The existence of positive solutions of nonlinear singular second-order boundary value problems
New results guaranteeing the existence of positive solutions to the boundary value problem \[ (1/p(t))(p(t) x'(t))'+ q(t) f(t,x(t),p(t) x'(t))= 0,\quad 0< t< 1,\quad \lim_{t\to 0+} p(t) x'(t)= 0= x(1), \] are given. Here, \(q\in C(0,1)\) and \(p\in C[0,1]\cap C^1(0, 1)\) are positive on \((0,1)\) and such that \[ \int^1_0 {dt\over p(t)}< \infty,\quad \int^1_0 p(t)q(t) dt< \infty, \] and \(f: [0,1]\times (0,\infty)\times (-\infty, 0]\to \mathbb{R}^0\) is continuous. The function \(f\) may be singular at \(x= 0\) and \(t=0\) or/and \(t=1\). To prove the existence theorems, the authors construct upper and lower solutions to the problem considered and use properties of the egress points and the consequent mapping. However, they apply the standard Schauder fixed-point theorem.
scientific article; zbMATH DE number 1755304