Existence of positive solutions for periodic boundary value problems of second-order impulsive differential equation with derivative in the nonlinearity

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 $$\begin{aligned} \left\{ \begin{array}{ll} -u''+\rho ^{2} u=f(t, u, u'), &{} t \in J', \\ -\left. \Delta u'\right| _{t=t_{k}}=I_{k}(u(t_{k})), &{} k=1,2, \ldots m, \\ u(0)=u(2\pi ),\quad u^{\prime }(0)=u^{\prime }(2\pi ), &{} \end{array}\right. \end{aligned}$$ where $$f:[0,2 \pi ] \times {\mathbb {R}}^{+} \times {\mathbb {R}} \rightarrow {\mathbb {R}}^{+}$$ is continuous, $${\mathbb {R}}^{+}=[0,+\infty )$$ , $$J=[0,2 \pi ]$$ , $$ \rho >0$$ , $$J^{\prime }=J \backslash \left\{ t_{1}, t_{2}, \ldots t_{m}\right\} .$$ 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 $$f\left( t, x_{1},x_{2}\right) $$ has superlinear or sublinear growth in $$x_{1}, x_{2}$$ .