Symmetric positive solutions for second-order singular differential systems with multi-point coupled integral boundary conditions (Q2792815)

The solvability of the following multi-point boundary value problem NEWLINENEWLINE\[NEWLINE\begin{cases} u''(t)+a(t)f(t,u(t),v(t))=0,\quad v''(t)+b(t)g(t,u(t),v(t))=0,\;\;t\in(0,1),\\ u(0)=\sum_{i=1}^{m}\alpha_i\int_{\eta_{i-1}}^{\eta_i}p(r)v(r)dr,\quad u(1)=\sum_{i=1}^{m}\alpha_i\int_{\overline{\eta}_{i-1}}^{\overline{\eta}_i}p(r)v(r)dr,\\ v(0)=\sum_{i=1}^{m}\beta_i\int_{\xi_{i-1}}^{\xi_i}q(r)u(r)dr,\quad v(1)=\sum_{i=1}^{m}\beta_i\int_{\overline{\xi}_{i-1}}^{\overline{\xi}_i}q(r)u(r)dr,\end{cases}NEWLINE\]NEWLINE NEWLINEis studied. Here, \(a,b:(0,1)\to[0,\infty)\) are continuous, symmetric on \((0,1)\) (that is, for example \(a(t)=a(1-t)\) for \(t\in(0,1)),\) may be singular at \(t=0\) and/or \(t=1\) and have the properties NEWLINENEWLINE\[NEWLINE0<\int_0^1s(1-s)a(s)ds<\infty,\quad 0<\int_0^1s(1-s)b(s)ds<\infty,NEWLINE\]NEWLINE NEWLINE\(f,g:[0,1]\times[0,\infty)\times[0,\infty)\to[0,\infty)\) are continuous and \(f(.,u.v),g(.,u,v)\) are symmetric on \([0,1]\) for all \(u,v\in[0,\infty)\), \(0=\eta_0<\eta_1<\eta_2<\ldots<\eta_{m-1}<\eta_m\leq0.5,\) \(0=\xi_0<\xi_1<\xi_2<\ldots<\xi_{m-1}<\xi_m\leq0.5,\) \(\eta_i+\overline{\eta}_{m-i}=1,\) \(\xi_i+\overline{\xi}_{m-i}=1,\) \(\alpha_i,\beta_i>0,\;\alpha_i=\alpha_{m+1-i},\beta_i=\beta_{m+1-i},\;i=0,1,\ldots,m,\) and \(p,q:[0,1)\to[0,\infty)\) are continuous and symmetric on \([0,1]\) with NEWLINENEWLINE\[NEWLINE0<\Bigl(\sum_{i=1}^{m}\alpha_i\int_{\eta_{i-1}}^{\eta_i}p(r)dr\Bigr)\Bigl(\sum_{i=1}^{m}\beta_i\int_{\xi_{i-1}}^{\xi_i}q(r)dr\Bigr)<1.NEWLINE\]NEWLINENEWLINENEWLINEUnder additional assumptions, the authors formulate results guaranteeing at least one or at least two symmetric positive solutions; a solution \((u,v)\in \Bigl(C[0,1]\cap C^2(0,1)\Bigr)^2\) is of this class if \(u\) and \(v\) are symmetric and positive on \([0,1].\) The proofs rely on a construction of a suitable cone and an application of fixed point theorem of cone expansion and compression of norm type.