pinecone/core-2020-05-10-deduplication huggingface dataset:
The Asymptotic Bounds of Solutions of Linear Delay Systems

AbstractOur aim is to obtain conditions under which every solutionyof the vector functional differential equation[formula]can be estimated in terms of a solution ω of a scalar functional equation[formula]wherea(x) is a suitable real positive function andbis a suitable real negative constant. Moreover, ifB(x) is diagonal matrix, we derive more precise estimates ofyin terms of a solution ν of a vector functional equation[formula] where the elements ofA+(x) are formed by the absolute values of the corresponding elements ofA(x). These asymptotic results enable us to obtain sharp estimates of solutions of certain linear delay systems
Generic Fréchet Differentiability of Convex Functions Dominated by a Lower Semicontinuous Convex Function

AbstractIn this paper, an extended real-valued proper lower semicontinuous convex functionfon a Banach space is said to have the Fréchet differentiability property (FDP) if every proper lower semicontinuous convex functiongwithg≤fis Fréchet differentiable on a denseGδsubset of intdomg, the interior of the effective domain ofg. We show thatfhas the FDP if and only if thew*-closed convex hull of the image of the subdifferential map offhas the Radon–Nikodým property. This is a generalization of the main theorem in a paper by Lixin and Shuzhong (to appear). According to this result, it also gives several new criteria of Asplund spaces
Gegenbauer Approximation and Its Applications to Differential Equations on the Whole Line

AbstractA Gegenbauer approximation is discussed. Several imbedding inequalities and inverse inequalities are obtained. Some approximation results are given. By variable transformation, differential equations on the whole line are changed to certain equations on a finite interval. Gegenbauer polynomials are used for their numerical solutions. The stabilities and convergences of proposed schemes are proved. The main idea and techniques used in this paper are also applicable to other multiple-dimensional problems in unbounded domains
On Spherical Convergence, Convexity, and Block Iterative Projection Algorithms in Hilbert Space

AbstractWe call a sequence {xn} in Hilbert space “spherical” if there existsusuch that lim‖xn−u‖ exists and is finite. If moreoveruis a weak accumulation point of the sequence, we call the sequence “spherically convergent.”We demonstrate that for large classes of nonexpansive (possibly nonstationary) discrete-time processes in Hilbert space the iterates are spherically convergent. Basic identities and orthogonality relations pertinent to this type of convergence are exhibited. Sufficient conditions for weak and spherical convergence, in terms of the “fullness” of the set of fixed points of the process, are established and compared. Spherical convergence of the general block iterative projection scheme in Hilbert space is established
Quadratic Convergence in Period Doubling to Chaos for Trapezoid Maps

AbstractThe trapezoid mapge(x) is defined for fixede∈(0,1) byge(x)=x/eforx∈[0,e],ge(x)=1 forx∈(e,2−e), andge(x)=(2−x)/eforx∈[2−e,2]. For a giveneand the associated one-parameter family {λge(x): 1<λ<2}, letting λn(e) be the smallest value of λ>1 for which a fixedx∈(e,2−e), sayxc, is a periodic point of period 2n, Beyer and Stein conjectured in 1982 that, for anye<1, the parameter sequence {λn(e)}∞1is quadratically convergent. In this paper the conjecture is proved. Further, the quadratic convergence is generalized to nonisosceles trapezoid maps
Separating Maps and Linear Isometries between Some Spaces of Continuous Functions

AbstractIn this paper we deal with some spaces of vector-valued continuous functionsCσ0(X,E) andCτ0(Y,F) on locally compact spacesXandY(among them the spaces of continuous functions vanishing at infinity). After giving some properties of these spaces, it is shown that the existence of a biseparating mapTbetween these spaces always implies the existence of a homeomorphism betweenXandY. Some results on automatic continuity are given whenTis just separating, linear and has closed range, as well as a description ofT. Finally these results are used to characterize the linear isometries between some of these spaces
Minimal Normalization of Wiener–Hopf Operators in Spaces of Bessel Potentials

AbstractA class of operators is investigated which results from certain boundary and transmission problems, the so-called Sommerfeld diffraction problems. In various cases these are of normal type but not normally solvable, and the problem is how to normalize the operators in a physically relevant way, i.e., not loosing the Hilbert space structure of function spaces defined by a locally finite energy norm. The present approach solves this question rigorously for the case where the lifted Fourier symbol matrix function is Hölder continuous on the real line with a jump at infinity. It incorporates the intuitive concept of compatibility conditions which is known from some canonical problems. Further it presents explicit analytical formulas for generalized inverses of the normalized operators in terms of matrix factorization
On a Diffusive Logistic Equation

AbstractWe study a reaction–diffusion version on all of RNof the logistic equation of population growth in which the birth rate depends on the spatial variable and may assume both positive and negative values. Our results which are obtained by the construction of sub- and supersolutions and the study of asymptotic properties of solutions show the interplay between the birth rate of the species and the extent of diffusion in determining the existence or nonexistence of nontrivial steady-state distributions of population
Formal Summation of Divergent Series

AbstractThe idea of telescoping a series is widely known, but is not widely trusted. It is often treated as a formalism with no meaning, unless convergence is already established. It is shown here that even for divergent series, the results of telescoping are self-consistent, and consistent with other well-behaved summation operations. Moreover, the summation operations obtained by telescoping are the strongest possible operations with these properties. Some Tauberian theorems are exhibited for telescoping
Parabolic Systems in Unbounded Domains. II. Equations with Time Delays

AbstractThis paper extends the results of an earlier article concerning the existence, uniqueness, comparison, and dynamics problems for a coupled system of parabolic equations in unbounded domains, including the whole space Rn. The present extension is for a system of functional parabolic-ordinary equations where the “reaction function” is allowed to depend on the unknown function with time delay, which may be discrete or continuous, finite or infinite. Applications are made to some model problems, with emphasis on the convergence fo the time-dependent solution to a steady-state solution and the decay property of the solution as |x|→∞
On a Suboptimal Filtration Method for Solving Convolution-Type Integral Equations of the First Kind

AbstractIn this paper a so-called local regularization method (Arsenin et al.,USSR Comput. Math. Math. Phys.28(3) (1988), 113–125; Arsenin and Timinov,Soviet Math. Dokl.32(2) (1985), 566–570; Tikhonov et al., “Mathematical Problems of Computer Tomography,” Nauka, Moscow, 1987) for solving convolution-type integral equations of the first kind is analyzed. It is shown that to increase the precision of the solution, it is appropriate to do only one approximation in the iterative scheme of this method. It is also shown that this method should not be interpreted as a method with variable regularization parameter α; one should view it as a method of suboptimal filtration. A new variant of a suboptimal filtration method is formulated
A Note on Approximating Fixed Points of Nonexpansive Mappings by the Ishikawa Iteration Process

AbstractLetXbe a uniformly convex Banach space that satisfies Opial's condition or whose norm is Fréchet differentiable, letCbe a bounded closed convex subset ofX, and letT:C→Cbe a nonexpansive mapping. It is shown that for any initial datax0∈C, the Ishikawa iterates {xn}, defined byxn+1=tnT(snTxn+(1−sn)xn)+(1−tn)xn,n≥0, with the restrictions thatlimnsnis less than 1, and for any subsequence {nk}∞k=0of {n}∞n=0, ∑∞k=0tnk(1−tnk) diverges, converge to a fixed point ofTweakly. Thus, such a result complements Theorem 1 of Tan and Xu (J. Math. Anal. Appl.178(1993), 301–308) and generalizes, to a certain extent, Theorem 2 of Reich (J. Math. Anal. Appl.67(1979), 274–276)
On Some Conjectures by Morriset al.about Zeros of an Entire Function

AbstractIt is known that the entire function ∑∞n=0(−z)nqn(n−1)/2/n!,q∈(0,1), has infinitely many positive but no other zeros in the complex plane. A number of conjectures on these zeros have been made by Morris, Feldstein, and Bowen and by Iserles. The main objective of this note is to establish some connection among these conjectures and to prove the following estimates:[formula]wheret0,t1,… are the zeros in monotonic increasing order. Some identities involving those zeros are proved by using Hadamard's factorisation theorem
On the Rate of Convergence of the Generalized Szász Type Operators for Functions of Bounded Variation

AbstractIn this article we introduce the Szász–Bézier operatorSn,αwhich is the modified form of the Szász–Mirakyan operator. By means of construction of suitable functions and the methods of Bojanic–Vuillemier [J. Approx. Theory31(1981), 67–79] and Cheng [J. Approx. Theory40(1984), 226–241], using some results and techniques of probability theory, we obtain the rate of convergence ofSn,αfor functions of bounded variation. Our results extend and sharpen the results of Cheng [J. Approx. Theory40(1984), 226–241], Guo and Khan [J. Approx. Theory58(1989), 90–101] for the Szász-Mirakyan operator
On the Distributional Fourier Duality and Its Applications

AbstractSampling theorems for bandlimited functions or distributions are obtained by exploiting the topological isomorphism between the space E′(R) of distributions of compact support on R and the Paley–Wiener spacePWof entire functions satisfying an estimate of the form |f(z)|≤A(1+|z|)NeB|Imz|for some constantsA,B,N≥0. We obtain sampling theorems forfinPWby expanding its Fourier transformTin a series converging in the topology of E′(R) and whose coefficients are samples taken fromf. By Fourier duality, we obtain a sampling theorem forfin the spacePW. These sampling expansions converge, in fact, uniformly on compact sets of C, since convergence in the topology ofPWimplies uniform convergence on compact sets of C. This procedure allows us to recover previous sampling theorems in a unified way. We also present further expansions of Paley–Wiener functions obtained by expanding their Fourier transform as a series involving Legendre or Hermite polynomials
Stability of the Mann and Ishikawa Iteration Procedures for φ-Strong Pseudocontractions and Nonlinear Equations of the φ-Strongly Accretive Type

AbstractWe study the stability of the Mann and Ishikawa iteration procedures for the class of Lipschitz φ-strongly pseudocontractive maps in arbitrary real Banach spaces. As a consequence we study the stability of these iteration procedures for the iterative approximation of solutions of nonlinear equations of the φ-strongly accretive type. Furthermore, we prove that theT-stability of the Mann iteration procedure leads to the strong convergence (to the fixed point ofT) of the so-called Mann iteration method with errors introduced by Liu [J. Math. Anal. Appl.194(1995), 114–125]
Robust Stability ofC0-Semigroups and an Application to Stability of Delay Equations

AbstractLetAbe a closed linear operator on a complex Banach spaceXand let λ∈ϱ(A) be a fixed element of the resolvent set ofA. LetUandYbe Banach spaces, and letD∈L(U,X) andE∈L(X,Y) be bounded linear operators. We definerλ(A;D,E) by[formula]and prove that[formula]We give two applications of this result. The first is an exact formula for the so-called stability radius of the generator of aC0-semigroup of linear operators on a Hilbert space; it is derived from a precise result about robustness under perturbations of uniform boundedness in the right half-plane of the resolvent of an arbitrary semigroup generator. The second application gives sufficient conditions on the norm of the operatorsBj∈L(X) such that the classical solutions of the delay equation[formula]are exponentially stable inLp([−h,0];X)
Invariance of a Partial Differential Equation of Fractional Order under the Lie Group of Scaling Transformations

AbstractIn this article a symmetry group of scaling transformations is determined for a partial differential equation of fractional order α, containing among particular cases the diffusion equation, the wave equation, and the fractional diffusion-wave equation. For its group-invariant solutions, an ordinary differential equation of fractional order with the new independent variablez=xt−α/2is derived. The derivative then is an Erdelyi–Kober derivative depending on a parameter α. Its complete solution is given in terms of the Wright and the generalized Wright functions
Type and Cotype of Musielak–Orlicz Sequence Spaces

AbstractLet φ=(φn) be a Musielak–Orlicz function andlφthe Musielak–Orlicz sequence space generated by φ. It is shown that φ satisfies condition δqiff φ is equivalent to aq-concave function ψ=(ψn). It is also proved thatlφis of cotypeq(typep) if and only if φ satisfies condition δq(δ2and δ*pconditions, respectively). As applications we obtain characterization of the type and cotype of various sequence spaces: Nakano, weighted Orlicz, and Orlicz sequence spaces. Furthermore, a relation betweenq-concavity (p-convexity), lowerq-estimate (upperp-estimate), and cotype (type) oflφis given
Limit Properties of Differential Mean Values

AbstractIn this paper, we establish some results that concern the behavior of two types of differential mean values as the interval length shrinks to zero. The first type involves divided difference and the second type is connected to a theorem due to Flett [Math. Gazette42(1958), 38–39]. We show that if a functionf(t) is of the formp(t)+(t−a)αg(t), wherep(t) is a polynomial, then the behavior of the limit is essentially independent of the functiong(t). The results established by Bao-lin [Amer. Math. Monthly104(1997), 561–562] and Jacobson [Amer. Math. Monthly89(1982), 300–301] are special cases of our results
Convexity and Openness with Linear Rate

AbstractThis paper presents conditions for openness with linear rate (or, equivalently, for metric regularity) of continuous mappings that possess certain convexity properties.Convex continuous functions on a Banach space are proved to be open with linear rate around each point that is not a minimum point. For continuous mappings that are convex with respect to a normal cone in a finite dimensional Banach space as image space, a sufficient condition for openness with linear rate is given. Special cases are treated: For Fréchet-differentiable cone–convex mappings, the surjectivity of the derivative is proved to be equivalent to openness with linear rate. Finitely generated cones lead to a sufficient condition for openness with linear rate that simplifies practical use.A tangency formula of Lyusternik-type is set up for mappings that are open with linear rate, and is applied to cone–convex mappings
On Bitangential Interpolation in the Time-Varying Setting for Hilbert–Schmidt Operators: The Continuous Time Case

AbstractThe Hilbert space of lower triangular Hilbert–Schmidt operators on the real line is a natural analogue of the Hardy space of a half-plane, where the complex numbers are now replaced by matrix-valued functions. One can associate with a bounded operator its “values” at a matrix-valued function [see Ballet al.,Oper. Theory Adv. Appl.56(1992), 52–89], and this allows [see Ballet al.,Integral Equations Operator Theory20(1994), 1–43] to define and solve the analogue of the two-sided Nudelman interpolation problem for bounded operators (which form an analogue ofH∞(C+)). In this paper we consider the two-sided interpolation problem with a Hilbert–schmidt norm constraint (rather than the more common operator-norm constraint) on the interpolant
The Study of Existence of Equilibria for Generalized Games without Lower Semicontinuity in Locally Topological Vector Spaces

AbstractThe aim of this paper is to establish general existence results of maximal elements for L-majorized mappings, which are, in turn, used to establish the general existence theorems of equilibria for generalized games (resp., abstract economics) without lower semicontinuity for both constraint and preference mappings (correspondences) and in which strategic (resp., commodity) spaces may not be compact, the set of players (resp., agents) may not be countable, and underlying spaces are either finite or infinite dimensional locally topological vector spaces. Our results unify or improve corresponding results in the literature
Some Examples of ConjugateP-Harmonic Differential Forms

AbstractThere has been new interest developing in the study of the Hodge theory andp-harmonic differential forms in recent years, largely pertaining to applications in quasi-conformal analysis, electromagnetic theory, relativity theory, mathematical physics, nonlinear potential theory, etc. In this paper, we develop an effective method to find conjugatep-harmonic differential forms inRnby applying Hodge theory in calculations with thep-harmonic equation. As applications of the method, we obtain some interesting examples of conjugatep-harmonic differential forms
An Example of Pure Stability for the Wave Equation with Moving Boundary

AbstractThis paper studies the behavior of the energy of the wave equation solution in a finite moving domain 0<x<a(t) witha(t) assumed to move slower than light and periodically. Moreover,ais continuous, piecewise linear with two independent parameters. The boundary conditions are of Dirichlet type. We show that the energy is always bounded and give results which involve only the independent parameters. The proof is based on the determination of the invariant measure of a homeomorphism of the circle related toa
Some Properties of the Second Conjugate of a Tauberian Operator

AbstractA bounded linear operatorT:X→Y(Banach spaces) is defined to be Tauberian provided whenever {xn}⊂Xis bounded and {T(xn)}⊂Yis weakly convergent, then {xn} has a weakly convergent subsequence. Hence, they appear as opposite to weakly compact operators. In 1991 a Tauberian operatorTbetween separable Banach spaces was found such that its second conjugateT** is not Tauberian. ThoughT** might not be Tauberian, in this paper we prove that it satisfies the following property whenXis separable: whenever {x*n*}⊂X** is bounded and {T**(x*n*)}⊂Y** is weakly convergent, then {x*n*} has aw*-convergent subsequence. Other properties ofT** are proved and the nonseparable case is also studied
Asymptotically Isometric Copies ofc0and Renormings of Banach Spaces

AbstractIf a Banach spaceXcontains an asymptotically isometric copy ofc0, thenXfails to have weak normal structure. Consequently, ifXis an infinite-dimensional subspace of (c0,‖·‖∞), thenXfails to have weak normal structure. Also, every equivalent renorming ofc0(Γ), for Γ uncountable, fails to have weak normal structure. Every separable Banach space can be equivalently renormed so as not to contain an asymptotically isometric copy ofc0. Every Banach space with the generalized Gossez-Lami Dozo (GGLD) property fails to contain a subspace isomorphic toc0
Tauberian Theorems via Statistical Convergence

AbstractThe concept of statistical convergence, which is related to the usual concept of convergence in probability, provides a regular summability method for abstract metric spaces. By using probabilistic tools, we provide some Tauberian theorems which have best possible order Tauberian conditions. Furthermore, these methods can be used to unify and improve the classical pointwise Tauberian theorems of summability theory for the random walk type methods as proved by Bingham, and Hausdorff methods as proved by Lorentz
Bifurcation from Zero or Infinity in Sturm–Liouville Problems Which Are Not Linearizable

AbstractWe consider the nonlinear Sturm–Liouville problem[formula][formula]whereai,biare real numbers with |ai|+|bi|>0,i=0,1, λ is a real parameter, and the functionspandaare strictly positive on [0,π]. Suppose that the nonlinearityhsatisfies a condition of the form[formula]as either |(ξ,η)|→0 or |(ξ,η)|→∞, for some constantsM0,M1. Then we show that there exist global continua of nontrivial solutions (λ,u) bifurcating fromu=0 or “u=∞,” respectively. These global continua have properties similar to those of the continua found in Rabonowitz' well-known global bifurcation theorem
Bochner–Schwartz Theorems for Ultradistributions

AbstractWe prove the Bochner–Schwartz theorem for the ultradistributions in the quasi-analytic case. In other words, every positive definite ultradistribution of class {Mp} is the Fourier transform of a positive {M}-tempered measure μ, that is, ∫exp[−M(ε|x|)]dμ<∞ for every ε>0, whereM(t) is the associated function ofMp. To prove this, we show that every positive elementuin S′{Mp}is a positive {M}-tempered measure, and that every positive definite ultradistribution of Roumieu type is nothing but a positive definite element in (SMpMp)′ and hence is the Fourier transform of a positive {M}-tempered measure. Our result includes the cases for Roumieu type and Beurling type and also both for all the non-quasianalytic cases and most of the quasi-analytic cases
Dissipative Quasi-geostrophic Dynamics under Random Forcing

AbstractThe quasi-geostrophic model is a simplified geophysical fluid model at asymptotically high rotation rate or at small Rossby number. We consider the quasi-geostrophic equation with dissipation under random forcing in bounded domains. We show that global unique solutions exist for appropriate initial data. Unlike the deterministic quasi-geostrophic equation whose well-posedness is well known, there seems to be no rigorous result on global existence and uniqueness of the randomly forced quasi-geostrophic equation. Our work provides such a rigorous result on global existence and uniqueness, under very mild conditions
Convergence Theorems for Strongly Pseudo-contractive and Strongly Accretive Maps

AbstractSupposeEis an arbitrary real Banach space andKis a nonempty closed convex and bounded subset ofE. SupposeT:K→Kis a uniformly continuous strong pseudo-contraction with constantk∈(0,1). Suppose {an}, {bn}, {cn}, {a′n}, {b′n}, and {c′n} are sequences in (0,1) satisfying the following conditions: (i)an+bn+cn=1=a′n+b′n+c′n∀integersn≥0; (ii) limbn=limb′n=limc′n=0; (iii) Σbn=∞; (iv) Σcn<∞. For arbitraryx0,u0,v0∈K, define the sequence {xn}∞n=0iteratively byxn+1=anxn+bnTyn+cnun;yn=a′nxn+b′nTxn+c′nvn,n≥0, where {un}, {vn} are arbitrary sequences inK. Then {xn} converges strongly to the unique fixed point ofT. Related results deal with the iterative solutions of nonlinear equations involving set-valued, strongly accretive operators
Dynamics of the Complex Standard Family

AbstractThe complexification of the standard family of circle maps θ↦θ+α+βsin(θ)mod(2π), whose parameter space contains the well-known Arnold tongues, is given byFαβ(ω)=ωeiαe(β/2)(ω−1/ω), a holomorphic map of C* with essential singularities at 0 and ∞. For real values of the parameters, we study the dynamical plane of the familyFαβ. Near the essential singularities we prove the existence ofhairsin the Julia set, an invariant set of curves organized by some symbolic dynamics, whose points (that are not endpoints) tend exponentially fast to 0 or ∞ under iteration. For β<1, we give a complex interpretation of the bifurcations of the family of circle maps. More precisely, we give a new characterization of the rational Arnold tongues in terms of some of the hairs attaching to the unit circle. For certain irrational rotation numbers, we show that the Fatou set consists exclusively of a Herman ring and its preimages. For β>1 we prove that, under certain conditions, all hairs end up attached to the unit circle as we increase the parameter