Disconjugacy and extremal solutions of nonlinear third-order equations
Ruyun Ma Yanqiong Lu
In this paper, we make an exhaustive study of the third order linear operators $u''' +Mu$, $u'''+Mu'$ and $u'''+Mu''$ coupled with $(k, 3-k)$-conjugate boundary conditions, where $k=1,2$. We obtain the optimal intervals on which the Green's functions are of one sign. The main tool is the disconjugacy theory. As an application of our results, we develop a monotone iteration method to obtain positive solutions of the nonlinear problem $u'''+Mu''+f(t,u)=0$, $u(0)=u'(0)=u(1)=0$.
keywords: lower and upper solutions. disconjugacy Third-order equations positive solutions
Unilateral global bifurcation for $p$-Laplacian with non-$p-$1-linearization nonlinearity
Guowei Dai Ruyun Ma
In this paper, we establish a unilateral global bifurcation result from interval for a class of $p$-Laplacian problems. By applying above result, we study the spectrum of a class of half-quasilinear problems. Moreover, we also investigate the existence of nodal solutions for a class of half-quasilinear eigenvalue problems.
keywords: Unilateral bifurcation nodal solutions half-quasilinear problems $p$-Laplacian.
Eigenvalues, bifurcation and one-sign solutions for the periodic $p$-Laplacian
Guowei Dai Ruyun Ma Haiyan Wang
In this paper, we establish a unilateral global bifurcation result for a class of quasilinear periodic boundary problems with a sign-changing weight. By the Ljusternik-Schnirelmann theory, we first study the spectrum of the periodic $p$-Laplacian with the sign-changing weight. In particular, we show that there exist two simple, isolated, principal eigenvalues $\lambda_0^+$ and $\lambda_0^-$. Furthermore, under some natural hypotheses on perturbation function, we show that $(\lambda_0^\nu,0)$ is a bifurcation point of the above problems and there are two distinct unbounded sub-continua $C_\nu^{+}$ and $C_\nu^{-}$, consisting of the continuum $C_\nu$ emanating from $(\lambda_0^\nu, 0)$, where $\nu\in\{+,-\}$. As an application of the above result, we study the existence of one-sign solutions for a class of quasilinear periodic boundary problems with the sign-changing weight. Moreover, the uniqueness of one-sign solutions and the dependence of solutions on the parameter $\lambda$ are also studied.
keywords: One-sign solutions. Eigenvalues Unilateral global bifurcation Periodic $p$-Laplacian
Traveling waves of a Lotka-Volterra strong competition system with nonlocal dispersal
Guo-Bao Zhang Ruyun Ma Xue-Shi Li

This paper is concerned with the traveling waves of a nonlocal dispersal Lotka-Volterra strong competition model with bistable nonlinearity. We first establish the asymptotic behavior of traveling waves at infinity. Then by applying the stronger comparison principle and the sliding method, we prove that the traveling waves with nonzero speed are strictly monotone. Moreover, the uniqueness of wave speeds is also obtained.

keywords: Nonlocal dispersal strong competition system traveling waves monotonicity uniqueness wave speed
Partial differential equations with Robin boundary condition in online social networks
Guowei Dai Ruyun Ma Haiyan Wang Feng Wang Kuai Xu
In recent years, online social networks such as Twitter, have become a major source of information exchange and research on information diffusion in social networks has been accelerated. Partial differential equations are proposed to characterize temporal and spatial patterns of information diffusion over online social networks. The new modeling approach presents a new analytic framework towards quantifying information diffusion through the interplay of structural and topical influences. In this paper we develop a non-autonomous diffusive logistic model with indefinite weight and the Robin boundary condition to describe information diffusion in online social networks. It is validated with a real dataset from an online social network, The simulation shows that the logistic model with the Robin boundary condition is able to more accurately predict the density of influenced users. We study the bifurcation, stability of the diffusive logistic model with heterogeneity in distance. The bifurcation and stability results of the model information describe either information spreading or vanishing in online social networks.
keywords: diffusive logistic model indefinite weight Robin boundary condition. Bifurcation online social networks stability
On the Bonheure-Noris-Weth conjecture in the case of linearly bounded nonlinearities
Ruyun Ma Tianlan Chen Yanqiong Lu
Let $B_1$ be the unit ball in $\mathbb{R}^N$ with $N \geq 2$. Let $f\in C^1([0, \infty), \mathbb{R})$, $f(0)=0$, $f(\beta) = \beta, \ f(s) < s$ for $s\in (0,\beta), \ f(s) > s$ for $s \in (\beta, \infty)$ and $f'(\beta)>\lambda^{r}_k$. D. Bonheure, B. Noris and T. Weth [Ann. Inst. H. Poincaré Anal. Non Linéaire 29(4) (2012)] proved the existence of nondecreasing, radial positive solutions of the semilinear Neumann problem $$ -\Delta u+u=f(u) \ \text{in}\ B_1,\ \ \ \ \partial_\nu u=0 \ \text{on}\ \partial B_1 $$ for $k=2$, and they conjectured that there exists a radial solution with $k$ intersections with $\beta$ provided that $f'(\beta) >\lambda^r_k$ for $k>2$, where $\lambda^r_k$ is the $k$-th radial eigenvalue of $\Delta + I$ in the unit ball with Neumann boundary conditions. In this paper, we show that the answer is yes in the case of linearly bounded nonlinearities.
keywords: Neumann problem oscillatory radial solutions bifurcation. Bonheure-Noris-Weth conjecture

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