Solvability of a nonlinear general third order two-point eigenvalue problem on time scales
The third order dynamic equation \[ y^{\triangle \triangle \triangle}(t)+\lambda f(t,y(t),y^{\triangle}(t),y^{\triangle \triangle}(t)) = 0, \quad a < t < b, \] is investigated subject to the boundary conditions \[ \alpha_{11}y(a)-\alpha_{12}y(b) = 0, \] \[ \alpha_{21}y^{\triangle}(a)-\alpha_{22}y^{\triangle}(b) = 0, \] \[ \alpha_{31}y^{\triangle \triangle}(a)-\alpha_{32}y^{\triangle \triangle}(b) = 0, \] where \(f\) is continuous and nonnegative valued and \(\alpha_{i,j} \geq 0\), \(i=1,2,3\), \(j=1,2\). The authors obtain conditions on the constants \(\alpha_{i,j}\) that ensure positivity of the Green function. As an underlying space, the authors choose \((C^3_{rd}[a,b],||\cdot||)\) with \(\|u\| = \max_{t \in [a,\sigma^3(b)]} |u(t)|\), which is claimed to be a Banach space. Since it is not a Banach space, the cone-theoretic method applied in this paper fails to work.
scientific article; zbMATH DE number 5821534