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We describe a reduction technique allowing one to combine an analysis of the existence of solutions with an efficient construction of approximate solutions for a state-dependent multi-impulsive boundary value problem which consists of non-linear system of differential equationsu^{\prime}(t)=f(t,u(t))\quad\text{for a.e. }t\in[a,b],subject to the state-dependent impulse conditionu(t+)-u(t-)=\gamma_{t}(u(t-))\quad\text{for }t\in(a,b)\text{ such that }g(t,u(% t-))=0,and the non-linear two-point boundary conditionV(u(a),u(b))=0.

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Singular differential equation(p(t)u′)′=p(t)f(u)is investigated. Herefis Lipschitz continuous on ℝ and has at least two zeros 0 andL>0. The functionpis continuous on [0,∞) and has a positive continuous derivative on (0,∞) andp(0)=0. An asymptotic formula for oscillatory solutions is derived.

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