## Positive solutions of even order system periodic boundary value problems.(English)Zbl 1195.34036

The authors consider the nonlinear boundary value problem
$(AD^2+ B)^n u= W(t) f(t,u),\quad t\in (0,\omega),\tag{1}$
$u^{(k)}(0)= u^k(\omega),\quad k= 0,1,\dots, 2n-1,\tag{2}$
where $$n\in\mathbb{N}$$, $$D={d\over dt}$$, $$u=(u_1,\dots, u_m)^T\in \mathbb{R}^m_+$$, $$A= \text{diag}(a_1,\dots, a_m)$$, $$B= \text{diag}(b_1,\dots, b_m)$$, with $$a_i= \pm1$$, $$b_i> 0$$; $$i= 1,\dots, m$$;
$W= (w_{ij})_{m\times m}\in C([0, \omega], \mathbb{R}^{m\times m}_+)$
such that $$\sum^m_{j=1} w_{i_j}(t)\not\equiv 0$$ on $$[0,\omega]$$ for all $$i= 1,\dots, m$$, and
$f= (f_1,\dots, f_m)^T\in C([0,\omega]\times \mathbb{R}^m_+, \mathbb{R}^m_+).$
The existence of one, two, any arbitrary number of positive solutions for problem (1), (2) is proved under some assumptions on the function $$f$$.

### MSC:

 34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations 34B15 Nonlinear boundary value problems for ordinary differential equations 34B27 Green’s functions for ordinary differential equations
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### References:

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