Conditions for the existence and uniqueness of a solution of the Cauchy problem $$u^{\text{'}}\left(t\right)=p\left(t\right)u\left(\tau \left(t\right)\right)+q\left(t\right),u\left(a\right)=c,$$ established in [2], are formulated more precisely and refined for the special case, where the function $\tau$ maps the interval $]a,b[$ into some subinterval $[{\tau }_0,{\tau }_1]\subseteq [a,b]$, which can be degenerated to a point.

In this paper we consider the existence and asymptotic behavior of solutions of the following problem: $$u_{tt}{(t,x)-(\alpha +\beta \parallel \nabla u(t,x)\parallel }_2^2+{\beta \parallel \nabla v(t,x)\parallel }_2^2)\Delta u(t,x)+\delta |u_t{(t,x)|}^{p-1}{u}_t(t,x)={\mu \left|u(t,x)\right|}^{q-1}u(t,x),x\in \Omega ,t\ge 0,v_{tt}{(t,x)-(\alpha +\beta \parallel \nabla u(t,x)\parallel }_2^2+{\beta \parallel \nabla v(t,x)\parallel }_2^2)\Delta v(t,x)+\delta |v_t{(t,x)|}^{p-1}{v}_t(t,x)={\mu \left|v(t,x)\right|}^{q-1}v(t,x),x\in \Omega ,t\ge 0,u(0,x)=u_0\left(x\right),u_t(0,x)=u_1\left(x\right),x\in \Omega ,v(0,x)=v_0\left(x\right),v_t(0,x)=v_1\left(x\right),x\in \Omega ,u{{|}_{{}_{\partial \Omega }}=v|}_{{}_{\partial \Omega }}=0$$ where $q>1$, $p\ge 1$, $\delta >0$, $\alpha >0$, $\beta \ge 0$, $\mu \in ℝ$ and $\Delta$ is the Laplacian in ${ℝ}^N$.

We study the existence of positive solutions of the integral equation $$x\left(t\right)=\mu {\int }_0^{1}k(t,s)f(s,x\left(s\right),x^{\text{'}}\left(s\right),...,x^{(n-1)}\left(s\right))ds,n\ge 2$$ in both $C^{n-1}[0,1]$ and $W^{n-1,p}[0,1]$ spaces, where $p\ge 1$ and $\mu >0$. Throughout this paper $k$ is nonnegative but the nonlinearity $f$ may take negative values. The Krasnosielski fixed point theorem on cone is used.