Pure imaginary solutions of the second Painlevé equation.
Some problems in differential equations evolve towards Topology from an analytical origin. Two such problems will be discussed: the existence of solutions asymptotic to the equilibrium and the stability of closed orbits of Hamiltonian systems. The theory of retracts and the fixed point index have become useful tools in the study of these questions.
Using a method developed by the author for an analysis of singular integral inequalities a stability theorem for semilinear parabolic PDEs is proved.
In the paper a sufficient condition for all solutions of the differential equation with -Laplacian to be proper. Examples of super-half-linear and sub-half-linear equations , are given for which singular solutions exist (for any , , ).
Sufficient conditions for the -th order linear differential equation are derived which guarantee that its Cauchy function , together with its derivatives , , is of constant sign. These conditions determine four classes of the linear differential equations. Further properties of these classes are investigated.