Sharp bounds on some distance-based graph invariants of $n$-vertex $k$-trees are established in a unified approach, which may be viewed as the weighted Wiener index or weighted Harary index. The main techniques used in this paper are graph transformations and mathematical induction. Our results demonstrate that among $k$-trees with $n$ vertices the extremal graphs with the maximal and the second maximal reciprocal sum-degree distance are coincident with graphs having the maximal and the second maximal reciprocal...

We formulate nonuniform nonresonance criteria for certain third order differential systems of the form $X^{{}^{\text{'}\text{'}\text{'}}}+A{X}^{{}^{\text{'}\text{'}}}+G(t,X^{{}^{\text{'}}})+CX=P\left(t\right)$, which further improves upon our recent results in [12], given under sharp nonresonance considerations. The work also provides extensions and generalisations to the results of Ezeilo and Omari [5], and Minhós [9] from the scalar to the vector situations.

In this paper, we shall give sufficient conditions for the ultimate boundedness of solutions for some system of third order non-linear ordinary differential equations of the form $$\stackrel{\text{t}o8pt...}{Xwidth0ptheight5.46pt}+F\left(\ddot{X}\right)+G\left(\dot{X}\right)+H\left(X\right)=P(t,X,\dot{X},\ddot{X})$$ where $X,F\left(\ddot{X}\right)$, $G\left(\dot{X}\right)$, $H\left(X\right)$, $P(t,X,\dot{X},\ddot{X})$ are real $n$-vectors with $F,G$, $H:{ℝ}^n\to {ℝ}^n$ and $P:ℝ\times {ℝ}^n\times {ℝ}^n\times {ℝ}^n\to {ℝ}^n$ continuous in their respective arguments. We do not necessarily require that $F\left(\ddot{X}\right),G\left(\dot{X}\right)$ and $H\left(X\right)$ are differentiable. Using the basic tools of a complete Lyapunov Function, earlier results are generalized.

The dynamical behaviour of a continuous time recurrent neural network model with a special weight matrix is studied. The network contains several identical excitatory neurons and a single inhibitory one. This special construction enables us to reduce the dimension of the system and then fully characterize the local and global codimension-one bifurcations. It is shown that besides saddle-node and Andronov-Hopf bifurcations, homoclinic and cycle fold bifurcations may occur. These bifurcation curves...

This paper is concerned with an SIR model with periodic incidence rate and saturated treatment function. Under proper conditions, we employ a novel argument to establish a criterion on the global exponential stability of positive periodic solutions for this model. The result obtained improves and supplements existing ones. We also use numerical simulations to illustrate our theoretical results.