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1.
Alexander G. Ramm Alexandra B. Smirnova Angelo Favini 《Annali di Matematica Pura ed Applicata》2003,182(1):37-52
A nonlinear operator equation F(x)=0, F:H→H, in a Hilbert space is considered. Continuous Newton’s-type procedures based on a construction of a dynamical system with
the trajectory starting at some initial point x
0 and becoming asymptotically close to a solution of F(x)=0 as t→+∞ are discussed. Well-posed and ill-posed problems are investigated.
Received: June 29, 2001; in final form: February 26, 2002?Published online: February 20, 2003
This paper was finished when AGR was visiting Institute for Theoretical Physics, University of Giessen. The author thanks
DAAD for support 相似文献
2.
D. R. Heath-Brown 《Proceedings Mathematical Sciences》1994,104(1):13-29
LetF(x) =F[x1,…,xn]∈ℤ[x1,…,xn] be a non-singular form of degree d≥2, and letN(F, X)=#{xεℤ
n
;F(x)=0, |x|⩽X}, where
. It was shown by Fujiwara [4] [Upper bounds for the number of lattice points on hypersurfaces,Number theory and combinatorics, Japan, 1984, (World Scientific Publishing Co., Singapore, 1985)] thatN(F, X)≪X
n−2+2/n
for any fixed formF. It is shown here that the exponent may be reduced ton - 2 + 2/(n + 1), forn ≥ 4, and ton - 3 + 15/(n + 5) forn ≥ 8 andd ≥ 3. It is conjectured that the exponentn - 2 + ε is admissable as soon asn ≥ 3. Thus the conjecture is established forn ≥ 10. The proof uses Deligne’s bounds for exponential sums and for the number of points on hypersurfaces over finite fields.
However a composite modulus is used so that one can apply the ‘q-analogue’ of van der Corput’s AB process.
Dedicated to the memory of Professor K G Ramanathan 相似文献
3.
R. I. Hovsepyan 《Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences)》2009,44(4):205-211
It is proved that any measurable, finite function f(x) has a smooth primitive F(x), i.e. there is a function F(x) such that F′(x) = f(x) almost everywhere, and particularly ω(δ; F) = o(δ ln δ). This is an improvement of N. N. Luzin’s theorem which states just the continuity of the primitive F(x). 相似文献
4.
Mohammad Ashraf Nadeem-ur-Rehman Shakir Ali Muzibur Rahman Mozumder 《Mathematica Slovaca》2012,62(3):451-460
The main purpose of this paper is to prove the following result: Let R be a 2-torsion free semiprime *-ring. Suppose that θ, φ are endomorphisms of R such that θ is onto. If there exists an additive mapping F: R → R associated with a (θ, φ)-derivation d of R such that F(xx*) = F(x)θ(x*) + φ(x)d(x*) holds for all x ∈ R, then F is a generalized (θ, φ)-derivation. Further, some more related results are obtained. 相似文献
5.
V. Metz 《Potential Analysis》2007,26(2):121-137
On the bounded Sierpinski gasket F we use the set of essential fixed points V
0 as a boundary and consider the fractal Brownian motion on F killed in V
0. The corresponding Dirichlet–Laplacian is described in terms of a kind of hyperbolic distance, a metric which explodes near
the boundary. We consider Harnack inequalities, Green’s function estimates and (random) products of matrices defining the
local energy of harmonic functions.
Supported by the DFG research group ‘Spektrale Analysis, asymptotische Verteilungen und stochastische Dynamik.’ 相似文献
6.
G. L. O’brien 《Israel Journal of Mathematics》1981,39(1-2):145-154
LetG be a finite directed graph which is irreducible and aperiodic. Assume each vertex ofG leads to at least two other vertices, and assumeG has a cycle of prime length which is a proper subset ofG. Then there exist two functionsr:G →G andb:G →G such that ifr(x)=y andb(x)=z thenx →y andx →z inG andy ≠z and such that some composition ofr’s andb’s is a constant function.
This work was supported in part by the Natural Sciences and Engineering Research Council of Canada. I am grateful to Cornell
University whose kind hospitality I enjoyed while working on this problem. 相似文献
7.
Jitsuro Sugie Kazuhisa Kita Naoto Yamaoka 《Annali di Matematica Pura ed Applicata》2002,181(3):309-337
Our concern is to solve the oscillation problem for the non-linear self-adjoint differential equation (a(t)x’)’+b(t)g(x)=0, where g(x) satisfies the signum condition xg(x)>0 if x≠0, but is not assumed to be monotone. Sufficient conditions and necessary conditions are given for all non-trivial solutions
to be oscillatory. The obtained results show that the number 1/4 is a critical value for this problem. This paper takes a
different approach from most of the previous research. Proof is given by means of phase plane analysis of systems of Liénard
type. Examples are included to illustrate the relation between our theorems and results which were given by Cecchi, Marini
and Villari.
Received: January 5, 2001?Published online: June 11, 2002 相似文献
8.
N. M. Timofeev 《Mathematical Notes》1999,66(4):474-488
Suppose thatg(n) is equal to the number of divisors ofn, counting multiplicity, or the number of divisors ofn, a≠0 is an integer, andN(x,b)=|{n∶n≤x, g(n+a)−g(n)=b orb+1}|. In the paper we prove that sup
b
N(x,b)≤C(a)x)(log log 10
x
)−1/2 and that there exists a constantC(a,μ)>0 such that, given an integerb |b|≤μ(log logx)1/2,x≥x
o, the inequalityN(x,b)≥C(a,μ)x(log logx(−1/2) is valid.
Translated fromMatematicheskie Zametki, Vol. 66, No. 4, pp. 579–595, October, 1999. 相似文献
9.
The main purpose of this paper is to prove a collection of new fixed point theorems and existence theorems for the nonlinear
operator equation F(x) =αx (α ≥ 1) for so-called 1-set weakly contractive operators on unbounded domains in Banach spaces. We also introduce the concept
of weakly semi-closed operator at the origin and obtain a series of new fixed point theorems and the existence theorems for
the nonlinear operator equation F(x) = αx (α ≥ 1) for such class of operators. As consequences, the main results generalize and improve the relevant results, which are
obtained by O’Regan and A. Ben Amar and M. Mnif in 1998 and 2009 respectively. In addition, we get the famous fixed point
theorems of Leray-Schauder, Altman, Petryshyn and Rothe type in the case of weakly sequentially continuous, 1-set weakly contractive
(μ-nonexpansive) and weakly semi-closed operators at the origin and their generalizations. The main condition in our results
is formulated in terms of axiomatic measures of weak compactness. 相似文献
10.
S. Sadiq Basha 《Optimization Letters》2011,5(4):639-645
Given non-void subsets A and B of a metric space and a non-self mapping T:A? B{T:A\longrightarrow B}, the equation T x = x does not necessarily possess a solution. Eventually, it is speculated to find an optimal approximate solution. In other words,
if T x = x has no solution, one seeks an element x at which d(x, T x), a gauge for the error involved for an approximate solution, attains its minimum. Indeed, a best proximity point theorem
is concerned with the determination of an element x, called a best proximity point of the mapping T, for which d(x, T x) assumes the least possible value d(A, B). By virtue of the fact that d(x, T x) ≥ d(A, B) for all x in A, a best proximity point minimizes the real valued function x? d(x, T x){x\longrightarrow d(x, T\,x)} globally and absolutely, and therefore a best proximity in essence serves as an ideal optimal approximate solution of the
equation T x = x. The aim of this article is to establish a best proximity point theorem for generalized contractions, thereby producing optimal
approximate solutions of certain fixed point equations. In addition to exploring the existence of a best proximity point for
generalized contractions, an iterative algorithm is also presented to determine such an optimal approximate solution. Further,
the best proximity point theorem obtained in this paper generalizes the well-known Banach’s contraction principle. 相似文献
11.
A new look at smoothing Newton methods for nonlinear complementarity problems and box constrained variational inequalities 总被引:17,自引:0,他引:17
In this paper we take a new look at smoothing Newton methods for solving the nonlinear complementarity problem (NCP) and the
box constrained variational inequalities (BVI). Instead of using an infinite sequence of smoothing approximation functions,
we use a single smoothing approximation function and Robinson’s normal equation to reformulate NCP and BVI as an equivalent
nonsmooth equation H(u,x)=0, where H:ℜ
2n
→ℜ
2n
, u∈ℜ
n
is a parameter variable and x∈ℜ
n
is the original variable. The central idea of our smoothing Newton methods is that we construct a sequence {z
k
=(u
k
,x
k
)} such that the mapping H(·) is continuously differentiable at each z
k
and may be non-differentiable at the limiting point of {z
k
}. We prove that three most often used Gabriel-Moré smoothing functions can generate strongly semismooth functions, which
play a fundamental role in establishing superlinear and quadratic convergence of our new smoothing Newton methods. We do not
require any function value of F or its derivative value outside the feasible region while at each step we only solve a linear system of equations and if
we choose a certain smoothing function only a reduced form needs to be solved. Preliminary numerical results show that the
proposed methods for particularly chosen smoothing functions are very promising.
Received June 23, 1997 / Revised version received July 29, 1999?Published online December 15, 1999 相似文献
12.
13.
Vivek V. Rane 《Proceedings Mathematical Sciences》2003,113(3):213-221
Euler-Maclaurin and Poisson analogues of the summations ε
a <n ≤b
χ(n)f(n), have been obtained in a unified manner, where (χ(n)) is a periodic complex sequence;d(n) is the divisor function andf(x) is a sufficiently smooth function on [a, b]. We also state a generalised Abel’s summation formula, generalised Euler’s summation formula and Euler’s summation formula
in several variables. 相似文献
14.
Let f and g be continuously differentiable functions on R
n
. The nonlinear complementarity problem NCP(f,g), 0≤f(x)⊥g(x)≥0, arises in many applications including discrete Hamilton-Jacobi-Bellman equations and nonsmooth Dirichlet problems. A
popular method to find a solution of the NCP(f,g) is the generalized Newton method which solves an equivalent system of nonsmooth equations F(x)=0 derived by an NCP function. In this paper, we present a sufficient and necessary condition for F to be Fréchet differentiable, when F is defined by the “min” NCP function, the Fischer-Burmeister NCP function or the penalized Fischer-Burmeister NCP function.
Moreover, we give an explicit formula of an element in the Clarke generalized Jacobian of F defined by the “min” NCP function, and the B-differential of F defined by other two NCP functions. The explicit formulas for generalized differentials of F lead to sharper global error bounds for the NCP(f,g). 相似文献
15.
S.-M. Jung 《Acta Mathematica Hungarica》2006,110(3):261-266
Summary We introduce a new characterization of linear isometries. More precisely, we prove that if a one-to-one mapping f:<span style='font-size:10.0pt;font-family:"Lucida Sans Unicode"'>ℝn→ℝn(2<span style='font-size:10.0pt;font-family:"Lucida Sans Unicode"'>≦n<<span style='font-size:10.0pt;font-family:"Lucida Sans Unicode"'>∞) maps every regular pentagon of side length a> 0 onto a pentagon with side length b> 0, then there exists a linear isometry I :<span style='font-size:10.0pt;font-family:"Lucida Sans Unicode"'>ℝn→ℝnup to translation such that f(x) = (b/a) I(x). 相似文献
16.
We obtain 3/2-condition for global attractivity to occur in the “food-limited” type functional differential equationx′ (t) + [1 +x(t)][1 −cx(t)]F(t, x(·)) = 0. These results contain and improve all corresponding theorems in literature. 相似文献
17.
Local conjugacy theorem, rank theorems in advanced calculus and a generalized principle for constructing Banach manifolds 总被引:6,自引:0,他引:6
MA Jipu 《中国科学A辑(英文版)》2000,43(12):1233-1237
Applications of locally fine property for operators are further developed. LetE andF be Banach spaces andF:U(x
0)⊂E→F be C1 nonlinear map, whereU (x
0) is an open set containing pointx
0∈E. With the locally fine property for Frechet derivativesf′(x) and generalized rank theorem forf′(x), a local conjugacy theorem, i. e. a characteristic condition forf being conjugate tof′(x
0) near x0,is proved. This theorem gives a complete answer to the local conjugacy problem. Consequently, several rank theorems in advanced
calculus are established, including a theorem for C1 Fredholm map which has been so far unknown. Also with this property the concept of regular value is extended, which gives
rise to a generalized principle for constructing Banach submanifolds. 相似文献
18.
H. G. Ghazaryan V. N. Margaryan 《Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences)》2011,46(4):200-211
This work is devoted to the study of two-dimensional, regular, almost hypoelliptic operators P(D) = P(D
2, D
2) with regular Newton polyhedrons. It is proved that all generalized (weak) solutions of the equation P(D)u = f from a several weighted Sobolev space are infinitely differentiable functions in the rectangle {x ∈ E
2: −a < x
1 < a, −b < x
2 < b} in the variable x
2, in which the function f is infinitely differentiable. 相似文献
19.
The paper is concerned with the ‘primal’ problem of maximizing a given quadratic pseudo-boolean function. Four equivalent
problems are discussed—the primal, the ‘complementation’, the ‘discrete Rhys LP’ and the ‘weighted stability problem of a
SAM graph’. Each of them has a relaxation—the ‘roof dual’, the ‘quadratic complementation,’ the ‘continuous Rhys LP’ and the
‘fractional weighted stability problem of a SAM graph’. The main result is that the four gaps associated with the four relaxations
are equal. Furthermore, a solution to any of these problems leads at once to solutions of the other three equivalent ones.
The four relaxations can be solved in polynomial time by transforming them to a bipartite maximum flow problem. The optimal
solutions of the ‘roof-dual’ define ‘best’ linear majorantsp(x) off, having the following persistency property: if theith coefficient inp is positive (negative) thenx
i=1 (0) in every optimum of the primal problem. Several characterizations are given for the case where these persistency results
cannot be used to fix any variable of the primal. On the other hand, a class of gap-free functions (properly including the
supermodular ones) is exhibited. 相似文献
20.
Andreas Fischer 《Mathematical Programming》1997,76(3):513-532
The paper deals with complementarity problems CP(F), where the underlying functionF is assumed to be locally Lipschitzian. Based on a special equivalent reformulation of CP(F) as a system of equationsφ(x)=0 or as the problem of minimizing the merit functionΘ=1/2∥Φ∥
2
2
, we extend results which hold for sufficiently smooth functionsF to the nonsmooth case.
In particular, ifF is monotone in a neighbourhood ofx, it is proved that 0 εδθ(x) is necessary and sufficient forx to be a solution of CP(F). Moreover, for monotone functionsF, a simple derivative-free algorithm that reducesΘ is shown to possess global convergence properties. Finally, the local behaviour of a generalized Newton method is analyzed.
To this end, the result by Mifflin that the composition of semismooth functions is again semismooth is extended top-order semismooth functions. Under a suitable regularity condition and ifF isp-order semismooth the generalized Newton method is shown to be locally well defined and superlinearly convergent with the
order of 1+p. 相似文献