共查询到20条相似文献,搜索用时 31 毫秒
1.
We prove that λ=0 is a global bifurcation point of the second-order periodic boundary-value problem (p(t)x′(t))′−λx(t)−λ2x′(t)−f(t,x(t),x′(t),x″(t));x(0)=x(1),x′(0)=x′(1). We study this equation under hypotheses for which it may be solved explicitly for x″(t). However, it is shown that the explicitly solved equation does not satisfy the usual conditions that are sufficient to conclude global bifurcation. Thus, we need to study the implicit equation with regard to global bifurcation. 相似文献
2.
Consider an operator equation B(u) − f = 0 in a real Hilbert space. Let us call this equation ill-posed if the operator B′(u) is not boundedly invertible, and well-posed otherwise. The dynamical systems method (DSM) for solving this equation consists of a construction of a Cauchy problem, which has the following properties: (1) it has a global solution for an arbitrary initial data, (2) this solution tends to a limit as time tends to infinity, (3) the limit is the minimal-norm solution to the equation B(u) = f. A global convergence theorem is proved for DSM for equation B(u) − f = 0 with monotone operators B. 相似文献
3.
Alemdar Hasanov 《Applied mathematics and computation》2003,140(2-3):501-515
An inverse polynomial method of determining the unknown leading coefficient k=k(x) of the linear Sturm–Liouville operator Au=−(k(x)u′(x))′+q(x)u(x), x(0,1), is presented. As an additional condition only two measured data at the boundary (x=0,x=1) are used. In absence of a singular point (u′(x)≠0,u″(x)≠0,x[0,1]) the inverse problem is classified as a well-conditioned . If there exists at least one singular point, then the inverse problem is classified as moderately ill-conditioned (u′(x0)=0,x0(0,1);u′(x)≠0,x≠x0;u″(x)≠0,x[0,1]) and severely ill-conditioned (u′(x0)=u″(x0)=0,x0(0,1);u′(x)≠0,u″(x)≠0,x≠x0). For each of the cases direct problem solution is approximated by corresponding polynomials and the inverse problem is reformulated as a Cauchy problem for to the first order differential equation with respect the unknown function k=k(x). An approximate analytical solution of the each Cauchy problems are derived in explicit form. Numerical simulations all the above cases are given for noise free and noisy data. An accuracy of the presented approach is demonstrated on numerical test solutions. 相似文献
4.
The existence, uniqueness and multiplicity of positive solutions of the following boundary value problem is considered: where λ>0 is a constant, f :[0,1]×[0,+∞)→[0,+∞) is continuous. 相似文献
u(4)(t)−λf(t,u(t))=0, for 0<t<1,u(0)=u(1)=u″(0)=u″(1)=0,
5.
Zaihong Wang 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2005,56(4):592-608
In this paper, we study the existence of periodic solutions of Rayleigh equation
where f, g are continuous functions and p is a continuous and 2π-periodic function. We prove that the given equation has at least one 2π-periodic solution provided that f(x) is sublinear and the time map of equation x′′ + g(x) = 0 satisfies some nonresonant conditions. We also prove that this equation has at least one 2π-periodic solution provided that g(x) satisfies
and f(x) satisfies sgn(x)(f(x) − p(t)) ≥ c, for t ∈R, |x| ≥ d with c, d being positive constants.Received: July 1, 2002; revised: February 19, 2003Research supported by the National Natural Science Foundation of China, No.10001025 and No.10471099, Natural Science Foundation of Beijing, No. 1022003 and by a postdoctoral Grant of University of Torino, Italy. 相似文献
6.
For a bounded linear injectionCon a Banach spaceXand a closed linear operatorA : D(A) X → Xwhich commutes withCwe prove that (1) the abstract Cauchy problem,u″(t) = Au(t),t R,u(0) = Cx,u′(0) = Cy, has a unique strong solution for everyx,y D(A) if and only if (2)A1 = AD(A2) generates aC1-cosine function onX1(D(A) with the graph norm), if (and only if, in caseAhas nonempty resolvent set) (3)Agenerates aC-cosine function onX. HereC1 = CX1. Under the assumption thatAis densely defined andC−1AC = A, statement (3) is also equivalent to each of the following statements: (4) the problemv″(t) = Av(t) + C(x + ty) + ∫t0 Cg(r) dr,t R,v(0) = v′(0) = 0, has a unique strong solution for everyg L1locandx, y X; (5) the problemw″(t) = Aw(t) + Cg(t),t R,w(0) = Cx,w′(0) = Cy, has a unique weak solution for everyg L1locandx, y X. Finally, as an application, it is shown that for any bounded operatorBwhich commutes withCand has range contained in the range ofC,A + Bis also a generator. 相似文献
7.
If F:H→H is a map in a Hilbert space H, , and there exists y such that F(y)=0, F′(y)≠0, then equation F(u)=0 can be solved by a DSM (dynamical systems method). This method yields also a convergent iterative method for finding y, and this method converges at the rate of a geometric series. It is not assumed that y is the only solution to F(u)=0. A stable approximation to a solution of the equation F(u)=f is constructed by a DSM when f is unknown but fδ is known, where fδ−f≤δ. 相似文献
8.
Entire functions that share a polynomial with their derivatives 总被引:1,自引:1,他引:0
Jian-Ping Wang 《Journal of Mathematical Analysis and Applications》2006,320(2):703-717
Let f be a nonconstant entire function, k and q be positive integers satisfying k>q, and let Q be a polynomial of degree q. This paper studies the uniqueness problem on entire functions that share a polynomial with their derivatives and proves that if the polynomial Q is shared by f and f′ CM, and if f(k)(z)−Q(z)=0 whenever f(z)−Q(z)=0, then f≡f′. We give two examples to show that the hypothesis k>q is necessary. 相似文献
9.
The psi function ψ(x) is defined by ψ(x)=Γ′(x)/Γ(x), where Γ(x) is the gamma function. We give necessary and sufficient conditions for the function ψ″(x)+[ψ′(x+α)]2 or its negative to be completely monotonic on (−α,∞), where . We also prove that the function [ψ′(x)]2+λψ″(x) is completely monotonic on (0,∞) if and only if λ1. As an application of the latter conclusion, the monotonicity and convexity of the function epψ(x+1)−qx with respect to x(−1,∞) are thoroughly discussed for p≠0 and . 相似文献
10.
Jinxi Ma 《Journal of Mathematical Analysis and Applications》1998,220(2):769
For 0 < p < 1, letSpdenote the class of functionsf(z) meromorphic univalent in the unit disk
with the normalizationf(0) = 0,f′(0) = 1, andf(p) = ∞. LetSp(a) be the subclass ofSpwith the fixed residuea. In this note we determine the extreme points of the classSp(a). As an application, we solve the problem of minimizing the outer area overSp(a), which was posed by S. Zemyan (J. Analyse Math.39, 1981, 11–23). 相似文献
11.
Mingliang Fang Lawrence Zalcman 《Journal of Mathematical Analysis and Applications》2003,280(2):273-283
There exists a set S with 3 elements such that if f is a non-constant entire function satisfying E(S,f)=E(S,f′), then f≡f′. The number 3 is best possible. The proof uses the theory of normal families in an essential way. 相似文献
12.
B. Z. Kacewicz 《Journal of Complexity》1988,4(4)
We deal with algorithms for solving systems z′(x) = f(x, z(x)), x ε [0, c], z(0) = η where f has r continuous bounded derivatives in [0, c) ×
s. We consider algorithms whose sole dependence on f is through the values of n linear continuous functionals at f. We show that if these functionals are defined by partial derivatives off then, roughly speaking, the error of an algorithm (for a fixed f) cannot converge to zero faster than n−r as n → +∞. This minimal error is achieved by the Taylor algorithm. If arbitrary linear continuous functionals are allowed, then the error cannot converge to zero faster than n−(r+1) as n → +∞. This minimal error is achieved by the Taylor-integral algorithm which uses integrals of f. 相似文献
13.
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
14.
We consider the Tikhonov regularizer fλ of a smooth function f ε H2m[0, 1], defined as the solution (see [1]) to We prove that if f(j)(0) = f(j)(1) = 0, J = m, …, k < 2m − 1, then ¦f − fλ¦j2 Rλ(2k − 2j + 3)/2m, J = 0, …, m. A detailed analysis is given of the effect of the boundary on convergence rates. 相似文献
15.
D. S. Chelkak 《Journal of Mathematical Sciences》2005,129(4):4053-4082
Asymptotics of spectral data of a perturbed harmonic oscillator −y″ + x2y + q(x)y are obtained for potentials q(x) such that q′, xq ∈ L2(ℝ). These results are important in the solution of the corresponding inverse spectral problem. Bibliography: 7 titles.__________Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 303, 2003, pp. 223–271. 相似文献
16.
In this paper, we study the existence of periodic solutions of the second order differential equations x″+f(x)x′+g(x)=e(t). Using continuation lemma, we obtain the existence of periodic solutions provided that F(x) () is sublinear when x tends to positive infinity and g(x) satisfies a new condition where M, d are two positive constants. 相似文献
17.
In this paper, we study the existence of periodic solutions for a fourth-order p-Laplacian differential equation with a deviating argument as follows:
[φp(u″(t))]″+f(u″(t))+g(u(t−τ(t)))=e(t).