## Impulsive periodic solutions of first-order singular differential equations.(English)Zbl 1144.34016

The paper proves the existence of impulsive periodic solutions of the first-order singular differential equation
$x'+a(t)x = f(t,x)+e(t), \quad t\in J',$
where $$J'=J\setminus \{t_1,\dots,t_p\}$$, $$J=[0,1]$$, $$0=t_0<t_1<\dots<t_p<t_{p+1}=1$$, and $$a,e$$ are continuous, 1-periodic functions. The nonlinearity $$f(t,x)$$ is continuous in $$(t,x)\in J'\times (0,\infty)$$ and 1-periodic in $$t$$, $$f(t_k^+,x)$$, $$f(t_k^-,x)$$ exist, $$f(t_k^-,x)=f(t_k,x)$$ and $$f(t,x)$$ can have a singularity at $$x=0$$. The model equation under interest is
$x'+a(t)x=\frac{1}{x^\alpha}+\mu x^\beta+e(t),$
where $$\alpha, \beta >0$$ and $$\mu$$ is a real parameter. Impulses $$I_k$$ are continuous functions. In particular, they are linear homogeneous. The proof is based on a nonlinear alternative principle of Leray-Schauder and a truncation technique. Some recent results in the literature are generalized and improved.

### MSC:

 34B37 Boundary value problems with impulses for ordinary differential equations 34B15 Nonlinear boundary value problems for ordinary differential equations 34B16 Singular nonlinear boundary value problems for ordinary differential equations
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