In this paper, we are mainly concerned with existence of positive solutions for periodic boundary value problem of second-order impulsive differential equation with derivative in the nonlinearity -u′′+ρ2u=f(t,u,u′),t∈J′,-Δu′t=tk=Ik(u(tk)),k=1,2,…m,u(0)=u(2π),u′(0)=u′(2π),\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \left\{ \begin{array}{ll} -u''+\rho ^{2} u=f(t, u, u'), &{} t \in J', \\ -\left. \Delta u'\right| _{t=t_{k}}=I_{k}(u(t_{k})), &{} k=1,2, \ldots m, \\ u(0)=u(2\pi ),\quad u^{\prime }(0)=u^{\prime }(2\pi ), &{} \end{array}\right. \end{aligned}$$\end{document}where f:[0,2π]×R+×R→R+\documentclass[12pt]{minimal}
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\begin{document}$$f:[0,2 \pi ] \times {\mathbb {R}}^{+} \times {\mathbb {R}} \rightarrow {\mathbb {R}}^{+}$$\end{document} is continuous, R+=[0,+∞)\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {R}}^{+}=[0,+\infty )$$\end{document}, J=[0,2π]\documentclass[12pt]{minimal}
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\begin{document}$$J=[0,2 \pi ]$$\end{document}, ρ>0\documentclass[12pt]{minimal}
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\begin{document}$$ \rho >0$$\end{document}, J′=J\t1,t2,…tm.\documentclass[12pt]{minimal}
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\begin{document}$$J^{\prime }=J \backslash \left\{ t_{1}, t_{2}, \ldots t_{m}\right\} .$$\end{document} Some inequality conditions on nonlinearity f and the spectral radius condition of linear operator are presented that guarantee the existence of positive solution to the problem by the theory of fixed point index. The conditions allow that ft,x1,x2\documentclass[12pt]{minimal}
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\begin{document}$$f\left( t, x_{1},x_{2}\right) $$\end{document} has superlinear or sublinear growth in x1,x2\documentclass[12pt]{minimal}
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\begin{document}$$x_{1}, x_{2}$$\end{document}.
