共查询到20条相似文献,搜索用时 31 毫秒
1.
G. V. Radzievskii 《Ukrainian Mathematical Journal》1994,46(3):290-303
For the equationL
0
x(t)+L
1x(t)+...+L
n
x
(n)(t)=O, whereL
k,k=0,1,...,n, are operators acting in a Banach space, we establish criteria for an arbitrary solutionx(t) to be zero provided that the following conditions are satisfied:x
(1–1) (a)=0, 1=1, ..., p, andx
(1–1) (b)=0, 1=1,...,q, for - <a< b< (the case of a finite segment) orx
(1–1) (a)=0, 1=1,...,p, under the assumption that a solutionx(t) is summable on the semiaxista with its firstn derivatives.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 3, pp. 279–292, March, 1994.This research was supported by the Ukrainian State Committee on Science and Technology. 相似文献
2.
In the case of existence the smallest numberN=Rakis called a Rado number if it is guaranteed that anyk-coloring of the numbers 1, 2, …, Ncontains a monochromatic solution of a given system of linear equations. We will determine Rak(a, b) for the equationa(x+y)=bzifb=2 andb=a+1. Also, the case of monochromatic sequences {xn} generated bya(xn+xn+1)=bxn+2 is discussed. 相似文献
3.
In this paper we study the existence of periodic solutions of the fourth-order equations uiv − pu″ − a(x)u + b(x)u3 = 0 and uiv − pu″ + a(x)u − b(x)u3 = 0, where p is a positive constant, and a(x) and b(x) are continuous positive 2L-periodic functions. The boundary value problems (P1) and (P2) for these equations are considered respectively with the boundary conditions u(0) = u(L) = u″(0) = u″(L) = 0. Existence of nontrivial solutions for (P1) is proved using a minimization theorem and a multiplicity result using Clark's theorem. Existence of nontrivial solutions for (P2) is proved using the symmetric mountain-pass theorem. We study also the homoclinic solutions for the fourth-order equation uiv + pu″ + a(x)u − b(x)u2 − c(x)u3 = 0, where p is a constant, and a(x), b(x), and c(x) are periodic functions. The mountain-pass theorem of Brezis and Nirenberg and concentration-compactness arguments are used. 相似文献
4.
We present the PFix algorithm for the fixed point problem f(x)=x on a nonempty domain [a,b], where d1,
, and f is a Lipschitz continuous function with respect to the infinity norm, with constant q1. The computed approximation
satisfies the residual criterion
, where >0. In general, the algorithm requires no more than ∑i=1dsi function component evaluations, where s≡max(1,log2(||b−a||∞/))+1. This upper bound has order
as →0. For the domain [0,1]d with <0.5 we prove a stronger result, i.e., an upper bound on the number of function component evaluations is
, where r≡log2(1/). This bound approaches
as r→∞ (→0) and
as d→∞. We show that when q<1 the algorithm can also compute an approximation
satisfying the absolute criterion
, where x* is the unique fixed point of f. The complexity in this case resembles the complexity of the residual criterion problem, but with tolerance (1−q) instead of . We show that when q>1 the absolute criterion problem has infinite worst-case complexity when information consists of function evaluations. Finally, we report several numerical tests in which the actual number of evaluations is usually much smaller than the upper complexity bound. 相似文献
5.
E. Preissmann 《Aequationes Mathematicae》1987,32(1):195-212
We solve independently the equations 1/θ(x)θ(y)=ψ(x)−ψ(y)+φ(x−y)/θ(x−y) and 1/θ(x)θ(y)=σ(x)−σ(y)/θ(x−y)+τ(x)τ(y), τ(0)=0. In both cases we find θ2=aθ4+bθ2+c. We deduce estimates for the spectral radius of a matrix of type(1/θ(x
r
−x
s
)) (the accent meaning that the coefficients of the main diagonal are zero) and we study the case where thex
r
are equidistant.
Dédié to à Monsieur le Professeur Otto Haupt à l'occasion de son cententiare avec les meilleurs voeux 相似文献
6.
Denote by xn,k(α,β) and xn,k(λ)=xn,k(λ−1/2,λ−1/2) the zeros, in decreasing order, of the Jacobi polynomial P(α,β)n(x) and of the ultraspherical (Gegenbauer) polynomial Cλn(x), respectively. The monotonicity of xn,k(α,β) as functions of α and β, α,β>−1, is investigated. Necessary conditions such that the zeros of P(a,b)n(x) are smaller (greater) than the zeros of P(α,β)n(x) are provided. A. Markov proved that xn,k(a,b)<xn,k(α,β) (xn,k(a,b)>xn,k(α,β)) for every n
and each k, 1kn if a>α and b<β (a<α and b>β). We prove the converse statement of Markov's theorem. The question of how large the function fn(λ) could be such that the products fn(λ)xn,k(λ), k=1,…,[n/2] are increasing functions of λ, for λ>−1/2, is also discussed. Elbert and Siafarikas proved that fn(λ)=(λ+(2n2+1)/(4n+2))1/2 obeys this property. We establish the sharpness of their result. 相似文献
7.
Bernard Epstein 《Israel Journal of Mathematics》1966,4(3):145-152
An analysis is presented of the equationf(x+a)−f(x)=e
−x
{f(x)−f(x−b)}. Herea andb denote arbitrary positive constants, and a solution is sought which satisfies the following conditions:f(−∞)=0,f(+∞)=1, 0≦f(x)≦1. Existence and uniqueness of solution are established, and then an analytical form of the solution is obtained by use
of bilateral Laplace transform.
Research supported by the National Science Foundation, Grant GP-2558. 相似文献
8.
Persistence, contractivity and global stability in logistic equations with piecewise constant delays
Yoshiaki Muroya 《Journal of Mathematical Analysis and Applications》2002,270(2):1532-635
We establish sufficient conditions for the persistence and the contractivity of solutions and the global asymptotic stability for the positive equilibrium N*=1/(a+∑i=0mbi) of the following differential equation with piecewise constant arguments: where r(t) is a nonnegative continuous function on [0,+∞), r(t)0, ∑i=0mbi>0, bi0, i=0,1,2,…,m, and a+∑i=0mbi>0. These new conditions depend on a,b0 and ∑i=1mbi, and hence these are other type conditions than those given by So and Yu (Hokkaido Math. J. 24 (1995) 269–286) and others. In particular, in the case m=0 and r(t)≡r>0, we offer necessary and sufficient conditions for the persistence and contractivity of solutions. We also investigate the following differential equation with nonlinear delay terms: where r(t) is a nonnegative continuous function on [0,+∞), r(t)0, 1−ax−g(x,x,…,x)=0 has a unique solution x*>0 and g(x0,x1,…,xm)C1[(0,+∞)×(0,+∞)××(0,+∞)]. 相似文献
9.
Let ga(t) and gb(t) be two positive, strictly convex and continuously differentiable functions on an interval (a, b) (−∞ a < b ∞), and let {Ln} be a sequence of linear positive operators, each with domain containing 1, t, ga(t), and gb(t). If Ln(ƒ; x) converges to ƒ(x) uniformly on a compact subset of (a, b) for the test functions ƒ(t) = 1, t, ga(t), gb(t), then so does every ƒ ε C(a, b) satisfying ƒ(t) = O(ga(t)) (t → a+) and ƒ(t) = O(gb(t)) (t → b−). We estimate the convergence rate of Lnƒ in terms of the rates for the test functions and the moduli of continuity of ƒ and ƒ′. 相似文献
10.
In this paper, a Galerkin type algorithm is given for the numerical solution of L(x)=(r(t)x'(t))'-p(t)x(t)=g(t); x(a)=xa, x'(a)=x'a, where r (t)>f0, and Spline hat functions form the approximating basis. Using the related quadratic form, a two-step difference equation is derived for the numerical solutions. A discrete Gronwall type lemma is then used to show that the error at the node points satisfies ek=0(h2). If e(t) is the error function on a?t?b; it is also shown (in a variety of norms) that e(t)?Ch2 and e'(t)?C1h. Test case runs are also included. A (one step) Richardson or Rhomberg type procedure is used to show that eRk=0(h4). Thus our results are comparable to Runge-Kutta with half the function evaluations. 相似文献
11.
Let Γ denote a distance-regular graph with diameter d≥3. By a parallelogram of length 3, we mean a 4-tuple xyzw consisting of vertices of Γ such that ∂(x,y)=∂(z,w)=1, ∂(x,z)=3, and ∂(x,w)=∂(y,w)=∂(y,z)=2, where ∂ denotes the path-length distance function. Assume that Γ has intersection numbers a
1=0 and a
2≠0. We prove that the following (i) and (ii) are equivalent. (i) Γ is Q-polynomial and contains no parallelograms of length 3; (ii) Γ has classical parameters (d,b,α,β) with b<−1. Furthermore, suppose that (i) and (ii) hold. We show that each of b(b+1)2(b+2)/c
2, (b−2)(b−1)b(b+1)/(2+2b−c
2) is an integer and that c
2≤b(b+1). This upper bound for c
2 is optimal, since the Hermitian forms graph Her2(d) is a triangle-free distance-regular graph that satisfies c
2=b(b+1).
Work partially supported by the National Science Council of Taiwan, R.O.C. 相似文献
12.
Let
be the classical middle-third Cantor set and let μ
be the Cantor measure. Set s = log 2/log 3. We will determine by an explicit formula for every point x
the upper and lower s-densities Θ*s(μ
, x), Θ*s(μ
, x) of the Cantor measure at the point x, in terms of the 3-adic expansion of x. We show that there exists a countable set F
such that 9(Θ*s(μ
, x))− 1/s + (Θ*s(μ
, x))− 1/s = 16 holds for x
\F. Furthermore, for μC almost all x, Θ*s(μ
, X) − 2 · 4− s and Θ*s(μ
, x) = 4− s. As an application, we will show that the s-dimensional packing measure of the middle-third Cantor set
is 4s. 相似文献
13.
For all integers m3 and all natural numbers a1,a2,…,am−1, let n=R(a1,a2,…,am−1) represent the least integer such that for every 2-coloring of the set {1,2,…,n} there exists a monochromatic solution to
a1x1+a2x2++am−1xm−1=xm.