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].

It will be proved that if $N$ is a bounded nilpotent operator on a Banach space $X$ of order $k+1$, where $k\ge 1$ is an integer, then the $\gamma$-th order Cesàro mean $C_t^{\gamma }:=\gamma t^{-\gamma }{\int }_0^t{(t-s)}^{\gamma -1}T\left(s\right)ds$ and Abel mean $A_{\lambda }:=\lambda {\int }_0^{\infty }{e}^{-\lambda s}T\left(s\right)ds$ of the uniformly continuous semigroup ${\left(T\left(t\right)\right)}_{t\ge 0}$ of bounded linear operators on $X$ generated by $iaI+N$, where $0\ne a\in ℝ$, satisfy that (a) $\parallel C_t^{\gamma }\parallel \sim t^{k-\gamma }(t\to \infty )$ for all $0<\gamma \le k+1$; (b) $\parallel C_t^{\gamma }\parallel \sim t^{-1}(t\to \infty )$ for all $\gamma \ge k+1$; (c) $\parallel A_{\lambda }\parallel \sim \lambda (\lambda \downarrow 0)$. A similar result will be also proved for the uniformly continuous cosine function ${\left(C\left(t\right)\right)}_{t\ge 0}$ of bounded linear operators on $X$ generated by ${(iaI+N)}^2$.

Let ${A_k}_{k=0}^{+\infty }$ be a sequence of arbitrary complex numbers, let α,β > -1, let Pₙα,βn=0+∞$betheJacobipolynomialsanddefinethefunctions$ $Hₙ(\alpha ,z)={\sum }_{m=n}^{+\infty }\left(A_m{z}^m\right)/(\Gamma (\alpha +n+m+1)(m-n)!)$, $G(\alpha ,\beta ,x,y)={\sum }_{r,s=0}^{+\infty }\left(A_{r+s}{x}^r{y}^s\right)/(\Gamma (\alpha +r+1)\Gamma (\beta +s+1)r!s!)$. Then, for any non-negative integer n, ${\int }_0^{\pi /2}G(\alpha ,\beta ,x²sin²\varphi ,y²cos²\varphi )P{ₙ}^{\alpha ,\beta }\left(cos²\varphi \right)si{n}^{2\alpha +1}\varphi co{s}^{2\beta +1}\varphi d=1/2Hₙ(\alpha +\beta +1,x²+y²)P{ₙ}^{\alpha ,\beta }((y²-x²)/(y²+x²))$. When $A_k={(-1/4)}^k$, this formula reduces to Bateman’s expansion for Bessel functions. For particular values of y and n one obtains generalizations of several formulas already known for Bessel functions, like Sonine’s first and second finite integrals and certain Neumann series expansions. Particular choices of ${A_k}_{k=0}^{+\infty }$ allow one to write all these type of formulas...