Asymptotic behavior of retarded differential equations (Q1820918)

This paper deals with a higher order retarded differential equation, \((1)\quad L_ ny(t)+f(t,y(g(t)))=h(t),\) \(t\geq 0\), \(n\geq 2\) where \(L_ n\) is an operator defined by \[ L_ 0y(t):=y(t)/r_ 0(t),\quad L_ iy(t):=\frac{1}{r_ i(t)}\frac{d}{dt}L_{i-1}y(t),\quad i=0,1,...,n,\quad r_ n(t):=1. \] Here \(r_ i(t)\in C^{n-1}[R_+,R]\) with \(r_ i(t)>0\) for \(i=0,1,...,n-1\). The following conditions are assumed to hold. (i) \(f\in C[R_+,R,R]\) and there exist two positive functions p(t), \(H(t)\in C[R_+,R_+]\) with H(t) nondecreasing and kH(t)\(\leq H(kt)\) for any \(k>0\) such that \(| f(t,u)| \leq p(t)H(| u|),\) (ii) \(g,h\in C[R_+,R]\), g(t)\(\leq t\), \(\lim_{t\to \infty}g(t)=\infty\), \[ (iii)\quad \liminf_{t\to \infty}\frac{1}{r_ 0(t)}>0,\quad \limsup_{t\to \infty}\frac{w_ i(t,u)}{w_{n-1}(t,u)}<\infty,\quad i=1,2,...,n-2, \] where \(w_ i(t,u)\) is defined by \[ w_ i(t,u):=\int^{t}_{u}r_ 1(s_ 1)\int^{s_ 1}_{u}r_ 2(s_ 2)...\int^{s_{i-1}}_{u}r_ i(s_ i)ds_ i...ds_ 2ds_ 1. \] The first main result is the following. Theorem 1. Let \((2a,b)\quad \int^{\infty}w_{n- 1}(t)p(t)<\infty,\int^{\infty}| h(t)| dt<\infty\) hold. If y(t) is a solution of (1), then \(y(g(t))=O(w_{n-1}(t,T))\) for some \(T>0\). On the basis of theorem 1, this paper gives the following theorem. Theorem 2. Let (2a,b) hold. Assume that for some \(T>0\) \[ \int^{\infty}_{T}r_ 1(s_ 1)\int^{\infty}_{s_ 1}r_ 2(s_ 2)... \] \[ ...\int^{\infty}_{s_{n-2}}r_{n-1}(s_{n- 1})\int^{\infty}_{s_{n-1}}p(s)H(cw_{n-1}(s,T))ds ds_{n- 1}...ds_ 1<\infty \] for any constant \(c>0\), and \[ \int^{\infty}_{T}r_ 1(s_ 1)\int^{\infty}_{s_ 1}r_ 2(s_ 2)...\int^{\infty}_{s_{n-2}}r_{n-1}(s_{n- 1})\int^{\infty}_{s_{n-1}}| h(s)| ds ds_{n-1}...ds_ 1<\infty \] hold. Then every oscillatory solution y(t) of (1) satisfies \(\lim_{t\to \infty}L_ iy(t)=0\) for \(i=1,2,...,n-1\).