## Positive periodic solutions of first-order singular systems.(English)Zbl 1300.34090

This paper deals with existence and the number of periodic solutions of the first order system $u'_i(t)=-a_i(t)u_i(t)+\lambda b_i(t)f_i(\vec u(t)),\quad i=1,\,\dots,\,n\eqno(1)$ where $$\vec u=(u_1,\,\dots,\,u_n)\in{\mathbb R}^n$$, $$a_i$$ and $$b_i$$ are continuous $$T$$-periodic functions with positive mean value in a period, $$f_i: {\mathbb R}^n\setminus0\to]0,\infty[$$ are continuous functions and $$\lambda$$ is a positive parameter.
In addition it is assumed that $$f_i(\vec u)\to +\infty$$ as $$\|\vec u\|\to0$$ and $$\frac{f_i(\vec u)}{\|\vec u\|}\to+\infty$$ as $$\|\vec u\|\to\infty$$.
The main result (showing that (1) mimics the behaviour of the autonomous single equation, which can be reduced to that of a real function) is the following: there exists $$\lambda^*>0$$ such that (1) has at least 2, at least 1, or no $$T$$-periodic solutions according to whether $$0<\lambda<\lambda^*$$, $$\lambda=\lambda^*$$ or $$\lambda>\lambda^*$$ respectively.
The proof makes use of lower and upper solutions and the usual degree-theoretic argument.
Related results had been given by H. Wang [J. Differ. Equations 249, No. 12, 2986–3002 (2010; Zbl 1364.34032); Appl. Math. Comput. 218, No. 5, 1605–1610 (2011; Zbl 1239.34043)].

### MSC:

 34C25 Periodic solutions to ordinary differential equations 47N20 Applications of operator theory to differential and integral equations

### Citations:

Zbl 1239.34043; Zbl 1364.34032
Full Text:

### References:

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