This paper is devoted to the investigation on the stability for two characteristic functions $f_1\left(z\right)=z^2+p{e}^{-z\tau }+q$ and $f_2\left(z\right)=z^2+pz{e}^{-z\tau }+q$, where $p$ and $q$ are real numbers and $\tau >0$. The obtained theorems describe the explicit stability dependence on the changing delay $\tau$. Our results are applied to some special cases of a linear differential system with delay in the diagonal terms and delay-dependent stability conditions are obtained.

This paper is concerned with the delay partial difference equation (1) $A_{m+1,n}+A_{m,n+1}-A_{m,n}+{\Sigma }_{i=1}^u{p}_i{A}_{m-k_i,n-l_i}=0$ where $p_i$ are real numbers, $k_i$ and $l_i$ are nonnegative integers, u is a positive integer. Sufficient and necessary conditions for all solutions of (1) to be oscillatory are obtained.

The paper considers a scalar differential equation of an advance-delay type $$\dot{y}\left(t\right)=-\left(a_0+\frac{a_1}{t}\right)y(t-\tau )+\left(b_0+\frac{b_1}{t}\right)y(t+\sigma ),$$ where constants $a_0$, $b_0$, $\tau$ and $\sigma$ are positive, and $a_1$ and $b_1$ are arbitrary. The behavior of its solutions for $t\to \infty$ is analyzed provided that the transcendental equation $$\lambda =-a_0{\mathrm{e}}^{-\lambda \tau }+b_0{\mathrm{e}}^{\lambda \sigma }$$ has a positive real root. An exponential-type function approximating the solution is searched for to be used in proving the existence of a semi-global solution. Moreover, the lower and upper estimates are given for such a solution.

In this article, we obtain sufficient conditions so that every solution of neutral delay differential equation $${(y\left(t\right)-\sum _{i=1}^k{p}_i\left(t\right)y\left(r_i\left(t\right)\right))}^{\left(n\right)}+v\left(t\right)G\left(y\left(g\left(t\right)\right)\right)-u\left(t\right)H\left(y\left(h\left(t\right)\right)\right)=f\left(t\right)$$ oscillates or tends to zero as $t\to \infty$, where, $n\ge 1$ is any positive integer, $p_i$, $r_i\in C^{\left(n\right)}([0,\infty ),ℝ)$  and $p_i$ are bounded for each $i=1,2,\cdots ,k$. Further, $f\in C\left(\right[0,\infty ),ℝ)$, $g$, $h$, $v$, $u\in C\left(\right[0,\infty ),[0,\infty \left)\right)$, $G$ and $H\in C(ℝ,ℝ)$. The functional delays $r_i\left(t\right)\le t$, $g\left(t\right)\le t$ and $h\left(t\right)\le t$ and all of them approach $\infty$ as $t\to \infty$. The results hold when $u\equiv 0$ and $f\left(t\right)\equiv 0$. This article extends, generalizes and improves some recent results, and further answers some unanswered questions from the literature. ...

In this work, necessary and sufficient conditions for the oscillation of solutions of 2-dimensional linear neutral delay difference systems of the form $$\Delta \left[\begin{array}{c}x\left(n\right)+p\left(n\right)x(n-m)\\ y\left(n\right)+p\left(n\right)y(n-m)\end{array}\right]=\left[\begin{array}{cc}a\left(n\right)& b\left(n\right)\\ c\left(n\right)& d\left(n\right)\end{array}\right]\left[\begin{array}{c}x(n-\alpha )\\ y(n-\beta )\end{array}\right]$$ are established, where $m>0$, $\alpha \ge 0$, $\beta \ge 0$ are integers and $a\left(n\right)$, $b\left(n\right)$, $c\left(n\right)$, $d\left(n\right)$, $p\left(n\right)$ are sequences of real numbers.

The aim of this paper is to establish an existence and uniqueness result for a class of the set functional differential equations of neutral type$$\left\{\begin{array}{c}{D}_{H}X\left(t\right)=F(t,X_t,D_H{X}_t),\hfill \\ {X|}_{[-r,0]}=\Psi ,\hfill \end{array}\right.$$ where $F:[0,b]\times {𝒞}_0\times {𝔏}_0^1\to K_c\left(E\right)$ is a given function, $K_c\left(E\right)$ is the family of all nonempty compact and convex subsets of a separable Banach space $E$, ${𝒞}_0$ denotes the space of all continuous set-valued functions $X$ from $[-r,0]$ into $K_c\left(E\right)$, ${𝔏}_0^1$ is the space of all integrally bounded set-valued functions $X:[-r,0]\to K_c\left(E\right)$, $\Psi \in {𝒞}_0$ and $D_H$ is the Hukuhara derivative. The continuous dependence of solutions on initial...

This paper deals with the existence of positive $\omega$-periodic solutions for the neutral functional differential equation with multiple delays $${(u\left(t\right)-cu(t-\delta ))}^{\text{'}}+a\left(t\right)u\left(t\right)=f(t,u(t-{\tau }_1),\cdots ,u(t-{\tau }_n)).$$ The essential inequality conditions on the existence of positive periodic solutions are obtained. These inequality conditions concern with the relations of $c$ and the coefficient function $a\left(t\right)$, and the nonlinearity $f(t,x_1,\cdots ,x_n)$. Our discussion is based on the perturbation method of positive operator and fixed point index theory in cones.

We show that every operator from ${\ell }_s$ to ${\ell }_p\otimes ̂{\ell }_q$ is compact when 1 ≤ p,q < s and that every operator from ${\ell }_s$ to ${\ell }_p\widehat{\otimes ̂}{\ell }_q$ is compact when 1/p + 1/q > 1 + 1/s.