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Sharp conditions are established for the existence of slowly varying solution and regularly varying solution of index 1 of the half-linear functional differential equation with both retarded and advanced arguments of the form (|x′(t)| α  sgn x′(t))′ ± p(t)|x(g(t))| α  sgn x(g(t)) ± q(t)|x(h(t))| α  sgn x(h(t)) = 0, where α &gt; 0 is a constant, p, q: [a,∞) → (0,∞), a ≧ 0 are continuous functions, g, h are continuous and increasing with g(t) &lt; t, h(t) &gt; t for t ≥ a and.

Regularly Varying Solutions of Half-Linear Diffferential Equations with Retarded and Advanced Arguments