Minimal positive solutions to some singular second-order differential equations
The author studies the boundary value problem \[ x''+ f(t, x,x')= 0,\quad 0< t< 1,\quad x(0)= x'(1)= 0, \] where the function \(f\) is continuous and satisfies the inequality \(f(t,x,y)\leq k(t) p(x) q(y)\) with functions \(k\), \(p\) and \(q\) satisfying certain conditions. A solution \(x(t)\) to this problem is said to be minimal positive if it belongs to \(C[0,1]\cap C^2(0,1)\), \(x(t)> 0\) for \(t\in (0,1]\) and no other positive solution is smaller than it. The author finds which condition the functions \(k\), \(p\) and \(q\) must satisfy to guarantee the existence of a minimal positive solution.
singular second-order differential equations
scientific article; zbMATH DE number 1744289