In this paper, we consider Seymour's Second Neighborhood Conjecture in 3-free digraphs, and prove that for any 3-free digraph $D$, there exists a vertex say $v$, such that $d$⁺⁺($v$) ≥ $⌊λd^+(v)⌋$, $λ$ = 0.6958 · · · . This slightly improves the known results in 3-free digraphs with large minimum out-degree.

In this paper, the existence and multiplicity of positive solutions for a class of non-resonant fourth-order integral boundary value problem

with two parameters are established by using the Guo-Krasnoselskii's fixed-point theorem, where $f∈C$([0,1]×[0,+∞)×(−∞,0], [0,+∞)), $q(t)∈L$¹[0,1] is nonnegative, $α, β ∈ R$ and satisfy $β<2π$², $α$>0, $α/π$⁴+$β/π$²<1, $λ$_1,2=(−$β$∓$\sqrt{β^2+4α}$)/2. The corresponding examples are raised to demonstrate the results we obtained.

Traditional May type cooperative model incorporating Michaelis-Menten type harvesting is proposed and studied in this paper. Sufficient conditions which ensure the extinction of the first species and the existence of a unique globally attractive positive equilibrium are obtained, respectively. Numerical simulations are carried out to show the feasibility of the main results.

Consider the $n$-dimensional incompressible Navier-Stokes equations

There exists a global weak solution under some assumptions on the initial function and the external force. It is well known that the global weak solutions become sufficiently small and smooth after a long time. Here are several very interesting questions about the global weak solutions of the Cauchy problems for the $n$-dimensional incompressible Navier-Stokes equations.
• Can we establish better decay estimates with sharp rates not only for the global weak solutions but also for all order derivatives of the global weak solutions?
• Can we accomplish the exact limits of all order derivatives of the global weak solutions in terms of the given information?
• Can we use the global smooth solution of the linear heat equation, with the same initial function and the external force, to approximate the global weak solutions of the Navier-Stokes equations?
• If we drop the nonlinear terms in the Navier-Stokes equations, will the exact limits reduce to the exact limits of the solutions of the linear heat equation?
• Will the exact limits of the derivatives of the global weak solutions of the Navier-Stokes equations and the exact limits of the derivatives of the global smooth solution of the heat equation increase at the same rate as the order $m$ of the derivative increases? In another word, will the ratio of the exact limits for the derivatives of the global weak solutions of the Navier-Stokes equations be the same as the ratio of the exact limits for the derivatives of the global smooth solutions for the linear heat equation?
The positive solutions to these questions obtained in this paper will definitely help us to better understand the properties of the global weak solutions of the incompressible Navier-Stokes equations and hopefully to discover new special structures of the Navier-Stokes equations.

This paper attempts to study the convergence of optimal values and optimal policies of continuous-time Markov decision processes (CTMDP for short) under the constrained average criteria. For a given original model $\mathcal{M}$_$∞$ of CTMDP with denumerable states and a sequence {$\mathcal{M}$_$n$} of CTMDP with finite states, we give a new convergence condition to ensure that the optimal values and optimal policies of {$\mathcal{M}$_$n$} converge to the optimal value and optimal policy of $\mathcal{M}$_$∞$ as the state space $S$_$n$ of $\mathcal{M}$_$n$ converges to the state space $S$_$∞$ of $\mathcal{M}$_$∞$, respectively. The transition rates and cost/reward functions of $\mathcal{M}$_$∞$ are allowed to be unbounded. Our approach can be viewed as a combination method of linear program and Lagrange multipliers.