Lipschitz-type Condition Research Articles

Almost Automorphic Solutions of Difference Equations

We study discrete almost automorphic functions (sequences) defined on the set of integers with values in a Banach space . Given a bounded linear operator defined on and a discrete almost automorphic function , we give criteria for the existence of discrete almost automorphic solutions of the linear difference equation . We also prove the existence of a discrete almost automorphic solution of the nonlinear difference equation assuming that is discrete almost automorphic in for each , satisfies a global Lipschitz type condition, and takes values on .

Extragradient algorithms extended to equilibrium problems¶

We make use of the auxiliary problem principle to develop iterative algorithms for solving equilibrium problems. The first one is an extension of the extragradient algorithm to equilibrium problems. In this algorithm the equilibrium bifunction is not required to satisfy any monotonicity property, but it must satisfy a certain Lipschitz-type condition. To avoid this requirement we propose linesearch procedures commonly used in variational inequalities to obtain projection-type algorithms for solving equilibrium problems. Applications to mixed variational inequalities are discussed. A special class of equilibrium problems is investigated and some preliminary computational results are reported. This article is dedicated to the Memory of W. Oettli.

COMMON FIXED POINTS UNDER LIPSCHITZ TYPE CONDITION

The aim of the present paper is three fold. Firstly, we obtain common fixed point theorems for a pair of selfmaps satisfying nonexpansive or Lipschitz type condition by using the notion of pointwise R-weak commutativity but without assuming the completeness of the space or continuity of the mappings involved (Theorem 1, Theorem 2 and Theorem 3). Secondly, we generalize the results obtained in first three theorems for four mappings by replacing the condition of noncompatibility of maps with the property (E.A) and using the R-weak commutativity of type <TEX>$(A_g)$</TEX> (Theorem 4). Thirdly, in Theorem 5, we show that if the aspect of noncompatibility is taken in place of the property (E.A), the maps become discontinuous at their common fixed point. We, thus, provide one more answer to the problem posed by Rhoades [11] regarding the existence of contractive definition which is strong enough to generate fixed point but does not forces the maps to become continuous.

Weakly dependent chains with infinite memory

We prove the existence of a weakly dependent strictly stationary solution of the equation X t = F ( X t − 1 , X t − 2 , X t − 3 , … ; ξ t ) called a chain with infinite memory. Here the innovations ξ t constitute an independent and identically distributed sequence of random variables. The function F takes values in some Banach space and satisfies a Lipschitz-type condition. We also study the interplay between the existence of moments, the rate of decay of the Lipschitz coefficients of the function F and the weak dependence properties. From these weak dependence properties, we derive strong laws of large number, a central limit theorem and a strong invariance principle.

TU‐C‐ValA‐02: A Simple Iterative Method to Invert a Deformation Field

Purpose: Inversion of a deformation field is applied frequently to map dose and regions of interest to a reference frame. A prevailing naïve approach that takes the opposite displacement of the forward deformation as the displacement of the inverse is mathematically wrong and can cause large errors for large or accumulative deformation. Inversion by “scattered data interpolation” has O(N2) complexity and is difficult to implement. We propose a simple iterative approach with O(N) complexity. Method: Instead of calculating the inverse, we calculate the displacement of the inverse. The displacement of the inverse is iteratively refined through the displacement of the forward map. We prove that such iterative scheme converges exponentially to the true solution when the deformation field is subject to a condition of the Lipschitz type. The Lipschitz type condition essentially states that the difference of the deformation of two points can not be too far. This is a mild restriction on the deformation field and is usually a valid assumption for any deformable registration method with regularization.Results: We tested the proposed method on both simulated 2D data and real 4D CT data of lung patient. The simulations showed that the proposed method has exponential convergences to the true inverse. For real 4D CT data, the forward deformation field constructed by deformable registration mapped the test phase to the reference phase and the inverse of that deformation field accurately map the reference phase to the test phase. Conclusions: A simple, accurate and fast method for inverting a deformation field is presented. Both the mathematical proof and the simulations showed its exponential convergence. Simulations and real data tests demonstrated its efficacy in medical image analysis and radiotherapy applications. Typically less than 10 iterations are needed to get an inverse deformation field with clinically relavent accuracy.

Network flow control under capacity constraints: A case study

In this paper, we demonstrate how tools from nonlinear system theory can play an important role in tackling “hard nonlinearities” and “unknown disturbances” in network flow control problems. Specifically, a nonlinear control law is presented for a communication network buffer management model under physical constraints. Explicit conditions are identified under which the problem of asymptotic regulation of a class of networks against unknown inter-node traffic is solvable, in the presence of control input and state saturation. The conditions include a Lipschitz-type condition and a “PE” condition. Under these conditions, we achieve either asymptotic or practical regulation for a single-node system. We also propose a decentralized, discontinuous control law to achieve (global) asymptotic regulation of large-scale networks. Our main result on controlling large-scale networks is based on an interesting extension of the well-known Young's inequality for the case with saturation nonlinearities. We present computer simulations to illustrate the effectiveness of the proposed flow control schemes.

Heat equation with singular potential and singular data

Nets of Schrödinger C0-semigroups (Sε)ε with the polynomial growth with respect to ε are used for solving the Cauchy problem (∂t − Δ)U + VU = f(t, U), U(0, x) = U0(x) in a suitable generalized function algebra (or space), where V and U0 are singular generalized functions while f satisfies a Lipschitz-type condition. The existence of distribution solutions is proved in appropriate cases by the means of white noise calculus as well as classical energy estimates.

Heat equation with singular potential and singular data

Nets of Schrödinger C0-semigroups (Sε)ε with the polynomial growth with respect to ε are used for solving the Cauchy problem (∂t − Δ)U + VU = f(t, U), U(0, x) = U0(x) in a suitable generalized function algebra (or space), where V and U0 are singular generalized functions while f satisfies a Lipschitz-type condition. The existence of distribution solutions is proved in appropriate cases by the means of white noise calculus as well as classical energy estimates.

A Note on the Harmonic Derivative

We characterize ordinary differentiability in terms on the harmonic derivative and a local Lipschitz type condition and apply the result to C−functions.

Geometric Models for Quasicrystals I. Delone Sets of Finite Type

This paper studies three classes of discrete sets X in $\Bbb R$ n which have a weak translational order imposed by increasingly strong restrictions on their sets of interpoint vectors X-X . A finitely generated Delone set is one such that the abelian group [X-X] generated by X-X is finitely generated, so that [X-X] is a lattice or a quasilattice. For such sets the abelian group [X] is finitely generated, and by choosing a basis of [X] one obtains a homomorphism $\varphi : [X] \rightarrow {\Bbb Z}^s$ . A Delone set of finite type is a Delone set X such that X-X is a discrete closed set. A Meyer set is a Delone set X such that X-X is a Delone set. Delone sets of finite type form a natural class for modeling quasicrystalline structures, because the property of being a Delone set of finite type is determined by ``local rules.'' That is, a Delone set X is of finite type if and only if it has a finite number of neighborhoods of radius 2R , up to translation, where R is the relative denseness constant of X . Delone sets of finite type are also characterized as those finitely generated Delone sets such that the map ϕ satisfies the Lipschitz-type condition ||ϕ (x) - ϕ (x')|| < C ||x - x'|| for x, x' ∈X , where the norms || . . . || are Euclidean norms on $\Bbb R$ s and $\Bbb R$ n , respectively. Meyer sets are characterized as the subclass of Delone sets of finite type for which there is a linear map $\tilde{L} : {\Bbb R}^n \rightarrow {\Bbb R}^s$ and a constant C such that ||ϕ (x) - $\tilde{L}$ (x)|| $\leq C$ for all x∈X . Suppose that X is a Delone set with an inflation symmetry, which is a real number η > 1 such that $\eta X \subseteq X$ . If X is a finitely generated Delone set, then η must be an algebraic integer; if X is a Delone set of finite type, then in addition all algebraic conjugates | η ' | $\leq$ η; and if X is a Meyer set, then all algebraic conjugates | η ' | $\leq$ 1.

A new semilocal convergence theorem for Newton's method

A new semilocal convergence theorem for Newton's method is established for solving a nonlinear equation F( x) = 0, defined in Banach spaces. It is assumed that the operator F is twice Fréchet differentiable, and F″ satisfies a Lipschitz type condition. Results on uniqueness of solution and error estimates are also given. Finally, these results are compared with those that use Kantorovich conditions.

Some estimates in the central limit theorem for φ-mixing random variables

Estimates of the discrepancy between the expectation of a function satisfying a Lipschitz type condition of sums of φ-mixing random variables and the expectation of the same function of a normal random variable are obtained.

A converse to the mean value property on homogeneous trees

The homogeneous tree T {\mathbf {T}} of degree q + 1 ( q ≥ 2 ) q + 1\quad (q \geq 2) may be considered as a discrete analogue of the open unit disc D {\mathbf {D}} . On D {\mathbf {D}} , every harmonic function satisfies the mean value property (MVP) at every point. Conversely, positive functions on D {\mathbf {D}} having the MVP with respect to a ball with specified radius at each point of D {\mathbf {D}} are harmonic under certain assumptions concerning the radius function: results of this type are due to J. R. Baxter, W. Veech and others. Here we consider harmonic functions on T {\mathbf {T}} with respect to a natural choice of a discrete Laplacian: the analogous MVP is true in this setting. We present a Lipschitz-type condition on the radius function (which now has integer values and refers to the discrete metric of T {\mathbf {T}} ) under which harmonicity holds for positive functions whose value at each point is the mean of its values over the ball of the radius assigned to this point. The method is based upon our previous results concerning the geometrical realization of Martin boundaries of certain transition operators as the space of ends of the underlying graph.

On the asymptotic behaviour of solutions of volterra equations in Hilbert space

u(0) =x, (1.1) where A is a nonlinear maximal monotone (possibly multi-valued) operator on a real Hilbert space H, f~Z&,([0, ~4); H), x E D(A) and G is a given mapping G:C([O, ~;D(A))-+~‘(O, T;H), VO<T< ~0. (1.2) By a recent result of Crandall & Nohel [5, theorem 11, the problem (1.1) has a unique generalized solution u E C([O, ~0); H), p rovided that G satisfies a Lipschitz type condition. The present note is concerned with the asymptotic behaviour, as t+ M of generalized solutions of (1.1). At the expense of appropriate assumptions on G [see (2.3), (2.4) below], we are able to derive natural analogs of results by several authors concerning the asymptotic properties of nonlinear contraction semigroups in Hilbert space. The plan of the paper is as follows: the results are presented in Section 2 and the proofs are given in Section 3. Although we deal with solutions of (l.l), we have always in mind the related nonlinear Volterra equation

Convergence Rates of the Ellipsoid Method on General Convex Functions

The ellipsoid method is applied to the unconstrained minimization of a general convex function. The method converges at a geometric rate, which depends only upon the dimension of the space but not on the actual function. This rate can be improved somewhat if the function satisfies some Lipschitz-type condition, or if the minimum set has dimension greater than zero. If the ellipsoid entirely contains the optimal set, equating the Steiner polynomial associated to the optimal set, and the volume of the ellipsoid at a given iteration, will give an upper bound on the minimum recorded function value.

Generic Differentiability of Lipschitzian Functions

It is shown that, in separable topological vector spaces which are Baire spaces, the usual properties that have been introduced to study the local “first order” behaviour of real-valued functions which satisfy a Lipschitz type condition are “generically” equivalent and thus lead to a unique class of “generically smooth” functions. These functions are characterized in terms of tangent cones and directional derivatives and their “generic” differentiability properties are studied. The results extend some of the well-known differentiability properties of continuous convex functions.

Generic differentiability of Lipschitzian functions

It is shown that, in separable topological vector spaces which are Baire spaces, the usual properties that have been introduced to study the local “first order” behaviour of real-valued functions which satisfy a Lipschitz type condition are “generically” equivalent and thus lead to a unique class of “generically smooth” functions. These functions are characterized in terms of tangent cones and directional derivatives and their “generic” differentiability properties are studied. The results extend some of the well-known differentiability properties of continuous convex functions.

Multiplier criteria of Marcinkiewicz type for Jacobi expansions

It is shown how an integral representation for the product of Jacobi polynomials can be used to derive a certain integral Lipschitz type condition for the Cesàro kernel for Jacobi expansions. This result is then used to give criteria of Marcinkiewicz type for a sequence to be multiplier of type (p, p), 1 &gt; p &gt; ∞ 1 &gt; p &gt; \infty , for Jacobi expansions.

Bounded solutions of perturbed Volterra integrodifferential systems

More recently (P) has been studied under the weaker assumption that (L) is admissible with respect to certain pairs of spaces. The author [2] proved under such admissibility assumptions that locally the set of bounded solutions of (P) is homeomorphic to the set of bounded solutions of (L), provided p satisfies a Lipschitz-type condition in X. Other authors [I, 6J have obtained similar and related results using a different approach for a restricted class of systems (A(S) = A, B(t, S) = B(t s)). Our purpose here is to extend our previous results in [2] to a broader class of perturbations; the expense of this generalization is less structure for the set of bounded solutions of (P) as related to the corresponding set for (L). This set will no longer necessarily be locally homeomorphic to that of (L), but nevertheless will be “at least as large;” we make this precise by using the notation of locally nonempty partitions with respect to an index set as defined below. Our main results appear in Theorem 1.

8.—Bifurcation and Asymptotic Bifurcation for Non-compact Nonsymmetric Gradient Operators

SynopsisThe first part of this paper is devoted to a study of the classical bifurcation problem in a Hilbert space, under the assumption that the operators involved are gradient operators, but not necessarily compact. Our approach to the problem was introduced by Krasnosel'skii, but here we show that his assumption about the compactness of the operators can be replaced by a much weaker Lipschitz type condition, without affecting the generality of his conclusions.The rest of the paper is concerned with the analogous problem when the operator is knownto be asymptotically linear rather than Fréchet differentiable. Indeed, we show that this question can always be reduced to the first case, after some manipulation. After this manipulation the new operator is found to be a Fréchet differentiable gradient operator, and so we can invoke the results of the first part. This manipulation is in the spirit of that of [11] but is necessarily different.