We show that if a real $n \times n$ non-singular matrix ($n \ge m$) has all its minors of order $m-1$ non-negative and has all its minors of order $m$ which come from consecutive rows non-negative, then all $m$th order minors are non-negative, which may be considered an extension of Fekete’s lemma.
We prove the existence of solutions of four-point boundary value problems under the assumption that $f$ fulfils various combinations of sign conditions and no growth restrictions are imposed on $f$. In contrast to earlier works all our results are proved for the Carathéodory case.
We will give an existence and uniqueness theorem for ordinary differential equations in Fréchet spaces using Lipschitz conditions formulated with a generalized distance and row-finite matrices.
In this paper, oscillation and asymptotic behaviour of solutions of
\[ y^{\prime \prime \prime} + a(t)y^{\prime \prime}+b(t)y^{\prime} + c(t)y=0 \] have been studied under suitable assumptions on the coefficient functions $a,b,c\in C([\sigma ,\infty),R)$, $ \sigma \in R$, such that $a(t)\ge 0$, $b(t) \le 0$ and $c(t) < 0$.
et $p\in (0,1)$ be a real number and let $n\ge 2$ be an even integer. We determine the largest value $c_n(p)$ such that the inequality
\[ \sum ^n_{i=1} |a_i|^p \ge c_n(p) \] holds for all real numbers $a_1,\ldots ,a_n$ which are pairwise distinct and satisfy $\min _{i\ne j} |a_i-a_j| = 1$. Our theorem completes results of Ozeki, Mitrinović-Kalajdžić, and Russell, who found the optimal value $c_n(p)$ in the case $p>0$ and $n$ odd, and in the case $p\ge 1$ and $n$ even.