Cascade second order ODEs on manifolds are defined. These objects are locally represented by coupled second order ODEs such that any solution of one of them can represent an external force for the other one. A generic saddle-node bifurcation theorem for 1-parameter families of cascade second order ODEs is proved.
Using a method developed by the author for an analysis of singular integral inequalities a stability theorem for semilinear parabolic PDEs is proved.
For the difference equation ,where is a Banach space, is a parameter and is a linear, bounded operator. A sufficient condition for the existence of a unique special solution passing through the point is proved. This special solution converges to the solution of the equation (0) as .
A sufficient condition for the nonoscillation of nonlinear systems of differential equations whose left-hand sides are given by -th order differential operators which are composed of special nonlinear differential operators of the first order is established. Sufficient conditions for the oscillation of systems of two nonlinear second order differential equations are also presented.
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