Regularly varying solutions of half-linear functional differential equations with retarded arguments
If \(f:\mathbb R^+\to\mathbb R^+\), then \(f\) is said to be regularly varying of index \(\rho\) if \(\lim_{t\to\infty}\frac{f(\lambda t)}{f(t)}=\lambda^\rho\) for any \(\lambda >0\). In case \(f\) is regularly varying with index \(0\), we say that \(f\) is slowly varying. The authors considers the equation \[ (| x'(t)| ^\alpha\text{sgn}\, x'(t))'=g(t)| x(g(t))| ^\alpha\text{sgn}\, x(g(t)), \] where \(\alpha >0\), \(q(t)>0\) and continuous, \(g(t)>0\) increasing and continuous such that \(g(t)<t\), \(\lim_{t\to \infty} g(t)=\infty\). Under the assumption \(\limsup_{t\to \infty}\frac{t}{g(t)}<\infty\), it is shown that the equation has a slowly varying solution and a regular solution of index 1 if and only if \(\lim_{t\to\infty}t^\alpha\) \(\int^\infty_t q(s)\,ds=0\).
half-linear functional differential equations
scientific article; zbMATH DE number 5355148