一类非线性奇异微分方程正解的存在性定理

设(i) f(t,u): (0,1)×(0,+∞)→[0,+∞)连续，关于u 单调增加; (ii) 存在函数g:[1,+∞)→(0,+∞),g(b)0,G(t,s)是相应问题的Green函数。     

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.     

一类奇异非线性三点边值问题的正解

应用锥上的不动点定理，建立了奇异非线性三点边值问题(u″(t)+a(t)f(u)=0,0＜t＜1,αu(0)-βu′(0)=0,u(1)-ku(η)=0)正解的一个存在性定理．这里η∈(0,1)是一个常数，a∈C( (0,1),［0,+∞)),f∈C(［0,+∞),［0,+∞))     

Robin型二阶m 点边值问题正解的存在性

设 a∈C［0,1］, b∈C(［0,1］,(-∞, 0)). 设\-1(t)为线性边值问题  u″+a(t)u′+b(t)u=0， u′(0)=0,\ u(1)=1  的唯一正解. 该文研究非线性二阶常微分方程m 点边值问题  u″+a(t)u′+b(t)u+h(t) f(u)=0，\= u′(0)=0, u(1)-∑[DD(]m-2[]i=1[DD)]α\-i u(ξ\-i)=d  正解的存在性. 其中 d 为参数, ξ\-i∈(0,1), α\-i∈(0,∞) 为满足 ∑[DD(]m-2[]i=1[DD)]α\-i\-1(ξ\-i)<1的常数, i∈{1,\:，m-2}. 在适当的条件下证得: 存在正常数 d\+*, 使 当0d\+*时无正解.     

p -级数域重排特征系统的(C, α)}和

令G_p 为p级数域.在文献[9]中, G.Gát 和 K.Nagy 已经证明p级数域的重排特征系统的(C,1)极大算子是强 (q, q)型(1 α_n f 几乎处处收敛于f.     

具有变号非线性项的奇异三点边值问题正解的存在性和不存在性

该文讨论奇异三点边值问题 y'(t)+a(t)f(t, y(t), y'(t))= 0, 0相似文献   

非线性二阶微分系统正解的存在性

考虑二阶微分系统边值问题[JB({]x″(t)+λ f(t,x(t),y(t))=0,\=y″(t)+μ g(t,x(t),y(t))=0,\ 00, f, g:［0,1］×［0,∞)×［0,∞)→R连续. 突破了以往文献要求非线性项 f, g非负的限制，运用锥上的一个不动点定理,在半正的情形下建立了问题正解的存在性     

微分方程f″+e^{az}f′+Q(z)f=F(z) 的复振荡

该文研究了线性微分方程f″+e^{az}f′+Q(z)f=F(z)的复振荡问题，其中Q(z)、F(z )( 0)是整函数，且σ(Q)=1,σ(F)<+∞,Q(z)=h(z)e^{bz},h(z)是多项式，b≠-1是复常数，那么上述线性微分方程的所有解f(z)满足~λ(f)=λ(f)=σ(f)=∞,~λ_2(f)=λ_2(f)=σ_2(f)=1.至多除去两个例外复数a及一个可能的有穷级例外解f_0(z)。     

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。     

正规族与复合亚纯函数

本文研究了亚纯函数族涉及复合有理函数与分担亚纯函数的正规性. 证明了一个正规定则:设 α(z) 和 F 分别是区域 D 上的亚纯函数与亚纯函数族, R(z) 是一个次数不低于 3 的有理函数.如果对族 F 中函数 f(z) 和 g(z), R○f(z) 和 R○g(z) 分担 α(z) IM,并且下述 条件之一成立:
(1) 对任何 z₀ ∈ D, R(z)-α(z₀) 有至少三个不同的零点或极点;
(2) 存在 z₀ ∈ D 使得 R(z)-α(z₀):=(z-β₀)^pH(z) 至多有两个零点(或极点) β₀,同时 k ≠ l|p|,其中 l 和 k 分别是 f(z)-β₀ 和 α(z)-α(z₀) 在 z₀ 处的零点重数, H(z) 是满足 H(β₀) ≠ 0, ∞ 的有理函数, α(z) 非常数并满足 α(z₀) ∈ C ∪{∞}.
那么 F 在 D 内正规.特别地,这个结果是著名的 Montel 正规定则的一种推广.     

亚纯函数在角域内的波莱耳方向

Suppose that f(z) is a meromorphic function of order λ(0<λ≤∞) and of lower order μ(0≤μ<∞) in the plane. Let ρ(μ≤ρ≤λ) be a finite positive number. B: arg z=θ₀(0≤θ₀ <2π) is called a Borel direction of order ρ of f(z), if for any complex number a, the equality holds, except at most for some a belonging to a set of linear measure zero. For the exceptional values a, we have ρ(θ₀, a)>ρ, except two possible values. With the above hypotheses on f(z), λ, μ and ρ, We have the following lemmas. Lemma 1. There exists a sequence of positive numbers (r_n) such that(?)=∞ and that Lemma 2. If f(z) has a deficient value a₀ with deficiency δ(a₀, f), then we have where (r_n) is the sequence defined in the Lemma 1 and when a_0=∞, we have to replace(?)by (?) in the left hand side of (*). Lemma 3. Suppose that B_1 : arg z =θ₁ and B₂ : arg z=θ₂ (0≤θ₁<θ₂<2π+θ₁) are two half straight lines from the origin and there are no Borel directions of order≥ρ(ρ>1/2) of f(z) in θ₁₀, the inequality holds as n is sufficiently large, where K₁ is a positive number not depending on n andεand when a₀=∞, it is necessary to replace we have θ₂-θ₁≤π/ρ. Theorem 1. Suppose that f(z) is a meromorphic function of order λ (1/2<λ≤+∞) and of lower order μ(0≤μ<+∞) in the plane. Let p be a number such that μ≤ρ≤λ and that 1/2<ρ<+∞If f~((k))(z) has p(1≤P<+∞) deficient values a_i (i=1,2,…,p) with deficiencies δ(a_i,f^(k)), then f(z) has a Borel direction of order ≥ρ in any angular domain, the magnitude of which is larger than It is convenient to consider Julia directions as Borel directions of order zero.Under this assumption, We have the following. Theorem 2. Suppose that f(z) is a meromorphic function of order λ and of finite lower order μ in the plane and that ρ(μ≤ρ≤λ) is a finite number. If p denotes the number of deficient values of f(z) and q denotes the number of Borel directions of order ≥p of f(z), then we have p≤q.     

极大高阶交换子的端点估计

该文主要确立了当b∈BMO 时, 极大高阶奇异积分算子交换子T_{b, m}* 满足如下不等式 |{y∈Rⁿ:T_{b, m}*f(y)>λ}|≤C||b||^m_BMO∫_Rⁿ|f(y)|/λ (1+log⁺|f(y)|/λ)^mdy 且T_{b, m}* 在L^p(Rⁿ)(1 < p <∞上有界.     

Oscillation criteria for second order nonlinear differential equations

Summary Various sufficient conditions are obtained which guarantee that all continuable solutions of (1.1) y″+q(t)y′+p(t)f(y)=0 are oscillatory. No explicit sign assumptions are made on p(t) although certain integral conditions are assumed to hold with regard to f(y), p(t) and q(t). Examples are given of the form p(t) = λ/t^μ + (βsint)/t^α, λ, β, μ, α>0. This research was supported by NRC Grant A-7673 and CMC Summer Research Grant. Entrata in Redazione il 4 settembre 1971.     

Asymptotic Behavior of Bifurcation Branch of Positive Solutions for Semilinear Sturm–Liouville Problems

We consider the nonlinear eigenvalue problem

Oscillation and nonoscillation of Hill's equation with periodic damping

We prove new results on the oscillation and nonoscillation of the Hill's equation with periodic damping:

y″+p(t)y′+q(t)y=0,t?0,     

Rectifiable oscillations in second-order half-linear differential equations

Second-order half-linear differential equation (H): on the finite interval I = (0,1] will be studied, where , p > 1 and the coefficient f(x) > 0 on I, , and . In case when p = 2, the equation (H) reduces to the harmonic oscillator equation (P): y′′ + f(x)y = 0. In this paper, we study the oscillations of solutions of (H) with special attention to some geometric and fractal properties of the graph . We establish integral criteria necessary and sufficient for oscillatory solutions with graphs having finite and infinite arclength. In case when , λ > 0, α > p, we also determine the fractal dimension of the graph G(y) of the solution y(x). Finally, we study the L ^p nonintegrability of the derivative of all solutions of the equation (H).      

Some inequalities concerning the characteristic frequencies of elastic continuum

Summary We consider the boundary value problem αz″(x)+m(x)y(x)=0, αy″(x)+p(x)z(x)=0, xε[0, 1], y(0)=y(1)=z(0)=0, where the functions m(x) and p(x) are assumed integrable and positive everywhere in [0, 1]. As the main result we obtain the inequalities for n=1, 2, ... where δ_n(m, p) stands for the product of the first n eigenvalues α_i(m, p) of the above system and where δ_n(m) abbreviates δ_n(m, m). Entrata in Redazione il 6 febbraio 1976.     

Solvability of boundary-value problems for nonlinear fractional differential equations

We consider the existence of nontrivial solutions of the boundary-value problems for nonlinear fractional differential equations

On FBDF5 Method for Delay Differential Equations of Fractional Order with Periodic and Anti-Periodic Conditions

In this paper, we study the fractional backward differential formula (FBDF) for the numerical solution of fractional delay differential equations (FDDEs) of the following form: \(\lambda _n {}_0^C D_t^{\alpha _n } y(t - \tau ) + \lambda _{n - 1} {}_0^C D_t^{\alpha _{n - 1} } y(t - \tau ) + \cdots + \lambda _1 {}_0^C D_t^{\alpha _1 } y(t - \tau ) + \lambda _{n + 1} y(t) = f(t), t \in [0,T]\), where \( \lambda _i \in \) \(\mathbb {R}\,(i = 1,\ldots ,n + 1)\,,\,\lambda _{n + 1} \ne 0,\,\, 0 \leqslant \alpha _1< \alpha _2< \cdots< \alpha _n < 1,\,\,T > 0,\) in Caputo sense. We find the Green’s functions for this equation corresponding to periodic/anti-periodic conditions in term of the Mittag-Leffler type. Our investigation is focused on stability properties of the numerical methods and we determine stability regions for the FDDEs. Finally, some numerical examples are given to show the effectiveness of the numerical method and the results are in excellent agreement with the theoretical analysis