Existence and uniqueness of positive solution for Sturm-Liouville singular boundary value problem (Q883218)

The authors consider the following singular boundary value problem: \[ -(Lu)(t) = f(t, u(t)), t \in (0, 1); \] \[ R_1 (u) =: \alpha_1u(0) + \beta_1 u'(0) = 0; R_2 (u) =: \alpha_2 u(1) + \beta_2 u'(1) = 0, \] where \[ (Lu)(t) = (p(t)u'(t))' + q(t)u(t), f(t, u) :(0, 1] \times (0, \infty) \rightarrow [0, \infty) \] is continuous and may be singular at \(t=0, t=1\), and \(u = 0\). Let the hypothesis (H1) be: \(p(t) \in C^1[0, 1], p(t) > 0, q(t) \in C[0, 1], q(t) \leq 0, \alpha_i \geq 0 (i = 1, 2), \beta_1 \leq 0, \beta_2 \geq 0, \alpha_1 \alpha_2 + \alpha_1 \beta_2 - \beta_1 \alpha_2 > 0\) such that the homogeneous boundary value problem \[ - (Lu) (t) = 0, t \in (0, 1); \] \[ R_1 (u) = R_2 (u) = 0 \] has only the trivial solution. Under (H1), the Green's function of the homogeneous BVP has been investigated in [\textit{D. Guo, J. Sun} and \textit{Z. Liu}, Functional methods in nonlinear ordinary differential equations, Jinan: Shandong Science and Technology Press (1995)]. Using the integral representation of solutions via the Green's function, under some control of \(f(t, u)\), the \(C\)-positive solution is proved to exist uniquely, where a \(C\)-positive solution means a solution which belongs to \(C^2 (0, 1) \cap C [0, 1]\). In the last section, some examples are given.