The nonlinear Sturm-Liouville boundary problem is considered of the form \[ -(py')'+qy=\lambda(1-f)ry, \]
\[ (a_j\lambda+b_j)y(j)=(c_j\lambda+d_j)(py')(j),\quad j=0,1. \] Here, \(a_0=c_0=0\), \(p>0\), \(r>0\) and real-valued \(q\) are functions depending on the independent variable \(x\) alone, while \(f\) depends on \(x\), \(y\) and \(y'\) and it is assumed to be continuous and zero when \(y=0\); \(a_1d_1-b_1c_1>0\), \(c_1\not=0\). This problem is presented in the operator form: \[ AY=\lambda (I-F(Y))Y, \tag{1} \] where the operator \(A\): \[ AY=\left(\begin{matrix} r^{-1}(-(py')'+qy) \\ by(1)-dp(1)y'(1)\end{matrix}\right), \] possesses the domain \[ D(A)=\left\{\left(\begin{matrix} y\\ -ay(1)+cp(1)y'(1)\end{matrix}\right): \begin{matrix} y, py'\in AC, \int_0^1r|r^{-1}(-(py')'+qy)|^2 dx<\infty, \\ y\in\{y\in C^1([0,1];\mathbb{R}): y(0)\cos\alpha=(py')(0)\sin\alpha\}\end{matrix} \right\} \] and \[ F(Y)(x)=\left[\begin{matrix} f(x,y(x),y'(x)) &0\\ 0 &0 \end{matrix}\right]. \] By an eigenpair is meant a pair \((\lambda,Y)\) with \(Y\not=0\) which solves equation (1). The eigenpairs of given \(\|Y\|\) and given oscillation count for \(y\) are investigated. Riesz basis properties of (normalized) eigenvectors are studied.