Global existence and asymptotic behavior of solutions of second-order nonlinear differential equations
The author considers the equation (1) \((p(t)y')'=f(t,y,y')\), \(t\geq 0\), under the assumptions: \((A_ 1)p: [0,\infty)\to (0,\infty)\) is continuous and \(P(t)=\int^{t}_{0}1/p(s)ds\to \infty\) as \(t\to \infty\); \((A_ 2)\) \(f: [0,\infty)\times R\times R\to (0,\infty)\) is continuous and f(t,u,v) is nondecreasing in u and v, \((A_ 3)\) Every Cauchy problem for (1) has a unique solution. Using the Schauder-Tychonov fixed point theorem and by the continuous dependence on initial data, the author proves the existence of proper solutions of (1) belonging to separate classes of solutions introduced in the paper. Also the existence of a singular solution of (1) is presented. Next, using the supersolution-subsolution method, the author shows that the above results for (1) can be applied to solve the elliptic exterior Dirichlet problem.
scientific article; zbMATH DE number 4050024