On a two-point boundary value problem for the second order ordinary differential equations with singularities
The authors consider the boundary value problem \[ u''= f(t, u,u'),\tag{1} \] \[ u(a+)= 0,\qquad u(b-)= 0,\tag{2} \] with \(f=\in K_{\text{loc}}(]a,b[\times\mathbb{R}^2;\mathbb{R})\). By a solution to problem (1), (2), the authors understand a function \(u\in\widetilde C_{\text{loc}}'(]a,b[; \mathbb{R})\) satisfying (1) almost everywhere in \(]a,b[\) and also conditions (2). By using a lower solution \(\alpha(t)\) to (1) and an upper solution \(\beta(t)\) to (1), the existence of at least one solution \(u(t)\) to problem (1), (2), is proved, where \(\alpha(t)\leq u(t)\leq \beta(t)\) for \(a\leq t\leq b\).
second-order singular differential equation
scientific article; zbMATH DE number 1880111