Positive decreasing solutions of systems of second order singular differential equations
Here, the singular differential systems \[ \bigl(p(t)|y' |^{\alpha-1} y'\bigr)'= \varphi(t)z^{-\lambda}, \quad\bigl(q(t)|z' |^{\beta-1} z'\bigr)'=\psi (t)y^{-\mu}, \tag{*} \] are considered, where \(\alpha, \beta,\lambda, \mu\) are positive constants in an interval \([a,\infty)\), and \(p,q,\varphi,\psi\) are positive continuous functions on \([a,\infty)\). A positive decreasing solution to system (*) is called singular according to whether it exists on \([a,\infty)\) or it ceases to exist at a finite point of \([a,\infty)\). First, conditions are given under which there does exist a singular solution to (*) and is called an extinct singular solution. Then conditions for the existence of proper solutions to system (*) are established. They are classified into moderately decreasing solutions and strongly decreasing solutions according to the rate of their decay as \(t\to\infty\) and called decreasing proper solutions. The basic results are presented in 5 theorems, and its significance is discovered in 3 examples. 2 remarkes and 3 applications.
scientific article; zbMATH DE number 1563280