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.

We study extension of $p$-trigonometric functions ${sin}_p$ and ${cos}_p$ to complex domain. For $p=4,6,8,\cdots$, the function ${sin}_p$ satisfies the initial value problem which is equivalent to (*) $$-{\left(u^{\text{'}}\right)}^{p-2}{u}^{\text{'}\text{'}}-u^{p-1}=0,u\left(0\right)=0,u^{\text{'}}\left(0\right)=1$$ in $ℝ$. In our recent paper, Girg, Kotrla (2014), we showed that ${sin}_p\left(x\right)$ is a real analytic function for $p=4,6,8,\cdots$ on $(-{\pi }_p/2,{\pi }_p/2)$, where ${\pi }_p/2={\int }_0^1{(1-s^p)}^{-1/p}$. This allows us to extend ${sin}_p$ to complex domain by its Maclaurin series convergent on the disc $\{z\in ℂ:|z|<{\pi }_p/2\}$. The question is whether this extensions ${sin}_p\left(z\right)$ satisfies (*) in the sense of differential equations in complex domain. This...

In this paper we show that from an estimate of the form $su{p}_{t\ge 0}\left|\right|C\left(t\right)-cos\left(at\right)I\left|\right|<1$, we can conclude that C(t) equals cos(at)I. Here ${\left(C\left(t\right)\right)}_{t\ge 0}$ is a strongly continuous cosine family on a Banach space.

Let $f_c(x,y)\equiv {\sum }_{j=1}^{\infty }{\sum }_{k=1}^{\infty }{a}_{jk}(1-cosjx)(1-cosky)$ with $a_{jk}\ge 0$ for all j,k ≥ 1. We estimate the integral ${\int }_0^{\pi }{\int }_0^{\pi }{x}^{\alpha -1}{y}^{\beta -1}\varphi \left(f_c(x,y)\right)dxdy$ in terms of the coefficients $a_{jk}$, where α, β ∈ ℝ and ϕ: [0,∞] → [0,∞]. Our results can be regarded as the trigonometric analogues of those of Mazhar and Móricz [MM]. They generalize and extend Boas [B, Theorem 6.7].