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Zbl 1144.34016
Chu, Jifeng; Nieto, Juan J.
Impulsive periodic solutions of first-order singular differential equations.
(English)
[J] Bull. Lond. Math. Soc. 40, No. 1, 143-150 (2008). ISSN 0024-6093; ISSN 1469-2120/e

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.
[Irena Rach\uu nková (Olomouc)]
MSC 2000:
*34B37 Boundary value problems with impulses
34B15 Nonlinear boundary value problems of ODE
34B16 Singular nonlinear boundary value problems

Keywords: impulsive periodic solution; existence; Leray-Schauder principle; positive solution

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