1956 China mathematician Guan published a paper "steepest descent method for solving nonlinear functional equations" in Journal of Mathematics. In this paper, it is proved that the steepest descent method for solving nonlinear equations in Hilbert space converges in norm in this space, and similar to linear problems, its convergence speed conforms to proportional series. This method can be used to solve some boundary value problems of nonlinear integral equations and nonlinear differential equations. Since then, the steepest descent method in infinite dimension has developed rapidly. In particular, the idea of monotone operator first appeared in this paper. The main assumption of this paper is that the derivative of the bit operator is positive definite. Guan pointed out that "it seems worth studying to prove the convergence of the method proposed in this paper under weak conditions" Later, through further in-depth study, people found that this weak condition is what we all know as the (strong) monotonicity condition.

The concept of monotone operator was formally put forward in the early 1960s. Monotone theory, including monotone operator, accretive operator, nonlinear semigroup and nonlinear evolution equation, has become an important branch of nonlinear functional analysis. Guan made a pioneering work on the development of monotone operator theory.