Abstract:
The article is a survey of work on non-linear monotone operators on Banach spaces. Let $F(x)$ be an operator acting from a Banach space into its adjoint space. If on the whole space the scalar product inequality $(F(x) - F(y), x - y)\ge 0$ holds, then $F(x)$ is said to be a monotone operator. It turns out that monotonicity, in conjunction with some other conditions, makes it possible to obtain existence theorems for solutions of operator equations. The results obtained have applications to boundary-value problems of partial differential equations, to differential equations in Banach spaces, and to integral equations.
Here is a list of questions touched upon in the article. General properties of monotone operators. Existence theorems for solutions of equations with operators defined on the whole space or on an everywhere dense subset of the space. Fixed point principles. Approximate methods of solution of equations with monotone operators. Examples that illustrate the possibility of applying the methods of monotonicity to some problems of analysis. In conclusion, the article gives a bibliography of over one hundred papers.
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