Positive solutions of functional-differential systems via the vector version of Krasnoselskii's fixed point theorem in cones (Q2891142)

The authors prove the existence of positive solutions to the functional-differential system NEWLINE\[NEWLINEu_1^{\prime \prime}(t)+a_1(t) f_1(u_1(g(t)),u_2(g(t)))=0,\;u_2^{\prime \prime}(t)+a_2(t) f_2(u_1(g(t)),u_2(g(t)))=0,\;0<t<1,NEWLINE\]NEWLINE subject to the boundary conditions NEWLINE\[NEWLINE\alpha_i u_i(0)-\beta_i u_i^\prime (0)=0,\;\gamma_i u_i(1)+\delta_i u_i^\prime (1)=0,\;u_i(t)=k_i,\;-\theta\leq t<0,NEWLINE\]NEWLINE where \(\theta\) is a given positive number, \(g\in C([0,1], [-\theta, 1])\), \(f_i\in C([0,\infty)^2, [0,\infty))\), \(a_i\in C([0,1], [0,\infty))\), all constants in the boundary conditions are nonnegative, and \(\gamma_i\beta_i+\alpha_i\gamma_i+\alpha_i\delta_i>0\). The paper is based on the previous paper by \textit{R. Precup} [J. Fixed Point Theory Appl. 2, No. 1, 141--151 (2007; Zbl 1134.47041)].