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.


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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