共查询到20条相似文献,搜索用时 62 毫秒
1.
R.E White 《Journal of Mathematical Analysis and Applications》1979,68(1):157-170
We consider weak solutions to the nonlinear boundary value problem (r, (x, u(x)) u′(x))′ = (Fu)′(x) with r(0, u(0)) u′(0) = ku(0), r(L, u(L)) u′(L) = hu(L) and k, h are suitable elements of [0, ∞]. In addition to studying some new boundary conditions, we also relax the constraints on r(x, u) and (Fu)(x). r(x, u) > 0 may have a countable set of jump discontinuities in u and r(x, u)?1?Lq((0, L) × (0, p)). F is an operator from a suitable set of functions to a subset of Lp(0, L) which have nonnegative values. F includes, among others, examples of the form (Fu)(x) = (1 ? H(x ? x0)) u(x0), (Fu)(x) = ∫xLf(y, u(y)) dy where f(y, u) may have a countable set of jump discontinuities in u or F may be chosen so that (Fu)′(x) = ? g(x, u(x)) u′(x) ? q(x) u(x) ? f(x, u(x)) where q is a distributional derivative of an L2(0, L) function. 相似文献
2.
We prove generalized Hyers-Ulam–Rassias stability of the cubic functional equation f(kx+y)+f(kx?y)=k[f(x+y)+f(x?y)]+2(k 3?k)f(x) for all \(k\in \Bbb{N}\) and the quartic functional equation f(kx+y)+f(kx?y)=k 2[f(x+y)+f(x?y)]+2k 2(k 2?1)f(x)?2(k 2?1)f(y) for all \(k\in \Bbb{N}\) in non-Archimedean normed spaces. 相似文献
3.
4.
Let A be an n × n matrix; write A = H+iK, where i2=—1 and H and K are Hermitian. Let f(x,y,z) = det(zI?xH?yK). We first show that a pair of matrices over an algebraically closed field, which satisfy quadratic polynomials, can be put into block, upper triangular form, with diagonal blocks of size 1×1 or 2×2, via a simultaneous similarity. This is used to prove that if , where g has degree 2, then for some unitary matrix U, the matrix U1AU is the direct sum of copies of a 2×2 matrix A1, where A1 is determined, up to unitary similarity, by the polynomial g(x,y,z). We use the connection between f(x,y,z) and the numerical range of A to investigate the case where f(x,y,z) has the form (z?αax? βy)r[g(x,y,z)]s, where g(x,y,z) is irreducible of degree 2. 相似文献
5.
G. Kutyniok 《Archiv der Mathematik》2002,78(2):135-144
Let
r\mathbbR \rho_{\mathbb{R}} be the classical Schrödinger representation of the Heisenberg group and let L \Lambda be a finite subset of
\mathbbR ×\mathbbR \mathbb{R} \times \mathbb{R} . The question of when the set of functions
{t ? e2 pi y t f(t + x) = (r\mathbbR(x, y, 1) f)(t) : (x, y) ? L} \{t \mapsto e^{2 \pi i y t} f(t + x) = (\rho_{\mathbb{R}}(x, y, 1) f)(t) : (x, y) \in \Lambda\} is linearly independent for all
f ? L2(\mathbbR), f 1 0 f \in L^2(\mathbb{R}), f \neq 0 , arises from Gabor analysis. We investigate an analogous problem for locally compact abelian groups G. For a finite subset L \Lambda of G ×[^(G)] G \times \widehat{G} and rG \rho_G the Schrödinger representation of the Heisenberg group associated with G, we give a necessary and in many situations also sufficient condition for the set {rG (x, w, 1)f : (x, w) ? L} \{\rho_G (x, w, 1)f : (x, w) \in \Lambda\} to be linearly independent for all f ? L2(G), f 1 0 f \in L^2(G), f \neq 0 . 相似文献
6.
Let N′(k) denote the number of coprime integral solutions x, y of y2 = x3 + k. It is shown that lim supk→∞N′(k) ≥ 12. 相似文献
7.
V. Maier 《Analysis Mathematica》1978,4(4):289-295
Оператор Канторович а дляf∈L p(I), I=[0,1], определяе тся соотношением $$P_n (f,x) = (n + 1)\sum\limits_{k = 0}^n {\left( {\begin{array}{*{20}c} n \\ k \\ \end{array} } \right)} x^k (1 - x)^{n - 1} \int\limits_{I_k } {f(t)dt,} $$ гдеI k=[k/(n}+1),(k+1)/(n+ 1)],n∈N. Доказывается, что есл ир>1 иf∈W p 2 (I), т.е.f абсол ютно непрерывна наI иf″∈L p(I), то $$\left\| {P_n f - f} \right\|_p = O(n^{ - 1} ).$$ Далее, установлено, чт о еслиf∈L p(I),p>1 и ∥P n f-f∥р=О(n ?1), тоf∈S, гдеS={f∶f аб-солютно непрерывна наI, x(1?x)f′(x)=∝ 0 x h(t)dt, гдеh∈L p(I) и ∝ 0 1 h(t)dt=0}. Если жеf∈Lp(I),p>1, то из условия ∥P n(f)?f∥pL=o(n?1) вытекает, чтоf постоянна почти всюду. 相似文献
8.
Let L be a lattice of finite length, ξ = (x 1,…, x k )∈L k , and y ∈ L. The remoteness r(y, ξ) of y from ξ is d(y, x 1)+?+d(y, x k ), where d stands for the minimum path length distance in the covering graph of L. Assume, in addition, that L is a graded planar lattice. We prove that whenever r(y, ξ) ≤ r(z, ξ) for all z ∈ L, then y ≤ x 1∨?∨x k . In other words, L satisfies the so-called c 1 -median property. 相似文献
9.
We present various new inequalities involving the logarithmic mean L(x,y)=(x-y)/(logx-logy) L(x,y)=(x-y)/(\log{x}-\log{y}) , the identric mean I(x,y)=(1/e)(xx/yy)1/(x-y) I(x,y)=(1/e)(x^x/y^y)^{1/(x-y)} , and the classical arithmetic and geometric means, A(x,y)=(x+y)/2 A(x,y)=(x+y)/2 and G(x,y)=?{xy} G(x,y)=\sqrt{xy} . In particular, we prove the following conjecture, which was published in 1986 in this journal. If Mr(x,y) = (xr/2+yr/2)1/r(r 1 0) M_r(x,y)= (x^r/2+y^r/2)^{1/r}(r\neq{0}) denotes the power mean of order r, then $ M_c(x,y)<\frac{1}{2}(L(x,y)+I(x,y)) {(x,y>0,\, x\neq{y})} $ M_c(x,y)<\frac{1}{2}(L(x,y)+I(x,y)) {(x,y>0,\, x\neq{y})} with the best possible parameter c=(log2)/(1+log2) c=(\log{2})/(1+\log{2}) . 相似文献
10.
We investigate the existence of positive solutions to the singular fractional boundary value problem: $^c\hspace{-1.0pt}D^{\alpha }u +f(t,u,u^{\prime },^c\hspace{-2.0pt}D^{\mu }u)=0$, u′(0) = 0, u(1) = 0, where 1 < α < 2, 0 < μ < 1, f is a Lq‐Carathéodory function, $q > \frac{1}{\alpha -1}$, and f(t, x, y, z) may be singular at the value 0 of its space variables x, y, z. Here $^c \hspace{-1.0pt}D$ stands for the Caputo fractional derivative. The results are based on combining regularization and sequential techniques with a fixed point theorem on cones. 相似文献
11.
E. S. Zhukovskiy 《Siberian Mathematical Journal》2018,59(6):1063-1072
The recent articles of Arutyunov and Greshnov extend the Banach and Hadler Fixed-Point Theorems and the Arutyunov Coincidence-Point Theorem to the mappings of (q1, q2)-quasimetric spaces. This article addresses similar questions for f-quasimetric spaces.Given a function f: R +2 → R+ with f(r1, r2) → 0 as (r1, r2) → (0, 0), an f-quasimetric space is a nonempty set X with a possibly asymmetric distance function ρ: X2 → R+ satisfying the f-triangle inequality: ρ(x, z) ≤ f(ρ(x, y), ρ(y, z)) for x, y, z ∈ X. We extend the Banach Contraction Mapping Principle, as well as Krasnoselskii’s and Browder’s Theorems on generalized contractions, to mappings of f-quasimetric spaces. 相似文献
12.
The oscillatory and asymptotic behavior of solutions of a class of nth order nonlinear differential equations, with deviating arguments, of the form (E, δ) Lnx(t) + δq(t) f(x[g1(t)],…, x[gm(t)]) = 0, where δ = ± 1 and L0x(t) = x(t), Lkx(t) = ak(t)(Lk ? 1x(t))., , is examined. A classification of solutions of (E, δ) with respect to their behavior as t → ∞ and their oscillatory character is obtained. The comparisons of (E, 1) and (E, ?1) with first and second order equations of the form y.(t) + c1(t) f(y[g1(t)],…, y[gm(t)]) = 0 and (an ? 1(t)z.(t)). ? c2(t) f(z[g1(t)],…, z[gm(t)]) = 0, respectively, are presented. The obtained results unify, extend and improve some of the results by Graef, Grammatikopoulos and Spikes, Philos and Staikos. 相似文献
13.
14.
《Mathematical and Computer Modelling》2000,31(2-3):17-29
In this paper, we consider the partial difference equation with continuous variables of the form P1z(x + a, y + b) + p2z (x + a, y) + p3z (x, y + b) − p4z (x, y) + P (x, y) z (x − τ, y − σ) = 0, where P ϵ C(R+ × R+, R+ − {0}), a, b, τ, σ are real numbers and pi (i = 1, 2, 3, 4) are nonnegative constants. Some sufficient conditions for all solutions of this equation to be oscillatory are obtained. 相似文献
15.
Yong-Zhuo Chen 《Positivity》2012,16(1):97-106
We apply the Thompson’s metric to study the global stability of the equilibium of the following difference equation
yn = \fracf2m+12m+1 (yn-k1r, yn-k2r, ..., yn-k2m+1r)f2m2m+1 (yn-k1r, yn-k2r, ..., yn-k2m+1r), n = 0,1,2, ?, y_{n} = \frac{f_{2m+1}^{2m+1} (y_{n-k_{1}}^r, y_{n-k_{2}}^r, \dots, y_{n-k_{2m+1}}^r)}{f_{2m}^{2m+1} (y_{n-k_{1}}^r, y_{n-k_{2}}^r, \dots, y_{n-k_{2m+1}}^r)}, \;\;\;\; n = 0,1,2, \ldots, 相似文献
16.
Denote by span {f 1,f 2, …} the collection of all finite linear combinations of the functionsf 1,f 2, … over ?. The principal result of the paper is the following. Theorem (Full Müntz Theorem in Lp(A) for p ∈ (0, ∞) and for compact sets A ? [0, 1] with positive lower density at 0). Let A ? [0, 1] be a compact set with positive lower density at 0. Let p ∈ (0, ∞). Suppose (λ j ) j=1 ∞ is a sequence of distinct real numbers greater than ?(1/p). Then span {x λ1,x λ2,…} is dense in Lp(A) if and only if $\sum\limits_{j = 1}^\infty {\frac{{\lambda _j + \left( {1/p} \right)}}{{\left( {\lambda _j + \left( {1/p} \right)} \right)^2 + 1}} = \infty } $ . Moreover, if $\sum\limits_{j = 1}^\infty {\frac{{\lambda _j + \left( {1/p} \right)}}{{\left( {\lambda _j + \left( {1/p} \right)} \right)^2 + 1}} = \infty } $ , then every function from the Lp(A) closure of {x λ1,x λ2,…} can be represented as an analytic function on {z ∈ ? \ (?∞,0] : |z| < rA} restricted to A ∩ (0, rA) where $r_A : = \sup \left\{ {y \in \mathbb{R}:\backslash ( - \infty ,0]:\left| z \right|< r_A } \right\}$ (m(·) denotes the one-dimensional Lebesgue measure). This improves and extends earlier results of Müntz, Szász, Clarkson, Erdös, P. Borwein, Erdélyi, and Operstein. Related issues about the denseness of {x λ1,x λ2,…} are also considered. 相似文献
17.
Let f be an L 2-normalized weight zero Hecke-Maaß cusp form of square-free level N, character χ and Laplacian eigenvalue λ≥1/4. It is shown that ‖f‖∞? λ N ?1/37, from which the hybrid bound ‖f‖∞?λ 1/4(N λ)?δ (for some δ>0) is derived. The first bound holds also for f=y k/2 F where F is a holomorphic cusp form of weight k with the implied constant now depending on k. 相似文献
18.
For $n \in \mathbb{N}$ , the n-order of an analytic function f in the unit disc D is defined by $$\sigma _{{{M,n}}} (f) = {\mathop {\lim \sup }\limits_{r \to 1^{ - } } }\frac{{\log ^{ + }_{{n + 1}} M(r,f)}} {{ - \log (1 - r)}},$$ where log+ x = max{log x, 0}, log + 1 x = log + x, log + n+1 x = log + log + n x, and M(r, f) is the maximum modulus of f on the circle of radius r centered at the origin. It is shown, for example, that the solutions f of the complex linear differential equation $$f^{{(k)}} + a_{{k - 1}} (z)f^{{(k - 1)}} + \cdots + a_{1} (z)f^{\prime} + a_{0} (z)f = 0,\quad \quad \quad (\dag)$$ where the coefficients are analytic in D, satisfy σ M,n+1(f) ≤ α if and only if σ M,n (a j ) ≤ α for all j = 0, ..., k ? 1. Moreover, if q ∈{0, ..., k ? 1} is the largest index for which $\sigma _{M,n} ( a_{q}) = {\mathop {\max }\limits_{0 \leq j \leq k - 1} }{\left\{ {\sigma _{{M,n}} {\left( {a_{j} } \right)}} \right\}}$ , then there are at least k ? q linearly independent solutions f of ( $\dag$ ) such that σ M,n+1(f) = σ M,n (a q ). Some refinements of these results in terms of the n-type of an analytic function in D are also given. 相似文献
19.
B. I. Golubov 《Mathematical Notes》2006,79(1-2):196-214
For functions from the Lebesgue space L(?+), we introduce the modified strong dyadic integral J α and the fractional derivative D (α) of order α > 0. We establish criteria for their existence for a given function f ∈ L(?+). We find a countable set of eigenfunctions of the operators D (α) and J α, α > 0. We also prove the relations D (α)(J α(f)) = f and J α(D (α)(f)) = f under the condition that $\smallint _{\mathbb{R}_ + } f(x)dx = 0$ . We show the unboundedness of the linear operator $J_\alpha :L_{J_{_\alpha } } \to L(\mathbb{R}_ + )$ , where L J α is its natural domain of definition. A similar assertion is proved for the operator $D^{(\alpha )} :L_{D^{(\alpha )} } \to L(\mathbb{R}_ + )$ . Moreover, for a function f ∈ L(?+) and a given point x ∈ ?+, we introduce the modified dyadic derivative d (α)(f)(x) and the modified dyadic integral j α(f)(x). We prove the relations d (α)(J α(f))(x) = f(x) and j α(D (α)(f)) = f(x) at each dyadic Lebesgue point of the function f. 相似文献
20.
Letf(x) ∈L p[0,1], 1?p? ∞. We shall say that functionf(x)∈Δk (integerk?1) if for anyh ∈ [0, 1/k] andx ∈ [0,1?kh], we have Δ h k f(x)?0. Denote by ∏ n the space of algebraic polynomials of degree not exceedingn and define $$E_{n,k} (f)_p : = \mathop {\inf }\limits_{\mathop {P_n \in \prod _n }\limits_{P_n^{(\lambda )} \geqslant 0} } \parallel f(x) - P_n (x)\parallel _{L_p [0,1]} .$$ We prove that for any positive integerk, iff(x) ∈ Δ k ∩ L p[0, 1], 1?p?∞, then we have $$E_{n,k} (f)_p \leqslant C\omega _2 \left( {f,\frac{1}{n}} \right)_p ,$$ whereC is a constant only depending onk. 相似文献
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