Twin positive symmetric solutions for Lidstone boundary value problems (Q1884560)

The authors are concerned with the existence of two positive solutions for the \(2n\)th-order Lidstone boundary value problem \[ (\Phi(y^{(2n-1)}))'(t) = f(t, y(t), y''(t), \dots, y^{(2(n-1))}(t)), \quad 0 \leq t \leq 1, \] \[ y^{(2i)}(0) = y^{(2i)}(1) = 0, \quad 0 \leq i \leq n-1, \] where \(f:[0,1] \times \mathbb{R}^n \to \mathbb{R}\) is continuous and \(\Phi(v) = | v| ^{p-2}v\), \(p > 1\). The authors allow the nonlinear term \(f\) to change signs. As such, they consider their results to be companion results to those found in [\textit{J. M. Davis}, \textit{P. W. Eloe} and \textit{J. Henderson}, J. Math. Anal. Appl. 237, 710--720 (1999; Zbl 0935.34020) and \textit{J. M. Davis}, \textit{J. Henderson} and \textit{P. J. Y. Wong}, J. Math. Anal. Appl. 251, 527--548 (2000; Zbl 0966.34023)]. In the second section of the paper, the authors develop a new fixed-point theorem for cone preserving operators. Their main result, in section 3, uses the Schauder fixed-point theorem in conjunction with the fixed-point theorem from section 2 to establish the existence of two positive solutions. This paper will be of interest to anyone doing research in Lidstone boundary value problems and those looking for a new cone preserving fixed-point theorem.