Multiple positive solutions of singularly perturbed differential systems with different orders (Q845001)

This paper deals with singular higher order perturbed systems with different orders of the form \[ \begin{aligned} &(-1)^nu^{(2n)}(t) = p_1(t)f_1(t,u(t),v(t))-q_1(t), \quad t\in (0,1)\\ &(-1)^mv^{(2m)}(t) = p_2(t)f_2(t,u(t),v(t))-q_2(t), \quad t\in (0,1),\\ &u^{(2i)}(0)=u^{(2i)}(1)=0, \quad 0\leq i\leq n-1,\\ &v^{(2j)}(0)=v^{(2j)}(1)=0, \quad 0\leq j\leq m-1, \end{aligned} \] where \(m,n\in {\mathbb N},\) \(f_1,f_2:[0,1]\times [0,\infty)\times [0,\infty)\to [0,\infty)\) are continuous and \(p_i, q_i: (0,1)\to [0,\infty)\) \((i=1,2)\) are Lebesgue integrable. Existence results for multiple positive solutions are given via a fixed point theorem in cones, in case when the nonlinearity can be sign changing.