A nonlinear Bessel differential equation associated with Cauchy conditions (Q1921237)

This paper is concerned with the existence and uniqueness of solutions of Bessel's nonlinear differential equation with Cauchy conditions, \(y''(x) + ({1 \over x}) y'(x) + y(x) = f(y(x))\), \(x > 0\), \(y(0) = 1\), \(y'(0) = y_0'\), where \(f \in C^1 (\mathbb{R},\mathbb{R})\), \(f(0) = 0\). By using a fixed point technique, the existence and uniqueness of solutions of Bessel's nonlinear problem is proved. In the case \(f(y) = y^2\), a case of some interest in applications, the authors show that \(y(x)\), the unique solution of Bessel's nonlinear differential equation, together with \(y'(x)\) and \(y''(x)\) tend to zero as \(x \to \infty\). At last, a numerical example of the above Cauchy problem on a very large interval is presented.