This article extends the classical ideas of convex contractions to multiplicative metric spaces by introducing multiplicative convex contraction mappings of order 2. Also, the existence and uniqueness of fixed points for convex contraction mappings of order 2 are established by introducing a generalization of these mappings. The premise of our theorems is illustrated with an example. As an extension of our main result, we also discuss the existence and uniqueness of solutions for a system of nonlinear equations arising in Fredholm integral equations.