Multiple positive solutions to a fourth-order boundary-value problem (Q2826991)

In this paper, the authors consider the \(4^{th}\)-order nonlinear ordinary differential equation: NEWLINE\[NEWLINE u^{(4)}(t)=f(t,u(t)),\quad 0<t<1, NEWLINE\]NEWLINE subject to homogeneous boundary conditions NEWLINE\[NEWLINE u(0) = u'(0) = u''(1) = u'''(1) = 0. NEWLINE\]NEWLINE The authors mix a fixed point approach both with a variational formulation to prove existence of positive solutions. More precisely, they seek critical points of some functional in a conical shell and then employ the compression-expansion Krasnosel'skii's fixed point theorem in cones of Banach spaces. Under positivity, monotonicity, and growth conditions on the nonlinearity \(f\), they prove existence of solutions first in the Sobolev space \(H^2\) and then in \(C^4\). Comparison estimates upon solutions are obtained. Illustrative examples of application are provided in this paper.