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In my former paper “A pre-order principle and set-valued Ekeland variational principle” (see [J. Math. Anal. Appl., 419, 904–937 (2014)]), we established a general pre-order principle. From the pre-order principle, we deduced most of the known set-valued Ekeland variational principles (denoted by EVPs) in set containing forms and their improvements. But the pre-order principle could not imply Khanh and Quy’s EVP in [On generalized Ekeland’s variational principle and equivalent formulations for set-valued mappings, J. Glob. Optim., 49, 381–396 (2011)], where the perturbation contains a weak τ-function, a certain type of generalized distances. In this paper, we give a revised version of the pre-order principle. This revised version not only implies the original pre-order principle, but also can be applied to obtain the above Khanh and Quy’s EVP. In particular, we give several new set-valued EVPs, where the perturbations contain convex subsets of the ordering cone and various types of generalized distances.

The purpose of this paper is to study a new viscosity iterative algorithm based on a generalized contraction for finding a common element of the set of solutions of a general variational inequality problem for finite inversely strongly accretive mappings and the set of common fixed points for a countable family of strict pseudo-contractions in uniformly smooth Banach spaces. We prove some strong convergence theorems under some suitable conditions. The results obtained in this paper improve and extend the recent ones announced by many others in the literature.

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.

The purpose of this paper is by using CSQ method to study the strong convergence problem of iterative sequences for a pair of strictly asymptotically pseudocontractive mappings to approximate a common fixed point in a Hilbert space. Under suitable conditions some strong convergence theorems are proved. The results presented in the paper are new which extend and improve some recent results of Acedo and Xu [Iterative methods for strict pseudo-contractions in Hilbert spaces. Nonlinear Anal., 67(7), 2258–2271 (2007)], Kim and Xu [Strong convergence of modified Mann iterations for asymptotically nonexpansive mappings and semigroups. Nonlinear Anal., 64, 1140–1152 (2006)], Martinez-Yanes and Xu [Strong convergence of the CQ method for fixed point iteration processes. Nonlinear Anal., 64, 2400–2411 (2006)], Nakajo and Takahashi [Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups. J. Math. Anal. Appl., 279, 372–379 (2003)], Marino and Xu [Weak and strong convergence theorems for strict pseudocontractions in Hilbert spaces. J. Math. Anal. Appl., 329(1), 336–346 (2007)], Osilike et al. [Demiclosedness principle and convergence theorems for k-strictly asymptotically pseudocontractive maps. J. Math. Anal. Appl., 326, 1334–1345 (2007)], Liu [Convergence theorems of the sequence of iterates for asymptotically demicontractive and hemicontractive mappings. Nonlinear Anal., 26(11), 1835–1842 (1996)], Osilike et al. [Fixed points of demi-contractive mappings in arbitrary Banach spaces. Panamer Math. J., 12 (2), 77–88 (2002)], Gu [The new composite implicit iteration process with errors for common fixed points of a finite family of strictly pseudocontractive mappings. J. Math. Anal. Appl., 329, 766–776 (2007)].

The paper investigates Lp convergence and Marcinkiewicz-Zygmund strong laws of large numbers for random elements in a Banach space under the condition that the Banach space is of Rademacher type p, 1 < p < 2. The paper also discusses Lr convergence and Lr bound for random elements without any geometric restriction condition on the Banach space.

In this paper, we study the limit cycles bifurcations of four fine focuses in Z 4-equivariant vector fields and the problems that its four singular points can be centers and isochronous centers at the same time. By computing the Liapunov constants and periodic constants carefully, we show that for a certain Z 4-equivariant quintic systems, there are four fine focuses of five order and five limit cycles can bifurcate from each, we also find conditions of center and isochronous center for this system. The process of proof is algebraic and symbolic by using common computer algebra soft such as Mathematica, the expressions after being simplified in this paper are simple relatively. Moreover, what is worth mentioning is that the result of 20 small limit cycles bifurcating from several fine focuses is good for Z 4-equivariant quintic system and the results where multiple singular points become isochronous centers at the same time are less in published references.

By using the positive linear functional and the monotone subhomogeneous functional, including the general means and Riccati technique, some new oscillation criteria are established for the second order linear matrix differential system $$ (P(t)X'(t))' + R(t)X'(t) + Q(t)X(t) = 0, t \geqslant t_0 > 0, $$ where P(t), R(t), Q(t) are n × n real continuous matrix functions, P(t) and R(t) are commutative. The results improve and generalize those given in some previous papers, which can be seen by the examples given at the end of this paper.

The purpose of this paper is to study the convergence problem of the iteration scheme xn+1 = λn+1y +(1–λn+1) Tn+1xn for a family of infinitely many nonexpansive mappings T1, T2, . . . in a Hilbert space. It is proved that under suitable conditions this iteration scheme converges strongly to the nearest common fixed point of this family of nonexpansive mappings. The results presented in this paper extend and improve some recent results.

This paper proves that, under the hypothesis g(t, 0, 0)≡0 and some natural assumptions, the generator g of a backward stochastic differential equation can be uniquely determined by the corresponding g-expectations with all terminal conditions. The main result of this paper also confirms and extends Peng Shige’s conjecture.

In this paper, we study the infinity behavior of the bounded subharmonic functions on a Ricci non-negative Riemannian manifold M. We first show that $$ \lim _{{r \to \infty }} \frac{{r^{2} }} {{V{\left( r \right)}}}{\int_{B{\left( r \right)}} {\Delta hdv{\kern 1pt} = {\kern 1pt} {\kern 1pt} 0} } $$ if h is a bounded subharmonic function. If we further assume that the Laplacian decays pointwisely faster than quadratically we show that h approaches its supremun pointwisely at infinity, under certain auxiliary conditions on the volume growth of M. In particular, our result applies to the case when the Riemannian manifold has maximum volume growth. We also derive a representation formula in our paper, from which one can easily derive Yau’s Liouville theorem on bounded harmonic functions.

In this paper, some new generalizations of inverse type Hilbert-Pachpatte integral inequalities are proved. The results of this paper reduce to those of Pachpatte (1998, J. Math. Anal. Appl. 226, 166–179) and Zhao and Debnath (2001, J. Math. Anal. Appl. 262, 411–418).

In this paper, we prove that for every symplectic matrix M possessing eigenvalues on the unit circle, there exists a symplectic matrix P such that P −1 MP is a symplectic matrix of the normal forms defined in this paper.

In this paper, we will show that every sub-Riemannian manifold is the Gromov–Hausdorff limit of a sequence of Riemannian manifolds.

This paper is devoted to the study of some fundamental properties of the sewing homeomorphism induced by a Jordan domain. In particular, using conformal invariants such as harmonic measure, extremal distance, and reduced extremal distance, we give several necessary and sufficient conditions for the sewing homeomorphism to be bi-Lipschitz or bi-Holder. Furthermore, equivalent conditions for a Jordan curve to be a quasicircle are also obtained.

In this paper, we investigate isometric extension problem in general normed space. We prove that an isometry between spheres can be extended to a linear isometry between the spaces if and only if the natural positive homogeneous extension is additive on spheres. Moreover, this conclusion still holds provided that the additivity holds on a restricted domain of spheres.

In this paper, we give a complete conformal classification of the regular space-like hypersurfaces in the de Sitter Space S m+1 1 with parallel para-Blaschke tensors.

In this paper, we classify the set of Lorentzian metrics on T2, analyse the topological structure of some subclasses and study whether a subclass could admit certain weak solutions to the eikonal equations.

In this paper, we will use the Schwarz lemma at the boundary to character the distortion theorems of determinant at the extreme points and distortion theorems of matrix on the complex tangent space at the extreme points for normalized locally biholomorphic quasi-convex mappings in the unit ball Bn respectively.

In this paper, let (Mn, g) be an n-dimensional complete Riemannian manifold with the m-dimensional Bakry–Emery Ricci curvature bounded below. By using the maximum principle, we first prove a Li–Yau type Harnack differential inequality for positive solutions to the parabolic equation $${u_t} = LF\left( u \right) = \Delta F\left( u \right) - \nabla f \cdot \nabla F\left( u \right),$$ on compact Riemannian manifolds Mn, where F ∈ C2(0,∞), F′ > 0 and f is a C2-smooth function defined on Mn. As application, the Harnack differential inequalities for fast diffusion type equation and porous media type equation are derived. On the other hand, we derive a local Hamilton type gradient estimate for positive solutions of the degenerate parabolic equation on complete Riemannian manifolds. As application, related local Hamilton type gradient estimate and Harnack inequality for fast dfiffusion type equation are established. Our results generalize some known results.

The non-uniqueness on geodesics and geodesic disks in the universal asymptotic Teichmuller space AT(D) are studied in this paper. It is proved that if μ is asymptotically extremal in [[μ]] with hζ*(μ) < h*(μ) for some point ζ ∈ ∂D, then there exist infinitely many geodesic segments joining [[0]] and [[μ]], and infinitely many holomorphic geodesic disks containing [[0]] and [[μ]] in AT(D).

In this paper, we obtain the estimates of all homogeneous expansions for a subclass of biholomorphic mappings which have parametric representation on the unit ball of complex Banach spaces. Meanwhile, we also establish the estimates of all homogeneous expansions for the above generalized mappings on the unit polydisk in C n . Especially, the above estimates are only sharp for biholomorphic starlike mappings and starlike mappings of order α under restricted conditions. Our derived results generalize many known results.

Let Sα* be the familiar class of normalized starlike functions of order α in the unit disk. In this paper, we establish the Fekete and Szego inequality for the class Sα*, and then we generalize this result to the unit ball in a complex Banach space or on the unit polydisk in Cn.

In this paper, we characterize the symbols for (semi-)commuting dual Toeplitz operators on the orthogonal complement of the harmonic Dirichlet space. We show that for φ, ψ ∈ W1,∞, SφSψ = SψSφ on (Dh)⊥ if and only if φ and ψ satisfy one of the following conditions: (1) Both φ and ψ are harmonic functions; (2) There exist complex constants α and β, not both 0, such that φ = αψ+β.

We conjecture that a Willmore torus having Willmore functional between 2π 2 and 2π 2 $$\sqrt 3 $$ is either conformally equivalent to the Clifford torus, or conformally equivalent to the Ejiri torus. Ejiri’s torus in S 5 is the first example of Willmore surface which is not conformally equivalent to any minimal surface in any real space form. Li and Vrancken classified all Willmore surfaces of tensor product in S n by reducing them into elastic curves in S 3, and the Ejiri torus appeared as a special example. In this paper, we first prove that among all Willmore tori of tensor product, the Willmore functional of the Ejiri torus in S 5 attains the minimum 2π 2 $$\sqrt 3 $$ , which indicates our conjecture holds true for Willmore surfaces of tensor product. Then we show that all Willmore tori of tensor product are unstable when the co-dimension is big enough. We also show that the Ejiri torus is unstable even in S 5. Moreover, similar to Li and Vrancken, we classify all constrained Willmore surfaces of tensor product by reducing them with elastic curves in S 3. All constrained Willmore tori obtained this way are also shown to be unstable when the co-dimension is big enough.

In this paper, we show that a closed convex subset C of a Banach space is strongly proximinal (proximinal, resp.) in every Banach space isometrically containing it if and only if C is locally (weakly, resp.) compact. As a consequence, it is proved that local compactness of C is also equivalent to that for every Banach space Y isometrically containing it, the metric projection from Y to C is nonempty set-valued and upper semi-continuous.

Let F and G be closed subspaces of the complex Hilbert space H, and U and V be closed subspaces of F and G, respectively. In this paper, using the technique of operator block, we present the necessary and sufficient conditions under which (U, V) is a pair of (strictly, non-degenerate) principal invariant subspaces for (F, G).

In this paper, we prove a local Holder estimate of (K1,K2)-quasiconformal mappings between n-dimensional hypersurfaces of Rn+1 under an assumption of bounded mean curvature of the original hypersurface M. With some new ingredients of the isoperimetric inequality and the co-area formula on manifolds, we extend Simon’s work of quasiconformal mappings on surfaces of ℝ3 to the setting of n-dimensional hypersurfaces of ℝn+1.

In this paper, we study a significant non-Riemannian quantity Ξ-curvature, which is defined by S-curvature. Firstly, we obtain the formula of Ξ-curvature for (α, β)-metrics. Based on it, we show that the Ξ-curvature vanishes for a class of (α, β)-metrics. In the end, we get the relation of Ξ-curvature for conformally related Finsler metrics, and classify conformally flat (α, β)-metrics with almost vanishing Ξ-curvature.

Let M 2 be an umbilic-free surface in the unit sphere S 3. Four basic invariants of M 2 under the Moebius transformation group of S 3 are Moebius metric g, Blaschke tensor A, Moebius second fundamental form B and Moebius form Φ. We call the Blaschke tensor is isotropic if there exists a smooth function λ such that A = λ g. In this paper, We classify all surfaces with isotropic Blaschke tensor in S 3.

In this paper, the behavior on the product of Lebesgue spaces is considered for the maximal operators associated with the bilinear singular integral operators whose kernels satisfy certain minimal regularity conditions.

In this paper, one considers the change of orbifold Gromov-Witten invariants under weighted blow-up at smooth points. Some blow-up formula for Gromov-Witten invariants of symplectic orbifolds is proved. These results extend the results of manifolds case to orbifold case., Comment: 24 pages. arXiv admin note: text overlap with arXiv:1307.2088 by other authors. To appear in Acta Math. Sinica

In this paper, we obtain a sufficient and necessary condition for a simply connected Riemannian manifold (Mn, g) to be isometrically immersed, as a submanifold with codimension p ≥ 1, into the product Sk × Hn+p−k of sphere and hyperboloid.

In this paper, we mainly study zeros and poles of the forward differences Δnf(z), where f(z) is a finite order meromorphic function with two Borel exceptional values.

In this paper, by introducing the space with weak mixed norms, weak type estimates of two kinds of multilinear fractional Hausdorff operators {ie1407-1} and {ie1407-2} on Lebesgue spaces are shown. By virtue of Marcinkiewicz interpolation, strong type estimates of these two operators on Lebesgue spaces are also obtained. Our methods shed some new light on dealing with the case of non-radial function Φ.

In this paper, we study the weighted estimates for multilinear pseudodifferential operators. We show that a multilinear pseudodifferential operator is bounded with respect to multiple weights whenever its symbol satisfies some smoothness and decay conditions. Our result generalizes similar ones from the classical Ap weights to multiple weights.

In this paper, we study Tingley’s problem on symmetric absolute normalized norms on ℝ2. We construct new methods for Tingley’s problem on two-dimensional spaces by using isosceles orthogonality, which does not make use of the notion of natural extension. Furthermore, using our methods, several sufficient conditions for Tingley’s problem on symmetric absolute normalized norms on ℝ2 are given. As applications, we present various new examples including the two-dimensional Lorentz sequence space d (2)(ω, q) and its dual d (2)(ω, q)* by simple arguments.

In this paper, the perturbations of the Moore-Penrose metric generalized inverses of linear operators in Banach spaces are described. The Moore-Penrose metric generalized inverse is homogeneous and nonlinear in general, and the proofs of our results are different from linear generalized inverses. By using the quasi-additivity of Moore-Penrose metric generalized inverse and the theorem of generalized orthogonal decomposition, we show some error estimates of perturbations for the single-valued Moore-Penrose metric generalized inverses of bounded linear operators. Furthermore, by means of the continuity of the metric projection operator and the quasi-additivity of Moore-Penrose metric generalized inverse, an expression for Moore-Penrose metric generalized inverse is given.

The aim of this paper is to establish a series of important properties of local Lipschitz-α mappings from a subset of a normed space into a normed space. These mappings include Lipschitz operators, Lipschitz-α operators and local Lipschitz functions. Some applications to the theory of sublinear expectation spaces are given.

In this paper, we study error bounds for lower semicontinuous functions defined on Banach space and linear regularity for finitely many closed subset in Banach spaces. By using Clarke’s subdifferentials and Ekeland variational principle, we establish several sufficient conditions ensuring error bounds and linear regularity in Banach spaces.

For α > −1, let dν α denote the weighted Lebesgue measure on the bidisk and μ a complex measure satisfying some Carleson-type conditions. In this paper, we show a sufficient and necessary condition for the Toeplitz operator $$T_{\bar \mu }^\alpha$$ to be bounded or compact on weighted Bergman space L 1 (dν α ).

In the paper, we investigate the complete convergence and complete moment convergence for the maximal partial sum of martingale difference sequence. Especially, we get the Baum-Katz-type Theorem and Hsu-Robbins-type Theorem for martingale difference sequence. As an application, a strong law of large numbers for martingale difference sequence is obtained.

In this paper, we establish the boundedness of parameterized Littlewood-Paley operator μ λ *,ρ and parameterized area integral μ Ω,S ρ with kernel satisfying the logarithmic type Lipschitz condition on the weak Hardy space.

Let λf(n) be the n-th normalized Fourier coefficient of a holomorphic Hecke eigenform f(z) ∈ Sk(Γ). In this paper, we established nontrivial estimates for $$\sum\limits_{n \leqslant x} {\lambda _f \left( {n^i } \right)\lambda _f \left( {n^j } \right),}$$ where 1 ≤ i < j ≤ 4.

In this paper, we obtain certain Lwp(ℝn)-mapping properties for the maximal operator associated with the commutators of quasiradial Bochner-Riesz means with index δ under certain surface condition on Σϱd, provided that δ > (n − 1)/2, b ∈ BMO(ℝn), 1 (n − 1)/2, then we prove that the above maximal operator admits weak type (Hw1(ℝn), Lw1(ℝn))-mapping properties for b ∈ BMO(ℝn) and w ∈ A1 under the surface condition on Σϱd.

In this paper, by using p-distances on uniform spaces, we establish a general vectorial Ekeland variational principle (in short EVP), where the objective function is defined on a uniform space and taking values in a pre-ordered real linear space and the perturbation involves a p-distance and a monotone function of the objective function. Since p-distances are very extensive, such a form of the perturbation in deed contains many different forms of perturbations appeared in the previous versions of EVP. Besides, we only require the objective function has a very weak property, as a substitute for lower semi-continuity, and only require the domain space (which is a uniform space) has a very weak type of completeness, i.e., completeness with respect to a certain p-distance. Such very weak type of completeness even includes local completeness when the uniform space is a locally convex topological vector space. From the general vectorial EVP, we deduce a general vectorial Caristi’s fixed point theorem and a general vectorial Takahashi’s nonconvex minimization theorem. Moreover, we show that the above three theorems are equivalent to each other. We see that the above general vectorial EVP includes many particular versions of EVP, which extend and complement the related known results.

In this paper, we study the normality of the family of meromorphic functions from the viewpoint of hyperbolic metric. Then, a new sufficient and necessary condition is obtained, which can determine a given family of meromorphic functions is normal or not.

In this paper, using an equivalent characterization of the Besov space by its wavelet coefficients and the discretization technique due to Maiorov, we determine the asymptotic degree of the Bernstein n-widths of the compact embeddings Open image in new window , where \(B_{q_0 }^{s + t} (L_{p_0 } (\Omega ))\) is a Besov space defined on the bounded Lipschitz domain Ω ⊂ ℝd. The results we obtained here are just dual to the known results of Kolmogorov widths on the related classes of functions.

In this paper, we apply a variant of the famous Mountain Pass Lemmas of Ambrosetti-Rabinowitz and Ambrosetti-Coti Zelati with (PSC)c type condition of Palais-Smale-Cerami to study the existence of new periodic solutions with a prescribed energy for symmetrical singular second order Hamiltonian conservative systems with weak force type potentials.

In this paper, the authors give the boundedness of the commutator of hypersingular integral T γ from the homogeneous Sobolev space $\dot L_\gamma ^p $ (ℝ n ) to the Lebesgue space L p (ℝ n ) for 1 < p < ∞ and 0 < γ < $\min \{ \tfrac{n} {2},\tfrac{n} {p}\} $ .

The purpose of this paper is to investigate the growth of meromorphic functions concerning Picard values with a radially distributed value. This generalizes a classic result due to Hayman [Hayman, W. K.: Picard values of meromorphic functions and their derivatives. Ann. of Math., 70, 9–42 (1959)].