappings having a mixed monotone property in partially ordered metric spaces. The main results of this paper are generalizations of the main results of T. G. Bhaskar
and V. Lakshmikantham
[Nonlinear Anal., Theory Methods Appl. 65, No. 7 (A), 1379–1393 (2006; Zbl 1106.47047
)]. As an application, we discuss the existence and uniqueness for a solution of a nonlinear integral equation.