Effective finding of the solutions of the form for the differential equations of the finite and infinite order
We recall the definition of Minimizing Movements, suggested by E. De Giorgi, and we consider some applications to evolution problems. With regards to ordinary differential equations, we prove in particular a generalization of maximal slope curves theory to arbitrary metric spaces. On the other hand we present a unifying framework in which some recent conjectures about partial differential equations can be treated and solved. At the end we consider some open problems.
In the paper, we obtain the existence of symmetric or monotone positive solutions and establish a corresponding iterative scheme for the equation , , where , , subject to nonlinear boundary condition. The main tool is the monotone iterative technique. Here, the coefficient may be singular at .