Existence of positive solutions of nonlinear second order Dirichlet problems perturbed by integral boundary conditions (Q6601628)

The subject of the paper is the existence of solutions to \N\[\Nu''(t)+\gamma u(t)\pm f(t,u(t))=0,\quad t\in [0,1]\N\]\Nunder the integral boundary conditions \N\[\Nu(0)=\delta_1 \int_0^1u(s)\,ds,\quad u(1)=\delta_2 \int_0^1u(s)\,ds\N\]\Nand assuming \(\gamma\le\pi^2\), \(\delta_i\ge0\).\N\NThe paper improves results from \textit{A. Cabada} and \textit{J. Iglesias} [Bound. Value Probl. 2021, Paper No. 66, 19 p. (2021; Zbl 1497.34039)].\N\NFirst the authors study the linear underlying problem where \(f\) is just a function \(\sigma(t)\), establishing the conditions about \(\gamma\), \(\delta_1\) and \(\delta_2\) under which there are nontrivial solutions. A symmetry property of the Green Function \(G\) is established when \(\gamma\) is not an eigenvalue relative to the two point conditions \(u(0)=u(1)=0\). The expression of \(G\) is given first for \(\gamma=0\), and significant properties of \(G\) are studied for general \(\gamma\). In particular, there is information about the sign of \(G\) in \([0,1]\times[0,1]\). In particular, \(G\) is found to be positive for \(\delta_1+\delta_2\) small, negative for \(\delta_1+\delta_2\) sufficiently large, and under some conditions it changes sign. It is proved that \(G\) satisfies some inequatities that allow to compare it to multiples of its restriction to a side of the square \([0,1]\times[0,1]\).\N\NFinally, on the basis of the properties established, and using the compression-expansion Krasnoselskii argument, the authors prove that the nonlinear problems have positive solutions. In terms of the assumptions on \(f\), bounds are given for the \(\|\cdot\|_\infty\) norm of that solution.\N\NFor the entire collection see [Zbl 1515.46001].