A Novel Study on the Non-Negative Solution of an Eighth-Order BVP

In this article, we investigate the existence of non-negative solutions for a boundary value problem associated with an eighth-order differential equation \(\lambda^{(8)} ( \varpi) = \psi( \varpi, \lambda(\varpi)\), \(\lambda^{(1)}( \varpi), \cdots, \lambda^{(7)} (\varpi))\) for \(0 < \varpi < 1\), under initial values \(\lambda(0)=\lambda^{'}(0) = \lambda^{''}(0) = \lambda^{'''}(0) =0\) and \(\lambda^{(4)}(1) = \lambda^{(5)}(1) = \lambda^{(6)}(1) = \lambda^{(7)}(1) = 0\), where \(\psi\) is non-negative continuous function. For this study, we use the nonlinear Leray-Schauder alternative and the Leray-Schauder fixed-point theorem to prove the existence of at least one non-negative solution. As a numerical application, we present an example to confirm the utility of the achieved results.