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

Chu, Jifeng; Nieto, Juan J.

Bull. Lond. Math. Soc. 40, No. 1, 143-150 (2008).

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.

Reviewer: Irena Rachůnková (Olomouc)

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

Keywords:

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