Two mappings related to semi-inner products and their applications in geometry of normed linear spaces

In this paper we introduce two mappings associated with the lower and upper semi-inner product (·, ·) _i and (·, ·) _S and with semi-inner products [·, ·] (in the sense of Lumer) which generate the norm of a real normed linear space, and study properties of monotonicity and boundedness of these mappings. We give a refinement of the Schwarz inequality, applications to the Birkhoff orthogonality, to smoothness of normed linear spaces as well as to the characterization of best approximants.     

On Minimax Fractional Optimality and Duality with Generalized Convexity

In this paper, we establish several sufficient optimality conditions for a class of generalized minimax fractional programming. Based on the sufficient conditions, a new dual model is constructed and duality results are derived. Our study naturally unifies and extends some previously known results in the framework of generalized convexity and dual models. Mathematics Subject Classifications: 90C25, 90C32, 90C47.     

Vector-valued reproducing kernel Banach spaces with applications to multi-task learning

Motivated by multi-task machine learning with Banach spaces, we propose the notion of vector-valued reproducing kernel Banach spaces (RKBSs). Basic properties of the spaces and the associated reproducing kernels are investigated. We also present feature map constructions and several concrete examples of vector-valued RKBSs. The theory is then applied to multi-task machine learning. Especially, the representer theorem and characterization equations for the minimizer of regularized learning schemes in vector-valued RKBSs are established.     

A Fourier series formula for energy of measures with applications to Riesz products

In this paper we derive a formula relating the energy and the Fourier transform of a finite measure on the -dimensional torus which is similar to the well-known formula for measures on .

We apply the formula to obtain estimates on the Hausdorff dimension of Riesz product measures. These give improvements on the earlier, classical results which were based on completely different techniques.

     

Generalized evolution variational inequalities and their applications to partial differential equations

We study the existence of solutions to generalized evolution vari ational inequalities by pseudomonotone mapping theory. We obtain results which generalize and extend previously known theorems. We then apply our results to parabolic partial differential equations with discontinuous nonlinearities.     

Generalized wreath products of table algebras determined by their character tables and applications to association schemes

Structures of table algebras whose character tables have a zero submatrix were studied in Blau and Xu (2014) and Chen and Xu (2018). In this paper we continue the research in this direction, and investigate the conditions under which the character table with a zero submatrix yields a generalized wreath product of table algebras. Applications to association schemes are also discussed.     

Generalized Jacobian for Functions with Infinite Dimensional Range and Domain

In this paper, locally Lipschitz functions acting between infinite dimensional normed spaces are considered. When the range is a dual space and satisfies the Radon–Nikodym property, Clarke’s generalized Jacobian will be extended to this setting. Characterization and fundamental properties of the extended generalized Jacobian are established including the nonemptiness, the β-compactness, the β-upper semicontinuity, and a mean-value theorem. A connection with known notions is provided and chain rules are proved using key results developed. This included the vectorization and restriction theorem, and the extension theorem. Therefore, the generalized Jacobian introduced in this paper is proved to enjoy all the properties required of a derivative like-set. Research of the first author is supported by the Hungarian Scientific Research Fund (OKTA) under grant K62316. Research of the second author is supported by the National Science Foundation under grant DMS-0306260.     

Generalized saddle point theorem and asymptotically linear problems with periodic potential

We prove a critical point theorem, which is an infinite dimensional generalization of the classical saddle point theorem of P. H. Rabinowitz. As an application we obtain solution of asymptotically linear Schrödinger equations with periodic potential.     

Invariant Representation for Stochastic Differential Operator by Bsdes with Uniformly Continuous Coefficients and its Applications

In this paper, we prove that a kind of second order stochastic differential operator can be represented by the limit of solutions of BSDEs with uniformly continuous coefficients. This result is a generalization of the representation for the uniformly continuous generator. With the help of this representation, we obtain the corresponding converse comparison theorem for the BSDEs with uniformly continuous coefficients, and get some equivalent relationships between the properties of the generator g and the associated solutions of BSDEs. Moreover, we give a new proof about g-convexity.     

Geometric Properties and Coincidence Theorems with Applications to Generalized Vector Equilibrium Problems

The present paper is divided into two parts. In the first part, we derive a Fan-KKM type theorem and establish some Fan type geometric properties of convex spaces. By applying our results, we also obtain some coincidence theorems and fixed-point theorems in the setting of convex spaces. The second part deals with the applications of our coincidence theorem to establish some existence results for a solution to the generalized vector equilibrium problems.     

Generalized KKM-type theorems for weakly generalized KKM mapping with some applications

In this paper, we establish some new generalized KKM-type theorems based on weakly generalized KKM mapping without any convexity structure in topological spaces. As applications, some minimax inequalities and an existence theorem of equilibrium points for an abstract generalized vector equilibrium problem are proved in topological spaces. The results presented in this paper unify and generalize some known results in recent literature.     

Generalized KKM type theorems in FC-spaces with applications (II)

This paper is a continuum of the preceding paper of author. By applying a coincidence theorem in noncompact FC-space without any convexity structure due to author, a new KKM type theorem is first proved under noncompact setting of FC-spaces. The equivalent relation between the coincidence theorem and the KKM type theorem is also established. As applications of the KKM type theorem, we establish some new existence theorems of solutions for three classes of generalized vector equilibrium problems under noncompact setting of FC-spaces. These theorems improve and generalize many known results in literature.     

Positive periodic solutions for impulsive differential equations with infinite delay and applications to integro-differential equations

Sufficient conditions for the existence of at least one positive periodic solution are established for a family of scalar periodic differential equations with infinite delay and nonlinear impulses. Our criteria, obtained by applying a fixed-point argument to an original operator constructed here, allow to treat equations incorporating a rather general nonlinearity and impulses whose signs may vary. Applications to some classes of Volterra integro-differential equations with unbounded or periodic delay and nonlinear impulses are given, extending and improving results in the literature.     

带有幂型的广义Zakharov方程组解的爆破率的下界估计

主要研究了关于R~2中一类带有幂型非线性的广义Zakharov方程组的Cauchy问题的有限时间爆破解的爆破率的下界估计.在α≤0和p≥3条件下,对于Cauchy问题任意给定的属于能量空间H~1(R~2)×L~2(R~2)×L~2(R~2)的有限时间的爆破解,得到了对于t靠近有限爆破时间T时的爆破率的最优下界估计.此外,给出了Cauchy问题维里等式的一个应用.     

Fixed point and maximal element theorems with applications to abstract economies and minimax inequalities

In this paper, we establish some fixed point theorems for a family of multivalued maps under mild conditions. By using our fixed point theorems, we derive some maximal element theorems for a particular family of multivalued maps, namely the Φ-condensing multivalued maps. As applications of our results, we prove some general equilibrium existence theorems in the generalized abstract economies with preference correspondences. Further applications of our results are also given to minimax inequalities for a family of functions.     

Generalized vector quasi-equilibrium problems with applications to common fixed point theorems and optimization problems

In this paper, we first establish the existence theorems of generalized vector quasi-equilibrium problems. From these results, we establish the existence theorems of common fixed point theorems for two multivalued maps and mathematical programs with an equilibrium constraint as applications.     

上下解方法给出具有时滞随机泛函微分方程解的存在性

讨论了具有不连续飘逸系数具有时滞的随机泛函微分方程,通过上下解方法和序拓扑空间中不动点原理给出了其解的存在性定理.