Three fixed-sign solutions of system model with Sturm--Liouville type conditions
This paper deals with existence and multiplicity of solutions for the system of ordinary differential equations, \[ u_{i}^{m_{i}}(t) + P_{i}(t,u_1(t), u_2(t), \cdots , u_n(t)) = 0, \] where \(0 \leq t \leq 1,~1 \leq i \leq n,~ m_{i} \geq 2\) and \(P_{i}:[0,1] \times \mathbb{R}^{n} \to \mathbb{R}\) is a suitable \(L^{1}\)-function, under the Sturm-Liouville-type boundary conditions \[ u_{i}^{(j)}(0) = 0,~0 \leq j \leq m_{i} - 3,\quad \xi u_{i}^{(m_i - 2)}(0) - \eta u_{i}^{(m_{i} - 1)}(0) = \omega u_{i}^{(m_i - 2)}(1) + \delta u_{i}^{(m_i - 1)}(1) = 0, \] with \(\eta, \delta \geq 0\), \(\eta + \xi > 0\), \(\delta + \omega > 0\) and \(\xi \omega + \xi \delta + \eta \omega > 0\). The author uses fixed-point arguments, namely both the Leggett-Williams fixed-point theorem and so called ``five-functional fixed-point theorem'' to show that the system above admits three solutions which, as a matter of fact, do not change sign in the interval \([0,1]\).
scientific article; zbMATH DE number 2117732
scientific article