Multiple solutions for \(2m\text{th}\)-order Sturm-Liouville boundary value problems
The authors consider the \(2m\)th-order ordinary differential equation \[ (-1)^{m}y^{(2m)} = f(t, y), \qquad 0\leq t\leq 1,\tag{1} \] with \(m\geq 1\) and satisfying the boundary conditions \[ \alpha_{i+1}y^{(2i)}(0) - \beta_{i+1}y^{(2i+1)}(0) = 0, \quad 0\leq i\leq m-1, \] \[ \gamma_{i+1}y^{(2i)}(1) - \delta_{i+1}y^{(2i+1)}(1) = 0, \quad 0\leq j\leq m-1, \tag{2} \] where \(f:\;[0, 1]\times [0, \infty) \to [0, \infty)\) is continuous, \(f(t, y) \neq 0\) on any subinterval of \([0, 1]\), and for all \(0<y<\infty \), and for each \(1\leq i\leq m\), \(\alpha_{i}, \beta_{i}, \gamma_{i}, \delta_{i} \geq 0\) such that \[ \rho_{i}=\gamma_{i} \beta_i +\alpha_i \gamma_i +\alpha_i \delta_i>0. \tag{3} \] The techniques for obtaining multiple solutions are involved in applications of Krasnosel'skii fixed-point theorem and properties of Green's function. The authors found by classical technique growth restrictions on \(f\) which yield the existence of positive and multiple solutions. Also, they show that any number of positive solutions to (1), (2) can be obtained when appropriate combinations of the proposed four assumptions are imposed on \(f\).
scientific article; zbMATH DE number 1528550
scientific article