二阶两点边值问题的多解存在性

本文讨论一类二阶两点边值问题$x^{\prime\prime}(t)+f(t,x(t),x^{\prime}(t))=0, t\in (0, 1)$, $a x(0)-b x^\prime(0)=0, ~~c x(1)+d x^\prime(1)=0$,~~其中 $f:[0,1]\times R^2\longrightarrow R$ 是连续的, $ a>0,b\ge 0,c>0,d\ge 0$. 通过运用上下解方法和 Leray-Schauder 度理论,得到了三个解的存在性结果.     

Nonoscillation theorems for certain second order nonlinear differential equations

Some necessary and sufficient conditions for nonoscillation are established for the second order nonlinear differential equation where p > 0 is a constant. These results are extensions of the earlier results of Hille, Wintner, Opial, Yan for second order linear differential equations and include the recent results of Li and Yeh, Kusano and Yoshida, Yang and Lo for half-linear differential equations. Authors’ address: School of Mathematical Sciences, South China Normal University, Guangzhou 510631, P.R. China     

Oscillation theorems for second order nonlinear dynamic equations

Some new criteria for the oscillation of nonlinear dynamic equations of the form $$\bigl(a(t)(x^{\Delta}(t))^{\alpha}\bigr)^{\Delta}+f(t,x^{\sigma}(t))=0$$ on a time scale $\mathbb{T}$ are established.     

Boundedness and oscillation for nonlinear dynamic equations on a time scale

We obtain some boundedness and oscillation criteria for solutions to the nonlinear dynamic equation

on time scales. In particular, no explicit sign assumptions are made with respect to the coefficient . We illustrate the results by several examples, including a nonlinear Emden-Fowler dynamic equation.

     

Boundedness of Solutions for Duffing Equation with Low Regularity in Time

It is shown that all solutions are bounded for Duffing equation x+ x~(2n+1)+2∑i=nPj(t)x~j= 0, provided that for each n + 1 ≤ j ≤ 2 n, P_j ∈ C~y(T~1) with γ 1-1/n and for each j with 0 ≤ j ≤ n, Pj ∈ L(T~1) where T~1= R/Z.     

Comparison and oscillatory behavior for certain second order nonlinear dynamic equations

We present some new necessary and sufficient conditions for the oscillation of second order nonlinear dynamic equation $$\bigl(a\bigl(x^{\Delta }\bigr)^{\alpha }\bigr)^{\Delta }(t)+q(t)x^{\beta }(t)=0$$ on an arbitrary time scale $\mathbb{T}$ , where α and β are ratios of positive odd integers, a and q are positive rd-continuous functions on $\mathbb{T}$ . Comparison results with the inequality $$\bigl(a\bigl(x^{\Delta }\bigr)^{\alpha }\bigr)^{\Delta }(t)+q(t)x^{\beta }(t)\leqslant 0\quad (\geqslant 0)$$ are established and application to neutral equations of the form $$\bigl(a(t)\bigl(\bigl[x(t)+p(t)x[\tau (t)]\bigr]^{\Delta }\bigr)^{\alpha }\bigr)^{\Delta }+q(t)x^{\beta }\bigl[g(t)\bigr]=0$$ are investigated.     

Oscillatory Behavior of Second-Order Nonlinear Differential Equations with a Nonpositive Neutral Term

We shall present new oscillation criteria of second-order nonlinear differential equations with a nonpositive neutral term of the form:

$$\begin{aligned} \left( (a(t)\left( \left( x(t)-p(t)x(\sigma (t) )^{\prime } \right) ^{\gamma } \right) ^{\prime }+q(t)x^{\beta }(\tau (t))=0,\right. \end{aligned}$$

with positive coefficients. The obtained results answer an open problem raised in Li et al. [Adv Differ Equ 35:7, 2015, Remark 4.3 (P2)]. Examples are given to illustrate the main results.

     

Existence and infinity of periodic solutions of some second order differential equations with isochronous potentials

In this paper, we deal with the existence and infinity of periodic solutions of differential equations, $$x^{\prime\prime}+f(x^{\prime})+V^{\prime}(x)+g(x)=p(t),$$ where V is a 2??/n-isochronous potential. When f, g are bounded, we give sufficient conditions to ensure the existence of periodic solutions of this equation. We also prove that the given equation has infinitely many 2??-periodic solutions under resonant conditions by using the topological degree approach.     

Asymptotic properties of Laguerre–Sobolev type orthogonal polynomials

In this contribution we consider the asymptotic behavior of sequences of monic polynomials orthogonal with respect to a Sobolev-type inner product

Monotonicity of some functions involving the gamma and psi functions

Let , where is Euler's gamma function. We determine conditions for the numbers so that the function is strongly completely monotonic on . Through this result we obtain some inequalities involving the ratio of gamma functions and provide some applications in the context of trigonometric sum estimation. We also give several other examples of strongly completely monotonic functions defined in terms of and functions. Some limiting and particular cases are also considered.

     

Stability of sets for impulsive functional differential equations via Razumikhin method

In this paper, we consider the functional differential equation with impulsive perturbations

Almost orthogonal operators on the bitorus II

We prove that the operator ${Tf(x,y)=\int^\pi_{-\pi}\int_{|x^{\prime}|<|y^{\prime}|} \frac{e^{iN(x,y) x^{\prime}}}{x^{\prime}}\frac{e^{iN(x,y) y^{\prime}}}{y^{\prime}}f(x-x^{\prime}, y-y^{\prime}) dx^{\prime} dy^{\prime}}$ , with ${x,y \in[0,2\pi]}$ and where the cut off ${|x^{\prime}|<|y^{\prime}|}$ is performed in a smooth and dyadic way, is bounded from L ² to weak- ${L^{2-\epsilon}}$ , any ${\epsilon > 0 }$ , under the basic assumption that the real-valued measurable function N(x, y) is “mainly” a function of y and the additional assumption that N(x, y) is non-decreasing in x, for every y fixed. This is an extension to 2D of C. Fefferman’s proof of a.e. convergence of Fourier series of L ² functions.     

Nonoscillation theorems for certain second order nonlinear differential equations

Some necessary and sufficient conditions for nonoscillation are established for the second order nonlinear differential equation (r(t)y(x(t))|x^￠(t)|^p-1x^￠(t))^￠+c(t)f(x(t))=0,    t 3 t₀,(r(t)\psi(x(t))\vert x^{\prime}(t)\vert^{p-1}x^{\prime}(t))^{\prime}+c(t)f(x(t))=0,\quad t\ge t_0,     

Wavelet Transforms Associated to Square Integrable Group Representations on L 2 ( C,H ; dz )

Let SL (2, C ) be the special linear group of 2 ‐ 2 complex matrices with determinant 1 and SU (2) its maximal compact subgroup. Then SL (2, C )/ SU (2) can be realized as the quaternionic upper half-plane $ {\cal H}^c $ . Let SL (2, C ) = NASU (2) be the Iwasawa decomposition and M the centerlizer of A in SU (2). Then P = NA and P a = NAM are the automorphism groups of $ {\cal H}^c $ . In this article, we define the unitary representations of P and P a on L 2 ( C , H ; dz ). From the viewpoint of square integrable group representations we discuss the wavelet transforms, and obtain the orthogonal direct sum decompositions for the function spaces $ L^2({\cal H}^c, \fraca {(dz\, d\rho)}{\rho ^3}) $ and $ L^2({\bf R}^2\times {\bf R}^2, \fraca {dx\, dy\, dx^{\prime }dy^{\prime }}{{({x^{\prime }}^2 + {y^{\prime }}^2)^{\fraca {3}{2}})}} $ .     

Boundary Values of Cauchy Transforms in Weighted Spaces of Integrable Distributions

Let $ k \in {\shadN} $ , $ w(x) = (1+x^2)^{1/2} $ , $ V^{\prime} _k = w^{k+1} {\cal D}^{\prime} _{L^1} = \{{ \,f \in {\cal S}^{\prime}{:}\; w^{-k-1}f \in {\cal D}^{\prime} _{L^1}}\} $ . For $ f \in V^{\prime} _k $ , let $ C_{\eta ,k\,}f = C_0(\xi \,f) + z^k C_0(\eta \,f/t^k)$ where $ \xi \in {\cal D} $ , $ 0 \leq \xi (x) \leq 1 $ $ \xi (x) = 1 $ in a neighborhood of the origin, $ \eta = 1 - \xi $ , and $ C_0g(z) = \langle g, \fraca {1}{(2i \pi (\cdot - z))} \rangle $ for $ g \in V^{\,\prime} _0 $ , z = x + iy , y p 0 . Using a decomposition of C 0 in terms of Poisson operators, we prove that $ C_{\eta ,k,y} {:}\; f \,\mapsto\, C_{\eta ,k\,}f(\cdot + iy) $ , y p 0 , is a continuous mapping from $ V^{\,\prime} _k $ into $ w^{k+2} {\cal D}_{L^1}$ , where $ {\cal D}_{L^1} = \{ \varphi \in C^\infty {:}\; D^\alpha \varphi \in L^1\ \forall \alpha \in {\shadN} \} $ . Also, it is shown that for $ f \in V^{\,\prime} _k $ , $ C_{\eta ,k\,}f $ admits the following boundary values in the topology of $ V^{\,\prime} _{k+1} : C^+_{\eta ,k\,}f = \lim _{y \to 0+} C_{\eta ,k\,}f(\cdot + iy) = (1/2) (\,f + i S_{\eta ,k\,}f\,); C^-_{\eta ,k\,}f = \lim _{y \to 0-} C_{\eta ,k\,} f(\cdot + iy)= (1/2) (-f + i S_{\eta ,k\,}f ) $ , where $ S_{\eta ,k} $ is the Hilbert transform of index k introduced in a previous article by the first named author. Additional results are established for distributions in subspaces $ G^{\,\prime} _{\eta ,k} = \{ \,f \in V^{\,\prime} _k {:}S_{\eta ,k\,}f \in V^{\,\prime} _k \} $ , $ k \in {\shadN} $ . Algebraic properties are given too, for products of operators C + , C m , S , for suitable indices and topologies.     

Positive eigenvalues of second order boundary value problems and a theorem of M. G. Krein

Conditions are given which guarantee that the least real eigenvalue is positive for certain boundary value problems for the vector-matrix equation . This leads to conditions which guarantee the stable boundedness, according to Krein, for solutions of with certain real values of . As a consequence, a result first stated by Krein is proven.

     

A parametric family of quintic Thue equations

For an integral parameter we investigate the family of Thue equations

originating from Emma Lehmer's family of quintic fields, and show that for the only solutions are the trivial ones with or . Our arguments contain some new ideas in comparison with the standard methods for Thue families, which gives this family a special interest.

     

Evans functions and bifurcations of standing wave fronts of a nonlinear system of reaction diffusion equations

Consider the following nonlinear system of reaction diffusion equations arising from mathematical neuroscience $\frac{\partial u}{\partial t}=\frac{\partial^2u}{\partial x^2}+\alpha[\beta H(u-\theta)-u]-w,~ \frac{\partial w}{\partial t}=\varepsilon(u-\gamma w).$ Also consider the nonlinear scalar reaction diffusion equation $\frac{\partial u}{\partial t}=\frac{\partial^2u}{\partial x^2}+\alpha[\beta H(u-\theta)-u].$ In these model equations, $\alpha>0$, $\beta>0$, $\gamma>0$, $\varepsilon>0$ and $\theta>0$ are positive constants, such that $0<2\theta<\beta$. In the model equations, $u=u(x,t)$ represents the membrane potential of a neuron at position $x$ and time $t$, $w=w(x,t)$ represents the leaking current, a slow process that controls the excitation.\\indent The main purpose of this paper is to couple together linearized stability criterion (the equivalence of the nonlinear stability, the linear stability and the spectral stability of the standing wave fronts) and Evans functions (complex analytic functions) to establish the existence, stability, instability and bifurcations of standing wave fronts of the nonlinear system of reaction diffusion equations and to establish the existence and stability of the standing wave fronts of the nonlinear scalar reaction diffusion equation.     

A nonlinear third order multi-point boundary value problem in the resonance case

In this paper, we discuss the following third order ordinary differential equation $$x^{\prime\prime\prime}(t)=f(t,x(t),x^{\prime}(t),x^{\prime\prime}(t))+e(t),\quad t\in (0,1)$$ with the multi-point boundary conditions $$x^{\prime}(0)=\alpha x^{\prime}(\xi),\qquad x^{\prime\prime}(0)=0,\qquad x(1)=\sum^{m-2}_{j=1}\beta_{j}x(\eta_{j})$$ where β _j (1≤j≤m?2), α∈R, 0<η ₁<η ₂<???<η _m?2<1, 0<ξ<1. When the β _j ’s have no same sign, some existence results are given for the nonlinear problems at resonance case. An example is provided in this paper.     

Stability analysis for delay differential equations with multidelays and numerical examples

In this paper we are concerned with the asymptotic stability of the delay differential equation

where are constant complex matrices, and 0$"> stand for constant delays . We obtain two criteria for stability through the evaluation of a harmonic function on the boundary of a certain region. We also get similar results for the neutral delay differential equation

where and are constant complex matrices and 0$"> stands for constant delays , . Numerical examples on various circumstances are shown to check our results which are more general than those already reported.