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Kong, Qingkai; Wang, Min

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

Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 72, No. 3-4, 1778-1791 (2010).
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 \(f\).
Reviewer: Anatolij Ivan Kolosov (Khar’kov)

Cited in 10 Documents

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

Keywords:

Green’s function; high order boundary value problems; positive solutions
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\textit{Q. Kong} and \textit{M. Wang}, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 72, No. 3--4, 1778--1791 (2010; Zbl 1195.34036)
Full Text: DOI

References:

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