A graph $G$ is a $\lbrace d,d+k\rbrace $-graph, if one vertex has degree $d+k$ and the remaining vertices of $G$ have degree $d$. In the special case of $k=0$, the graph $G$ is $d$-regular. Let $k,p\ge 0$ and $d,n\ge 1$ be integers such that $n$ and $p$ are of the same parity. If $G$ is a connected $\lbrace d,d+k\rbrace $-graph of order $n$ without a matching $M$ of size $2|M|=n-p$, then we show in this paper the following: If $d=2$, then $k\ge 2(p+2)$ and (i) $n\ge k+p+6$. If $d\ge 3$ is odd and $t$ an integer with $1\le t\le p+2$, then (ii) $n\ge d+k+1$ for $k\ge d(p+2)$, (iii) $n\ge d(p+3)+2t+1$ for $d(p+2-t)+t\le k\le d(p+3-t)+t-3$, (iv) $n\ge d(p+3)+2p+7$ for $k\le p$. If $d\ge 4$ is even, then (v) $n\ge d+k+2-\eta $ for $k\ge d(p+3)+p+4+\eta $, (vi) $n\ge d+k+p+2-2t=d(p+4)+p+6$ for $k=d(p+3)+4+2t$ and $p\ge 1$, (vii) $n\ge d+k+p+4$ for $d(p+2)\le k\le d(p+3)+2$, (viii) $n\ge d(p+3)+p+4$ for $k\le d(p+2)-2$, where $0\le t\le \frac{1}{2}{p}-1$ and $\eta =0$ for even $p$ and $0\le t\le \frac{1}{2}(p-1)$ and $\eta =1$ for odd $p$. The special case $k=p=0$ of this result was done by Wallis [6] in 1981, and the case $p=0$ was proved by Caccetta and Mardiyono [2] in 1994. Examples show that the given bounds (i)–(viii) are best possible.
Making use of a modified Hadamard product, or convolution, of analytic functions with negative coefficients, combined with an integral operator, we study when a given analytic function is in a given class. Following an idea of U. Bednarz and J. Sokó l, we define the order of convolution consistence of three classes of functions and determine a given analytic function for certain classes of analytic functions with negative coefficients.
In this paper we consider the third-order nonlinear delay differential equation (∗) (a(t) x ′′(t) ) γ ) ′ + q(t)x γ (τ (t)) = 0, t ≥ t0, where a(t), q(t) are positive functions, γ > 0 is a quotient of odd positive integers and the delay function τ (t) 6 t satisfies lim t→∞ τ (t) = ∞. We establish some sufficient conditions which ensure that (∗) is oscillatory or the solutions converge to zero. Our results in the nondelay case extend and improve some known results and in the delay case the results can be applied to new classes of equations which are not covered by the known criteria. Some examples are considered to illustrate the main results.
In this paper we study the oscillation of the difference equations of the form Δ 2 xn+PnΔxn + f(n, Xn-ff, Δx n-h) = 0, in comparison with certain difference equations of order one whose oscillatory character is known. The results can be applied to the difference equation Δ 2 xn+pnΔxn +q n |x-_g|λ|Δxn -h |μ sgnx„-9 = 0, where A and \i are real constants, λ > 0 and μ ≥ 0.
Various new criteria for the oscillation of nonlinear neutral difference equations of the form Δi (xn — x n - h) + qn\xn~g\c sgns n -9 =0 , i = 1,2,3 and c > 0, are established.