Extreme and moderate solutions of nonoscillatory second order half-linear differential equations

An existence and asymptotic theory is built for second order half-linear differential equations of the form $${\rm (A)}\qquad (p(t)|x^{\prime }|^\alpha {\rm sgn} \,x^{\prime })^{\prime } =q(t)|x|^\alpha {\rm sgn}\, x, \quad t \geq a \gt 0, $$ where $\alpha \gt 0$ is constant and $p(t)$ and $q(t)$ are positive continuous functions on $[a,\infty )$, in which a crucial role is played by a pair of the generalized Riccati differential equations $${\rm (R1)}\qquad u^{\prime }=q(t)-\alpha p(t)^{-{1}/{\alpha }}|u|^{1+{1}/{\alpha }},$$ $${\rm (R2)}\qquad v^{\prime }=p(t)^{-{1}/{\alpha }}-\frac {1}{\alpha }q(t)|v|^{1+\alpha }$$ associated with (A). An essential part of the theory is the construction of nonoscillatory solutions $x(t)$ of (A) enjoying explicit exponential-integral representations in terms of solutions $u(t)$ of (R1) or in terms of solutions $v(t)$ of (R2).