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

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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 $\alph