EXISTENCE OF A POSITIVE SOLUTION OF THE BOUNDARY-VALUE PROBLEM FOR ONE NONLINEAR FUNCTIONAL DIFFERENTIAL EQUATION OF THE SECOND ORDER

摘要

Many methods have been developed for the investigation of questions related to positive solutions of various nonlinear equations. Natural tools for such studies are the methods of functional analysis based on the theory of semiordered spaces. This theory is associated with the names of F. Riesz, M.G.Krein, L.V. Kantorovich, H. Preudenthal, G.Birkhoff, and others. Many authors applied methods of semiordered spaces to problems of positive solutions in various aspects. That is why the general results obtained in terms of functional analysis were applied to the first boundary-value problem for quasilinear elliptic equations, to nonlinear integral equations, nonlinear oscillations, the problem of bifurcation points, the theory of the Monge-Ampere equations. These applications are based on special constructions and use the properties of the Green functions of various differential operators. In addition, the methods of semiordered spaces are widely used in wave theory, elasticity theory, etc.