Integral criteria are established for $dim{V}_i\left(p\right)=0$ and $dim{V}_i\left(p\right)=1,i\in \{0,1\}$, where $V_i\left(p\right)$ is the space of solutions $u$ of the equation $$u^{\text{'}\text{'}}+p\left(t\right)u=0$$ satisfying the condition $${\int }^{+\infty }\frac{u^2\left(s\right)}{s^i}ds<+\infty .$$

Let $T$ be a $\gamma$-contraction on a Banach space $Y$ and let $S$ be an almost $\gamma$-contraction, i.e. sum of an $\left(\epsilon ,\gamma \right)$-contraction with a continuous, bounded function which is less than $\epsilon$ in norm. According to the contraction principle, there is a unique element $u$ in $Y$ for which $u=Tu.$ If moreover there exists $v$ in $Y$ with $v=Sv$, then we will give estimates for $\parallel u-v\parallel .$ Finally, we establish some inequalities related to the Cauchy problem.

The paper deals with the periodic boundary value problem (1) $L_{4}x\left(t\right)+a\left(t\right)x\left(t\right)\in F(t,x\left(t\right))$, $t\in J=[a,b]$, (2) $L_{i}x\left(a\right)=L_{i}x\left(b\right)$, $i=0,1,2,3$, where $L_{0}x\left(t\right)=a_{0}x\left(t\right)$, $L_{i}x\left(t\right)=a_i\left(t\right)L_{i-1}x\left(t\right)$, $i=1,2,3,4$, $a_0\left(t\right)=a_4\left(t\right)=1$, $a_i\left(t\right)$, $i=1,2,3$ and $a\left(t\right)$ are continuous on $J$, $a\left(t\right)\ge 0$, $a_i\left(t\right)>0$, $i=1,2$, $a_1\left(t\right)=a_3\left(t\right)·F(t,x):J\times R\to${nonempty convex compact subsets of $R$}, $R=(-\infty ,\infty )$. The existence of such periodic solution is proven via Ky Fan’s fixed point theorem.

The higher order neutral functional differential equation $$\frac{{\mathrm{d}}^n}{\mathrm{d}{t}^n}\left[x\left(t\right)+h\left(t\right)x\left(\tau \right(t\left)\right)\right]+\sigma f\left(t,x\left(g\right(t\left)\right)\right)=0\left(1\right)$$ is considered under the following conditions: $n\ge 2$, $\sigma =\pm 1$, $\tau \left(t\right)$ is strictly increasing in $t\in [t_0,\infty )$, $\tau \left(t\right)<t$ for $t\ge t_0$, ${lim}_{t\to \infty }\tau \left(t\right)=\infty$, ${lim}_{t\to \infty }g\left(t\right)=\infty$, and $f(t,u)$ is nonnegative on $[t_0,\infty )\times (0,\infty )$ and nondecreasing in $u\in (0,\infty )$. A necessary and sufficient condition is derived for the existence of certain positive solutions of (1).