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In this paper we prove the optimal upper bound λnλm≤n2m2λn>λm≥11supx∈[0,1]q(x)for one‐dimensional Schrodinger operators with a nonnegative differentiable and single‐barrier potential q(x), such that ∣q′(x)∣≤q∗, where q∗=215min{q(0),q(1)}. In particular, if q(x) satisfies the additional condition supx∈[0,1]q(x)≤π211, then λnλm≤n2m2 for n>m≥1. For this result, we develop a new approach to study the monotonicity of the modified Prufer angle function.

The purpose of this paper is to provide a detailed description of the spaces that can be specified as $L^2$ domains for the operators of a first order elliptic complex on a compact manifold with conical singularities. This entails an analysis of the nature of the minimal domain and of a complementary space in the maximal domain of each of the operators. The key technical result is the nondegeneracy of a certain pairing of cohomology classes associated with the indicial complex. It is further proved that the set of choices of domains leading to Hilbert complexes in the sense of Br\"uning and Lesch form a variety, as well as a theorem establishing a necessary and sufficient condition for the operator in a given degree to map its maximal domain into the minimal domain of the next operator.

The paper concerns the uniqueness problem of Riemann zeta-function. It is showed that the Riemann zeta-function is uniquely determined in terms of the preimages of three complex values a,b,0 except possibly a set G with n(r,G)=o(r), where G is called an exceptional set.

In this paper, the 2D Navier-Stokes-Voight equations with 3 delays in R2 is considered. By using the Faedo-Galerkin method, Lions-Aubin lemma, and Arzela-Ascoli theorem, we establish the global well-posedness of solutions and the existence of pullback attractors in H1.

In this paper, we study hypersurfaces of the homogeneous nearly Kahler manifold S3×S3 with typical properties. We first show that in the NK S3×S3 there exist neither totally umbilical hypersurfaces nor hypersurfaces of parallel second fundamental form. Then we investigate hypersurfaces of S3×S3 such that its shape operator A and induced almost contact structure ϕ satisfy the condition Aϕ=ϕA, and as the main result, a complete classification of this remarkable family of hypersurfaces in S3×S3 is presented.

This paper presents 2 new classes of the Bessel functions on a compact domain [0,T] as generalized-tempered Bessel functions of the first- and second-kind which are denoted by GTBFs-1 and GTBFs-2. Two special cases corresponding to the GTBFs-1 and GTBFs-2 are considered. We first prove that these functions are as the solutions of 2 linear differential operators and then show that these operators are self-adjoint on suitable domains. Some interesting properties of these sets of functions such as orthogonality, completeness, fractional derivatives and integrals, recursive relations, asymptotic formulas, and so on are proved in detail. Finally, these functions are performed to approximate some functions and also to solve 3 practical differential equations of fractionalorders.

Let Bp,qs(Ω) be the usual Besov spaces in bounded Lipschitz domains Ω in Rn,n∈N (bounded intervals if n=1). The paper clarifies under which conditions the continuous embedding between two such spaces with 1

In the paper, abstract interpolation of Hardy spaces on circular domains is studied. In particular, an extension of Peter Jones's result on the interpolation of Hardy spaces on the disc to multiply-connected domains is presented. Moreover, some applications in approximation theory are given.

In this paper, we study the nth-order nonlinear dynamic equation with Laplacians and a deviating argument (x[n−1])Δ(t)+p(t)φγ(x(g(t)))=0 on an above-unbounded time scale, where n⩾2, x[i](t):=ri(t)φαi(x[i−1])Δ(t),i=1,2,…,n−1,withx[0]=x. New oscillation criteria are established for the cases when n is even and odd and when α > γ,α = γ, and α

In this paper, by the method of J. F. Li and X. J. Xu (Differential Harnack inequalities on Riemannian manifolds I: Linear heat equation, Adv. in Math., 226 (2011), 4456–4491), we shall consider the nonlinear parabolic equation Δ−∂∂tu(x,t)+h(x,t)uα(x,t)=0 on Riemannian manifolds with Ricci(M)≥−k, k≥0. First of all, we shall derive the corresponding Li–Xu type gradient estimates of the positive solutions for α≤1 . As applications, we deduce Liouville type theorem and Harnack inequality for some special cases. Besides, when α=1, our results are different from Li and Yau's results. We also extend the results of J. F. Li and X. J. Xu, and the results of Y. Yang.

The aim of this paper is twofold. We first establish the Besov spaces on metric spaces endowed with a doubling measure, via the remarkable orthonormal wavelet basis constructed recently by T. Hytonen and O. Tapiola, and characterize the dual spaces of these Besov spaces. Second, we prove the T1 type theorem for the boundedness of Calderon–Zygmund operators on these Besov spaces. Finally, we introduce a new class of Lipschitz spaces and characterize these spaces via the Littlewood–Paley theory. Copyright © 2016 John Wiley & Sons, Ltd.

The “oldest cubic” functional equation was introduced and solved by the second author of this paper (see: Glas. Mat. Ser. III 36(56) (2001), no. 1, 63-72). which is of the form: f(x + 2y) = 3f(x + y) + f(x − y) − 3f(x) + 6f(y). For further research in various normed spaces, we are introducing new cubic functional equations, and establish fundamental formulas for the general solution of such functional equations and for the “Ulam stability” of pertinent cubic functional inequalities.

In this paper, we investigate the locally uniformly non-square point of Orlicz–Bochner sequence spaces endowed with Luxemburg norm. Analysing and combining the generating function M and properties of the real Banach space X, we get sufficient and necessary conditions of locally uniformly non-square point, which contributes to criteria for locally uniform non-squareness in Orlicz–Bochner sequence spaces. The results generalize the corresponding results in the classical Orlicz sequence spaces.

The centroid of a subset of Rd with positive volume is a well-known characteristic. An interesting task is to generalize its definition to at least some sets of zero volume. In the presented paper we propose two possible ways how to do that. The first is based on the Hausdorff measure of an appropriate dimension. The second is given by the limit of centroids of e-neighbourhoods of the particular set when e goes to 0. For both generalizations we discuss their existence and basic properties. Then we focus on sufficient conditions of existence of the second generalization and on conditions when both generalizations coincide. It turns out that they can be formulated with the help of the Minkowski content, rectifiability, and self-similarity. Since the centroid is often used in stochastic geometry as a centre function for certain particle processes, we present properties that are needed for both generalizations to be valid centre functions. Finally, we also show their continuity on compact convex m-sets with respect to the Hausdorff metric topology.

In this paper, we study the well-posedness of the degenerate differential equations with fractional derivative Dα(Mu)(t)=Au(t)+f(t),(0≤t≤2π) in Lebesgue–Bochner spaces Lp(T;X), periodic Besov spaces Bp,qs(T;X) and periodic Triebel–Lizorkin spaces Fp,qs(T;X), where A and M are closed linear operators in a complex Banach space X satisfying D(A)⊂D(M), α>0 and Dα is the fractional derivative in the sense of Weyl. Using known operator-valued Fourier multiplier results, we completely characterize the well-posedness of this problem in the above three function spaces by the R-bounedness (or the norm boundedness) of the M-resolvent of A.

In this paper, we consider a class of nonlinear fractional differential equations on the infinite interval D0+αu(t)+ft,u(t),D0+α−1u(t)=0,t∈(0,+∞), with the integral boundary conditions u(0)=0,D0+α−1u(∞)=∫0τg1(s)u(s)ds+a,D0+α−2u(0)=∫0τg2(s)u(s)ds+b. By using Krasnoselskii fixed point theorem, the existence results of positive solutions for the boundary value problem in three cases τ=0,τ∈(0,+∞) and τ=+∞, are obtained, respectively. We also give out two corollaries as applications of the existence theorems and some examples to illustrate our results. Copyright © 2016 John Wiley & Sons, Ltd.

In the present paper, our aim is to establish several formulas involving integral transforms, fractional derivatives, and a certain family of extended generalized hypergeometric functions. As corollaries and consequences, many interesting results are shown to follow from our main results. A probability density function involving the extended generalized hypergeometric function is introduced, and its properties are studied. The corresponding properties of some of the classical probability distributions and their associated probability density functions are easily derivable as special cases of our general results. Copyright © 2016 John Wiley & Sons, Ltd.

The main objective of this paper was to study the global stability of the positive solutions and the periodic character of the difference equation xn+1=axn−l+bxn−k+cxn−sdxn−s−e,n=0,1,..., where the parameters a, b, c, d, and e are positive real numbers and the initial conditions x−t,x−t + 1,...,x−1, x0 are positive real numbers where t = max{l,k,s}. Some numerical examples will be given to explicate our results. Copyright © 2016 John Wiley & Sons, Ltd.

We give a sharp estimate for the codimension of the poly-Bergman space in the poly-Bergman space over the punctured domain. It is established the behaviour at the infinity point of polyanalytic Bergman functions on the complement of closed disks. In the main result of the paper, we prove that for and the j-polyanalytic Bergman space over the domain U is trivial precisely when the complement of U has at most one point and at most two points or three points lying in a circle, respectively. We point out the differences between the domains over which the Bergman space and the non-analytic poly-Bergman space are trivial.

The main object of this paper is to extend some known approximation results for the complex beta operators of the first kind attached to analytic functions in strips of compact disks. The extensions of these approximation properties, which are presented here, involve bicomplex beta operators in compact -disk. Several other related results are also considered. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, we give the sufficient and necessary conditions for the boundedness of fractional integral operators on the α-modulation spaces. The main theorem substantially extends and improves some known results.

In this paper we present new results on two-weight Hardy, Hardy–Poincare and Rellich type inequalities with remainder terms on a complete noncompact Riemannian Manifold M. The method we use is flexible enough to obtain more weighted Hardy type inequalities. Our results improve and include many previously known results as special cases.

In this paper, we study the class of Banach spaces with James constant . It is shown that, for a Banach space of three or more dimensions, the James constant becomes if and only if the norm is induced by an inner product. Moreover, the symmetric absolute norms on with James constant are completely characterized in terms of convex functions on the unit interval, which provides many new examples of such norms other than the Euclidean or regular octagonal norms. However, it is also shown that there exist two-dimensional normed spaces with James constant outside of the family of symmetric absolute norms.

In this paper, we obtain the existence of infinitely solutions for a class of nonlocal elliptic systems of (p1(x),⋯,pn(x))-Kirchhoff type. Our main results are new. Our approach are based on general variational principle because of B. Ricceri and the theory of the variable exponent Sobolev spaces. Copyright © 2015 John Wiley & Sons, Ltd.

In this paper, we investigate the initial value problem (IVP henceforth) associated with the generalized Ostrovsky equation as follows: with initial data in the modified Sobolev space Hs~(R). Using Fourier restriction norm method, Tao's [k,Z]−multiplier method and the contraction mapping principle, we show that the local well-posedness is established for the initial data u0∈Hs~(R) with s≥−74(k = 2) and is established for the initial data u0∈Hs~(R) with s≥−14(k = 3). Using these results and conservation laws, we also prove that the IVP is globally well-posed for the initial data u0∈Hs~(R) with s = 0(k = 2,3). Finally, using complex variables technique and Paley–Wiener theorem, we prove the unique continuation property for the IVP benefited from the ideas of Zhang ZY. et al., On the unique continuation property for the modified Kawahara equation, Advances in Mathematics (China), http://advmath.pku.edu.cn/CN/10.11845/sxjz.2014078b. Copyright © 2015 John Wiley & Sons, Ltd.

In this paper, we study the homoclinic solutions of the following second-order Hamiltonian system where , and . Applying the symmetric Mountain Pass Theorem, we establish a couple of sufficient conditions on the existence of infinitely many homoclinic solutions. Our results significantly generalize and improve related ones in the literature. For example, is not necessary to be uniformly positive definite or coercive; through is still assumed to be superquadratic near , it is not assumed to be superquadratic near .

In this paper, we study the existence of infinitely many solutions to p-Kirchhoff-type equation (0.1) where f(x,u) = λh1(x)|u|m − 2u + h2(x)|u|q − 2u,a≥0,μ > 0,τ > 0,λ≥0 and . The potential function verifies , and h1(x),h2(x) satisfy suitable conditions. Using variational methods and some special techniques, we prove that there exists λ0>0 such that problem (0.1) admits infinitely many nonnegative high-energy solutions provided that λ∈[0,λ0) and . Also, we prove that problem (0.1) has at least a nontrivial solution under the assumption f(x,u) = h2|u|q − 2u,p < q< min{p*,p(τ + 1)} and has infinitely many nonnegative solutions for f(x,u) = h1|u|m − 2u,1 < m < p. Copyright © 2015 John Wiley & Sons, Ltd.

This paper deals with the quasilinear Keller–Segel system with rotation where is a bounded domain with smooth boundary, D(u) is supposed to be sufficiently smooth and satisfies D(u)≥D0um − 1(m≥1) and D(u)≤D1(u + 1)K − mum − 1(K≥1) for all u≥0 with some positive constants D0 and D1, and f(u) is assumed to be smooth enough and non-negative for all u≥0 and f(0) = 0, while S(u,v,x) = (sij)n × n is a matrix with and with l≥2, where is nondecreasing on [0,∞). It is proved that when , the system possesses at least one global and bounded weak solution for any sufficiently smooth non-negative initial data. Copyright © 2015 John Wiley & Sons, Ltd.

In this paper, we prove the existence and uniqueness of the local mild solution to the Cauchy problem of the n-dimensional (n≥3) Wigner–Poisson–BGK equation in the space of some integrable functions whose inverse Fourier transform are integrable. The main difficulties in establishing mild solution are to derive the boundedness and locally Lipschitz properties of the appropriate nonlinear terms in the Wiener algebra. Copyright © 2015 John Wiley & Sons, Ltd.

In this paper, first, we consider the existence of a positive solution for the nonlinear fractional differential equation boundary value problem CDαu(t)+f(t,u(t))=0,0

In this paper we consider the singularities of the varieties parameterizing stable vector bundles of fixed rank and degree with sections on a smooth curve of genus at least two. In particular, we extend results of Y. Laszlo, and of the second author, regarding the singularities of generalized theta divisors. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

This paper is devoted to the proof of almost global existence results for Klein-Gordon equations on Zoll manifolds (e.g. spheres of arbitrary dimension) with Hamiltonian nonlinearities, when the Cauchy data are smooth and small. The proof relies on Birkhoff normal form methods and on the specific distribution of eigenvalues of the laplacian perturbed by a potential on Zoll manifolds.

The theory of mappings of finite distortion has arisen out of a need to extend the ideas and applications of the classical theory of quasiconformal mappings to the degenerate elliptic setting where one finds concrete applications in non-linear elasticity and the calculus of variations. In this paper we initiate the study of extremal problems for mappings with finite distortion and extend the theory of extremal quasiconformal mappings by considering integral averages of the distortion function instead of its supremum norm. For instance, we show the following. Suppose that $f_o$ is a homeomorphism of the circle with $f_{o}^{-1} \in {\cal W}^{1/2, 2}$. Then there is a unique extremal extension to the disk which is a real analytic diffeomorphism with non-vanishing Jacobian determinant. The condition on $f_o$ is sharp. Classically the mapping $f_o$ is assumed to be quasisymmetric. Then there is an extremal quasiconformal mapping with boundary values $f_o$, but it is not always unique and it is seldom smooth. Indeed, even when $f_o$ is quasisymmetric, the ${\cal L}^1$-minimiser for the distortion function will almost never be quasiconformal. We further find that there are many new and unexpected phenomena concerning existence, uniqueness and regularity for these extremal problems where the functionals are polyconvex but typically not convex. These seem to differ markedly from phenomena observed when studying multi-well type functionals.

Let X be a Banach space, let K be a non–empty closed subset of X and let T : K X be a non–self mapping. The main result of this paper is that if T satisfies the contractive–type condition (1.1) below and maps ∂K (∂K the boundary of K) into K then T has a unique fixed point in K.