共查询到20条相似文献,搜索用时 593 毫秒
1.
K.R. Kazmi M.I. Bhat Naeem Ahmad 《Journal of Computational and Applied Mathematics》2009,233(2):361-371
In this paper, we give the notion of M-proximal mapping, an extension of P-proximal mapping given in [X.P. Ding, F.Q. Xia, A new class of completely generalized quasi-variational inclusions in Banach spaces, J. Comput. Appl. Math. 147 (2002) 369–383], for a nonconvex, proper, lower semicontinuous and subdifferentiable functional on Banach space and prove its existence and Lipschitz continuity. Further, we consider a system of generalized implicit variational inclusions in Banach spaces and show its equivalence with a system of implicit Wiener–Hopf equations using the concept of M-proximal mappings. Using this equivalence, we propose a new iterative algorithm for the system of generalized implicit variational inclusions. Furthermore, we prove the existence of solution of the system of generalized implicit variational inclusions and discuss the convergence and stability analysis of the iterative algorithm. 相似文献
2.
Let X, Y be Banach spaces and M be a linear subspace in X × Y = {{x, y}|x ∈ X, y ∈ Y }. We may view M as a multi-valued linear operator from X to Y by taking M (x) = {y|{x, y} ∈ M }. In this paper, we give several criteria for a single-valued operator from Y to X to be the metric generalized inverse of the multi-valued linear operator M . The principal tool in this paper is also the generalized orthogonal decomposition theorem in Banach spaces. 相似文献
3.
In Geoffroy et al, Acceleration of convergence in Dontchev's iterative method for solving variational inclusions Serdica Math. J. 29 (2003), pp. 45–54] we showed the convergence of a cubic method for solving generalized equations of the form 0 ∈ f(x) +- G(x) where f is a function and G stands for a set-valued map. We investigate here the stability of such a method with respect to some perturbations. More precisely, we consider the perturbed equation y ∈ f(x) +- G(x) and we show that the pseudo-Lipschitzness of the map (f +- G)−1 is closely tied to the uniformity of our method in the sense that the attraction region does not depend on small perturbations of the parameter y. Finally, we provide an enhanced version of the convergence theorem established by Geoffroy, et al. 相似文献
4.
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
5.
In this paper, we consider a space-time Riesz–Caputo fractional advection-diffusion equation. The equation is obtained from
the standard advection-diffusion equation by replacing the first-order time derivative by the Caputo fractional derivative
of order α ∈ (0,1], the first-order and second-order space derivatives by the Riesz fractional derivatives of order β
1 ∈ (0,1) and β
2 ∈ (1,2], respectively. We present an explicit difference approximation and an implicit difference approximation for the equation
with initial and boundary conditions in a finite domain. Using mathematical induction, we prove that the implicit difference
approximation is unconditionally stable and convergent, but the explicit difference approximation is conditionally stable
and convergent. We also present two solution techniques: a Richardson extrapolation method is used to obtain higher order
accuracy and the short-memory principle is used to investigate the effect of the amount of computations. A numerical example
is given; the numerical results are in good agreement with theoretical analysis. 相似文献
6.
We consider the Dirichlet problem for a class of quasilinear degenerate elliptic inclusions of the form ?div(𝒜(x, u, ?u)) + f(x)g(u) ∈ H(x, u, ?u), where 𝒜(x, u, ?u) is allowed to be degenerate. Without the general assumption that the multivalued nonlinearity is characterized by Clarke's generalized gradient of some locally Lipschitz functions, we prove the existence of bounded solutions in weighed Sobolev space with the superlinear growth imposed on the nonlinearity g and the multifunction H(x, u, ?u) by using the Leray-Schauder fixed point theorem. Furthermore, we investigate the existence of extremal solutions and prove that they are dense in the solutions of the original system. Subsequently, a quasilinear degenerate elliptic control problem is considered and the existence theorem based on the proven results is obtained. 相似文献
7.
Floris Takens 《Bulletin of the Brazilian Mathematical Society》2002,33(2):231-262
The reconstruction theorem deals with dynamical systems which are given by a map ψ : M → M together with a read out function 𝒻 : M → ℝ. Restricting to the cases where ψ is a diffeomorphism, it states that for generic (ψ, 𝒻 ) there is a bijection between
elements x ∈ M and corresponding sequences (𝒻(x), 𝒻 (ψ(x)), . . . , 𝒻 (ψ
k
-1(x))) of k successive observations, at least for k sufficiently big. This statement turns out to be wrong in cases where ψ is an endomorphism.
In the present paper we derive a version of this theorem for endomorphisms (and which is equivalent to the original theorem
in the case of diffeomorphisms). It justifies, also for dynamical systems given by endomorphisms, the algorithms for estimating
dimensions and entropies of attractors from obervations.
Received: 20 June 2002 相似文献
8.
In this paper we study set-valued optimization problems with equilibrium constraints (SOPECs) described by parametric generalized
equations in the form 0 ∈ G(x) + Q(x), where both G and Q are set-valued mappings between infinite-dimensional spaces. Such models particularly arise from certain optimization-related
problems governed by set-valued variational inequalities and first-order optimality conditions in nondifferentiable programming.
We establish general results on the existence of optimal solutions under appropriate assumptions of the Palais-Smale type
and then derive necessary conditions for optimality in the models under consideration by using advanced tools of variational
analysis and generalized differentiation.
Dedicated to Jiří V. Outrata on the occasion of his 60th birthday.
This research was partly supported by the National Science Foundation under grants DMS-0304989 and DMS-0603846 and by the
Australian Research Council under grant DP-0451168. 相似文献
9.
I. Ginchev A. Guerraggio M. Rocca 《Journal of Optimization Theory and Applications》2009,143(1):87-105
The present paper studies the following constrained vector optimization problem: min
C
f(x), g(x)∈−K, h(x)=0, where f:ℝ
n
→ℝ
m
, g:ℝ
n
→ℝ
p
and h:ℝ
n
→ℝ
q
are locally Lipschitz functions and C⊂ℝ
m
, K⊂ℝ
p
are closed convex cones. In terms of the Dini set-valued directional derivative, first-order necessary and first-order sufficient
conditions are obtained for a point x
0 to be a w-minimizer (weakly efficient point) or an i-minimizer (isolated minimizer of order 1). It is shown that, under natural assumptions (given by a nonsmooth variant of the
implicit function theorem for the equality constraints), the obtained conditions improve some given by Clarke and Craven.
Further comparison is done with some recent results of Khanh, Tuan and of Jiiménez, Novo. 相似文献
10.
An Application of a Mountain Pass Theorem 总被引:3,自引:0,他引:3
We are concerned with the following Dirichlet problem:
−Δu(x) = f(x, u), x∈Ω, u∈H
1
0(Ω), (P)
where f(x, t) ∈C (×ℝ), f(x, t)/t is nondecreasing in t∈ℝ and tends to an L
∞-function q(x) uniformly in x∈Ω as t→ + ∞ (i.e., f(x, t) is asymptotically linear in t at infinity). In this case, an Ambrosetti-Rabinowitz-type condition, that is, for some θ > 2, M > 0,
0 > θF(x, s) ≤f(x, s)s, for all |s|≥M and x∈Ω, (AR)
is no longer true, where F(x, s) = ∫
s
0
f(x, t)dt. As is well known, (AR) is an important technical condition in applying Mountain Pass Theorem. In this paper, without assuming
(AR) we prove, by using a variant version of Mountain Pass Theorem, that problem (P) has a positive solution under suitable
conditions on f(x, t) and q(x). Our methods also work for the case where f(x, t) is superlinear in t at infinity, i.e., q(x) ≡ +∞.
Received June 24, 1998, Accepted January 14, 2000. 相似文献
11.
From Scalar to Vector Optimization 总被引:3,自引:0,他引:3
Initially, second-order necessary optimality conditions and sufficient optimality conditions in terms of Hadamard type derivatives
for the unconstrained scalar optimization problem ϕ(x) → min, x ∈ ℝ
m
, are given. These conditions work with arbitrary functions ϕ: ℝ
m
→ ℝ, but they show inconsistency with the classical derivatives. This is a base to pose the question whether the formulated
optimality conditions remain true when the “inconsistent” Hadamard derivatives are replaced with the “consistent” Dini derivatives.
It is shown that the answer is affirmative if ϕ is of class
(i.e., differentiable with locally Lipschitz derivative).
Further, considering
functions, the discussion is raised to unconstrained vector optimization problems. Using the so called “oriented distance”
from a point to a set, we generalize to an arbitrary ordering cone some second-order necessary conditions and sufficient conditions
given by Liu, Neittaanmaki, Krizek for a polyhedral cone. Furthermore, we show that the conditions obtained are sufficient
not only for efficiency but also for strict efficiency. 相似文献
12.
Chmielinski has proved in the paper [4] the superstability of the generalized orthogonality equation |〈f(x), f(y)〉| = |〈x,y〉|. In this paper, we will extend the result of Chmielinski by proving a theorem: LetD
n be a suitable subset of ℝn. If a function f:D
n → ℝn satisfies the inequality ∥〈f(x), f(y)〉| |〈x,y〉∥ ≤ φ(x,y) for an appropriate control function φ(x, y) and for allx, y ∈ D
n, thenf satisfies the generalized orthogonality equation for anyx, y ∈ D
n. 相似文献
13.
In this paper we give a criterion for a given set K in Banach space to be approximately weakly invariant with respect to the differential inclusion x′(t) ∈ Ax(t) + F(x(t)), where A generates a C
0-semigroup and F is a given multi-function, using the concept of a tangent set to another set. As an application, we establish the relation
between approximate solutions to the considered differential inclusion and solutions to the relaxed one, i.e., x′(t) ∈ Ax(t) + [`(co)]\overline {co}
F(x(t)), without any Lipschitz conditions on the multi-function F. 相似文献
14.
In the paper, a global optimization problem is considered where the objective function f (x) is univariate, black-box, and its first derivative f ′(x) satisfies the Lipschitz condition with an unknown Lipschitz constant K. In the literature, there exist methods solving this problem by using an a priori given estimate of K, its adaptive estimates, and adaptive estimates of local Lipschitz constants. Algorithms working with a number of Lipschitz
constants for f ′(x) chosen from a set of possible values are not known in spite of the fact that a method working in this way with Lipschitz
objective functions, DIRECT, has been proposed in 1993. A new geometric method evolving its ideas to the case of the objective
function having a Lipschitz derivative is introduced and studied in this paper. Numerical experiments executed on a number
of test functions show that the usage of derivatives allows one to obtain, as it is expected, an acceleration in comparison
with the DIRECT algorithm.
This research was supported by the RFBR grant 07-01-00467-a and the grant 4694.2008.9 for supporting the leading research
groups awarded by the President of the Russian Federation. 相似文献
15.
Let M be a closed surface, orientable or non-orientable, and letf be a C° flow onM of which all singular points are isolated. Thenf has the pseudo-orbit tracing property if and only if (i) for anyx ∈ M, both the ωlimit set ω(x) and the α-limit set α(x) ofx contain only one orbit; (ii) for any regular pointx off, if ω(x) is not quasi-attracting, then α(x) is quasi-exclusive; (iii) every saddle point off is strict, and at most 4-forked.
Project supported by the National Natural Science Foundation of China. 相似文献
16.
The Lipschitz classes Lip(α, M) , 0 α≤ 1 are defined for Orlicz space generated by the Young function M, and the degree of approximation by matrix transforms of f ∈ Lip (α, M) is estimated by n-α . 相似文献
17.
Sophia Th. Kyritsi Nikolaos Matzakos Nikolaos Papageorgiou 《Czechoslovak Mathematical Journal》2005,55(3):545-579
In this paper we study two boundary value problems for second order strongly nonlinear differential inclusions involving a
maximal monotone term. The first is a vector problem with Dirichlet boundary conditions and a nonlinear differential operator
of the form x ↦ a(x, x′)′. In this problem the maximal monotone term is required to be defined everywhere in the state space ℝN. The second problem is a scalar problem with periodic boundary conditions and a differential operator of the form x ↦ (a(x)x′)′. In this case the maximal monotone term need not be defined everywhere, incorporating into our framework differential
variational inequalities. Using techniques from multivalued analysis and from nonlinear analysis, we prove the existence of
solutions for both problems under convexity and nonconvexity conditions on the multivalued right-hand side. 相似文献
18.
The main goal of the article is to show that Paley-Wiener functions ƒ ∈ L
2(M) of a fixed band width to on a Riemannian manifold of bounded geometry M completely determined and can be reconstructed
from a set of numbers Φi (ƒ), i ∈ ℕwhere Φi
is a countable sequence of weighted integrals over a collection of “small” and “densely” distributed compact subsets. In particular, Φi, i ∈ ℕ,can be a sequence of weighted Dirac measures δxi, xi ∈M.
It is shown that Paley-Wiener functions on M can be reconstructed as uniform limits of certain variational average spline
functions.
To obtain these results we establish certain inequalities which are generalizations of the Poincaré-Wirtingen and Plancherel-Polya
inequalities.
Our approach to the problem and most of our results are new even in the one-dimensional case. 相似文献
19.
P. V. Tsynaiko 《Ukrainian Mathematical Journal》1998,50(9):1478-1482
We study a periodic boundary-value problem for the quasilinear equation u
tt
−u
xx
=F[u, u
t
, u
x
], u(x, 0)=u(x, π)=0, u(x + ω, t) = u(x, t), x ∈ ℝ t ∈ [0, π], and establish conditions that guarantee the validity of a theorem on unique solvability.
Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 9, pp. 1293–1296, September, 1998. 相似文献
20.
We consider the parametric programming problem (Q
p
) of minimizing the quadratic function f(x,p):=x
T
Ax+b
T
x subject to the constraint Cx≤d, where x∈ℝ
n
, A∈ℝ
n×n
, b∈ℝ
n
, C∈ℝ
m×n
, d∈ℝ
m
, and p:=(A,b,C,d) is the parameter. Here, the matrix A is not assumed to be positive semidefinite. The set of the global minimizers and the set of the local minimizers to (Q
p
) are denoted by M(p) and M
loc
(p), respectively. It is proved that if the point-to-set mapping M
loc
(·) is lower semicontinuous at p then M
loc
(p) is a nonempty set which consists of at most ?
m,n
points, where ?
m,n
= is the maximal cardinality of the antichains of distinct subsets of {1,2,...,m} which have at most n elements. It is proved also that the lower semicontinuity of M(·) at p implies that M(p) is a singleton. Under some regularity assumption, these necessary conditions become the sufficient ones.
Received: November 5, 1997 / Accepted: September 12, 2000?Published online November 17, 2000 相似文献