n阶非线性常微分方程组边值问题的正解

运用不动点指数理论,研究以下$n$阶非线性常微分方程组边值问题正解的存在性和多重正解的存在性\[\left\{\ay\begin{array}{l}-u^{(n)}=f_1(x,u,v),\q-v^{(n)}=f_2(x,u,v),\\[2mm]u^{(i)}(0)=u^{(p)}(1)= v^{(i)}(0)=v^{(p)}(1)=0.\end{array}\right. \] 这里$n\geq 2$, $i = 0,1,2,\cdots,n-2$, $p \in \{1,2,\cdots,n-1\}$, $f_i\in C([0,1]\times\mathbb R^+\times\mathbb R^+,\mathbb R^+)~(i=1,2)$. 用凹函数刻画非线性项$f_1,f_2$的耦合行为, 因而非线性项 $f_i(i=1,2)$ 既可以都是超线性的, 也可以都是次线性的,还可以是混合非线性的(即其中一个是超线性的, 另一个是次线性的).     

带阻尼项的二阶奇异耦合系统的正周期解

We establish the existence of positive periodic solutions of the second-order singular coupled systems{x′′+ p_1(t)x′+ q_1(t)x = f_1(t, y(t)) + c_1(t),y′′+ p_2(t)y′+ q_2(t)y = f_2(t, x(t)) + c_2(t),where pi, qi, ci ∈ C(R/T Z; R), i = 1, 2; f_1, f_2 ∈ C(R/T Z ×(0, ∞), R) and may be singular near the zero. The proof relies on Schauder's fixed point theorem and anti-maximum principle.Our main results generalize and improve those available in the literature.     

拟线性抛物方程的弱解

该文给出了拟线性退化抛物方程pa_t{u}+pa_x{f(u)}=pa_xx{A(u(x,t))}∈R^2_+×(0,+∞) ,u(x,0)=u_0(x),x∈R 一种弱解的新定义, 利用Div Curl引理证明了解的存在性．     

一维$P$-Laplace二阶差分系统奇异边值问题正解的存在性

应用锥压缩锥拉伸不动点定理和Leray-Schauder 抉择定理研究了一类具有P-Laplace算子的奇异离散边值问题$$\left\{\begin{array}{l}\Delta[\phi (\Delta x(i-1))]+ q_{1}(i)f_{1}(i,x(i),y(i))=0, ~~~i\in \{1,2,...,T\}\\\Delta[\phi (\Delta y(i-1))]+ q_{2}(i)f_{2}(i,x(i),y(i))=0,\\x(0)=x(T+1)=y(0)=y(T+1)=0,\end{array}\right.$$的单一和多重正解的存在性,其中$\phi(s) = |s|^{p-2}s, ~p>1$,非线性项$f_{k}(i,x,y)(k=1,2)$在$(x,y)=(0,0)$具有奇性.     

在Banach空间中推广的 Roper-Suffridge算子(III)

假定 $X$ 是具有范数$\|\cdot\|$的复 Banach 空间, $n$ 是一个满足 $\dim X\geq n\geq2$的正整数. 本文考虑由下式定义的推广的Roper-Suffridge算子 $\Phi_{n,\beta_2, \gamma_2, \ldots , \beta_{n+1}, \gamma_{n+1}}(f)$: \begin{equation} \begin{array}{lll} \Phi _{n, \beta_2, \gamma_2, \ldots, \beta_{n+1},\gamma_{n+1}}(f)(x) &;\hspace{-3mm}=&;\hspace{-3mm}\dl\he{j=1}{n}\bigg(\frac{f(x^*_1(x))}{x^*_1(x)})\bigg)^{\beta_j}(f''(x^*_1(x))^{\gamma_j}x^*_j(x) x_j\\ &;&;+\bigg(\dl\frac{f(x^*_1(x))}{x^*_1(x)}\bigg)^{\beta_{n+1}}(f''(x^*_1(x)))^{\gamma_{n+1}}\bigg(x-\dl\he{j=1}{n}x^*_j(x) x_j\bigg),\nonumber \end{array} \end{equation} 其中 $x\in\Omega_{p_1, p_2, \ldots, p_{n+1}}$, $\beta_1=1, \gamma_1=0$ 和 \begin{equation} \begin{array}{lll} \Omega_{p_1, p_2, \ldots, p_{n+1}}=\bigg\{x\in X: \dl\he{j=1}{n}| x^*_j(x)|^{p_j}+\bigg\|x-\dl\he{j=1}{n}x^*_j(x)x_j\bigg\|^{p_{n+1}}<1\bigg\},\nonumber \end{array} \end{equation} 这里 $p_j>1 \,( j=1, 2,\ldots, n+1$), 线性无关族 $\{x_1, x_2, \ldots, x_n \}\subset X $ 与 $\{x^*_1, x^*_2, \ldots, x^*_n \}\subset X^* $ 满足 $x^*_j(x_j)=\|x_j\|=1 (j=1, 2, \ldots, n)$ 和 $x^*_j(x_k)=0 \, (j\neq k)$, 我们选取幂函数的单值分支满足 $(\frac{f(\xi)}{\xi})^{\beta_j}|_{\xi=0}= 1$ 和 $(f''(\xi))^{\gamma_j}|_{\xi=0}=1, \, j=2, \ldots , n+1$. 本文将证明: 对某些合适的常数$\beta_j, \gamma_j$, 算子$\Phi_{n,\beta_2, \gamma_2, \ldots, \beta_{n+1}, \gamma_{n+1}}(f)$ 在$\Omega_{p_1, p_2, \ldots , p_{n+1}}$上保持$\alpha$阶的殆$\beta$型螺形映照和 $\alpha$阶的$\beta$型螺形映照.     

A Remark on the Existence of Entire Large and Bounded Solutions to a(k_1, k_2)-Hessian System with Gradient Term

In this paper, we study the existence of positive entire large and bounded radial positive solutions for the following nonlinear system{S_k_1(λ(D_(u1)~2)) + a_1(|x|) |▽_(u_1) |~(k_1)= p_1(|x|) f_1(u_2) for x ∈ R~N,S_k_2(λ(D_(u_2)~2)) + a_2(|x|) |▽_(u2) |~(k_2)= p_2(|x|) f_2(u_1) for x ∈ R~N.Here S_k_i(λ(D_(u_i)~2) is the k_i-Hessian operator, a_1, p_1, f_1, a_2, p_2 and f_2 are continuous functions.     

一类分数阶微分方程多点边值问题解的存在性

本文讨论下面一类分数阶微分方程多点边值问题 $$\align &D^{\alpha}_{0+}u(t) = f(t, u(t),~D^{\alpha-1}_{0+}u(t), D^{\alpha-2}_{0+}u(t), D^{\alpha-3}_{0+}u(t)),~~t\in(0,1), \\&I^{4-\alpha}_{0+}u(0) = 0, ~D^{\alpha-1}_{0+}u(0)=\displaystyle{\sum_{i=1}^{m}}\alpha_{i}D^{\alpha-1}_{0+}u(\xi_{i}),\\&D^{\alpha-2}_{0+}u(1)=\sum\limits_ {j=1}^{n}\beta_{j} D^{\alpha-2}_{0+}u(\eta_{j}),~D^{\alpha-3}_{0+}u(1)-D^{\alpha-3}_{0+}u(0)=D^{\alpha-2}_{0+}u(\frac{1}{2}),\endalign$$其中$3<\alpha \leq 4$是一个实数.通过应用Mawhin重合度理论和构建适当的算子,得到了该边值问题解的存在性结果.     

一类两点边值问题的多重非负解

该文考虑两点边值问题[1/q(t)][q(t)y′(t)]′+p(t)f(y(t))= 0,λ_1 y(α)-λ_2y′(α)=0 and y(β)=B非负解的存在性, 其中p(t)可能在t=α或t=β附近具有奇异性, f(0)≥0, lim_(y→+∞)f(y)/y=+∞, 并且存在y>0, 使得f(y)<0.      

半无穷区间广义Sturm-Liouville边值问题的多个正解存在性

该文利用Leggett-Williams 不动点定理, 研究半无穷区间边值问题 (p(t)x'(t))'+Φ(t) f (t, x(t), x'(t))=0, t∈[0,+∞), α₁x(0)-β₁lim_t→0⁺ p(t) x'(t)=a₁, α₂lim_t→+∞ x'(t)+β₂lim_t→+∞ p(t) x'(t)=a₂. 多个正解的存在性.     

关于奇异非线性多调和方程的正整体解

该文主要研究形如Δ((Δ\+nu)\+\{p-1*\}) = f(|x|, u, |u|)u\+\{-β\},\ x∈R\+2的奇异非线性多调和方程在R\+2上的正整体解，此处p>1，β≥0是常数，n是自然数，f: [AKR-]\-+×R\-+×[AKR-]\-+→R\-+是 一个连续函数,ξ\+\{α*\}:=|ξ|\+\{α-1\}ξ,ξ∈R，α>0 . 证明了这种解 u必无界且其渐进阶（当n→∞时u作为无穷大量的阶）不低于|x|\+\{2n\}log|x| ，给 出了该方程具有无穷多个其渐进阶刚好为 |x|\+\{2n\}log|x| 的正整体解的充分与充分必要条件. 这些结论可以推广到更一般的方程中去.      

Chemotaxis-fluild 系统的全局弱解

We investigate the existence of the global weak solution to the coupled Chemotaxisfluid system ■in a bounded smooth domain ??R~2. Here, r≥0 and μ 0 are given constants,?Φ∈L~∞(?) and g∈L~2((0, T); L_σ~2(?)) are prescribed functions. We obtain the local existence of the weak solution of the system by using the Schauder fixed point theorem. Furthermore, we study the regularity estimate of this system. Utilizing the regularity estimates, we obtain that the coupled Chemotaxis-fluid system with the initial-boundary value problem possesses a global weak solution.     

多维非退化扩散过程的象集与图集的一致Hausdorff维数

设X(t)=X(0)+∫^t_0α(X(s))dB(s)+∫^t_0β( X(s))ds为一d(d≥3)维非退化扩散过程。令X(E)={X(t): t∈E}, GRX(E)={(t,X(t)): t∈E}，该文证明了：对几乎所有ω：E B([0,∞)),有dimX(E,ω)=dimGRX(E,ω)=2dimE,这里dimF表示F的Hausdorff维数。     

A remark on the existence and multiplicity result for a nonlinear elliptic problem involving the p-Laplacian

In this work, motivated by Wu (J Math Anal Appl 318:253–270, 2006), and using recent ideas from Brown and Wu (J Math Anal Appl 337:1326–1336, 2008), we prove the existence of nontrivial nonnegative solutions to the following nonlinear elliptic problem:

On the first simultaneous sign change for Fourier coefficients associated with Hecke–Maass forms

The Ramanujan Journal - Let f and g be two distinct Hecke–Maass cusp forms of weight zero for $$SL(2,\mathbb {Z})$$ with Laplacian eigenvalues $$\frac{1}{4}+u^{2}$$ and $$\frac{1}{4}+v^{2}$$...     

强耗散KDV型方程的指数吸引子

In this paper,we consider the strong dissipative KDV type equation on an unbounded domain R1.By applying the theory of decomposing operator and the method of constructing some compact operator in weighted space,the existence of exponential attractor in phase space H2（R1） is obtained.     

The critical exponent for a weakly coupled system of the generalized Fujita type reaction-diffusion equations

We study nonnegative solutions of the initial value problem for a weakly coupled system

Multilinear Marcinkiewicz-Zygmund Inequalities

We extend to the multilinear setting classical inequalities of Marcinkiewicz and Zygmund on \(\ell ^r\)-valued extensions of linear operators. We show that for certain \(1 \le p, q_1, \dots , q_m, r \le \infty \), there is a constant \(C\ge 0\) such that for every bounded multilinear operator \(T:L^{q_1}(\mu _1) \times \cdots \times L^{q_m}(\mu _m) \rightarrow L^p(\nu )\) and functions \(\{f_{k_1}^1\}_{k_1=1}^{n_1} \subset L^{q_1}(\mu _1), \dots , \{f_{k_m}^m\}_{k_m=1}^{n_m} \subset L^{q_m}(\mu _m)\), the following inequality holds

$$\begin{aligned} \left\| \left( \sum _{k_1, \dots , k_m} |T(f_{k_1}^1, \dots , f_{k_m}^m)|^r\right) ^{1/r} \right\| _{L^p(\nu )} \le C \Vert T\Vert \prod _{i=1}^m \left\| \left( \sum _{k_i=1}^{n_i} |f_{k_i}^i|^r\right) ^{1/r} \right\| _{L^{q_i}(\mu _i)}. \end{aligned}$$ (1)

In some cases we also calculate the best constant \(C\ge 0\) satisfying the previous inequality. We apply these results to obtain weighted vector-valued inequalities for multilinear Calderón-Zygmund operators.