ESTIMATES OF POSITIVE NONTRIVIAL SOLUTIONS OF A DIFFERENTIAL EQUATION WITH POWER NONLINEARITY

Differential equations in paper with power nonlinearity are considered. Solutions which are defined in some neighborhood of plus infinity are called proper solutions. It is proved that propersolution to the equation is kneser solution, which means that solution and it’s quasiderivatives change their signs and tend to zero. The integral representation for proper solutions is proved. Upper estimates for solution and it’s quasiderivatives for proper solutions with maximal interval of existence is positive semiaxis to the equation with quasiderivative are proved. Upper and lowerestimates of solution and it’s derivatives for proper solutions with maximal interval of existence is positive semiaxis to the equation with derivative are provedDifferential equationsy[n] = rn(x)ddx(rn

Oscillatory and monotone solutions of first-order nonlinear delay differential equations

For first order nonlinear delay differential equations, necessary and sufficient conditions are established for the oscillation of all proper solutions as well as for the existence of at least one vanishing at infinity proper Kneser solution.