pinecone/core-2020-05-10-deduplication huggingface dataset:
Log-Convex Solutions to the Functional Equationf(x+1)=g(x)f(x): Γ-Type Functions

AbstractIn this paper we discuss log-convex solutionsf:R+→R+to the functional equation with initial condition given byf(x+1)=g(x)f(x)forx>0 andf(1)=1, (*)whereg:R+→R+. Our main result, a generalization of the Bohr–Mollerup–Artin classical characterization of the gamma function, is that ifgis eventually log- concave with the property that, for eachw>0,g(x+w)/g(x)→1 asx→∞, then (∗) has a unique eventually log-convex solution, determined by the formulaf(x)=limn→∞g(n)…g(1)gx(n)g(n+x)…g(x)forx>0.A functionfarising thus is called a Γ-type function, two examples of which are the gamma function Γ and theq-gamma function Γq(0<q<1) generated, respectively, by the functionsg(x)=xandg(x)=(1−qx)/(1−q). We establish for Γ-type functions analogues ofLegendre's Duplication Formula,Gauss' Multiplication Formula,Stirling's Formula,Euler's constant, andWeierstrass' infinite productfor the gamma function, and we use the theory of Γ-type functions to find log-convex solutionsf:R+→R+to certain functional equations of the typef(x)f(x+1m)…f(x+m−1m)h(x)=1.The backdrop of Γ-type functions serves to place classical theory into perspective. Throughout, the concepts of log-convexity and log-concavity play central roles
A Nonlocal Parabolic System Modelling Axially Symmetric Thermoelastic Contact of Two Discs

AbstractA system of parabolic integro-differential equations modelling homogeneous and isotropic elastic thin discs subject to axially symmetric temperature fields and displacements is studied. Under some appropriate assumptions the questions of existence, uniqueness, and continuous dependence of the solutions upon the data are demonstrated. The main feature of our method in this paper is that the difficulties associated with this physical problem can be seen clearly and overcome easily via the decoupling procedures and transformations
Riesz Type Kernels over the Ring of Integers of a Local Field

AbstractLetOdenote the ring of integers of a local field. In this note we prove an approximation theorem for the Riesz type kernels {γμ,λ,n}n=1∞overO. The proof requires a sharp estimate of theDirichlet kernelDn(x) onP−1\O, which may also have independent interest. As a consequence we solve the local field analog of1concerning rates of convergence of theFejér sumon the Walsh system. Our approach has potential applications to other operators such as Cesáro means and Abel–Poisson means of “type I.
Inverse and Implicit Function Theorems for Nonsmooth Maps in Banach Spaces

AbstractWe extend the classical inverse and implicit function theorems, the implicit function theorems of Lyusternik and Graves, and the results of Clarke and Pourciau to the situation when the given function is not smooth, but it has a convex strict prederivative whose measure of noncompactness is smaller than its measure of surjectivity. The proof of the main results requires Banach's open mapping theorem, Michael's selection theorem, Ekeland's variational principle, and Kakutani's fixed point theorem
Topological Additively Representable Semigroups

AbstractWe analyze the structure of nontrivial totally ordered connected topological semigroups (S,+,≲), and prove that they are homeomorphic, algebraically isomorphic, and isotonic to one of the following continua: ((a,∞),+,≤), ([a,∞),+,≤), (a≥0) ((−∞,b),+,≤), ((−∞,b],+,≤), (b≤0) or (R,+,≤), all considered as subsets of the totally ordered group of additive real numbers endowed with the usual (Euclidean) topology. We conclude with a general study about the continuity in the representation of topological totally ordered semigroups through additive real-valued order-preserving homomorphisms
Generalized Games and Non-compact Quasi-variational Inequalities

AbstractIn this paper, by developing an approximation approach which is originally due to Tuleca in 1986, we prove the existence of equilibria for generalized games in which constraint mappings (correspondences) are lower (resp., upper) semicontinuous instead of having lower (resp., upper) open sections or open graphs in infinite dimensional topological spaces. Then, existence theorems of solutions for quasi-variational inequalities and non-compact generalized quasi-variational inequalities are also established. Finally, existence theorems of constrained games with non-compact strategy sets are derived. Our results unify and generalize many well known results given in the existing literature. In particular, we answer the question raised by Yannelis and Prabhakar in 1983 in the affirmative under more weaker conditions
On the Periodic Lotka–Volterra Competition Model

AbstractWe analyze the existence, stability, and multiplicity ofT-periodic coexistence states for the classical nonautonomous periodic Lotka–Volterra competing species model. This is done by treating the average values of the birth rates of species as parameters, and studying the global structure of the set of coexistence states as these parameters vary. As a result of this analysis, we can explain the interesting differences between the results for the periodic case and the associated autonomous model
Mann and Ishikawa Type Perturbed Iterative Algorithms for Generalized Quasivariational Inclusions

AbstractIn this paper, we introduce a class of generalized quasivariational inclusions and show its equivalence with a class of fixed point problems by making use of the properties of proximal maps. Using this equivalence, we develop the Mann and Ishikawa type perturbed iterative algorithms for this class of generalized quasivariational inclusions. Further, using fixed point techniques, we prove the existence of solutions for the class of generalized quasivariational inclusions and discuss the convergence criteria for the perturbed algorithms. Our algorithms and results improve and generalize many known corresponding algorithms and results
A New Green's Function Method for Solving Linear PDE's in Two Variables

AbstractA new spectral method for solving linear and integrable nonlinear PDE's in two variables has recently been developed. This method is based on a Lax pair formulation, and generates an elegant integral representation of the solution to a given initial-boundary value problem that is particularly useful for determining long-time asymptotics. In this paper, it is shown that the integral representation obtained using Lax pairs can also be obtained using Green's function techniques. This new Green's function method is illustrated by solving a variety of initial-boundary value problems for the diffusion equation, the linearized Schrödinger equation, and the linearized KdV equation
Perturbed Proximal Point Algorithms for Generalized Quasivariational Inclusions

AbstractIn this paper, we study a class of generalized quasivariational inclusions. By using the properties of the resolvent operator associated with a maximal monotone mapping in Hilbert space, we have established an existence theorem of solutions for generalized quasivariational inclusions, suggesting a new iterative algorithm and a perturbed proximal point algorithm for finding approximate solutions which strongly converge to the exact solution of the generalized quasivariational inclusions. As special cases, some known results in this field are also discussed
On the Iterates of Some Bernstein-Type Operators

AbstractIn this paper, we establish two basic functional-type identities between the iterates of the Bleimann–Butzer–Hahn operator and those of the Bernstein operator, on the one hand, and the iterates of the (modified) Meyer–König and Zeller operator and those of the Baskakov operator, on the other. These identities allow us to transfer the properties of these operators from one to another. Attention is focused on the limit behavior of the iterates and the linear combinations of iterates of Fejér–Korovkin type
Convergence of the Viscosity Solutions for the System of Nonlinear Elasticity

AbstractIn this paper some special entropy pairs of Lax type are constructed for the system of nonlinear elasticity, in which the progressive terms are functions of a single variable. The necessity estimates for the major terms are obtained by using the theory of singular perturbation of the ordinary differential equations. The special entropy pairs are used to prove that the family of Young measures uniquely determined by the viscosity solution sequence is a family of Dirac measures. Hence, a convergence theorem on the viscosity solutions is established by applying the method of compensated compactness
The Mixed Dirichlet–Neumann–Cauchy Problem for Second Order Hyperbolic Operators

AbstractA hyperbolic mixed initial boundary-value problem is investigated in which the Neumann condition and the Dirichlet condition are given on complementary parts of the boundary. An existence and uniqueness result in Sobolev spaces with additional differentiation in the tangential directions to the interface is proved by obtaining energy estimates and applying a duality argument. The goal is the eventual analysis by the Wiener–Hopf method of the asymptotic behavior of the solution near the interface
Optimality Conditions and Duality Models for a Class of Nonsmooth Constrained Fractional Optimal Control Problems

AbstractBoth parametric and nonparametric necessary and sufficient optimality conditions are established for a class of nonsmooth constrained optimal control problems with fractional objective functions and linear dynamics. Moreover, using the forms and contents of these optimality principles, four parametric and eight parameter-free duality models are constructed and weak, strong, and strict converse duality theorems are proved. These optimality and duality results contain, as special cases, similar results for fractional optimal control problems containing square roots of positive semidefinite quadratic forms in their objective and constraint functions. The optimality and duality criteria presented in this paper generalize a number of existing results for optimal control problems and subsume a fairly large number of cognate results obtained previously in the areas of finite-dimensional linear, fractional, and nonlinear programming
On the Characterization of Properly Relaxed Delayed Controls

AbstractIn previous papers Warga and Zhu have proven the conjecture of Rosenblueth and Vinter that a particular abstract model characterizes the setR̄of relaxed controls that are limits of ordinary controls with (possibly time-dependent) delays. Rosenblueth and Vinter have also provided a simple and concrete characterization ofR̄for the case of constant delays that are consecutive multiples of a common divisorh. In the present paper we provide a concrete characterization ofR̄for the case of delays that are arbitrary (and not necessarily consecutive) multiples of someh. We also exhibit a simple compactification of the set of solutions of controlled differential equations with such delays in control
The Generalization of Paraproducts and the Full T1 Theorem for Sobolev and Triebel–Lizorkin Spaces

AbstractWe study the boundedness of generalized Calderón–Zygmund operators acting on Sobolev and, more generally, Triebel–Lizorkin spaces of arbitrary order of smoothness. We are able to relax the assumptionsT(xγ)=0 and/orT*(xγ)=0, which have been required in earlier results by other authors. To do this, we consider generalized paraproduct operators. We obtain the sharpness of our assumptions for some special cases. Using the same technique, we also obtain some sharper results for “norming” and “molecular” families
Trapped Rays in Cylindrically Symmetric Media and Poles of the Analytically Continued Resolvent

AbstractThe acoustic (stratified) propagator is defined as the positive self-adjoint operatorK≡−c2(y)ρ(y)∇zρ−1(y)∇zon L2(Rn,(c2ρ)−1dnz), wherez=(x,y)x∈Rk,y∈Rm. Bounds on the weighted resolvent ofK, defined as〈z〉−α(K−ζ)−1〈z〉−αfor α>1/2 and 〈z〉=(1+|z|2)1/2, are obtained at high energies (i.e., in the semi-classical limit Re(ζ)⪢0) using the Virial conditionc−y·∇c≥εc>0. This implies energy decay estimates for trapped waves in fiber optics. With further smoothness assumptions oncand ρ the resolvent can be analytically extended onto the second Riemann sheet. The failure of the Virial condition on some radial interval, form≥2, implies the existence ofresonant-traveling modesfor the wave equation (∂t2+K)ψ=0. In particular it is shown that ifn>m≥2 thenKhas resonance states at high energies. No such states exist ifm=1. This is explained using a shape resonance picture
An Uncertainty Principle for Ultraspherical Expansions

AbstractMotivated by Heisenberg–Weyl type uncertainty principles for the torusTand the sphereS2due to Breitenberger, Narowich, Ward, and others, we derive an uncertainty relation for radial functions on the spheresSn⊂Rn+1and, more generally, for ultraspherical expansions on [0,π]. In this setting, the “frequency variance” of aL2-function on [0,π] is defined by means of the ultraspherical differential operator, which plays the role of a Laplacian. Our proof is based on a certain first-order differential-difference operator on the doubled interval [−π,π]. Moreover, using the densitiesftof “Gaussian measures” on [0,π] with the timettending to 0, we show that the bound of our uncertainty principle is optimal
A Generalization of Walsh's Two-Circle Theorem to Vector Spaces

AbstractIn this article we investigate the location of the null sets of generalized pseudo-derivatives of the product (or quotient) of abstract polynomials. Some special cases of this general problem were studied by Walsh as geometry of polynomials in the complex plane. One of our two results deduces an improved version of the first part of the two-circle theorem due to Walsh on the quotient of two polynomials and the other furnishes an improved version of the two-circle theorem on the product of two polynomials
On Critical Exponents for the Heat Equation with a Mixed Nonlinear Dirichlet–Neumann Boundary Condition

AbstractIn this paper we consider the heat equationut=Δuin an unbounded domain Ω⊂RNwith a partly Dirichlet conditionu(x,t)=0 and a partly Neumann conditionuν=upon the boundary, wherep>1 and ν is the exterior unit normal on the boundary. It is shown that for a sectorial domain inR2and anorthantdomain inRNthere exists an explicit critical exponentpc(Ω)>1 such that all positive solutions blow up in finite time whenp∈(1,pc] while there exist positive global solutions ifp>pcand initial data are suitably small. All our blowup results include the critical case
Synthesis of Time-Variant Optimal Control with Nonquadratic Criteria

AbstractUnder the assumption of convexity of nonquadratic time-variant criteria for a linear time-variant control system, it is proved in this paper that the closed-loop synthesis of the optimal control is given by a nonlinear feedbacku(t)=−R−1(t)B*(t)P(t,x(t)),in whichP(t,x) is the normal solution of a quasi-Riccati operator equation. It is also shown that the nonlinear feedback operatorP(t,x) can be explicitly expressed by solutions of the associated nonlinear algebraic equation and nonlinear integral equations respectively in three cases corresponding to Mayer problems, Lagrange problems, and Bolza problems
Webs and Bounded Finitely Additive Measures

AbstractLetM={μs:s∈S} be a family of scalar bounded finitely additive measures defined on a σ-algebra A. The Nikodym–Grothendieck boundedness theorem states that ifMis simply bounded in A thenMis uniformly bounded in A. In this paper we prove that if V={Ascr;n1,n2,…,np:p,n1,n2…np∈N} is an increasing web in A, then there is a strand {An1n2…ni:i∈N} such that ifMis simply bounded in one An1n2…nithenMis uniformly bounded in A (Theorem 3.1). This result is deduced from the fact that if W={En1n2…np:p,n1,n2,…,np∈N} is a linear increasing web inl0∞(X,A), then there exists a strand {En1n2…ni:i∈N} such that everyEn1n2…niis barrelled and dense inl0∞(X,A) (Theorem 2.7). From this strong barrelledness condition previous results of the author jointly with J. C. Ferrando are improved here. These results are related to the classical result of Diestel and Faires in vector measures
The Supremum and Infimum of the Set of Fuzzy Numbers and Its Application

AbstractIn this paper, we prove that the bounded set of fuzzy numbers must exist supremum and infimum and give the concrete representation of supremum and infimum. As an application, we obtain that the continuous fuzzy-valued function on a closed interval exists supremum and infimum and give the precise representation. We also show that the bounded fuzzy-valued function on a closed interval can define the lower and upper sums and the lower and upper integrals of Riemann and Riemann–Stieltjes by the usual way
Existence and Regularity of a Class of Weak Solutions to the Navier–Stokes Equations

AbstractWe construct a class of weak solutions to the Navier–Stokes equations, which have second order spatial derivatives and one order time derivatives, ofppower summability for 1<p≤5/4. Meanwhile, we show thatu∈Ls(0,T;W2,r(Ω)) with 1/s+3/2r=2 for 1<r≤5/4.rcan be relaxed not to exceed 3/2 if we consider only in the interior of Ω. In the end, we extend the classical regularity theorem. Our results show thatuis a regular solution if ∇u∈Ls(0,T;Lr(Ω)) with 1/s+3/2r=1 for Ω satisfying (1.3), with 1/s+1/r=5/6 for arbitrary domain inR3and 1<s≤2. For Ω=Rnwithn≥3, this result was previously obtained byH. Beirão da Veiga (Chinese Ann. Math. Ser. B16, 1995, 407–412)
Cauchy Problem with Subcritical Nonlinearity

AbstractThe semilinear parabolic system onRnwith subcritical nonlinearity is studied as an abstract evolutionary equation in a Banach spaceX≔Lp(Rn). As in the case of bounded domains the existence of a strongly continuous semigroup of globalXα-solutions and its dissipativeness is shown to follow from a single estimate of solutions. Despite the lack of compactness of the Sobolev embeddings inRnthe compactness of trajectories of the semigroup is proved using the auxiliary estimate of solutions in a weighted space Lp(Rn,(1+|x|2)ν), and the existence of a global attractor is also shown. Thus this paper generalizes earlier considerations and provides a basis for further applications
Fritz-John Optimality and Duality for Non-convex Programs

AbstractIn this paper we have obtained Fritz-John type necessary optimality criteria for non-linear programs under the hypotheses that the right differentials, at the optimal point, of the objective and the active constraint functions with respect to an arc are convex and the inactive constraint functions are continuous. A Mond-Weir type dual is associated and weak and strong duality results are established by assuming the functions involved to be locallyP-connected. These results are further extended and necessary as well as sufficient optimality conditions are obtained for a weakly efficient solution of a multiobjective non-linear program. A proper weak minimum is defined and duality results are established by using this concept
Invariance Principles for Hyperbolic Random Walk Systems

AbstractReaction random walk systems are hyperbolic models for the description of spatial motion (in one dimension) and reaction of particles. In contrast to reaction diffusion equations, particles have finite propagation speed. For parabolic systems invariance results and maximum principles are well known. A convex set is positively invariant if at each boundary point an outer normal is a left eigenvector of the diffusion matrix, and if the vector field defined by the pure reaction equation “points inward” at the boundary. Here we show a corresponding result for random walk systems. The model parameters are the particle speeds, the rates of change in direction, and the reaction vector field. A convex domain is invariant if at each boundary point an outer normal is a left eigenvector of the “speed matrix” and if a vector field given by the reaction equation combined with the turning rates points inward. Finally a positivity result is shown
Nonadditive Set Functions on a Finite Set and Linear Inequalities

AbstractA set function is a function whose domain is the power set of a set, which is assumed to be finite in this paper. We treat a possibly nonadditive set function, i.e., a set function which does not satisfy necessarily additivity, ϕ(A)+ϕ(B)=ϕ(A∪B) forA∩B=∅, as an element of the linear space on the power set. Then some of the famous classes of set functions are polyhedral in that linear space, i.e., expressed by a finite number of linear inequalities. We specify the sets of the coefficients of the linear inequalities for some classes of set functions. Then we consider the following three problems: (a) the domain extension problem for nonadditive set functions, (b) the sandwich problem for nonadditive set functions, and (c) the representation problem of a binary relation by a nonadditive set function, i.e., the problem of nonadditive comparative probabilities
Linear Second Order Ordinary Differential Equations with Nonsmooth Coefficients

AbstractThis paper deals with a Dirichlet boundary value problem for a linear second order ordinary differential operator, whose coefficients belong to certainLp-spaces. Its solution is to be understood in the sense of Sobolev, so that the Fredholm alternative holds. The main purpose of this paper is, in case of unique solvability, to introduce a Green's function by means of which the solution can be given explicitly by integrals. We give the precise definition of the Green's function via Riesz' Representation Theorem and establish some of its basic properties. As a preliminary tool the Cauchy initial value problem is considered
Ishikawa and Mann Iteration Methods with Errors for Nonlinear Equations of the Accretive Type

AbstractLetEbe anarbitrary Banach spaceandT:E→Ea Lipschitz strongly accretive operator. It is proved that for a givenf∈E, the Ishikawa and the Mann iteration methods with errors introduced by18converge strongly to the solution of the equationTx=f. Furthermore, ifEis a uniformly smooth Banach space andT:E→Eis demicontinuous and strongly accretive, it is also proved that both the Ishikawa and the Mann iteration methods with errors converge strongly to the solution of the equationTx=f. Related results deal with the iterative approximation of fixed points of strongly pseudocontractive operators, and the solution of the equationx+Tx=f,f∈EwhenT:E→Eis m-accretive
C-Cosine Functions and the Abstract Cauchy Problem, II

AbstractFor 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=A∣D(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=C∣X1. 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)+∫t0Cg(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