Takaŝi Kusano and Tomoyuki Tanigawa

Abstract

Singular differential systems of the type (p(t)|y′|α−1y′)′=φ(t)z−λ,  (q(t)|z′|β−1z′)′=ψ(t)y−μ  (∗) are considered in an interval [a,∞), where α, β, λ, μ are positive constants and p, q, φ,ψ are positive continuous functions on [a,∞). A positive decreasing solution of (∗) is called proper or singular according to whether it exists on [a,∞) or it ceases to exist at a finite point of (a,∞). First, conditions are given under which there does exist a singular solution of (∗). Then, conditions are established for the existence of proper solutions of (∗) which are classified into moderately decreasing solutions and strongly decreasing solutions according to the rate of their decay as t→∞.