Positive solutions and eigenvalue regions of two-delay singular systems with a twin parameter (Q878529)

The authors consider, on the interval \([0,1]\), the nonlinear differential system with parameters \[ (\varphi(u'_1))'+\lambda_1 h_1(t) f_1(u_1(t-\tau_1), u_2(t-\tau_2))=0, \] \[ (\varphi(u'_2))'+\lambda_2 h_2(t) f_2(u_1(t-\tau_1), u_2(t-\tau_2))=0 \] with boundary conditions \[ u_i(t)=0,\qquad -\tau_i\leq t\leq 0,\qquad i=1,2, \] \[ u_i(1)=0,\qquad i=1,2. \] Here, \(\varphi:\mathbb R\to \mathbb R\) is an odd, increasing homeomorphism, \(f_i:\mathbb R^2_+\to \mathbb R_+\) are continuous functions, \(h_i:(0,1)\to \mathbb R_+\) are measurable functions, and \(\lambda_i>0\). In the paper, conditions are established guaranteeing the existence of at least one, respectively two, positive solution(s) to the problem considered. Moreover, criteria on the absence of positive solutions to the problem under consideration are given. The proofs of the main results are based on the Krasnosel'skii fixed-point theorem.