Half eigenvalues and the Fučík spectrum of multi-point boundary value problems (Q413460)

The paper is concerned with the nonlinear multi-point boundary value problem NEWLINE\[NEWLINE-u''=f(u)+h \quad \quad \text{a.e. on}\,\,\,(-1,1),\tag{1}NEWLINE\]NEWLINE NEWLINE\[NEWLINEu(\pm 1)=\displaystyle\sum_{i=1}^{m^{\pm}}\alpha_i^{\pm}u(\eta_i^{\pm}),\tag{2}NEWLINE\]NEWLINE where \(h \in L^1(-1,1)\), \(m^{\pm}\geq 1\) are integers, \(\alpha^{\pm}=(\alpha_1^{\pm},\ldots, \alpha_{m^{\pm}}^{\pm})\in\mathbb{R}_+^{m^{\pm}}\), \(\eta^{\pm}\in (-1,1)^{m^{\pm}}\)NEWLINENEWLINE and \(\displaystyle\sum_{i=1}^{m^{\pm}}\alpha_i^{\pm}<1\). The equation NEWLINE\[NEWLINE-u''=\lambda(a u^+-b u^-)\,\,\,\text{on}\,\,\,(-1,1)\tag{3}NEWLINE\]NEWLINE with the boundary conditions \((2)\), where \(a,\,b>0\) and \(u^{\pm}(x)=\max\{\pm u(x),0\}\) for \(x\in [-1,1]\) is also investigated. The authors prove the existence of a sequence of half-eigenvalues \(\lambda=\lambda(a,b)\) for the problem \((2)-(3)\). Some spectral and degree theoretic properties of the set of half-eigenvalues, and specified nodal properties of the corresponding half-eigenfunctions are also presented. Based on these properties, the authors obtain solvability and non-solvability results, a global bifurcation theorem and nodal solutions for the problem \((1)-(2)\).